PH3710 Semiconductors and Superconductors

James Nicholls

Physics Department, Royal Holloway, University of London

These notes cover most of the material on semiconductors.

Contents

Course Description 1Lectures 1Syllabus: Semiconductors 1Teaching, Learning and Assessment Strategy 2Books 2

I. Introduction 3A. Four Classes of Conductor 3

1. Metals/semimetals 32. Semiconductors 33. Insulators 44. Superconductors 4

B. Which elements/compounds are semiconductors? 4C. Crystal structure of semiconductors 5D. Doping in semiconductors 6

II. Electrons in Semiconductors 7A. Free electron model 7B. Electrons in a periodic potential 7C. The effective mass approximation 8D. Metals vs Semiconductors 9

III. Carrier concentrations 10A. Intrinsic semiconductors 10B. Extrinsic semiconductors 11

1. Intrinsic ↔ saturated regions 112. Saturated ↔ freeze-out regions 11

IV. Conduction in Semiconductors 14A. The Drude Model 14B. The mobility µ 14

1. Charged impurities 142. Lattice vibrations 14

C. Hall effect and magnetoresistivity tensor (Drudemodel) 14

D. The Hall effect for two carriers 15E. Electrical measurements 16

1. Four-terminal measurements 162. Four-probe measurements 173. Corbino disk 17

V. Optical Properties of Semiconductors 18A. Optical properties 18B. Optical constants 18C. A microscopic model 18D. Phonon absorption 20E. Shallow donors and acceptors 20F. Band-gap absorption 21G. Excitons 22H. Free carrier absorption 22I. Cyclotron resonance 23

VI. Solid State Electronics 25A. Inhomogeneous semiconductors 25B. Diffusion currents 25C. Diffusion length 25D. The p-n junction in equilibrium 26E. A biased p-n junction 27F. Applications of p-n junctions 28

1. Varactor diode 282. Photodiodes (or p-i-n diodes) 283. Solar cells 284. Light-emitting diodes (LEDs) 29

G. Appendix: Calculation of capacitance of pn-junction 30

VII. Growth and Doping of Semiconductors 31A. Growth of bulk crystals 31B. Epitaxial growth 31C. Diffusion doping of semiconductors 32D. Ion implantation 33E. Chip production 33

VIII. Heterostructures and Nanoelectronics 34A. Introduction to low-dimensional systems 34B. The two-dimensional electron gas (2DEG) 34

1. Density of states (DOS) in 2D 34C. Quantum wells 34D. High electron mobility transistors 35

1. Nanostructures 362. Quantum dots and the single-electron transistor 363. Low-dimensional physics 36

E. Density of free electron states in 1D, 2D, and 3D 361. 3D DOS: g3D(E) 372. 2D DOS: g2D(E) 373. 1D DOS: g1D(E) 374. Summary 38

Course Description

Lectures

The second part of this course consists of approximatelysix double lectures and three problem classes. The lec-tures are scheduled for

12:00 - 14:00 Friday T125,

and the problem classes will take place

15:00 -16:00 Monday T125,

but not every week. Problem sheets will be issued everytwo weeks. The aim of this half of the course is to presentthe physics of semiconductors, and to give some examplesof applications.

Syllabus: Semiconductors

Elemental and compound semiconductors; doping; ex-trinsic and intrinsic semiconductors; carrier concentra-tion; transport properties; optical properties; the p-njunction; MBE semiconductors; applications.

