The Physics of Industrial Crystalline Silicon Solar Cells · CHAPTER ONE The Physics of Industrial...

CHAPTER ONE

The Physics of IndustrialCrystalline Silicon Solar CellsOtwin BreitensteinMax Planck Institute of Microstructure Physics, Halle, Germany

Contents

1. Introduction and Chapter Methodology 12. Basic Theory of Solar Cells 4

2.1 Solar cell in thermal equilibrium 42.2 Biased solar cell 82.3 Analysis of the bulk lifetime 132.4 Depletion region recombination 182.5 Illuminated solar cell 202.6 Reverse current 23

3. Theory Versus Experiment 244. Origins of Nonideal Characteristics 26

4.1 The depletion region recombination (second diode) current 274.2 The diffusion (first diode) current 374.3 The ohmic current 444.4 The reverse current 484.5 Relation between dark and illuminated characteristics 59

5. Summary and Outlook 67Acknowledgments 70References 70

1. INTRODUCTION AND CHAPTER METHODOLOGY

Solar cells made from silicon wafers are the oldest type of solar cells,

which were developed in Bell Laboratories in the 1950s for space applica-

tions. While the first silicon solar cell made in 1953 had an energy conver-

sion efficiency of 6%, already in 1958 the “Vanguard 1” satellite was

powered by 108 silicon solar cells having an efficiency of 10.5% (http://

en.wikipedia.org/wiki/Solar_cell#History_of_solar_cells). Today, the

Semiconductors and Semimetals, Volume 89 # 2013 Elsevier Inc.ISSN 0080-8784 All rights reserved.http://dx.doi.org/10.1016/B978-0-12-381343-5.00001-X

1

Author's personal copy

http://en.wikipedia.org/wiki/Solar_cell#History_of_solar_cells

http://en.wikipedia.org/wiki/Solar_cell#History_of_solar_cells

http://dx.doi.org/10.1016/B978-0-12-381343-5.00001-X

world efficiency record for crystalline silicon solar cells is at 25% (Green

et al., 2012), and typical industrial cells are already approaching 20%

(Song et al., 2012). This impressive advancement was only possible based

on a deep understanding of the physics underlying these solar cells. Note that

semiconductor physics is a relatively young science. The theory of a p–n

junction was developed only in 1949 (Shockley, 1949), the papers describ-

ing the Shockley–Read–Hall (SRH) recombination statistics appeared in

1952 (Hall, 1952; Shockley and Read, 1952), and in 1957 the diode theory

became extended to generation and recombination processes in the deple-

tion region (Sah et al., 1957). Until now, these are the basic papers for

understanding the physics of solar cells. Today, this theory is an integral part

of textbooks on semiconductor physics and technology (see, e.g., Sze and

Ng, 2007). This chapter will not replace such a textbook. For understanding

it, basic knowledge in solid state and semiconductor physics is required. This

chapter basically consists of two parts. In the first part, the established theory

of the operation of solar cells is reviewed. Here the most important relations

describing a solar cell are derived and made physically clear. Then the pre-

dictions of this theory are compared with typically measured solar cell char-

acteristics, which reveal significant deviations from the theory. The main

focus of the second part of this chapter is to point on the reasons for these

deviations and explain their physical origins.

The widely accepted model electrically describing silicon solar cells is the

so-called two-diodemodel, which will be discussed in the following section.

However, as mentioned above, the current–voltage (I–V ) characteristics of

industrial silicon solar cells show significant deviations from the classical

two-diode model predictions. This holds particularly for cells made from

multicrystalline material, which contain high concentrations of crystal

defects like grain boundaries, dislocations, and precipitates, fabricated by

the so-called vertical gradient freeze (Trempa et al., 2010) or Bridgman

method (Muller et al., 2006). Even the characteristics of industrial mono-

crystalline cells, which do not contain these crystal defects, deviate from

the theoretical predictions. In particular, the so-called depletion region

recombination current or second diode current is usually several orders of

magnitude larger than expected, and its ideality factor is significantly larger

than the expected value of two. This nonideal behavior was observed already

very early and tentatively attributed to the existence of metallic precipitates

or other defects in the depletion region (Queisser, 1962). In that work

(Queisser, 1962) it was already suspected that local leakage currents could

be responsible for the nonideal diode behavior, and it was speculated that

2 Otwin Breitenstein


the edge region of a cell could significantly contribute to these nonideal cur-

rents. Later on, this nonideal behavior was attempted to be explained also

under the assumption of a homogeneous current flow by attributing it to

trap-assisted tunneling (Kaminski et al., 1996; Schenk and Krumbein,

1995). However, in crystalline silicon solar cells, the defect levels responsible

for this effect could never be identified. There were attempts to explain the

large ideality factors solely by the influence of the series resistance

(McIntosh, 2001; van der Heide et al., 2005). As will be shown in

Section 4.1, this explanation is not sufficient for interpreting large ideality

factors in well-processed cells.

It has turned out that the key for a detailed understanding of the dark

characteristic of solar cells is the spatially resolved mapping of the local cur-

rent density of solar cells in the dark. Until now, all textbooks dedicated to

solar cells still generally assume that a solar cell behaves homogeneously, e.g.,

(Green, 1998; Wurfel, 2005). Until 1994, there was no experimental

technique available that could map the dark forward current of a solar cell

with sufficient accuracy. In principle, this current can be mapped by infrared

(IR) thermography (Simo andMartinuzzi, 1990). However, since silicon is a

good conductor of heat, the thermal signals are generally weak and the

images appear blurred. Therefore, conventional IR thermography is only

able to image breakdown currents under a reverse bias of several Volts,

and the obtained spatial resolution is very poor (several mm, see

Simo and Martinuzzi, 1990). The first method enabling a sensitive imaging

of the dark forward current with a good spatial resolution was the “Dynamic

Precision Contact Thermography” (DPCT) method (Breitenstein et al.,

1994, 1997). Here a very sensitive miniature temperature sensor was

probing the cell surface point-by-point in contact mode, and in each

position the cell bias was square-pulsed and the local surface temperature

modulation was measured and evaluated over some periods according to

the lock-in principle. This technique already reached a sensitivity in the

100 mK range (standard thermography: 20–100 mK), and, due to its

dynamic nature, the spatial resolution was well below one mm. Its only

limitation was its low speed; taking a 100!100 pixel image took several

hours. Therefore, DPCT was later replaced by IR camera-based lock-in

thermography (LIT). This technique was developed already before it was

introduced to photovoltaics (Kuo et al., 1988), and since then it was mainly

used in nondestructive testing, hence for “looking below the surface of

bodies” (Busse et al., 1992). In the following, LIT was also used for inves-

tigating local leakage currents in integrated circuits (Breitenstein et al., 2000)

3The Physics of Industrial Crystalline Silicon Solar Cells


and in solar cells (Breitenstein et al., 2001). Meanwhile, LIT is a widely

used standard imaging method for characterizing solar cells, which is

commercially available. Details to its basics, realization, and application

are given in (Breitenstein et al., 2010a). Since the illuminated I–V

characteristic of a solar cell is closely related to its dark characteristic, LIT

can even be used for performing a detailed local analysis of the efficiency

of inhomogeneous solar cells (Breitenstein, 2011, 2012). In the last years,

in addition to LIT, also camera-based electroluminescence (EL) and

photoluminescence (PL) imaging methods have been developed for the

local characterization of inhomogeneous solar cells. An overview over these

methods and their comparison to LIT-based methods can be found in

Breitenstein et al. (2011a).

