Investigation of single crystal germanium

pn-junctions for use in tandem CdTe/Ge solar cells

by

James Ross Sharp

B.E. (Hons.), University of Melbourne, 2006

This thesis is submitted to the

Faculty of Engineering, Computing and Mathematics

of the University of Western Australia in fulfillment

of the requirement for the degree of

Doctor of Philosophy

School of Electrical, Electronic and Computer Engineering

2016

This thesis entitled:Investigation of single crystal germanium pn-junctions for use in tandem

CdTe/Ge solar cellswritten by James Ross Sharp

has been approved for the School of Electrical, Electronic and ComputerEngineering

Winthrop Professor Lorenzo Faraone

Winthrop Professor and Dean John Dell

Date

The final copy of this thesis has been examined by the signatories, and we findthat both the content and the form meet acceptable presentation standards of

scholarly work in the above mentioned discipline.

3

Sharp, James Ross (B.E. (Hons.))

Investigation of single crystal germanium pn-junctions for use in tandem CdTe/Ge

solar cells

Thesis supervised by Winthrop Professor Lorenzo Faraone

Abstract

Thin film cadmium telluride solar cells are a viable renewable energy tech-

nology, due to low manufacturing cost, fast energy pay-back times, and an energy

gap well matched to the solar spectrum. The technology is already mature and

has commercial application, with more than 10GW of installed thin film CdTe

modules world wide. As with any commercial photovoltaic technology, research

is perpetually focussed on how to boost module efficiency, improve process yield,

and lower production costs. Although startlingly rapid progress has been made

in the field of CdTe photovoltaics in recent years, with commercial CdTe module

manufacturer First Solar reporting record breaking research module efficiencies,

there will inevitably come a point where the performance of CdTe single junction

modules cannot be improved any further, as the conversion efficiency approaches

the Shockley-Queisser limit. At such time, higher efficiencies can only be achieved

by means of adopting multijunction tandem configurations.

This work aims to investigate a possible material combination, namely CdTe

and germanium (Ge), for the advancement of CdTe technology into the realm of

multijunction and concentrated photovoltaics. Germanium is a favourable material

for this purpose due to its narrow energy gap located in a suitable area of the

solar spectrum, availability of large area Ge substrates in epi-ready format, and

idealised electrical and mechanical properties. This thesis investigates germanium

4

processing in a low-cost and manufacturable manner in order to develop a process

for the formation of the lower cell of a multijunction photovoltaic device. Novel

techniques for germanium doping, passivation and contacting are expounded and

a complete methodology for germanium device fabrication is presented. This is of

interest not only to the photovoltaics sector but more generally the techniques are

applicable to a wide range of germanium opto-electronic devices.

In order to predict performance and optimise device structures, simulation

and modelling is undertaken in both a commercial device simulator (Synopsys

Sentaurus Device) as well as in a custom developed analytical/numerical simu-

lation framework. The goal of simulation is to investigate both monolithic and

mechanically stacked configurations and determine which device structure would

be optimal in terms of photocurrent matching and also in terms of optical proper-

ties to minimise optical reflection losses from device active layers. A mechanically

stacked configuration featuring CdTe grown epitaxially on sapphire is considered in

this work and its possible performance compared to a monolithic CdTe/Ge struc-

ture. It is shown that such a structure could contribute an efficiency improvement

of 5.03% absolute over a single junction CdTe solar cell, whereas a monolithic

tandem would boost single junction solar cell efficiency by a mere 3.6% absolute.

Subsequently, doped layers of single crystal germanium were prepared from

bulk germanium wafers utilising spin on dopants, either by directly spinning on a

thin film of dopant, or by vapour transport in the “proximity” doping technique, or

by the novel “sandwich-stacked diffusion” technique developed in this work. These

layers were processed into electronic and photovoltaic devices using standard pro-

cessing techniques, passivated and contacted using the technologies demonstrated

within, and finally characterised. The result is a high quality process for germanium

opto-electronic device fabrication. Optoelectronic devices are shown with surface

recombination velocities as low as 21 cm/s and with specific contact resistivities

5

as low as 1.26 ×10−7 Ω · cm2 . This highlights the quality of the passivation and

contacting procedures developed in this work.

An investigation of germanium doping for device active region formation

is undertaken. It is concluded that both proximity doping and sandwich-stacked

diffusion yield degenerate p-type doping of germanium and surface concentrations

of up to 1020 atoms cm−3 can be achieved, but degenerate n-type doping can only

be achieved by means of direct spin-on doping. The reason is most likely the high

vapour pressure of phosphorus and its oxides at the processing temperature. Direct

spin on doping gave a maximum donor concentration of 4e19 cm−3, in contrast

with a maximum concentration of 6e18 cm−3 for proximity doping.

Germanium pn-junction devices with ideality factors equal to 1 and showing

breakdown due to Zener effect are presented, as well as a 5.4% efficient solar cell.

The solar cell illustrates the complete germanium diode fabrication process includ-

ing contacting and passivation and the device is shown to be stable in efficiency

when remeasured after eight months. The solar cell was capped with a combined

passivation/anti-reflection solution shown to reduce reflection losses to 6.47%.

Finally, CdTe was grown on both germanium and sapphire substrates and

the results were characterised by a variety of methods including RHEED, XRD,

and optical transmission measurements. CdTe thin films grown on sapphire are

presented with double crystal rocking curve (DCRC) full-width at half maxima

(FWHM) as low as 59 arc seconds as rocked about the 〈111〉 diffraction plane.

CdTe grown on germanium was processed into heterojunction CdTe/Ge P/n junc-

tions and the IV and CV characteristics were measured to elucidate the electronic

properties of the heterojunction. A CdTe/Ge diode with ideality factor n=1.65 is

presented demonstrating reasonable quality material growth and device processing

utilising this novel combination of materials.

Dedication

To my father, Peter, and my mother, Angela and to the love of my life,

Hitomi.

7

Contents

Chapter

1 Introduction 13

1.1 Harnessing the power of the sun . . . . . . . . . . . . . . . . . . 13

1.1.1 Standard solar reference spectrum . . . . . . . . . . . . . 15

1.1.2 A CdTe/germanium solar cell . . . . . . . . . . . . . . . 17

1.1.3 What is cadmium telluride? . . . . . . . . . . . . . . . . 18

1.1.4 What is germanium? . . . . . . . . . . . . . . . . . . . . 19

1.1.5 The Shockley-Queisser limit . . . . . . . . . . . . . . . . 20

1.1.6 Detailed balance limit of efficiency of tandem solar cells . 22

1.1.7 Mechanically stacked vs. monolithic combination . . . . . 24

1.1.8 Development of Cadmium Telluride technology in research

circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.1.9 Germanium standalone solar cell research . . . . . . . . . 29

1.1.10 Growth of CdTe on germanium . . . . . . . . . . . . . . 31

1.2 Research outcomes and thesis outline . . . . . . . . . . . . . . . 33

2 Analytical and numerical techniques for optoelectronic device modelling 37

2.1 Elementary theory of solar cells . . . . . . . . . . . . . . . . . . 37

2.2 Derivation of an analytical model . . . . . . . . . . . . . . . . . 39

2.2.1 Recombination . . . . . . . . . . . . . . . . . . . . . . . 43

8

2.2.2 Carrier absorption/generation . . . . . . . . . . . . . . . 44

2.2.3 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.4 Total photocurrent . . . . . . . . . . . . . . . . . . . . . 52

2.2.5 Depletion region width . . . . . . . . . . . . . . . . . . . 52

2.2.6 Dark current . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.7 Device parasitics . . . . . . . . . . . . . . . . . . . . . . 56

2.2.8 Summary analytical model for solar cells . . . . . . . . . 58

2.2.9 Key device characteristics . . . . . . . . . . . . . . . . . 60

2.2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3.2 Equation set in continuous form . . . . . . . . . . . . . . 65

2.3.3 Dependent variables . . . . . . . . . . . . . . . . . . . . 66

2.3.4 Discretisation of Poisson equation in 1D . . . . . . . . . 67

2.3.5 Discretisation of electron and hole drift-diffusion and con-

tinuity equations in 1D . . . . . . . . . . . . . . . . . . 67

2.3.6 Variable scaling . . . . . . . . . . . . . . . . . . . . . . 69

2.3.7 Solution of discretised equation set over a finite mesh . . 70

2.3.8 Physical models . . . . . . . . . . . . . . . . . . . . . . 88

2.3.9 Test case - Germanium Solar Cell . . . . . . . . . . . . . 98

2.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 100

3 Device modelling, design and optimisation of tandem II-VI and germanium

solar cells 102

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.1.1 CdTe homojunction on crystalline germanium substrate . 106

3.1.2 Conventional CdTe heterojunction . . . . . . . . . . . . 113

9

3.2 Mechanically stacked tandem solar cell . . . . . . . . . . . . . . 116

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4 Low-cost techniques for germanium device active region formation 121

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2 Spin-on dopants . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3 Degenerate p-type doping and sandwich stacked diffusion . . . . 122

4.4 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.4.1 Sample preparation . . . . . . . . . . . . . . . . . . . . 123

4.4.2 Sandwich-stacked diffusion . . . . . . . . . . . . . . . . 124

4.5 Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5.1 Sheet resistance measurements . . . . . . . . . . . . . . 128

4.5.2 SIMS profiling . . . . . . . . . . . . . . . . . . . . . . . 130

4.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.6.1 Fitting procedure to extract diffusivities . . . . . . . . . . 131

4.6.2 Thermal activation energy and pre-exponential factor ex-

traction . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.7 Proximity doping . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7.2 Experimental - Proximity doping investigation using SIMS 139

4.7.3 Proximity doping with Sb3m Antimony spin-on film . . . 140

4.7.4 Proximity doping with GaB260 spin-on film: Gallium diffusion144

4.7.5 Proximity doping with phosphorus . . . . . . . . . . . . . 147

4.7.6 Direct spin-on doping with phosphorus . . . . . . . . . . 148

4.7.7 Fit to phosphorus depth profiles using non-linear model . 150

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

10

5 Passivation, antireflection, and contacting technologies for germanium opt-

electronic devices 155

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.2 Passivation of germanium . . . . . . . . . . . . . . . . . . . . . 156

5.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . 158

5.2.2 Characterisation . . . . . . . . . . . . . . . . . . . . . . 161

5.2.3 Anti-reflection coatings . . . . . . . . . . . . . . . . . . 165

5.3 Contacts to Germanium . . . . . . . . . . . . . . . . . . . . . . 173

5.3.1 Derivation of specific contact resistivity for some contact

metals . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6 Germanium pn-junction devices 185

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.2 Germanium wafer selection . . . . . . . . . . . . . . . . . . . . 186

6.3 Diode fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.3.1 Fabrication process . . . . . . . . . . . . . . . . . . . . . 187

6.3.2 Ge n+/p diodes . . . . . . . . . . . . . . . . . . . . . . 192

6.3.3 Ge n+/p+ “tunnel” diode . . . . . . . . . . . . . . . . . 195

6.4 Investigation of a defective sample using scanning electron microscopy198

6.5 Germanium solar cells . . . . . . . . . . . . . . . . . . . . . . . 199

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7 II-VI/Germanium Materials Growth and Heterojunction Optoelectronic De-

vices 208

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.2 Choice of materials and crystallography . . . . . . . . . . . . . . 209

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 209

11

7.2.2 Materials science of CdTe/ZnTe/Ge/sapphire . . . . . . . 211

7.3 Material growth techniques . . . . . . . . . . . . . . . . . . . . 219

7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 219

7.3.2 Thermal Evaporation . . . . . . . . . . . . . . . . . . . 220

7.3.3 Molecular Beam Epitaxy . . . . . . . . . . . . . . . . . . 222

7.4 Characterisation of thin films . . . . . . . . . . . . . . . . . . . 225

7.4.1 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . 225

7.4.2 Reflection high-energy electron diffraction (RHEED) . . . 230

7.4.3 Nomarski contrast microscopy . . . . . . . . . . . . . . . 232

7.4.4 Optical constants from transmission measurements . . . . 233

7.5 Electronic properties of the CdTe/Ge interface . . . . . . . . . . 236

7.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 236

7.5.2 Fabrication Process . . . . . . . . . . . . . . . . . . . . 236

7.5.3 CdTe Epilayer sheet resistance . . . . . . . . . . . . . . . 239

7.5.4 CdTe / Ge heterojunction current-voltage characteristics . 240

7.5.5 CdTe/Ge heterojunction capacitance-voltage profiling . . 241

7.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 243

8 Conclusions 247

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8.2 Research outcomes . . . . . . . . . . . . . . . . . . . . . . . . . 250

8.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

8.4 Final Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 254

12

Bibliography 256

Appendix

A Decoupled solution - Gummel’s method 267

B Fully coupled solution 270

C List of original contributions in this work 279

D List of publications arising from this work 280

D.1 Conference presentations . . . . . . . . . . . . . . . . . . . . . . 280

Chapter 1

Introduction

1.1 Harnessing the power of the sun

Australia’s potential for photovoltaic power generation ranks high among

that of the world’s nations. With 1.5 million square kilometers of desertified land,

and insolation equivalent to some of the sunniest African nations, there lies in

wait a vast untapped resource. The question then becomes how to harness this

energy and in what form. Solar thermal concentrators seem perfectly suited to the

harsh environment of the Australian desert. These devices concentrate sunlight to

heat up water and create steam. On the other hand, highly efficient, concentrated

photovoltaics (CPV) and solar tracking systems could be the answer. These sys-

tems concentrate sunlight by manipulating a lens to follow the sun throughout the

day and focus it onto small photovoltaic (PV) receivers. These installations can

be highly efficient - over 40 %. With so much land area available, on the other

hand, efficiency may not be so critical so perhaps thin film photovoltaic cells - with

their lower production costs due to reduced material usage - could be a winner to

deliver Australia’s growing energy needs. With much talk at time of writing in the

political sphere about increasing renewable energy targets, perhaps it is a matter

of time before Australia’s uninhabitable areas become centers for national power

generation.

Fig. 1.1 shows the insolation across Australia for a day in summer. With

14

most of the uninhabited areas of Australia receiving 36 MJ/m2, the total amount

of received energy from the sun is unimaginable. For example, If only 0.03% of

Australia’s uninhabited land were utilised for PV power generation, at an efficiency

of just 10%, a peak power of almost 50 GW could be realised, enough to meet all

of Australia’s energy needs for some time to come. In one day in summer around

16 petajoules - that is, 16× 1015 J - would be harvested.

Figure 1.1: Typical Insolation on an Australian summer’s day.Source: Bureau of Meteorology

According to the office of the chief economist, Australia’s current energy

consumption is almost 6,000 petajoules (PJ) per year, of which renewables only

make up 345.7 PJ. Of the renewables, solar PV accounts for a mere 17.5 PJ.

What stimulus is required to tip the balance in favour of renewables? Once again,

motivation in the political sphere may be the answer.

15

In this thesis, we do not attempt to solve Australia’s energy supply problems

once and for all. Instead, we focus on a technology that could one day help bridge

the gap between fossil fuels and renewable energy sources, working in harmony with

other renewable energy technologies for a greener, cleaner future. The technology

we attempt to develop sits somewhere between the concentrated photovoltaics and

the thin film solutions mentioned in the opening paragraph. To set the scene for

our technology, we must first understand the nature of solar energy as it reaches

the Earth’s surface.

1.1.1 Standard solar reference spectrum

0 500 1000 1500 2000 2500 3000 3500 4000

Wavelength, nm

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Sp

ectr

al

Irra

dia

nce

, W

m−

2n

m−

1

AM1.5G spect rum according to ASTM G173-03

Ge

Si

CdTe

ZnTe

Figure 1.2: AM-1.5G Spectrum

16

The radiation from the sun approximates a 5800K blackbody. The radiation

incident on Earth as measured outside the Earth’s atmosphere is termed the air

mass 0, or AM0, spectrum. This radiation strongly resembles standard blackbody

radiation. As the radiation passes through the atmosphere it becomes absorbed

by the molecules that make up the air, particularly water vapour, O2 and CO2.

The radiation spectrum that is incident on the equator is known as AM1, whereas

radiation which passes through the atmosphere at a solar zenith angle of 48.19 is

known as AM1.5. A standardised reference spectrum produced by the American

Society for Testing and Materials (ASTM) is the AM1.5G standard. The “G”

stands for “global tilt” as both direct illumination and diffuse atmospheric reflec-

tions are taken into account by assuming the receiving plane is at a 37 tilt toward

the equator.

The total power density as integrated from the reference spectrum for AM1.5G

is roughly 1000 Wm−2. The spectrum is depicted in Fig. 1.2 and shows the po-

sition in the solar spectrum of the energy gaps of materials under consideration

in this work. Most of the energy is concentrated in the UV-visible portion of the

spectrum, from 400 to 800 nm. A material like Si is capable of absorbing the

bulk of that energy, out to 1100 nm. Germanium however has extended infra-red

absorption and can absorb the radiation out to roughly 1800nm. CdTe on the

other hand only absorbs the red, green and blue / UV portions of the spectrum,

omitting the infrared region entirely. Hence, from the point of view of efficiently

harvesting the solar spectrum with readily available solar materials, a combination

of the two materials could be posited. This would have the advantage of absorb-

ing the solar spectrum efficiently deep into the infrared, with minimal unabsorbed

(hence wasted) energy.

17

1.1.2 A CdTe/germanium solar cell

In this work we propose combining the materials cadmium telluride and

germanium to form a hybrid photovoltaic device known as a tandem solar cell.

CdTe and Ge are both semiconducting materials which have been studied and

used extensively since the advent of the modern era. There are many reasons why

it is desirable to combine materials to form a tandem solar cell. A tandem solar

cell may be more efficient as a whole than any one of its components. This means

that more useful energy is generated from the incident radiation. Although we

have seen that sunlight is abundant in Australia and that a comparatively small

area of land is required to fulfill all of Australia’s energy needs, the cost of raw

materials and production costs to process them need to be brought down to a level

whereby renewable energy technologies can compete with existing fossil fuel based

energy generation technologies. By increasing efficiency more energy is created

per individual photovoltaic module and the production and raw material costs are

thereby offset.

In addition to offsetting the costs, there is potential for raw material reduc-

tion. Reduced usage of scarce raw material and processing is required for energy

generation if module efficiency can be improved. This means, in effect, that less

energy needs to be expended to produce the same power output, effectively short-

ening energy payback times.

By combining materials in a stacked structure (tandem), more of the incident

light can be harnessed and there is less energy that cannot be converted. Since

a solar cell can only convert energy above the energy gap of the materials it

is made from, any sub-band gap radiation is lost; a tandem structure however

features lower-energy gap subcells which absorb this energy. The result is that the

incident spectrum can be more efficiently converted by dividing the spectrum into

18

absorption windows amongst the cells making up the structure.

1.1.3 What is cadmium telluride?

Cadmium is bluish-white metal and element 48 of the periodic table. Cad-

mium is usually produced as a by-product of zinc refining since cadmium is found in

zinc ores. Principal uses for cadmium include cadmium based pigments, favoured

historically due to their lasting characteristics (cadmium red and yellow pigments

do not fade over time in comparison to other pigments), nickel-cadmium (NiCad)

batteries, corrosion resistant plating for steel components, and CdTe solar cells.

Due to the toxicity of Cd, NiCad batteries and cadmium based pigments have

fallen out of favour and non-toxic replacements are usually preferred.

Tellurium is a rare, brittle metalloid with atomic number 52. Tellurium was

first discovered in 1782 by Franz Joseph Muller von Reichenstein. Tellurium is

scarce and has an abundance in the earth’s crust similar to platinum, around 1

ppb, with world reserves of tellurium limited to 24,000 metric tonnes. Tellurium

is mostly obtained from anode sludges produced during copper refinement, but

is also present in some naturally occurring gold ores. Uses for tellurium include

addition to copper to improve machineability without impairing conductivity, to

steel also to improve machineability, to cast iron to help improve the depth of chill,

and to malleable iron as a carbide stabiliser [1]. It has many other industrial uses

including as a vulcanising agent for rubber production and as an additive to glass

(pigment).

CdTe is a compound semiconductor created from the stoichiometric com-

bination of cadmium and tellurium. It has an energy gap of around 1.45 eV [2],

making it a highly suitable solar material. It has the added benefit of being a

potential use for cadmium, which as noted is a by-product of refinement of more

useful zinc. When combined with zinc, the resultant alloy Cd1−xZnxTe is useful

19

for X and gamma ray detection, and when combined with mercury, HgxCd1−xTe

is a strategic material for use in high performance infrared detectors. The primary

industrial use of cadmium telluride is in CdTe solar cells.

1.1.4 What is germanium?

Germanium is a lustrous metalloid with atomic number 32. Germanium was

first predicted by chemist Dmitri Mendeleyev 17 years before its initial discovery.

Mendeleyev was able to predict in advance its mass number, density, oxide and

chloride density and other properties. Mendeleev had named the element ekasil-

icon. Germanium was first discovered in 1885 by the chemist Clemens Winkler

who isolated it from samples of a mineral named argyrodite.

Zinc ores containing germanium are the principal source of germanium pro-

duction worldwide. Germanium is also found in silver, lead and copper ores and

can also be present in some coal deposits. Refinement is usually by roasting the ore

to create germanium oxides followed by conversion to germanium chloride using

hydrochloric acid and finally distillation of the resultant chlorides.

Germanium finds use in modern day electronic applications as part of SiGe

RF-CMOS processes for cutting edge RF technology in laptops, mobile phones,

PDAs and other mobile devices. SiGe is rapidly replacing GaAs based micro-

electronics technology due to lower production costs for comparable devices and

integrability with traditional silicon CMOS processing.

Germanium is a key component in contact formation for GaAs devices for

use in optoelectronics and high frequency applications. Germanium is also used as

a substrate for growth of GaAs-based multijunction solar cells. These are used for

space applications due to their high efficiency (>40%) and for terrestrial concen-

trated photovoltaic (CPV) installations.

20

1.0 2.0 3.0

10

20

30

Ge

SiCdTe

ZnTe

Energy Gap

Effi

cie

ncy

Figure 1.3: Detailed balance limit of efficiency annotated withsome common photovoltaic materials for 1 sun illumination [3]

1.1.5 The Shockley-Queisser limit

Thermodynamic considerations place an upper bound on the efficiency of a

single junction solar cell. By appealing to the principle of detailed balance, i.e.

that each process (in this case absorption of energy) should be equilibrated by its

reverse process, William Shockley and Hans Quisser [3] set out the theoretical limit

for solar cells of a single junction. Fig. 1.3 depicts the detailed balance efficiency

limit for one sun illumination as a function of energy gap and is annotated with

21

some common materials under consideration in this work.

Fig. 1.3 shows that the ultimate efficiency of a CdTe or silicon single

junction solar cell is roughly 30% whereas for a germanium or ZnTe solar cell the

ultimate efficiency is roughly 19%. Obviously from this chart we can gauge the

how far away from the peak efficiency point (roughly 1.3-1.4 eV) we can stray

before suffering a performance degradation. ZnTe and Ge are therefore simply too

distant in energy from the maximum efficiency point to be considered for single

junction operation, but are suitable for use as a component of a multijunction

tandem arrangement.

A multijunction solar cell is a n-junction structure whereby sunlight is har-

vested by two or more solar cells of different energy gaps. Fig. 1.5 gives some

examples of possible tandem solar cell configurations. By way of example, simply

by inspection of Fig. 1.3, the combination of silicon or CdTe in tandem with ger-

manium should allow the germanium solar cell, with its lower ultimate efficiency,

to harvest any photons not collected by the upper cell. The total efficiency of

the device should be roughly equal to the combined individual efficiencies, once

shading of the bottom cell is taken into account. Hence even a naive analysis

shows that the Shockley-Queisser limit for single junction cells can be overcome

by adding more cells to the stack. It must be stressed that thermodynamic laws

have not been violated; but rather the incoming energy is now being converted

more efficiently through spectral splitting between multiple absorbing cells. Solar

cells of 3 or more junctions have been fabricated and are used in space applica-

tions, where the insolation outside earths atmosphere (AM0) is more intense, and

in terrestrial concentrator arrangements, where light is concentrated by a lens to

focus the incoming radiation onto a small photovoltaic device.

Increasing the concentration ratio (the number of “suns”) is another way

to overcome the Shockley-Queisser limit for 1 sun illumination. Again, this does

22

not imply that we have violated thermodynamic laws, but rather that the ultimate

efficiency of a photovoltaic device is actually a function of the power density of

incoming radiation. By increasing the concentration ratio we increase the power

density and shift the detailed balance limit. This is known as concentrated photo-

voltaics. There are some obvious drawbacks to concentrated photovoltaics. Firstly,

the photovoltaic device must be robust enough to handle high solar concentration

ratios without degrading or failing. Secondly, a large lens assembly is required to

focus light onto the concentrated PV device, and this usually requires a tracking

system to follow the sun throughout the day. This adds cost and complexity to

the system as a whole.

1.1.6 Detailed balance limit of efficiency of tandem solar cells

Detailed balance limit of efficiency computations can be performed for tan-

dem solar cells. This yields the efficiency limit as a function of energy gaps for a

particular concentration ratio. For a two junction tandem, a simple contour plot

can be devised which allows the data to be visualised. An excellent reference for

this is De Vos [4]. Such a contour plot is presented in Fig. 1.4.

Fig. 1.4 places the CdTe and Germanium tandem combination on the 40%

isoefficiency curve. While not optimal, the difference between the optimal tandem

combination in terms of ultimate efficiency is not particularly high (2%). Hence

despite being a suboptimal set of energy gaps, purely out of thermodynamic con-

siderations, CdTe and Ge are still well situated in energy space. In this way the

thermodynamic limit of a single junction CdTe solar cell may be circumvented by

the addition of germanium as a lower cell. Hence at a time when CdTe technology

approaches thermodynamic limits, with commercial modules performing close to

23

3

2

1

0 1 2 3

20

30

35

40

4235

30

20

20

CdTe/Ge

tandem

Figure 1.4: Detailed balance limit of efficiency for a two junctiontandem solar cell under 1 sun illumination [4]

theoretical upper bounds, research into tandem cells will become a viable option

to overcome this boundary.

A tandem combination with germanium may also be beneficial from a raw

materials point of view. As mentioned, Te is a rare material and its reserves are

limited. By raising the efficiency of a CdTe module by tandem cell technology, less

scarce tellurium is required to effect the same photovoltaic energy conversion. For

example, since the maximum efficiency of the tandem combination is, in relative

24

terms, greater by more than 25%, less than 3/4 the amount of tellurium is required

for the same power output, lowering the amount of raw material required to achieve

the same end result.

n+ p+p-Ge

Sapphire

Passivation /Anti-reflectionlayer

Grid bars

bottom cell

top cell

Illumination

Buffer layer

p-CdTe absorber

ITO

In grid barAu grid bar

Mechanically stacked tandem

p+ p+p-Ge

Rear contacts

Illumination

Monolithic tandem

p-CdTe

Front contacts Front passivation /

AR coating

CdTe n+ emitter

Tunnel junction +

Ge n+ emitter

Rear passivation

Rear metallisation

Figure 1.5: Mechanically stacked vs. monolithic solar moduleconfigurations

1.1.7 Mechanically stacked vs. monolithic combination

In this work we consider two possible configurations for a tandem CdTe/Ge

photovoltaic device. These are the mechanically stacked and monolithic tandem

configurations, and are depicted in Fig. 1.5. In the mechanically stacked con-

figuration, the two photovoltaic cells forming the tandem device are physically

created separately and then mechanically stacked and interconnected to form a

25

complete electronic device. This has the advantage that device processing is sepa-

rated for both cells which can be of advantage for materials which cannot be easily

combined. Issues which preclude monolithic combination could be low tempera-

ture alloying preventing high temperature processing, lattice mismatch, and other

physical properties that may render the combination awkward from a materials

science perspective.

In the monolithic structure, however, we treat Ge as a substrate for thin

film deposition and grow active device layers on the substrate using a thin film

growth technology like molecular beam epitaxy (MBE) or metal-organic chemical

vapour deposition (MOCVD). This necessitates a certain amount of compatibility

between substrate and epi-layers. For example, heteroepitaxy is difficult if the

lattice constants are not well matched. It is also possible for the substrate to react

with the epilayer material during growth or subsequent processing. Care must

therefore be taken when choosing materials.

Another key difference between monolithic and mechanically stacked tandem

structures is the interconnect between the cells of the tandem. In a mechanically

stacked tandem, the two cells may simply be connected in series provided the

photocurrents are well matched, or the device may not be interconnected at all,

yielding a four terminal device. In a monolithic structure, the interconnect between

the two cells could also be by two or four wire external connection, or by internal

connection consisting of a tunneling/recombination junction (TRJ).

A TRJ is a region of the device where a heavily doped p+/n+ junction

is formed where majority carriers recombine either by band-to-band tunneling

(BTBT) or trap-assisted tunneling (TAT). In BTBT, carriers from the conduc-

tion band on the n+-doped side of the junction tunnel to the valence band of the

p+-doped side and vice versa. In TAT, carriers recombine through traps and defect

states at the interface of the two materials. The key motivation for this type of

26

interconnect is to allow majority carriers of the two series connected sub-cells to

recombine but prevent minority carriers from recombining, allowing a current to

flow through the tandem device. If minority carriers recombine at this interface,

efficiency will be impaired.

1.1.8 Development of Cadmium Telluride technology in research

circles

Cadmium telluride was identified in the 1970’s as a possible solar cell material

due to the existence of low cost fabrication techniques and its high absorption

coefficient [5]. In practical terms this means that thin films of only a micron or

so are sufficiently absorbent of solar radiation for useful photovoltaic conversion.

It was shown that the conversion efficiency of CdTe/CdS solar cells could reach

17% [6], which has been reached and even exceeded in recent years. More modern

estimates show the maximum efficiency to be in excess of 29% [7]. Early research

focused on growing CdS layers on single crystal CdTe substrates using chemical

vapour deposition (CVD) and vacuum evaporation, for which efficiencies peaked

at 10% and 8%, respectively [8].

Chu et al. [9] demonstrated CdTe pn-junctions formed by ion implantation in

1978. The cells demonstrated 3% solar efficiency. It must be noted that the cells

involved were a homojunction type cell, which may explain the poor efficiencies

due to excessive surface recombination loss [7]. Werthen et al. [10] demon-

strated 10.5% efficient CdTe buried homojunctions formed from crystalline CdTe

substrates coated with ITO deposited using electron beam (e-beam) evaporation.

CdS/CdTe heterojunctions formed in an analogous manner were also realized with

efficiencies approaching 7.5%. Matsumoto et al. [11] fabricated 12.8% efficient

CdS/CdTe solar cells using screen printing in 1984. Front contacts were fashioned

from Ag/In and rear contacts from carbon and silver. A small amount of copper

27

(50 ppm) was added to the carbon electrode followed by annealing in nitrogen

atmosphere for 30 minutes at 400C. This was seen to improve efficiency, which

was attributed to p+ doping of the CdTe layer by the copper.

In 1991 Woodcock et al. [12] reported a 10.1% efficient n-CdS/p-CdTe

heterojunction solar cell deposited on tin oxide coated glass. CdS was deposited

from an aqueous solution of cadmium ions with thiourea acting as sulphiding agent

[12]. The CdTe layer was electrodeposited from a cadmium rich electrolyte solution

and was 1.7µm thick. Interestingly, it was found that heat treatment in air in the

range 400-500C converts the CdTe layer to p-type. This is interesting since from

other reports we note that a slight non-stoichiometry alone is enough to cause type

conversion [13]. Large-area cells were fabricated using conventional interconnects

based on laser scribing [12]. The back contact was formed on a CdTe surface

which was modified to be tellurium rich, thus lowering contact resistance.

In 1993 Britt and Ferekides [14] reported highly efficient CdTe solar cells

manufactured using chemical bath deposition (CBD) of the cadmium sulphide

(CdS) window layer and close spaced sublimation (CSS) of the p-type cadmium

telluride layer. The efficiency was reported to be 15.8% under AM1.5 illumination.

In the first step of manufacture, fluorine doped SnO2 films were deposited on glass

using MOCVD to provide a low resistance contact to the CdS layer. Next the CdS

window layer was formed using CBD. The thickness of the films was typically 0.07

- 0.1 µm. Prior to CdTe deposition, the structures were annealed for 5-20 minutes

in a hydrogen atmosphere. This was found to improve fill factor in the final device

structure. Next, CdTe was deposited using CSS from a 99.999% purity cadmium

telluride source. Typically, 5 µm thick CdTe was deposited.

In 2001, Wu et al [15] presented high efficiency polycrystalline thin film so-

lar cells with efficiencies of 16.5% and fill factors higher than 77% percent. The

structures used departed significantly from the more conventional SnO2/CdS/CdTe

28

makeup and instead created window/top contact layers from CTO (cadmium stan-

nate) to improve fill factor and ZTO (zinc stannate) to improve reproducibility of

the cells. The CTO and ZTO films were first deposited using RF magnetron sput-

tering and varied from 100-300 nm in thickness. The cadmium sulphide layer was

deposited using chemical bath deposition (CBD) and the cadmium telluride layer

was deposited using CSS, as in conventional methods for CdTe solar cell fabrica-

tion. The CdTe was deposited at 570-625C for 3-5 minutes in O2/He atmosphere

[15], and after CdTe deposition the cells were subjected to CdCl2 treatment for

15 minutes at 400-430C to promote re-crystallisation of the CdTe. This is a typ-

ical step in CdTe solar cell processing, often referred to as activation. The back

contact was formed by a layer of CuTe:HgTe-doped graphite paste, followed by a

layer of silver paste [15].

In 1996, Shao et al [16] demonstrated an 11.6% efficient RF magnetron-

sputtered thin-film CdTe solar cell. The same group [17] demonstrated improved

formation of CdTe solar cells using RF-magnetron sputtering in 2004. In this

case, the transparent conducting oxide (TCO) was aluminium doped zinc oxide

(ZnO:Al), since it shows excellent transparency over the entire visible spectrum

[17]. This TCO is not normally used since it reacts with CdS under the high

temperatures required for close spaced sublimation or due to environmental inter-

actions during electroplating. RF sputtering, being a low temperature deposition

technique does not require such extreme temperatures and hence allows ZnO:Al

to be explored as a possible TCO. Hall measurements showed ZnO:Al to have

higher mobility than the more usual fluorine-doped tin oxide (SnO2:F). To further

compare the two TCOs, cells were created on ZnO:Al/aluminosilicate glass (ASG)

and SnO2:F soda lime glass (SLG) by depositing 0.13 µm CdS and 2.3 µm CdTe

by RF magnetron sputtering at 250C. Cells were activated using CdCl2 treatment

at 387C. Efficiency was 14% for the ZnO:Al cell and 12.6% for the SnO2:F cell.

29

ZnO:Al cells however showed higher performance degradation (42.7% as opposed

to 27.1%) under stressing/light soak conditions, which was attributed to lower

thermal stability or interdiffusion across the ZnO/CdS interface.

The most spectacular advancement in CdTe solar cell technology came as

CdTe solar panel manufacturer First Solar demonstrated a string of record module

efficiencies over the past five years culminating in a 21.5% efficient cell first shown

in early 2015 [18]. Unfortunately, the manufacturer is very secretive about techni-

cal details and the precise mechanism by which the efficiency of their modules was

improved. For example, did they alter deposition processes to improve lifetimes,

or did they focus instead on doping the CdTe with a suitable acceptor to gain on

open circuit voltage, or perhaps they invented a novel technology for back con-

tact formation to reduce contact resistance and improve fill factor, or finally did

they concentrate on window and buffer layer optimisation to allow for better light

penetration to the absorber layer, or perhaps even some combination of these? In

any case, the result demonstrates how far CdTe technology has come in over 35

years of continuous development as illustrated in Fig. 1.6 in terms of efficiencies

achieved.

1.1.9 Germanium standalone solar cell research

Stand-alone germanium solar cells have received considerable research inter-

est for their use in tandem/concentrator cells, usually in combination with GaAs

top absorbers for space applications. They have also received attention for use in

thermophotovoltaic (TPV) systems.

Venkatasubramanian et al. [19] presented a 9% efficient (AM0) germanium

solar cell as a part of an investigation of Ge and Si0.07Ge0.93 solar cells for bottom

30

3

5

7

9

11

13

15

17

19

21

Effi

cie

ncy

1980

1983

1986

1989

1992

1995

1998

2001

2004

2007

2010

2013

2016

Chu et al.

Wethen et a

l.Matsu

moto

et al.

Woodcock et al.

Britt &

Fere

kides

Shao et al.

Wu et al.

Shao et al.

First

Solar inc.

Year

Figure 1.6: CdTe solar cell research milestones

cells in tandem structures for space applications in 1991. The junctions were

grown by chemical vapour deposition (CVD) at reduced pressures. The germanium

devices were grown at temperatures ranging from 600 to 800C. The junctions were

mostly p-on-n structures. The base regions were 6 µm and the emitters ranged

from 0.2 - 1µm in thickness. An AlGaAs passivation layer was used, which led to

an almost 20% improvement in external quantum efficiency.

Khvostikov et al. [20] demonstrated 5% efficient (AM1.5D, 100 suns) ger-

manium solar cells realised with ZnS/MgF2 antireflection coatings for thermopho-

tovoltaic applications. Junctions were formed by diffusion of zinc into n-type

germanium substrates. A layer of LPE grown p-type GaAs was used for surface

passivation which improved output voltage and efficiency.

Posthuma et al. [21] demonstrated a 6.7% efficient (AM1.5) stand-alone

germanium solar cell in 2003. The shallow emitter was realised using a phosphorus-

containing spin-on dopant source. Diffusion time was kept short to reduce surface

roughening and realise a shallow emitter, and in a similar fashion the diffusion

31

temperature was optimised. For passivation, a thin layer of amorphous silicon was

deposited using plasma-enhanced chemical vapour deposition (PECVD). A silver

finger pattern served as front contact and was diffused through the passivation

layer. This innovation was necessary to circumvent the lack of a uniform wet etch

with high selectivity between the amorphous silicon and germanium. Finally an

anti reflective-coating of ZnS and MgF2 was applied.

The same authors [22] reported an improved efficiency of 7.8% (AM1.5G)

in 2007. The fabrication process was essentially identical, however it featured

contacts formed from a thin layer of palladium and a thick layer of silver (as

opposed to the aluminium used in earlier studies) which helped to improve cell fill

factor reproducibility.

These results are summarised in Fig. 1.7.

1.1.10 Growth of CdTe on germanium

Cadmium telluride has previously been grown on germanium substrates using

molecular beam epitaxy (MBE). Matsumura et al. [23] prepared CdTe 〈111〉 and

CdTe 〈100〉 on 〈111〉 and 〈100〉 Ge substrates. The growth temperature ranged

between 150 C and 400 C .

It was noted that as crystallinity improved with growth temperature, the

films became milky in appearance. Two effusion cells were used during growth,

Cd at 290 C and Te at 405 C . From these nominal temperatures, flux ratio was

varied between 1/5 to 5/1. It was found that at ratios 1/2 - 1/5 (Cd:Te) the film

was polycrystalline. Twinning was observed on 〈111〉 substrates but not on 〈100〉

substrates. X-ray diffraction showed decrease of the rocking curve full-width at

half maximum (FWHM) for increasing substrate temperature, up to 350 C .

Zanatta et al. [24] presented 〈211〉B CdTe films grown on 〈211〉 Ge sub-

strates at a growth temperature of 250 C . During growth the temperature was

32

ramped to 320 C to optimize the crystal quality. The growth rate was 45 nm

per minute. Double crystal rocking curves (DCRC) for x-ray diffraction gave the

average FWHM of 89 arc seconds with a std. deviation of 6 arc seconds.

5

6

7

8

9

Effi

cie

ncy

1990

1993

1996

1999

2002

2005

2008

Venkatasu

bram

anian

(AM0)

Year

Khvostiko

v

Posthum

aPosth

uma

Figure 1.7: Ge solar cell research milestones

The efforts of Zanatta et al. came about in the context of HgCdTe growth on

CdTe buffer layers. HgCdTe is a high performance infrared detector material that,

despite its excellent characteristics for IR detector applications, has the unfortunate

disadvantage that it lacks a high quality large area lattice matched substrate.

HgCdTe is lattice matched to Cd0.95Zn0.05Te but substrates are difficult to fabricate

in large area format due to brittle mechanical characteristics [24, 25]. Hence

alternative substrates such as Si, GaAs and Ge have been considered for HgCdTe

growth. Ge has a better coefficient of thermal expansion and lattice constant

match with HgCdTe in comparison to silicon, hence the choice as a potential

substrate candidate.

33

In this work we consider the use of Ge not only as a potential substrate but

also as an active device layer with photovoltaic devices formed prior to CdTe epitax-

ial growth. This would necessitate altered growth conditions to prevent damage to

the underlying germanium device during epitaxial growth. As a substrate, Ge still

has the unfortunate disadvantage of lattice mismatch (14%) with CdTe, however

this is more favourable than silicon (19%). Germanium is chosen predominantly for

more efficient spectral splitting as opposed to lattice matching. Since Ge has the

narrower energy gap, it can absorb the longer wavelength photons out to roughly

1800 nm, whereas silicon can only absorb out to roughly 1100nm. This extra ab-

sorption region should allow for better matching of photocurrents between the two

cells of the tandem stack. The ability to absorb longer wavelength photons as well

as match the photocurrents between the two cells should allow for the creation of

an efficient two junction solar cell, with potentially higher efficiency than its single

junction CdTe counterpart.

1.2 Research outcomes and thesis outline

From the discussion in Section 1.1.8 it is evident that good progress has been

made over the past 35 years in CdTe processing technology. With the efficiency

of single junction CdTe solar cells now having been demonstrated at 21.5%, the

pathway towards the ultimate efficiency (≃30%) of single junction CdTe seems

comparatively free of encumbrance. However when that ultimate efficiency is to

all intents and purposes reached, progress can only be made by expanding into the

realm of multi junction solar cells. The aim of this work is not to advance single

junction CdTe processing any further, since this sphere of research is being under-

taken at commercial scale by manufacturers with significant research expertise and,

due to their vested interest in the development of CdTe technology, commensurate

research budgets. We assume in this work that CdTe single junction technology

34

has already developed to the point where a tandem configuration is justified, and

we will concentrate on development of the Ge bottom cell and interconnect.

In Section 1.1.9 the progress of standalone germanium solar cells was out-

lined. The conversion efficiencies at one-sun are quite low in the results hitherto

published for germanium solar cells; it begs the question how much of an efficiency

improvement can be expected from the addition of a germanium bottom cell to

form a multijunction solar cell. Device processing techniques would need to be

advanced further until single junction germanium solar cells yielded efficiencies of

10% or more to add substantial improvement to CdTe solar cell efficiency in a tan-

dem configuration. For example, while theoretically any efficiency improvement

is significant, the extra expense may not be justifiable. The research challenges

are therefore how to reduce surface recombination in Ge standalone solar cells

with adequate surface passivation, how to make low resistivity contacts to germa-

nium, and how to form device regions in a low cost manner without any unwanted

contamination which may impair device efficiency.

This investigation therefore expands upon the work of Posthuma et al. [21,

26, 22, 27, 28] to improve upon device active region formation, passivation and

contacting technologies developed by this group in order to further germanium

standalone solar cell research, as well as investigating properties of the CdTe/Ge

heterojunction. Materials growth draws on research by Zanatta et al. [24, 29] to

develop a process for growing CdTe epitaxially on Ge.

We now turn our attention to investigating these research challenges in the

following chapters. In Chapter 2, we explore analytical and numerical simulation of

photovoltaic devices and create a simple simulation framework for the evaluation

of single junction solar cells. This framework consists of both well-known closed

form solutions of the drift-diffusion equations as well as a compact numerical drift-

diffusion solver that discretises and solves these equations over a small mesh. The

35

purpose of developing this framework is to add insight into numerical simulation

and complement more complicated commercial device simulators.

In chapter 3 we consider, simulate and optimise the device structures. We

do so by employing a commercial simulator application and our self-developed

simulation framework to simulate devices by solving the drift-diffusion equations for

solar cells illuminated by the solar spectrum. Of particular importance is matching

the photocurrent between the two cells, particularly in a monolithic configuration.

A comparison is made between monolithic and mechanically stacked tandem solar

cells. For the case of a mechanically stacked tandem, photocurrents are matched

so that the individual sub-cells can be series connected to form a two terminal

device.

Chapter 4 considers low cost techniques for junction formation for germa-

nium optoelectronic devices. Doping of bulk germanium is considered using spin-on

dopants, silica or polymer films which are spun onto a target wafer to deliberately

introduce impurity atoms into device regions. Three techniques are presented for

device active region formation, these being “sandwich-stacked” diffusion, proximity

doping, and direct spin-on doping. The three methods give the three possible con-

figurations for doping germanium in a low cost manner by either direct contact or

vapour transport. Hence a proof-of-concept of all three techniques demonstrates

possible avenues for low cost manufacture of germanium sub-cells for multijunction

tandem application.

Chapter 5 examines passivation, antireflection and contacting techniques for

germanium optoelectronic devices. We consider a range of chemical pre-treatments

and passivation layers and compare their efficacy by measuring the lifetime using

a simple photoconductive decay apparatus. Wet and dry chemical pre-treatments

are necessary to prepare the sensitive germanium surface prior to passivation layer

deposition. This is done so by terminating dangling bonds at the highly chemi-

36

cally reactive germanium surface so that when the passivation layer is deposited

the complete structure is comparatively free of recombination centers and defects.

This is critical to high performance photovoltaic devices. Investigated passiva-

tion layers are inductively coupled plasma enhanced chemical vapour deposition

(ICPECVD) grown thin films. The advantage of using an ICPECVD reactor to

deposit the passivation layer is the potential for in-situ dry pre-treatments with re-

active gases, and it is found that in-situ ammonia pretreatment increases minority

carrier lifetime.

In Chapter 6, we investigate materials growth of CdTe and ZnTe on sapphire

and germanium, and investigate the properties of a CdTe/Ge heterojunction. CdTe

and ZnTe thin films on germanium and sapphire are prepared using molecular

beam epitaxy (MBE) and thermal evaporation, respectively, and are characterised

by a variety of methods including RHEED, optical transmission methods, and

X-ray diffraction (XRD). A sample of CdTe grown on a germanium substrate is

processed into a heterojunction device and the electronic properties are investigated

by measuring the IV and CV characteristics.

Chapter 7 summarizes and makes conclusions about the work as a whole.

Chapter 2

Analytical and numerical techniques for optoelectronic device modelling

2.1 Elementary theory of solar cells

2.1.0.1 Theory of pn junctions

A pn junction is a semiconducting device that consists of two doped regions,

one doped p-type, that is to say, doped with acceptors, and the other doped n-

type, that is, with donors. By bringing these two semiconducting regions together,

a potential barrier known as the built-in potential is formed at the metallurgical

junction, which can be used to separate charge carriers, and hence produce a

photocurrent when illuminated.

Fig. 2.1 shows the pn junction at equilibrium. The symbols used are ex-

plained in Tab. 2.1. Diagram a) shows the space-charge distribution, b) the

electric field distribution, c) the potential and d) the energy band diagram. At

equilibrium, the differing signs of the charge on either side of the junction brings

about the space-charge region. Here, local electric fields cause carriers to cross

the metallurgical junction at x = 0 to cancel out the imbalance of carriers. The

space-charge or depletion region then becomes depleted of carriers, and an electric

field develops across the junction, with a built-in potential ψbi . This forces the

Fermi-level to sit level within the device.