2

Teaching, Learning and Assessment Strategy

1. There is no set book for this course which, as a thirdyear optional course, contains some advanced materialand some topics related to recent research. Thereforean extensive set of notes has been produced covering allof the syllabus. These notes will handed out, usually afew lectures in advance, thereby giving you the oppor-tunity to study the material at your own pace, if youdesire, before the lectures. The notes themselves are self-contained, however in addition an extensive book list fol-lows and you are encouraged to read around the subject(in order to experience a different approach to the mate-rial and to aid your learning process). This notes-basedteaching method is appropriate for a subject where a sin-gle text book containing all of the required informationis not available.2. All the basic material on the semiconductor part of thecourse will be provided in the handouts. The handoutsdo not give all steps in a particular derivation, and youwill be expected to work these out yourself, or by lookingin reference books. A course summary will be handed outat the end of the course as an aid to revision, with thesame structure as the notes.3. Problem sheets will be distributed every two weeks.The three problem sets on semiconductors are designedto bring the course together as a whole. They (a) rein-force the course material, (b) contribute to assessmentand feedback, (c) act as a focus for private study. Acoursework mark of 10% is allocated on the basis of an-swers submitted. The numerical answers will be postedon moodle after marking.4. There is a two-hour examination for PH3710. Theexam paper is made up of two sections: Section A willconsist of three questions on semiconductors, and, Sec-tion B will consist of three questions on superconductors.You will be asked to answer three questions, including atleast one from each of sections A and B. No credit willbe given for attempting any further questions. The examresult accounts for 90% of the marks for the course.

Books

There are many good general books on solid state physicsand specialised books on semiconductors and supercon-ductors. The Library classifications are:

Solid State Physics 530.41Semiconductors 537.622

1. A good starting point is the book for PH2710: SolidState Physics J R Hook and H E Hall, 1st Ed., J Wiley,1991. Library class: 530.14.HOO (6 copies: 2 normalloan, 4 one week loan).2. Semiconductor science: growth and characterizationby Tudor E Jenkins, Prentice Hall, 1995. Library class:

537.622 JEN (4 copies - all one week loan). This is an ex-cellent book about the properties of real semiconductorsand their growth and characterisation. It also includes agood treatment of the physics of semiconductors and theprinciples and techniques used.3. Introduction to Solid State Physics by C Kittel, Wiley,7th edition, 1996. Library class: 530.41 KIT (3 copies -all one week loan). There is also an 8th edition of thisclassic text (3 copies - all one week loan) which is ≈ £60.It covers a much wider range of solid state physics thanin PH371 and is very useful for more advanced coursesand as a reference book should you work in this area aftergraduation.

3

I. INTRODUCTION

A. Four Classes of Conductor

To answer the question what is a semiconductor? welook at the resistivity ρ of materials, which is observedto vary by almost 30 orders of magnitude.Four classes of conductor can be identified:1

1. Metals/semimetals

+ Resistivity at room temperature 10−8 < ρ(T ) <10−6 Ωm. Example: copper ρ = 1.7× 10−8 Ωm.

+ Impurities increase ρ(T ).

+ ρ(T ) decreases as T decreases.

+ ρ(0) = 0 for an ideal (pure) metal.

+ ρ(0) > 0 for a metal with impurities (residual re-sistivity).

+ Carrier concentration about 1 electron/atom (>1028 m−3). Metals are classified as conductors be-cause their outer electrons are not tightly bound.

+ A semimetal has a small overlap in the energy of theconduction band and valence bands. However, thebottom of the conduction band is typically situatedin a different part of momentum space than the topof the valence band. Typical example, graphite:ρ = 1012 Ωm

2. Semiconductors

+ Resistivity at room temperature ρ(T ) = 10−4 −1 Ωm.

+ Impurities decrease ρ (in general).

+ ρ(T ) increases (usually) as T decreases.

+ ρ(0) = ∞ for lightly doped semiconductor.

+ Carrier concentration ¿ 1 electron/atom (<1023 m−3).

+ Carriers can be +ive (holes) or -ive.

+ Basic characteristic is an energy gap. Evidencefor a gap: Variation of optical absorption coefficientwith photon energy - see Figs. 1 and 2. Some energygaps are given in Table I; a typical value is 1-3 eV.