The topics covered in this chapter are as follows:

• Section 2: Basic Theory of Solar Cells

– Section 2.1: Solar cell in thermal equilibrium

– Section 2.2: Biased solar cell

– Section 2.3: Analysis of the bulk lifetime

– Section 2.4: Illuminated solar cell

– Section 2.5: Reverse current

• Section 3: Theory Versus Experiment

• Section 4: Origins of Nonideal Characteristics

– Section 4.1: The depletion region recombination (second diode)

current

– Section 4.2: The diffusion (first diode) current

– Section 4.3: The ohmic current

– Section 4.4: The reverse current

– Section 4.5: Relation between dark and illuminated characteristics

• Section 5: Summary and Outlook

2. BASIC THEORY OF SOLAR CELLS2.1. Solar cell in thermal equilibrium

Figure 1.1A shows qualitatively the band scheme of an n"–p junction,

including its ohmic contacts, as it is present in usual industrial solar

cells, in thermal equilibrium. Particularly, the x-axis is not to scale, in reality

the emitter thickness is a factor of 500 smaller than the base thickness.

“n"–p” means that the n-side is much more highly doped (up to 1020 cm#3)



Figure 1.1 Schematic band diagram (A), profile of the space charge density (B),and profile of the electric fields (C) in an n"–p junction, assuming homo-geneous nondegenerate emitter doping, including its ohmic contacts (ME$metal),not to scale.



than the p-side of the junction (typically 1016 cm#3), which holds for typical

P-diffused p-base solar cells. Moreover, for keeping the explanations simple,

the n"-type emitter in Fig. 1.1 is assumed to be homogeneously and

nondegenerately doped, in contrast to real diffused emitters. Figure 1.1B

shows the local space charge densities and (C) the electric fields in this

device, both also not to scale. For homogeneous doping, the volume of

the n- and of the p-material does not contain any space charge or any fields.

Therefore, these regions are called the neutral material. In reality, the

inhomogeneously doped emitter contains certain fields. At the metallurgical

p–n junction, holes have been diffused into the n-material and electrons

into the p-material, which has led to an electrostatic potential difference

between both regions. The electrostatic potential in n-material is more pos-

itive compared to that in the p-material. It is sometimes hard to

understand why this potential difference cannot be equilibrated by closing

an electric contact between both sides. Here it must be known that the elec-

trostatic potential in a semiconductor is not a measurable voltage, as it is in a

metal. In a semiconductor, the measurable voltage is the position of the

chemical potential of the electrons, which is the Fermi level. By definition,

a slope of the Fermi level is equivalent to a current flow in the device (Sze

and Ng, 2007). Therefore, in thermal equilibrium, the Fermi level (dash-

dotted line in Fig. 1.1A) crosses the device horizontally, hence no current

flows. In the two metal contacts, the Fermi level is at the same energy as

in the semiconductor, therefore closing these contacts does not lead to

any current flow and particularly not to an equilibration of the different

electrostatic potentials in the n- and in the p-region. As long as the

Fermi level is lying sufficiently deep (several kT) in the band gap (non-

degenerate doping condition), the electron resp. hole concentrations n

and p can be described by

n$Nc expEF#Ec

kT

! ", p$Nv exp

Ev#EF

kT

! "%1:1&

Here Nc and Nv are the effective densities of states in the conduction

resp. valence band, EF is the Fermi energy position, Ec and Ev are the energy

positions of the conduction resp. valence band, and kT is the thermal energy.

At room temperature, the shallow donors and acceptors are completely ion-

ized, hence, in the absence of any compensation, in the neutral n-region

n$ND and in the neutral p-region p$NA holds, with ND being the donor

doping concentration at the n-side and NA the acceptor doping



concentration at the p-side. Thus, via Eq. (1.1) the doping concentrations

govern the position of the Fermi level in the neutral p-material:

EF#Ev$ xp$ kT lnNv

NA

! "%1:2&

For a typical base doping concentration of 1016 cm#3, Eq. (1.2) leads to

xp'190 meV at RT. Note that Eqs. (1.1) and (1.2) only hold for the

p-material, since the n-doped emitter is usually degenerately doped

(metallic-like behavior). Here Fermi statistics has to be applied, leading

for ND$2!1020 cm#3 to xn'#50 meV, hence the Fermi level is actually

lying in the conduction band there.

In the depletion region of the p–n junction, n and p are negligibly small;

therefore, the two doping concentrationsND andNA govern the positive resp.

negative ionic space charge densities there. According to the Poisson equation

(Sze andNg, 2007), the space charge density is proportional to the second deriv-

ative of the potential to x, hence to the bending of the potential. Therefore, in

the depletion region, for homogeneous doping concentration, the bands have a

parabolic shape. IfND(NA holds, as shown here,most of the potential drop in

equilibrium and under reverse bias occurs within the more lowly doped

p-region, leading to the relation for the depletion region width:

W $

###########################2ee0 Vr"Vd% &

eNA

s

%1:3&

Here Vr is the applied reverse voltage (zero in Fig. 1.1), Vd is the equi-

librium barrier height, also called the diffusion voltage, ee0 is the permittivity

of the material, and e is the electron charge. Note that (1.3) only holds in

total depletion approximation, hence by neglecting the so-called edge

regions of the junction, where the free carrier concentration gradually

decreases toward zero. According to the discussion above concerning the

Fermi energy position, the diffusion voltage Vd equals the gap energy Eg

minus the sum of the energy distances between the Fermi energy and the

neighbored band edges in the p- and the n-material xp and xn, divided by

e for converting the energy into a voltage:

Vd $Eg#xp#xn

e%1:4&

Also at the ohmic contacts depletion regions exist. In some older text-

books, ohmic contacts are still described as carrier accumulation regions,



based on the idea that the barrier height of a metal contact is just the differ-

ence between the electron affinities of the semiconductor and the metal. It is

claimed that, for certain metal–semiconductor combinations, this could lead

to a negative barrier height and thus to an accumulation contact. This theory

is wrong, see e.g., Sze and Ng (2007). In reality, the barrier height is always

dominated by interface states, and the electron affinity plays only a minor

role. These interface states capture free carriers and are therefore in

n-material negatively and in p-material positively charged (see Fig. 1.1B).