It is this built-in potential that sweeps carriers out of the junction if they

enter it, and in particular separates electron-hole pairs into their constituent parts,

38

i.e. electron and hole. In this way, minority carriers can be swept across the

junction to become majority carriers and be collected at the contact. This is the

basic principle by which pn-junction solar cells operate.

d)

c)

b)Area = built in potential,

a)

+

-

0

0

depletion region

Depletion charge

p-region n-region

Donor density

Acceptor density

Figure 2.1: pn junction at equilibrium, after [30]

39

ND the donor doping density

NA the acceptor doping density

WDp the width of the depletion region in the p-type material

WDn the width of the depletion region in the n-type region

E the electric field

Em the maximum electric field

ψbi the built-in potential

ψp the potential in the p-type region

ψn the potential in the n-type region

ψBp

the energy difference between the

intrinsic level and the Fermi level

in the p type region

ψBn

the energy difference between the

intrinsic level and the Fermi level in

the n type region

φp

the energy difference between the

valence band and the Fermi level in

the p-type region

φn

the energy difference between the

conduction band and the Fermi level

in the n-type region

Table 2.1: Symbols for Fig. 2.1

2.2 Derivation of an analytical model

Whilst solar cells are becoming increasingly complicated in terms of device

structure, necessitating complex numerical simulation techniques for their analysis,

simplified analytical models can be derived subject to certain assumptions. An

40

analytical model can be used as an adjunct to a more thorough numerical model,

and has the following advantages:

• An analytical model will yield results much faster than a numerical simu-

lation (which may take hours to converge), facilitating experimentation.

• An analytical model may help to verify results from numerical simulation.

• Analytical models give rise to closed form expressions for determining the

effect of parameter variation on key device metrics.

Analytical solutions can be obtained for the light and dark currents in the

three regions of a p-n junction solar cell, subject to certain simplifying assumptions.

These are well known and can be found in many references, such as [31]. These

solutions are derived in the following section to lay the foundation for the analytical

model. The pn-junction structure under consideration is depicted in Fig 2.2. Here,

x is defined to be the position within the cell, with x = 0 set to be at the

metallurgical junction. H is the total width of the cell, and Hp and Hn are the

widths of the p and n type regions respectively. Wn and Wp are the widths of

the depletion region in the n and p-type regions respectively and W is the total

depletion region width.

Figure 2.2: Cell considered for derivation of an analytical model[31]

41

In deriving an analytical model, we begin with the following assumptions

[31]:-

• The analysis is restricted to one dimension.

• Light is incident normal to the surface, and we neglect scattering and

internal reflection.

• Both regions of the pn junction are non-degenerate and donors/acceptors

are fully ionized.

• There are no hot carrier effects and a photon excites a single electron-hole

pair.

• Minority carrier recombination is pseudo-first order in the bulk and at the

surfaces.

• Low level minority carrier injection/diffusion is the operative transport

mechanism.

• Device parasitics are ignored (we will consider them later using a circuit

analysis approach).

By assuming that minority carrier injection and diffusion are the only opera-

tive modes of transport, that we can ignore device parasitics, and further that the

cell remains in low injection throughout the bias/optical excitement range [31],

we can appeal to the principle of superposition and essentially decouple the light

and dark current densities, so that they can be modelled separately and the results

superposed, greatly simplifying the results.

When modelling the electrical properties of semiconductors, we solve the

following set of equations:-

42

∇2φ(x) = −∇E(x) = − 1

ǫǫ0ρ(x) (2.1)

Je(x) = q · (n(x) · µe · E(x) + Dn · ∇n(x)) (2.2)

Jh(x) = q · (p(x) · µh · E(x)− Dp · ∇p(x)) (2.3)

1

q∇ · Je(x)− re(x) + ge(x) = 0 (2.4)

−1

q∇ · Jh(x)− rh(x) + gh(x) = 0 (2.5)

Where φ(x) is the potential at x, E(x) is the electric field at x, ρ is the

space charge, ǫ is the relative permittivity of the material, ǫ0 is the vacuum per-

mittivity, Je and Jh are the electron and hole current densities, respectively, q is

the elementary charge, n(x) and p(x) are the electron and hole concentrations

respectively, re and rh are the electron and hole recombination rates respectively,

and ge and gh are the electron and hole generation rates, respectively. Equation

2.1 is the Poisson equation, Eqns. 2.2, 2.3 are the electron and hole drift-diffusion

equations, and Eqns. 2.4/2.5 are the electron and hole continuity equations.

By restricting ourselves to one dimension, we can differentiate Eqn. 2.2 and

substitute it into Eqn. 2.4. The procedure is then repeated for holes, yielding the

following set of equations which can be solved to yield the carrier concentrations

in the device [31]:-

Dn

d2n

dx+ µeE

dn

dx+ nµe

dE

dx− re(x) + ge(x) = 0 (2.6)

Dp

d2p

dx− µhE

dp

dx− pµh

dE

dx− rh(x) + gh(x) = 0 (2.7)

43

2.2.1 Recombination

For the recombination terms in Eqns. 2.6 and 2.7, we assume low injection

conditions and hence that recombination in the semiconductor is pseudo-first order

[31]. Hence the recombination rates may be written as [31]:-

re(x) =np − n0p

τe=

De(np − n0p)

L2e0 ≤ x ≤ Hp (2.8)

rh(x) =pn − p0

n

τh=

Dh(pn − p0n)

L2h−Hn ≤ x ≤ 0 (2.9)

Where np is the electron concentration in the p region, pn is the electron

concentration in the n region, De and Dh are the electron and hole diffusivities

respectively, Le , Lh are the electron and hole diffusion lengths respectively, and n0p

and p0n are the dark minority carrier densities.

The diffusion length for electrons and holes is given by [32]:-

Le =√

Deτ (2.10)

Lh =√

Dhτ (2.11)

where τ is the bulk lifetime for electrons and holes.

2.2.1.1 Surface recombination velocity

Surface recombination velocity is the rate at which carriers recombine at sur-

faces. These surfaces include the front and back surfaces of a solar cell as well as

any grain boundaries within the cell if it consists of polycrystalline material. Grain

boundaries act as recombination centers because the lattice is unterminated and

there are a great many defects and dangling bonds at such sites. The interfaces

between layers within the solar cell also act as surfaces, with a certain surface re-

combination velocity used to express the recombination rate due to surface effects.

44

Passivation is necessary to adequately terminate the crystal lattice at such

sites to reduce recombination rates and hence lower surface recombination velocity.

By passivating defects such as dangling bonds at surfaces and grain boundaries,

that is, rendering them inert, recombination can be prevented in such areas. This

will serve to increase charge collection probability, since carriers now have a lower

probability of recombining at interfaces and grain boundaries and hence more

chance of being collected at contacts.

In the context of analytical models, surface recombination velocity is usually

a boundary condition imposed on surfaces, i.e.

Dh

dp

dx

∣

∣

∣

∣

x=surface

= Sp · p|x=surface (2.12)

In analytical simulations of a single dimension, this parameter is used to

model the effectiveness of contacts to allow majority carriers to recombine in

preference to minority carriers, which determines the charge collection probability.

Hence in a single dimension, where the surface beyond the contact region cannot

be accounted for, surface recombination velocity at the contact itself is used to

encompass surface effects at cell front and back surfaces. Although Ohmic con-

tacts are usually considered to be sites of infinite recombination, the use of this

parameter in a one-dimensional simulation serves to factor in surface effects and

model the quality of device passivation for a particular cell.

2.2.2 Carrier absorption/generation

We now consider the generation of carriers in the semiconductor. These are

given by the so called Beer-Lambert expression, as follows [31]:

45

g ne (x) =g n

h (x) = αnλφ

emitterλ exp[−αn

λ(Hn + x)] − Hn ≤ x ≤ 0 (2.13)

g pe (x) =g

ph (x) = αp

λφbaseλ exp[−αp

λ(x −Wp)] 0 ≤ x ≤ Hp (2.14)

where αnλ and αp

λ are the absorption coefficients in the n and p regions,

respectively, and φemitterλ and φbase

λ are the photon fluxes into the emitter and base

region, respectively, as given by [31]:

φemitterλ = φ0

λ(1− rλ) photons ·m−2s−1 (2.15)

φbaseλ = φemitter

λ exp(−αnλHn)exp(−αp

λWp) photons ·m−2s−1 (2.16)

where φ0λ is the illumination, and rλ is the reflectivity for the wavelength

under consideration.

The absorption coefficients ανλ where ν ∈ (n, p) can be taken from tables

of the complex refractive index of the material, since the absorption coefficient is

related to the imaginary part (or k-value) of the complex refractive index [30]:

α =4πkrλ

(2.17)

2.2.3 Reflection

When two media of differing refractive index meet, light incident on the

interface will be partially reflected, partially absorbed, and partially transmitted.

This leads to a decrease in efficiency in solar cell devices, since any light reflected

from the air-semiconductor interface cannot contribute to the photocurrent.

To this end, solar cells front surfaces are usually capped with an antireflection

coating, which may consist of multiple layers of different materials. Antireflection

46

coatings are designed to give the lowest possible reflectance for the widest possible

region of the solar spectrum, in order to maximize cell efficiency.

The propagation of light in the system can be modelled using the direct

matrix method [33], [34]:-

Meq =

n∏

j=1

cos(φj)iηjsin(φj)

iηj sin(φj) cos(φj)

(2.18)

where Meq is the characterisation matrix of the thin film stack, nj is the

refractive index of the j th layer, φj =2πλneffj dj ,ηj = Y0njcos(θ) for parallel polari-

sation, ηj = Y0nj

cos(θ)for perpendicular polarisation, θj is the angle of incidence in

layer j , Y0 is the admittance of free space, and dj is the thickness of the j th layer.

From this, the characteristic matrix of the assembly can be written down:

B

C

= Meq

1

ηs

(2.19)

where ηs is the effective complex refractive index of the substrate similarly

defined as above.

The reflectance, R , transmittance, T , and absorption, A of the assembly of

j thin film layers can be obtained as follows [34]:

R =

(

η0B − C

η0B + C

)(

η0B − C

η0B + C

)∗(2.20)

T =4η0Re(ηs)

(η0B + C )(η0B + C )∗(2.21)

A = 1− R − T =4η0Re(BC

∗ − ηs)

(η0B + C )(η0B + C )∗(2.22)

To compute the reflectance at the top active layer (usually emitter) of a

solar cell, a matrix stack of all thin films from the illumination source to the

active layer is assembled for all wavelengths of the spectrum. The transmission

47

and reflectance can then be computed using the above relations. This yields the

incident light for all wavelengths of the spectrum for the topmost active region of

the solar cell. From there, the solar cell’s efficiency can be calculated. Note: this

does not incorporate reflection from the cell’s bottom contact and hence multiple

passes through the device. This is because the analytical models presented in this

work only compute photocurrents for a single pass of illumination.

2.2.3.1 Emitter (n-type) quasi-neutral region

In the emitter region we consider the current from minority carrier holes.

To obtain the hole concentration throughout the emitter (−Hn ≤ x ≤ −Wn), we

first substitute Eqn. 2.9 into Eqn. 2.7, then based on our assumption that the

carrier concentrations obey the principle of superposition, we subtract the terms

involving the dark hole concentration [31], yielding a differential equation solely

for the photo-generated holes in the n region (=pphn ) [31]:

Dh

d2pphn

dx2− Dhp

phn

L2h+ g n

λ (x) = 0 (2.23)

Note that the terms involving E are set to zero since there is no electric field

in the quasi-neutral region. Solutions to this equation will yield the carrier gener-

ation profile as a function of x for a particular wavelength of light, λ. Solutions

can be obtained subject to the following boundary conditions [31]:

pphn (−Wn) = 0 (2.24)

Dh

dpphn

dx

∣

∣

∣

∣

x=−Hn

= Sppphn (−Hp) (2.25)

These boundary conditions state that the hole concentration is zero at the

edge of the space charge region, and that the rate at which holes leave through

48

the front contact is equal to the front hole surface recombination velocity Sp times

the hole concentration at that point.

Equation 2.23 is an inhomogeneous second order differential equation, hence

its solution is of the form [31]:-

pphn (x ,λ) = CF + PI (2.26)

CF, the complementary function, is the solution of 2.23 with g nλ(x) set to

zero, and is of the form [31]:-

CF = Aphin cosh

(

x

Le

)

+ Bphin sinh

(

x

Le

)

(2.27)

The particular integral PI is some constant C times Eqn. 2.14. To find the

constant C, we simple set the CF to zero and substitute Eqn. 2.26 into Eqn. 2.23

[31]:-

PI =Cφemitterλ exp[−αn

λ(Hn + x)] = C · y (x)

⇒Dh

d2C · y (x)dx2

− DhC · y (x)L2h

+ y (x) = 0

⇒C

(

Dh(αnλ)

2 − Dh

L2h

)

+ 1 = 0

∴ C =− L2hDh[(αn

λ)2L2h − 1]

With some algebra, the solution for the hole density can be obtained [31]:-

49

pphn (x ,λ) =

φemitterλ αn

λL2h

Dh[(αnλ)

2L2h − 1]exp(−αn

λQn)

×

cosh[(Hn + x)/Lh] + (ShLh/Dh) sinh[(Hn + x)/Lh]

+(αnλLh + ShLh/Dh) sinh[−(Wn + x)/Lh]exp(α

nλQn)

cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)− exp[−αn

λ(Wn + x)]

(2.28)

where Qn is the width of the emitter region, Hn −Wn.

Differentiation gives the hole current profile in the emitter region [31]:-

jphemitter (x ,λ) =qDh

dpphn (x)

dx

=− qφemitterλ αn

λLh

[(αnλ)

2L2h − 1]exp(−αn

λQn)

×

− sinh[(Hn + x)/Lh]− (ShLh/Dh) cosh[(Hn + x)/Lh]

+(αnλLh + ShLh/Dh) cosh[−(Wn + x)/Lh]exp(α

nλQn)

cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)

−αnλLh exp[−αn

λ(Wn + x)]

(2.29)

The total current entering the junction from the emitter is given by jphemitter (−Wn,λ)

[31]:-

50

jphemitter (−Wn,λ) =− qφemitter

λ αnλLh

[(αnλ)

2L2h − 1]exp(−αn

λQn)

×

− sinh[Qn/Lh]− (ShLh/Dh) cosh[Qn/Lh]

+(αnλLh + ShLh/Dh) exp(α

nλQn)

cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)

−αnλLh

(2.30)

2.2.3.2 Base (p-type) quasi-neutral region

The solution for the photocurrent in the base quasi-neutral region is found

in a similar way, by solving electron continuity equation for the minority carrier

electron concentration in the region Wp ≤ x ≤ Hp [31]:-

De

d2nphp

dx2−

Denphp

L2p+ g

pλ (x) = 0 (2.31)

The solution takes the following form [31]:-

jphbase(x ,λ) =− qφbase

λ αpλLe

(αpλ)

2L2e − 1

×

− sinh[(Hp − x)/Le]− (SeLe/De) cosh[(Hp − x)/Le]

−(αnλLe − SeLe/De) cosh[(x −Wp)/Le ]exp(−αp

λQp)

cosh(Qp/Le) + SeLe/De sinh (Qp/Le)

+αpλLe exp[−α

pλ(x −Wp)]

(2.32)

where Qp is the width of the base region, Hp−Wp. The photocurrent density

flowing into the junction is given by jphbase(Wp,λ) [31]:-

51

jphbase(Wp,λ) =− qφbase

λ αpλLe

(αpλ)

2L2e − 1

×

− sinh[Qp/Le ]− (SeLe/De) cosh[Qp/Le]

−(αnλLe − SeLe/De) exp(−αp

λQp)

cosh(Qp/Le) + SeLe/De sinh (Qp/Le)

+αpλLe

(2.33)

2.2.3.3 Space-charge region

In deriving the current in the space charge region, we may consider either

electrons or holes; here a choice is made in favour of electrons. The assumption in

the space-charge region is that carriers are swept out by the built-in electric field

sufficiently quickly that no recombination occurs [31]. Hence, Eqn. 2.4 reduces

to:-

1

q

dJe

dx+ ge(x) = 0 (2.34)

This can be solved using simple integration to yield jphscr . To simplify the

integration, the problem can be made symmetric about the origin, x = 0. Defining

the illumination at this point as:-

φjnλ = φemitter

λ exp(−αnλHn) (2.35)

and defining the generation of carriers in the SCR:-

gλ(x) =

αnλφ

jnλ exp(−αn

λx) −Wn ≤ x ≤ 0

αpλφ

jnλ exp(−α

pλx) 0 ≤ x ≤ Wp

(2.36)

We can now integrate the carrier generation in the SCR to find the photocurrent:-

52

jphscr =− q

∫ Wp

−Wn

gλ(x)dx

=− q

(∫ 0

−Wn

αnλφ

jnλ exp(−αn

λx)dx +

∫ Wp

0

αpλφ

jnλ exp(−α

pλx)dx

)

=− qφjn [exp(αnλWn)− exp(−αn

λWp)] (2.37)

2.2.4 Total photocurrent

The total photocurrent for a single wavelength λ is simply the sum of the

previously derived photocurrents for the emitter, base, and space-charge regions

[31]:-

jph(λ) = jphemitter (−Wn,λ) + jphscr (λ) + j

phbase(Wp,λ) (2.38)

The total photocurrent is then the integral over all wavelengths of the illu-

mination spectrum [31]:

jph =

∫

AM1.5G

jph(λ) dλ (2.39)

2.2.5 Depletion region width

The depletion region width for a homojunction can be found by adopting the

exhaustion layer (Schottky) approximation. This allows the following expressions

to be derived [31]:-

Wn =NA

NA + ND

W (2.40)

Wp =ND

NA + ND

W (2.41)

W =

[

2ǫ0ǫps ǫ

nsψbi(NA + ND)

2

qNAND(ǫpsNA + ǫnsND)

]1/2

(2.42)

53

The built-in potential ψbi is given by [30]:-

ψbi = kT lnNDNA

n2i(2.43)

For a heterojunction, the depletion region width is given by [30]:-

WD1 =

[

2NA2ǫs1ǫs2(ψbi − V )

qND1(ǫs1ND1 + ǫs2NA2)

]1/2

(2.44)

WD2 =

[

2ND1ǫs1ǫs2(ψbi − V )

qNA2(ǫs1ND1 + ǫs2NA2)

]1/2

(2.45)

where WDn is the width of depletion region n, ǫsn is the dielectric constant

of region n, NDn is the donor concentration in region n, NAn is the acceptor

concentration in region n, ψbi is the built-in potential, and V the applied potential.

The built-in potential of the heterojunction is given by [35]:-

qψbi =∆EC −∆EV

2+ kTln

NdNa

ni ,n + ni ,p(2.46)

where ∆EC and ∆EV are the conduction and valence band discontinuities

respectively, Nd and Na are the doping densities either side of the heterojunction,

and ni ,p and ni ,n are the intrinsic carrier concentrations in the p and n regions,

respectively.

2.2.6 Dark current

2.2.6.1 Emitter dark current

Minority carrier holes are considered when computing the emitter dark cur-

rent in the n-type emitter region, in an analogous manner to that of section

2.2.3.1. Substituting Eqn. 2.9 into Eqn. 2.7 and dropping the electric field and

carrier generation terms, yields [31]:-

54

Dp

d2pDKn

dx− Dh(p

DKn − p0

n)

L2h= 0 (2.47)

which can be solved subject to the following boundary conditions [31]:-

De

dpDKn

dx

∣

∣

∣

∣

∣

x=−Hn

= Sh[pDKn (−Hn)− p0

n] (2.48)

pn(−Wn) = p0n exp(qVj/kT) (2.49)

where Eqn. 2.48 states that the flux of minority carriers at the emitter surface

(i.e. the front contact) is equal to the surface recombination velocity times the

number of carriers present at the surface. Equation 2.49 arises from the so-called

low-level injection conditions [31] and gives the minority carrier concentration at

the edge of the junction for some junction bias Vj .

The solution can be shown to take the following form [31]:-

pDKn (x) =p0

n + p0n[exp(qVj/kT)− 1]

×

cosh[(Hn + x)/Lh] + (ShLh/Dh) sinh[(Hn + x)/Lh]

cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)

−Hn ≤ x ≤ −Wn

(2.50)

Substituting into Eqn. 2.3 gives the hole current density profile in the emitter

[31]:-

jDKemitter (x) =

qDhp0n

Lh[exp(qVj/kT)− 1]

×

sinh[(Hn + x)/Lh] + (ShLh/Dh) cosh[(Hn + x)/Lh]

cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)

(2.51)

At the edge of the junction, i.e. at x = −Wn, the current flowing into the

emitter can be expressed as [31]:-

55

jDKemitter (−Wn) =

qDhp0n

Lh[exp(qVj/kT)− 1]Ξn (2.52)

where Ξn is the emitter width factor, given by [31]

Ξn =sinh[Qn/Lh] + (ShLh/Dh) cosh[Qn/Lh]

cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)(2.53)

2.2.6.2 Base dark current

The expressions for the dark electron density and electron current density

profile are derived in an identical manner to that of section 2.2.6.1. They are as

follows [31]:-

nDKp (x) =n0p + n0p[exp(qVj/kT)− 1]

×

cosh[(Hp − x)/Le ] + (SeLe/De) sinh[(He − x)/Le ]

cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)

Wp ≤ x ≤ Hp

(2.54)

jDKbase(x) =

qDen0p

Le[exp(qVj/kT)− 1]

×

sinh[(Hp − x)/Le] + (SeLe/De) cosh[(Hp − x)/Le]

cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)

(2.55)

As in section 2.2.6.1, the current flowing into the base from the junction at

x = Wp can be expressed as [31]:-

jDKbase(Wp) =

qDen0p

Le[exp(qVj/kT)− 1]Ξp (2.56)

where Ξp is the base width factor, given by [31]:-

Ξp =sinh[Qp/Le ] + (SeLe/De) cosh[Qp/Le ]

cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)(2.57)

56

2.2.6.3 Space-charge region dark current

In the dark space charge region Eqn. 2.6, and Eqn. 2.7 become [31]:-

Dn

d2n

dx+ µeE

dn

dx+ nµe

dE

dx= 0 (2.58)

Dp

d2p

dx− µhE

dp

dx− pµh

dE

dx= 0 (2.59)

Their solution is quite involved and can be found in [31]. However, since in

the dark space charge region there is (effectively) no generation or recombination,

it follows that the dark current density in the space charge region is constant.

On the assumption that all majority carriers injected at one end of the junction

become minority carriers when emerging from the space-charge region, the dark

current in the space-charge region can be expressed as [31]:-

jDKscr = jDK

emitter (−Wn) + jDKbase(Wp)

=

[

qDhp0n

LhΞn +

qDen0p

LeΞp

]

[exp(qVj/kT)− 1] (2.60)

Which reduces to the Shockley equation for Ξn = Ξp = 1 [31].

2.2.7 Device parasitics

Device parasitics can be conveniently modelled as two parasitic resistors, one

in series and one in parallel. The series resistance is the lumped total of contact

resistance, which itself can be resolved into contributions from the semiconduc-

tor/metal interface as well as the metallisation itself, and any bondout resistance.

The shunt resistance encompasses a variety of processes that act like a simple

resistance in parallel with the device. This could be due to pin holes and defects

that shunt out the emitter, or lateral defects causing junction shunting current to

flow at the edges of the mesa.

57

Figure 2.3: Two diode model showing shunt (RP) and seriesresistance (RS), effective diodes (D1 and D2), load resistance(RL), and the current source representing the short circuit currentdensity (JSC)

The model includes two diodes, having different ideality factors to account

for two regimes in the device characteristics. These are space-charge region (SCR)

generation and recombination (GR) currents, and quasi-neutral region GR currents.

The quasi-neutral region GR currents dominate at higher forward biases and the

diode which models this behaviour usually has an ideality factor close to one. The

space-charge region GR currents dominate at low forward bias and, depending

on material qualities, this part of the diode model usually has an ideality factor

closer to 2. The key parameters determining the ideality factor are the radiative,

Shockley-Read-Hall (SRH) and Auger parameters (see Section 2.3).

The SRH formalism describes recombination through a trap level. In regions

where the lattice is unterminated, such as grain boundaries, dislocations and bulk

defects, and surfaces, the SRH lifetime is significantly shorter. These defect sites

are accounted for by the higher ideality factor of the second diode. A device with

comparably fewer grain boundaries, bulk defects, and well passivated surfaces may

have a lower ideality factor for this second diode or the device may be well-behaved

enough to be described by a single diode model.

We can analyze the circuit in Fig. 2.3 fairly simply and write out the circuit

58

equations for it as follows:-

IL(Vj) = Id1(Vj) + Id2(Vj) + Ishunt(Vj) (2.61)

= J0,1A

(

exp

(

Vj

n1kT

)

− 1

)

+ J0,2A

(

exp

(

Vj

n2kT

)

− 1

)

+Vj

RP

(2.62)

VL(Vj) = Vj + RS IL(Vj) (2.63)

where IL, VL are the load current and voltage, J0,i , ni are the saturation

current densities of the ith diode, kT is the thermal voltage, and Vj is the junction

voltage.

We note that the factor[

qDhp0n

LhΞn +

qDen0p

LeΞp

]

has been encapsulated into

J0,1, diode 1’s saturation current density. J0,2 on the other hand may be either set

to zero or determined empirically.

2.2.8 Summary analytical model for solar cells

The complete analytical model, including photocurrent, dark current, and

device parasitics is summarized below for convenience.

2.2.8.1 Photocurrent

jph =

∫

AM1.5G

jph(λ) dλ (2.64)

jph(λ) = jphemitter (−Wn,λ) + jphscr (λ) + j

phbase(Wp,λ) (2.65)

59

jphemitter (−Wn,λ) =− qφemitter

λ αnλLh

[(αnλ)

2L2h − 1]exp(−αn

λQn)

×

− sinh[Qn/Lh]− (ShLh/Dh) cosh[Qn/Lh]

+(αnλLh + ShLh/Dh) exp(α

nλQn)

cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)

−αnλLh

(2.66)

jphscr =− q

∫ Wp

−Wn

gλ(x)dx

=− q

(∫ 0

−Wn

αnλφ

jnλ exp(−αn

λx)dx +

∫ Wp

0

αpλφ

jnλ exp(−α

pλx)dx

)

=− qφjn [exp(αnλWn)− exp(−αn

λWp)] (2.67)

jphbase(Wp,λ) =− qφbase

λ αpλLe

(αpλ)

2L2e − 1

×

− sinh[Qp/Le ]− (SeLe/De) cosh[Qp/Le]

−(αnλLe − SeLe/De) exp(−αp

λQp)

cosh(Qp/Le) + SeLe/De sinh (Qp/Le)

+αpλLe

(2.68)

2.2.8.2 Darkcurrent

jDK (Vj) =

[

qDhp0n

LhΞn +

qDen0p

LeΞp

]

[exp(qVj/kT)− 1] (2.69)

jDK ,parasitics(Vj) = J0,1

(

exp

(

Vj

n1kT

)

− 1

)

+ J0,2

(

exp

(

Vj

n2kT

)

− 1

)

+vj

RP

(2.70)

VL(Vj) = Vj + RS IL(Vj) (2.71)

60

2.2.8.3 Total current

jtotal(Vj) =

∫

AM1.5G

jph(λ) dλ− jDK ,parasitics(Vj) (2.72)

2.2.9 Key device characteristics

There are several key parameters that can be extracted from solar cell device

characteristics that allow the performance of a cell to evaluated and compared with

other devices. These can be summarised as follows:-

• Short Circuit Current

• Open Circuit Voltage

• Fill Factor

• Efficiency

2.2.9.1 Short circuit current

The short circuit current (i.e. when the load is effectively a short circuit) is

roughly equivalent to the photocurrent, since at zero bias the dark current should

be zero. In practice, due to recombination effects which are not included in the

idealised diode model (Auger recombination, GR currents, trap-assisted and band

to band tunneling, shunting effects due to process related defects etc.) this may

not be the case and a very small leakage current may exist, as evidenced in the

dark IV characteristics. In general, however, at zero bias these effects are much

smaller than the photocurrent and can be neglected.

Short circuit current is therefore related to the number of photons absorbed

by the device. This necessitates adequate collection of incoming photons by en-

suring they are absorbed in the device and not lost due to reflection or inefficient

61

absorption. Hence the absorber layer must account for reflections at the front

surface and prevent photons from being lost due to optical mismatch between the

front surface and the atmosphere, as well as ensuring that any unabsorbed photons

do not exit from the rear surface, but are, in fact, reflected at the rear surface and

hence absorbed on a second or further pass through the device. Better yet would

be a light trapping design that traps photons and keeps them confined within the

body of the device due to total internal reflection.

From Eqn. 2.66 and Eqn. 2.68, it is evident that diffusion length, diffusivity,

and surface recombination velocity play a vital role in determining the probability

of photo-generated electron-hole pairs reaching a contact for collection. In order

to maximise photocurrent, surface recombination velocities must be minimised,

and diffusivity (i.e. mobility) must be maximised, in so doing improving diffusion

length. To lower surface recombination velocities, front and rear surfaces must

be adequately passivated. This necessitates combined passivation/antireflection

coatings that terminate the semiconductor surface as well as providing light trap-

ping properties. To improve diffusion lengths, mobilities must be optimised. This

necessitates high quality materials with good electronic properties. Single crystal

semiconducting devices will always outperform polycrystalline and amorphous thin

films in this regard, at the expense of increased material cost. The same can

be said of lifetimes, also a component in determining diffusion lengths. Hence

there is a trade off between material cost and electronic properties when choosing

materials to optimise photocurrents.

2.2.9.2 Open circuit voltage

Open circuit voltage conditions occur when the photocurrent exactly cancels

the dark current, yielding no net current. For this case we can write:-

62

jph =

[

qDhp0n

LhΞn +

qDen0p

LeΞp

]

[exp(qVoc/kT)− 1]

⇒ exp(qVoc/kT) = 1 +jph

qDhp0n

LhΞn +

qDen0pLe

Ξp

⇒ Voc =kT

qln

1 +jph

qDhp0n

LhΞn +

qDen0pLe

Ξp

(2.73)

giving the open circuit voltage of the cell, Voc . Since the open circuit voltage

is essentially determined by the ratio of the light and dark currents, it is necessary to

maximise the photocurrent whilst minimising the dark current to optimise the open

circuit voltage. This means that many of the the material parameters affecting

the photocurrent also determine the open circuit voltage, i.e. diffusion length,

diffusivity, and surface recombination velocity. In addition, open circuit voltage is

determined by the emitter and base doping densities. By increasing the doping

densities, dark current can be reduced, improving open circuit voltage. It can be

seen from equation Eqn. 2.73 that if the photocurrent were directly scaled, for

example by using a lens to concentrate the incident radiation, the open circuit

voltage would increase. In this way, the power increases with concentration of the

incoming solar radiation. This power increase is the principle by which concentrated

photovoltaics (CPV) is feasible.

2.2.9.3 Efficiency

Efficiency is defined as the ratio of the output power to the incident power,

or

η =Pout

Pin

× 100% (2.74)

Pin for AM1.5G illumination is 100 mW/cm2. Efficiency is determined by

the maximum output power of the device, at the device maximum power point,

63

since power output is determined by the load. To optimise efficiency therefore, the

position of the maximum power point must be optimised. This means optimising

not only short circuit current and open circuit voltage, but also fill factor, as

discussed in the next section. This is where device parasitics come into play as the

series and shunt resistance of the device shift the position of the maximum power

point, affecting the efficiency.

2.2.9.4 Fill factor

Voltage, Volts

Cell PV

Curr

ent

Area b

Area a

Pow

er

Cell JV

FF=Area b / Area a x 100%

Figure 2.4: Graphical depiction of maximum power point, opencircuit voltage, short circuit current, and fill factor and theirrelationship.

Fill factor is the geometric ratio of the area bounded by the maximum power

point and the open circuit voltage/short circuit current. This relationship is de-

picted graphically in Fig. 2.4.

FF =Jmpp · Vmpp

Jsc · Voc

× 100% =η × Pin

Jsc × Voc

(2.75)

64

Factors determining the fill factor include recombinative losses, since both

current and voltage are determined by diffusion length and surface recombination

velocity, and series resistance. Series resistance shifts the position of the maximum

power point back toward the current axis. Shunt resistance shifts the position of

the maximum power point down towards the voltage axis. Hence we need low

recombination losses, low series resistance and high shunt resistance to optimise

fill factor and hence efficiency.

2.2.10 Summary

A review of analytical solutions to the drift-diffusion equations has been

presented. From this, an analytical model for solar cells has been built up including

a discussion of device parasitics. The closed form nature of the analytical model

and governing equations means that a device can be modelled quickly. This allows

back-of-the-envelope calculations for novel device structures to be carried out

quickly, as a preparation for more detailed numerical calculations, which solve

the drift-diffusion equations numerically using finite element analysis rather than

rely on closed-form solutions allowing more complicated models to be evaluated.

We turn our attention to such solutions in the next section.

65

2.3 Numerical simulation

2.3.1 Introduction

In this section, numerical solution of the drift-diffusion equations is pre-

sented. This involves discretisation and solution of the drift-diffusion equations,

as well as discussion of various models of interest. The purpose of this section is

to build up a stand-alone unidimensional drift-diffusion solver for numerical simu-

lation of solar cells. This is a useful exercise that allows one to gain insight into

the function of numerical simulators, which are very complex and quite difficult to

understand at first.

2.3.2 Equation set in continuous form

The drift-diffusion set of equations belongs to a class of problems collectively

known as diffusion-advection-reaction problems. Their numerical solution is known

to be problematic, and this is particularly the case when dealing with semiconductor

problems, as the tight coupling between potential and carrier concentrations intro-

duces non-linearities which impair numerical stability. In this section we present the

unidimensional discretisation and possible solution methods for the drift-diffusion

equations.

The drift-diffusion set of equations for the steady state behaviour of an

electronic device were presented in Section 2.2.

The discretisation of the electron and hole drift-diffusion and continuity equa-

tions (equations 2.2 - 2.5) over a mesh is itself problematic since a non-linear in-

terpolation is usually necessary to determine mesh points on the odd grid (for first

differences) as required for solution of the electron and hole continuity equations.

66

2.3.3 Dependent variables

We will mainly consider the set of dependent variables (φ, n, p) for electron

and hole concentration and potential, respectively. However, it is possible to

work with other sets of dependent variables. One particular example is the set of

variables (φ, u, v ) where u and v are given by:-

n = ni · exp(

φ

kT

)

· u (2.76)

p = ni · exp(

− φ

kT

)

· v (2.77)

Although it is fairly obvious that the dynamic range required by this variable

set is very large, to the extent that practical implementation even on modern

computers is very difficult [36].

Another such variable set is (φ, φn, φp), where raw carrier concentrations

are replaced by quasi-Fermi potentials according to the following relation:-

n = ni · exp(

φ− φn

kT

)

(2.78)

p = ni · exp(

φp − φ

kT

)

(2.79)

Finally another set of dependent variables are the Slotboom variables, (φ,

Φn, Φp) where:-

Φn = ni · exp(

− φn

kT

)

(2.80)

Φp = ni · exp(

φp

kT

)

(2.81)

however, it is noted that this formulation is difficult to extend to degenerate

conditions. Although the variables (φ, n, p) seem the most natural choice for

67

solving semiconductor problems, a change of variables is often advantageous to

circumvent the problems associated with the non-linearity and tight coupling of

the drift-diffusion model.

2.3.4 Discretisation of Poisson equation in 1D

The Poisson equation is comparatively easily discretised, since the Laplacian

operator requires only physical mesh points (not odd grid, or interpolated, mesh

points) since it is discretised by a second difference, and so the Poisson equation

(2.1) becomes (see e.g. [36]):-

Fφ(φ, n, p)|i = ǫǫ0

(

φxi−1− φxi

xi − xi−1

+φxi+1

− φxi

xi+1 − xi

)

2

xi+1 − xi−1

−e·(n − p − ND + NA) = 0

(2.82)

where the subscripts i , i − 1 etc. refer to the mesh point.

2.3.5 Discretisation of electron and hole drift-diffusion and continuity

equations in 1D

After combining Eqns. 2.2 and 2.4, the following expression can be discre-

tised to yield a function Fn(φ, n, p).

∇ · (n(x) · µe · E(x) + Dn · ∇n(x))− re(x, n, p,φ) + ge(x) = 0 (2.83)

The difficulty in discretisation lies in the terms in n, the electron concentra-

tion, which require first differences. This requires construction of an ”odd grid”,

with interleaved mesh points, or as more commonly used, a non-linear interpola-

tion. If the first differences are naively computed this will lead to poor convergence

and generation of artifacts. The most common interpolation is the scheme due

68

to Scharfetter and Gummel, and involves solution of the drift-diffusion equations

between physical mesh points under the assumption that the partial derivative in

φ is constant [36], which can be assumed valid for arbitrarily fine meshes. This

gives rise to the following so called growth function for interpolation [36]:-

g(x ,φ) =1− exp

(

φi+1−φi

kT· x−xixi+1−xi

)

1− exp(

φi+1−φi

kT

) (2.84)

The electron concentration at the mesh midpoint is therefore given by the

following relation [36]:-

Jn,i+1/2 =µn(xi) · kT

q·(

ni+1 · B(φi+1−φi

kT)− ni · B(φi−φi+1

kT)

xi+1 − xi

)

(2.85)

The analogous relation for the hole concentration can be similarly derived.

By approximating the divergence operator with a simple first difference, the 1D

discretisation can be completed as follows.

Fn(φ, n, p)|i =2µn(xi) · kT

q · (xi+1 − xi−1)

·(

ni+1 · B(φi+1−φi

kT)− ni · B(φi−φi+1

kT)

xi+1 − xi− ni · B(φi−φi−1

kT)− ni−1 · B(φi−1−φi

kT)

xi − xi−1

)

− R(xi ,φ, n, p) + G (xi) = 0 (2.86)

Here B(y ) is the Bernoulli function defined by [36]

B(y ) =y

ey − 1(2.87)

and G (x) and R(x ,φ, n, p) are the local generation and recombination, re-

spectively.

Similarly, the discretisation of the analagous relations for holes can be derived

[36].

69

Fp(φ, n, p)|i = − 2µp(xi) · kTq · (xi+1 − xi−1)

·(

pi+1 · B(φi−φi+1

kT)− pi · B(φi+1−φi

kT)

xi+1 − xi− pi · B(φi−1−φi

kT)− pi−1 · B(φi−φi−1

kT)

xi − xi−1

)

− R(xi ,φ, n, p) + G (xi) = 0 (2.88)

These discretisations can be readily extended to two and three dimensions,

although for higher dimensionality alternate discretisations are possible using box

integration or the finite volume method.

2.3.6 Variable scaling

Variable scaling is important for solving semiconductor problems since scalars

have very different magnitudes if raw SI (or cgs) units are used. For example,

potential in eV and carrier concentration in cm−3 differ on average by 10-20 orders

of magnitude.

Furthermore, it can be shown from singular perturbation theory that for

small λ (where λ is the characteristic Debye length of the device region of interest)

the coupled system becomes well conditioned in the variable set (φ, u, v ) [37, 36].

Hence scaling is required to reduce the λ as small as possible. This can be achieved

by scaling voltage units by kT , the thermal voltage.

Many sets of scaling units have been proposed. The system adopted in this

work for numerical scaling is based on cgs units and is simple and effective. It has

been shown to bring about rapid convergence for the solvers used herein, and is

presented in Tab. 2.2.

Scaling of lengths (e.g. cm) can be either by a constant or in terms of

maximum doping concentration (Nmax), multiplied by 1e3. The latter has the

advantage that for low doped devices the residual is comparable, since it has

70Parameter Unit Scale ValueLength cm 1e8 or 1e3× 3

√Nmax

Second s 1e8Potential V 1/kTCharge C 1/q

Temperature K 1/300

Table 2.2: Scaling parameters for numerical simulation used inthis work

approximately the same weight irrespective of doping concentration, so that a

constant definition of the maximum residual for convergence (i.e. tolerance) can

be used (e.g. 1e-16) for different types of devices.

2.3.7 Solution of discretised equation set over a finite mesh

2.3.7.1 Iterative methods

The above set of non-linear discretised equations in one dimension can only

be solved by means of an iterative procedure, such as Newton’s method [36].

f(y) = 0 (2.89)

For the system described by Eqn. 2.89, Newton’s method involves iteratively

stepping from an initial guess of the solution toward a local minimum by means

of the following relation [36]:

yk+1 = yk −f(yk)

f ′(yk)(2.90)

where xk is the solution vector at the k th iteration step. Reformulating Eqn.

2.89 to suit solution Eqn. 2.82, Eqn. 2.86 and Eqn. 2.88, we may write:

71

yk+1 =

φk+1

nk+1

pk+1

(2.91)

=

φk

nk

pk

−

∂Fφ

∂φ

∂Fφ

∂n

∂Fφ

∂p

∂Fn

∂φ∂Fn

∂n∂Fn

∂p

∂Fp

∂φ∂Fp

∂n∂Fp

∂p

−1

Fφ(φk , nk , pk)

Fn(φk , nk , pk)

Fp(φk , nk , pk)

(2.92)

= yk − J−1F(yk) (2.93)

Iteration is performed until the residue given by ||F(yk)|| is less than some

predetermined tolerance value.

The difficulty in solution using the fully-coupled Newton’s method lies in

the tight coupling of the equations, specifically since carrier concentrations are a

non-linear function (exponential) of the potential term φ. Furthermore, Newton’s

method for a system of coupled non-linear equations in three variables (which

are themselves row vectors with entries for each mesh point) reduces to a matrix

inversion problem which is computationally expensive since the Jacobian has 3N×

3N entries.

Furthermore, the computation of the Jacobian matrix, let alone its inverse,

is complex and cumbersome. It is possible to compute an exact Jacobian matrix

by manually calculating analytical functions to generate the Jacobian matrix en-

tries as well as estimating the Jacobian using finite differences (inexact Jacobian).

Nevertheless, this approach is computationally expensive in and of itself.

It is possible to employ decoupled solution methods to ease the burden of

these computations. We now introduce Gauss-Seidel iteration, an iterative solu-

tion method for the decoupled solution of coupled non-linear PDEs, and present

Gummel’s method, a simple but effective solution method for semiconductor prob-

72

lems.

2.3.7.2 Gauss-Seidel iteration

Gauss-Seidel iteration can be used to decouple the drift-diffusion model equa-

tions. If the Jacobian matrix can be assumed definite and all blocks on its main

diagonal are non-singular [36] we may decompose the Jacobian into an upper and

lower triangular matrix in order to decouple the system as follows [36]:

∂Fφ

∂φ0 0

∂Fn

∂φ∂Fn

∂n0

∂Fp

∂φ∂Fp

∂n∂Fp

∂p

k

·

δφk

δnk

δpk

m+1

= −

Fφ(φk , nk , pk)

Fn(φk , nk , pk)

Fp(φk , nk , pk)

−

0∂Fφ

∂n

∂Fφ

∂p

0 0 ∂Fn

∂p

0 0 0

k

·

δφk

δnk

δpk

m

(2.94)

where the column vector

([

δφk δnk δpk

]m)T

represents the mth “inner”

iterate for solution for the correction vector,

([

δφk δnk δpk

])T

, from which

the state vector over the mesh can be computed by the relation [36]:

φk+1 = φk + δφk

nk+1 = nk + δnk

pk+1 = pk + δpk

The system Eqn. 2.94 can be decoupled and the partial derivatives resub-

stituted to yield a set of three equations that are solved separately using iterative

techniques [36]:

73

∂F kφ

∂φ· δφk,m+1 = −Fφ(φk , nk + δnk,m, pk + δpk,m) (2.95)

∂F kn

∂n· δnk,m+1 = −Fn(φk + δφk,m+1, nk , p

k + δpk,m) (2.96)

∂F kp

∂p· δpk,m+1 = −Fp(φk + δφk,m+1, nk + δnk,m+1, pk) (2.97)

The problem again reduces to solution of a set of linear equations for the vec-

tor(

[

δφk δnk δpk

]m+1)T

. The process is repeated until ||(Fφ, Fn, Fp)||k,m+1 < ǫ,

i.e. less than some pre-determined tolerance figure.

However, the precise formulation of the functions Fφ, Fn, Fp will doubtless

have a big impact on convergence. Performance is usually quite poor if the variable

set is chosen as the natural variables φ, n, p, however better convergence may be

achieved by reformulating and linearising the variable set φ, u, v introduced in

Section 2.3.3.

In the next section we consider an implementation of Gauss-Seidel iteration

in the form of Gummel’s method.

2.3.7.3 Gummel’s method

Derivation

In order to decouple the system of equations, Gummel’s method relies on

reformulation of the Poisson equation to include non-linear terms in the carrier

concentrations, which helps to bring about better convergence since the carrier

concentrations vary wildly during iterative solution of the Poisson equation. These

non-linear terms arise from the reformulation of the Poisson equation for the de-

pendent variable set (φ, u, v ) [36]:

74

∇2φ− e

ǫǫ0·(

exp

(

φ

kT

)

· u − exp

(−φkT

)

· v − ND + NA

)

= 0 (2.98)

If Eqn. 2.98 is solved using an iterative procedure, i.e. Newton’s method,

then this equation itself can be decoupled to solve for φk+1 (the potential term

for the k + 1th iterate) in terms of uk , vk . Furthermore, resubstituting the natural

variables nk , pk in terms of φk and collecting terms gives the following non-linear

equation:-

∇2φ− e

ǫǫ0· (nk ·

(

exp

(

φk+1 − φk

kT

)

− 1

)

+ nk

− pk ·(

exp

(

−φk+1 − φk

kT

)

− 1

)

− pk − ND + NA) = 0 (2.99)

This equation can be linearized and greatly simplified using a truncated series

expansion of the exponential, i.e.,

exp(x) = 1 + x +x2

2!+

x3

3!... ≃ 1 + x (2.100)

which gives:

∇2φ − e

ǫǫ0·((

nk + pk

kT· (φk+1 − φk)

)

+ nk − pk − ND + NA

)

= 0 (2.101)

Discretisation in one dimension

Following from this we may write out the set of discretised equations to be

solved in turn as part of a decoupled iterative solution algorithm.