1 Hook and Hall Chapter 5, Kittel Chapter 8.

FIG. 1 Evidence for a gap: the variation of the absorptioncoefficient α with photon energy. Note that GaAs has a directgap, whereas Si and Ge are indirect gap semiconductors.

valence band edge

conduction band edgeωg

E

k0

valence band edgeω

E

k0

Ω

kc-kc

onset of directphoton transition

Adsorption

Transparentregion

photon energy ħωħωg

onset of indirectphoton transition

Adsorption

photon energy ħωE +g ħΩ Evert

onset ofdirect transition

FIG. 2 Direct and indirect energy gaps.

TABLE I Band gaps (in eV) of common semiconductors.

Group IV Group III-V Group II-VI Group IV-VI

Si 1.11 GaAs 1.43 ZnTe 2.15 PbS 0.43

Ge 0.66 InP 1.35 CdTe 1.50 PbTe 0.29

SiC 2.31 GaP 2.26 ZnSe 2.67

AlAs 2.16 CdSe 1.70

GaSb 0.73 ZnO 3.2

InSb 0.81 CdS 2.42

GaN 3.39

4

3. Insulators

+ No appreciable conduction at room temperature(like semiconductors with a very large gap). Mostmaterials are insulators, because even the outer-most electrons are so tightly bound that there isessentially zero electron flow through them withordinary voltages. Examples: quartz (fused), ρ =5 × 1016 Ωm; glass, ρ = 1012 Ωm, teflon (PTFE),ρ = 1022-1024 Ωm.

4. Superconductors

+ Zero resistivity below a transition temperature Tc.

+ Energy gap.

+ Meissner Effect (exclusion of magnetic flux).

+ Many special properties (see other part of PH3710).

FIG. 3 Periodic table with outer electron configurations.

B. Which elements/compounds are semiconductors?

• Group I, II, III: metals (and superconductors).

• Group VII and VIII: insulators.

• Semiconductors:Group IV elements: C (diamond), Si, Ge, α-SnIII-V compounds: GaAs, GaP, InSb, InAsII-VI compounds: CdTe, CdS, ZnSe, etc.Ternary compounds: AlxGa1−xAs

5

TABLE II Crystal structure and lattice constant a (innanometres) of some common semiconductors.

Diamond Zinc blende

Si 0.5431

Ge 0.5657

GaP 0.5447

AlAs 0.5639

InP 0.5869

GaAs 0.6554

GaSb 0.6095

InSb 0.6479

C. Crystal structure of semiconductors

Table II shows that the elemental Group IV semicon-ductors adopt the diamond structure. The diamondstructure, see Fig. 4(a), is a cubic (Bravais lattice is face-centred cubic, fcc) with a basis of 2 atoms. There are2 atoms per primitive unit cell, and 8 atoms per cubicunit cell. The centre-to-centre distance between atomsis d = a

√3/4, where a is the side of the cubic unit

cell. From the Periodic Table in Fig. 3, we see that theelectronic structure of Si, a typical semiconductor withZ = 14, is 1s22s22p63s23p2. The group IV elements arecovalent bonded with 4 valence electrons/atom. Elec-trons are “shared” between neighbouring atoms to formcovalent bonds, see Fig. 5, hence the electrons are tightlybound and there are no free carriers at low temperatures.Because of the covalent bonding, each atom has fournearest neighbours in a tetrahedral coordination. Thediamond structure is very open, the densities of Si andGe are 2.33 g/cc and 5.32 g/cc, respectively.

The zinc blende structure2 shown in Fig. 4(b) isadopted by many of the compound semiconductors. InGaAs the atoms occupy positions similar to those in Si,except they alternate between Ga and As. The III-V andII-VI compounds are partly covalent and partly ionic incharacter, this can be seen in the valence charge densitycontours of GaAs compared to those of Si, both shownin Fig. 6.

Other semiconductors, for example II-VI compoundssuch as ZnS, CdS, and CdSe adopt the wurtzite struc-ture, which is a hexagonal rather than cubic crystal sys-tem.

2 Hook and Hall p. 18, Kittel p. 20.

FIG. 4 (a) Diamond lattice; (b) zinc blende lattice. In (a)the atoms marked R make up a puckered hexagonal ring.