Therefore in both cases the metal–semiconductor contact implies a deple-

tion region, the metal type and the doping concentration only govern the

barrier height. According to Eq. (1.3), which also holds for metal–

semiconductor contacts, the doping concentration influences the depletion

region width W. For highly doped material, W comes into the nm region.

Then the carriers may tunnel through the barrier, which is the mechanism

how ohmic contacts work. Note that the p-contact region is also highly

doped, which has two reasons: first, only the high doping concentration

enables ohmic contact formation, as for the n" contact described above. Sec-

ond, the higher electrostatic potential in the p-contact region reduces under

illumination the electron (minority carrier) concentration in this region,

thereby reducing the recombination rate at the contact. At the junction

between the p- and the p"-region, holes have diffused from the p"- into

the p-region. This leads to a positive space charge at the p- and a negative

at the p"-edge. This is the origin of the so-called back surface field (BSF) in

this region, which repels minority carriers (see Fig. 1.1C). For a diffused BSF

region, due to the concentration depth profile, this field extends up to the

p-contact. The same holds for a highly doped diffused emitter region.

2.2. Biased solar cellIn the following section, the equations describing the current–voltage (I–V)

characteristic of a p–n junction will be derived. Figure 1.2 shows schemat-

ically the band structure of a p–n junction (a) in thermal equilibrium, (b)

under reverse, and (c) under forward bias. The dashed line in the middle

of the gap symbolizes a mid-gap SRH recombination center, which governs

the excess carrier lifetime t in the neutral material. The physics of a p–n

junction can only be understood by considering horizontal and vertical ther-

mally induced processes, which are symbolized in Fig. 1.2 by arrows. For

clarity only electron processes are indicated, the same processes also hold

for holes, where the energy scaling is inverted. Even in thermal equilibrium



there is spontaneous thermal carrier generation (upward arrows) and recom-

bination (downward arrows), and there is horizontal carrier movement.

Note that the free carriers not only exist close to the band edges, as it is often

displayed in such schemes, but also deep in the bands. They follow the Fermi

statistics, which, if the Fermi level is lying within the band gap, corresponds

to Maxwell–Boltzmann statistics. These electrons deep in the band are char-

acterized by a large kinetic energy. Therefore, they may be called high-

energy or “hot” electrons, though they are in thermal equilibrium with

all other electrons and with the lattice. With increasing energy distance

DE to the band edges, the free carrier concentration decreases essentially

proportional to exp(#DE/kT) (in reality also the density of states plays a

role). Since the Fermi energy is going horizontally through Fig. 1.2A

(not shown there, but in Fig. 1.1), the concentration of electrons in the

p-side (in Fig. 1.2 left) essentially equals that in the n-side (right) having

an energy above the position of the conduction band edge in the p-side.

Only these “hot” electrons have sufficient kinetic energy to overcome

the decelerating electric field in the depletion region and to enter the

p-side. The two driving forces for horizontal carrier movement are the con-

centration gradient, leading to the so-called diffusion current, and the elec-

tric field, leading to the field current. The consideration of these two current

contributions independently, with only the sum of both being a measurable

net current, is called the detailed balance principle. In thermal equilibrium,

across the whole depletion region, these two horizontal currents balance

each other (Sze and Ng, 2007). Then also the net horizontal current across

the p–n junction is zero. A similar detailed balance principle holds for

recombination and thermal generation. In any position, under thermal equi-

librium, thermal carrier generation is balanced by carrier recombination. For

Figure 1.2 Schematic band diagram of a p–n junction (A) in thermal equilibrium, (B)under reverse bias, and (C) under forward bias, only electron currents are shown Thedashed line represents a deep SRH recombination center governing the excess carrierlifetime in the p-region. After Breitenstein (2013), by courtesy of Springer.



any kind of homogeneous carrier generation, the equilibrium electron

(minority carrier) concentration at the p-side np can be expressed by the

generation rate G (given in units of generated carriers per cm3 and second)

multiplied by the excess carrier lifetime t. This relation is the base of all

quasi-static lifetime measurement techniques. It also holds for the equilib-

rium thermal carrier generation in the neutral volume sketched in

Fig. 1.2. On the other hand, the electron concentration in the p-material

can be expressed by np$ni2/NA (ni$ intrinsic carrier concentration,

NA$ acceptor concentration), leading to an expression for the thermal gen-

eration rate G:

np$Gt$ n2iNA

, G$ n2itNA

%1:5&

An interesting point here is to understand why the equilibrium minority

carrier concentration is independent of the lifetime. It might be expected

that, in low lifetime regions, the excess carrier concentration is lower, as

it holds, for example, under light excitation condition. However, in these

regions the rate of spontaneous thermal carrier generation is also corre-

spondingly higher, leading to the same excess carrier concentration as in

high lifetime regions, if the net doping concentration is the same.

Under reverse bias (Fig. 1.2B), the concentration of “hot” electrons at

the n-side, which have sufficient kinetic energy to overcome the barrier,

is reduced. Therefore, the diffusion current of electrons from the n- to

the p-side becomes negligibly small. Now the current across the p–n junc-

tion is dominated by the flow of thermally generated electrons from the

p-side to the n-side. The electric field of the junction drains all electrons

generated within one diffusion length Ld $########Det

p(De$electron diffusion

constant in the p-region). This horizontal current density can be expressed

regarding (1.5) as

J01$GeLd$en2i Ld

tNA$ n2i e

######De

p

NA

###t

p %1:6&

Since this current density is independent of the reverse bias, it is called a

saturation current density. Under zero bias, this thermally generated current

is exactly balanced by the diffusion current of electrons running from the n-

to the p-side, see the horizontal arrows in Fig. 1.2A. When a forward bias is

applied (Fig. 1.2C), the magnitude of this diffusion current rises exponen-

tially with increasing forward bias V, since correspondingly more electrons



have enough kinetic energy to overcome the energy barrier. Since the dif-

fusion current at 0 V equals (1.6), its bias dependence can be described as

Jdiff $ J01 expeV

kT

! "%1:7&

The net dark current is Eq. (1.7) minus the thermal generation current

(1.6), leading to Shockley’s diode equation (Shockley, 1949):

J $ J01 expeV

kT

! "#1

! "%1:8&

This net current is traditionally also called “diffusion current” (Sah et al.,

1957), since for V>kT/e (the thermal voltage VT, about 26 mV at

room temperature) it is dominated by Eq. (1.7). The electrons, which are

injected under forward bias into the p-region, recombine there basically

within one diffusion length Ld, indicated by the thick downward arrows

in Fig. 1.2C. This is the same region, which is responsible for the generation

current (1.6). This means that J01 is a measure of the bulk recombination rate

within one diffusion length; the stronger the bulk recombination (low t),the larger is J01.