75

Fφ(φ, n, p)|i ,k+1 = ǫǫ0

(

φi−1,k+1 − φi ,k+1

xi − xi−1

+φi+1,k+1 − φi ,k+1

xi+1 − xi

)

2

xi+1 − xi−1

− e ·(

n1,k + p1,k

kT· (φi ,k+1 − φi ,k)

)

− e · (ni ,k − pi ,k − ND + NA) = 0 (2.102)

Fn(φ, n, p)|i ,k+1 =2µn(xi) · kTxi+1 − xi−1

· (ni+1,k+1 · B(φi+1,k+1−φi ,k+1

kT)− ni ,k+1 · B(φi ,k+1−φi+1,k+1

kT)

xi+1 − xi−

ni ,k+1 · B(φi ,k+1−φi−1,k+1

kT)− ni−1,k+1 · B(φi−1,k+1−φi ,k+1

kT)

xi − xi−1

)

− R(xi ,φk+1, nk+1, pk) + G (xi) = 0 (2.103)

Fp(φ, n, p)|i ,k+1 = −2µp(xi) · kTxi+1 − xi−1

· (pi+1,k+1 · B(φi ,k+1−φi+1,k+1

kT)− pi ,k+1 · B(φi+1,k+1−φi ,k+1

kT)

xi+1 − xi−

pi ,k+1 · B(φi−1,k+1−φi ,k+1

kT)− pi−1,k+1 · B(φi ,k+1−φi−1,k+1

kT)

xi − xi−1

)

− R(xi ,φi ,k+1, ni ,k+1, pi ,k+1) + G (xi) = 0 (2.104)

As stated, each equation is solved in turn until the norm of the RHS (right

hand side) of all equations is below some threshold. First Eqn. 2.102 is solved

to yield φk+1 given φk , n and p. Next Eqn. 2.103 is solved to give nk+1 for

the given φk+1, n and p. Finally, Eqn. 2.88 is solved to give pk+1 for the given

φk+1, nk+1, pk+1.

Computer implementation

76

After arranging the linearised Eqn. 2.102 into canonical matrix form Ax+b =

0 and using appropriate linear algebra techniques to compute the solution, the

vector φ can be obtained. The N × N matrix A takes this explicit form:

fφ1 (i) =

2ǫǫ0(xi − xi−1)(xi+1 − xi−1)

(2.105)

fφ2 (i) =

−2ǫǫ0(xi − xi−1)(xi+1 − xi−1)

− 2ǫǫ0(xi+1 − xi)(xi+1 − xi−1)

− e(ni ,k − pi ,k)

kT

(2.106)

fφ3 (i) =

2ǫǫ0(xi+1 − xi)(xi+1 − xi−1)

(2.107)

gφ(i) = −e · (ni ,k + pi ,k) · φi ,k

kT+ e(ni ,k − pi ,k − ND + NA) (2.108)

1 0 0 . .

fφ1 (1) f

φ2 (1) f

φ3 (1) . .

0 fφ1 (2) f

φ2 (2) f

φ3 (3) 0 .

. .

. .

fφ1 (N − 1) f

φ2 (N − 1) f

φ3 (N − 1)

0 1

φ0,k+1

φ1,k+1

φ2,k+1

φ3,k+1

.

φN−1,k+1

φN,k+1

=

gφ(0)

gφ(1)

gφ(2)

.

.

gφ(N − 1)

gφ(N)

(2.109)

The first and last matrix rows are essentially identity elements since these in-

corporate boundary conditions on the mesh. We note that the resultant matrix for

77

solution of this block is tridiagonal in form, making solution of the matrix equation

reasonably straight forward using e.g. LU decomposition/back substitution.

The discretised electron and hole drift-diffusion-continuity equations can be

similarly arranged into matrix form.

f n1 (φ, i) =2 · µn(xi) · kT

(xi+1 − xi−1)(xi − xi−1)B

(

φi−1,k+1 − φi ,k+1

kT

)

(2.110)

(2.111)

f n2 (φ, i) = −2 · µn(xi) · kT(xi+1 − xi−1)

(

B(φi ,k+1−φi+1,k+1

kT)

(xi+1 − xi)+

B(φi ,k+1−φi−1,k+1

kT)

(xi − xi−1)

)

(2.112)

f n3 (φ, i) =2 · µn(xi) · kT

(xi+1 − xi−1)(xi+1 − xi)B

(

φi+1,k+1 − φi ,k+1

kT

)

(2.113)

g n(φ, n, p, i) = −R(xi ,φk+1, nk+1, pk) + G (xi) (2.114)

fp1 (φ, i) = − 2 · µp(xi) · kT

(xi+1 − xi−1)(xi − xi−1)B

(

φi ,k+1 − φi−1,k+1

kT

)

(2.115)

fp2 (φ, i) =

2 · µp(xi) · kT(xi+1 − xi−1)

(

B(φi+1,k+1−φi ,k+1

kT)

(xi+1 − xi)+

B(φi−1,k+1−φi ,k+1

kT)

(xi − xi−1)

)

(2.116)

fp3 (φ, i) = − 2 · µp(xi) · kT

(xi+1 − xi−1)(xi+1 − xi)B

(

φi ,k+1 − φi+1,k+1

kT

)

(2.117)

g p(φ, n, p, i) = −R(xi ,φk+1, nk+1, pk+1) + G (xi) (2.118)

78

1 0 0 . .

fn,p1 (1) f

n,p2 (1) f

n,p3 (1) 0 . .

0 fn,p1 (2) f

n,p2 (2) f

n,p3 (3) 0 . .

. .

. .

fn,p1 (N − 1) f

n,p2 (N − 1) f

n,p3 (N − 1)

. . 0 0 1

·

n, p0,k+1

n, p1,k+1

n, p2,k+1

n, p3,k+1

.

n, pN−1,k+1

n, pN,k+1

=

g n,p(0)

g n,p(1)

g n,p(2)

.

.

g n,p(N − 1)

g n,p(N)

(2.119)

Strictly speaking, there should be additional terms in ni ,k+1 and pi ,k+1 sepa-

rated from g n,p(i) and reexpressed fn,p2 , since ni ,k+1,pi ,k+1 are not constant. These

modifications are necessary to ensure convergence for high recombination rates [36]

and depend on the types of recombination models in use.

2.3.7.4 Fully-coupled non-linear solver

Since in this work we are interested in modelling tunnel-recombination re-

gions for high-efficiency monolithic tandem solar cells, we require solutions of the

basic drift diffusion model at high injection levels and with regions with degen-

erate doping levels. These conditions are traditionally held to bring about poor

convergence with a decoupled scheme.

79

However, it is possible to solve the basic drift diffusion model using a fully

coupled Newton solution algorithm. The advantage being the superlinear (best

case, quadratic) convergence of Newton’s method, as well as better convergence

when the coupling between equations becomes more pronounced (i.e. with greater

iteration to iteration variation).

In a series of papers, Bank et al. developed both a discretisation and solution

formalism [38] and iterative update algorithm [39] for the solution of non-linear

PDEs that arise in semiconductor problems.

In particular, Bank et al. highlight the dangers of the hitherto used ap-

proach of computation of the Jacobian matrix for Gauss-Seidel-Jacobi iteration

directly from the discretised equation set. Instead, they propose computation of

the Jacobian matrix entries from the equation set directly and then discretising the

resultant “operators”. As in the previous section, the equations are recast using a

change of variables to the variable set φ, u, v .

In this section we derive expressions for the entries in the Jacobian matrix

by means of a similar but slightly less complicated pathway.

Discretised linearisation of the electron and hole drift-diffusion-continuity

equations for use in fully-coupled Newton solution

Starting out with the Scharfetter-Gummel discretisation for current density

on the odd grid, i.e. Eqn. 2.85, we make the change of variables from the natural

variables (φ, n, p) to the variables (φ, u, v ),

Jn,i+1/2 =µn(xi) · kT

q·(

ni+1 · B(φi+1−φi

kT)− ni · B(φi−φi+1

kT)

xi+1 − xi

)

=µn(xi) · kT

q · (xi+1 − xi)exp

(

φi

kT

)

B

(

φi − φi+1

kT

)

[ui+1 − ui ] (2.120)

Implicit differentiation gives:-

80

dJn,i+1/2 =µn(xi) · kT

q · (xi+1 − xi)[B ′(

φi − φi+1

kT

)

·(

dφi − dφi+1

kT

)

·exp(

φi

kT

)

(ui+1 − ui) +

B

(

φi − φi+1

kT

)

exp

(

φi

kT

)

· [dui+1 − dui ] + exp

(

φi

kT

)

· dφi · [ui+1 − ui ]

]

(2.121)

Inspection of Eqn. 2.121, and differentiation with respect to the variable set

(φ, u, v ), has shown a number of hidden dependencies on φ which would not have

been seen otherwise. Resubstitution of the natural variable set gives the following:-

dJn,i+1/2 =µn(xi) · kT

q · (xi+1 − xi)

[

B

(

φi+1 − φi

kT

)

· dni+1 − B

(

φi − φi+1

kT

)

· dni]

+

[

dφi

kT+ GB

(

φi − φi+1

kT

)

·

dφi − dφi+1

kT

]

· Jn+1/2 (2.122)

Where GB is defined as follows:-

GB(y ) =B ′(y )

B(y )=

exp(y ) · (1− y )− 1

y · (exp(y )− 1)(2.123)

Similarly for the current density at the previous odd-grid point,

dJn,i−1/2 =µn(xi) · kT

q · (xi − xi−1)

[

B

(

φi − φi−1

kT

)

· dni − B

(

φi−1 − φi

kT

)

· dni−1

]

+

[

dφi

kT+ GB

(

φi − φi−1

kT

)

·

dφi − dφi−1

kT

]

· Jn−1/2 (2.124)

By substituting into Eqn. 2.125, we can derive the expression for the discre-

tised linearisation of the electron drift-diffusion-continuity equations.

−∇·(dJn)+∂R(xi ,φk,i , nk,i , pk,i)

∂n·dn+∂R(xi ,φk,i , nk,i , pk,i)

∂p·dp = ∇·Jn−R(xi ,φk,i , nk,i , pk,i)

(2.125)

81

q∇ · (dJn)−∂R(φi , ni , pi)

∂n· dn − ∂R(φi , ni , pi)

∂p· dp

=2 · q

(xi+1 − xi−1)

[

dJn,i+1/2 − dJn,i−1/2

]

− ∂R(φi , ni , pi)

∂n· dn − ∂R(φi , ni , pi)

∂p· dp

= dni−1 · f 1n (φ, n, p) + dni · f 2n (φ, n, p) + dni+1 · f 3n (φ, n, p)

+dφi

kT·[

ni−1 · f 1n (φ, n, p) + ni · f 2n (φ, n, p) + ni+1 · f 3n (φ, n, p)]

+2 · q

kT · (xi+1 − xi−1)·dφi ·

[

GB

(

φi − φi+1

kT

)

· Jn,i+1/2 − GB

(

φi − φi−1

kT

)

· Jn,i−1/2

]

− 2 · qkT · (xi+1 − xi−1)

· dφi+1 ·[

GB

(

φi − φi+1

kT

)

· Jn,i+1/2

]

+2 · q

kT · (xi+1 − xi−1)·dφi−1·

[

GB

(

φi − φi−1

kT

)

· Jn,i−1/2

]

−∂R(φi , ni , pi)

∂n·dn−∂R(φi , ni , pi)

∂p·dp

= −q∇ · Jn + R(φi , ni , pi) = −Fn(φ, n, p) (2.126)

A similar procedure can be followed for holes, to yield the following relation

for the linearised hole current density.

− q∇ · (dJp)−∂R(φi , ni , pi)

∂n· dn − ∂R(φi , ni , pi)

∂p· dp

=2 · q

(xi+1 − xi−1)

[

dJp,i+1/2 − dJp,i−1/2

]

− ∂R(φi , ni , pi)

∂n· dn − ∂R(φi , ni , pi)

∂p· dp

= dpi−1 · f 1p (φ, n, p) + dpi · f 2p (φ, n, p) + dpi+1 · f 3p (φ, n, p)

− dφi

kT·[

pi−1 · f 1p (φ, n, p) + pi · f 2p (φ, n, p) + pi+1 · f 3p (φ, n, p)]

− 2 · qkT · (xi+1 − xi−1)

·dφi ·[

GB

(

φi+1 − φi

kT

)

· Jp,i+1/2 − GB

(

φi−1 − φi

kT

)

· Jp,i−1/2

]

+2 · q

kT · (xi+1 − xi−1)· dφi+1 ·

[

GB

(

φi+1 − φi

kT

)

· Jp,i+1/2

]

− 2 · qkT · (xi+1 − xi−1)

·dφi−1·[

GB

(

φi−1 − φi

kT

)

· Jp,i−1/2

]

−∂R(φi , ni , pi)

∂n·dn−∂R(φi , ni , pi)

∂p·dp

= q∇ · Jn + R(φi , ni , pi) = −Fp(φ, n, p) (2.127)

82

Mobility scaling

Poor choice of variable scaling can lead to convergence problems and nu-

merical errors, resulting from high dynamic range between adjacent entries in the

Jacobian matrix.

In addition to scaling voltages by the thermal voltage (so that the system,

from a singular perturbation point of view, is well conditioned), it is also necessary

to scale mobilities appropriately. This is not critical in a decoupled (i.e. Gauss-

Seidel) solution method since each variable is solved independently, so relative

magnitudes of potential and carrier concentration and partial derivatives are not

an issue.

A simple method of mobility scaling is simply to divide through by mobility

in the discretised electron-hole continuity equations, such that inverse mobility

terms appear only as a scale for inhomogeneous terms (e.g. recombination and

generation terms). This ensures rapid and stable convergence of the coupled

system.

Computer implementation

The discretisation derived in Section 2.3.7.4 for the linearised electron and

hole continuity equations, in combination with the discretisation of the Poisson

equation given in Section 2.3.4 can be used to write out the entries for the Jacobian

matrix and RHS as follows.

F ∗φ |i =

2λ

(xi − xi−1)(xi+1 − xi−1)· φi−1

−[

2λ

(xi − xi−1)(xi+1 − xi−1)+

2λ

(xi+1 − xi)(xi+1 − xi−1)

]

· φi

+2λ

(xi+1 − xi)(xi+1 − xi−1)· φi+1

− (nk − pk − ND + NA) (2.128)

83

∂F ∗φ

∂φi−1=

2λ

(xi − xi−1)(xi+1 − xi−1)(2.129)

∂F ∗φ

∂φi

= −[

2λ

(xi − xi−1)(xi+1 − xi−1)+

2λ

(xi+1 − xi)(xi+1 − xi−1)

]

(2.130)

∂F ∗φ

∂φi+1=

2λ

(xi+1 − xi)(xi+1 − xi−1)(2.131)

∂F ∗φ

∂ni= − q

kT(2.132)

∂F ∗φ

∂pi=

q

kT(2.133)

∂F ∗φ

∂ni−1=

∂F ∗φ

∂ni+1=

∂F ∗φ

∂pi−1=

∂F ∗φ

∂pi+1= 0 (2.134)

F ∗n |i =

2

q (xi+1 − xi−1)·[

Jn,i+1/2 − Jn,i−1/2

]

− 1

µn

[R(xi ,φk+1, nk+1, pk)− G (xi)] = 0 (2.135)

J ′n,i+1/2 =

Jn,i+1/2

µn

=kT

q·

[

ni+1 · B(

φi+1−φi

kT

)

− ni · B(

φi−φi+1

kT

)]

xi+1 − xi(2.136)

J ′n,i−1/2 =

Jn,i−1/2

µn

=kT

q·

[

ni · B(

φi−φi−1

kT

)

− ni−1 · B(

φi−1−φi

kT

)]

xi − xi−1

(2.137)

∂F ∗n

∂φi−1

=2q

kT · (xi+1 − xi−1)· GB

(

φi − φi−1

kT

)

· J ′n,i−1/2 (2.138)

∂F ∗n

∂φi

=2q

kT · (xi+1 − xi−1)·

[

J ′n,i+1/2 − J ′

n,i−1/2 + GB

(

φi − φi+1

kT

)

· J ′n,i+1/2 − GB

(

φi − φi−1

kT

)

· J ′n,i−1/2

]

(2.139)

84

∂F ∗n

∂φi+1= − 2q

kT · (xi+1 − xi−1)· GB

(

φi − φi+1

kT

)

· J ′n,i+1/2 (2.140)

∂F ∗n

∂ni−1=

1

µn

· f n1 (φ, n, p) (2.141)

∂F ∗n

∂ni=

1

µn

· f n2 (φ, n, p) +1

µn

·[

∂R

∂n− ∂G

∂n

]

(2.142)

∂F ∗n

∂ni+1=

1

µn

· f n3 (φ, n, p) (2.143)

∂F ∗n

∂pi−1= 0 (2.144)

∂F ∗n

∂pi=

1

µn

·[

∂R

∂p− ∂G

∂p

]

(2.145)

∂F ∗n

∂pi+1= 0 (2.146)

F ∗p |i = − 2

q (xi+1 − xi−1)·[

Jp,i+1/2 − Jp,i−1/2

]

− 1

µp

[R(xi ,φk+1, nk+1, pk)− G (xi)] = 0 (2.147)

J ′p,i+1/2 =

Jp,i+1/2

µp

=kT

q·

[

pi+1 · B(

φi−φi+1

kT

)

− pi · B(

φi+1−φi

kT

)]

xi+1 − xi(2.148)

J ′p,i−1/2 =

Jp,i−1/2

µp

=kT

q·

[

pi · B(

φi−1−φi

kT

)

− pi−1 · B(

φi−φi−1

kT

)]

xi − xi−1(2.149)

∂F ∗p

∂φi−1

=2q

kT · (xi+1 − xi−1)· GB

(

φi − φi−1

kT

)

· J ′p,i−1/2 (2.150)

85

∂F ∗p

∂φi

=2q

kT · (xi+1 − xi−1)·

[

J ′p,i+1/2 − J ′

p,i−1/2 + GB

(

φi − φi+1

kT

)

· J ′p,i+1/2 − GB

(

φi − φi−1

kT

)

· J ′p,i−1/2

]

(2.151)

∂F ∗p

∂φi+1

=2q

kT · (xi+1 − xi−1)· GB

(

φi − φi+1

kT

)

· J ′p,i+1/2 (2.152)

∂F ∗p

∂ni−1

= 0 (2.153)

∂F ∗p

∂ni=

1

µp

[

∂R

∂n− ∂G

∂n

]

(2.154)

∂F ∗p

∂ni+1

= 0 (2.155)

∂F ∗p

∂pi−1

= fp1 (φ, n, p) (2.156)

∂F ∗p

∂pi= f

p2 (φ, n, p) +

1

µp

[

∂R

∂p− ∂G

∂p

]

(2.157)

∂F ∗p

∂pi+1= f

p3 (φ, n, p) (2.158)

The partial derivatives are therefore organized into the Jacobian matrix, in

order to solve for the update vector as follows:-

86

1 0 0 ...

0 1 0 ...

0 0 1 ...

∂Fφ

∂φ0

∂Fφ

∂n0

∂Fφ

∂p0

∂Fφ

∂φ1

∂Fφ

∂n1

∂Fφ

∂p1

∂Fφ

∂φ2

∂Fφ

∂n2

∂Fφ

∂p2...

∂Fn

∂φ0

∂Fn

∂n0

∂Fn

∂p0

∂Fn

∂φ1

∂Fn

∂n1

∂Fn

∂p1

∂Fn

∂φ2

∂Fn

∂n2

∂Fn

∂p2...

∂Fp

∂φ0

∂Fp

∂n0

∂Fp

∂p0

∂Fp

∂φ1

∂Fp

∂n1

∂Fp

∂p1

∂Fp

∂φ2

∂Fp

∂n2

∂Fp

∂p2...

. .

. .

. .

·

dφ0

dn0

dp0

dφ1

dn1

dp1

dφ2

dn2

dp2

.

.

.

=

−Fφ|0−Fn|0−Fp|0−Fφ|1−Fn|1−Fp|1−Fφ|2−Fn|2−Fp|2

.

.

.

(2.159)

2.3.7.5 Summary and conclusion

In order to compare the solvers (i.e. decoupled Gauss-Seidel vs. fully coupled

Newton iteration), implementations of the solution procedures presented in Section

2.3.7.3 and Section 2.3.7.4 were used to solve a simple silicon pn-junction problem

with the following characteristics:- SRH recombination through a single, mid-gap

trap level with a bulk lifetime of 10ns, emitter p+doping 5e19 cm−3, base doping

1e16 cm−3, device width 1µm, mesh size 100 points, temperature 300K. Material

parameters (intrinsic carrier concentration, dielectric constant etc.) were set to

nominal values for single-crystal silicon at 300K. Mobilities were set to 100 and 40

cm2/V /s for electrons and holes, respectively.

The problem was so posed in order to bring about the conditions (abrupt

junction, high recombination rates, large step change in applied potential, high

87

injection levels) under which Gauss-Seidel iteration is known to perform poorly.

In this way we quantatively compare the performance of the solution algorithms.

The IV trace is presented in Tab. 2.3.

Fully coupled Newton Gauss-Seidel iterationV J RHS n Time J RHS n Time

mA/cm2 s mA/cm2 s0.0 1.12E-06 3.5E-16 6 1.87 6.74E-06 1.1E-13 6 1.140.2 1.76E-02 2.1E-16 7 1.09 1.76E-02 1.6E-16 13 0.640.4 1.86E+00 7.6E-15 8 1.32 1.86E+00 7.7E-15 13 0.660.6 1.31E+03 6.1E-15 8 1.31 1.31E+03 7.3E-15 13 0.650.7 3.60E+04 1.71E-15 8 1.23 3.60E+04 1.4E-15 9 0.5

Total 35 6.55 53 3.62Av. time / iteration 0.19 0.07

Table 2.3: Comparison of solver algorithms for drift diffusionmodel

From Tab. 2.3 it is obvious that the fully coupled solution converges more

quickly for comparable residual for all bias voltages. However, the additional time

required for Jacobian matrix assembly and system solution means that the fully-

coupled solver actually takes longer than the decoupled solver. This is indicated

by the roughly 3 fold increase in time per iteration.

It is possible to offset the cost of Jacobian computation in some cases by

reusing the Jacobian and its factorisation for as many iterations as possible (i.e until

the error does not decrease), and then recomputing the Jacobian and repeating

until convergence. However, such a solution procedure is almost certainly slower

to converge iteration-wise, but taking far less time per iteration step. Indeed,

this method is certainly applicable to very large matrices where recomputing the

Jacobian at each iteration would be computationally expensive.

Table 2.3 indicates that the decoupled solver is numerically stable and rapidly

converges even under high injection/recombination conditions, contrary to other

88

implementations. Although in commercial simulation packages such as Synopsys

Sentaurus Device often use a fully coupled solution, for the kinds of simple nu-

merical models developed herein to characterise and design optoelectronic devices,

Tab. 2.3 tends to suggest that Gauss-Seidel iteration performs more than ade-

quately. In addition, the source code for the decoupled solution is smaller by a

factor of 4, making it much simpler to understand and work with, so it is by far

the most useful solution method for the didactic purposes of this work.

The python source code is presented in Appendix A and B.

2.3.8 Physical models

2.3.8.1 Boundary conditions

The boundary conditions imposed on the discretisation functions Fφ, Fn, Fp

and its derivatives at the edges of the mesh and interfaces between regions are

as significant in defining the electronic properties of the device as the parameters

of the semiconductor themselves. Strictly speaking, boundary conditions on nu-

merical problems fall into 3 categories: Dirichlet, where the dependent variable

is specified at the boundary, von Neumann, where the derivative of the depen-

dent variable is specified, and mixed, a combination of both, often called a Robin

boundary condition. Boundary conditions on the mesh are applied both in the

right hand side (RHS) of the discretised drift diffusion equations (Dirichlet), or

incorporated into the Jacobian itself (von Neumann).

It is common in solving semiconductor problems to impose boundary condi-

tions on the current density at a contact, i.e.

~Jn · n|contact = BC (2.160)

Semiconductor-vacuum interface

89

The semiconductor-vacuum interface is rarely handled by simulators, espe-

cially since in most practical applications the semiconductor is never truly in contact

with free space. However, a very important concept in simulation of optoelectronic

devices such as solar cells is the notion of surface recombination velocity. This

is simply the rate, S, at which excess carriers recombine at the surface, and has

units cm/s, hence the name surface recombination velocity. Although sophisti-

cated models for surface recombination are possible, in this work we consider the

following simple von Neumann boundary condition [40]:-

~Jn · n|contact = S · n (2.161)

Note that in 1D simulations, surface recombination velocity is usually com-

bined with the Ohmic contact boundary conditions (for fairly obvious reasons).

Under these circumstances, the surface recombination velocity is usually used to

model recombination of minority carriers at the contact.

Ohmic metal-semiconductor interface

Ohmic metal-semiconductor junctions are usually modelled as sites of infinite

generation and recombination, in the sense that surface recombination velocities

for carriers of both type are usually assumed to be infinitely high. In addition, the

electron and hole concentration are forced to their equilibrium values, given by the

following balance equations [36]:-

n0 + NA = p0 + ND (2.162)

n0 · p0 = n2i (2.163)

∴ n0 =

√

(ND − NA)2

4+ n2i ,eff +

ND − NA

2(2.164)

p0 =

√

(ND − NA)2

4+ n2i ,eff − ND − NA

2(2.165)

90

The potential can be computed from the applied potential Vapp, and the

above relations, assuming Boltzmann statistics,

φ0 = Vapp +kT

qasinh

(

ND − NA

2 · ni ,eff

)

− Eg

2− χ0 (2.166)

Additionally, we may want to reduce the recombination rate at the contact

for either electrons or holes as the case may be to some surface recombination

velocity Sn or Sp. This enables modelling of blocking contacts, and the effect of

front/back surface fields on the semiconductor.

Hence the expressions for the functions Fφ, Fn, Fp at the contact become:-

Fp|contact = pcontact − (p0 +D

Sp

· ∇p) = 0 (2.167)

Fn|contact = ncontact − (n0 −D

Sn

· ∇n) = 0 (2.168)

Fφ|contact = φcontact − φ0 (2.169)

Degenerate ohmic metal-semiconductor interface

For the case of degenerate doping beneath the contact region, Fermi-Dirac

statistics are necessary. We intend to solve the following equation [36] for the

conduction band energy, Ec ,

Nv · F1/2

(

Ev − Ef ,p

kT

)

− Nc · F1/2

(

Ef ,n − Ec

kT

)

+ N+D − N−

A = 0 (2.170)

However, this equation is not in and of itself soluble, since it is a single

equation in 3 unknowns. To solve this equation, we make both quasi-Fermi levels

equal at the contact, since we assume it is a perfectly Ohmic contact and hence

a site of infinite generation/recombination, therefore the carrier concentrations

91

at the contact should take on their equilibrium values. Assuming we are in a

degenerate p-type region, we can rewrite Eqn. 2.170 as follows:-

Nv · F1/2 (η)− Nc · F1/2 (−η − Eg ) + N+D − N−

A = 0 (2.171)

where η = Ev − Ef ,p. This equation can simply be solved graphically.

Having solved for the quantity η, which is in effect equal to the built-in

potential, we can obtain the potential φ, assuming a reference potential for the

Fermi energies, i.e. Ef ,n = Ef ,p = 0eV . This would place, in effect, the Fermi

level in the semiconductor at 0eV under thermal equilibrium. Hence the potential

at the semiconductor can be defined as follows:-

φ = Vapp + η − Eg − χ0 (2.172)

Schottky metal-semiconductor interface

Although many models have been presented for the metal-semiconductor

interface [36], we consider here a simple model useful for numerical simulation

[41].

φ = Vapp − ΦB +kT

qln

(

NC

ni ,eff

)

− Eg

2− χ0 (2.173)

~Jn · n = q · vn(

n − nB0)

(2.174)

~Jp · n = −q · vp(

p − pB0

)

(2.175)

nB0 = NC exp

(

−qΦB

kT

)

(2.176)

pB0 = NV exp

(−Eg ,eff + qΦB

kT

)

(2.177)

Where ΦB is the Schottky barrier height, NC ,NV are the conduction and

valence band density of states, respectively, and vn, vp are the thermionic recom-

bination (emission) velocities.

92

Since Schottky barrier devices do not play a significant role in this work,

higher order effects such as image-force barrier lowering are not considered. Tun-

nelling (field-emission) is also not considered in the context of Schottky devices,

but models often used to predict barrier tunneling currents are discussed later to

model tunnel junction devices.

2.3.8.2 Recombination models

Recombination processes in semiconductors are typified by three main for-

malisms. These processes include direct band-to-band recombination involving ra-

diative transitions (radiative recombination), recombination via electron and hole

capture in traps, and the inverse process of generation via impact ionization, known

as the Auger process. Recombination rates are computed from simple analytical

expressions involving the carrier concentrations and hence introduce additional

terms into the left hand side expressions (i.e. Jacobian matrix).

Radiative recombination

Radiative recombination can be modelled using the following simple expres-

sion [40]:-

Rrad = C · (np − n2i ) (2.178)

Where the radiative coefficient, C in cm3/s expresses the rate at which an

excess of carriers recombine radiatively.

Shockley-Read-Hall recombination

In indirect bandgap semiconductors such as silicon and germanium, direct

radiative transitions are unlikely and recombination is predominantly through elec-

tron and hole capture in traps. Analysis of possible capture processes gives rise to

the Shockley-Read-Hall formalism, which is usually expressed as follows [36]:-

93

RSRH =n · p − n2i

τp · (n + n1) + τn · (p + p1)(2.179)

Where n1, p1 are the density of occupied electron and hole traps for a given

temperature, as given by:-

n1 = ni · exp(

Etrap

kT

)

(2.180)

p1 = ni · exp(

−Etrap

kT

)

(2.181)

where Etrap is the energy level of the trap.

In practice, these formulae are difficult to implement in the discretisation

directly, but for low recombination rates it is entirely possible to consider a single

inhomogeneous recombination term. For increasing recombination rates however,

particularly in e.g. Gummel solvers, it is advisable to decouple at least the numer-

ator for inclusion into the LHS (e.g. Fφ, Fn, Fp) to solve for the next iterate.

Additionally, it is important to realise that the lifetimes τn, τp show a strong

dependence on the doping concentration. As such, these lifetimes ought to be

deduced from the bulk lifetime and doping density according to a simple relation,

e.g., that due to Scharfetter [41]:-

τdop(NA,0 + ND,0) = τmin +τmax − τmin

1 +(

NA,0+ND,0

Nref

)γ (2.182)

Auger recombination

Auger recombination can be thought of in general terms as the inverse pro-

cess of impact ionization, discussed in Section 2.3.8.3. Although there are several

possible combinations collectively referred to as Auger processes, the sum total

can be conveniently described by the following model [36]:-

94

RAU =(

CAUn · n + CAU

p · p)

·(

n · p − n2i)

(2.183)

Material Auger coefficient, cm6/s ReferenceElectron Hole

Si 2.8e-31 9.9e-32 [36]Ge 1e-30 1e-30 [42]

CdTe 1e-29 1e-29 [43]

Table 2.4: Typical Auger parameters for materials used in thiswork at T=300K

where CAUn and CAU

p are the Auger coefficients for electrons and holes, re-

spectively. Some nominal values for these parameters for materials of interest in

this work are tabulated in Tab. 2.4.

2.3.8.3 Generation models

Impact Ionization

Impact ionization is the generation of electrons and holes due to collisions

with the lattice. In this work we consider only a fairly simple impact ionization

model, that due to Chynoweth [36]. This basic model should be valid for the fields

of interest (< 1e5 V/cm):-

αn = α∞n · exp

(

−[

E critn

E

]βn)

(2.184)

αp = α∞p · exp

−[

E critp

E

]βp

(2.185)

Optical Generation

Optical generation is modelled using the Beer-Lambert law, which predicts

an exponential fall off of photon concentration with distance through the semicon-

95Material αn E crit

n βn ReferenceSi 1.28e6 2.54e6 1 [36]Ge 1.55e7 1.56e6 1 [36]

Table 2.5: Typical impact ionization parameters for electrons inmaterials used in this work

Material αp E critp βp Reference

Si 1.0e6 2.22e6 1 [36]Ge 1.0e7 1.28e6 1 [36]

Table 2.6: Typical impact ionization parameters for holes in ma-terials used in this work

ductor due to absorption. This expression was first introduced in Section 2.2.2.

Specifically, the generation is given by Eqn. 2.13 and Eqn. 2.14.

Non-local tunnelling Models

Non-local tunneling models are necessary to accurately model transport in

regions of devices such as tunnel diodes, monolithic multi-junction solar cells (with

a tunnel-recombination junction interconnect), Schottky barrier devices and Zener

diodes. The term ”non-local” is used to indicate that the tunnelling process occurs

between mesh points (i.e. at dissimilar points in k-space), rather than between

bands at a single mesh point (or location in k-space).

In this work, we consider the non-local tunnelling model due to Ieong et

al. [44]. Although this model was presented in the context of tunneling currents

through Schottky barriers for use in MOSFET simulations, it is comparatively

straight forward to extend to band-to-band tunneling in bulk pn junction devices,

as is relevant to this work.

The basis of Ieong’s model is as follows. Consider the following expression

(Eqn. 2.186) for the tunneling current density at point v (metal contact) due to

96

Metal

n-type

semiconductor

Figure 2.5: Barrier Tunnelling, after Ieong et al. [44]

electrons tunneling into available states from point u in a Schottky barrier device,

as depicted in Fig. 2.5:-

Jtun = Jtun,M→S − Jtun,S→M

= A∗ · T 2 ·∫ ∞

ǫ

Γ(u) ln

[

1

1 + exp ((ǫ′−EF ,n)/kT)

]

dǫ′−

A∗ · T 2 ·∫ ∞

ǫ

Γ(u) ln

[

1

1 + exp ((ǫ′−EF ,m)/kT)

]

dǫ′ (2.186)

since by Ieong’s arguments, the generation rate due to tunnelling can be

related to the tunnelling current as follows:-

Gtun(u) =dJtun

dǫ· E (2.187)

=A∗ · T

k· E · Γ(u) · ln

[

1 + exp (−(φ−φn)/kT)

1 + exp (−(φ−φm)/kT)

]

(2.188)

where E is the local electric field at point u, Γ(u) is the tunnelling probability

at point u, A∗ is the electron Richardson constant, and k is the Boltzmann con-

stant. The tunnelling probability must be computed by solving the Schroedinger

97

equation in one dimension along the tunnelling path. As in Ieong’s original model,

this can be most easily achieved using a WKB approximation to solve for the

wavevectors required to compute the tunneling path. WKB refers to Wenzel-

Kramers-Brillouin, the originators of this technique, which is most useful for solv-

ing second order differential equations with slowly varying spatial inhomogeneities,

and is particularly applicable to tunnelling problems. The tunnelling probability as

given by the WKB approximation is simply:-

Γ(u) = exp

[

−2

~

∫ u

v

√

(2m∗ (φB + φm − φ))dx

]

(2.189)

98

2.3.9 Test case - Germanium Solar Cell

Parameter ValueGe bulk lifetime 100ns

Electron mobility, µe 200cm2V−1s−1

Hole mobility, µh 100cm2V−1s−1

Emitter doping, ND 1e18 cm−3Base doping, NA 1e17 cm−3Emitter width 1 µmBase width 170 µm

ARC MgF2 90 nm,SiNx 60 nm,a-Si:H 50 nm

SRV 1e4 cm/s

Table 2.7: Ge solar cell simulation parameters

The following is a simulation of a germanium solar cell made with the code

developed in this chapter. We compare both the analytical simulation code devel-

oped in Section 2.2 and the numerical simulation code developed in Section 2.3.

The parameters of the simulation are tabulated in Tab. 2.7. The simulations yield

both the JV (current vs. voltage) and PV (power vs. voltage) data for the solar

cell under AM1.5G illumination.

The analytical and numerical simulators derived in this section agree well

with each other although both simulators give conflated efficiencies and unrealistic

open circuit voltages. This is because of unrealistic recombination models. The

analytical simulator reports JSC = 40 mA/cm2, Voc = 276 mV, FF = 70% and

η = 7.8%, whereas the numerical simulator reports JSC = 38.3 mA/cm2, Voc =

281 mV, FF = 68.9% and η = 7.4%. Both efficiencies are quite high given the

input lifetime and surface recombination velocity, indicating the idealised nature

of the simulations.

Despite the difference in the simulation code (the analytical simulator relies

99

on closed-form expressions to evaluate the solutions to the drift-diffusion equations,

whereas the numerical simulator solves these equations numerically using a 1D

version of finite element analysis) the results are very similar. The numerical

simulator reports lower short circuit current density. This is due to more realistic

recombination models, for example Shockley-Read-Hall statistics, as opposed to

the pseudo first-order model assumed in deriving the analytical model. However

the difference is negligible. The results confirm and verify both models. The only

noticeable difference between the two simulations is the time taken: the analytical

simulator took one tenth the time to complete. This demonstrates the power of

analytical modelling.

100

2.4 Summary and Conclusions

In this chapter an analytical model for single junction solar cells was ex-

pounded. This involved computation of the dark current, photocurrent and device

parasitics. Analytical models are seen to yield answers more rapidly due to the

analytical nature of output expressions for a given input.

A numerical simulation framework was also developed. This consisted of

discretisation of the drift-diffusion equations, and numerical solution using either

a decoupled or a fully-coupled non-linear solver. The decoupled solver was seen

to outperform the fully-coupled solver since the increased size of the fully coupled

matrix bore a computation time penalty that outweighed the benefit of fewer

iterations.

Finally, comparing both the numerical and analytical simulators, performance

was seen to be similar in the sense that outputs agreed within reasonable limits,

however the analytical simulator was the fastest to yield answers. This makes one

question the need for numerical simulators for back of the envelope computations,

and suggests that analytical calculations are faster and sufficiently accurate to

render more complicated models unnecessary. Although the example given is for

the case of a single junction germanium solar cell under AM1.5G illumination, the

modelling framework developed in this chapter is applicable to all kinds of single

and multi junction solar cell structures.

101

0.00 0.05 0.10 0.15 0.20 0.25 0.30Voltage, V

0

10

20

30

40

50Current, mA/cm−2

Current-Voltage characteristics

AnalyticalNumerical

0.00 0.05 0.10 0.15 0.20 0.25 0.30Voltage, V

0

1

2

3

4

5

6

7

8

Power, mW/cm−2

Power-Voltage characteristics

AnalyticalNumerical

Figure 2.6: Comparison of simulators developed in this chapter

Chapter 3

Device modelling, design and optimisation of tandem II-VI and

germanium solar cells

3.1 Introduction

Thin film photovoltaic (PV) cells have begun to establish themselves as viable

alternatives to the dominant PV technology, crystalline silicon, which is a proven,

robust and efficient technology. At present, commercial thin-film technologies,

comprising cadmium telluride (CdTe), copper indium gallium diselenide (CIGS)

and amorphous silicon (a-Si:H) cannot achieve the peak efficiencies of crystalline

silicon solar modules, but rather are competitive for economical reasons, offering

a lower cost per peak watt to end customers [45].

CdTe solar cells have been previously fabricated using a wide variety of meth-

ods, including ion-implantation [9], molecular beam epitaxy (MBE) [46], close-

spaced sublimation (CSS) [14, 47, 48], spray pyrolysis [8], close-spaced vapour

transport (CSVT) [49], metal-organic chemical vapour deposition (MOCVD) [50,

51], chemical vapour deposition (CVD) [52], and RF-magnetron sputtering [16,

17, 53]. Although in almost all cases CdTe solar cells are a heterojunction formed

between a naturally n-type CdS window layer and a naturally p-type CdTe layer,

homojunctions have also been investigated [9, 52]. It must be noted that the

efficiencies reported thus far for homojunctions (roughly 3-10%) are not compa-

rable with those commonly reported for heterojunctions. There is some evidence

103

to suggest that the CdS layer reduces electron leakage through the junction due

to tunnelling [54], however this alone does not explain the difference between re-

ported results of device efficiency, which could be attributable to a range of issues

including surface recombination, difficulty with doping CdTe, or poor fill factors

relating to difficulties in forming contacts to the devices.

Germanium solar cells have been the subject of some research for application

in thermophotovoltaics [27] and as bottom cells in multijunction solar modules for

space and terrestrial concentrators [55]. Germanium wafers may be doped using

spin-on dopants [21, 22, 27] or gaseous diffusion [56]. Back contact formation

usually consists of deposition of aluminium and annealing above the eutectic tem-

perature [22]. Polycrystalline germanium thin-films have been deposited using

RF-magnetron sputtering [57] and found to have higher absorption coefficients

and optical band gaps. Polycrystalline thin-films are of interest since the combina-

tion of thin-film CdTe and thin-film polycrystalline germanium in tandem promises

a cost effective, highly efficient solar cell [58]. In this section we employ Synopsys

Sentaurus Device to examine such a solar cell structure.

While CdTe thin-film growth on germanium substrates has been the subject

of research as a buffer layer for growth of HgCdTe [24, 29] for infrared detector ap-

plications, very little attention has been given to the electronic (and photovoltaic)

properties of this combination of materials. In contrast, Katnani et al. investigated

overlayers of silicon and germanium on a variety of different substrates (including

CdTe) and determined band offsets by synchrotron-radiation photoemission [59].

From this, the valence band offset of CdTe on germanium was determined to be

0.85 eV. Despite these investigations, the electronic properties of the CdTe/Ge

interface remain relatively unknown.

Although the bandgaps (1.5eV and 0.67eV for CdTe and Ge, respectively)

are not optimal for a two-junction tandem cell [4, 60], the photocurrents are

104

reasonably well matched and simulation has shown that respectable conversion

efficiencies (approaching or exceeding 20%) can be achieved, if the best practi-

cal CdTe and Ge cells are combined in tandem [58].There is perhaps a point of

concern of the performance of the lower band-gap germanium cell at elevated

temperatures; however it must be noted that germanium has performed well in

thermophotovoltaic applications [27] and as the bottom cell of multijunction III-V

concentrator cells [55].

Thin film multijunction solar cells have been investigated previously. For

example, amorphous silicon devices with up to 3 junctions have been presented

with stable conversion efficiencies reaching 13% [61]. Multijunction amorphous

silicon devices are usually a series of stacked p-i-n structures with interconnect

consisting of tunnel-recombination junctions (TRJ) where trap assisted tunnelling

(TAT) and increased recombination rates are employed to allow majority carriers to

recombine and hence exit the sub-cells [62]. On the other hand, Compaan et al.

present an RF-magnetron sputtered CdTe/HgCdTe tandem with a tunnel junction

interconnect consisting of ZnTe:N and ZnO:Al [53]. The interconnect between

series connected monolithic tandem cells is very important to device performance;

high recombination rates at the interface for majority carriers exiting from each

cell are required. To this end, III-V concentrator cells usually rely on band to band

tunneling (BTBT) as the main mechanism for carrier recombination in the TRJ

to accommodate the high current densities required [63, 64].

The monolithic structures presented in this section feature tunnel junctions

which rely on BTBT. It must be conceded that this makes a number of assumptions

about the CdTe/Ge hetero-interface that have not yet been experimentally veri-

fied. For this reason, and to ease the requirements for high doping, an alternative

structure is presented herein which consists of an all germanium tunnelling/re-

combination region, where BTBT can readily be achieved [65]. Device structures

105

relying entirely on TAT in the tunnel junction region are not considered in this

section, as too little is known about the electronic properties of the CdTe/Ge

interface to produce meaningful simulations of this effect.

Simulation has shown that optimisation of the tunnel junction is of crucial

importance to realising high device efficiencies. In particular, minority carrier flow

must be prevented; in III-V concentrator cells, barriers can be imposed to block

minority carrier flow [64], since this is achievable within the III-V material system.

In combining CdTe and Ge, however, such a luxury does not exist, and instead

regions must be doped appropriately (forming front and back surface fields as

appropriate) to reduce minority carrier flow.

Three monolithic tandem cell structures are now considered and their per-

formance evaluated. The first two structures are CdTe homojunctions intended to

be grown by MBE on doped germanium wafers, and differ in their approach to

the tunnel junction. MBE has been chosen as the growth technology due the ease

of growing complex multilayered structures of high quality with good control over

interfaces and dopant incorporation. It must be stressed that MBE is not intended

for the long term development of thin-film tandem cells; but rather intended to

fulfill short term needs to prove the concept and obtain necessary parameters to

improve the accuracy of simulations. Long term, the structure presented in section

3.1.2 is to be pursued which can be fabricated using a combination of chemical

bath deposition (CBD), CSS, and/or sputtering. This structure is intended for low

cost production, and is essentially a thin-film on thin-film tandem solar cell.

Additionally, a mechanically stacked tandem structure is also considered for

completeness to compare the optical matching and device efficiency with the mono-

lithic counterparts. Some advantages of the mechanically stacked configuration

are outlined.

106

Figure 3.1: CdTe homojunction on Ge substrate device structure.Tunnel junction is a heterojunction formed at the CdTe/Ge in-terface.

3.1.1 CdTe homojunction on crystalline germanium substrate

3.1.1.1 Heterostructure tunnel junction

Two approaches to the problem of sub-cell interconnection using a tunnel

diode are under consideration. In the first approach, a tunnel diode is formed at the

heterointerface of CdTe and Ge. In this structure, both materials are very heavily

doped (≥ 2 × 1019 cm−3) in a narrow (∼ 30 nm) region about the metallurgical

junction. Simulation (performed using Synopsys Sentaurus Device) results of the

tunnel junction, where the valence band offset in germanium is taken to be 0.88 eV

[58] and the electron and hole effective masses are 0.08 and 0.117 m0 respectively,

are depicted in Fig. 3.2 [58]. It is evident that the junction is more than capable

of allowing sufficient majority carrier flow (recombination of majority carriers in

the vicinity of the metallurgical junction) by tunnelling to support functional cells,

107

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16Bias, Volts

0

200

400

600

800

1000

1200

1400

Curr

ent

Densi

ty, J, mA/cm

2

Figure 3.2: CdTe/Ge heterojunction tunnel diode current-voltagecharacteristics. Peak tunneling current density more than satis-fies the requirements for cells operating under 1 sun illumination(≃ 26mA/cm−2) [58].

Parameter Description Valueµe,CdTe CdTe electron mobility 1260 cm2/Vs [2]µh,CdTe CdTe hole mobility 104 cm2/Vs [2]Eg,CdTe CdTe band gap 1.5 eV [2]χ0,CdTe CdTe electron affinity 4.05 eV [58]τSRH,CdTe CdTe SRH lifetime 1 ns [66]µe,Ge Ge electron mobility 3800 cm2/Vs [67]µh,Ge Ge hole mobility 1800 cm2/Vs [67]Eg,Ge Ge band gap 0.67 eV [42]χ0,Ge Ge electron affinity 4.0 eV [58]τSRH,Ge Ge SRH lifetime 1 ms [42]

Table 3.1: Simulation parameters for crystalline CdTe homojunc-tions on crystalline Ge substrates

since the peak tunneling current density (≃ 1200 mA/cm−2) is much greater than

the device short circuit current density (≃ 24− 26 mA/cm−2).

The full device structure and energy band diagram are presented in Fig.

108

Figure 3.3: Proposed energy band diagram for a CdTe homo-junction on Ge substrate. Note desirable overlap of valence andconduction bands in the tunnelling / recombination junction.Not to scale.

Ge FSF doping Ge emitter doping Jsc (mA/cm2) η3e19 cm−3 3e17 cm−3 25.60 19.76 %3e19 cm−3 2e18 cm−3 25.54 19.75 %4e19 cm−3 3e17 cm−3 25.63 20.09 %5e19 cm−3 3e17 cm−3 25.68 20.32 %5e19 cm−3 2e18 cm−3 25.66 20.31 %

Table 3.2: Efficiency vs. Ge FSF/emitter doping, under 1-sun,AM1.5G illumination. Note efficiency improvement with increas-ing front surface field (FSF:n+germanium tunnel diode region)doping.