FIG. 5 (a) Face-centred cubic unit cell of silicon (2 atoms percell). Each atom is bound to four neighbours, e.g., (0) to (1),(2), (3) and (4). (b) Diagrammatic representation of the fourcovalent bonds each with a pair of electrons.

FIG. 6 Top: valence charge density in Si - experiment andtheory. Bottom: valence charge density in GaAs.

6

TABLE III Static relative dielectric constant of semiconduc-tors.

Crystal ε Crystal ε

Diamond 5.5 GaSb 15.7

Si 11.7 GaAs 13.13

Ge 15.8 AlAs 10.1

InSb 11.7 AlSb 10.3

InAs 14.55 SiC 10.2

D. Doping in semiconductors

A pure, or undoped semiconductor is known as an in-trinsic semiconductor. If the semiconductor is dopedwith substitutional impurities, it is then known as anextrinsic semiconductor.

Acceptors from Group III: B, Al, Ga, InDonors from Group V: P, As, Sb

Electrons/holes are bound to these charged impuritiesand have hydrogenic energy levels and ionisation ener-gies Ry×m∗/m× 1/ε2, where the Rydberg energy (Ry)is given by

Ry =me4

8ε20h

2= 13.6 eV;

m∗ is the effective mass, and ε is the dielectric constant.ε = 11.7 for Si and ε = 15.8 for Ge, and Table III givesthe relative dielectric constant of other semiconductors.For P doped into Si (written as Si:P) the ionisation en-ergy is 45 meV, which is much less than the band gapEg. This ionisation energy is equivalent to a tempera-ture T = 540 K, and so at room temperature most ofthe impurity states will be ionised. ND is defined asthe donor concentration, and NA as the acceptor con-centration. Semiconductors with both donors and ac-ceptors are compensated. If all the levels are ionisedthen the electron concentration will be n = ND −NA (ifND > NA) or the hole concentration will be p = NA−ND

(if NA > ND). The compensation ratio K is definedas

K = ND/NA.

Majority carrier denotes the carrier which is primar-ily responsible for current transport in a piece of semi-conductor. If an intrinsic semiconductor is doped withn-type impurities then the majority carriers are electronsand the minority carriers are holes. If the semicon-ductor is doped with p-type impurities then the majoritycarriers are holes and the minority carriers are elec-trons.

Table IV gives the ionisation energies of some impu-rities in Si, Ge and GaAs - these impurities have smallionisation energies (< 0.1 eV) and are known as shallow

FIG. 7 Left figure: (a) Qualitative temperature dependenceof the concentration n of electrons in the conduction band ofan n-type semiconductor for two different donor concentra-tions. (b) Qualitative temperature dependence of the Fermienergy EF (T ) in the same semiconductor. Right figure: Theconcentration n of the free electrons in n-type Ge measuredusing the Hall effect (Sec. IV.C). For samples (1)-(6), thedonor concentration varies between 1018 and 1013 cm−3. Thetemperature dependence of the electron concentration in theintrinsic region is shown by the dashed line.

TABLE IV Ionisation energies in eV of some impurities in Si,Ge and GaAs. The acceptor (a) ionisation energy is measuredfrom the valence band edge EV , and the donor (d) from theconduction band edge EC .

Si Ge GaAs

Li (d) 0.033 Li (d) 0.0095 Si (d) 0.0058

P (d) 0.045 Sb (d) 0.096 Si (a) 0.035

As (d) 0.049 P (d) 0.012 Ge (d) 0.006

Sb (d) 0.039 As (d) 0.013 Ge (a) 0.04

impurity levels. There are also deep levels whichlie much closer to the centre of the gap - these can begroup II or VI impurities or transition metal atoms suchas chromium, Cr. These can play an important role inoptical properties as recombination centres, and also canbe used to make highly resistive substrates.