The name “diffusion current” suggests that this current contribution

would be governed by transport properties, which is actually misleading.

It is wrong to imagine the p–n junction as a kind of valve, where the mag-

nitude of the current flow is only governed, for example, by the barrier

height. Instead, it must be considered that, both under zero and under for-

ward bias, the lateral carrier exchange between the n- and the p-side due to

the thermal carrier movement is so strong that the magnitude of the net cur-

rent is determined by the speed at which the carriers are recombining on the

other side. Therefore, in some research groups, Eq. (1.8) is called “recom-

bination current,” which is traditionally used for the depletion region

recombination current, which will be described below. This still increases

the linguistic confusion. Therefore, throughout this chapter, Eq. (1.8) will

be called “diffusion current.” The correct way to imagine this current is to

consider the quasi Fermi levels of electrons and holes as chemical potentials,

which essentially horizontally cross the p–n junction, as it is described, for

example, byWurfel (2005). Then it can easily be understood that each single

recombination channel (e.g., bulk and surface recombination) leads to a sep-

arate and independent contribution to the corresponding total J01. The same

physics works for the hole exchange between the emitter and the base,



which is not shown in Fig. 1.2, leading to J01e (“e” for emitter). Since the

emitter (donor) doping concentration ND in the denominator of

Eq. (1.5) is very high, the emitter contribution to J01 is often neglected com-

pared to the base contribution. It will be shown in the next section and in

Section 4.2 that this is not generally justified.

Equation (1.6) actually only holds for a cell having a thickness much

larger than the minority carrier diffusion length in the material. This is

the case, for example, in the positions of recombination-active grain bound-

aries in multicrystalline solar cells. In good regions of these cells and also in

monocrystalline cells, the base thickness is not large but rather small com-

pared to the diffusion length, which is the presupposition for collecting most

of the generated excess carriers. Then also the recombination at the back

surface and/or the back contact contributes significantly to the bulk recom-

bination and thus influences J01. In the limit of infinite diffusion length, the

quasi Fermi level crosses the bulk horizontally, leading for the bulk contri-

bution of J01 to the simple relation:

J01 $ Jbulk01 " Jback01 $ en2i d

NAtbulk" en2i sback

NA$ en2i d

NAteff%1:9&

Here sback is the recombination velocity of the backside and d is the bulk

thickness. For the general case of arbitrary diffusion length, other expressions

for J01 can be derived, which contain the influence of a finite bulk thickness

and the front and back surface recombination velocities, see appendix of the

PVCDROM (http://pveducation.org/pvcdrom). The use of these expres-

sions is often avoided by replacing t in Eq. (1.6) or (1.9) by an effective bulklifetime teff, which also includes the back surface recombination. A typical

average value for teff of a monocrystalline silicon solar cell in today’s standard

technology implying a full-area Al back contact is about 160 ms, and for a

multicrystalline cell it is about 40 ms, leading after Eq. (1.6) to expected

values of the base contribution of J01 of about 500 and 1000 fA/cm2, respec-

tively. It must be reminded that the use of teff in combination with Eq. (1.6)

only holds for low lifetime regions like grain boundaries. It suggests that the

diffusion current is proportional to the inverse square root of the lifetime.

For an infinitely thick cell, this square root dependence only stems from

the fact that the recombination volume is proportional to the diffusion

length Ld )###t

p. Within this volume, the recombination rate is propor-

tional to 1/t. In the same way, if the cell is thinner than Ld and the recom-

bination volume is constant, the recombination rate and thus also the



http://pveducation.org/pvcdrom

diffusion current is proportional to 1/teff and not to 1=#######teff

p, as Eq. (1.9)

shows. A saturation density of 1000 fA/cm2 corresponds according to

Eq. (1.9) for d$200 mm and NA$1016 cm#3 to an effective lifetime of

32 ms, compared to 40 ms according to Eq. (1.6).

As Fig. 1.2C shows, the diffusion current under forward bias involves

only the high-energy fraction of the electrons on the n-side. When these

carriers arrive at the p-side, they have lost their kinetic energy due to the

decelerating field, without having dissipated any heat. Due to this current

flow, the mean temperature of the electron gas decreases, which instantly

leads to a decrease of the crystal temperature. This is the physical reason

for the Peltier cooling effect, which occurs at the p–n junction under for-

ward bias (Breitenstein and Rakotoniaina, 2005).

2.3. Analysis of the bulk lifetimeExcess carrier lifetime measurements are the base of semiconductor material

characterization. The most popular method for doing this is quasi-steady-

state photoconductance (QSSPC) (Sinton et al., 1996). Here a wafer or a

device is exposed to a light pulse with slowly varying intensity, and the

change of the conductance is measured inductively as a function of the light

intensity. Alternatively, for high lifetimes, also the transient of the conduc-

tance after short pulse excitation may be measured and converted into a life-

time. If the electron and hole mobilities are known, the conductivity change

may be converted into an excess carrier density, which is, for a given gen-

eration rate, proportional to the excess carrier lifetime. The dependence of

this lifetime on the excess carrier density delivers valuable information on

the recombination mechanisms. As a rule, this technique is used to investi-

gate the bulk lifetime in surface-passivated wafers. However, it can also be

used to investigate devices containing a p–n junction, if these devices are not

metallized. If applied to wafers, the result of a QSSPC analysis describes the

recombination properties of the bulk material and the surfaces. If QSSPC is

applied to devices containing a p–n junction, the results also deliver infor-

mation on the emitter-part of the diffusion current J01e , which describes the

recombination in the emitter. Since this type of measurement is a standard

tool for characterizing bulk and emitter recombination, its physical base will

be reviewed here.

A SRH recombination center is basically characterized by three param-

eters, which are its energy position in the gap, usually expressed as the energy

distance to the nearest band edge, and the capture coefficients for electrons



and for holes cn and cp, respectively. These capture coefficients, having the

unit of cm3/s, are often, in analogy to particle physics, expressed as products

of a capture cross section and the thermal velocity. Knowing these param-

eters, the thermal emission rates (probabilities) for electrons and holes may

be derived (Sze and Ng, 2007). For a center concentration ofNt, the capture

coefficients cn and cp govern the lifetime of excess electrons or holes, under

the condition that the center is occupied by a hole or an electron,

respectively:

1

tn$ cnNt,

1

tp$ cpNt %1:10&

For many device analysis methods [e.g., in the popular solar cell

simulation software PC1D (http://www.pv.unsw.edu.au/info-about/

our-school/products-services/pc1d)], instead of Nt, cn, and cp, only the

two lifetimes tn and tp defined by Eq. (1.10) are given. Note, however, that

these are not always real excess carrier lifetimes. The real lifetime still

depends on the occupancy state of the center, as it will be described below.