3.1 and Fig. 3.3, respectively. The device is modelled using optical databases

109

(n and k values) from Tousek [68] for CdTe and Palik [69] for Ge. Minority

carrier lifetimes are set to 1 ns [66] and 1 ms [42] for CdTe and germanium

respectively. Front and back surface recombination velocities are set to 5e5 cm/s

and 1e3 cm/s [58], respectively. Surface recombination at the interface is set

to 1e2 cm/s; however simulation results are relatively insensitive to the value

assumed (since recombination in the tunnel junction is desirable). Simulation

takes into account band gap narrowing and doping dependent mobility in the

germanium cell. The doping dependent mobility data is from Palankovski [67].

Such effects are neglected for the CdTe cell, due to a lack of experimental data.

Incomplete ionization is also included. TAT is simulated using the Hurkx model

[41]. Simulation parameters for this cell are given in Tab. 3.1.

As noted above, simulation has shown that majority carrier flow across the

tunnel junction is also possible and this imposes limits on ultimate device efficiency.

Electron flow from the CdTe absorber region to the Ge emitter is negligible; this

is owing to the tunnel junction p+ region acting as a back surface field (BSF).

The same cannot be said for hole flow from the Ge emitter to the CdTe absorber;

however, this may be mitigated by reducing the Ge emitter doping and doping

the tunnel diode n+ region more heavily, producing a front surface field (FSF).

The emitter doping ought not to be reduced too drastically, as this will result in

lower open circuit voltage for the lower cell (and hence the tandem structure). By

reducing minority carrier flow accross the tunnel diode we may make some gains

on open circuit voltage due to reduced recombination but there will inevitably be

a cross over point where any potential advantages are out weighed by diminishing

bottom cell Voc . Results of overall cell efficiency vs. tunnel junction n+ region

doping are presented in Tab. 3.2 for devices with a 2.5 µm thick CdTe absorber

layer and a 15 µm thick Ge absorber layer. The device structure is simulated under

1 sun, AM1.5G illumination.

110

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Wavelength, m

0.2

0.0

0.2

0.4

0.6

0.8

1.0a.u

EQE and Reflection

EQE CdTeEQE GeReflection

Figure 3.4: CdTe homojunction on Ge substrate: external quan-tum efficiency and reflectance plot versus wavelength. Note thereflection losses caused at the CdTe/Ge interface. Data obtainedfrom Sentaurus simulation using transmission matrix method.

Although the heterostructure tunnel junction performs well in simulation,

there are definite concerns about physical realizability, since p type doping of CdTe

to the levels required is non-trivial. Although hole concentrations of 5× 1019 have

been reported using ion-implantation and pulsed electron beam annealing [70], ion

implantation is not particularly amenable to commercial manufacture of large-scale

modules. Nevertheless, this approach can be used as a starting point to investigate

whether or not such a structure can be physically realised.

Optical modelling using the transfer matrix method (TMM) shows significant

111

reflection at the interface. The external quantum efficiency (EQE) and reflectance

plot for the two cells of the tandem is shown in Fig. 3.4 as a function of wavelength.

The optical losses at the CdTe/Ge interface, combined with the non-idealities of

the tunnel junction, present difficulties in matching photocurrents between the two

cells of the tandem. Note that the the bi-layer antireflection coating of the device

structure consists of 60 nm of TiOx covered by 100 nm of MgF2 and serves to

minimize reflection losses at the front surface.

3.1.1.2 All germanium tunnel junction

In order to relax the doping requirements for p-type CdTe somewhat, and cir-

cumvent the problem of potentially unfavourable band alignment at the CdTe/Ge

interface, an alternative structure is presented in Fig. 3.5, which features a tunnel

diode realised entirely in germanium. The energy band diagram for this structure

is depicted in Fig. 3.6. This is the second approach to the problem of intercon-

nection of the two tandem cells. In this configuration, the underlying Ge substrate

wholly contains the tunnel diode which is defined using e.g. spin on diffusants.

In order to improve majority carrier injection while reducing minority carrier flow

across the heterointerface, a back surface field is realised in the CdTe absorber.

The doping density required for the back surface field is considerably less than for

the above heterojunction tunnel diode p+ region. Results are presented in Tab.

3.3.

The disadvantages of this structure are the limitations imposed on the doped

substrate in terms of high temperature processing. For example, it is usual during

MBE growth on Ge to outgas the native oxide at 650C [24] which would cause

significant diffusion of the dopants in the germanium substrate [22]. The time for

oxide outgassing therefore must be kept to a minimum.

Another disadvantage is that there is now a p+-CdTe / p+-Ge region formed,

112

Figure 3.5: CdTe homojunction on Ge substrate featuring all-germanium tunnel junction. Tunnel diode is formed entirelywithin the underlying germanium substrate.

CdTe BSF doping Jsc (mA/cm2) FF η2e18 cm−3 24.74 74.06 % 19.48 %3e18 cm−3 24.74 77.9 % 20.43 %6e18 cm−3 24.74 78.2 % 20.47 %

Table 3.3: Efficiency vs. CdTe BSF doping for structure withall-Ge tunnel diode under 1 sun, AM1.5G illumination. Notethe efficiency improvement with increasing BSF doping. Theshort circuit current density is limited by the germanium bottomcell and hence is unaffected by CdTe BSF doping. This can beimproved slightly by optimising CdTe thickness, see Tab. 3.5.

and interdiffusion of dopants from either side of this junction will cause parasitic

doping which will affect the net doping of each region detrimentally (for example

Sb is a p-type dopant in CdTe but an n-type dopant in Ge).

113

Figure 3.6: Proposed energy band diagram for all-Ge-TRJ con-taining structure

Parameter Description Valueµe,CdTe CdTe electron mobility 320 cm2/Vs [?]µh,CdTe CdTe hole mobility 40 cm2/Vs [?]Eg,CdTe CdTe band gap 1.5 eV [?]τSRH,CdTe CdTe SRH lifetime 1 ns [66]µe,Ge Ge electron mobility 350 cm2/Vsµh,Ge Ge hole mobility 150 cm2/VsEg,Ge Ge band gap 0.88 eV [57]τSRH,Ge Ge SRH lifetime 1 µs

Table 3.4: Simulation parameters for thin film CdS/poly-CdTe/poly-Ge heterostructures

3.1.2 Conventional CdTe heterojunction

Unlike the previous structures, which are intended to serve as proof of con-

cept cells to be grown on single crystal germanium substrates by MBE, in this

114

Figure 3.7: CdS/CdTe/Ge tandem solar cell structure. 100 nmthick MgF2 is deposited on the glass front surface as an anti-reflection coating.

CdTe absorber Jsc FF Voc ηwidth (µm) (mA/cm2) (Volts)

0.5 22.66 78.06 % 1.088 19.24 %0.6 23.52 76.76 % 1.08 19.51 %0.8 24.29 74.14 % 1.078 19.28 %

Table 3.5: CdS/CdTe/Ge tandem cell simulation results. Notethe optimum CdTe absorber width.

structure, a CdS/CdTe heterojunction cell is to be first deposited on ITO glass,

with a MgF2 antireflection coating, as usually found in the literature [48, 14].

The polycrystalline germanium cell is then to be formed using a low-temperature

deposition process, e.g. sputtering. The layers must be doped appropriately to

realise the tunnel junction and emitter/absorber regions. The structure is depicted

in Fig. 3.7 and its energy band diagram in Fig. 3.8.

In order to model polycrystalline germanium, the band gap, mobility, minority

115

Figure 3.8: Proposed energy band diagram for conventional CdTeheterojunction thin film tandem cell

carrier lifetimes and optical database are modified accordingly. The n and k values

were taken from Tsao [71] and the electron mobilities were set to 350 and 150

cm2/Vs, respectively, in order to be indicative of polycrystalline material [72].

Likewise, lifetimes were set to 1 µs. These figures were chosen as being plausible

in the context of the literature. Simulation parameters are summarized in Tab.

3.4.

Simulation results are presented in Tab. 3.5. The Ge tunnel diode n+ region

doping was set to 3e19 cm−3 and the Ge emitter doping was set to 2e18 cm−3.

116

The width of the Ge absorber was set to 20 µm. Incomplete ionization, doping

dependent mobility, bandgap narrowing and TAT were taken into account.

It must be noted that the optimal CdTe absorber width as shown in Tab 3.5

is only 0.6 µm, which is significantly less than normally deposited for commercial

CdTe solar cells [73]. If this is a realistic value, then this would be of advantage

in manufacture, requiring less scarce tellurium per module.

On the other hand, simulation has shown that thicker germanium films are

required than first anticipated [58]. Around 20 µm is required to optimize device

performance by attempting to better match photocurrents. Beyond this value,

negligible improvement in device efficiency results.

3.2 Mechanically stacked tandem solar cell

We now examine the potential for a mechanically stacked CdTe/Ge tandem

solar cell. This structure has a few potential advantages over the three monolithic

structures already discussed. Firstly, the device may be bonded in a four wire con-

figuration. In this configuration, both cells are independently bonded and there

is no interconnect between cells. This has the advantage that photocurrents do

not need to be matched, and there is no tunnelling/recombination junction which,

as we have seen, is problematic in the case of CdTe and Ge. Secondly, optical

matching between the CdTe and Ge may be better achieved since the device is no

longer monolithic and A/R interlayers are now feasible. One example of such a

structure is CdTe on sapphire, which is almost refractive index matched to silicon

nitride (SiNx). If CdTe is grown on sapphire and mechanically stacked with a

germanium subcell coated in a SiNx based A/R coating, we would conjecture that

the reflection losses at the interface could potentially be reduced in comparison to

the monolithic case. The third advantage is that we overcome some of the limita-

tions of the materials science of CdTe and Ge by separating the substrates. CdTe

117

and Ge can now be processed independently at elevated temperatures without any

detrimental affect on one another and without risk of alloy formation. Also, we

now have flexibility in the choice of sub-/super- strate for CdTe deposition, and

can choose a material with more favourable material properties, such as lattice

constant.

Disadvantages of the mechanically stacked tandem include higher processing

costs due to added complexity, and more complicated interconnect, especially in

the case of a four-wire connected tandem. However, with difficult materials such

as CdTe and Ge, these disadvantages may be outweighed by the benefits gained

from separating the two cells mechanically.

Front contacts

Rear contacts

Figure 3.9: Diagram of proposed mechanically stacked tandemstructure

The proposed mechanically stacked tandem structure is depicted in Fig. 3.9.

The main features are the CdTe heterojunction solar cell deposited on sapphire,

with buried contacts to the CdTe layer bought out to the front surface, and an

aluminium doped zinc oxide (AZO) transparent conducting oxide. AZO is chosen

due to better infrared transparency than indium tin oxide (ITO). The germanium

118

cell is a back-junction design with contacts to the device protruding from the rear

surface. This allows for a flat interface between the sapphire surface and the SiNx

passivation/AR layer, so there is no air gap. In fact the two cells could be adhered

to one another using some kind of wafer bonding process, however what impact

that would have on each subcell is not known.

0 500 1000 1500 2000Wavelength, nm

0

20

40

60

80

100

Transm

ission / reflectance (%)

Transmission / reflectance for mechanically stacked tandem solar cell

TR

Figure 3.10: Reflectance and transmission at the Ge cell emittersurface for structure of Fig. 3.9

The optical transmission and reflectance at the Ge bottom cell emitter is

shown in Fig. 3.10. When compared with Fig. 3.4, the reflectance is not sig-

nificantly better, weakening our argument that the mechanically stacked tandem

should have better optical performance. The interference is mostly due to the

interface of the sapphire substrate/SiNx layer and is due to slight refractive index

mismatch. Experimental tuning of the SiNx layer thickness did not yield significant

performance gains. The best course of action to obtain a better match would be

SiNx refractive index tuning. This is discussed in Chapter 5.

Simulation results for the lower cell in the stacked (which can be simulated

independently in the case of a mechanically stacked tandem) are tabulated in Tab.

3.6. Since photocurrents no longer need to be matched, the CdTe can be assumed

119Sun Jsc FF Voc η

(mA/cm2) (Volts)1 22.6 72.16 % 0.310 5.03 %100 2,260 77.28 % 0.425 7.47 %

Table 3.6: Simulation results for Ge bottom cell of mechanicallystacked tandem

to be at least 5 µm thick meaning that the top cell can be optimised to absorb

almost all radiation above the energy gap of CdTe. This means that energy can

be most efficiently converted in the top cell leaving only the unabsorbed photons

to permeate to the lower cell. This highlights the advantage of a mechanically

stacked configuration with four-wire interconnect. Simulation shows that the Ge

bottom cell could add 5.03% efficiency in this configuration for 1 sun illumination

and 7.47% for 100× concentration. Since the device is 4-wire connected, the

efficiencies simply add. If the top CdTe cell could be made as efficient as leading

CdTe solar cells of today, i.e. 21.5% efficient, the total efficiency would be 26.5%

at 1 sun. This would be a very promising photovoltaic device indeed.

3.3 Conclusion

A novel thin film tandem device, consisting of the combination of thin film

cadmium telluride with either crystalline or polycrystalline germanium, has been

modelled and simulated in Synopsys Sentaurus Device. Three such monolithic

structures have been considered. It has been shown that device efficiencies ap-

proaching or exceeding 20% are achievable assuming a nominally 15-16% efficient

CdTe top cell. An alternative structure, whereby the tunnel junction is realised

entirely in germanium, thus relaxing the constraints for extremely high p-type dop-

ing concentrations in CdTe, has been considered, and has been shown to function

adequately. It must be noted that the doping concentrations required to form

120

a BSF capable of achieving high efficiencies are still quite difficult to achieve in

CdTe, even under idealised circumstances.

A more commerically feasible structure, a CdS/CdTe heterojunction with a

polycrystalline germanium bottom cell, was also considered, and performed quite

well in simulation, in spite of poorer lifetimes and mobilities chosen to be indicative

of polycrystalline material. It is assumed that such a structure can be fabricated

using CSS or sputtering on glass superstrates, essentially fabricated from the top

down. This is in contrast to the first two structures proposed herein, which are

to be constructed from the bottom up by deposition of material on crystalline

germanium substrates.

Finally, a mechanically stacked tandem structure was investigated. This

offers the advantage of easing the requirements for photocurrent matching, allevi-

ating the difficulties of materials processing, particularly for high temperature pro-

cessing, and potentially improving optical matching and reducing reflection losses.

In practice, the structure considered in this section did not significantly improve

optical losses but did deliver a more efficient lower cell. At 100× concentration

ratio, as much as 7.47% efficiency could be achieved for the lower cell, boosting

the efficiency of a standalone CdTe cell. This requires specialised processing of

CdTe solar cells, and in particular inverting the structure to realise it in substrate

rather than superstrate configuration. Nevertheless, the mechanically stacked tan-

dem structure promises a great deal for the case when monolithic structures are

deemed unrealisable due to difficult material properties.

Chapter 4

Low-cost techniques for germanium device active region formation

4.1 Introduction

In this chapter we present some low cost methods for germanium pn-junction

device active region formation. These techniques allow one to form the crucial

device region, the pn-junction itself, in a low cost manner and most importantly,

in a manner amenable to mass-manufacture. We consider the use of spin-on

dopants to introduce impurities into the device regions for junction formation. We

present, in addition to the more conventional spin-on technique, a comparison of

both “proximity doping”, a variation of doping from solid sources, and “sandwich-

stacked diffusion”, a novel technique to introduce the impurities from a spin-on

source.

4.2 Spin-on dopants

Spin-on dopants are predominantly silica based solutions that can be spun

onto a wafer and outgassed to form a uniform dopant impregnated layer to act as

a diffusion source. The dopant products can be obtained from a variety of manu-

facturers or created in-house using sol-gel techniques. For example, spin on glass

can be manufactured using TEOS (tetra-ethyl orthosilicate), water, isopropanol,

and hydrochloric acid [74]. The result is a transparent, thin solution of solvents

and embedded silica nanoparticles, following the polymerisation of silica during the

122

reaction. By adding into the mix a proportion of dopant material, the solution will

contain a certain amount of dopant and will be able to act as a diffusion source

when spun onto a wafer.

4.3 Degenerate p-type doping and sandwich stacked diffusion

1. Si wafer spin

coated with

spin-on dopant

film

("Source Wafer")

2. dopant

"source"

wafer out-

gassed

at 100 - 200 Deg. C

in cold end of furnace

3. "Sandwich" diffusion

stack assembled

Silicon wafer: back-side diffusion barrier

Germanium"Target wafer"

Silicon"Source wafer" Spin-on

dopant film

4. "Sandwich"

diffusion

stack

loaded into

quartz boat

for

introduction

into furnace

("pre-dep")

Figure 4.1: “Sandwich-stacked diffusion”

In this section, we explore the use of Ga doping of Ge using a novel technique

- “sandwich-stacked diffusion” - which yields doping densities ≫ 5 × 1019 cm−3

with the potential for shallow junctions, and which is amenable to rapid thermal

processing (RTP). This process is depicted in Fig. 4.1.

We note that the sandwich-stacked diffusion method differs from proximity

doping [75] by the fact that the sol-gel film (as spin coated on the silicon dopant

source wafer) is in physical contact with the target wafer (in this case, a piece of

123

Ge). The sandwich-stacked method has the advantage of providing better control

of diffusion into Ge of impurities which do not have high vapour pressure at the

diffusion temperature, e.g. Ga and its oxides.

Gallium is a particularly suitable dopant for degenerate doping of Ge by

solid-state diffusion due to reasonable diffusivity, high solubility (> 1020 cm−3)

and natural affinity for Ge. This is in contrast to B which displays very high

activation energies [76, 77] and lower solubility [77]. Gallium diffusion in Ge has

been investigated by diffusion from solid sources in sealed ampoules [76] and from

ion implantation [78]. Solid state diffusion of Ga into Ge from spin-on dopants

(SODs), however, appears to be a fairly novel concept.

In this work, we present this novel technique and employ it to degenerately

dope n and p-type Ge wafers. Sheet resistance and SIMS profiling are used to

characterise the dopant profile, from which thermal activation energies and pre-

exponential factors are extracted, and a simple analytical model is proposed for the

post-“drive-in” profile. It is envisaged that this technique can be applied in a range

of electronic and optoelectronic applications, in particular for use in back-surface

fields for multijunction photovoltaic cells.

4.4 Experimental

4.4.1 Sample preparation

A 2” p-type Ga doped (3.3 Ω · cm, 1× 1015 cm−3) Ge wafer purchased

from University Wafer [Boston, MA, USA] and a 3” n-type Antimony doped

(40 Ω · cm,∼ 1× 1014 cm−3) Ge wafer purchased from Umicore [Olen, Belgium]

were diced into 1cm × 1cm samples and cleaned in a 3 step solvent clean. This

consisted of 3 min. in each of boiling trichloroethylene, acetone, and methanol,

followed by 1 minute of ultrasonic cleaning after each step, to remove fine Ge

124

Samples Supplier Conductivity type Resistivity (Ω · cm)

GE219:1-4 University Wafer p-type 3.3

GE220:1-4 Umicore n-type 40

Table 4.1: Sample summary

particles left by the cleaving process. The samples were labeled as documented in

Tab. 4.1.

These samples were then cleaned in 1:1:20 NH4OH :H2O2 :DI water for 30s

to etch ∼ 100nm of the surface, followed by a 1 min. soak in 1:1:3 HCl:HBr :DI

to strip any surface oxide and passivate the surface. This passivation process is

known to be relatively air-stable [79, 80].

Silicon wafers were cleaved into 1cm × 1cm squares and cleaned using the

standard RCA cleaning procedure. Half the Si squares were then spin-coated with

Ga/B spin-on dopant solution GaB 260 from Filmtronics [Butler, PA, USA]. The

dopant films are not spun onto the Ge directly, since after diffusion HF insoluble

films form [81] which cannot easily be removed without damage to the Ge itself,

due to the high chemical reactivity of Ge.

The Si and Ge samples were packaged and labelled ready for diffusion in a

horizontal 2” tube furnace. First the Si dopant source wafers were loaded into the

quartz boat and outgassed in the cool part of the furnace tube at 100 - 200C to

cure the dopant film and drive off organic solvents and other contaminants within

the film [82, 83].

4.4.2 Sandwich-stacked diffusion

To diffuse the dopant into the Ge, a sandwich-stack is formed with the

dopant source at the bottom, covered by the target Ge wafer, which is then itself

125

covered by a spare piece of clean silicon wafer, to prevent unintentional diffusion

from volatile dopant compounds and impurities, e.g. B, which was present in the

GaB spin-on dopant film. This is depicted in Fig. 4.1.

The samples were diffused in flowing nitrogen ambient, with high gas flow

(10 L/min). This removes any residual, volatile organic contaminants from the

sample, which are present in the sol-gel dopant films. High nitrogen gas flow

also prevents oxidation of the sensitive germanium (which oxidizes quite rapidly at

elevated temperatures).

Samples were loaded into the quartz boat and introduced into the furnace.

After diffusion, the samples must not be retracted rapidly; i.e. air quenched, as this

will introduce defects, particularly in the low doped n-type wafers, in which defects

can even introduce type conversion to p-type [84]. To alleviate this problem, the

sample is left inside the furnace tube and the furnace is switched off, allowing it

to cool. The time constant for cooling (defined as time for temperature to decay

to 63% of the final value) is discussed below; the average cooling rate is roughly

2 C/minute from 800 C.

Due to the low diffusivity of all acceptors in Ge, practical constraints place

700C as a lower bound on the diffusion temperature, since below this temperature

no appreciable diffusion will occur, even after several hours. Above 850C, the

crystallinity of the Ge will be severely compromised introducing a great many

defects. Hence, a practical diffusion temperatures lie in the range of be 750 - 800

C.

The heating and cooling dynamics of the sample need to be accounted for

when characterising the diffusion process. Firstly, the quartz boat and the quartz

rod handle will limit the maximum heating rate of the sample. Secondly, cooling

under gas flow, which has a non-negligible time constant, effectively prolongs the

diffusion. In order to derive a time-dependent model for heating within the furnace,

126

Process time constant (s)

Sample cool down 3529

Sample rise time 48

Table 4.2: Furnace time constants, defined as time taken toreach 63% of the final value

Sample Nominal Time (s) Temp. Eff. time (s) Eff.

set point (C ) temp. (C )

GE0219:2 1380 845 1143 845

GE0219:3 600 866 362 865

GE0219:4 2400 880 2160 880

Table 4.3: Effective drive in times and temperatures for samplesGE0219:2-4.

these two time constants needed to be extracted. To obtain the approximate rise

time within the boat, a quartz rod of the same dimensions (500mm length ×

10mm OD) with quad 1mm bore holes was threaded with a k-type thermocouple.

This was then introduced into the furnace. The thermocouple voltage was sampled

at 1 Hz with a digital multimeter with GPIB attachment and recorded on a PC.

When the temperature had settled to the set-point, the furnace was then switched

off. The respective time constants are tabulated in Tab. 4.2.

The samples in this work were annealed as follows. First, the sandwich-stack

was prepared and introduced into the furnace for the “pre-dep” step. In this step,

a small amount of dopant is diffused to create a heavily doped region with a high

surface concentration of dopant. This can be carried out at lower temperatures to

coat the surface, with the effect that there is minimal diffusion into the bulk. Each

matching pair (e.g. GE0219:1 with GE0220:1 etc.) is pre-deposited simultaneously

127

Sample Eff. time (s) Eff. temp. (C )

GE0219:1 124 749

GE0219:2 360 700

GE0219:3 360 762

GE0219:4 120 772

Table 4.4: Effective predep times and temperatures for samplesGE0219:1-4.

at temperatures ranging from 700 - 800 C.

Then the Si dopant wafer is removed from the stack and each matching

pair of Ge wafers is then sandwiched together, so that the doped regions touch.

This is intended to prevent out-diffusion of dopant. However, as discussed in this

work, n-type 〈211〉 and p-type 〈100〉 wafers have slightly different diffusivities at

a given temperature, so some degree of out-diffusion will still occur. By plotting

the temperature profile using the time constants extracted above, we can estimate

an average temperature and time for use in fitting analytical models to extract

diffusivities and construct the Arrhenius plots necessary to extract the thermal

activation energy and pre-exponential factors, which are necessary to characterise

the diffusion process. We do so by computing the time that the diffusivity (from

D ≃ exp(− 3kT

)) is above 90% of its final value, and computing the average

temperature over this period. Note that an activation energy of 3 eV is taken

from the literature [76]. These effective drive in times and temperatures, based

on the nominal times and the temperature set point, are summarized in Tab. 4.3.

Predep effective times and temperatures are shown in Tab. 4.4.

Samples GE0219:1 and GE0220:1 were not subjected to a drive-in step, and

instead were used to characterise the dose in-diffused during the “pre-dep” step.

128

Sample Sheet Resistance (Ω/2)

GE0219:1 113

GE0219:2 12.4

GE0219:3 11.7

GE0219:4 6.5

Table 4.5: Extracted sheet resistances of GE0219 after drive-in

4.5 Characterisation

The samples were then characterised by measuring sheet resistances and

performing secondary ion mass spectroscopy (SIMS) profiling. From these two

techniques, and assuming complete ionization of acceptors, it is possible to recon-

struct the depth profile and fit a model according to diffusion theory.

4.5.1 Sheet resistance measurements

Sheet resistances were measured using a four-point probe and voltmeter to

measure the sample using a four-wire (Kelvin) measurement. The sheet resistance

is given by the correction factor [85]:-

Rsh =π

ln(2)R4W =

ρ

t(4.1)

where Rsh is the sheet resistance, R4W is the four wire resistance, ρ is the

resistivity, and t is the thickness. The sheet resistances for GE0219 are summarized

in Tab. 4.5. For the case of GE220, however, the sheet resistances were measured

both before and after the drive-in step. These are tabulated in Tab. 4.6.

Under the assumption of complete ionization for dopants, we can calibrate

the SIMS secondary ion yield, as presented in the following section, by using the

129

Sample Sheet Resistance after pre-dep after drive-in

(Ω/2) (Ω/2)

GE0220:1 172.14 -

GE0220:2 135.9 19.5

GE0220:3 36 14.4

GE0220:4 52 5.9

Table 4.6: Sheet resistances of GE0220

following relationship [85]:-

σsh = q

∫ t

0

(µn(n)n(x) + µp(p)p(x)) dx =1

Rsh

(4.2)

For the n-type wafers (samples 220:1-4), this process would be relatively

straight forward, since we may assume in this case we have a high quality p-on-n

junction with minimal injection into the underlying n-type layer. However, for the

p-type wafers (samples 219:1-4) we must first extract the sheet resistance of the

heavily doped region by assuming a simple model of two resistances in parallel.

From this, we can compute the sheet resistance of the doped layer, and then

normalize the SIMS profiles accordingly. We do this as follows:-

Rsh =1

1

R2−

1

R1

(4.3)

where R1 is the measured sheet resistance before and R2 is the measured

sheet resistance after diffusion. Note that some error will be introduced by this

method but it is assumed this will not produce any noticeable effect on the carrier

concentration.

130

We cannot simply scale the SIMS secondary ion yield in proportion to the

quotient of the measured Rsh and the computed sheet resistance from Eqn. 4.2,

since the doping dependence of the mobilities in this integral are non-linear. In-

stead, we use an iterative procedure, where we initialize the scale constant α to

some value, e.g. 1014 counts−1 · cm−3. Then the sheet resistance (Rsh,comp.)

is computed and compared to Rsh. If the absolute value is greater than some

threshold, e.g, if the following relation is not satisfied:-

|Rsh,comp. − Rsh| < ǫ (4.4)

where ǫ may be a few milliohms, then the scale parameter α is adjusted as

follows:-

αn+1 = αn

(

1 + log

(

Rsh,comp.

Rsh

))

(4.5)

The process is then repeated until Eqn. 4.4 is satisfied. This iterative

procedure was found to rapidly converge to a suitable scale constant α yielding

consistency between SIMS depth profiles and sheet resistance measurements.

4.5.2 SIMS profiling

Samples GE0219:1-4 were sent to the National University of Singapore, Sur-

face Sciences Laboratory, for SIMS profiling. Profiling was performed using positive

SIMS using a Cameca IMS6f with a O2+ primary beam at 12.5keV. The raster

area was 250 × 250µm2 . The data was first analysed with a high sputtering rate,

≃ 140 nm/min, and then again with a lower sputtering rate to analyze impurity

content (Fe, B). Interestingly, B was only found at the surface of the samples,

although it is present in equal quantities within the spin-on diffusant film as the

131

Ga impurity (2%). It is evident that B does not easily diffuse at the temperatures

involved. Early studies noted that B does not easily “alloy” with Ge [76], and more

recent studies have called into question previously held results about B diffusion

in Ge [77], which was traditionally held to be a vacancy mediated process. In

fact, results concerning radiation-enhanced diffusivity of B and self-diffusion of Ge

in Ge [86] tend to suggest that B diffusion is in fact due to self-interstitials, the

concentration of which would be quite low at the temperatures investigated in this

work.

In contrast, the Fe content was more pronounced and was seen to diffuse in

one case as far as 300nm into the sample. This Fe is most likely from trace metal

content of the dopant films themselves, and has been noted previously [82].

4.6 Analysis

4.6.1 Fitting procedure to extract diffusivities

Diffusion can be analysed by means of Fick’s second law [87]:-

∂N

∂t= D

∂2N

∂x2(4.6)

where D is the diffusivity of the material, x , t are position (in 1 dimension)

and time, respectively, and N is the concentration of the impurity at x at time t.

For a diffusion from an infinite source with surface concentration of N0 we

have the following well known solution [87]:-

N(x , t) = N0erfc

(

x

2√Dt

)

(4.7)

Whereas if the surface concentration is effectively a delta function, we have

the following solution (constant total dopant problem) [87]:-

132

N(x , t) =Q0√πDt

exp

(−x2

4Dt

)

(4.8)

where Q0 is the total dose at the surface.

In practice neither model is adequate to describe the process discussed in

this work and in fact both models can be fitted to Fig. 4.2. This can be attributed

to the surface deposited layer of Ga compounds (e.g. GaO) or Ga:Ge alloy (which

is evident for example in the SIMS profile of sample GE0219:1) which continues

to supply dopant during the drive-in step, and to out-diffusion between the two

samples of the sandwich-stack during drive-in. Furthermore, the temperature is

not constant throughout diffusion due to heating and cooling effects of the quartz

boat etc.

Since these effects are difficult to account for, we first assume a constant

average temperature and effective diffusion time as computed in Section 4.4.2,

and fit both erfc and Gaussian profiles. Interestingly, the erfc profile provides a

better fit to the tail of the diffusion, whereas the Gaussian profile better accounts

for the first diffusion length of the profile.

The parameters extracted from the model fits are tabulated in Tab. 4.7.

Sample 1 differs from the other samples in that the profile is that of the predepo-

sition step (this sample was not driven-in). In this profile, we see a surface delta

of Ga rich material, which is a residue from the predeposition step, and a shallow

diffused region (∼ 80nm depth).

The three remaining samples were driven-in at higher temperatures for longer

times, in order to tune the surface concentration and extract the model for the

profile, and hence follow a more reproducible pattern. The profile for the region

0 ≤ x ≤ 2√Dt, where 2

√Dt is the diffusion length, is described best by a

Gaussian fit with a slightly lower diffusivity than the tail of the diffusion, which

133

0 10 20 30 40 50 60 70 80 90 depth, nm

1015

1016

1017

1018

1019

1020

1021

Atoms ·cm

−3

GE0219:1

erfc fitGaussian fitSIMS Data

0.0 0.5 1.0 1.5 2.0 depth, µm

1015

1016

1017

1018

1019

1020

1021

Atoms ·cm

−3

2√Dt = 0.48 µm

GE0219:2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 depth, µm

1015

1016

1017

1018

1019

1020

1021

Atoms ·cm

−3

2√Dt = 0.34 µm

GE0219:3

0.0 0.5 1.0 1.5 2.0 2.5 3.0 depth, µm

1015

1016

1017

1018

1019

1020

1021

Atoms ·cm

−3

2√Dt = 1.1 µm

GE0219:4

Figure 4.2: SIMS data with erfc/Gaussian model fits

134

fits very well to an erfc profile with higher (1.2 - 1.4 ×) diffusivity.

We therefore propose the following simple analytical model to describe dop-

ing profiles from diffusion using this method:-

NGa(x , t) ≃

Q0√π0.8Dt

exp(

−x2

3.2Dt

)

x ≪ 2√Dt

N0erfc(

x

2√Dt

)

x ≫ 2√Dt

(4.9)

where the dose, Q0, is set to remove the discontinuity at x = 2√Dt:-

N0erfc(1) =Q0√π0.8Dt

exp

(

− 1

0.8

)

(4.10)

⇒ Q0 ≃ 0.77× 2√DtN0√π

(4.11)

and N0, the effective surface concentration, is roughly (1 − 3)×1020 cm−3

depending on time and temperature.

In order to extract corresponding parameters for the n-type wafers (sample

GE0220), we fit the model presented in Eqn. 4.9 to the sheet resistance values after

drive-in, and a simple erfc model to the values of the sheet resistance measured

before drive-in, as given in Tab. 4.6. The fitting parameters are summarized in

Tab. 4.8.

4.6.2 Thermal activation energy and pre-exponential factor extrac-

tion

An Arrhenius plot of the data presented in Section 4.6.1 is shown in Fig. 4.3.

The uncertainty in temperature measurement is assumed to be ± 10 C, and the

uncertainty in the extracted diffusivities (which is essentially due to the uncertainty

in diffusion time) is assumed to be 10%. This gives a very large uncertainty in

the extracted thermal activation energy and pre-exponential factors. This high

135

SampleN0 [erfc fit]

cm−3

D [erfc fit]

cm2/s

D [Gaussian fit]

cm2/s

GE0219:1 9.9e+19 1.7e-14 1.7e-15

GE0219:2 1.8e+20 5.5e-13 4.2e-13

GE0219:3 2.9e+20 8.6e-13 5.8e-13

GE0219:4 1.1e+20 1.3e-12 1e-12

Table 4.7: Extracted diffusivities for GE0219:1-4

Sample Temperature (C ) Time (s) D (cm2/s)

GE0220:1

Pre-dep720 120 4.4e-15

GE0220:2

Pre-dep700 360 2.3e-15

GE0220:2

Drive-in845 1141 3.2e-13

GE0220:3

Pre-dep762 360 2.2e-14

GE0220:3

Drive-in865 360 5.6e-13

GE0220:4

Pre-dep772 120 3.2e-14

GE0220:4

Drive-in880 2160 9e-13

Table 4.8: Extracted diffusivities for GE0220:1-4

uncertainty is due to difficulties in measuring the precise sample temperature within

the furnace tube in the presence of high gas flow.

136

In a similar manner, we can fit the parameters given in Tab. 4.8 to a linear

regression of log(D) vs. 1kT

in order to obtain the thermal activation energy and

pre-exponential factor for the diffusion of Ga into 〈211〉 oriented n-type Ge wafers.

It is insightful to compare these results, despite their limited accuracy, with

published data in order to draw a few conclusions about these results. These values

are presented in Tab. 4.9. In general, we see reasonable agreement between the

extracted thermal activation energies and pre-exponential factors between this

and other published work (see Tab. 4.9). However, the large uncertainty in the

extracted values reported in this work are of some concern, particularly for the

purposes of making any general conclusions about the diffusivity of Ga. For the

case of the values extracted for p-type material from analysis of SIMS data, the

uncertainty is comparable to [78], however, since the n-type data is analyzed by

means of model fits, the uncertainty is much higher. Nevertheless the agreement

seems reasonable.

0.80 0.85 0.90 0.95 1.00 1.05 1000 / T, mK−1

10-15

10-14

10-13

10-12

10-11

Diffusivity, cm

2/s

GE0219 H = 3.4 ± 0.2 eV, D0 = 600 ± 5000 / 100 cm2 /s

Ge self diffusion (Hueger et al.)

Arrhenius plot of GE0219

Figure 4.3: Arrhenius plot of extracted diffusivities

137

SIMS profiles of the p-type samples (GE0219) in this work showed no evi-

dence of concentration dependent diffusivity. However there was indeed a slight

discrepancy between the n-type (GE0220) and p-type (GE0219) doped wafers in-

vestigated in this thesis, but this appears to be within experimental error. This

could also be attributed to the differing orientation of the wafers, causing slight

changes in the extracted Arrhenius parameters.

It has recently been proposed that the diffusion of Group III acceptors In

and Ga (and potentially Al) are mediated by a singly negatively charged vacancy-

impurity complex due to the reported independence of the diffusivity of Ga on

the substrate doping [78], and further that the activation energies of the diffusion

processes of these acceptors is much higher than that of Ge self-diffusion [88]. We

indeed see no evidence to contradict these findings, which is perhaps lamentable,

as extrinsic diffusion can in some cases lead to more favourable profiles for opto-

electronic devices, as is found in phosphorus diffusion from spin-on dopant sources

[22]. However, this can be a mixed blessing as the concomitant diffusion enhance-

ment of donor diffusion in Ge gives rise to difficulty in forming shallow junctions,

even at comparatively low temperatures.

Since a comprehensive explanation of the precise diffusion mechanism (or

mechanisms) of Ga in Ge was not the intention of this work, and the uncertainties

in extracting the diffusivities were considered to be too high to draw any firm

conclusion, we do not attempt to resolve this issue. Nevertheless, a comprehensive

survey of the dynamics of diffusion from Ga, In, and Al (which are all applicable to

the diffusion technique presented herein) is warranted, in particular in comparison

to faster diffusing donors [86].

138

Reference

Substrate

Material

type /

doping cm−3

Ga

Surface

concen-

tration

cm−3 or

implant

dose

cm−2

H, eV D0, cm2/s

1 This workp-type

1e15

≃ 2e20

cm−3

3.4 ±0.2

600 ±5000 / 100

1 This workn-type

1e14

≃ 2e20

cm−3

3.2

± 0.4

100 ±7140 / 10

2 Dunlap [76] n-type (≃ 1e17) 3.0 10

3Riihimaki

et al. [78]

p-type

2e19

1e14

cm−2

Ga66 (implant)

3.4 ± 0.2800 ±

7000 / 700

4Riihimaki

et al. [78]intrinsic

1e14

cm−2

Ga66 (implant)

3.21

± 0.07

80 ±90 / 40

5 Huger et al. [89] intrinsic

Ge

self

diffusion

3.13 ± 0.03 25.4

Table 4.9: Comparison of Arrhenius parameters

4.7 Proximity doping

4.7.1 Introduction

While the previously introduced technique, sandwich-stacked diffusion, pro-

vides a simple method to degenerately dope Ge, we turn our attention now to

139IR lamps

IR lamps

Quartz susceptor

Quartz process tube

Source wafer

Target wafer

support

wafer

struts

Sample

Gas flow

Thermocouple

Figure 4.4: Proximity doping technique

doping Germanium using the proximity doping method. The technique is illus-

trated in Fig. 4.4.

In this method, the source and target wafer are held in close proximity during

the doping process to ensure transfer of dopant material from source to target

wafer. In this case, the silicon source wafer is covered with a spin-on dopant film.

The doping process takes place in a rapid thermal processing (RTP) furnace, inside

a 4” process tube. The source and target wafers are both heated from above and

below by IR lamps and the process is temperature controlled by a thermocouple.

The source and target wafer sit inside a quartz susceptor inside the furnace and the

thermocouple touches the wafer stack through the bottom of the quartz susceptor.

The dopant is transferred by thermal out-diffusion and convection by the carrier

gas, an inert gas such as nitrogen. The dopant then in-diffuses and dopes the

target wafer as required.

4.7.2 Experimental - Proximity doping investigation using SIMS

In order to investigate proximity doping, secondary ion mass spectrometry

(SIMS) was employed as in the previous section. It was arranged to use the Univer-

sity of Western Sydney’s Cameca IMS5f microprobe on this occasion, as opposed

140

to the National University of Singapore’s ion microprobe. To calibrate the ion mi-

croprobe, a set of ion-implanted standard reference samples were prepared. These

were implanted at the Australian National University’s Department of Electronic

Materials Engineering. These calibrated reference samples were prepared according

to the following implant parameters, as tabulated in Tab. 4.10. The implants were

performed at room temperature and at a slight angle to minimize ion channelling

effects.

Material Dose (cm−2) Implant Energy (keV)

Ga 1e15 60

Sb 1e15 60

P 1e15 30

Table 4.10: Implant specifications for calibrated standards

4.7.3 Proximity doping with Sb3m Antimony spin-on film

An ion implanted standard was prepared and profiled in order to calibrate

the ion microprobe. The SIMS depth profile and TRIM simulation are compared

in Fig. 4.5 in order to calibrate and convert the secondary ion yield (counts, a.u)

to atomic concentration (atoms cm−3) and the sputtering time (s) to depth (nm).

This was achieved by matching the simulation with the depth profile.

The samples were prepared by cleaving and cleaning large pieces of germa-

nium, roughly 1” square. The source wafers had the spin-on dopant film spun on

at 3k RPM and was outgassed at 300 C for 10 minutes. Then the source and

target wafers were placed one on top of the other with spacers separating the two

as shown in Fig. 4.4. The spacer thickness was 170 micron. The assembly was

then heated to the desired temperature for the desired time with flowing nitrogen

141

0 50 100 150 200 250 300Depth, nm

1016

1017

1018

1019

1020

1021

Co

nce

ntr

ati

on

, cm

−3

TRIM Simulation

121Sb (60keV, 1e+15 cm−2 )

121Sb depth profile for ion-implanted standard

Figure 4.5: SIMS depth profile of Sb ion-implanted standard

ambient with a gas flow rate of 2 L/min. The temperature ramp rate was such

that full temperature was reached within 6 minutes. This limited ramp rate was to

ensure that the spin on dopant films did not crack. Film cracking is an issue [22]

due to CTE (coefficient of thermal expansion) mismatch and high tensile stress in

the as-spun SOD film [74].

In Fig. 4.6, we see results of the depth profiling for Sb as diffused in proximity

from the Sb3m spin on dopant source for 3 target wafers at temperatures from

600 to 700 C .

Although the data are very noisy, it is still possible to fit a sensible erfc

profile to investigate the diffusivity and surface concentration as a function of

temperature. In this way, data can be extracted from the fitted curves to help

formulate a model for the proximity doping with Sb.

By plotting the extracted diffusivities versus 1000/T, we can fit the diffusivity

142

0 50 100 150 200 250 300Depth, nm

1017

1018

1019Conce

ntration, ato

ms

/ cc

T, Sb27. 10 min, 700T, Sb27. 10 min, 700, fit.T, Sb29. 10 min, 650T, Sb29. 10 min, 650, fit.T, Sb31. 10 min, 600T, Sb31. 10 min, 600, fit.

Sb concentration in p-Ge @ 20nA beam current

Figure 4.6: SIMS data of proximity doping with antimony withmodel fits to data

to an Arrhenius law and hence extract the thermal activation energy and the pre-

exponential factor from the plot. These data are then useful to model accurately

future processes. We extract from this plot an thermal activation energy of 1.2 ±

0.3 eV and a preexponential factor of 1.5× 10−7 ± 3.6× 10−6/3.7× 10−9 cm2/s

indicating quite low activation energy of this process indeed. In this notation, the

two figures given for the plus and minus values of the preexponential factor are

the high and low values for the corresponding values of activation energy (since

the relationship is exponential high and low values are given instead of plus/minus

for the uncertainties). This is compared with H = 2.4 eV and D0 = 4.2 cm2/s as

reported by Dunlap [76]. This lower thermal activation energy (H) indicates that

Sb diffuses more easily in the proximity configuration.

143

1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.161000 / T, mK−1

10-14

10-13

10-12

Diffusivity, D, cm

2s−

1

H = 1.2 ± 0.3 eV, D0 = 1.5e-07 ± 3.6e-06 / 3.7e-09 cm2 /s

Arrhenius plot of diffusivity

Figure 4.7: Arrhenius plot of extracted diffusivities for Sb in Ge

This may be due to partial evaporation of Ge itself at elevated temperatures,

facilitating dopant in-diffusion. Another issue may be that the target wafer itself

undergoes some stress as a function of heating in the table like configuration,

where the sample is supported by support wafer fragments. This stress may also

facilitate the diffusion process, lowering the thermal activation energy. Finally, the

wafers themselves, being fairly poor quality VGF wafers, may have a high defect

density and this may have affected the diffusivity, enabling the dopant to diffuse

more readily for a given temperature.

Under the assumption that surface concentration of dopant at the surface of

the wafer in the proximity configuration should also follow Arrhenius law, we can

plot the surface concentration as a function of inverse temperature for the data.

This is depicted in Fig. 4.8.

In Fig. 4.8, we can see a reasonably linear relationship between log of

144

1.00 1.05 1.10 1.15 1.201000 / T, mK−1

1017

1018

1019

Surface

conce

ntration, N

0, cm

−3

H = 1.1 ± 0.3 eV

Arrhenius plot of surface concentration N0

Figure 4.8: Arrhenius plot of extracted surface concentrationsfor Sb in Ge

N0, the surface concentration, and 1000/T , the inverse temperature. This tends

to suggest that the surface concentration for a given time period (in this case,

10 minutes) is a thermally activated process following an Arrhenius law. This

information may be used to model the process at a later stage, by providing a

model for the surface concentration and diffusivity for proximity doped antimony

processes of comparable time and temperatures.

4.7.4 Proximity doping with GaB260 spin-on film: Gallium diffusion

In an identical manner to the previous section, Ga proximity doped Ge sam-

ples were examined using the ion-microprobe at the University of Western Sydney.

Depicted in Fig. 4.9 is the data for 4 target wafers diffused from GaB260 (the spin-

on dopant product) source wafers at temperatures from 800 to 875 C . This shows

145

that quite a large proportion of gallium was diffused, with surface concentrations

ranging from ≃ 1e19 − 1e20 cm−3 as a function of temperature. This indicates

that as in the previous case with degenerate gallium doping having been achieved

using the “sandwich-stacked” diffusion technique, for the higher temperatures,

degenerate doping can be achieved using the proximity doping technique. This

also reaffirms and supports the previous measurements for “sandwich-stacked”

diffusion.

0 200 400 600 800 1000 1200Depth, nm

1015

1016

1017

1018

1019

1020

1021

Concentration, atoms / cc

T, Ga18, 875 Deg. C, 10 minT, Ga18, 875 Deg. C, 10 min, fit.T, Ga20, 850 Deg. C, 10 minT, Ga20, 850 Deg. C, 10 min, fit.T, Ga22, 825 Deg. C, 10 minT, Ga22, 825 Deg. C, 10 min, fit.T, Ga24, 800 Deg. C, 10 minT, Ga24, 800 Deg. C, 10 min, fit.

Gallium concentration in n-Ge @ 20nA beam current

Figure 4.9: Proximity doping of Ge with gallium, showing SIMSempirical data and model fits

By comparing Fig. 4.10 and Fig. 4.3, we see a huge difference in the ther-

mal activation energy between the proximity diffused samples and the “sandwich-

stacked” diffusion samples (and published data [78, 76]). With a thermal activa-

tion energy of only 1.9 ± 0.7 eV, the diffusion has occurred much more rapidly at

146

lower temperatures than expected by the previous model. Again the explanation

may be a combination of Ge outdiffusion of the target wafer and mechanical stress

undergone in the diffusion chamber. In any case, it is handy to know that diffusion

can occur at lower energies (i.e. more rapidly for a given temperature) as this may

be of advantage for some processes.