7

II. ELECTRONS IN SEMICONDUCTORS

A. Free electron model

Currents are carried by electrons, and the free electronmodel3 (FEM) is a good starting point for discussingtransport properties. In the FEM the electrons are non-interacting point charges in a box where the potential en-ergy is zero (or constant). Each electron state is specifiedby its energy E (which is just kinetic energy for free elec-trons) and its momentum p = hk, where k = 2π/λ is thewavevector. For non-relativistic free electrons p = mv,where v is the velocity. Hence the energy E is

E(k) =mv2

2=

p2

2m=

h2k2

2m, (2.1)

and the dispersion curve E(k) is parabolic, as shown inFig. 8(a). For free electrons in a sample of volume V , thenormalised wavefunction is

ψk =1√V

eikr. (2.2)

The number of electron states (including spin) in anenergy range from E → E + dE is g(E) dE, where g(E)is the density of states per unit energy range per unitvolume, and is given by

g(E) =1

π2h3 (2m3E)1/2 = CE1/2. (2.3)

Electrons are fermions and the occupation probability fora state of energy E is given by the Fermi-Dirac distribu-tion function

f(E) =1

exp[(E − EF )/kBT ] + 1. (2.4)

For N free electrons at T = 0 K, the electron states areoccupied up to the Fermi energy

EF =h2k2

F

2m= kBTF , (2.5)

where kF = (3π2n)1/3, the electron concentration isn = N/V , and TF is the Fermi temperature. The Fermienergy at any temperature T is defined by the condition(see Fig. 9)

n =∫ ∞

−∞N(E) dE =

∫ ∞

−∞f(E) g(E)dE. (2.6)

3 Hook and Hall, Chapter 3, Kittel, p. l46-151.

FIG. 8 (a) Plot of E(k) for free electrons. (b) E(k) for anelectron in a monatomic linear lattice of lattice constant a.The energy gap Eg is associated with the first Bragg reflectionat k = ±π/a.

FIG. 9 Calculation of the density of occupied electron statesN(E). (a) The Fermi-Dirac distribution f(E), (b) the densityof states g(E), and (c) N(E) = f(E)× g(E).

B. Electrons in a periodic potential

Electrons in crystals move in the potential of theatoms, which is periodic.4 The electrons interact with thelattice and can be Bragg reflected. The electron statesare described by the dispersion relation (or band struc-ture) E(k), which can depart from free electron parabolicbehaviour (see Fig. 8(b)). One approach5 to this is toconsider the atomic energy levels for each atom. As acollection of atoms are brought together each energy leveldevelops into a band of levels within an energy range orbandwidth. These bands may or may not overlap, andmay be full or partly full.

In semiconductors, there is an energy gap betweenthe valence band and the conduction band. How-ever, the actual band structure is quite complicated. Thesituation is similar to the phonon spectrum m(q), or dis-

4 Hook and Hall, Chapter 4, Kittel, Chapter 7.5 The tight binding approximation.

8

FIG. 10 The first Brillouin zone of the fcc lattice. The pointsof high symmetry, Γ, X, and L, are indicated.

persion relation, for lattice vibrations. Just as for thephonons the band structure for a 1D crystal is plotted inthe Brillouin zone, −π/a < k < π/a, where a is the lat-tice spacing. On the Brillouin zone boundary, k = ±π/a,the wavelength of the phonon or electron is λ = 2a andstrong Bragg reflection occurs. For the face-centred cubic(fcc) crystal structure, the three-dimensional Brillouinzone is the truncated octahedron shown in Fig. 10.6

At T = 0 K, the valence band states are all full and theconduction band states are empty. Therefore no conduc-tion can occur and the conductivity is zero, σ = 0. Thewavefunction of an electron in state E(k) can be writtenas

ψk(r) =1√V

uk(r) eikr, (2.7)

where uk(r) is a function which is periodic in the crystalstructure and repeats in each unit cell. The result thatthe wavefunction can be written in the form of Eq. 2.7for a periodic system is called Bloch’s theorem.