Since a center may only be occupied either by a hole or by an electron, the

lifetimes (1.10) never hold both at the same time. For example, in p-type

material all deep levels are occupied by holes. Hence, an incoming hole finds

the center already occupied by a hole and this center does not reduce the

hole lifetime. Therefore, in p-material, the hole lifetime is close to infinite,

even if the material contains a high concentration of recombination centers

and is described by a low tp according to Eq. (1.10).

It is usually assumed that a center has two charge states, one if occupied

by an electron and one if not, or one if not occupied by a hole and one if

occupied, which is the same. This duality of electron and hole occupancy is

only a definition, which also holds for energy bands: A band totally occupied

by electrons (e.g., a valence band in an n-doped semiconductor) does not

contain a significant amount of holes, and a band totally occupied by holes

(e.g., a conduction band in a p-doped semiconductor) does not contain a

significant amount of electrons. Independent of the energy position of an

SRH center (in the upper or in the lower half of the gap) the charge state

of the two possible occupation states governs whether a center is donor- or

acceptor-like. If the charge state changes between 0 and " (for being occu-

pied by an electron or not), it is a donor, and if the charge state changes

between 0 and # (for being occupied by a hole or not), it is an acceptor.

Hence, donor-like levels are, in ionized state (if occupied by a hole),



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electrostatically attractive for electrons, hence for them cn( cp holds, and

acceptor-like levels are, if occupied by an electron, attractive for holes, lead-

ing to cp( cn. Therefore, the often-made assumption cn$ cp (or tn$tp) for amid-gap level (e.g., made in McIntosh, 2001) is actually unrealistic. Deep

centers may also have more than one occupancy state and correspondingly

several energy levels, which will not be considered here. Since for shallow

levels, like B and P in Si, at room temperature the thermal emission rates are

higher than the capture rates, they are ionized at room temperature. For

so-called deep levels (lying more than about 200 meV distant to the band

edge), at room temperature and in the neutral material, the thermal emission

may be neglected compared to thermal carrier capture. In the following sec-

tion, we will concentrate on deep levels, hence we will neglect all terms

related to thermal emission. In thermal equilibrium, the occupancy state

of a center is given by the energy position relative to the Fermi level: All

levels lying below the Fermi level are occupied by an electron (at 0 K),

and all lying above the Fermi level are occupied by a hole. Under steady-

state excitation condition, only the centers lying above the electron

quasi-Fermi level are generally occupied by a hole and that lying below

the hole quasi-Fermi level are generally occupied by an electron. For the

deep centers lying between the quasi-Fermi levels, the electron and hole

occupancy factors !n and !p depend on the ratio of the capture rates for elec-trons and holes and on the electron and hole concentrations n and p:

!n$cnn

cnn" cpp, !p$

cpp

cnn" cpp, !n"!p$ 1 %1:11&

For optical excitation always Dn$Dp holds, since electron–hole-pairs

are generated. Taking p0$NA, Eqs. (1.10) and (1.11) lead to the carrier

dependence of the excess carrier lifetime in p-material if governed by a single

deep SRH level (Sze and Ng, 2007):

1

tSRH$ cnNt!p$

NA"DntpDn" tn NA"Dn% & %1:12&

This formula only holds for deep levels, where the thermal emission

probability can be neglected. If also more shallow levels are considered,

an analog formula also contains the concentrations n1 and p1, which are

the electron and hole concentrations if the Fermi level coincides with the

energy position of the levels (Sze and Ng, 2007). For deep levels, these con-

centrations are negligibly small. In the limit of small excess carrier



concentrationsDn, tSRH$tn always holds. This is the low excitation limit of

tSRH. It coincides with Eq. (1.10) because for low excitation intensity any

deep level in p-material is occupied by a hole. If Dn increases, the behaviordepends on the ratio of tp/tn$ cn/cp. For tn(tp (acceptor-like center) inp-material also for high excitation levels tSRH'tn holds, since the center

remains mainly occupied by a hole. For tp(tn (donor-like center), how-ever, with increasing Dn the hole occupancy factor according to Eq. (1.11)

decreases. Then tSRH increases with increasing Dn. If the lifetimes strongly

deviate from each other, in a certain excess carrier concentration regime

tSRH increases proportional to Dn. In the limit of high excitation, if

tSRH$tp"tn holds, then the lifetime is governed by the larger of the

two lifetimes. This makes the center less recombination-active to incoming

electrons. This effect is called “saturation of a SRH center.”

Another interesting point is the influence of a p–n junction on the effec-

tive bulk lifetime. For example, the lifetime may be investigated in a wafer

directly coming out of the POCl3-diffusion, which is completely sur-

rounded by a p–n junction. Alternatively, a complete solar cell prior to met-

allization may be investigated. In both cases, the electrically floating p–n

junction influences the measured lifetimes. Since the emitter is very thin,

such a QSSPC lifetime measurement only reflects the bulk properties. As

a rule such lifetime investigations are performed as a function of the excess

carrier concentration Dn (in p-type material). The effective bulk lifetime

may be influenced by SRH recombination, by radiative and Auger recom-

bination, by a recombination-active surface, and by the presence of a p–n

junction:

1

teff$ 1

tSRH" 1

trad" 1

tAuger" 1

tsurf" 1

tp#n%1:13&

The injection-level dependence of tSRH has been discussed above and

trad plays only a minor role in silicon. However, since lifetime investigations

are usually performed up to high-injection levels, tAuger has to be consid-

ered. For this mechanism, several parameterizations are available, which

have been compared for example in Reichel et al. (2012a). The surface

recombination is described by the surface recombination velocity s, which

is often assumed to be independent of Dn and is related to the lifetime via

(d$bulk thickness):

1

tsurf$ s

d%1:14&



Wewill now discuss how an electrically floating p–n junction influences

the bulk lifetime for various excess carrier concentrations. Under illumina-

tion, excess electrons are attracted by the emitter and flow into it, leading to

a bias-independent photocurrent Jph; see Section 2.5. If the p–n junction is

not shunted, this biases the emitter into forward direction until Voc-

conditions are established. Then the emitter injects most of the electrons

back to the base, which reduces the net electron current into the base. This

is the reason why floating p–n junctions have already been used for passiv-

ating surfaces. However, not all electrons are injected back. The photocur-

rent is balanced by the complete dark current, which is the sum of the bulk

diffusion current described by Eq. (1.8) and the emitter diffusion current

characterized by J01e:

Jediff $ Je01 expV

VT#1

! "%1:15&

Only the bulk diffusion current injects electrons back into the base, but the

emitter diffusion current injects holes into the base, where they recombine.