0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.941000 / T, mK−1

10-14

10-13

10-12

Diffusivity, D, cm

2s−

1

H = 1.9 ± 0.7 eV, D0 = 8.4e-05 ± 0.14 / 2.8e-06 cm2 /s

Arrhenius plot of diffusivity

Figure 4.10: Arrhenius plot of extracted diffusivities for Ga in Geusing proximity doping

The Arrhenius plot of surface concentration as a function of temperature

in Fig. 4.11 shows again a reasonably linear curve. These data may be used to

model the process in process models to determine the surface concentration and

diffusivity as a function of temperature.

147

0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.941000 / T, mK−1

1019

1020

Surface concentration, N

0, cm

−3 H = 3.2 ± 0.5 eV

Arrhenius plot of surface concentration N0

Figure 4.11: Arrhenius plot of extracted surface concentrationsfor Ga in Ge using proximity doping

4.7.5 Proximity doping with phosphorus

In this experiment, proximity doping was investigated from phosphorus sources

(P507, P509 SOD films from Filmtronics, [Butler, PA, United States]). The differ-

ence between the two spin-on dopant source films is the phosphorus concentration,

4% for P507 and 15% for P509. The samples were profiled with a 50 nA SIMS

beam current as this gave the best results for most profiles. The profiles are de-

picted in Fig. 4.12. The profiles cannot be fit to a standard erfc diffusion profile

since phosphorus diffuses extrinsically in Ge [90], that is, the diffusivity is a function

of the intrinsic carrier concentration and the dopant concentration, viz. [36]:-

Di = D intri ·

1 + β−i · n

ni+ β=

i ·(

nni

)2

+ β+i · p

ni

1 + β−i + β=

i + β+i

(4.12)

where Di is the effective diffusivity, D intri is the intrinsic diffusivity β−

i is the

148

0 100 200 300 400 500 600 700Depth, nm

1017

1018

1019

1020Concentration, atoms / cc

3T. 800 Deg. C, 6 min,P50911T. 800 Deg. C, 6 min,P5075T. 750 Deg. C, 6 min,P507

P concentration in p-Ge @ 50nA beam current

Figure 4.12: Proximity doping from phosphorus spin-on dopantsource

weighting of singly negatively charged vacancies, β=i is the weighting of neutral

vacancies, and β+i is the weighting of singly positively charged vacancies.

This has the effect that the diffusivity is much higher than usual if the free

electron concentration significantly outweighs the intrinsic carrier concentration.

This tends to distort the profiles and render them fairly abrupt. The profiles are

fit using a non-linear model in Section 4.7.7.

4.7.6 Direct spin-on doping with phosphorus

Lastly, we consider directly spin-on doping the target wafer using phospho-

rus films (specifically, the P507 spin-on dopant product). This is in general to

be avoided since the dopant films themselves are inhomogeneous and prone to

149

cracking. This creates non-uniformly doped layers within the semiconductor. The

spin-on dopant films themselves may contain trace elements of contaminants that

are not advisable to bring into contact with the wafer surface. This may contam-

inate the wafer and bring about lifetime degradation. Nevertheless, direct spin-on

doping with phosphorus brings about the best looking depth profiles, with abrupt

junctions and box-like morphology.

0 200 400 600 800 1000 1200 1400Depth, nm

1016

1017

1018

1019

1020

1021

1022

1023

Concentration, atoms / cc

SOD matrix

Ge matrix

4S. 750 Deg. C, 6 min,P5076S. 700 Deg. C, 6 min,P5078S. 655 Deg. C, 6 min,P507

P concentration in p-Ge @ 50nA beam current

Figure 4.13: Direct spin-on doping with phosphorus spin-ondopants.

In Fig. 4.13 we see an initial ≃ 340nm region which is the spin-on dopant

matrix itself. Thereupon the ion microprobe cuts its way through the SOD matrix

into the underlying Ge wafer (Ge matrix). This is due to the fact that the source

wafers did not have the spin-on dopant films removed prior to SIMS analysis, so

that the makeup of the SOD matrix itself could be studied. In all 3 cases, the SOD

150

Sample Temperature (C ) Time (s) Approx. junction depth (nm)

4S 750 360 923

6S 700 360 593

8S 650 360 333

Table 4.11: Approximate junction depth of spin-on doped sam-ples

matrix appears to have the same shape, and hence the phosphorus concentration

is consistent sample to sample. However obtaining accurate results from the ion

microprobe with insulating coatings is difficult, so not much can be quantatively

assessed from these data. This is due to charging affects which interfere with the

ion beam and distort results.

The abrupt profiles are due, once again, to extrinsic diffusion, the mechanism

by which the diffusivity becomes proportional to the quotient of the free electron

concentration and the intrinsic carrier concentration. The approximate junction

depths for the three samples investigated in Fig. 4.13 are tabulated in table Tab.

4.11.

4.7.7 Fit to phosphorus depth profiles using non-linear model

In order to extract the diffusivity of phosphorus, the diffusion equation (Fick’s

second law, Eqn. 4.6) must be solved numerically with a non-linear model for the

diffusivity at each mesh point, viz.:-

Dext ≃ D

(

Cphos

ni

)2

(4.13)

since we assume that the term involving neutral vacancies of Eqn. 4.12

dominates. Profiles are first normalised to remove any initial portions of the SIMS

151

data that contain sputtering of surface materials such as SOD films or surface

coatings from the diffusion process. In the case of the direct spin-on samples, the

concentration is clipped at the solubility limit of the material at that temperature

as determined from the SIMS data itself. This produces a ”squaring-off” of the

profiles in the fitted data.

0 200 400 600 800 1000 1200 1400Depth, nm

1017

1018

1019

1020

Concentration, atoms / cc

Fits to depth profiles using non-linear model

8S. 655 Deg. C, 6 min,P507Model fit6S. 700 Deg. C, 6 min,P507Model fit4S. 750 Deg. C, 6 min,P507Model fit5T. 750 Deg. C, 6 min,P507Model fit3T. 800 Deg. C, 6 min,P509Model fit11T. 800 Deg. C, 6 min,P507Model fit

Figure 4.14: Fit to depth profiles using non-linear model

It is difficult to fit to the data from samples which are directly spun-on. This

is most likely due to the extremely high surface concentration of dopant above

the solid solubility limit of the film [22]. This has an effect on the surface of the

152

sample, warping the crystal lattice and introducing higher-order effects not taken

into account by this simple model. Nevertheless, the model fits are of reasonable

accuracy. The model fits to data for both phosphorus “proximity”- and direct

spin-on doping are shown in Fig. 4.14.

0.90 0.95 1.00 1.05 1.10 1.151000 / T, mK−1

10-15

10-14

10-13

10-12

10-11

Diffusivity, D, cm

2s−

1

H = 3.1 ± 1.1 eV, D0 = 1.3e+03 ± 0.3 / 3.9e+08 cm2 /s

Arrhenius plot of phosphorus diffusivity

Figure 4.15: Arrhenius plot of phosphorus diffusion in Ge

An Arrhenius plot of extracted diffusivities is shown in Fig. 4.15. The

extracted activation energy is 3.1 ± 1.1 eV, which is higher than reported in the

literature, 2.4 eV [76]. This may be due to difficulties in fitting the data and the

inaccuracy of the fits, but in all likelihood this is a reasonable figure, particularly

when extracted under extrinsic conditions (the data of W.C Dunlap Jr. [76] is

for the case of intrinsic diffusion). Nevertheless the published data is within this

uncertainty range, which is quite large due to conflicting measurements at the

same temperature.

153

4.8 Conclusions

A novel technique, “sandwich-stacked diffusion”, for the degenerate doping

of Ge, has been presented. The thermal activation energies and pre-exponential

factors have been extracted for both diffusion into low doped p-type and n-type Ge

wafers, and are found to be in agreement with published data, within experimental

error. While thorough conclusions could not be drawn about the nature of the Ga

diffusion process in this work, this technique provides a simple, low cost, method

to further investigate diffusion of acceptors in Ge, which for practical reasons can

be problematic.

An approximate analytical model has been derived to describe the doping

profile, which may be used to design and fabricate devices. Processes may be

tuned by simple sheet resistance measurements in conjunction with this analytical

model.

This technique is suitable for low-cost processing of Ge devices for a wide

range of applications, including degenerate doping for tunnel FETs and Esaki

diodes for high frequency and low power devices, and degenerate doping for back

surface fields in photovoltaic and thermophotovoltaic devices.

In this section we have also presented data for “proximity” diffusion and

direct spin-on doping using phosphorus. We have compared “sandwich-stacked

diffusion”, the novel technique developed in this work, with “proximity doping”

and direct spin-on doping. Direct spin-on doping should be avoided due to peeling

and cracking of the film due to CTE (coefficient of thermal expansion) mismatch

between the silica films and the germanium wafers, and inhomogeneities and con-

taminants in the spin-on dopant films. Nevertheless, we saw that direct spin-on

doping had a more abrupt and hence advantageous profile as compared to proxim-

ity doping for phosphorus spin-on dopant sources. This could be useful for creating

154

tunnel diodes and other such devices.

It was shown that degenerate p-type doping is possible both with sandwich

stacked diffusion as well as with proximity doping with gallium. However the

thermal activation energy was seen to be much lower for the proximity doping

case. This was attributed to a combination of mechanical stress on the wafer

during the process as well as Ge out-diffusion and partial evaporation.

Some concerns may be raised about the discrepancy in results recorded for

Ga diffusion by the two techniques. Firstly, different equipment was used to char-

acterise the two sets of samples. Since SIMS profiling is destructive, it is difficult

to profile the same sample on both sets of equipment to ensure that discrepancies

are not due to equipment related error. Hence such comparisons were not per-

formed. However, since depth is measured on the resultant craters using a stylus

profilometer, in spite of differences in sputtering rate on the two pieces of equip-

ment, the depth profile data should be accurate. The ion yield was calibrated on

the UWS equipment by means of ion implanted samples so should be more trust-

worthy than the data from the National University of Singapore, which had to be

calibrated using sheet resistances. The two sets of equipment were employed in

this study purely for reasons of cost; it was less expensive to profile a large number

of samples at the University of Western Sydney. However, the equipment at UWS

was in poorer condition than that of the National University of Singapore. This

resulted in noisy data for low ion yield samples such as the antimony samples.

Nevetheless it is concluded that the discrepancy in data obtained for Ga diffusion

is purely a function of the above mentioned effects and not equipment related.

Since the techniques described herein were applicable to rapid thermal pro-

cessing, we anticipate that they would be readily amenable to mass manufacture

and hence low cost in a manufacturing environment, as is necessary for devices in

the solar cell manufacturing sector.

Chapter 5

Passivation, antireflection, and contacting technologies for germanium

optelectronic devices

5.1 Introduction

In this chapter we present the critical passivation, antireflection and con-

tacting technologies developed for this work. Device passivation has three main

purposes: to terminate the sensitive semiconductor lattice and passivate any dan-

gling bonds at the surface which may impair device performance, to protect sen-

sitive device regions such as junctions from impurities and inclusions which may

cause catastrophic device failure, and to protect the device from moisture and

atmospheric impurities during subsequent device processing or storage. Antireflec-

tion coatings are used to account for the mismatch in refractive index between

germanium and any material onto which it may interface along the optical path

to the photon source; an example would be the mismatch between the refractive

index of air and the semiconductor itself. Our simple passivation/antireflection

coating technology is shown to reduce reflection losses of a single junction germa-

nium solar cell under AM1.5G illumination to under 10%, which is adequate, since

surface texturing can be used to further reduce reflection losses (assuming light

trapping texturing). A reflectivity of 10% is a fair trade-off between an optimal

structure and a practical structure in terms of cost and complexity. For instance,

the emphasis in this work is on practical materials that are readily available to solar

156

cell processes and which are low in cost, which is a higher priority than absolutely

optimal optical matching using more exotic materials.

Contact metallisation is discussed in this section and a variety of contact

metals are evaluated. The purpose is to engineer low resistivity contacts to ger-

manium to lower losses due to series resistance.

5.2 Passivation of germanium

Germanium electronic and optoelectronic devices have historically suffered

due to lack of a stable native oxide as passivation layer [91, 92]. In this work, we

are careful to distinguish between the many different processes which are generally

lumped together under the umbrella of passivation - that is, wet chemical treat-

ment/dry chemical pretreatment (i.e. pretreatment for dangling bond termination

and to prevent oxidation of the surface), passivation layers (e.g an insulator or

wide band gap semiconducting overlayer to prevent surface leakage currents, and

act as a template for deposition of further layers) and finally environmental passi-

vation (to prevent issues such as moisture/contaminant ingress). It must be noted

however, that effective passivation methods for formation of high quality interfaces

for gate stack formation (e.g. for high frequency CMOS applications) are not con-

sidered in this work, which is intended solely to address the need for stable, high

quality passivation/anti-reflection (AR) coating for optoelectronic applications.

With respect to the first of the three above passivation processes, i.e. chem-

ical pretreatments, the following have been shown to be effective in terminating

dangling bonds at the Ge surface: hydrogenation [26, 93, 94, 95], passivation

with halides HCl or HCl/HBr [79, 96, 97, 98], sulphidation with H2S [99] or

(NH4)2S [80, 100, 101], and nitridation with in-situ NH3 or N2 (or both) as gas

(at high temperature) or plasma (at lower temperatures) [102, 103, 104, 105].

Hydrogenation can be performed using wet treatments of HF and DI water and

157

using in-situ H2 plasma [26]. There is also evidence of organic compounds pas-

sivating the germanium surface, yielding low surface recombination velocities (∼

20 cm/s) [106].

Regarding passivation layers, germanium has been capped with phosphorosil-

icate glass (PSG) [107], III-V materials such as InGaAs/GaAs [56], a-Si:H [26],

chemical vapour deposited (CVD) SiNx, Ge3N4 [105] and high-κ dielectrics such

as TaN, HfO2, sometimes in combination with thermally grown GeO2 [105]. MgO

has also been investigated [101], in combination with sulphur wet treatment, to

de-pin the Fermi level at the surface of the Ge. A CVD-deposited silicon interlayer

between Ge and the passivation layer has shown to reduce interface traps and yield

good MOS characteristics [108].

These technologies however have their drawbacks, in terms of processing

temperature for passivation layer deposition (i.e. 600C for SiNx deposition, or

oxide growth [105]), which would destroy device junctions (due to diffusion), or in

terms of the fact that they cannot be heated to high temperatures post passivation

(as in a diffusion/drive-in/implant activation step, or subsequent high temperature

treatment) without damage to the passivation layer. In this section, we compare

published passivation technologies for optoelectronic devices and present a novel

low temperature, low cost inductively-coupled plasma enhanced chemical vapour

deposition (ICPECVD) based passivation technology which shows excellent perfor-

mance for use in optoelectronic devices. We review surface wet/dry pre-treatment

methods and demonstrate the importance of adequate surface pre-treatment to

well passivated devices. The anti-reflection properties of the passivation layer stack

are considered, and we show that it is possible to optimise both, giving a three

layer anti-reflection coating (ARC) consisting of a-Si:H/SiNx/MgF2 which shows

excellent chemical and environmental robustness, low reflectance, high transmis-

sion and low surface recombination velocity (SRV).

158

5.2.1 Experimental

5.2.1.1 Sample preparation

Photoconductors were prepared for photoconductive decay measurements

from low doped (∼40 Ohm·cm) n-type Ge<211> wafers [Umicore, Olen, Belgium]

by cleaning (soak in warm trichloroethylene, acetone, methanol plus ultrasonic

bath), pre-treatment and passivation layer deposition, and finally depositing Cr/Au

Schottky contacts after stripping the passivation layer using reactive ion etching

(RIE) from the contact pad regions.

It has been noted that surface pre-treatment prior to passivation is important

to reducing SRVs and reducing interface trap densities. Two main wet treatments

are investigated in this work: HF/H2O, and HCl/HBr. HF/H2O treatment has

been shown to strip native oxides and terminate the Ge surface with hydrogen

[109]. HCl/HBr has been shown to passivate dangling bonds and leave the surface

terminated with Cl [110] or potentially Br. This treatment is carried out at room

temperature, since at elevated temperatures, this solution can etch the germanium

surface and potentially consume device active regions. Each treatment is preceded

with a 30s dip in 1:1:5 NH4OH : H2O2 : DI to remove approx 100nm of the surface

damaged region of the wafer (i.e. due to sawing) and any diffused impurities at

the surface. The chemical treatments are tabulated in Tab. 5.1.

5.2.1.2 Dry pre-treatment

Two main plasma treatments are considered in this work, namely H2, and

NH3 plasmas. The purpose of the dry plasma treatment is much the same as the

wet treatment, i.e. to desorb any native oxides and passivate dangling bonds. It

has been suggested that ammonia plasma treatment leaves the surface terminated

with nitrogen [4]. Whilst the ammonia pre-treatment can be carried out in-situ

159

Treatment Steps

A - Hydrogenation

1. Dip 1:1:5 NH4OH:H2O2:DI 30s

2. HF dip (1:3 BOE), 1 min

3. DI soak, 3 min

B - Halide Passivation1. Dip 1:1:5 NH4OH:H2O2:DI 30s

2. 1:1:5 HCl:HBr:DI soak, 1 min

Table 5.1: Wet pre-treatments for Germanium passivation

Treatment Gas flow rate Plasma Power

A. H2 plasma (ex situ) 10 sccm RF, 15W 30s

B. NH3 plasma (in situ) 10 sccm ICP, 200W 30s

Table 5.2: Dry pre-treatments for Germanium passivation

within the ICPEVCD reactor, the hydrogen plasma treatment was performed in a

separate tool, a Plasmalab System 100 RIE from Oxford Instruments [Austin, TX,

USA]. It must be noted that the brief exposure to atmosphere when transferring

to the ICPECVD after H2 plasma treatment may negate some of its beneficial

effects. The basic recipes are described in Tab. 5.2. The ammonia treatment

took place at 250 C , whereas the H2 plasma treatment was performed at room

temperature.

5.2.1.3 Passivation layer deposition

In this work we consider SiNx, a-Si:H/SiNx, and SiO2 as potential passivation

layers. These are deposited in a Sentech SI500D ICPECVD [Sentech Instruments

GmbH, Berlin, Germany]. The recipes are shown in Tab. 5.3. It was noted in

previous experiments that low ICP power favours higher quality SiO2 and a-Si:H

160

MaterialTable

temp. ( C)Flow rates ICP Power

SiO2 120 SiH4 = 6.5 sccm 450W

He = 123 sccm

Ar = 126 sccm

O = 70 sccm

a-Si:H 300 SiH4 = 5 sccm 35W

Ar = 95 sccm

SiNx 100 SiH4 = 7.3 sccm 800W

Ar = 139 sccm

NH3 = 10 sccm

Table 5.3: Passivation layer film deposition parameters

with good optical and mechanical properties and resilience to high temperature

processing, however for SiNx, the reverse is true. Here SiNx is deposited at very

high (∼ 800 W) ICP power, which furnishes very high quality, chemically resistant

SiNx with wide area uniformity (> 5” diameter uniformity area).

5.2.1.4 Anti-reflection coatings

In order to investigate the potential for combining a-Si:H, SiNx and MgF2

as both passivation and anti-reflection (AR) coating, films were first deposited

individually on germanium in order to obtain optical constants using spectroscopic

ellipsometry. This data was then used to optimise AR film parameters. Whereas

a-Si:H and SiNx were deposited in the Sentech ICPECVD, MgF2 was evaporated

in a bell-jar thermal evaporator at a rate of 0.5-1 A/S. Optical constants (complex

refractive index) and thicknesses were extracted by spectroscopic ellipsometry using

a Woollam M2000D [J. A. Woollam Co., Lincoln, NE, USA].

161

5.2.1.5 Contact deposition

Contacts were deposited using thermal evaporation of 5nm Cr and 50nm Au

at a rate of 0.5 A/s (Cr) and 10 A/s (Au), respectively, in a bell jar evaporator

at a vacuum of 1e-6 Torr. The contacts were patterned using photo-lithography

and lift-off patterning, and lift-off was performed in boiling acetone, with residual

resist removed using a swab dipped in hot acetone.

5.2.2 Characterisation

5.2.2.1 Photoconductive decay

PMOS

White LED

Sample under test

To power supply

To square

wave generator

probes

To parameter

analyser

To parameter

analyser

Test structure viewed from above

Contact

pads

Mesa

Bulk

+ -Applied bias

Figure 5.1: Measurement setup for photoconductive decay mea-surements

Photoconductive decay measurements were undertaken by illuminating the

device with a white LED broadband source pulsed at 100 Hz. The test setup is

shown in Fig. 5.1, which also shows a photoconductor viewed from above. The

162

mesa region of the photoconductors was 40 µm wide and between 96 and 900

µm long. Devices were biased at 30V, in order to ensure significant photocurrent.

The white LED was switched using a p-channel MOSFET with switching times

several orders of magnitude shorter than the minority carrier lifetime. Since the

surface recombination velocity (SRV) is essentially a boundary condition on the

drift diffusion equations (i.e. for holes dp

dx|surface = S ·p where S is the SRV, p is the

hole concentration) a useful expression can be derived by solution of the diffusion

equation [111]. If we make the assumption that S is much less than the diffusivity

of electrons divided by half the wafer thickness (= 500 µm ), in this case roughly

4000 cm/s, then we can make use of the following expression relating the effective

lifetime to the bulk lifetime and S [26]:

1

τeff=

1

τbulk+

1

τSRV=

1

τbulk+

2S

d(5.1)

where τeff is the effective lifetime, τSRV is the surface recombination dom-

inated lifetime, τbulk is the bulk lifetime, S is the surface recombination velocity

(total for both sides) and d is the wafer thickness. The effective lifetime is taken

to be the dominant decay curve of a potentially multi-exponential curve [112].

The bulk lifetime was assumed to be roughly 3.8 ms, a value taken from

the literature [26]. In [26] this value was determined for wafers of comparable

quality from the same manufacturer as were used in these experiments so should

be approximately valid. The usual method for determining the bulk lifetime and

surface recombination velocity is to perform measurements on samples of varying

thickness. A plot of 1/τeff vs. 1/d is then constructed; the y intercept is equal to

1/τbulk and the slope is equal to 2S [26, 112]. Due to budget and time constraints,

wafers of only a single thickness were used in these experiments, hence a value for

the bulk lifetime is assumed, rather than experimentally determined.

163

EntryWet

treatment

Dry

treatmentPassivation Layer τSRV S (cm/s)

1 A A 40nm SiO2 150 µs 173

2 A A40nm a-Si:H

75nm SiNx171 µs 150

3 A A 75nm SiNx 60 µs 423

4 B B 40nm SiO2 150 µs 173

5 B B40nm a-Si:H

75nm SiNx1.7 ms 21

6 B B 75nm SiNx 60 µs 423

7 B -40nm a-Si:H

75nm SiNx78 µs 327

8 B - 75nm SiNx 40 µs 631

9 B - Unpassivated 200 µs 131

Table 5.4: Passivation layer film deposition parameters, showingwet pretreatment (see table Tab. 5.1) and dry pretreatment(see table Tab. 5.2), measured lifetime and extracted surfacerecombination velocity.

The photoconductive decay results are shown in table Tab. 5.4.

From the photoconductive decay measurements, we may draw a few infer-

ences about the various passivation processes. Firstly, the HCl/HBr pretreatment,

which has been shown previously to reduce interface state density can successfully

terminate the lattice so as to reduce surface recombination velocity, which is evi-

denced by entry 9 in table Tab. 5.4. Secondly, in situ dry treatment is preferable

to ex situ or wet treatment alone, which may be attributed to native oxide re-

moval. Finally, insulating wide band gap (optical band gap ≃ 2.2eV) amorphous

silicon is a superior passivation layer in combination with SiNx, reducing interfacial

164

recombination/tunneling, which leads to longer surface recombination dominated

lifetimes and hence reduced surface recombination velocities. The photoconduc-

tive decay measurement of the NH3 plasma treated sample is depicted in Fig. 5.2

showing extraction of the dominant τeff .

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time,ms

10-2

10-1

100

101

Photocurrent, m

A

Photoconductive decay of NH3 treated Ge:a-Si:H:SiNx

Fit, (τeff = 1.7 ms)

Meas.

Figure 5.2: Photoconductive decay of NH3 plasma treated Gecapped with ICPECVD a-Si:H/SiNx

5.2.2.2 Summary

It is insightful to compare these results with published values for comparable

technologies in germanium. This compares the results obtained in this work to

two references from the literature which employed the microwave reflected pho-

toconductive decay (µPCD) method as opposed to the simple constant voltage

photoconductive decay method employed herein. In the µPCD method, the pho-

165

toconductivity is measured by microwave reflection after carriers are injected by

optical pulses from a pulsed laser [85].

These results are set out in Tab. 5.5. Note that the results of Posthuma et

al. [26], while impressive, are derived from extrapolation from curves of 1/τ vs.

1/d, which gives very optimistic values for the surface recombination velocity and

bulk lifetime. Indeed the bulk lifetime seems quite high, even for very pure CZ

germanium, which tend to contradict lifetime studies as published in the literature

[113, 114], which indicate a maximum lifetime of ∼ 2ms for the bulk lifetime.

Nevertheless, this might be attributable to advances in Ge material processing,

since the first measurements of lifetime were made many decades ago. On the

other hand, the results of Swain et al. [115] seem a great deal more realistic

and serve to highlight the affinity the germanium surface has for both hydrogen

and chlorine. However, we must concede that wet treatment alone is simply not

enough to adequately passivate germanium, as it does not deal with the issues

of necessary A/R coatings nor environmental passivation. Hence, we can surmise

that wet treatment (with its reasonable stability in air) can be used to prevent

oxidation of the sample prior to loading into an ICPECVD tool to deposit capping

and A/R layers. Finally, we must point out that since the PCD technique as used in

this work involves making contacts, and that contacts are recombination centers,

the PCD results will always give shorter lifetimes than the µPCD technique, which

is a contactless microwave-reflected photoconducive decay method. Nevertheless,

we can conclude that the SRVs presented in this thesis are comparable to those

reported in the literature.

5.2.3 Anti-reflection coatings

In addition to passivation (which lowers surface recombination velocities,

thus improving photocurrent in optoelectronic devices), anti-reflection coatings are

166

Ref.Passivation

Layer stackMaterial

τeff

τbulkS (cm/s)

Meas.

techn.

Posthuma

et al. [26]

a-Si:H

∼ 100nm

p-type

CZ Ge

550µs

3.8ms17 µPCD

Swain

et al. [115]

Wet treatment

(HF

HCl)

p-type

CZ Ge

∼ 475, 370µs

1.17ms23, 37 µPCD

This

work

a-Si:H

∼ 40nm :

SiNx

100nm

n-type

CZ Ge

1.7ms

3.8ms21 PCD

Table 5.5: Summary of Germanium passivation technologies

necessary to further maximize photoresponse or efficiency of power transfer. In

this work, we consider the use of a-Si:H/SiNx/MgF2 as combined passivation/anti-

reflection (A/R) coating. This is depicted in Fig. 5.3.

Figure 5.3: Schematic diagram of proposed 3 layer A/R coating.Ei is the incident wave, Er the reflected wave. dj is the layerthickness; ηj is the refractive index of each layer.

It is informative to calculate what the optimal refractive indices ought to be

in order to totally cancel any reflections at the interfaces between layers, in order

167

to compare them with the chosen materials’ properties. In this case, we must

satisfy [33, 34]:-

η1η0

=η2η1

=η3η2

=ηSη3

(5.2)

with dj set to a quarter wavelength. A possible solution is to have η3 : η2 : η1

= 2.8 : 2 : 1.4, which is a reasonable match for a-Si:H (n ∼ 3.5) : SiNx (n ∼ 2)

: MgF2 (n ∼ 1.4).

In order to optimise this AR structure, the optical constants were first ex-

tracted for each material used in the coating. In the case of SiNx, it is known

that the refractive index can be tuned by adjusting the ammonia flow rate. For

the SiNx recipe in Tab. 5.3, refractive index as a function ammonia flow rate is

depicted in Fig. 5.4.

Figure 5.4: Refractive index (@ 632nm) vs. ammonia flow rate

Since the higher refractive index is required, the lowest ammonia flow rate

(10 SCCM) is used; lower flow rates were not considered since it was assumed

that this would disturb the governing reactions adversely and give rise to poor

quality films. Another property that is necessary to tailor is the hydrogen content

of the SiNx film; high hydrogen content will cause absorption throughout the

168

visible and infrared regions such that any anti-reflection properties would be more

than negated. High ICP power and low growth rate were seen to ensure low

hydrogen content and hence low absorption, as evidenced by the imaginary part

of the complex refractive index (i.e. k) being less than 10−5 for all wavelengths

longer than 400 nm.

This is in contrast to a-Si:H, which is deposited with low ICP power and

higher substrate temperatures to ensure high quality films. Under these limit-

ing conditions, the deposition technology approaches chemical vapour deposition

proper. These low hydrogen content films are quite robust and well suited to high

temperature processing, surviving 10-15 minute treatment at 750C without any

change in optical thickness, and without evidence of bubbling (which can occur if

H content is substantial). Optical constants of all 3 films are shown in Fig. 5.5.

Using the optical constants, 3 layer anti-reflection coatings may be designed

and optimised for a particular wavelength or spectrum. In this work, we consider

optimisation of the AR coatings for the solar (AM1.5G) spectrum for use in pho-

tovoltaic applications. The AM1.5G spectrum was chosen in order to improve the

efficiency of stand-alone Ge solar cells, as presented in chapter 6. In this work,

we do not choose to optimise for longer wavelengths, such as for the irradiation

expected in a four terminal tandem cell configuration with CdTe top cell. This is

because we the precise thickness of the CdTe top cell was not firmly determined at

time of writing. However, an identical procedure can be followed to optimise the

structure for longer wavelengths. As we shall see, the below structure shows rea-

sonable response at longer wavelengths so we wouldn’t expect much would change

if we were to optimise for the infra-red portion of the spectrum. Note that no anti-

reflection coating is required for germanium in the case of a monolithic tandem

cell; instead, we would like a reflective back contact to reflect any light reaching

the back surface back through the device. Nonetheless, adequate passivation of

169

0 500 1000 1500 2000

Wavelength (nm)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5R

efr

acti

ve index, n

SiNx a-Si:H MgF2

Optical properties of films

Figure 5.5: Refractive indices of films used for passivation/ARcoating

the back surface is crucial to reducing back surface recombination velocities so

the structure presented below is still applicable to a monolithic tandem to improve

electronic properties of the back surface.

The optical stack is optimised by evaluating the usual expressions for the

reflectance (R), transmittance (T) and absorption (A) of the multilayer film [33,

34]:

R =

(

η0B − C

η0B + C

)(

η0B − C

η0B + C

)∗(5.3)

T =4η0Re(ηS)

(η0B + C )(η0B + C )∗(5.4)

A = 1− R − T =4η0Re(BC

∗ − ηS)

(η0B + C )(η0B + C )∗(5.5)

170

where:

B

C

= Meq

1

ηS

(5.6)

Meq =n∏

j=1

cos(φj)iηjsin(φj)

iηj sin(φj) cos(φj)

(5.7)

η0 = Y0 = ǫ0 · c0 (5.8)

ηS = Y0(nS − ikS) (5.9)

ηj = Y0(nj − ikj) (5.10)

(5.11)

φj =2πηj · dj

λ(5.12)

where Y0 is the permittivity of free space and nl , kl are the refractive index

and extinction coefficient of medium l respectively (l = 0 = air, l = S = substrate,

l = j = layer j, etc.) and λ is the wavelength.

Although dispersion in the films is comparably small, it cannot be neglected

entirely, necessitating solution for the optimal structure via numerical (i.e. global

solution), rather than analytical means (i.e. for one particular wavelength). This

allows one to generate a 3 dimensional dataset to optimize the optical properties

of the film stack, as weighted over the AM1.5G spectrum, in order to maximize

energy transfer to the underlying device.

Fig. 5.7 shows the slice through the data cube for d1 = 90 nm, showing

reflection, transmission, and absorption characteristics as a function of the two

remaining free parameters, d2 and d3.

The amorphous silicon films are highly absorptive. This is due in part to the

above band gap absorption of the films (the optical bandgap of the amorphous

171

400 800 1200 1600

Wavelength, nm

0

20

40

60

80

100

%

Reflectance, RTransmission, T

Reflectance and Transmission for A/R coating

Figure 5.6: Transmission/reflectance of as-deposited a-Si:H/SiNx/MgF2 A/R stack

silicon as deposited in this work is roughly 2.2eV) and partly due to absorption

due to excess hydrogen. By neglecting the absorption of the amorphous silicon,

we may optimise the AR stack and minimise reflection. This yields a minimum

reflectivity of the stack of 6.47%, with d1:d2:d3 = 89:58:76 nm, as shown in Fig.

5.8. However, this does not maximise the transmission. The optical transmission

of the coating is maximised for d1:d2:d3 = 89:61:10 nm, where the amorphous

silicon is made as thin as possible, as shown in Fig. 5.7. Figure 5.6 shows the

transmission and reflectance of the latter, transmission-optimised, AR stack.

As part of future investigations, annealing should be experimented with to

reduce the absorption of these films and maximise transmission. However, an-

172

75 80 85 90 95 100 105 110 MgF2 thickness, nm

79.20

79.25

79.30

79.35

79.40

79.45

79.50

79.55

79.60

Tra

nsm

issi

on,

%

T = 79.57 % MgF2 = 90 nm SiNx = 61 nm a-Si:H = 10 nm

Transmission vs. MgF2 thickness

0.02 0.04 0.06 0.08 0.10 SiNx thickness, m

0.02

0.04

0.06

0.08

0.10

a-S

i:H t

hic

kness

,

m

45.0%

55.0%

65.0%

75.0%

Transmission vs. layer thickness MgF2 = 90 nm

0.02 0.04 0.06 0.08 0.10 SiNx thickness, m

0.02

0.04

0.06

0.08

0.10

a-S

i:H t

hic

kness

,

m

8.0%

8.0%

10.0%

10.0%

12.0%

14.0

%

14.0%

16.0%

16.0

%

18.0%

Relectivity

0.02 0.04 0.06 0.08 0.10 SiNx thickness, m

0.02

0.04

0.06

0.08

0.10

a-S

i:H t

hic

kness

,

m

16.0%

26.0%

36.0%

Absorption

Figure 5.7: Optimisation of as-deposited a-Si:H/SiNx/MgF2 A/R stack

75 80 85 90 95 100 105 110 MgF2 thickness, nm

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

Reflect

ivit

y,

%

R = 6.47 % MgF2 = 89 nm SiNx = 58 nm a-Si:H = 76 nm

Reflectivity vs. MgF2 thickness

0.02 0.04 0.06 0.08 0.10 SiNx thickness, m

0.02

0.04

0.06

0.08

0.10

a-S

i:H t

hic

kness

,

m

8.0%

8.0%

10.0%

10.0%12.0%

14.0

%

14.0%

16.0%

16.0

%

18.0%

Reflectivity vs. layer thickness MgF2 = 89 nm

Figure 5.8: As-deposited a-Si:H/SiNx/MgF2 A/R stack: optimalreflectivity

173

nealing will cause the silicon to recrystallise, narrowing the optical band gap and

altering optical characteristics. Such final optimisations are beyond the scope of

this work.

5.3 Contacts to Germanium

A number of contact metals have been considered for germanium since the

inception of germanium semiconductor research and device processing. For exam-

ple, the traditional method of contacting to n-Ge used antimony (≃ 1%) doped

gold, with the antimony acting as a dopant to heavily dope the region beneath

the contact, lowering contact resistance and improving Ohmicity.

One difficulty with making reliable, Ohmic or Schottky contacts to germa-

nium is Fermi-level pinning [116, 117, 80]. Interface states pin the Fermi level

at the Ge surface, typically resulting in Schottky behaviour on n-type material or

at the very least poor contact resistivity, and always ohmic behaviour on p-type

material [117]. This has a major impact on the quality of contacts that can reliably

be formed.

More recently, Ni has been investigated [116] as a contact to heavily doped

n-Ge. This process involved annealing the contacts to form an interlayer of NiGe

at temperatures of 340 C. It was thought that this interlayer allowed better

tunnelling through the Schottky barrier at the Ge/Ni interface [116]. Lieten et

al. [117] investigated a number of contact materials to Ge in conjunction with

an interlayer of Ge3N4 to mitigate Fermi-level pinning at the Ge surface. Metals

investigated include Al, Cr, Co, Au, and Pt. Thathachary et al. [80] reported on

the use of aqueous ammonium sulphide ([NH4]2S) treatment to perform Fermi-

level depinning for metallic contacts featuring Al, Zr, Ta, W, Ni and Pt.

174

5.3.1 Derivation of specific contact resistivity for some contact met-

als

In this work, we investigated a range of contact metals available in the UWA

clean room including Au, Pt, Al, In, Ni and Ti to heavily n-type doped material.

Indium (In) did not form a very reliable contact to Ge so no data could be

collated. The material itself formed a non-uniform film on the Ge surface which

scratched when the device was probed and displayed rectifying properties. Similar

difficulties arose when trying to use Pt as the contact material. Pt itself should

be reasonably compatible with the Ge surface, however it may have required an

interlayer for adhesion, for example Ti or Cr, in order to bond to the Ge surface

adequately.

5.3.1.1 Mask for contact resistivity measurements and diode fabri-

cation

The mask design for contact resistivity measurements and diode fabrication is

shown in Fig. 5.9. It features a number of test structures such as linear and circular

transfer length method structures, Kelvin resistance structures, diodes, JFETs,

hall bars, and fixed resistances. These test structures were designed to enable

characterisation on a single die of a wide variety of parameters for a particular

process. These include hall mobility, diode IV characteristics, resistance, specific

contact resistivity, sheet resistance, etc. Hence this particular mask set is well

suited to process characterisation.

5.3.1.2 Transfer length method

The linear transfer length method is a simple technique for determining

the specific contact resistivity and sheet resistance of a specially designed contact

175Circular TLM Structures

Linear Transmission Lines

Fixed value resistanceDiode structures

Hall bars

Kelvin resistance structures

JFETs

Figure 5.9: Mask set design

structure. This structure is a linear strip of contacts separated by unequal distances

d1, d2 ... dn but w wide and L long. The structure is detailed in Fig. 5.10.

L

w

Figure 5.10: Transmission line construction

176

Contact separation, d (cm)

Figure 5.11: Transmission line graph

The resistance between each successive rung on the ladder is measured and

plotted against the distance between contacts (see Fig. 5.11 for an example of the

plot). Three pieces of information can be deduced from this plot [85]. The sheet

resistance is given by the slope of the plot, with RSH = slope× w. The intercept at

d=0 gives the contact resistance; at d = 0, RT = 2RC [85]. The transfer length is

then given by the intercept at RT = 0, −d = 2LT, from which the specific contact

resistivity ρc can be computed using the following [85]:-

LT =√

ρc/RSH (5.13)

5.3.1.3 Circular Transfer Length Method

The above linear transmission line method is readily adapted to circular

structures where it can be more easily processed and corrected for (since there are

no current crowding effects in circular structures that occur at the edges of the

linear structures, since no mesa is required as the structures are self-isolating [116]).

The circular structures are created as follows. A disc of radius r is surrounded by

an annulus separated by gap spacing d , as shown in Fig. 5.12.

177

R

r d

Figure 5.12: Circular transmission line construction

The total resistance, RT , of the structure can then be approximated by the

following [85]:-

RT =Rsh

2π

[

ln

(

R

r

)

+ LT

(

1

R+

1

r

)]

(5.14)

where Rsh is the sheet resistance of the structure, and LT is the transfer

length.

The transfer length is given by extrapolation from a logarithmic fit to the

experimental data, i.e. [85]

LT =RT |x=0

Rsh

πr (5.15)

The specific contact resistivity, the desired quantity, is then given by

ρc = (LT )2 × Rsh (5.16)

5.3.1.4 Cr / Au contacts to Ge

Here we investigate the 5nm/50nm Cr/Au contact using the circular transfer

length method. A Ge substrate, with a heavily doped phosphorus layer realized

178

1.0 1.1 1.2 1.3 1.4 1.5Normalized Gap Spacing (r/R), a.u

200

400

600

800

1000

1200Resistance, mΩ

ρ ≃ 2.053e-5 Ω ·cm2 , Rsh ≃ 16.6 Ω/sq.

Circular Transmission Line Data for Cr/Au contacts to n+ Ge

FitMeas.

Figure 5.13: Cr-Au contact CTLM measurement data

using spin-on dopants, was patterned using the AZ-2035 photo resist (for lift-off

processing) prior to deposition of the Cr and Au using a simple thermal evapora-

tion technique in a standard bell-jar evaporator. The pattern was then “lifted-off”

by soaking in a beaker of warm acetone for 15-20 mins. This removed the pho-

tolithographic mask and hence patterned the active device regions.

The resistance of each CTLM structure was measured using a HP 4156A

precision parameter analyser. This showed ohmic contacts with fairly low resis-

tance. The resistance was then computed from the conductivity extracted from

the slope of the current-voltage (IV) plot in each case. This was the most accurate

method to extract resistance in the presence of noise. The resistance measure-

ments were made using a four probe (Kelvin) configuration to eliminate spurious

179

resistance due to probes and cabling. From the raw resistance measurements, the

transfer length, sheet resistance, and specific contact resistivity were extracted in

a spreadsheet using the formulae in section 5.3.1.3. The data are presented in Fig.

5.13.

The specific contact resistivity of ρ ≃ 2.053 × 10−5 Ω · cm2 indicates a

reasonably low resistivity contact. The sheet resistance of the layer of 16.6 Ω/

indicates high doping of the n-type layer - approximately 3×1019cm−3 for a nominal

depth of 0.5 µm.

5.3.1.5 Ni Contact

1.0 1.1 1.2 1.3 1.4 1.5Normalized Gap Spacing (r / R), a.u

0

100

200

300

400

500

Re

sist

an

ce,

mΩ

ρ ≃ 1.26e-7 Ω ·cm2 , Rsh ≃ 11 Ω/sq.

Circular Transmission Line Data for Ni contacts to n+ Ge

FitMeas.

Figure 5.14: Ni contact CTLM measurement data

Devices with nickel contacts were fabricated in an analogous manner. The

180

nickel was deposited directly onto the germanium, this time with e-beam evap-

oration as the deposition technology from pellets of 6N purity Ni using a BOC

Edwards E-beam evaporation system. There followed the usual lift-off patterning

to form the CTLM structures.

These proved to be of such low resistance that the parameter analyzer had

great difficulty performing the measurements accurately. Hence the measured data

does not agree with the model particularly well and the specific contact resistivity

extracted is only of limited accuracy.

Nevertheless the data were processed and the key parameters extracted in an

entirely analogous manner to the foregoing section, which gave a specific contact

resistivity ρ ≃ 1.26 × 10−7 Ω · cm2. This seems a little optimistic, although

difficulties in fitting the data may be responsible. Additionally, the sheet resistance

of 11 Ω/ would yield a carrier concentration of 6×1019cm−3 for a nominal depth

of 0.5 µm - this seems quite high but still in the range of carrier concentration of

these heavily doped devices that can be reasonably expected. The data are given

in Fig. 5.14.

5.3.1.6 Al contact

Aluminium was deposited using a standard bell-jar evaporation system from

4N Al wire. The devices were processed using lift-off patterning, as in previous

sections.

The rather high specific contact resistivity, ρ ≃ 3.98×10−3 Ω ·cm2, is rather

surprising. This indicates that Al did not form a very good contact to the n+ doped

Ge. This may be due to the fact that Al acts as a p-type dopant in Ge. This

could have formed a small barrier at the Al-Ge interface, increasing specific contact

resistivity. The IV characteristics showed slight rectifying characteristics. The sheet

resistance of 18 Ω/ gives an n+ doping concentration of around 2.5×1019cm−3

181

1.0 1.1 1.2 1.3 1.4 1.5Normalized Gap Spacing (r / R), a.u

3400

3450

3500

3550

3600

3650

3700

3750

3800

Resistance, mΩ

ρ ≃ 3.98e-3 Ω ·cm2 , Rsh ≃ 18 Ω/sq.

Circular Transmission Line Data for Al contacts to n+ Ge

FitMeas.

Figure 5.15: Al contact CTLM measurement data

for the nominal sheet thickness 0.5 µm. Again, this seems reasonable. The data

are depicted in Fig. 5.15.

5.3.1.7 Ti contact

Titanium contacts were deposited using the e-beam evaporation system de-

scribed above and patterned using lift-off patterning. The specific contact resis-

tivity of ρ ≃ 1.01×10−6 Ω ·cm2 is quite low, which indicates a favourable contact.

On the other hand, the sheet resistance of 14.2 Ω/ is again, within the range

expected. The data are shown in Fig. 5.16.

182

1.0 1.1 1.2 1.3 1.4 1.5Normalized Gap Spacing (r / R), a.u

0

100

200

300

400

500

600

700R

esi

sta

nce

, mΩ

ρ ≃ 1.01e-6 Ω ·cm2 , Rsh ≃ 14.2 Ω/sq.

Circular Transmission Line Data for Ti contacts to n+ Ge

FitMeas.

Figure 5.16: Ti contact CTLM measurement data

5.3.1.8 Summary - contact metals

The results obtained are summarised in the table Tab. 5.6 for the as-

deposited metal contacts on n+ doped germanium with some reference values

taken from the literature. The author was unable to find any published data for

the Cr/Au and Al contacts on germanium; perhaps contacting germanium has not

been previously considered using these materials. In the case of Ti and Ni, the

values reported in this thesis are of the same order of magnitude as reported in the

literature, which is highly encouraging. In this case of ni contacts, as deposited

the value is lower than the annealed reports in [116]; this seems a little surprising

and is possibly due to the uncertainty of this measurement for the reasons noted

above.

183

Metal

Specific

contact

resistivity,

Sheet

resistance,

Reference

valueRef.

ρ (Ω · cm2) Rsh (Ω/) (Ω · cm2)

Cr/Au 2.053× 10−5 16.6 - -

Ni 1.26× 10−7 11 (2.3± 1.8)× 10−7 [116]

Al 3.98× 10−3 18 - -

Ti 1.01× 10−6 14.2 7× 10−7 [118]

Table 5.6: Summary of contact metal parameters

5.4 Conclusions

In this section we investigated the use of ICPECVD films and a combination

of wet and dry treatments to help passivate the sensitive and highly reactive Ge

surface, the use of ICPECVD films and thermally evaporated MgF2 to provide

antireflection capacity, and the use of 4 readily available contact metals to make

low resistance contacts to Ge.

We noted that of the wet and dry treatments, the combination of halide

wet treatment and NH3 dry treatment produced the best results with the lowest

surface recombination velocity for the devices. This indicated that dangling bonds

and interface states were best accounted for with these techniques. This was in

contrast to wet treatment with HF/H2O and hydrogen dry treatment, which did

not perform as well. Interestingly, only the dry NH3 treated, a-Si:H/SiNx capped

sample outperformed the control sample which was only treated with halide wet

chemical treatment. This tends to indicate that other dry pre-treatments either

did not passivate the surface states or even caused surface damage which created

further interface states.

184

Of the AR coatings, the combination of a-Si:H, SiNx and MgF2 was opti-

mised to yield reflection losses below 10%, or to be precise, with minimum reflec-

tivity of 6.47%.