C. The effective mass approximation

Fortunately a very simple approximation can be used,based on parabolic energy bands.7 For GaAs the energyEe of an electron state just above the conduction bandedge at EC can be written as

Ee(k) = EC +h2

2 me(k2

x + k2y + k2

z) = EC +h2k2

2 me, (2.8)

where ~k = (kx, ky, kz), and me is the electron effectivemass whose value depends on the band structure. The

6 See Kittel, p. 437 Hook and Hall p. 131, Kittel p. 212.

FIG. 11 Typical E(k) of a III-V semiconductor showing lighthole, heavy hole, and split-off hole bands. The valence bandstructure for the elemental semiconductors is similar.

FIG. 12 Dispersion curves E(k) for (a) GaAs and (b) Si. InGaAs, the minimum of the conduction band and the maxi-mum of the valence band are at the centre of the Brillouinzone; it is a direct gap semiconductor. Si has an indirectband gap.

contours of constant energy are spherical.The energy of an electron state just below EV , the top

of the valence band, can be written as

Ee(k) = EV − h2k2

2 mh(2.9)

which is equivalent to a state with negative effectivemass! Since a full band does not contribute to conduc-tion it is better to work with the empty states, calledholes, which can be regarded as positive carriers withpositive effective mass. In the semiconductors consid-ered here there are actually three hole bands, known aslight holes, heavy holes and split-off holes - each with itsown effective mass (mlh, mhh, and mso).

In silicon, the situation for electrons is more compli-cated, as the minimum in the conduction band occurs

9

FIG. 13 Constant energy surfaces in the conduction bands ofGe and Si. For Ge the dashed parts of the ellipses of revolutionare outside the first Brillouin zone. For Si the six equivalententire conduction band minima are inside the first Brillouinzone.

TABLE V Electron and hole effective masses near band ex-trema for some common semiconductors.

Si Ge GaAs InP GaP

mL 0.916 1.59 - - 7.25

mT 0.1905 0.0815 - - 0.21

me - - 0.067 0.079 -

mhh 0.54 0.28 0.45 0.65 0.67

mlh 0.15 0.04 0.082 0.12 0.17

mso 1.06 0.09 0.15 0.12 -

away from the band centre, along the 〈100〉 directions.8We can write the electron energy near the conductionband minimum along 〈100〉 as

Ee(k) = EC +h2

2 mL(kx−kmin)2+

h2

2 mT(k2

y+k2z), (2.10)

where kmin is the position of the energy minimum alongthe kx axis. The effective mass depends on the direc-tion of motion of the electron and is anisotropic: mL

is the longitudinal mass and mT is the transverse mass.Table V gives values for other semiconductors. In ger-manium, the minima in the conduction band energies liealong the 〈111〉 directions in k-space.9

D. Metals vs Semiconductors

The key difference between a metal and a semiconduc-tor is the presence of a gap in density of states (DOS)in the latter. In metals the DOS is continuous and atT = 0 K electronic states are filled up to the Fermi en-ergy EF , and for finite temperatures the product of the

8 Kittel p. 214.9 Kittel p. 215.

FD distribution and the DOS (see Fig. 9) describes howthe states near EF become available.

In an intrinsic semiconductor such a description is com-plicated by the presence of the gap Eg between the va-lence and conduction bands. At T=0 K, all the states inthe valence band are completely full, while those in theconduction band are completely empty; the Fermi energylies mid gap, that is at EF = EV +Eg/2 = EC−Eg/2. Atfinite temperature the picture (see Fig. 9) which showshow states are counted in the FEM fails because elec-tron (hole) states become available close to EC (EV ),but far away from EF . Section III shows that in an in-trinsic semiconductor electrons and holes are thermallyactivated in equal quantities.

Note: in a semiconductor the band gap Eg ∼ 1 eV ismuch bigger than the thermal energy (kBT ∼ 25 meVat 300 K). Intrinsic: for an undoped semiconductor theFermi energy stays roughly mid gap for all tempera-tures. Extrinsic: for an n-type (p-type) semiconductorthe Fermi energy lies close to the bottom (top) of theconduction (valence) band.