Therefore, for a floating p–n junction, this current contribution Jdiffe repre-

sents the loss of excess carriers due to recombination in the emitter. For

the double emitter (sandwich) geometry or the single emitter plus passivated

backside geometry mentioned above, it is usually assumed that the excess car-

rier concentration Dn is essentially homogeneous across the bulk thickness.

Then, for V>VT, the exponential term containing V in Eq. (1.15) may be

expressed by the excess carrier concentration, leading to

np$Dn NA"Dn% &$ n2i expV

VT, Jediff $ Je01

Dn NA"Dn% &n2i

%1:16&

Now the excess electron loss at the floating p–n junction may be

expressed in terms of a recombination velocity sp–n:

Jloss$ Jediff $ eDn sp#n, sp#n $ Je01NA"Dn

en2i%1:17&

This leads together with Eq. (1.14) to the final result:

1

tp#n$sp#n

d$ Je01

NA"Dnedn2i

%1:18&

This means that, for low excess carrier concentrations, the influence

of the emitter on the lifetime is independent of Dn and is described by



1/tp–n$ J01e NA/(edni

2). IfDn comes into the order ofNA (toward high injec-

tion), 1/tp–n increases proportional toDn; hence tp–n decreases proportionalto 1/Dn with the proportionality factor being proportional to J01

e according

to Eq. (1.18). In Section 4.2, an example of a lifetime analysis will be intro-

duced. Equation (1.18) is the base of the so-called Kane–Swanson method,

which is the most common method for measuring J01e independent of

J01b (Kane and Swanson, 1985). Note that, if the lifetime depends on Dn,the measured excess carrier transient is not exponential anymore. Then

the definition of the lifetime is the slope of the Dn-transient related to Dnat this time:

t t% &$ @Dn t% &=@tDn t% &

%1:19&

This definition is used in most time-dependent (non-steady-state)

methods for measuring lifetimes, and it is also the base for quasi-steady-state

lifetime measuring methods like QSSPC and PL imaging. In the limit of

high injection (Dn(NA), a 1/t transient forms instead of an exponential

one (Kane and Swanson, 1985). In any case, already the evaluation of

one single Dn(t) transient allows the measurement of J01e according to

Eqs. (1.18) and (1.19), but the quasi-steady-state methods are equivalent.

2.4. Depletion region recombinationIn Fig. 1.2, generation and recombination are considered not only in the

bulk, but also in the depletion region. This generation and recombination

is most effective for mid-gap levels and is then locally confined to a narrow

region in the middle of the depletion region, where in thermal equilibrium

the Fermi level crosses the defect level (Sze and Ng, 2007). The effective

width of this region should be w. As Fig. 1.2A shows, at zero bias in this

region recombination and thermal generation occur at the same time in

the same place, as in the neutral material. However, they occur with a sig-

nificantly higher rate per volume, since the electron occupancy state of the

mid-gap level is 1/2 here, whereas in the neutral p-material it is very

small. Therefore, this current is often called “recombination-generation

current” (Sah et al., 1957). As explained above, in this sense also J01 is a

“recombination-generation current” of the neutral material and the surfaces.

Again, under reverse bias (Fig. 1.2B), the depletion region generation cur-

rent dominates over recombination, and under forward bias (Fig. 1.2C)

recombination dominates over generation. The thermal generation rate



in the middle of the depletion regionGdr is calculated in analogy to Eq. (1.5)

by replacing np by ni. The saturation current density J02 for the depletion

region current is then calculated in analogy to Eq. (1.6), leading to:

Gdr $ nitbulk

, J02 $Gdrew$ eniw

tbulk%1:20&

Here tbulk is the bulk excess carrier lifetime, which should be larger than

teff since it does not contain any surface contribution. Under forward bias,

the quasi Fermi energies in a silicon cell are usually crossing the p–n junction

horizontally; therefore, in any position of the junction np$ni2exp(eV/kT)

holds. This recombination occurs in the middle of the depletion region,

therefore in this region n$p$ #####np

p $ni exp(V/2VT) holds. Since the deple-

tion region recombination current is proportional to n resp. p in this region,

this leads together with Eq. (1.20) to the expression for the depletion region

current:

Jrec$ J02 expeV

2kT

! "#1

! "%1:21&

The number “2” in the denominator of Eq. (1.21) is called the ideality

factor of the depletion region current. Following the originally given name

(Sah et al., 1957) and the convention in most textbooks (e.g., Sze and Ng,

2007), throughout this chapter we will call Eq. (1.21) the “recombination

current” and Eq. (1.8) the “diffusion current” contribution of the dark

current.

Unfortunately, the effective recombination layer width w in Eq. (1.20) is

not exactly known. In fact, the occupancy state of the mid-gap level is

strongly position-dependent, and the extension of the recombination-

generation region also depends on the bias V. This is the reason why, even

for a mid-gap level, the ideality factor of the recombination current is

expected to be slightly smaller than 2 (McIntosh, 2001; McIntosh et al.,

2000; Sah et al., 1957). In McIntosh (2001), the graph shown in Fig. 1.3

was published ( J02 called J0DR here), which is based on realistic numerical

device simulations using the same assumption of a mid-gap level as done

here. It shows that, for a bulk conductivity of r bulk$1.5 O cm, which is

typical for solar cells, and for a lifetime of about 40 ms, the expected value

of J02 should be about 5!10#11 A/cm2. Note that this is significantly larger

than J01, which is expected to be 10#12 A/cm2 here (1000 fA/cm2). There-

fore, at low forward bias, the recombination current always dominates over

the diffusion current, but at higher forward bias the diffusion current



dominates. In the absence of ohmic currents, the expected effective ideality

factor at low voltages should be about two, and at higher voltages it should

be unity, as long as the base stays in low-injection condition, hence as long as

n*p holds there, and if the series resistance does not play a role yet. The

bias, at which this transition occurs, strongly depends on the magnitudes

of J01 and J02. In our example, it is expected to be about 0.2 V. Hence, at

the maximum power point (mpp) of a solar cell, which typically is close

to 0.5 V, the theoretically expected characteristic should not be influenced

by the recombination current anymore.