Out of all contact materials to n-type Ge, Ni gave the lowest as-deposited

specific contact resistivity, with Ti and Cr/Au both attractive other options for

low resistance Ge contacts. Al was not a good choice for n+ doped material

due to unintentional doping and consequential barrier formation giving rise to

highly resistive contacts. Calculations for the solar cells presented in the following

chapter assumed a specific contact resistivity of 2e-5 Ohm · cm2 and 100 µm thick

contacts to give a power loss of 0.22% due to I2R losses in the grid bars; this tends

to indicate that all contact materials considered in this chapter with the exception

of aluminium are suitable for use with germanium solar cells, however since lowest

cost would be obtained with a nickel contact, this would be the material of choice.

Chapter 6

Germanium pn-junction devices

6.1 Introduction

In this chapter we investigate germanium pn junction devices. Since the

passivation, antireflection and contacting processes were developed in previous

chapters, these can be combined into a fully featured diode process capable of

creating high quality diode devices with good reproducibility and high yield.

The process is based on mesa etching as opposed to a planar process since

germanium lacks a stable native oxide and, furthermore, there are very few selective

etches between the available field oxides and nitrides. The only material available

to fulfill the role of field oxide is PECVD or spin-on silicon dioxide, which is not

a particularly good diffusion mask. Field oxide or nitride deposition also adds

additional processing steps and extra complexity, which is not desirable for a solar

cell process. Correspondingly it was decided in favour of a mesa etch for the edge

isolation step when designing the process, which is the more usual case for solar

cell fabrication.

One interesting feature of the germanium process is the use of combined AR

coating and reactive ion etching (RIE) etch mask. It was found that the top layer of

the passivation/AR coating (MgF2) was quite resilient to RIE and could be used as

a mask, which removes the necessity of another mask material for this processing

step. Instead, the MgF2 is pre-patterned with lift-off patterning and then the

186

whole sample etched in an RIE to open contact pads prior to contact deposition.

By removing the additional processing step, the process is made simpler and more

manufacturable.

Difficulties processing germanium include its high chemical reactivity and its

brittle mechanical properties. Germanium reacts with most wet etches and espe-

cially with O2/CF4 RIE. Hence wafer cleaning times must be kept to a minimum

and RIE etch rates carefully determined to avoid overetching through device active

regions. There are very few selective etches for Ge which compounds the problem.

The solution is to exercise caution when etching germanium. One must also exer-

cise caution with thin (170 µm thick) solar grade wafers since they are extremely

brittle and prone to shatter or crack if mishandled.

Care must also be taken with exposed mesas since contamination could result

in an inclusion on the edge of the mesa which would cause the device to break

down prematurely. Cleanliness is required during processing and the devices are

given a preclean/etch before passivation to ensure that the mesa edges are free

from defects and inclusions.

6.2 Germanium wafer selection

In this chapter, germanium wafers from three suppliers are considered. There

are several reasons for the choice of wafers. Firstly, the wafers from Sylarus were

the lowest in cost. They were available in 4” format and could be cleaved to

suitably sized samples. However, these wafers were only 170 µm thick; this made

this material very difficult to work with as the samples were quite brittle and prone

to cracking. Hence thicker wafers were sought after. Wafers from University

Wafer were of reasonable cost and much thicker (500 µm) so these were used in

some experiments. However the doping of these wafers (1e15 cm−3) was too low

to be used for solar cell applications (we require based doping of at least 1e17

187

Wafer supplier Thickness Doping Cost Experiments

Sylarus 170 µm 5e17 cm−3 $70 GE095, GE0125

University Wafer 500 µm 1e15 cm−3 $120 GE099

JMP Technology Sarl 300 µm 1e17 cm−3 $250 GE0314, GE0313-2

Table 6.1: Wafers used in this chapter

cm−3 for reasonable open circuit voltage of single junction Ge solar cells) so a

third batch of wafers, from JMP Technology Sarl, was obtained. These were VGF

wafers that sadly despite their cost were of poor quality and contained a great

many crystallographic defects, impairing some experiments. The choice of wafers

for the experiments detailed in this chapter is presented in Tab. 6.1.

6.3 Diode fabrication

6.3.1 Fabrication process

Small samples are first prepared from bulk Ge wafers by scribing with a

diamond scribe and breaking along the scribe lines. Thin Ge wafers (170 µm

thick) cleave very easily but are brittle and prone to cracking. In addition, wafers

supplied with a miscut for III-V solar cell epitaxy were found to cleave along non-

preferred crystallographic axes due to the misorientation. This made things quite

difficult; nevertheless the cleaving process allowed the larger wafers to be divided

into smaller, more manageable, pieces could be handled in the laboratory.

Cleaved samples must be degreased in boiling trichloroethylene, acetone,

and methanol, and subject to ultrasonic bathing to remove any traces of grease

from the wafer sawing process or dust and contaminants from the cleaving process.

Germanium does not react with the solvents but is partially soluble in water so

deionised water rinsing time is kept to a minimum.

1881. Device active region

formed using proximity

doping or sandwich

stacked diffusion in RTP

or tube furnace

2. Mesa etched by

first patterning with

thick photoresist (AZ4562)

3. Mesa cleaned with

chemical clean and

passivation layer

deposited

(PECVD a-Si:H and SiNx)

4. deposited and

patterned using liftoff

patterning

6. Contacts deposited and

patterned using lift-off

patterning

5. used as etch mask

for Reactive Ion Etch

(RIE)

Gas flow

Dopant source

wafer

Susceptor

Target wafer

Spacer

Photoresist

Mesa

p-Ge

n+

a-Si:H / SiNx conformal coating

p-Ge

n+

p-Ge

n+

p-Ge

n+

p-Ge

n+

Contact metallisation

Figure 6.1: Diode fabrication process

Samples are then chemically cleaned by etching in H2O2:NH4OH:H2O 1:1:5

for 10s and in HCl:HBr:H2O 1:1:3 for 30s. The first step oxidizes the germanium

surface to remove any surface damaged areas from the wafer. The oxide is stripped

in the second step, and the chemically reactive germanium surface is passivated

with halide ions (Cl and Br). This prepares the sensitive germanium sample for

the diffusion step.

The next processing step is to diffuse dopant into the top layer of the ger-

189

manium sample to form the device active region. For example, this could be n+

dopant diffused into a p-type wafer for an n-on-p process. Diffusion is from spin-on

dopant sources and can be performed using either direct spin-on doping, proximity

doping, or “sandwich-stacked” diffusion.

If the sample is directly spun with dopant solution, an extra processing step

is required to strip the spin-on dopant film. This adds extra processing time and

complexity. Additionally, if the spin-on dopant film is directly adhered to the

wafer, any contaminants in the spin-on dopant matrix can diffuse into the sample,

impairing lifetime. For example the spin-on dopants may contain lifetime impairing

impurities such as iron and copper.

Another issue impairing the direct spin-on approach is small defects in the as-

spun layer. These are caused by aging of the phosphorus spin-on dopant film which

over time starts to form lumps of silica gel. These defects and inhomogeneities

will propagate to the doped layer forming an inhomogeneous active region. This

has the potential to cause premature breakdown or complete failure in devices and

is to be avoided.

Correspondingly the preferred method for emitter formation is by the prox-

imity doping method. In this method the extra processing step to remove spin-on

dopant film is no longer required, however there is some additional sample prepa-

ration necessary to prepare the SOD source wafer. Since the SOD source wafer

is pre-outgassed to remove contaminants prior to diffusion, the issue of contam-

ination is circumvented. Since the doping occurs by a vapour transport process,

this issue of striations and defects in the SOD film causing an inhomogeneous ac-

tive region is also circumvented. Hence the proximity doping process is the more

reliable process for device fabrication.

To isolate the edges of the active region and define the device, a mesa etch

step follows. The isolation mesa etches are performed using H2O2:NH4OH:H2O

190

1:1:20. This consumes Ge at a rate of 233 nm/min. To prevent potential degra-

dation of the photoresist in this solution (which is basic and acts like photoresist

developer) a thick photoresist such as AZ4562 is necessary. This allows for the

photoresist to maintain its integrity throughout the etch process even allowing

for gradual etching of the resist mask. The mesa etches are performed for long

enough to completely isolate the device, i.e. at least 5 minutes, which removes

over a micron of material.

After mesa etch, the samples are prepared for passivation deposition by a

chemical clean. This consists of a dip etch in NH4OH:H2O2:DI 1:1:5 for 30s to

remove any surface damage and clean away any potential impurities and inclusions

on the sensitive mesa boundary, and a soak for 1 minute in 1:1:5 HCl:HBr:DI to

remove any surface oxide and terminate the surface with Cl and Br ions.

The diode under construction is then loaded into the ICPECVD chamber and

exposed to ammonia gas at 250 C . This is the dry pre-treatment stage of the

passivation process outlined in chapter 5. The ammonia gas pre-treatment serves

to passivate in situ in the deposition chamber any dangling Ge bonds with either

nitrogen or hydrogen. The advantage of in-situ treatment is obviously the process

is performed under vacuum with no exposure of the Ge surface to the atmosphere

prior to passivation layer deposition.

Next, 10 nm of a-Si:H followed by 60 nm high quality SiNx are then confor-

mally deposited on the sample which passivates the mesa boundary and conformally

coats the entire sample surface. The sample should be terminated with H and N

so the surface should feature Ge-N-Si and Ge-H bonds. This should adequately

passivate the sensitive germanium surface and yield a robust device which can

handle subsequent processing without risk of damage.

MgF2 is now deposited by a simple thermal evaporation process. Care must

be taken when thermally evaporating MgF2 since it undergoes a phase transition at

191

1263 C to liquid phase. This temperature is similar to the temperature reached

by the boat during evaporation. As a liquid, the rate of MgF2 evaporation is

significantly higher than as a crystalline solid. Hence the deposition thickness

may become difficult to control if the deposition rate spontaneously increases by

a large amount. This necessitates careful control of the deposition rate and time

to minimize the chance of a spontaneous phase transition.

The MgF2 is patterned using liftoff patterning. This is the only available

option for MgF2 patterning since MgF2 is chemically resistant to most acids, bases,

and RIE. Since the melting point of MgF2 is quite low the deposition should not

occur at elevated temperatures (i.e. due to substrate heating radiatively from the

boat) so the use of a thin negative photoresist like AZ2035 is possible.

The next processing step is RIE. RIE is carried out in an Oxford Instruments

Plasmalab Plasma 100 reactive ion etch. The etch used is O2/CF4 which consumes

both SiNx and a-Si:H but leaves MgF2 intact. However, the RIE process also

consumes Ge, and more importantly, at an extremely high etch rate. Hence the

process has to be appropriately timed to prevent over-etching the germanium and

damaging the device active layer. Since the RIE process does not attack the

MgF2 significantly, the sample can simply be taken from the lift-off patterning

step directly to the RIE since there is no need for additional masking. This saves

a processing step.

Contacts are deposited again using thermal evaporation and lift-off pattern-

ing. The lowest resistance contact to Ge is with nickel metal. This can be thermally

evaporated or alternatively e-beam evaporated. Contacts can be annealed at 250

C to further improve contact resistance.

192

6.3.2 Ge n+/p diodes

Ge n+/p diodes were fabricated on p-type wafers from Sylarus [St. George,

UT, USA] and from University Wafer [Boston, MA, USA]. The substrate doping

of the Sylarus wafers was roughly 5e17 cm−3 and of University Wafer roughly

1e15 cm−3. Since the substrate doping was highest in the Sylarus wafers, devices

fabricated from these wafers showed premature breakdown in reverse bias due to

the Zener effect.

−1.5 −1.0 −0.5 0.0 0.5Bias, V

−150

−100

−50

0

50

100

150

200

Curr

ent,

mA

Dark IV characteristics for GE095

Meas., 300KFit

Figure 6.2: Dark n+/p IV characteristics for GE095

Fig. 6.2 depicts GE095, an n+/p diode fabricated on Sylarus wafer material,

and we can clearly see the onset of Zener tunneling in reverse bias. This is due to

the high base (≃ 5e17cm−3) and emitter (≃ 2e19cm−3) doping. The combination

of the two creates band overlap as the bands are pulled up abruptly in reverse bias.

This causes reverse breakdown due to the Zener effect. A semilogarithmic plot is

given in Fig. 6.3.

193

The model fit to the data in Fig. 6.2 is a simple two diode model. The

parameters of the fit are n1 = 1.0, n2 = 2.0 (ideality factors of the two diodes), A

= 0.01131 cm2 (area), J0,1 = 4e-6 A/cm2, J0,2 = 3e-4 A/cm2 (saturation current

densities), Rs = 0.21 Ω (series resistance), and Rp = 3e3 Ω (shunt resistance).

The ideality factor n1 being exactly 1.0 indicates a reasonably high quality

crystalline wafer with a long bulk lifetime was used. Consequently the diode itself is

of reasonable quality. If the device were fabricated on a poor quality wafer we would

expect to see a higher saturation current for the second diode of our two diode

model and the ideality factor of the first diode higher than 1.0. An ideality factor

higher than one indicates a low quality junction and is indicative of recombination

losses [119]; an ideality factor approaching 2.0 is found if generation-recombination

in the space charge region is significant and shallow traps close to the intrinsic

level are present [119]. In our case we can see that generation-recombination is

not significant and there is not a high density of mid-gap defects.

In contrast, the n+/p junction GE099 was fabricated on low doped University

Wafer material. This gave no premature Zener breakdown in reverse bias since

the substrate doping (roughly 1e15 cm−3) is simply too low. Reverse breakdown

occurred instead at -25V or greater (i.e. more negatively biased). This breakdown

may be due to avalanche effect or potentially Zener tunneling; it is unclear which

from the characteristics themselves but Zener tunneling seems more likely since

the breakdown characteristics are not particularly abrupt. This device is shown in

Fig. 6.4 and Fig. 6.5.

Considering the log-linear plot of the IV characteristics of GE099, we see a

reasonable fit to the measurements with a simple two diode model. The fitting

parameters were n1 = 1.0, n2 = 1.16, A = 0.01131 cm2, J0,1 = 1e-6 A/cm2, J0,2

= 4.8e-4 A/cm2, Rs = 1.1 Ω, and Rp = 1e5 Ω. The comparatively high shunt

resistance and low ideality factor (n1 = 1.0) highlight the quality of the junction.

194

−0.4 −0.2 0.0 0.2 0.4Bias, V

10-6

10-5

10-4

10-3

10-2

10-1

100

Curr

ent,

A

Dark IV characteristics for GE095

Meas., 300KFit

Figure 6.3: Dark n+/p IV characteristics for GE095 - logarithmicscale

−1.5 −1.0 −0.5 0.0 0.5Bias, V

−20

0

20

40

60

80

100

Curr

ent,

mA

Dark IV characteristics for GE099

Meas., 300KFit

−25 −20 −15 −10 −5 0 5−10

0

10

20

30

40

50

60

Figure 6.4: Dark IV characteristics for GE099

195

We may compare the fitted results for GE095 and GE099 to published data.

Wang et al. [120] report a Si:Ge solar cell fabricated with over 90% Ge content;

they report n1 = 1.0, n2 = 2.0, J0,1 = 1.3e-4 A/cm2, J0,2 = 1.0e-7 A/cm2, Rs =

0.5 Ω·cm2, and Rp = 1e5 Ω·cm2. The higher saturation current density (by almost

two orders of magnitude) is indicative of the poorer quality material resulting from

thin film RPCVD (reduced pressure chemical vapour deposition) which no doubt

will yield poorer lifetimes as compared to our bulk wafers. Our devices show better

series resistance characteristics but poorer shunt resistance. The shunt resistance

is no doubt due to pin hole defects in the spin-on dopant films causing small

conductive paths through the junction. Since our extracted parameters are for

the most part better than those reported in [120] we can expect higher conversion

efficiency for solar cells using this diode process than that reported in [120] (0.79%

efficiency at 30 suns). To further improve efficiency we would need to increase

bulk lifetime (reducing J0,1 and J0,2) and increase Rp.

6.3.3 Ge n+/p+ “tunnel” diode

A heavily doped n+/p+ diode was fabricated by diffusing into pre-prepared

Ga p+ doped material using standard direct spin-on phosphorus SOD film. The Ga

doping was performed using the sandwich-stacked diffusion technique. Gallium was

diffused and driven in until the profile had a surface concentration of approximately

1e19 cm−3 as estimated from the model derived in section Section 4.6.1. The

resultant diode had very early onset of Zener tunneling, which showed that both

emitter and base doping were quite high. However, not high enough, or abrupt

enough, to realise an Esaki diode. The diode was measured at 300K and 80K.

Diode IV characteristics are shown in Fig. 6.6.

The hump of excess current shown at 80K is most likely the onset of tunneling

processes within the diode at 80K as the band alignment becomes more favourable

196

−0.4 −0.2 0.0 0.2 0.4Bias, V

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Curr

ent,

A

Dark IV characteristics for GE099

Meas., 300KFit

Figure 6.5: Dark IV characteristics for GE099, logarithmic plot

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8Bias, V

10-5

10-4

10-3

10-2

10-1

100

101

102

Curr

ent,

A

Dark IV characteristics for GE0313-2

Meas., 300KMeas., 80K

Figure 6.6: Dark IV characteristics for GE0313-2, logarithmicplot

197

due to kT becoming smaller [65]. However, there is not enough tunneling current

to allow the device to be called a true tunnel diode.

Since the device is passivated with a PECVD silicon dioxide film, it is unlikely

that lateral defects or inclusions play a role in the breakdown characteristics of this

“tunnel diode”. Its characteristics are purely due to the Zener effect due to heavy

doping either side of the junction, since CV profiling confirmed a narrow depletion

width and heavy doping of the base region.

198

6.4 Investigation of a defective sample using scanning electron

microscopy

Mesa

boundary

Metallisation

Contact

Defect

Figure 6.7: Scanning electron micrograph of fully processed diodewafer

Scanning electron microscopy was utilised to examine a defective sample that

199

had been fabricated from vertical gradient freeze (VGF) wafers supplied by JMP

Technology Sarl [Porrentruy, Switzerland]. It is possible that the high temperature

ramp rate and associated mechanical stress resulted in recrystallisation of the VGF

wafers during the diffusion process, which caused a great many defects to nucleate

in the wafer. These defects are visible under the scanning electron microscope at

high magnification.

Fig. 6.7 shows a close up of the diode mesa. The defects are clearly visible,

both on the surface, where they are approximately 1 micron in size and square-

shaped, and in the etched trough outside the mesa boundary, where they are

diamond shaped and closer to 3-4 microns in size, and also on the mesa itself,

where the defect leaves two etch pits in its wake along the mesa edge.

Staining is also present down the edge of the mesa. This is due to high

phosphorus content staining the germanium during etching due to the dissimilar

etch rates between gallium and phosphorus doped germanium. This staining can

be used to identify the junction depth. For example, as viewed directly from above

the stained region is 2.5 microns in distance. This projected down the 54.76 degree

slope made by the < 111 > glide plane gives a junction depth of roughly 1.8 µm.

6.5 Germanium solar cells

Before the process described Section 6.3.1 was developed, a number of trial

processes were experimented with. The process in Section 6.3.1 was arrived at

through these experiments, which sought to optimise process parameters to obtain

higher yield and better quality devices. One such initial process was used to create

a solar cell of rather poor efficiency.

This solar cell (GE0125) was prepared by cleaving and cleaning a 1cm x 1cm

piece of Ge wafer supplied by Sylarus [St. George, UT]. The sample was then

masked using spin-on glass which was patterned using HF etching. This formed

200

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Bias, V

−14

−12

−10

−8

−6

−4

−2

0C

urr

ent

densi

ty, J, mA/cm

2Illuminated IV and PV characteristics for GE0125

J, Meas.−1.0

−0.8

−0.6

−0.4

−0.2

0.0

Pow

er,

P, m

W

Voc: 159 mV Jsc: 12 mA/cm2 FF: 54.50% η: 1.0%

P, Meas.

Figure 6.8: GE0125, illuminated IV and PV characteristics

the diffusion mask by which the active region was formed. The device active region

was then formed by direct spin-on doping the sample and diffusing at 650C for

10 minutes. The spin-on glass was not removed after the process but instead left

on the device as a combined passivation/antireflection coating. This layer was

then patterned for contact formation, and contacts were deposited and defined via

lift-off.

The IV and PV characteristics of this cell are depicted in Fig. 6.8. Without

the many optimisations leading up to the process outlined in Section 6.3.1, the

results are fairly poor. The device is 1% efficient and has a short circuit current

density of 12 mA/cm2. This poor short circuit current density is most likely due to

the phosphorus doped spin-on glass film used as a passivation layer reflecting and

201

absorbing most of the light. The poor open circuit voltage is a direct consequence

of the low short circuit current density.

0.00 0.05 0.10 0.15 0.20Bias, V

−60

−50

−40

−30

−20

−10

0

10

Curr

ent

densi

ty, J, mA/cm

2

Illuminated IV characteristics for GE0314

J, Meas., June 2013, 300KJ, Meas., Feb 2014, 300K

Figure 6.9: GE0314, illuminated IV characteristics

In contrast, a solar cell fabricated using the process outlined in Section 6.3.1

had much improved efficiency. This solar cell was 0.25 cm2 in area and fabricated

on a 1 cm × 1 cm die cut from a 3” VGF wafer supplied by JMP Technology Sarl

[Porrentruy, Switzerland]. The process used to fabricate the device was identical

to the previous section. This consisted of initial wafer clean, followed by proximity

diffusion at 800 C of phosphorus to realise the n+ emitter, followed by mesa etch,

clean, passivate, and A/R coat. Once again, the MgF2 acted as an RIE etch mask

to open up contact areas. The back contact was made from Al; this was annealed

at 700 C for 30s to realise a back surface field (BSF), that is, a heavily Al doped

202

0.00 0.05 0.10 0.15 0.20Bias, V

−9

−8

−7

−6

−5

−4

−3

−2

−1

0Pow

er

densi

ty, P, mW/cm

2

June 2013, Voc: 194 mV Jsc: 54 mA/cm2 FF: 51.71% η: 5.4%

Feb 2014, Voc: 190 mV Jsc: 52 mA/cm2 FF: 54.32% η: 5.3%

Illuminated PV characteristics for GE0314

P, Meas., June 2013, 300KP, Meas., Feb 2014, 300K

Figure 6.10: GE0314, illuminated PV characteristics

Ge region that serves to reflect minority carriers and only allow majority carrier

holes to recombine at the back surface. Front contacts were made from Cr and

thick gold, however, the gold could not be made thick enough (only 200-300 nm)

for low contact resistance of the finger pattern. Hence the contact resistance was

higher than expected (this manifested itself as a series resistance of ≃ 1 Ohm).

The measured parameters for this cell were JSC = 54 mA, Voc = 194 mV, η =

5.4%, FF = 51.7%. The IV characteristics for this cell are shown in Fig. 6.9 and

the PV characteristics are depicted in Fig. 6.10.

It was a pleasing result that the device had identical characteristics when

remeasured again almost 9 months later. When remeasured in February 2014, the

device performed almost identically with measured parameters JSC = 52 mA, Voc

203

= 190 mV, η = 5.3 %, FF = 54.3%. The device had not been kept in a controlled

environment nor subjected to any particular environmental controls (such as tem-

perature, humidity, etc). This indicates that the storage stability of the devices

is quite good and the passivation technology serves as an excellent environmental

passivant protecting the sensitive devices from moisture and atmospheric contam-

inants.

Given what was seen in Section 6.4, the high number of defects in the wafers

supplied by JMP Technology Sarl are the most likely cause of the low efficiency of

this device. These defects act as recombination centers and lower the bulk lifetime

in the device, impacting on open circuit voltage, short circuit current density, and

fill factor. The fill factor is also adversely affected by series and shunt resistance

in this device. The shunt resistance may be attributable to defects in the dopant

source film, which become more significant due to the larger area of the device,

or potentially to lateral defects and inclusions (such as the precipitates seen in

Section 6.4). It is unclear which mechanism is responsible for the shunt resistance.

Series resistance on the other hand is marred by thin contact films for the top

contact. These films need to be microns thick, however using a simple thermal

evaporation process, only hundreds of nanometers are practical. Technologies for

thicker contact deposition such as screen printing or electrodeposition need to be

investigated.

It is interesting to compare the two solar cell devices and attribute the effi-

ciency gain to various key process technologies developed in the previous chapters.

Firstly, GE0304 is 10 times the area of GE0125. This shows the improvement in

processing due to the use of proximity doping that allows the area to be scaled.

The improvement in short circuit current density (54 as opposed to 12 mA/cm2) is

due to the improvement of the passivation and antireflection technology developed

in chapter 5.

204

6.6 Conclusions

In this section a fully featured process for diode fabrication was presented.

The process consists of 6 basic processing steps and can move through a small

laboratory scale foundry, such as the A.G Nassibian Nanofabrication Facility at the

University of Western Australia, in under a week.

The process was well suited to use of proximity doping or sandwich stacked

diffusion for device fabrication. It was concluded that either of these techniques

were preferable to direct spin-on doping of the substrate to prevent diffusion of

impurities as well as process related defects such as influence of precipitates, small

bubbles, and radial striations which can come about as the dopant films approach

their shelf-life.

Process novelties included the use of MgF2 as a combined AR coating and

etch mask due to the high chemical inertness of MgF2 and its resistance to reac-

tive ion etching. This allowed a processing step to be removed saving time and

simplifying the process.

Out of the many diodes fabricated, two of particular quality were chosen to

be indicative of the potential for this process. These diodes had reasonable IV

characteristics including ideality factors close to 1 showing high quality crystalline

substrates with few defects and long bulk lifetimes. Diodes fabricated on epi-

ready solar-grade wafers showed early breakdown due to the Zener effect. Since

these substrates were intended for solar cell application, the substrate doping was

a nominal 5 × 1017. This is suitably high for early onset of Zener tunneling in

a moderate bandgap material such as germanium. Devices fabricated on lower

doped substrates showed much higher breakdown, at around 25V. This breakdown

could be due to the avalanche effect due to the high field at such voltages, or

Zener tunneling. It is difficult to discern which effect causes the breakdown from

205

the characteristics directly.

SEM was used to investigate precipitate-like defects on a fully processed

diode die. The size of the defects was quite considerable and the defects propa-

gated with the mesa etch process and became enlarged with etch depth. These

defects are thought to be the result of partial recrystallisation of the germanium

wafer during the doping process, which occurs at high temperatures with fast tem-

perature ramp rates. It can be concluded that the VGF wafers were simply of too

poor quality to realise high-performance devices and alternative wafer suppliers

should be considered.

Utilising both n and p type doping, a “tunnel” diode was fabricated. This

did not display the onset of tunnelling in forward bias but did show temperature

dependent changes. Unfortunately, the diode was not heavily doped enough on

the p-side, due to difficulty matching the highly degenerate Ga doping and the

phosphorus spin-on doping, which have different solubilities. For example, the Ga

sample had to be driven in to give the desired doping profile, i.e. surface concen-

tration of 1 × 1019 cm−3, so that the phosphorus doping could fully compensate

and exceed this concentration. However since the Ga doping level at the cross-over

point (where the n-type doping concentration no longer exceeds the p-type doping

concentration) may not necessarily be high enough due to the curvature of the

Gaussian driven-in profile and the unpredictable depth of the phosphorus diffusion,

the overall band bending at the junction may not be sufficient to realise a true

Esaki tunnel diode. This necessitates careful process control and optimisation in

order to balance the two simultaneous doping processes.

A solar cell was fabricated using the diode process. This was a large area

device, with an active area 0.25 of cm2. This demonstrates the applicability of

the process to larger area devices, the use of proximity doping, and the high

quality passivation technology working together to allow for high yield in spite

206

of the larger diode area. In general, the probability of catastrophic failure of a

device scales with area since there is a higher probability of a defect affecting

a critical part of the device, such as mesa or contact pad. The solar cells had

reasonable efficiency for germanium devices, however the efficiency was quite low

compared to other results in the literature. We may conclude that the wafer used

for the device was sub-optimal. Since the VGF wafers were examined by SEM

and found to be highly defective, particularly after partial recrystallisation during

the diffusion process. These defects most likely caused severe degradation of the

bulk lifetime in the device, as well as affected surface recombination velocities,

since these defects propagated to the surface, and cannot be passivated due to

the nature of the defect. This would explain the poor efficiency of the device.

This is somewhat saddening, since budget constraints and the high cost of bulk

germanium forced this particular choice of wafer. It is therefore possible that future

work could investigate the use of higher quality wafers and this process to yield

higher efficiency devices.

The solar cell GE0314 had near identical characteristics when remeasured,

almost 9 months after initial fabrication. This highlighted the storage stability

of the passivated diodes and the effective environmental passivation of the a-

Si:H/SiNx/MgF2 coating. This is due in part to the high chemical inertness of the

MgF2 which does not react with atmospheric contaminants.

It is regrettable that some of the most important experiments in this chapter

were performed with poor quality wafers. The poor efficiency and fill factor of

GE0314 illustrate the poor nature of the JMP Technology Wafers. Unfortunately

time and budget did not permit fabrication of solar cells with better quality wafers

to fully investigate the efficiency improvement offered by the passivation and anti-

reflection technology developed in this work. This is left to future work to elucidate

further. It is recommended that wafers from one supplier are used in comparison

207

experiments in future work to draw the most meaningful conclusions.

It can be concluded that the diode process is highly successful with high

yield and excellent storage lifetime of devices. It is a versatile process that is

equally suited to small signal Ge diode creation as well as photodiode and solar

cell fabrication. It is envisaged that a variety of germanium optoelectronic devices

could be fabricated with this process, and since both n and p type doping has

been demonstrated in this work using a low cost technology, it may be possible to

extend the process further, adding processing steps for formation of three terminal

devices such as bipolar junction transistors.

Chapter 7

II-VI/Germanium Materials Growth and Heterojunction Optoelectronic

Devices

7.1 Introduction

In the context of this work, several growth techniques were developed to

investigate the potential for combination of single crystal germanium with either

single crystal or polycrystalline II-VI materials in either monolithic or mechanically

stacked configuration for use in optoelectronic devices. However, it needs to be

noted that no II-VI materials are directly compatible with germanium, neither in

terms of lattice spacing, nor materials compatibility (since germanium is a compar-

atively reactive substance), nor for that matter in terms of favourable heterojunc-

tion electronic properties. This is of course, in direct contrast to III-V materials,

which are very similar in lattice constant, have very favourable electronic properties

for forming heterojunction devices, and for which materials growth technologies

are mature and well established.

To ameliorate these compatibility issues, we discuss herein the use of an-

other material, namely sapphire, as substrate for materials growth. This substrate

has the advantage of high optical transparency, comparatively low cost, excellent

mechanical properties and a high degree of chemical inertness.

In this section we present the results of materials growth of ZnTe and CdTe

on both sapphire and germanium using thermal evaporation and molecular beam

209

epitaxy (MBE). We also present a CdTe/Ge heterostructure and measure both

current-voltage (IV) and capacitance-voltage (CV) characteristics in order to de-

termine the electronic properties of the interface. The purpose is to pave the way

for monolithic CdTe/Ge tandem solar cells by addressing the materials challenges

associated with group II-VI/IV heteroepitaxy and confirming electronic properties

to assist in further modelling of CdTe/Ge solar cells.

7.2 Choice of materials and crystallography

7.2.1 Introduction

Group II-VI compound semiconductor materials have found application for

use in optoelectronic devices such as infrared detectors, radiation detectors, and

LEDs. However, despite numerous possible applications, the II-VI material sys-

tem is in general less utilized and correspondingly less well understood than other

materials systems such as group IV (Si, Ge, SiC) and group III-V and III-nitride

semiconductors (InGaAs, InGaP, AlGaAs, GaAs, GaN, InGaN etc. etc.) which

have widespread commercial application. Out of all possible group II and group

VI materials (Hg, Cd, Mg, Mn, Te, Se, S, O etc.) we choose to focus on stoichio-

metric ZnTe and CdTe for simplicity. In principal, an alloy of these two materials

(i.e. Zn1−xCdxTe) could be used to tune lattice spacing and electronic properties

(such as forbidden gap) for a particular application over a wide range of values, as

discussed herein.

We now consider some basic principles of crystal systems in order to under-

stand the crystal structure of the materials we are interested in (CdTe, ZnTe, Ge

and sapphire) and introduce fundamental properties needed to characterise these

materials using x-ray diffraction (XRD), presented later in this chapter.

210

7.2.1.1 Space groups and Crystal Systems

In order to create a semiconducting thin film or bulk crystal, atoms must

be arranged in a periodic formation which gives rise to the optical and electronic

properties desirable for a particular application. There are only a limited number

of distinct kinds of crystallographic arrangements possible, which are called crystal

systems. Furthermore, there are a limited number of unit cell configurations termed

Bravais lattices (14 in all). These Bravais lattices are given initials to describe the

class of lattice (e.g. F = face centred cubic, B = body centred cubic, P = primitive,

etc.). In this chapter we primarily concern ourselves with face centred cubic (F)

Bravais lattices, although the rather special case of sapphire is also considered.

Space groups describe a set of crystallographic operations which when ap-

plied to a crystal lattice produce the lattice itself. That is, space groups define a

series of operations such as translations, rotations, and mirroring, which no matter

how many times they are applied simply reproduce or extend the original lattice.

Hence the operations of an infinite lattice’s space group on itself would in fact pro-

duce no effect, i.e., the lattice would be invariant with respect to such operations

[121].

7.2.1.2 The Stereographic Projection

The stereographic projection, or cartographer’s projection, is of particular

usefulness to visualise space groups. It is also invaluable in working with miscut

or misaligned wafers to determine the correct pole, as evidenced below in our

discussion of Ge substrates.

The projection is computed by placing directions on the unit sphere (i.e. by

normalization) and tracing a line from the pole through the direction as located

on the surface of the sphere through the imaging plane as described in Fig. 7.1.

211

S

N

P1

O

P2

N

P1'

P2'

O

Equatorial

plane of

Projection

Small circle on sphere

Small circle in

plane of projection

Small circle in

plane of projection

Figure 7.1: Stereographic projection, after Hammond [122]

7.2.2 Materials science of CdTe/ZnTe/Ge/sapphire

7.2.2.1 Crystal system, phase, structure, and lattice spacings

In this chapter we consider 4 main materials, namely CdTe and ZnTe, as

potential low-cost wide bandgap semiconductor materials for use in optoelectronic

devices, and Ge and sapphire, as potential substrates for thin film deposition. The

crystal systems, phases, structures and lattice parameters are tabulated in Tab.

7.1.

We consider now each material in turn and elucidate the precise nature of

212

Material Phase Crystal system Space GroupLattice

Parameters

Sapphire α− Al2O3 Trigonal - Hexagonal R 3ca = 4.76 A,

c = 12.99 A

CdTe Cubic Cubic (Zincblende) F 43m a = 6.48 A

ZnTe Cubic Cubic (Zincblende) F 43m a = 6.1 A

Ge Cubic Cubic (Diamond) Fd3m a = 5.66 A

Table 7.1: Materials properties for materials in this work ([2],[123])

each as relevant to this work.

7.2.2.2 Sapphire - R-plane

As noted in Tab. 7.1, sapphire belongs to the trigonal-rhombohedral crystal

system, and possesses a rhombohedral unit cell. However, rather paradoxically,

symmetry within the lattice produce a number of other possible ”facets”, of which

there are two hexagonal prisms. The coordinate system within the sapphire lattice

is therefore hexagonal, with two axes of equal length a with angular separation

120 and a third axis of length c perpendicular to the other axes. The lattice

constants a and c are tabulated in Tab. 7.1.

Although when dealing with other kinds of lattices (i.e. cubic), a set of

3 Miller indices is all that is required, in order to promote interchangeability of

crystallographically equivalent directions when dealing with hexagonal lattices, a

set of 4 indices is often used, with the introduction of a fourth crystallographic

axis, as described in Fig. 7.2 [124]. The four indices 〈hki · l〉 are chosen such that,

by geometry,

213

Figure 7.2: Crystallographic axes of sapphire, in the C-plane

i = −(h + k) (7.1)

where 〈hk · l〉 are conventional Miller indices. This ensures that, for example,

the directions 〈101 · 2〉, 〈101 · 2〉, 〈110 · 2〉, 〈110 · 2〉, 〈011 · 2〉, 〈011 · 2〉 and

〈110 · 2〉 which are crystallographically equivalent have interchangeable indices

(and are simply equivalent to 〈01 · 2〉 in rhombohedral coordinates). This system

is sometimes referred to as ”hexagonal Miller indices”.

The principal crystallographic planes of sapphire (or indeed any hexagonal

crystal system) are the C, a, and R planes, as noted in Fig. 7.3.

Figure 7.4 is the stereographic projection for R-plane sapphire.

7.2.2.3 CdTe / ZnTe

Cadmium telluride (CdTe) and zinc telluride (ZnTe) belong to the II-VI

family of semiconducting materials, by virtue of the fact that Cd and Zn both

belong to group II of the periodic table, with Te belonging to group VI. Both

materials have cubic and hexagonal phases, however in this work we only consider

zincblende (cubic) structures.

We concern ourselves with three orientations in this work. These are 〈100〉,

〈111〉, and 〈211〉. The 〈211〉 orientation is important for growth of the material

214C

R-plane

A-plane

C-plane

Figure 7.3: Crystallographic planes of sapphire, after [123]

HgCdTe and is often used. It is thought that this orientation prevents twinning of

the HgCdTe ternary alloy, and is the orientation of choice for such a material [29].

The stereographic projection for CdTe/ZnTe 〈111〉 (space group F 43m) is

shown in Fig. 7.5. In a zincblende cubic crystal, 〈111〉 exhibits three-fold rota-

tional symmetry, and the perimeter of the projected unit circle (〈111〉 equatorial

zone) is populated with repeated interchangeable 〈110〉 and 〈422〉 directions. This

symmetry is useful for identifying diffractograms and even RHEED patterns.

7.2.2.4 Germanium

In this work we consider bulk germanium wafers from 4 manufacturers.

These wafers were all purchased on the basis of supply, i.e. available material,

rather than made to order, mostly due to the paucity and cost prohibitive nature

215

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

1.0

<0,0,0,-1> <1,0,-1,0>

<1,-1,0,0>

<0,1,-1,0>

<0,-1,1,0>

<1,0,-1,-1>

<1,-1,0,-1>

<0,1,-1,-1>

<0,-1,1,-1>

<-1,1,0,-1>

<-1,0,1,-1> <1,0,-1,-2>

<1,-1,0,-2>

<0,1,-1,-2>

<0,-1,1,-2>

<-1,1,0,-2>

<-1,0,1,-2>

<2,-1,-1,0>

<1,1,-2,0>

<1,-2,1,0>

<2,-1,-1,1>

<2,-1,-1,-1>

<1,1,-2,-1>

<1,-2,1,-1>

<-1,2,-1,-1>

<-1,-1,2,-1>

<-2,1,1,-1>

<2,-1,-1,-2>

<1,1,-2,-2>

<1,-2,1,-2>

<-1,2,-1,-2>

<-1,-1,2,-2>

<-2,1,1,-2>

<2,-2,0,1>

<2,-2,0,-1>

<0,2,-2,-1>

<0,-2,2,-1>

<-2,2,0,-1>

<2,1,-3,0>

<1,2,-3,0>

<2,1,-3,1>

<2,1,-3,-1>

<1,2,-3,-1>

<-1,-2,3,-1>

<-2,-1,3,-1>

<2,1,-3,-2>

<1,2,-3,-2>

<-1,-2,3,-2>

<-2,-1,3,-2>

<2,2,-4,1>

<2,2,-4,-1>

<-2,-2,4,-1>

Figure 7.4: Stereographic projection of 〈1012〉 Sapphire

of bulk germanium as an electronic material. The wafer manufacturers, nomi-

nal orientation, actual orientation, doping, and approximate cost per wafer are

tabulated in Tab. 7.2.

Referring to Tab. 7.2, in many cases the wafers purchased deviated consid-

erably from the nominal orientation. This is also true of wafer resistivity, however

since in general wafer resistivities are “binned” and not intended to be accurate,

this is to a certain extent to be expected.

216

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

1.0

<1,1,0>

<1,0,1>

<1,0,-1>

<1,-1,0>

<0,1,1>

<0,1,-1>

<0,-1,1>

<-1,1,0>

<-1,0,1>

<1,1,1>

<1,1,-1>

<1,-1,1>

<-1,1,1><2,0,0>

<0,2,0>

<0,0,2>

<3,1,0>

<3,0,1>

<3,0,-1>

<3,-1,0>

<1,3,0>

<1,0,3>

<0,3,1>

<0,3,-1>

<0,1,3>

<0,-1,3>

<-1,3,0>

<-1,0,3>

<3,1,1>

<3,1,-1>

<3,-1,1>

<3,-1,-1>

<1,3,1>

<1,3,-1>

<1,1,3>

<1,-1,3>

<-1,3,1>

<-1,3,-1>

<-1,1,3>

<-1,-1,3>

<3,3,1>

<3,3,-1>

<3,1,3>

<3,1,-3>

<3,-1,3>

<3,-3,1>

<1,3,3>

<1,3,-3>

<1,-3,3>

<-1,3,3>

<-3,3,1>

<-3,1,3>

<4,2,0>

<4,0,2>

<4,0,-2>

<4,-2,0>

<2,4,0>

<2,0,4>

<0,4,2>

<0,4,-2>

<0,2,4>

<0,-2,4>

<-2,4,0>

<-2,0,4>

<4,2,2>

<4,2,-2>

<4,-2,2>

<4,-2,-2>

<2,4,2>

<2,4,-2>

<2,2,4>

<2,2,-4>

<2,-2,4>

<2,-4,2>

<-2,4,2>

<-2,4,-2>

<-2,2,4>

<-2,-2,4>

<-4,2,2>

<5,1,0>

<5,0,1>

<5,0,-1>

<5,-1,0>

<1,5,0>

<1,0,5>

<0,5,1>

<0,5,-1>

<0,1,5>

<0,-1,5>

<-1,5,0>

<-1,0,5>

<5,1,1>

<5,1,-1>

<5,-1,1>

<5,-1,-1>

<1,5,1>

<1,5,-1>

<1,1,5>

<1,-1,5>

<-1,5,1>

<-1,5,-1>

<-1,1,5>

<-1,-1,5>

<5,3,0>

<5,0,3>

<5,0,-3>

<5,-3,0>

<3,5,0>

<3,0,5>

<0,5,3>

<0,5,-3>

<0,3,5>

<0,-3,5>

<-3,5,0>

<-3,0,5>

<4,4,2>

<4,4,-2>

<4,2,4>

<4,2,-4>

<4,-2,4>

<4,-4,2>

<2,4,4>

<2,4,-4>

<2,-4,4>

<-2,4,4>

<-4,4,2>

<-4,2,4>

<5,3,3>

<5,3,-3>

<5,-3,3>

<3,5,3>

<3,5,-3>

<3,3,5>

<3,3,-5>

<3,-3,5>

<3,-5,3>

<-3,5,3>

<-3,3,5>

<-5,3,3>

<5,5,1>

<5,5,-1>

<5,1,5>

<5,1,-5>

<5,-1,5>

<5,-5,1>

<1,5,5>

<1,5,-5>

<1,-5,5>

<-1,5,5>

<-5,5,1>

<-5,1,5>

<5,5,3>

<5,5,-3>

<5,3,5>

<5,3,-5>

<5,-3,5>

<5,-5,3>

<3,5,5>

<3,5,-5>

<3,-5,5>

<-3,5,5>

<-5,5,3>

<-5,3,5>

Figure 7.5: Stereographic projection of 〈111〉 CdTe

As noted in Section 7.2.1.2, the stereographic projection is the most simple

way to study the misorientation specified by the supplier. The actual wafer orien-

tation, where significantly different from the nominally specified orientation, was

confirmed by X-ray diffraction, as noted in the sections below.

Fig. 7.6 shows the stereographic projection of wafers supplied by Sylarus

Technologies [St. George, UT]. This tends to show that the wafer despite its

misorientation is still a nominal 〈100〉 Ge wafer, i.e. the misorientation has not

217

Suppli- Nominal Nominal Actual Doping Approx.

er orient- mis- orient- cost/unit

ation orientation ation

Sylarus 〈100〉 6 about the ≃ 〈100〉 5e17 cm−3 $70

〈110〉

axis toward

the nearest

〈111〉

Umicore 〈211〉 4 about the ≃ 〈533〉 1e14 cm−3 $500

〈110〉

axis toward

the nearest

〈111〉

JMP Tech. 〈100〉 Not specified ≃ 〈311〉 1e17 cm−3 $250

Sarl

University 〈100〉 - - 1e15 cm−3 $120

Wafer

Table 7.2: Ge wafers considered in this work

realigned another orientation to the pole position. However, the difficulty then

becomes imaging the misaligned substrate; particularly in RHEED, where the mis-

orientation gives rise to a peculiar surface reconstruction.

Figure 7.7, on the other hand, demonstrates the stereographic projection

of the misoriented 〈211〉 Ge wafers supplied by Umicore [Olen, Belgium]. The

transformation having been applied, the 〈533〉 now appears almost directly at the

218

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

1.0

<0,0,-1>

<0,-1,0>

<-1,0,0><1,0,-1>

<1,-1,0>

<0,1,-1>

<0,-1,-1>

<-1,1,0>

<-1,0,-1>

<-1,-1,0>

<1,1,-1>

<1,-1,-1>

<-1,1,-1>

<-1,-1,-1>

<2,0,-1><1,0,-2>

<1,-2,0>

<0,2,-1>

<0,1,-2>

<0,-1,-2>

<0,-2,-1>

<-1,0,-2>

<-1,-2,0>

<-2,1,0>

<-2,0,-1>

<-2,-1,0>

<2,1,-1>

<2,-1,-1>

<1,2,-1>

<1,1,-2>

<1,-1,-2>

<1,-2,-1>

<-1,2,-1>

<-1,1,-2>

<-1,-1,-2>

<-1,-2,-1>

<-2,1,-1>

<-2,-1,-1>

<2,2,-1>

<2,1,-2>

<2,-1,-2>

<2,-2,-1>

<1,2,-2>

<1,-2,-2>

<-1,2,-2>

<-1,-2,-2>

<-2,2,-1>

<-2,1,-2>

<-2,-1,-2>

<-2,-2,-1>

<3,0,-1><1,0,-3>

<1,-3,0>

<0,3,-1>

<0,1,-3>

<0,-1,-3>

<0,-3,-1>

<-1,0,-3>

<-1,-3,0>

<-3,1,0>

<-3,0,-1>

<-3,-1,0>

<3,1,-1>

<3,-1,-1>

<1,3,-1>

<1,1,-3>

<1,-1,-3>

<1,-3,-1>

<-1,3,-1>

<-1,1,-3>

<-1,-1,-3>

<-1,-3,-1>

<-3,1,-1>

<-3,-1,-1>

<3,0,-2><2,0,-3>

<2,-3,0>

<0,3,-2>

<0,2,-3>

<0,-2,-3>

<0,-3,-2>

<-2,0,-3>

<-2,-3,0>

<-3,2,0>

<-3,0,-2>

<-3,-2,0>

<3,2,-1>

<3,1,-2>

<3,-1,-2>

<3,-2,-1>

<2,3,-1>

<2,1,-3>

<2,-1,-3>

<2,-3,-1>

<1,3,-2>

<1,2,-3>

<1,-2,-3>

<1,-3,-2>

<-1,3,-2>

<-1,2,-3>

<-1,-2,-3>

<-1,-3,-2>

<-2,3,-1>

<-2,1,-3>

<-2,-1,-3>

<-2,-3,-1>

<-3,2,-1>

<-3,1,-2>

<-3,-1,-2>

<-3,-2,-1>

<3,2,-2>

<3,-2,-2>

<2,3,-2>

<2,2,-3>

<2,-2,-3>

<2,-3,-2>

<-2,3,-2>

<-2,2,-3>

<-2,-2,-3>

<-2,-3,-2>

<-3,2,-2>

<-3,-2,-2>

<3,3,-1>

<3,1,-3>

<3,-1,-3>

<3,-3,-1>

<1,3,-3>

<1,-3,-3>

<-1,3,-3>

<-1,-3,-3>

<-3,3,-1>

<-3,1,-3>

<-3,-1,-3>

<-3,-3,-1>

<3,3,-2>

<3,2,-3>

<3,-2,-3>

<3,-3,-2>

<2,3,-3>

<2,-3,-3>

<-2,3,-3>

<-2,-3,-3>

<-3,3,-2>

<-3,2,-3>

<-3,-2,-3>

<-3,-3,-2>

Figure 7.6: Stereographic projection of 〈100〉 Ge misoriented 6degrees about the 〈110〉 axis toward the nearest 〈111〉

pole of the projection, indicating that the true wafer orientation is indeed 〈533〉 as

opposed to the nominal 〈211〉. This appears to be a strange miscut indeed; there

is no indication that this orientation would be of benefit to any materials growth,

let alone growth of II-VI materials. None the less, it is most likely to be old stock

stored by the manufacturer that was simply available at the time.

219

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

1.0

<1,1,0>

<1,0,1>

<1,0,-1>

<1,-1,0>

<0,1,1>

<0,1,-1>

<0,-1,1>

<1,1,1>

<1,1,-1>

<1,-1,1>

<-1,1,1><2,0,0>

<0,2,0>

<0,0,2>

<3,1,0>

<3,0,1>

<3,0,-1>

<3,-1,0>

<1,3,0>

<1,0,3>

<0,3,1>

<0,3,-1>

<0,1,3>

<0,-1,3>

<-1,3,0>

<-1,0,3>

<3,1,1>

<3,1,-1>

<3,-1,1>

<3,-1,-1>

<1,3,1>

<1,3,-1>

<1,1,3>

<1,-1,3>

<-1,3,1>

<-1,3,-1>

<-1,1,3>

<-1,-1,3>

<3,3,1>

<3,3,-1>

<3,1,3>

<3,1,-3>

<3,-1,3>

<3,-1,-3>

<3,-3,1>

<3,-3,-1>

<1,3,3>

<1,3,-3>

<1,-3,3>

<-1,3,3>

<4,2,0>

<4,0,2>

<4,0,-2>

<4,-2,0>

<2,4,0>

<2,0,4>

<0,4,2>

<0,4,-2>

<0,2,4>

<0,-2,4>

<-2,4,0>

<-2,0,4>

<4,2,2>

<4,2,-2>

<4,-2,2>

<4,-2,-2>

<2,4,2>

<2,4,-2>

<2,2,4>

<2,2,-4>

<2,-2,4>

<2,-4,2>

<-2,4,2>

<-2,2,4>

<5,1,0>

<5,0,1>

<5,0,-1>

<5,-1,0>

<1,5,0>

<1,0,5>

<0,5,1>

<0,5,-1>

<0,1,5>

<0,-1,5>

<-1,5,0>

<-1,0,5>

<5,1,1>

<5,1,-1>

<5,-1,1>

<5,-1,-1>

<1,5,1>

<1,5,-1>

<1,1,5>

<1,-1,5>

<-1,5,1>

<-1,5,-1>

<-1,1,5>

<-1,-1,5>

<5,3,0>

<5,0,3>

<5,0,-3>

<5,-3,0>

<3,5,0>

<3,0,5>

<3,0,-5>

<3,-5,0>

<0,5,3>

<0,5,-3>

<0,3,5>

<0,-3,5>

<4,4,2>

<4,4,-2>

<4,2,4>

<4,2,-4>

<4,-2,4>

<4,-2,-4>

<4,-4,2>

<4,-4,-2>

<2,4,4>

<2,4,-4>

<2,-4,4>

<-2,4,4>

<5,3,3>

<5,3,-3>

<5,-3,3>

<5,-3,-3>

<3,5,3>

<3,5,-3>

<3,3,5>

<3,3,-5>

<3,-3,5>

<3,-5,3>

<-3,5,3>

<-3,3,5>

<5,5,1>

<5,5,-1>

<5,1,5>

<5,1,-5>

<5,-1,5>

<5,-1,-5>

<5,-5,1>

<5,-5,-1>

<1,5,5>

<1,5,-5>

<1,-5,5>

<-1,5,5>

<5,5,3>

<5,5,-3>

<5,3,5>

<5,3,-5>

<5,-3,5>

<5,-3,-5>

<5,-5,3>

<5,-5,-3>

<3,5,5>

<3,5,-5>

<3,-5,5>

<-3,5,5>

Figure 7.7: Stereographic projection of 〈211〉 Ge misoriented 4degrees about the 〈110〉 axis toward the nearest 〈111〉 (≃ 〈533〉Germanium)

7.3 Material growth techniques

7.3.1 Introduction

In this section we outline two physical vapour deposition (PVD) based tech-

niques which are of interest to II-VI materials growth. On one end of the spectrum,

molecular beam epitaxy (MBE) is an extremely versatile crystal growth technol-

220

ogy, but consequently extremely complicated and expensive. On the other end

of the spectrum, perhaps the most simplest PVD technology, thermal evapora-

tion, is comparatively economical. In this section we discuss the merits of the two

techniques.

7.3.2 Thermal Evaporation

Thermal evaporation is simply the evaporation of a solid under medium

vacuum (i.e. 10−5 - 10−7 Torr). Materials can be evaporated by heating charges

in a crucible or boat, or combination of the two. The most common configuration

for a thermal evaporation system is the conventional bell jar evaporator depicted

in Fig. 7.8.

Rotating shutter

Bell jar

Evap. source boats

To vacuum pump

Shutter control

Substrate

holder

Heater

block

Figure 7.8: Conventional bell-jar evaporation system

The main advantage of a bell-jar evaporation system is the low complexity

and hence low maintenance. The disadvantages however are very poor control

221

over film deposition rates and poor uniformity. This often leads to poor quality

films, and it would be very difficult indeed to grow a high quality crystal under

such circumstances. Nevertheless, thermal evaporation is considered in this work

for growth of low-quality ZnTe films on sapphire 〈1012〉 and Ge 〈311〉 substrates.

For these experiments, ZnTe was prepared from 6N purity Zinc and tellurium shot,

by mixing a stoichiometric amount of each material by weight, and melting under

vacuum in a crucible. Although elemental Zn and Te are both metallic in ap-

pearance (Zn a reasonably lustrous material, Te a dull, brittle grey) stoichiometric

ZnTe is red in appearance. Thus the colour change was used to confirm conversion

to roughly stoichiometric ZnTe within the crucible.

After ZnTe charge preparation, evaporation was performed at rates ranging

from 1− 3A/s with a substrate heater. The temperature for deposition was varied

between 200 − 400C . Below roughly 200C , an optically transparent film did

not form, indicating that the material was deposited non-stoichiometrically in a

metallic phase. Within the 200 − 400C range, however, homogeneous, optically

transparent ZnTe films were deposited with reasonable uniformity.

222

7.3.3 Molecular Beam Epitaxy

Growth chamber Analysis chamber

Rotating shutters

Effusion cells

To cryogenic / ion

pumps

To ellipsometer

Well cryoshroud

Cells cryoshroud

Substrate holder

(molybdenum block)

Sample manipulator

Viewing

window

To introduction

chamber

To ion

pump

Ports for in situ

analysis tools

(e.g. AES cannon)

Figure 7.9: RIBER 32 MBE system

Figure 7.9 depicts a RIBER 32 MBE system, indicative of the MBE systems

used in this work. Shown are the effusion cells, in which charges of materials of

interest are heated in crucibles made of pyrolytic boron nitride. Heating of the

source materials causes effusion of source material from the crucible, giving rise to

a flux. This molecular flux (or beam) is directed toward a substrate. Beam fluxes

can be interrupted using the rotating shutter feeds as shown on the diagram. The

rotating shutters block the beam path preventing arrival of the beam flux at the

substrate. This can be used to engineer abrupt interfaces or material regions. The

evaporation of source material occurs in ultra-high vacuum, as the chamber is

pumped from either ion or cryogenic pumps or a combination of both. The choice

of pump is determined by the materials used in the growth process. For example

ion pumps do not work well with mercury, a high vapour pressure element that is

223

better suited to cryogenic pumping.

In addition to ion and cryogenic pumps, most modern MBE systems are

equipped with cryogenic shrouds around the effusion cells and in the main chamber

well area. These shrouds adsorb atoms and further serve to reduce vacuum in the

growth chamber during growth. Cryogenic shrouds (“cryopanels”) are particularly

necessary when dealing with high vapour pressure evaporation materials such as

phosphorus or mercury. The cryopanels in the vicinity of the cells further serve

to force directionality of the molecular fluxes by adsorbing any stray molecules.

In this way the molecular beams remain highly directional even when high vapour

pressure materials are used.

The use of ultra-high vacuum (UHV) serves multiple purposes. Firstly, the

crystals can be grown with extremely high purity and minimal contamination from

the chamber. This is in contrast to other vacuum technologies such as PECVD in

which the reactions occur under much weaker vacuums. Secondly, UHV allows for

in-situ growth characterisation technologies such as reflected high-energy electron

diffraction (RHEED). In an UHV environment it is possible to operate an electron

gun and hence fire electrons at the substrate and observe the diffractogram. This

can yield a wealth of information about the crystal during growth such as orien-

tation, crystal quality, and even lattice parameters. This allows growth experts to

diagnose fault conditions during growth and observe the quality of the crystal at

critical stages during its development. Ultra-high vacuum also allows the use of

ion-gauge based flux monitoring prior to growth to recalibrate growth parameters

and ensure reproducibility.

Figure 7.9 shows a connecting port for an ellipsometer. In situ spectroscopic

ellipsometry is a powerful analysis technique based on the polarisation of light.

Light is first bounced off the substrate and reflected to a receiving viewport on the

other side of the growth chamber where it is received by an analyzer. The analyzer

224

detects the change in polarisation and amplitude of the light spectroscopically

(from the UV through to the short-wave infrared) which is used by sophisticated

software to identify the optical constants of the film as it is deposited. This can

yield film thickness and, in the case of alloyed materials such as Hg1−xCdxTe,

material composition.

Most production MBE systems also feature an analysis chamber. This cham-

ber sits between the introduction chamber and the main chamber. Commonly

employed analysis tools include Auger electron spectroscopy (AES) and X-ray pho-

toelectron spectroscopy (XPS). AES is a technique whereby the sample surface

is bombarded with electrons causing Auger electrons to be emitted. An elec-

tron energy analyzer then records the electron energy which can be matched to

a particular element; Auger electron energies are characteristic of the sample and

independent of the primary beam energy [85]. AES usually involves sputtering

with an argon source to yield depth profiles. XPS, on the other hand, is a high

energy version of the photoelectric effect [85]. Electrons are ejected from core

shells by incident X-ray photons. These electrons are then analysed as the energy

of the ejected electrons can determine the element from which the electron was

ejected. Both these analytical techniques can be used to characterise as-grown

materials and calibrate growth processes. An advantage of having the analysis

chamber present is that the samples do not need to be removed from the system

for characterisation, speeding up calibration procedures and training runs.

In this work, CdTe was grown on germanium and sapphire with CdTe and

tellurium fluxes of BEP 1.1e-6 Torr and 1.5e-6, respectively. Substrates were

outgassed at elevated temperatures (650 C ) prior to growth to remove native

oxide and improve RHEED pattern. Substrate temperatures ranged from 300 C to

320 C , with temperatures in the range 300 C - 305 C seen to give the best

results. Growth rate was roughly 13 nm / minute.

225

7.4 Characterisation of thin films

7.4.1 X-ray diffraction

X-ray diffraction, or XRD, is a useful means of material characterisation.

Its main advantage is ease of sample preparation (i.e. almost none) and non-

destructive nature, since x-rays of typical energies used in XRD do not harm

semiconducting crystals. XRD can be used to identify the phases present (i.e.

the orientations of crystallites in the sample) and structure of crystalline solids.

While it is possible to produce 2-dimensional X-ray diffractograms, in the

form of pole figures and reciprocal space maps, in this work we do not delve deeply

into these techniques, for the most part due to the fact that the structure of the

thin films is known a priori. Instead, we focus on the fairly standard 2θ−ω scan in

Bragg-Brentano configuration, which is useful for identifying crystallite orientation

and differentiating between thin film and substrate. To identify crystallinity, ω−2θ

or rocking curves, are employed. The difference between these two configurations

is depicted in Fig. 7.10.

In terms of the theory of XRD, it is the interaction between X-ray plane waves

and crystallographic planes that give rise to diffraction maxima in accordance with

the well-known Bragg condition [122]:-

2d sin θ = nλ (7.2)

where d is the interplanar spacing, or d -spacing, θ is the X-ray angle of inci-

dence with respect to the sample surface, n is an integer, and λ is the wavelength

of the X-ray radiation.

Figure 7.11 shows the 2θ − ω scan of GE0020, 〈211〉 CdTe grown on 〈533〉

Ge. The wafer, as mentioned previously, is miscut from the 〈211〉, bringing the

surface orientation close to the 〈533〉 orientation; this is clearly evidenced by the

226

X-ray source X-ray detector

Sample stage

Bragg-Brentano configuration

X-ray source X-ray detector

Sample stage

sample rocking

Rocking curve configuration

Figure 7.10: Configuration for X-ray diffraction

〈533〉 Ge reflection from Cu k−beta radiation, shown at approximately 107. The

〈533〉 for Cu k−alpha−1 radiation is rocked (i.e. ω−2θ scan) to determine quality

of the crystal; this is shown to have a FWHM of roughly 6 arc. sec. ω, indicative

of an extremely high quality crystal. The apparatus for this measurement involved

two 4-bounce crystals to ensure a very narrow X-ray beam and was performed at

227

0 20 40 60 80 100 120Degrees 2θ

Counts, a.u

CdTe <1 1 1>

CdTe <2 2 2>

CdTe <4 2 2>

CdTe <4 0 0>

Ge <4 0 0>C

dTe <4 2 2> - Cu k-β

Ge <5 3 3> - Cu k-β

GE0020 2θ/ω scan

35.200 35.300 35.400 35.500 35.600 35.700 35.800 35.900Degrees ω

FWHM = 389 arc seconds

CdTe <4 2 2>

63.23663.23863.24063.24263.24463.24663.248Degrees ω

FWHM = 6 arc seconds

Ge <5 3 3>

Figure 7.11: X-ray diffraction for GE0020, MBE grown 〈211〉CdTe on 〈533〉 Ge

the University of Illinois at Chicago. The rocking curve of the epilayer 〈422〉 is

also shown and gives a FWHM of 389 arc. sec. ω, indicating a fairly poor quality

epilayer. In this experiment, the growth temperature (250C ) was too low and far

from optimum; giving rise to an inhomogeneous epilayer. This temperature was

selected on the basis of published results, however subsequent experiments showed

that a higher growth temperature is beneficial to crystal quality. In addition, the

surface pretreatment, which consisted of exposure to Te flux, served to roughen

the surface. Finally, it is not altogether clear that 〈533〉 oriented germanium is

useful for growing 〈211〉 oriented CdTe under any circumstances; this was simply

the orientation provided by the manufacturer as an off the shelf product, saving

the expense of growing and sawing a custom ingot.

Figure 7.12 shows the 2θ − ω and ω − 2θ scans of SAPH001, CdTe 〈111〉

grown on 〈1012〉 sapphire using MBE at a substrate temperature of 305 C. The

228

0 20 40 60 80 100 120Degrees 2θ

Counts

, a.u

CdTe

<1 1

1>

Sapphire <

0 2

4>

- C

u k

-Beta

Sapphire <

0 1

2>

Sapphire <

0 2

4>

CdTe

<1 1

1>

- C

u k

-Beta

SAPH001 2θ/ω scan

11.800 11.820 11.840 11.860 11.880 11.900 11.920Degrees ω

FWHM = 72 arc seconds

DCRC of CdTe <1 1 1>

Figure 7.12: X-ray diffraction for SAPH001, MBE grown 〈111〉CdTe on 〈1012〉 Sapphire

2θ − ω scan clearly shows the presence of sapphire 〈012〉 (〈1012〉) and 〈024〉

(〈2024〉) reflections, which confirm substrate orientation as nominal 〈1012〉. The

difference in relative intensities may simply be explained by a slight miscut of

the substrate causing better fulfillment of the Bragg condition at 〈024〉. The

DCRC shown in the inset of Fig. 7.12 is performed using a Panalytical Empyrean

powder diffractometer with 4-bounce Ge 〈220〉 monochromator with 3-bounce

〈220〉 symmetrical analyzer for Cu. The sample is rocked about the CdTe 〈111〉.

The FWHM is roughly 72 arc. sec., which is an impressive result, given the film is

no thicker than 1 µm. For example, Badano et al. [125] report FHWM of roughly

360 arc seconds for comparable thickness films on Ge 〈211〉. This tends to indicate

a high quality crystal with comparatively few defects, and even better crystallinity

can be expected for thicker films [125].

On the other hand, Fig. 7.13, grown under almost identical conditions, but

229

0 20 40 60 80 100 120Degrees 2θ

Counts

, a.u

CdTe

<1 1

1>

CdTe

<1 1

1>

- C

u k

-Beta

SAPH002 2θ/ω scan

11.800 11.820 11.840 11.860 11.880 11.900 11.920Degrees ω

FWHM = 59 arc seconds

DCRC of CdTe <1 1 1>

Figure 7.13: X-ray diffraction for SAPH002, MBE grown 〈111〉CdTe on 〈1012〉 Sapphire

with a substrate temperature of 302 C, shows a much narrower full-width at half

maximum. This is partly due to a thicker film and partly due to more careful prepa-

rations during MBE growth and crystal nucleation. Finally, it could be argued that

the slight change in substrate temperature gave rise to more favourable outcomes.

Again, as an inset, the DCRC is plotted with a FWHM of roughly 59 arc. secs.,

a very high quality crystal indeed. Although difficult to see due to low X-ray yield

(in general, 2 bounce Ge 〈220〉 or 4-bounce Ge 〈440〉 monochromators/analyzers

are preferred for rocking curves, which reduce drastically x-ray counts), side-lobes

are clearly visible on the rocking curve.

Figure 7.14 depicts the X-ray diffraction of polycrystalline ZnTe on R-plane

sapphire. Since the films prepared by thermal evaporation are reasonably thin (≃

300 nm), it is not feasible to scan using the Bragg-Brentano configuration, or for

that matter, perform rocking curves. The reason being partially that ZnTe is itself

230

10 20 30 40 50 60 70 80 90Degrees 2θ

Co

un

ts,

a.u

Zn

Te <

1 1

1>

Zn

Te <

2 2

0>

Zn

Te <

3 1

1>

Zn

Te <

2 2

2>

Zn

Te <

4 2

0>

ZnTe / Sapphire 2θ/ω scan

12.30012.40012.50012.60012.70012.80012.90013.000Degrees ω

FWHM = 449 arc seconds

ZnTe <1 1 1>

Figure 7.14: X-ray diffraction for ZnTe thermally evaporated on〈1012〉 Sapphire

not sufficiently dense to stop a significant proportion of X-rays at that thickness,

so reflections from the substrate itself will dominate. Additionally, in this case, the

d -spacing of both R-plane sapphire and ZnTe is very similar, further necessitating

suppression of substrate reflections. However, it is still possible to analyze these

very thin films using glancing angle scans. In such a scan, the source is positioned

with a fixed angle ω which is very small, i.e. < 4 degrees. Then the angle the

detector makes with the substrate (i.e. θ) is varied as in a normal 2θ−ω scan, and

diffraction maxima are still observed at corresponding values of 2θ (the detector

angle) as if the scan were indeed in Bragg-Brentano configuration.

7.4.2 Reflection high-energy electron diffraction (RHEED)

Reflection high-energy electron diffraction (RHEED) is related to transmis-

sion electron diffraction and indeed low-energy electron diffraction (LEED). It

231

10 20 30 40 50 60 70Degrees 2θ

Counts, a.u

ZnTe <1 1 1>

ZnTe <2 2 0>

ZnTe <3 1 1>

ZnTe <2 2 2>

Ge <3 1 1>

ZnTe <4 2 0>

ZnTe / Ge 2θ/ω scan

12.30012.40012.50012.60012.70012.80012.90013.000Degrees ω

FWHM = 471 arc seconds

ZnTe <1 1 1>

Figure 7.15: X-ray diffraction for ZnTe thermally evaporated on〈100〉 Germanium

(a) CdTe on germanium, After 4 minutes

(b) CdTe on germanium, After 1 hour

Figure 7.16: RHEED patterns and XRD of CdTe grown on Ge

232

differs mainly in the use of very low angles of incidence, i.e. < 10, and very

high energies, i.e. > 10keV. This allows the condition of the surface, rather than

the bulk, to be imaged, which gives clues to the surface reconstruction, or said

otherwise, the arrangement of atoms on the surface, which may be different to

that of the bulk.

RHEED is particularly suited to MBE growth chambers since it can be used

to accurately image the surface of a growing crystal, i.e. in situ. This is due

to the ultra-high vacuum of the MBE growth chamber which facilitates in-situ

characterisation techniques.

Figure 7.16 depicts the RHEED patterns during growth for CdTe grown on

Ge at various stages of the growth. The RHEED pattern shows a series of dots

indicating reasonable quality nucleation and 2D growth. The dots then become

streaks as growth continues and the surface becomes smooth [25].

7.4.3 Nomarski contrast microscopy

233

50um

Figure 7.17: Normarski contrast micrograph of SAPH001

Nomarski contrast microscopy shows a comparatively featureless growth for

SAPH001, the CdTe grown on R-plane sapphire. The sample shows no obvious

defects but a corrugated surface with small hillocks of around 1 micron size. These

indicate a reasonable quality film.

7.4.4 Optical constants from transmission measurements

Since sapphire is 80 - 90% transparent in the short wave infrared (SWIR), UV

(ultra violet) and visible ranges, spectroscopic transmission measurements are pos-

sible to determine optical properties, film thickness, and to examine the absorption

tail around the band edge as a means to qualitatively assess the crystallinity.

The transmission data show a high quality CdTe 1 micron thick film on

sapphire with sharp cut-off indicating high quality material. The measurements

do not agree well with the model; the measured sample has a higher energy gap

(roughly 810 nm or 1.53 eV as opposed to 820 nm or 1.51 eV). This wider energy

234

gap may be due to the lattice constant not being exactly as assumed by the

model data due to the heteroepitaxy, i.e. a lattice strain effect. The compressively

strained layer may warp the energy bands producing the wider optical band gap.

It seems that additionally the optical properties differ between the model and the

measured thin film.

The 300 nm ZnTe film on sapphire however shows poor optical properties.

This is partially due to the the film thickness, but mostly due to the poor crys-

tallinity of the film. Again the optical properties differ between the model and the

measured data.

235

200 400 600 800 1000 1200 1400 1600 1800wavelength, nm

intensity, a.u

Optical transmission of CdTe ⟨111

⟩ on sapphire (SAPH002)

ModelMeas.

200 400 600 800 1000 1200 1400 1600 1800wavelength, nm

intensity, a.u

Optical transmission of polycrystalline ZnTe ⟨111

⟩ on sapphire

Figure 7.18: Optical transmission measurements of CdTe andZnTe thin films on sapphire

236

7.5 Electronic properties of the CdTe/Ge interface

7.5.1 Overview

Electronic properties of the CdTe/Ge heterointerface are not particularly

well known, most likely due to lack of interest in these materials, or disfavourable

material properties as discussed above. Nevertheless, since it has been shown

that it is indeed possible to grow material of sufficient quality for characterisation

purposes, we present here a process for the fabrication and characterisation of a

CdTe/Ge heterojunction.

7.5.2 Fabrication Process

CdTe material grown on Ge by molecular beam epitaxy (as described in

Section 7.3.3) was first doped using a copper spin-on dopant product (Cu950 from

Filmtronics, [Butler, PA, USA]) by spinning the silica based film onto a ≃ 1cm

× 1cm square sample which had been cleaned for 15s in a solution of Br/MeOH,

with care taken not to expose the sample to water (MeOH/isopropanol rinse and

dry). The Cu acceptor was diffused at 300 degrees for 1 hour to ensure uniform

diffusion within the CdTe epilayer (Cu diffusivity in Ge is comparatively low and the

solubility is below the substrate n-doping level, so diffusion of Cu acceptor ions into

the substrate may be neglected [126]). This has the effect of converting the CdTe

epilayer to p-type, whereas the n-type germanium substrate (donor concentration

≃ 1×1014cm−3) remains n-type. The reason for the low temperature is to prevent

Cd and Ge from alloying at the surface and, most importantly, to prevent surface

roughening of the CdTe due to Cd outdiffusion, which can occur at elevated

temperatures, especially around cracks and defects in the silica dopant film.

Subsequently, the dopant film is removed in dilute (≤ 10%) HF solution,

which does not affect the CdTe surface, at least if etching time is kept to a

237

minimum (e.g. 10-20s). The device region, e.g. Mesa, is then defined using

photolithograpy. Since the CdTe epilayer was in these cases quite thick (roughly

6 µm) this requires a fairly long etching time in 1% Br in HBr, which has an etch

rate of roughly 1 micron per minute at room temperature. Hence, to fully etch the

sample at least seven minutes etching time is required to ensure the all unwanted

material is fully removed, which requires fairly thick, robust photoresists. Of the

three positive photoresists, AZ4562, the thickest resist, was found to give best

results. Since etch rates vary as etchant is consumed, visual inspection using a

microscope was used to confirm successful termination of the etch process.

Contact Deposition

Photolithography Contact metal (Cr/Au) deposition Liftoff

Mesa Etch

Photolithography

Epilayer Doping

Spin-on Cu SOD

Mesa etch in 1% Br/HBr Photoresist strip

Diffuse on Hot Plate SOD film strip

Figure 7.19: CdTe/Ge heterojunction fabrication process

Finally, contacts could be made to the CdTe/Ge device using antimony

238

Au contacts

Cu doped

CdTe epilayer MESA

Ge wafer

Figure 7.20: Optical micrograph of CdTe / Ge heterojunctiondevice at 5x magnification

doped gold to form Ohmic contacts to both CdTe and Ge device layers. Although

thermally evaporated 1% Sb doped Au wire has traditionally been the preferred Ge

contact material, it was found that much comparable performance can be achieved

by first evaporating a thin layer of Cr to improve adhesion and prevent peeling (≃

5 nm), followed by a thin layer (5 - 20 nm) of Sb, and then evaporating a thicker

layer of Au (50 - 100 nm) which is subsequently annealed and hence alloyed.

Since alloying takes place at relatively high temperatures (400 - 500 C.) this

is not entirely practical with CdTe devices. Hence, these devices (GE020), were

processed without antimony, that is, using a pure Cr / Au contact, which was then

annealed for 30 minutes at 200 C on a hotplate to improve contact characteristics.

The contacts were patterned using lift-off patterning with a negative resist

(AZ2035) with release in warm acetone and ultrasonic bathing. The full device

fabrication process is tabulated in Fig. 7.19, with an optical micrograph of the

completed devices in Fig. 7.20.

239

1.00 1.05 1.10 1.15 1.20 1.25 1.30Normalized Gap Spacing (r / R), a.u

0

10

20

30

40

50

Resistance, kΩ

Circular Transmission Line Data for Cr/Au contacts to p CdTe

FitMeas.

Figure 7.21: CTLM measurements of p-CdTe on Ge

7.5.3 CdTe Epilayer sheet resistance

To determine the specific contact resistivity and sheet resistance of the CdTe

device layer, a set of CTLM structures were fabricated on the same die as described

in Section 7.5.2. The CTLM method was described in Section 5.3.1.3. Although

there are only several data points (the test structures did not all turn out as

intended and gave strange measurements) the data fits well with expectations.

The data were extracted from a parameter analyzer fit to IV curves to extract

conductivity. The contacts were reasonably ohmic but of very high resistivity.

Despite this, the usual analysis of CTLM measurements gave an epilayer sheet

240

resistance 1 MΩ/ and a resistivity of 600 Ω · cm. The CTLM data is depicted

in Fig. 7.21.

7.5.4 CdTe / Ge heterojunction current-voltage characteristics

−1.0 −0.5 0.0 0.5 1.0Corrected bias, Volts - I ·RS , V

10-5

10-4

10-3

10-2

10-1

100

101

Curr

ent,

mA

IV Characteristics of CdTe / Ge rectifying heterojunction

I=Is ·(exp(

V−I ·Rs

1.6 ·kT )−1)

I=Is ·(exp(

V−I ·Rs

1.6 ·kT )−1)+ V

Rp

Meas.

Figure 7.22: CdTe / Ge heterojunction electronic properties

The current-voltage characteristics of the heterojunction were measured us-

ing a HP-4156 parameter analyzer using a set of adjustable gold probes between

±1 Volt. The results are plotted in Fig. 7.22. The device clearly displays rectifying

characteristics, but has a high series resistance of 10 kΩ. This series resistance

is due to the high resistivity on both sides of the junction. As we have seen, the

resistivity of the CdTe was approximately 600 Ω · cm, the resistivity of the Ge was

approximately 40 Ω · cm. From these resistivities and knowledge of the dimen-

241

sions of the device, a figure of 10kΩ seems unattributable to the bulk alone, and

a significant proportion of the series resistance must be attributable to contact

resistance. This illustrates the difficulty making contacts to both low doped n-Ge

and p-CdTe.

The device also exhibits shunt resistance of the order of 1.25MΩ, indicating

a reasonable quality interface with minimal leakage through the junction. This

means that the quality of the crystal growth is reasonable, with few pin-hole like

defects and inhomogeneities at the interface. The ideality factor is roughly 1.65

showing a reasonable quality diode, again highlighting the comparative quality of

the crystal growth.

7.5.5 CdTe/Ge heterojunction capacitance-voltage profiling

The capacitance of the heterojunction is given by the following relation [30]:-

Chj = A

√

q · ǫnǫp2

NaNd

(Naǫp + Ndǫn)(φbi − VA)(7.3)

A more useful quantity to consider is the inverse square of the capacitance,

since this transforms the complicated relation in Eqn. 7.3 into a simple linear

relationship.

1

C 2hj

=1

A2

2

q · ǫnǫp(Naǫp + Ndǫn)(φbi − VA)

NaNd

(7.4)

By inspection of Eqn. 7.4, it is evident that the x-intercept is equal to the

built in potential of the device. Furthermore, it is possible to determine the doping

profile as a function of depth in an unknown region of the device from the slope

of the 1C2hj

versus VA plot.

242

d

dV

(

1

C 2hj

)

= − 1

A2

2

q · ǫnǫp(Naǫp + Ndǫn)

NaNd

(7.5)

If the doping in one region of the Pn heterojunction is significantly higher

than in the other region, it is possible to solve Eqn. 7.5 analytically. In other cases,

graphical methods can be employed to determine the carrier concentration from

the transcendental equation. Since in this case the doping concentrations on both

sides of the heterointerface are comparable, this technique must be employed.

The capacitance-voltage profiling of the CdTe/Ge heterojunction is shown

in Fig. 7.23. Doping densities some distance from the junction are extracted

using the above techniques. The doping density for the germanium substrate is

computed from wafer resistivity (40 Ω · cm ≃ 1e14 cm−3). The values of the

dielectric constants for CdTe and Ge are taken to be 10.4 and 16 ǫ0 respectively

[2]. The built in potential is extracted by extrapolation of the 1C2hj

versus VA plot,

and is around 1.42V. This built in potential has components attributable to the

doping densities either side of the interface and the band offsets themselves, as

illustrated by equation Eqn. 2.46. If one band offset were known, the other

could be computed from this equation. The doping concentration extracted is

the acceptor concentration in the p-type CdTe and is fairly constant. From the

literature [127] we would expect a Gaussian profile of Cu acceptor ions at the

surface of the CdTe (a distance 6 µm from the junction as indicated in Fig. 7.23).

We do not see such a feature. This could be attributable to the limited accuracy

of the CV profiling at such a distance from the junction. The doping density

extracted, however, may not be completely due to the introduction of copper ion

but may be due to a native defect that arises during the diffusion process, which

serves to anneal and activate the sample in air, a common step in CdTe solar cell

processing.

243

−10 −8 −6 −4 −2 0 2Bias, V

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.071

/ C

apaci

tance

square

d, 1/pF2

1/C2 −V Plot of CdTe/Ge heterojunction

0

5

10

15

20

25

30

35

40

Capaci

tance

, pF

Built in potential = 1.42±0.1 V

−6 −5 −4 −3 −2 −1 0Distance from junction, µm

1013

1014

1015

Dopin

g c

once

ntr

ati

on, cm

−3

Emitter doping profile of CdTe/Ge heterojunction

Figure 7.23: CdTe / Ge heterojunction capacitance profiling

7.6 Summary and conclusions

In this chapter we computed the true orientation of substrates used in this

work using the stereographic projection. We noted the misorientation of wafers

244

can lead to a dramatic change in wafer orientation, for instance nominally <2 1

1> wafers were in fact oriented <5 3 3>. This is important to take notice of,

particularly when performing XRD on these samples. This also effects the growth

orientation for crystals grown on such wafers.

ZnTe thin films were prepared using a simple thermal evaporation method,

based on preparing stoichiometric ZnTe from Zn and Te and mixing in a crucible

to form a stoichiometric source. The ZnTe was deposited at temperatures ranging

from 200 to 400C . Optically transparent and reasonably uniform films were

deposited within this temperature range.

Molecular Beam Epitaxy was employed to deposit high quality CdTe films

on both germanium and sapphire substrates. MBE is the growth technology of

choice for high quality semiconducting films with good control over interfaces

and material composition. This growth technology yielded high quality layers as

compared to thermal evaporation. The success of the sapphire grown samples

illustrates that MBE is a viable technology for experimentation with mechanically

stacked multijunction CdTe solar cells utilising sapphire substrates.

Optical contrast microscopy allowed the surface morphology of the as-deposited

epilayer to be examined. The surface morphology was smooth and did not feature

any noticeable defects. The surface was slightly corrugated with small hillocks of

1 micron in size. The overall appearance was seen to indicate a reasonable quality

film.

In situ RHEED was used during growth to characterise the film in situ and

showed high quality nucleation followed by a transition to smooth morphology with

a streaky appearance of the RHEED pattern. RHEED was also used during initial

preparations to determine the growth temperature by adjusting the temperature

prior to nucleation iteratively according to the RHEED pattern. RHEED is a

useful characterisation tool and one of the many advantages of ultra-high vacuum

245

technologies such as MBE.

Optical transmission measurements showed high quality films with good op-

tical transparency. The films did not match up with the models particularly well

with differing optical bandgaps and optical properties. Nonetheless, the films

themselves showed good properties highlighting the quality of the material.

X-ray diffraction was used to investigate the crystallinity of samples of CdTe

and ZnTe prepared on Ge and Sapphire. The samples prepared on sapphire had the

best crystallinity as shown by double crystal rocking curves (DCRC) full width at

half maximum (FWHM) as low as 59 arc seconds about the < 1 1 1 > diffraction

plane, indicating a very high quality crystal. The samples prepared on < 5 3 3 >

Ge was less crystalline owing to suboptimal growth conditions. The ZnTe samples

were polycrystalline due to the simpler, lower quality deposition technology.

A CdTe/Ge heterojunction was fabricated using copper spin-on dopants to

dope the CdTe region p-type with Cu acceptors. Current-voltage (IV) and capac-

itance voltage (CV) measurements were made of the heterojunction diode. The

diode showed rectifying characteristics, a series resistance of approximately 10kΩ,

a shunt resistance of 1.25MΩ, and an ideality factor of 1.65. This indicated a

reasonable quality diode had been formed. The CV profiling showed a built-in

potential of 1.42V and the extracted doping concentration indicated the doping

profile was fairly uniform throughout the epilayer. This may be attributable to

doping due to a native defect in the CdTe which was activated during the diffusion

process.

The CdTe/Ge heterojunction shows that the electronic properties of the

CdTe/Ge interface are suitable to create a heterojunction device such as a tandem

solar cell, since the device showed good characteristics over a certain voltage

range and obeyed theory well enough to admit a simple curve fit. The fitted

curve had ideality factor 1.65, showing the presence of defects in the device most

246

likely due to the lattice mismatch, the skewed orientation of the substrate and the

suboptimal growth conditions. Future investigations of CdTe/Ge heterojunctions

could include deep-level transient spectroscopy (DLTS) to determine the energy

levels of the defects at the CdTe/Ge interface. This technique can yield a wealth

of information about the electronic properties of CdTe and Ge which can feed back

into device models. Knowledge of the trap levels in the interface could help design

an optimised tunneling/recombination junction and improve solar cell efficiency.

In summary, it was shown that it is possible to grow high quality II-VI films

on sapphire, an optically transparent, mechanically hard, and insulating substrate.

This shows that mechanically stacked multijunction solar cells featuring high qual-

ity II-VI thin films grown on sapphire are physically realisable and should be consid-

ered. Future work should include in situ doping of the II-VI films during epitaxial

growth to realise complete tandem structures on sapphire. These structures can

then easily be combined with the germanium pn-junction diode and solar cell pro-

cess developed in this work and presented in Chapter 6. It was also shown that the

electronic properties of the CdTe/Ge heterojunction were favourable and a device

was fabricated to show this. This tends to indicate that a monolithic device should

be possible, however further work is needed to devise processes for heavily doping

cadmium telluride, since the doping density achieved with copper spin on doping

was not particularly high.

Chapter 8

Conclusions

8.1 Summary

Throughout this work we have been concerned with demonstrating the feasi-

bility of germanium as a bottom cell for monolithic or mechanically stacked tandem

solar cells with cadmium telluride. We have found that monolithic growth of the

materials is possible but troublesome and plagued with difficulties, particularly re-

lating to high temperature processing. Since Ge and CdTe form a eutectic alloy

at around 319 C , there is a restriction on the temperatures that can be used

on CdTe devices deposited on Ge. Also, the lattice mismatch makes it difficult to

grow high quality CdTe epilayers on top of Ge, even using a high-cost, low deposi-

tion rate technology like MBE. Nevertheless, a CdTe/Ge heterojunction device was

created that allowed electronic properties of the heterostructure to be explored.

This device was fabricated using MBE grown CdTe on Ge grown at the University

of Illinois at Chicago, and processed at the UWA clean room (A. G. Nassibian

Nanofabrication Facility). The device was a P/n heterojunction and exhibited

good rectifying characteristics, with ideality factor of 1.65 indicating reasonable

quality material and a shunt resistance of 1.25 MΩ suggesting very few shunt like

defects at the material interface.

Work was undertaken to develop low cost technologies for device active re-

gion formation. The techniques investigated were direct spin-on doping, proximity

248

doping, and sandwich stacked diffusion. Direct spin-on doping is seen to give the

best profile for phosphorus doped devices in terms of abruptness, however, direct

spin on doping was to be avoided due to issues with inhomogeneities and defects

causing poor yield. Proximity doping was shown to give favourable results but

lacked the surface concentration for P and Sb necessary for degenerate doping.

Proximity doping was shown to work quite well for Ga as well, and degenerate

doping could be achieved. Finally, sandwich stacked diffusion, the novelty intro-

duced in this thesis, was used to degenerately dope Ge with Ga acceptors. These

low-cost techniques are all amenable to mass manufacture and adaptable to the

mass manufacturing environment. These techniques could one day be used to

form the Ge junction part of a CdTe/Ge tandem solar cell.

To demonstrate the viability of the low cost doping technologies, Ge diodes

were fabricated using Ge single crystal wafers from Sylarus inc. [St. George,

Utah, USA] and University Wafer [Boston, MA, USA]. These were seen to be high

quality diodes with ideality factors close to 1. These devices did suffer from Zener

breakdown in reverse bias. However, on the lower doped substrate, the reverse

breakdown occurred at -25V which is acceptable reverse breakdown performance.

These devices illustrated the use of the low cost doping techniques.

Passivation technologies were investigated and it was found that a pre-

treatment using ammonia followed by a-Si:H/SiNx passivation layer deposition

gave the most favourable surface recombination velocities, with a surface recom-

bination velocity as low as 21 cm/s reported in this work. This was as compared

with unpassivated but chemically treated control samples as well as a variety of

other passivation techniques, including hydrogen pretreatement and SiO2 passiva-

tion layers. These passivation technologies are again low cost and amenable to

mass-manufacture.

Antireflection coatings to Ge were optimised and a final design of a-Si:H,

249

SiNx and MgF2 of thicknesses 10-80 nm, 50-60 nm, and 80-90 nm, respectively,

were found to minimise reflectivity. Specifically, for thicknesses 76nm (a-Si:H),

58nm (SiNx) and 89nm (MgF2) the reflectivity was reduced to 6.47%. However,

thicker a-Si:H implies more absorption and hence less transmission to the underly-

ing Ge layer. Hence it is favourable to trade off reflectivity for transmissivity and

thus incorporate a thinner amorphous silicon layer.

Contacts were fabricated and optimised to germanium. It was found that

Ni had the best specific contact resistivity but was hard to measure with the test

structures in question since the resistances involved were so low the measurements

had impaired accuracy. A specific contact resistivity of 1.26 ×10−7 Ω · cm2 is

reported. Other metals investigated included Al, Cr/Au and Ti. With the exception

of Al, all metals made good quality ohmic contacts to the germanium.

Ge solar cells were fabricated, with a peak efficiency of 5.4%. Although this

is not world changing in and of itself, it nevertheless demonstrates the technology.

These cells were fabricated from single crystal germanium by using the techniques

developed in this work: that is, diffusion by low cost proximity doping technique,

passivation and anti-reflection coating, and contacts. Hence it is fair to say that

the solar cell symbolises the culmination of all the thesis’ work in one single device.

The device was stable 8-9 months after initial testing which proved the potential

efficacy of the passivation layer as an environmental passivant. However, a lack of

environmental testing means the conclusions which may be drawn are limited.

Single crystal CdTe was grown of very good quality on R-plane sapphire.

The best X-ray DCRC FWHM was 59 arc seconds, indicating very high quality

material. Since this is a viable option, it may be possible to create mechanically

stacked tandem solar cells with top cell featuring CdTe grown on sapphire. This

could be a possible future research direction.

250

8.2 Research outcomes

The key research aims of this work were to reduce surface recombination in

Ge standalone solar cells with adequate surface passivation, to make low resistivity

contacts to germanium, and form device regions in a low cost manner without

any unwanted contamination which may impair device efficiency. These aims

were addressed sufficiently by the research in this work. Further research aims

included to investigate the feasibility of combining CdTe and Ge together to form

an electronic device and determine the electronic properties of the heterostructure.

This was also addressed and a viable heterojunction was demonstrated.

The following can be concluded about the combination of CdTe with Ge:-

• MBE growth yields good results, but heteroeptiaxy is difficult due to lattice

mismatch

• The optimum growth MBE temperature lies somewhere between 300 and

305C , and the optimum fluxes are beam equivalent pressure (BEP) 1.1e-

6 Torr for CdTe, and 1.5e-6 Torr for Te.

• Care must be taken to always keep the temperature of the sample below

the eutectic point (319 C ) to prevent alloying of CdTe and Ge. This is

particularly necessary if annealing in the presence of Cd vapour.

Provided that low temperature processing is used, monolithic CdTe and Ge

devices can be adequately fabricated with good electronic properties. If elevated

processing temperatures are required, a mechanically stacked tandem should be

pursued, and for this case, sapphire can be used as an optically transparent sub-

strate.

251

8.3 Future work

Due to time and budget constraints, sandwich-stacked diffusion was only

investigated for Ga spin-on dopants and only with diffusion followed by drive-in.

It would be highly recommended to research all dopants investigated in this work

using this technique to determine its applicability to a wide range of dopant sources.

Furthermore, models for dopant profiles should be compiled so that devices can

be designed. It would be interesting to see if higher surface concentrations can be

achieved with sandwich-stacked diffusion for the high vapour pressure phosphorus

dopant sources. If so, this will overcome some of the limitations of phosphorus

proximity doping which cannot achieve a high surface concentration due to the

high vapour pressure and volatility of phosphorus.

Process models need to be focussed on to improve the prospects of realising

a tunnel-diode. In particular, co-doping with phosphorus and gallium needs to

be investigated to determine the effects one dopant has on the diffusivity of the

other. Such a study should make extensive use of secondary ion mass spectrometry

(SIMS) depth profiles to adequately ascertain the secondary effects of codoping.

These models then need to be utilised to design a tunnel diode, which can then be

used to realise the structure introduced in Chapter 3, which consisted of an all-Ge

tunnel/recombination junction.

Work needs to be undertaken to improve the efficiency of the standalone

germanium solar cells. Their efficiency was hampered by two main factors: poor

quality VGF wafers from JMP technology Sarl, which contained a great many

defects and dislocations, and poor fill factor attributed to high series resistance due

to insufficiently thick front contacts. A technology needs to be developed for thick

plating of contacts to micron scale thickness. One possible technique would be

a simple nanoparticle paste which is spread on the contact material and annealed

252

to form a thick contact. Waste material should not adhere to the passivation

layer and should instead “ball up” or migrate to the contacts. An example nano-

particle paste would be simple Sn/Pb solder paste. Other nanoparticle pastes

might be suitable. Another possible technology could be electroplating. Thick

layers of contact material could be deposited by passing a current through the

device itself in a chemical bath. Finally, inkjet printing with nanoparticle inks is

a novel technology that has only recently entered the research arena. This could

be used combined with annealing with a nanoparticle solution of, for example, Ni,

to make thick low resistivity contacts to the Ge. These techniques should lower

series resistance and improve fill factor.

Open circuit voltage can be improved in the solar cells by using slightly

more heavily doped wafers. The wafers from JMP Technology Sarl were nominally

1e17 cm−3 doped. By raising the substrate doping to 5e17 or 1e18 cm−3 a few

tens of millivolts open circuit voltage can be gained. In addition, open circuit

voltage is limited by shunt resistance. The device presented in chapter 6 had a

suboptimal shunt resistance which can be attributed to defects during the doping

process which are more prominent in this device due to its larger area. This means

the proximity doping technique is still affected by defects and inhomogeneities

present in the spin-on dopant matrix. Or alternatively, the shunt resistance may

be attributable to lateral defects due to the dislocations on the wafer itself as

evidenced by SEM on a similarly fabricated diode. Again, due to large area, the

perimeter is considerable and will contain a great many of these defects, some

located within the junction. These may be the cause of the shunt resistance, in

which case simply changing wafer manufacturers should solve this problem.