2.5. Illuminated solar cellUntil now only the current in the dark was considered. If a solar cell is illu-

minated and light is absorbed, electron–hole pairs are generated, which are

excess carriers. This optical carrier generation acts exactly like the spontane-

ous thermal carrier generation considered for Fig. 1.2, except that it is many

orders of magnitude more intense and is independent of the excess carrier

lifetime. Just as for the thermal generation current (1.6), the photocurrent

is independent of the bias V. In the case of an infinitely thick solar cell

and a homogeneous optical carrier generation rateG, the photocurrent also

10-710-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6 10-5

Carrier lifetime (s)

rbulk = 0.2 W. cm

rbulk = 1 W. cm

rbulk = 5 W. cm

J 0D

R (

A/c

m2 )

10-4 10-3

Figure 1.3 Numerical simulation of a diffused silicon junction. J0DR was only slightlyaffected by variation in the emitter profile. From McIntosh (2001). By courtesy ofK.R. McIntosh.



can be described for a thick cell as Jph$GeLd. Like the thermal generation

current (1.6), the photocurrent Jph is a reverse current. It superimposes on

the bias-dependent dark current described by Eqs. (1.8) and (1.21), which is

called the superposition principle, leading to:

J $ J01 expeV

kT

! "#1

! "" J02 exp

eV

2kT

! "#1

! "# Jph %1:22&

Here the first diode term with the ideality factor of 1 describes the dif-

fusion current, which is finally due to the recombination in the bulk and

emitter material and at the surfaces, and the second diode term with the

ideality factor of 2 describes the recombination current, which is due to

recombination in the depletion region. If the cell is under short circuit

(V$0), these two dark current contributions are zero and J$ Jph holds.

Therefore, Jph is called the short-circuit current density Jsc. If the cell is at

open circuit, as a rule the first diode term in Eq. (1.22) dominates the dark

current. Neglecting the second diode term in Eq. (1.22) and the “#1” in the

first diode term, the condition J$0 leads to the relation for the open-circuit

voltage:

Voc $kT

eln

JscJ01

! "%1:23&

This equation shows that, for obtaining a high open-circuit voltage, J01must be as small as possible. This underlines the importance of the dark cur-

rent for maximizing the efficiency of a solar cell: By minimizing recombi-

nation in the cell, the dark current in a solar cell has to be as small as possible.

In fact, the dark current is one of the major enemies of the solar cell maker.

The I–V characteristic of real solar cells is also influenced by an inevitable

series resistance Rs of the device (being the second major enemy of the solar

cell maker), which leads to the fact that the so-called “local voltage” directly

at the p–n junction deviates from the voltage V applied to the device.

Though the diode theory outlined above does not explain any ohmic con-

ductivity, experience has shown that all solar cells show a noninfinite parallel

resistance Rp. Typical values of Rp are between some or some 10 O cm2

(heavily shunted cells) and some 104 O cm2 (faultless cells, see, e.g.,

Kaminski et al., 1996). The reasons for this ohmic conductivity will be dis-

cussed in Section 4.3. It will be shown in Sections 3 and 4.1 that the ideality

factor of the second diode is often larger than two and therefore expressed as



a variable named n2. Thus, the final two-diode equation, which is widely

accepted for describing real solar cells, reads:

J $ J01 expV #Rs J

VT

! "#1

! "" J02 exp

V #Rs J

n2VT

! "#1

! "" eV #Rs J

Rp# Jsc

%1:24&

AgainVT$kT/e is the thermal voltage being 25.69 mV atT$25 +C. As

mentioned above, in many cases n2$2 is assumed in Eq. (1.24). Note also

that, since Eq. (1.24) is a current density, the resistances Rs and Rp are

expressed here area-related in units of O cm2. The implications of this

approach will be discussed in Section 4.5. Note also that Eq. (1.24) is an

implicit equation for J(V), which complicates practical calculations. There-

fore, in a limited bias range (usually between the mpp and Voc), it is often

simplified to the empirical “one-diode” solar cell equation, again neglecting

Rs and containing effective values for J0 and the ideality factor n:

J V% &$ Jeff0 expV

neffVT#1

! "# Jsc %1:25&

In this equation, the influence of ohmic and recombination (second diode)

current contributions is contained in J0eff and neff. This effective ideality factor

neff is that of the whole current and not only of the recombination current. If a

real cell characteristic is fitted to Eq. (1.25) for each biasV separately, this leads

to the bias-dependent ideality factor n(V), which is very useful for analyzing

the conduction mechanism of solar cells (see Section 4.1).

The series resistance Rs in Eq. (1.24) contains contributions from the grid

lines, fromthecontact resistances, fromthehorizontal current flow in theemitter

layer, and from the current flow in the base. In a module, contributions from

thebusbar connection and the interconnectionwiringmust also be added. Itwill

be shown in Section 4.5 that the use of a constant value for Rs, with the same

value in the dark and under illumination, is actually wrong and represents only

a coarse approximation.Nevertheless, this approximation is oftenmade.Typical

values for Rs are between 0.5 and 1 O cm2 (see, e.g., Kaminski et al., 1996).

Note that for correctly measuring Rp of a solar cell, the two-diode equa-

tion (1.24) has to be fitted to a measured dark or illuminated I–V character-

istic. It is not sufficient to evaluate only the linear part of the dark

characteristic for low voltages and to interpret the slope as the inverse of

Rp, as this is often being done. For low voltages, the two exponential terms

in Eq. (1.24) may be developed in a power series. For small values ofV, both



lead to a linear characteristic nearV$0. ForRp$1 the apparent (effective)

parallel resistance is:

Reffp $ 1

J01VT

" J02n2VT

' n2VT

J02%1:26&

The latter relation holds due to the fact that always J02( J01 holds. It is

also not correct to measure Rp from the slope of the illuminated character-

istic close toV$0, as this is also often being done. In this case, certain depar-

tures from the superposition principle may lead to erroneous results, which

will be discussed in Section 4.5.

2.6. Reverse currentUnder large reverse bias, Eq. (1.24) is not valid anymore, since any p–n junc-

tion breaks down at a certain reverse bias. Moreover, since J02( J01 holds,

the thermal carrier generation in the depletion region governs the reverse

current. For this case, the second exponential diode term in Eq. (1.24) is only

an approximation. Under reverse bias, the generation region widens and

becomes nearly homogeneous within the whole depletion widthW, which

increases with increasing reverse bias according to Eq. (1.3). Therefore, over

a rather large bias range, as long as there is no avalanche multiplication yet,

considering Eqs. (1.3) and (1.20), the reverse current Jr should increase

according to Sze and Ng (2007).