It was seen that the optical transparency of the a-Si:H films was not partic-

ularly good. There are two reasons for this, one is that the optical band gap of

the film lies within the visible range, and hence absorbs energy (a-Si:H has a rea-

253

sonably high absorption coefficient, hence its use in thin film solar cells); the other

is the presence of excess hydrogen. It should be possible to anneal out some of

the excess hydrogen without adversely affecting the device. Of critical importance

is that the passivating interactions at the surface when prepared with ammonia

pretreatment are not interfered with during the annealing process. For example, it

is thought that nitrogen plays a role in terminating the lattice and passivating Ge

dangling bonds. If so, the passivation should be relatively unaffected by thermal

treatment. If however hydrogen itself is responsible in part for the passivation of

the surface, then it is unlikely that the structure will be temperature-stable and

hence may degrade during the process. This warrants further investigation.

Future work should encompass doping studies of CdTe. Since spin-on Cu

solution worked quite well to form a simple heterojunction diode, this avenue may

be followed. To increase doping density, higher temperatures need to be consid-

ered. This means the experiments need to take place in saturated Cd environment

with Cd overpressure supplied from a solid charge to prevent Cd out-diffusion and

damage to the epilayer structure. This necessitates annealing in sealed ampoules.

This technique would be limited to sapphire substrates since Ge will dissolve in

the Cd environment forming a eutectic alloy. A much better alternative would be

to develop in situ doping in the MBE system itself. As for choice of dopants, it

would be difficult indeed to find suitable materials since in situ doping of CdTe is

quite difficult. It may be possible to use iodine or indium as donors, and antimony

or arsenic as acceptors but would require a great amount of work to perfect this

technology.

Growth of CdTe on Ge needs to be improved to reduce the full width at half

maximum of double crystal X-ray rocking curves. Perhaps a better orientation of

wafer should be chosen since growth on <5 3 3> germanium was not particularly

successful. A better orientation is <1 0 0> but if miscut for epitaxy then X-ray

254

peaks can be difficult to find.

8.4 Final Conclusions

This thesis has demonstrated a low cost fabrication process for germanium

optoelectronic devices including solar cells, that features optimised passivation and

contacting technologies. Low cost techniques for device active region formation

were demonstrated. The process as a whole was amenable to mass manufacture.

The devices were stable over time indicating that the passivation was adequate.

These key elements were satisfied in the context of developing the technology

for monolithic and mechanically stacked tandem solar cells. It was shown that

the doping densities demanded in the modelling and design of the tandem solar

cells could be achieved using the low cost technologies developed in this work, at

least in the case of Ge. Heavy doping of CdTe could not be demonstrated, but

moderate doping was achieved using a Cu spin-on dopant source. A solar cell

of moderate efficiency was demonstrated using the diode process. This solar cell

could be improved by many of the techniques discussed in the previous section.

Having satisfied the main research aims, this thesis has been moderately

successful. It is perhaps lamentable that a full tandem solar cell could not be

fabricated, however the amount of work involved in fabricating such a device and

developing from scratch the technology to do so is well outside the realm of a single

PhD thesis. Several recommendations for future work in the previous section tackle

the work involved and break it down into smaller pieces.

Since the author chose to carry out this study at a research group with a

background in infrared detectors, it is not surprising that the main outcome has

been a high-quality process for photodiodes as opposed to large area solar modules.

This is simply a question of research know-how within a research group. Since this

was the first solar project that this group has ever undertaken, the outcome of

255

working solar cells and high quality diodes can be seen as a major success. Should

the research into solar cells at the Microelectronics Research Group continue, there

should be no reason why efficiencies cannot match those of world leading research

groups, as intellectual property is developed over time.

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Appendix A

Decoupled solution - Gummel’s method

1 # minimalist implementation of Gummel ’s method for Silicon pn junction

IV curve trace

2 impor t math

3 impor t scipy . linalg

4 impor t numpy as np

5 impor t matplotlib . pyplot as plt

6 from pylab impor t ∗7 impor t time

8 c l a s s units: # Units and scaling

9 cm = 1e8 ; s = 1e8 ; V = 1.0 / 0 . 02585 ;10 C = 1.0 /1 . 602176462 e−19; K = 1.0 /300 ; m = 1e2∗cm ;

11 um = 1e−4∗cm ; J = C∗V ; eV = 1.602176462e−19∗J ;12 A = C/s ; mA = 1e−3∗A ; kb = 1.3806503 e−23∗J/K ;

13 e = 1 ; eps0 = 8.854187818e−12∗C/V/m ;

14 eps = 11.7∗ units . eps0 ; affinity = 4.17∗ units . V ; Eg = 1.12∗ units . V; ni = 1.45 e10 ∗pow ( units . cm, −3.0) ;

15 mun = 100∗ units . cm∗units . cm/units . V/units . s ; mup = 40∗ units . cm∗units . cm/units . V/units . s

16 tau = 1e−8∗units . s ; kT = 1 ; Nd = 1e16 ∗ pow ( units . cm,−3) ; Na = 5e19

∗ pow ( units . cm,−3)17 npoints = 100 ; dx = (1e−4∗units . cm ) / npoints # Mesh definition: 1um

device width

18 de f B ( y ) : # Bernoulli function w/ range checking

19 i f y == 0: r e t u r n 1

20 i f y > 100: r e t u r n 021 r e t u r n y / ( math . exp ( y ) − 1)

22 # Discretization functions (x_i_1 = x_i -1, x_i_2 = x_i etc .). All

functions have same form for reuse of matrix construction etc .

23 f_phi_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x2−x1 ) ∗( x3−x1 ) ) # x_i -1 term in discretised linearised non -linear Poisson

24 f_phi_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −2∗eps/ ( ( x2−x1 ) ∗( x3−x1 ) ) − 2∗ eps/ ( ( x3−x2 ) ∗( x3−x1 ) ) − units . e ∗( nk + pk ) /kT # x_i term

25 f_phi_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x3−x2 ) ∗( x3−x1 ) ) # x_i +1 term

26 g_phi = lambda nk, pk, N_D, N_A, phi: −(units . e ∗( nk + pk ) ∗phi ) /kT +units . e∗( nk − pk − N_D + N_A ) # inhomogeneous terms

27 f_n_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ mun ∗kT/ ( ( x3−x1

) ∗( x2−x1 ) ) ) ∗B ( ( phi1 − phi2 ) /kT ) # Electron continuity equations

28 f_n_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗mun ∗kT/ ( x3−x1 )

) ∗( B ( ( phi2−phi3 ) /kT ) / ( x3−x2 ) + B ( ( phi2 − phi1 ) /kT ) / ( x2−x1 ) ) + pk/ ( tau ∗nk + tau ∗pk + 2∗tau ∗ni )

29 f_n_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ mun ∗kT/ ( ( x3−x1

) ∗( x3−x2 ) ) ) ∗B ( ( phi3 − phi2 ) /kT )

268

30 g_n = lambda nk, pk, N_D, N_A, phi: ni∗ni/ ( tau ∗nk + tau ∗pk + 2∗tau ∗ni)

31 f_p_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗mup ∗kT/ ( ( x3−x1

) ∗( x2−x1 ) ) )∗B ( ( phi2 − phi1 ) /kT ) # Hole continuity equations

32 f_p_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ mup ∗kT/ ( x3−x1 )

) ∗( B ( ( phi3−phi2 ) /kT ) / ( x3−x2 ) + B ( ( phi1 − phi2 ) /kT ) / ( x2−x1 ) ) + nk/ ( tau ∗nk + tau ∗pk + 2∗tau ∗ni )

33 f_p_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗mup ∗kT/ ( ( x3−x1

) ∗( x3−x2 ) ) )∗B ( ( phi2 − phi3 ) /kT )

34 g_p = lambda nk, pk, N_D, N_A, phi: ni∗ni/ ( tau ∗nk + tau ∗pk + 2∗tau ∗ni)

35 f_phi_0 = lambda n, p, N_D, N_A, V_app: V_app + kT∗math . asinh ( ( N_D −N_A ) / (2 ∗ ni ) ) − Eg / 2 − affinity # Ohmic boundary condition

36 f_n_0 = lambda n, p, N_D, N_A, V_app: n − ( math . sqrt ( ( N_D − N_A ) ∗(N_D − N_A ) /4 + ni∗ni ) + ( N_D − N_A ) /2 )

37 f_p_0 = lambda n, p, N_D, N_A, V_app: p − ( math . sqrt ( ( N_D − N_A ) ∗(N_D − N_A ) /4 + ni∗ni ) − ( N_D − N_A ) /2 )

38 # Matrix building

39 RHS = lambda i, v1, f1, v2, f2, v3, f3, g: v1∗f1 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1)

,mesh_phi [ i−1],mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )+\40 v2∗f2 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1],mesh_phi [ i ]

,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )+\41 v3∗f3 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1],mesh_phi [ i ]

,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )−\42 g ( mesh_n [ i ] , mesh_p [ i ] ,doping ( i ) [ 0 ] ,doping ( i ) [ 1 ] ,mesh_phi [

i ] )

43 f_phi_RHS = lambda i: RHS ( i,mesh_phi [ i−1] ,f_phi_1,mesh_phi [ i ]

,f_phi_2,mesh_phi [ i+1] ,f_phi_3,g_phi )44 f_n_RHS = lambda i: RHS ( i,mesh_n [ i−1], f_n_1,mesh_n [ i ] ,f_n_2,mesh_n [ i

+1],f_n_3,g_n )

45 f_p_RHS = lambda i: RHS ( i,mesh_p [ i−1], f_p_1,mesh_p [ i ] ,f_p_2,mesh_p [ i

+1],f_p_3,g_p )

46 M_int = lambda f1, f2, f3: [ ( [ 0 ] ∗ ( i−1) ) + [ f1 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1],mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] ) ,\

47 f2 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1]

,mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ]

,mesh_p [ i ] ) ,\48 f3 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1]

,mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ]

,mesh_p [ i ] ) ]+\49 [ 0 ] ∗ ( npoints − i − 2) f o r i i n range (1 ,npoints

−1) ]

50 M = lambda j: [ [ 1 ] + [ 0 ] ∗ ( npoints −1) ] + j + [ [ 0 ] ∗ ( npoints −1) + [ 1 ] ]51 de f doping ( x ) :

52 i f x > npoints/2: r e t u r n ( Nd,0 )

53 else: r e t u r n (0 ,Na )

54 mesh_phi = np . array ( [ f_phi_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i n

range (0 ,npoints ) ] ) #Initial guess for solution

55 mesh_n = np . array ([−f_n_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i n

range (0 ,npoints ) ] )

56 mesh_p = np . array ([−f_p_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i n

range (0 ,npoints ) ] )

57 ion ( ) ; p = plt . subplot (221) ; p . set_title ( "n,p vs. x" ) ; p . set_xlabel ( "x, $\

mu m$" ) ; p . set_ylabel ( "$cm ^-3 $" ) ;

58 N, = plt . semilogy ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, mesh_n ∗pow (units . cm,3 ) ,’-’ )

59 P, = plt . semilogy ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, mesh_p ∗pow (units . cm,3 ) ,’-’ )

60 p = plt . subplot (222) ; p . set_title ( "$E_C, E_V vs. x$" ) ; p . set_xlabel ( "x, $\

mu m$" ) ; p . set_ylabel ( "eV" )

61 Ec, = plt . plot ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, −affinity/units . V

− mesh_phi/units . V,’-’ )

269

62 Ev, = plt . plot ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, −affinity/units . V

− mesh_phi/units . V − Eg/units . V,’-’ )

63 p = plt . subplot (223) ; p . set_title ( "Log current density vs. diode bias" ) ; p

. set_xlabel ( "Volts " ) ; p . set_ylabel ( "$mA / cm ^2$" )

64 I, = plt . semilogy ( [ 0 . 0 ,1 . 0 ] , [ 1 e−10,20000 ] ,’-o’ )

65 p = plt . subplot (224) ; p . set_title ( "Current density vs. diode bias" ) ; p .

set_xlabel ( "Volts" ) ; p . set_ylabel ( "$mA / cm^2$" )

66 I2, = plt . plot ( [ 0 . 0 ,1 . 0 ] , [−0.1 ,20000 ] ,’-o’ )67 voltages = [ ] ; currents = [ ] ; start = time . clock ( )

68 p r i n t "Bias, V\t\tJ, mA/cm2 \t\t||RHS ||\ t\tIterations \t\tTime (s)"

69 f o r Vapp i n np . array ( [ 0 . 0 ,0 . 2∗ units . V,0 . 4∗ units . V,0 . 6∗ units . V,0 . 7∗ units .

V,0 . 8∗ units . V ] ) : # Solver loop

70 r = 1000; old_mesh_phi = mesh_phi ; old_mesh_n = mesh_n ; old_mesh_p =mesh_p ; k = 0

71 wh i l e r > 1e−12: # Solve equations using Gummel ’s method

72 prev_r = r ; k += 1

73 A = M ( M_int ( f_phi_1,f_phi_2,f_phi_3 ) ) # Assemble Phi matrix

74 b = [ f_phi_0 ( mesh_n [ 0 ] ,mesh_p [ 0 ] ,doping (0) [ 0 ] ,doping (0) [ 1 ] ,Vapp ) ]+ \

75 [ g_phi ( mesh_n [ i ] , mesh_p [ i ] ,doping ( i ) [ 0 ] ,doping ( i ) [ 1 ] ,mesh_phi

[ i ] ) f o r i i n range (1 ,npoints−1) ]\76 + [ f_phi_0 ( mesh_n [ npoints −1],mesh_p [ npoints −1],doping ( npoints

−1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ] # RHS for Phi

77 mesh_phi = scipy . linalg . solve ( A,b ) # Update new Phi vector

78 A = M ( M_int ( f_n_1,f_n_2,f_n_3 ) ) # Assemble n matrix

79 b = [−f_n_0 (0 ,0,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ] + \80 [ g_n ( mesh_n [ i ] ,mesh_p [ i ] ,0,0,0 ) f o r i i n range (1 ,npoints−1) ] +

[−f_n_0 (0 ,0,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ]81 mesh_n = scipy . linalg . solve ( A,b ) # Solve for new n vector

82 A = M ( M_int ( f_p_1,f_p_2,f_p_3 ) )

83 b = [−f_p_0 (0 ,0,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ] +\84 [ g_p ( mesh_n [ i ] ,mesh_p [ i ] ,0,0,0 ) f o r i i n range (1 ,npoints−1) ]

+\85 [−f_p_0 (0 ,0,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ]

86 mesh_p = scipy . linalg . solve ( A,b ) # Solve p-block

87 r = scipy . linalg . norm ( [ f_phi_RHS ( i ) f o r i i n range (1 ,npoints−1) ] +

\88 [ f_n_RHS ( i ) f o r i i n range (1 ,npoints−1) ] + \89 [ f_p_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ) / (3∗ npoints

) #compute residual

90 i f r>prev_r and k>3: break

91 old_mesh_phi = mesh_phi ; old_mesh_n = mesh_n ; old_mesh_p = mesh_p

# if we didn’t break due to res increase update old mesh

92 N . set_ydata ( old_mesh_n ∗ pow ( units . cm,3 ) ) ; P . set_ydata ( old_mesh_p ∗pow ( units . cm,3 ) ) ; # Plot carrier concentrations

93 Ec . set_ydata (−affinity/units . V − old_mesh_phi/units . V ) ; Ev . set_ydata (−affinity/units . V − old_mesh_phi/units . V − Eg/units . V ) # Plot

energy bands

94 ph = ( old_mesh_phi [1]− old_mesh_phi [ 0 ] ) /kT # Compute current

density at contact

95 j = (( kT ∗ mun ) ∗ ( ( old_mesh_n [ 1 ] ∗ B ( ph ) − old_mesh_n [ 0 ] ∗ B(−ph ) ) /dx )

+\96 ( kT ∗ mup ) ∗ ( ( old_mesh_p [ 0 ] ∗ B ( ph ) − old_mesh_p [ 1 ] ∗ B(−ph ) ) /dx ) ) ∗

pow ( units . cm,2 ) ∗1 e3/units . A # Scale to mA/cm2

97 voltages . append ( Vapp/units . V + 1e−6∗j ) ; currents . append ( j ) # +

Vapp/units .V/50e3 ) # series spec. resistivity: 1mOhm.cm2, 10 kOhm.

cm2 shunt

98 I . set_xdata ( voltages ) ; I . set_ydata ( currents ) ; I2 . set_xdata ( voltages ) ;I2 . set_ydata ( currents ) ; draw ( ) # update plots

99 stop = time . clock ( ) ; p r i n t "%g\t\t%g\t\t%g\t\t%d\t\t%g" % ( Vapp/units .

V + 1e−6∗j, j, r, k, stop − start ) ; start = time . clock ( ) ;

100 i f j > 20000 : break # break on overcurrent

101 ioff ( ) ; plt . show ( ) # Freeze on plot, print solve time

Appendix B

Fully coupled solution

1 # minimalist implementation of fully -coupled Newton ’s method for Silicon

pn junction IV curve trace

2 impor t math

3 impor t scipy . linalg

4 impor t numpy as np

5 impor t matplotlib . pyplot as plt

6 from pylab impor t ∗7 impor t time

8 c l a s s units: # Units and scaling

9 cm = 1e8 ; s = 1e8 ; V = 1.0 / 0 . 02585 ;10 C = 1.0 /1 . 602176462e−19; K = 1.0 /300 ; m = 1e2∗cm ;

11 um = 1e−4∗cm ; J = C∗V ; eV = 1.602176462e−19∗J ;12 A = C/s ; mA = 1e−3∗A ; kb = 1.3806503 e−23∗J/K ;

13 e = 1 ; eps0 = 8.854187818e−12∗C/V/m ;

14 eps = 11.7∗ units . eps0 ; affinity = 4.17∗ units . V ; Eg = 1.12∗ units . V; ni = 1.45 e10 ∗pow ( units . cm, −3.0) ;

15 mun = 100∗ units . cm∗units . cm/units . V/units . s ; mup = 40∗ units . cm∗units . cm/units . V/units . s

16 tau = 1e−8∗units . s ; kT = 1 ; Nd = 1e16 ∗ pow ( units . cm,−3) ; Na = 5e19

∗ pow ( units . cm,−3)17 npoints = 100 ; dx = (1e−4∗units . cm ) / npoints # Mesh definition: 1um

device width

18

19 p r i n t units . cm∗units . cm/units . V/units . s

20

21 de f B ( y ) : # Bernoulli function w/ range checking

22 i f y == 0: r e t u r n 1

23 i f y > 100 : r e t u r n 0

24 r e t u r n y / ( np . exp ( y ) − 1)

25

26 de f G ( y ) : # B ’(y)/B(y)

27 i f y == 0: r e t u r n 0

28 i f y > 100 : r e t u r n 0

29 r e t u r n ( np . exp ( y )∗(1−y )−1)/ ( y∗( np . exp ( y )−1) )

30

31 de f C ( y ) :

32 ’’’Derivative of the Bernoulli function ’’’

33 i f x == 0:

34 r e t u r n −0.5

35 i f x > 100 :36 r e t u r n 0

37 r e t u r n −((np . exp ( x ) ∗(x−1) + 1) / ( ( np . exp ( x )−1)∗∗2) )

38

271

39 # Discretization functions (x_i_1 = x_i -1, x_i_2 = x_i etc .). All

functions have same form for reuse of matrix construction etc .

40

41 f_phi_1_0 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x2−x1 )

∗( x3−x1 ) ) # x_i -1 term in discretised linearised non -linear

Poisson

42 f_phi_2_0 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −2∗eps/ ( ( x2−x1 ) ∗(x3−x1 ) ) − 2∗ eps/ ( ( x3−x2 ) ∗( x3−x1 ) ) − units . e∗( nk + pk ) /kT # x_i term

43 f_phi_3_0 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x3−x2 )

∗( x3−x1 ) ) # x_i +1 term

44 g_phi_0 = lambda nk, pk, N_D, N_A, phi: −(units . e∗( nk + pk ) ∗phi )/kT + units . e∗( nk − pk − N_D + N_A ) # inhomogeneous terms

45

46 f_phi_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x2−x1 ) ∗( x3−x1 ) ) # x_i -1 term in discretised linearised non -linear Poisson

47 f_phi_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −2∗eps/ ( ( x2−x1 ) ∗( x3−x1 ) ) − 2∗ eps/ ( ( x3−x2 ) ∗( x3−x1 ) ) # x_i term

48 f_phi_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: 2∗ eps/ ( ( x3−x2 ) ∗( x3−x1 ) ) # x_i +1 term

49 g_phi = lambda nk, pk, N_D, N_A, phi: ( nk − pk − N_D + N_A ) #

inhomogeneous terms

50 f_n_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ kT/ ( ( x3−x1 ) ∗(x2−x1 ) ) ) ∗B ( ( phi1 − phi2 ) /kT ) # Electron continuity equations

51 f_n_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗kT/ ( x3−x1 ) ) ∗( B( ( phi2−phi3 ) /kT ) / ( x3−x2 ) + B ( ( phi2 − phi1 ) /kT ) / ( x2−x1 ) ) + (1 /mun ) ∗pk/ (tau ∗nk + tau ∗pk + 2∗tau ∗ni )

52 f_n_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ kT/ ( ( x3−x1 ) ∗(x3−x2 ) ) ) ∗B ( ( phi3 − phi2 ) /kT )

53 g_n = lambda nk, pk, N_D, N_A, phi: (1 /mun ) ∗ ni∗ni/ ( tau ∗nk + tau ∗pk+ 2∗tau ∗ni )

54 f_p_1 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗kT/ ( ( x3−x1 ) ∗(x2−x1 ) ) ) ∗B ( ( phi2 − phi1 ) /kT ) # Hole continuity equations

55 f_p_2 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: (2∗ kT/ ( x3−x1 ) ) ∗( B( ( phi3−phi2 ) /kT ) / ( x3−x2 ) + B ( ( phi1 − phi2 ) /kT ) / ( x2−x1 ) ) + (1 /mup ) ∗nk/ (tau ∗nk + tau ∗pk + 2∗tau ∗ni )

56 f_p_3 = lambda x1, x2, x3, phi1, phi2, phi3, nk, pk: −(2∗kT/ ( ( x3−x1 ) ∗(x3−x2 ) ) ) ∗B ( ( phi2 − phi3 ) /kT )

57 g_p = lambda nk, pk, N_D, N_A, phi: (1 /mup ) ∗ ni∗ni/ ( tau ∗nk + tau ∗pk+ 2∗tau ∗ni )

58 f_phi_0 = lambda n, p, N_D, N_A, V_app: V_app + kT∗math . asinh ( ( N_D −N_A ) / (2 ∗ ni ) ) − Eg / 2 − affinity # Ohmic boundary condition

59 f_n_0 = lambda n, p, N_D, N_A, V_app: n − ( math . sqrt ( ( N_D − N_A ) ∗(N_D − N_A ) /4 + ni∗ni ) + ( N_D − N_A ) /2 )

60 f_p_0 = lambda n, p, N_D, N_A, V_app: p − ( math . sqrt ( ( N_D − N_A ) ∗(N_D − N_A ) /4 + ni∗ni ) − ( N_D − N_A ) /2 )

61

62 # ----- derivatives .

63

64 # ----- Potential

65

66 d_f_phi_d_phi_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_phi_1 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

67 d_f_phi_d_phi_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_phi_2 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 ) − units . e∗( n2 +

p2 ) /kT

68 d_f_phi_d_phi_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_phi_3 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

69

70 d_f_phi_d_n_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

71 d_f_phi_d_n_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: −units . e

272

72 d_f_phi_d_n_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

73

74 d_f_phi_d_p_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

75 d_f_phi_d_p_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: units . e

76 d_f_phi_d_p_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

77

78 # ----- Electron concentration

79

80 # i + 1/2

81 J_n_plus_1_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3: ( ( n3∗B ( ( phi3 − phi2 ) ) − n2∗B ( ( phi2 − phi3 ) ) ) / ( x3−x2 ) )

82 # i - 1/2

83 J_n_minus_1_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3: ( ( n2∗B ( ( phi2 − phi1 ) ) − n1∗B ( ( phi1 − phi2 ) ) ) / ( x2−x1 ) )84

85 G1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3: G

( ( phi2 − phi1 ) ) ∗ J_n_minus_1_2 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 )

86 G2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3: G

( ( phi2 − phi3 ) ) ∗ J_n_plus_1_2 ( x1, x2, x3, phi1, phi2, phi3, n1, n2, n3,

p1, p2, p3 )

87

88 #G1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3: (

mun *kT/units .e) * ((n2*C(( phi2 - phi1)/kT ) - n1*B(( phi1 - phi2)/kT )*G

(( phi2 - phi1)/kT ))/(x2 -x1))

89

90 d_f_n_d_phi_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: ( 2 . 0 / ( x3−x1 ) ) ∗( G1 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 ) )91

92 d_f_n_d_phi_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: ( 2 . 0 / ( x3−x1 ) ) ∗(\93 J_n_plus_1_2 ( x1, x2, x3, phi1, phi2,

phi3, n1, n2, n3, p1, p2, p3 ) −\94 J_n_minus_1_2 ( x1, x2, x3, phi1, phi2,

phi3, n1, n2, n3, p1, p2, p3 ) +\95 + G2 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 ) − G1 ( x1, x2, x3,

phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3 ) )

96

97 d_f_n_d_phi_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: −(2.0/ ( x3−x1 ) )∗G2 ( x1, x2, x3, phi1, phi2, phi3, n1,

n2, n3, p1, p2, p3 )98

99 d_f_n_d_n_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_n_1 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

100 d_f_n_d_n_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_n_2 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 ) + (1 /mun ) ∗p2/ ( tau ∗n2 + tau ∗p2 + 2∗tau ∗ni )

101 d_f_n_d_n_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_n_3 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

102

103 d_f_n_d_p_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

104 d_f_n_d_p_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: (1 /mun ) ∗n2/ ( tau ∗n2 + tau ∗p2 + 2∗tau ∗ni )105 d_f_n_d_p_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

273

106

107 # ----- Hole concentration

108

109 # i + 1/2

110 J_p_plus_1_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3: −((p3∗B ( ( phi2 − phi3 ) ) − p2∗B ( ( phi3 − phi2 ) ) ) / ( x3−x2 ) )

111 # i - 1/2

112 J_p_minus_1_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3: − ( ( p2∗B ( ( phi1 − phi2 ) ) − p1∗B ( ( phi2 − phi1 ) ) ) / ( x2−x1 ) )

113

114 G4 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3: G

( ( phi3 − phi2 ) )∗ J_p_plus_1_2 ( x1, x2, x3, phi1, phi2, phi3, n1, n2, n3,

p1, p2, p3 )115 G3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3: G

( ( phi1 − phi2 ) )∗ J_p_minus_1_2 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 )

116

117 d_f_p_d_phi_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: −(2.0/ ( x3−x1 ) ) ∗( G3 ( x1, x2, x3, phi1, phi2, phi3, n1,

n2, n3, p1, p2, p3 ) )

118

119 d_f_p_d_phi_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: −(2.0/ ( x3−x1 ) ) ∗(\120 J_p_plus_1_2 ( x1, x2, x3, phi1, phi2,

phi3, n1, n2, n3, p1, p2, p3 ) −\121 J_p_minus_1_2 ( x1, x2, x3, phi1, phi2,

phi3, n1, n2, n3, p1, p2, p3 ) +\122 G4 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 ) − G3 ( x1, x2, x3,

phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3 ) )

123 d_f_p_d_phi_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: ( 2 . 0 / ( x3−x1 ) )∗G4 ( x1, x2, x3, phi1, phi2, phi3, n1, n2,

n3, p1, p2, p3 )

124

125 d_f_p_d_p_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_p_1 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

126 d_f_p_d_p_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_p_2 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 ) + (1 /mup ) ∗n2/ ( tau∗n2 + tau ∗p2 + 2∗tau ∗ni )

127 d_f_p_d_p_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: f_p_3 ( x1,x2,x3,phi1,phi2,phi3,n2,p2 )

128

129 d_f_p_d_n_1 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

130 d_f_p_d_n_2 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: (1 /mup ) ∗p2/ ( tau ∗n2 + tau ∗p2 + 2∗tau ∗ni )131 d_f_p_d_n_3 = lambda x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2,

p3, N_D, N_A: 0

132

133 # Matrix building

134 RHS = lambda i, v1, f1, v2, f2, v3, f3, g: v1∗f1 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1)

,mesh_phi [ i−1],mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )+\135 v2∗f2 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1],mesh_phi [ i ]

,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )+\136 v3∗f3 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1],mesh_phi [ i ]

,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] )−\137 g ( mesh_n [ i ] , mesh_p [ i ] ,doping ( i ) [ 0 ] ,doping ( i ) [ 1 ] ,mesh_phi [

i ] )

138 f_phi_RHS = lambda i: RHS ( i,mesh_phi [ i−1] ,f_phi_1,mesh_phi [ i ]

,f_phi_2,mesh_phi [ i+1] ,f_phi_3,g_phi )

139 f_phi_RHS_0 = lambda i: RHS ( i,mesh_phi [ i−1] ,f_phi_1_0,mesh_phi [ i ]

,f_phi_2_0,mesh_phi [ i+1] ,f_phi_3_0,g_phi_0 )

274

140 f_n_RHS = lambda i: RHS ( i,mesh_n [ i−1], f_n_1,mesh_n [ i ] ,f_n_2,mesh_n [ i

+1],f_n_3,g_n )

141 f_p_RHS = lambda i: RHS ( i,mesh_p [ i−1], f_p_1,mesh_p [ i ] ,f_p_2,mesh_p [ i+1],f_p_3,g_p )

142

143 M_int = lambda f1, f2, f3: [ ( [ 0 ] ∗ ( i−1) ) + [ f1 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1)

,mesh_phi [ i−1],mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ] ,mesh_p [ i ] ) ,\144 f2 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1]

,mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ]

,mesh_p [ i ] ) ,\145 f3 ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),mesh_phi [ i−1]

,mesh_phi [ i ] ,mesh_phi [ i+1],mesh_n [ i ]

,mesh_p [ i ] ) ]+\146 [ 0 ] ∗ ( npoints − i − 2) f o r i i n range (1 ,npoints

−1) ]

147 M = lambda j: [ [ 1 ] + [ 0 ] ∗ ( npoints −1) ] + j + [ [ 0 ] ∗ ( npoints −1) + [ 1 ] ]

148

149

150 de f f_RHS ( ) :

151

152 b1 = [−(f_phi_0 ( mesh_n [ 0 ] ,mesh_p [ 0 ] ,doping (0) [ 0 ] ,doping (0) [ 1 ] ,Vapp ) −mesh_phi [ 0 ] ) , \

153 f_n_0 ( mesh_n [ 0 ] ,0,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ,\154 f_p_0 (0 ,mesh_p [ 0 ] ,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ]

155 b2 = [ j f o r k i n zip ( [ f_phi_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ,\156 [ f_n_RHS ( i ) f o r i i n range (1 ,npoints−1) ] , \157 [ f_p_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ) f o r j i n k ]

158 b3 = [−(f_phi_0 ( mesh_n [ npoints −1],mesh_p [ npoints −1],doping ( npoints −1)[ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) − mesh_phi [−1] ),\

159 f_n_0 ( mesh_n [−1] ,0,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ,\160 f_p_0 (0 ,mesh_p [−1] ,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ]

161

162

163 r e t u r n b1 + b2 + b3

164

165

166 de f J ( ) :

167

168 # Assemble the Jacobian matrix .

169

170 A = [ [ 1 ] + [ 0 ] ∗ ( ( npoints ∗3)−1),

171 [ 0 ,1 ] + [ 0 ] ∗ ( ( npoints ∗3)−2),

172 [ 0 ,0,1 ] + [ 0 ] ∗ ( ( npoints ∗3)−3) ]173

174 f o r i i n range (1 ,npoints − 1) :

175

176 # compute jacobian entries .

177

178 # x1, x2, x3, phi1, phi2, phi3, n1, n2, n3, p1, p2, p3, N_D, N_A

179

180 vec = [ ]

181 f o r func i n [ d_f_phi_d_phi_1, d_f_phi_d_n_1, d_f_phi_d_p_1, \182 d_f_phi_d_phi_2, d_f_phi_d_n_2, d_f_phi_d_p_2, \183 d_f_phi_d_phi_3, d_f_phi_d_n_3, d_f_phi_d_p_3, \184 d_f_n_d_phi_1, d_f_n_d_n_1, d_f_n_d_p_1, \185 d_f_n_d_phi_2, d_f_n_d_n_2, d_f_n_d_p_2, \186 d_f_n_d_phi_3, d_f_n_d_n_3, d_f_n_d_p_3, \187 d_f_p_d_phi_1, d_f_p_d_n_1, d_f_p_d_p_1, \188 d_f_p_d_phi_2, d_f_p_d_n_2, d_f_p_d_p_2, \189 d_f_p_d_phi_3, d_f_p_d_n_3, d_f_p_d_p_3 ] :

190

191 vec . append ( func ( dx ∗(i−1),dx ∗( i ) ,dx ∗( i+1),\192 mesh_phi [ i−1], mesh_phi [ i ] , mesh_phi [ i+1],\

275

193 mesh_n [ i−1], mesh_n [ i ] , mesh_n [ i+1],\194 mesh_p [ i−1], mesh_p [ i ] , mesh_p [ i+1],\195 doping ( i ) [ 0 ] , doping ( i ) [ 1 ] ) )196

197 d_phi = [ 0 . 0 ] ∗ ( ( i−1)∗3) + vec [ 0 :9 ] + [ 0 . 0 ] ∗ ( npoints ∗3 − (i−1)∗3 −9)

198 d_n = [ 0 . 0 ] ∗ ( ( i−1)∗3) + vec [ 9 :18 ] + [ 0 . 0 ] ∗ ( npoints ∗3 − (i−1)

∗3 − 9)199 d_p = [ 0 . 0 ] ∗ ( ( i−1)∗3) + vec [ 18 :27 ] + [ 0 . 0 ] ∗ ( npoints ∗3 − (i−1)

∗3 − 9)

200

201 #

202

203 A . append ( d_phi )

204 A . append ( d_n )

205 A . append ( d_p )

206

207 A . append ( [ 0 . 0 ] ∗ ( ( npoints ∗3)−3) + [ 1 . 0 ,0 . 0 ,0 . 0 ] )208 A . append ( [ 0 . 0 ] ∗ ( ( npoints ∗3)−2) + [ 1 . 0 ,0 . 0 ] )

209 A . append ( [ 0 . 0 ] ∗ ( ( npoints ∗3)−1) + [ 1 . 0 ] )

210

211 r e t u r n A

212

213

214 de f doping ( x ) :

215 i f x > npoints/2: r e t u r n ( Nd,0 )

216 else: r e t u r n (0 ,Na )

217

218 mesh_phi = np . array ( [ f_phi_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i n

range (0 ,npoints ) ] ) #Initial guess for solution

219 mesh_n = np . array ([−f_n_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i n

range (0 ,npoints ) ] )

220 mesh_p = np . array ([−f_p_0 (0 ,0,doping ( x ) [ 0 ] ,doping ( x ) [ 1 ] ,0 ) f o r x i nrange (0 ,npoints ) ] )

221 mesh = [ j f o r i i n zip ( mesh_phi, mesh_n, mesh_p ) f o r j i n i ]

222

223 ion ( ) ; p = plt . subplot (221) ; p . set_title ( "n,p vs. x" ) ; p . set_xlabel ( "x, $\

mu m$" ) ; p . set_ylabel ( "$cm ^-3 $" ) ;224 N, = plt . semilogy ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, mesh_n ∗pow (

units . cm,3 ) ,’-’ )

225 P, = plt . semilogy ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, mesh_p ∗pow (units . cm,3 ) ,’-’ )

226 p = plt . subplot (222) ; p . set_title ( "$E_C, E_V vs. x$" ) ; p . set_xlabel ( "x, $\

mu m$" ) ; p . set_ylabel ( "eV" )

227 Ec, = plt . plot ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, −affinity/units . V

− mesh_phi/units . V,’-’ )

228 Ev, = plt . plot ( np . arange (0 ,npoints ) ∗dx / units . cm ∗ 1e4, −affinity/units . V

− mesh_phi/units . V − Eg/units . V,’-’ )229 p = plt . subplot (223) ; p . set_title ( "Log current density vs. diode bias" ) ; p

. set_xlabel ( "Volts " ) ; p . set_ylabel ( "$mA / cm ^2$" )

230 I, = plt . semilogy ( [ 0 . 0 ,1 . 0 ] , [ 1 e−10,20000 ] ,’-o’ )

231 p = plt . subplot (224) ; p . set_title ( "Current density vs. diode bias" ) ; p .

set_xlabel ( "Volts" ) ; p . set_ylabel ( "$mA / cm^2$" )232 I2, = plt . plot ( [ 0 . 0 ,1 . 0 ] , [−0.1 ,20000 ] ,’-o’ )

233 voltages = [ ] ; currents = [ ] ; start = time . clock ( )

234

235 Vapp = 0

236

237 k = 0

238

239 old_mesh_phi = mesh_phi ; old_mesh_n = mesh_n ; old_mesh_p = mesh_p ;

old_mesh = mesh

240 omega = 1.0

276

241

242 r = 1000

243

244 wh i l e k<3:

245 # Solve equations using Block Newton

246

247 k += 1

248

249 #

250

251 A = M ( M_int ( f_phi_1_0,f_phi_2_0,f_phi_3_0 ) ) # Assemble Phi matrix

252

253 # -f_phi_0 (mesh_n [0] ,mesh_p [0] ,doping (0) [0] ,doping (0) [1] ,Vapp) +

mesh_phi [0]

254 # f_phi_RHS (i)

255 # -f_phi_0 (mesh_n [npoints -1] ,mesh_p [npoints -1] ,doping (npoints -1)

[0] ,doping (npoints -1) [1] ,0) + mesh_phi [-1]

256

257 b = [ ( f_phi_0 ( mesh_n [ 0 ] ,mesh_p [ 0 ] ,doping (0) [ 0 ] ,doping (0) [ 1 ] ,Vapp )

− mesh_phi [ 0 ] ) ] + \258 [−f_phi_RHS_0 ( i ) f o r i i n range (1 ,npoints−1) ]\259 + [ ( f_phi_0 ( mesh_n [ npoints −1] ,mesh_p [ npoints −1],doping (

npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) − mesh_phi [−1] ) ] #

RHS for Phi

260

261 dphi = scipy . linalg . solve ( A,b )

262 mesh_phi = old_mesh_phi + omega∗dphi # Update new Phi vector

263

264 A = M ( M_int ( f_n_1,f_n_2,f_n_3 ) ) # Assemble n matrix

265

266 b = [ f_n_0 ( mesh_n [ 0 ] ,0,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ] + \267 [ f_n_RHS ( i ) f o r i i n range (1 ,npoints−1) ] + [ f_n_0 ( mesh_n [−1]

,0,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ] ,0 ) ]268

269 dn = scipy . linalg . solve ( A,b )

270 mesh_n = old_mesh_n − omega ∗dn # Solve for new n vector

271

272 A = M ( M_int ( f_p_1,f_p_2,f_p_3 ) )273

274 b = [ f_p_0 (0 ,mesh_p [ 0 ] ,doping (0) [ 0 ] ,doping (0) [ 1 ] ,0 ) ] +\275 [ f_p_RHS ( i ) f o r i i n range (1 ,npoints−1) ] +\276 [ f_p_0 (0 ,mesh_p [−1] ,doping ( npoints −1) [ 0 ] ,doping ( npoints −1) [ 1 ]

,0 ) ]277

278 dp = scipy . linalg . solve ( A,b )

279 mesh_p = old_mesh_p − omega ∗dp # Solve p-block

280

281

282 r = scipy . linalg . norm ( [ f_phi_RHS_0 ( i ) f o r i i n range (1 ,npoints−1) ]

+ \283 [ f_n_RHS ( i ) f o r i i n range (1 ,npoints−1) ] + \284 [ f_p_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ) /

(3∗ npoints ) #compute residual

285

286 old_mesh_phi = mesh_phi ; old_mesh_n = mesh_n ; old_mesh_p = mesh_p

# if we didn’t break due to res increase update old mesh

287

288 k = 0289

290 mesh = [ j f o r i i n zip ( old_mesh_phi, old_mesh_n, old_mesh_p ) f o r j i n i ]

291 mesh_phi = mesh [ 0 ::3 ]

292 mesh_n = mesh [ 1 ::3 ]

293 mesh_p = mesh [ 2 ::3 ]

277

294

295 old_mesh_phi = mesh [ 0 ::3 ]

296 old_mesh_n = mesh [ 1 ::3 ]297 old_mesh_p = mesh [ 2 ::3 ]

298

299 p r i n t "Bias, V\t\tJ, mA/cm2 \t\t||RHS ||\ t\tIterations \t\tTime (s)"

300

301

302 f o r Vapp i n np . array ( [ 0 . 0 ∗ units . V,0 . 2∗ units . V,0 . 4∗ units . V,0 . 6∗ units . V,0 . 7∗units . V,0 . 8∗ units . V ] ) : # Solver loop

303

304 #print "Solving for Vd = %g V" % (Vapp/units .V);

305 r = 1000; old_mesh_phi = mesh_phi ; old_mesh_n = mesh_n ; old_mesh_p =mesh_p ; old_mesh = mesh

306

307 r = scipy . linalg . norm ( f_RHS ( ) ) /npoints

308

309 k = 0310 omega = 1

311 prev_r = r

312

313 mesh = [ j f o r i i n zip ( old_mesh_phi, old_mesh_n, old_mesh_p ) f o r j i n

i ]314 mesh_phi = mesh [ 0 ::3 ]

315 mesh_n = mesh [ 1 ::3 ]

316 mesh_p = mesh [ 2 ::3 ]

317

318 k = 0319

320 tk = 1.0

321

322 wh i l e r>5e−15 or k<3: # Solve equations using fully coupled method

323

324 k += 1

325 prev_r = r ;

326 A = J ( ) # Assemble Jacobian matrix

327 b = np . array ( f_RHS ( ) )

328 dmesh = scipy . linalg . solve ( A,b ) # Update new Phi vector

329 i = 0

330 pp_r = r

331 wh i l e 1:

332 prev_mesh = mesh

333 mesh = old_mesh − tk ∗ dmesh

334

335 mesh_phi = mesh [ 0 ::3 ]

336 mesh_n = mesh [ 1 ::3 ]

337 mesh_p = mesh [ 2 ::3 ]

338

339 r = scipy . linalg . norm ( f_RHS ( ) ) /npoints/3

340

341 i f k <5:

342

343 break344

345 i f r > prev_r:

346 tk ∗= 0.99

347 else:

348 tk = 1.0349 break

350

351 i f r > pp_r and i>5:

352 mesh = prev_mesh

353 break

278

354 i+=1

355 pp_r = r

356 i f i> 5:357 mesh = prev_mesh

358 break

359

360 i f (r>prev_r and k>5):

361 r = prev_r

362 break

363

364 i f k > 20:

365 break

366

367 old_mesh = mesh ; # if we didn ’t break due to res increase update

old mesh

368

369 old_mesh_phi = mesh [ 0 ::3 ]

370 old_mesh_n = mesh [ 1 ::3 ]371 old_mesh_p = mesh [ 2 ::3 ]

372

373 i f (r>0.9∗ prev_r and k>3):

374 break

375

376 try:

377 N . set_ydata ( old_mesh_n ∗ pow ( units . cm,3 ) ) ; P . set_ydata ( old_mesh_p

∗ pow ( units . cm,3 ) ) ; # Plot carrier concentrations

378 Ec . set_ydata (−affinity/units . V − old_mesh_phi/units . V ) ; Ev .

set_ydata (−affinity/units . V − old_mesh_phi/units . V − Eg/units .V ) # Plot energy bands

379 ph = ( old_mesh_phi [1]− old_mesh_phi [ 0 ] ) /kT # Compute

current density at contact

380 j = (( kT ∗ mun ) ∗ ( ( old_mesh_n [ 1 ] ∗ B ( ph ) − old_mesh_n [ 0 ] ∗ B(−ph ) )

/dx ) +\381 ( kT ∗ mup ) ∗ ( ( old_mesh_p [ 0 ] ∗ B ( ph ) − old_mesh_p [ 1 ] ∗ B(−ph ) ) /dx

) ) ∗pow ( units . cm,2 ) ∗1 e3/units . A # Scale to mA/cm2

382

383 #j = ((kT * mun ) * ((old_mesh_n [1]*B(ph) - old_mesh_n [0]*B(-ph))

/dx )*pow (units .cm,2))*1 e3/units .A # Scale to mA/cm2

384

385 voltages . append ( Vapp/units . V + 1e−6∗j ) ; currents . append ( j ) # +

Vapp/units .V/50e3 ) # series spec. resistivity: 1mOhm.cm2, 10

kOhm.cm2 shunt

386 I . set_xdata ( voltages ) ; I . set_ydata ( currents ) ; I2 . set_xdata (voltages ) ; I2 . set_ydata ( currents ) ; draw ( ) # update plots

387

388 stop = time . clock ( ) ;

389

390 p r i n t "%g\t\t%g\t\t%g\t\t%d\t\t%g" % ( Vapp/units . V + 1e−6∗j, j, r,

k, stop − start )

391

392 start = time . clock ( ) ;

393

394 except:

395 pas s

396 i f j > 20000 : break # break on overcurrent

397 #print "J = %g mA/cm2 " % j

398 ioff ( ) ; plt . show ( ) # Freeze on plot, print solve time

Appendix C

List of original contributions in this work

• proximity doping of germanium using phosphorus, antimony and gallium

• Comprehensive investigation of contacts to germanium of low specific

resistivity

• passivation using ammonia pretreatment and a a-Si:H/SiNx passivation

stack

• numerical simulation of the monolithic CdTe / Ge device structure

• original presentation of fully coupled solution to drift diffusion equation

• diode process utilising MgF2 as etch mask

Appendix D

List of publications arising from this work

• James Sharp, Won Jae Lee, Klaus Ploog, Gilberto A. Umana-Membreno,

Lorenzo Faraone, John M. Dell, ”A novel technique for degenerate p-type

doping of germanium”, Solid-State Electronics, Volume 89, November

2013, Pages 146 - 152

• James Sharp, David Pulfrey, Gilberto A. Umana-Membreno, Lorenzo Faraone,

John M. Dell, ”Modeling and design of a thin-film CdTe / Ge tandem solar

cell”, Journal of Electronic Materials, Volume 41, October 2012, Pages

2759-2765

D.1 Conference presentations

• James Sharp, David Pulfrey, Gilberto Umana-Membreno, Lorenzo Faraone,

John Dell, ”Toward a CdTe/Ge thin film tandem solar cell: Modelling and

Design”, 2011 US Workshop on the physics and chemistry of II-VI mate-

rials, Chicago, IL, October 2011

• J. Sharp, D. Pulfrey, F. Troni, L. Faraone, J. Dell, ”Improving the effi-

ciency of CdTe solar cells: the case for a thin film CdTe / Ge tandem

solar cell”, International Conference on Nanoscience and Nanotechnology,

ICONN, Perth WA, January 2012

281

• J.Sharp, W. J. Lee, G. A. Umana-Membreno, J.Dell and L. Faraone, ”Sin-

gle crystal and amorphous Ge for use in stand-alone and thin film tandem

solar cells”, Conference on Optoelectronic and Microelectronic Materials

and Devices, COMMAD, Melbourne VIC, December 2012