Jr Vr% &$ eGdrW $ eniW

tbulk$ni

#############################2eee0 Vr"Vd% &

p

tbulk#######NA

p %1:27&

This means that, under reverse bias, the reverse current should be in the

order of J02 and should increase sub-proportionally to Vr. If the electric field

in the depletion region exceeds a certain limit, the carriers are multiplied by

the avalanche effect, leading to a steep increase of the reverse current (break-

down). According to Miller (1957) the avalanche multiplication factor can

be described by:

MC V% &$ 1

1# Vr=Vb% &m%1:28&

Here Vb is the breakdown voltage, and m is the Miller exponent, often

assumed to be m$3. For Vr$Vb, MC$1 holds, which is the basic def-

inition of Vb. For a typical base doping concentration of 1016 cm#3 and a



plane silicon junction, Vb is expected to be about 60 V (Sze and Gibbons,

1966; Sze andNg, 2007). Thus, the theoretically expected reverse current of

a solar cell should be the product of Eqs. (1.27) and (1.28). Band-to-band

tunneling under reverse bias (internal field emission, Zener effect) should

not play any role for silicon solar cells, since it dominates over avalanche

multiplication only for a base doping concentration above 5!1017 cm#3

(Sze and Ng, 2007), which is significantly higher than that used for typical

solar cells. However, trap-assisted tunneling may be considered to be

responsible for certain pre-breakdown phenomena (see Section 4.4).

3. THEORY VERSUS EXPERIMENT

Based on the theoretical predictions summarized in Section 2, now the

theoretically expected dark and illuminated I–V characteristics of a typical

multicrystalline silicon solar cell with an effective bulk lifetime of 40 ms,corresponding to J01$1000 fA/cm2, will be calculated and compared with

experimentally measured characteristics of a typical industrial cell. Since the

classic diode theory does not explain any parallel resistance, Rp$1 will be

assumed here. The results are presented in Fig. 1.4. This cell is a typical

156!156 mm2 sized cell made in an industrial production line by the pres-

ently (2012) dominating cell technology (50 O/sq emitter, acidic

texturization, full-area Al back contact, 200 mm thickness) from Bridgman-

type multicrystalline solar-grade silicon material. The same cell is used for

the comparison between dark and illuminated characteristics in Section 4.5.

For calculating the theoretical illuminated characteristic, the value of

Jsc$33.1 mA/cm2 from the experimentally measured illuminated character-

istic of this cell was used. The series resistance of Rs$0.81 O cm2 was calcu-

lated from the voltage difference between the measured open-circuit voltage

(0.611 V) and the dark voltage necessary for a dark current equal to the short-

circuit current (0.638 V), which is an often used procedure for measuring Rs:

Vdark Jsc% &#Rs Jsc $Voc %1:29&

In this case, Rs$0.81 O cm2 fulfilled condition (1.29). For the bulk life-

time in Eq. (1.27), as a lower limit the assumed effective bulk lifetime of

40 ms was used.It is visible in Fig. 1.4 that, in the dark forward characteristic (A), the low

voltage range (V<0.5 V) shows the strongest deviation between theory and

experiment. The measured current in this bias range is governed by the sec-

ond diode and by ohmic shunting. In the theoretical curve, there was no



ohmic shunting assumed, and the second diode contribution is so small that

it is not visible in the displayed data range. Also in the cell used for these

characteristics the ohmic shunting is very low. It will be demonstrated in

Section 4.5 that the shown experimental dark characteristic can be described

by values of Rp$44.4 kO cm2, J02$5.17!10#8 A/cm2 and n2$2.76.

Hence, there is some nonnegligible ohmic conductivity in this cell, J02 is

several orders of magnitude larger than the predicted value of

5!10#11 A/cm2, and its ideality factor n2 is larger than the expected max-

imum value of two. The transition between the J02- and the J01-dominated

part of the dark characteristic is close to the mpp near 0.5 V. This proves that

in this cell the recombination current already influences the fill factor of this

cell, even at full illumination intensity. This result is typical for industrial

solar cells and has often been published (see, e.g., Kaminski et al., 1996).

The reasons for these discrepancies to the theoretically expected behavior

will be discussed in Sections 4.1 and 4.3. Also the experimental value of

J01, which governs the dark characteristic for V>0.5 V, is somewhat larger

than theoretically expected. The reason for this discrepancy will be discussed

Figure 1.4 Comparison of experimentally measured and theoretically predicted(A) dark forward, (B) illuminated, and (C) dark reverse I–V characteristics. AfterBreitenstein (2013), by courtesy of Springer.



in Section 4.2. Since the dark current was underestimated by theory, also the

illuminated characteristics in Fig. 1.4B significantly deviate between theory

and experiment. Both the open-circuit voltage Voc and the fill factor are in

reality smaller than theoretically estimated. This graph also contains an illu-

minated characteristic, which was simulated based on the experimental dark

characteristic by applying the superposition principle and regarding a con-

stant series resistance of 0.81 O cm2, both in the dark and under illumina-

tion. In the dotted curve in Fig. 1.4B Voc is correctly described (it was

used to calculate Rs), but the simulated fill factor appears too large. This dis-

crepancy will be resolved in Section 4.5. Finally, Fig. 1.4C shows the the-

oretical and experimental dark reverse characteristics. Also these curves

deviate drastically. The theoretical dark current density is negligibly small

in the displayed current range for Vr<50 V (in the nA/cm2 range, subli-

nearly increasing with Vr), and the breakdown occurs sharply at

Vr$60 V. In the experimentally measured curve, on the other hand, the

reverse current increases linearly up to Vr$5 V and then increases super-

linearly, showing a typical “soft breakdown” behavior. A sublinear increase,

as predicted by theory, is not visible at all. It will be described in Section 4.4

how this reverse characteristic can be understood.

In the following sections, the present state of understanding the different

aspects of the nonideal behavior of industrial crystalline silicon solar cells, in

particular of cells made from multicrystalline material, will be reviewed, and

a selection of experimental results leading to this understanding will be pres-

ented. All results regarding the edge region or technological problems (e.g., Al

particles at the surface, scratches) hold both for mono- and multicrystalline

cells. On the other hand, all results dealing with crystal lattice defects or pre-

cipitates only hold for multicrystalline cells. In the last years, “quasi-mono” or

“quasi-single crystalline” solar silicon material has also appeared (Gu et al.,

2012). This material does not contain large angle grain boundaries, but it

may contain a high concentration of dislocations and also low angle grain

boundaries, which are basically rows of dislocations. This material is obviously

lying anywhere between multi and mono material; therefore, the conclusions

from this chapter should also be valid for this kind of material.

4. ORIGINS OF NONIDEAL CHARACTERISTICS

In the following subsections, the different aspects of the nonideal

behavior of industrial solar cells are separately discussed and the physical ori-

gins for this behavior are revealed. It will be shown that all these deviations



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