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III. CARRIER CONCENTRATIONS

A. Intrinsic semiconductors

The effective mass approximation can be used to modela semiconductor; it is based on spherical conduction andvalence bands, with effective masses me and mh, and anenergy gap Eg = EC − EV .10 In this picture the freeelectron density of states (Eq. 2.3) is modified so thatm → me, and the energy is measured from the bottomof the conduction band at EC :

ge(E) =√

2 m3/2e

π2 h3 (E − EC)1/2

= Ce (E − EC)1/2.

(3.1)

The Fermi-Dirac function, which applies to electrons, is

fe(E) = [exp[(E − EF )/kBT ] + 1]−1

≈ exp[−(E − EF )/kBT ],(3.2)

where the latter approximation is made in the non-degenerate regime, |E − EF | À kBT .The total electron concentration n is given by11

n =∫ ∞

−∞fe(E) ge(E) dE

= Ce

∫ ∞

EC

(E − EC)1/2 exp[−(E − EF )/kBT )] dE

≈ NC exp[−(EC − EF )/kBT ],(3.3)

where NC = 2(

mekBT2 π h2

)3/2is the effective density of

electron states. For the hole states, the probability offinding an empty state at an energy E in the valenceband is given by fh, where

fh = 1− fe = 1− 1exp[(E − EF )/kBT ] + 1

≈ exp[(E − EF )/kBT ].(3.4)

Hence the hole concentration p is given by

p = NV exp[−(EF − EV )/kBT ], (3.5)

where NV is the effective density of hole states.Figure 14 shows a model where the conduction band

is replaced by NC states at energy EC , and the valenceband is replaced by NV states at energy EV , where

NC = 2(

mekBT

2 π h2

)3/2

NV = 2(

mhkBT

2 π h2

)3/2

10 Hook and Hall p. 139, Kittel p. 216.11 Use x2 = (E − EC)/kBT and

∫∞0 x2 e−x2

dx =√

π/4.

FIG. 14 Intrinsic semiconductor at T = 0 K, where EF liesexactly midway between the valence and the conduction band.The band gap is Eg = EC − EV .

and

NCNV = 4(

kBT

2 π h2

)3

(memh)3/2. (3.6)

Note that NC and NV are temperature dependent. Theexpressions for n and p in Eqs. 3.3 and 3.5 can be multi-plied to obtain

np = NCNV exp[−Eg/kBT ]. (3.7)

This is a very important equation, relating the electronand hole concentrations in a semiconductor. It is truefor both intrinsic and extrinsic semiconductors (exceptat very high carrier concentrations in degenerate semi-conductors) and is known as the law of mass action.

At room temperature, T = 300 K, with me = mh = mwe find

NC = NV = 2.5× 1025 m−3.

In any semiconductor there must be charge neutrality,for an intrinsic semiconductor this gives

ni = pi = (NCNV )1/2 exp[−Eg/2kBT ], (3.8)

where ni and pi are the intrinsic electron and hole con-centrations. Note that ni and pi depend on T , me, mh

and the energy gap Eg. The intrinsic carrier concentra-tions for Si, Ge and GaAs at room temperature are:

Eg (eV) ni (m−3)

Ge 0.67 2.4× 1019

Si 1.1 1.5× 1016

GaAs 1.43 1.8× 1013

Making Eqs. 3.3 and 3.8 equal to each other, the po-sition of the Fermi level in an intrinsic semiconductor isgiven by

EF =12(EC + EV ) +

34kBT ln

(mh

me

). (3.9)

11

FIG. 15 Band diagram, density of states, Fermi function andcarrier concentration as functions of energy for an intrinsicsemiconductor.

At T = 0 K, the Fermi energy lies exactly midway be-tween the valence and conduction bands, see Fig. 15. Ifme = mh, then EF = (EC + EV )/2 at all temperatures.For a semiconductor with heavy and light holes,

NV = 2(mhh + mlh)3/2

(kBT

2πh2

)3/2

. (3.10)

For Si, with six conduction band minima (MC = 6)

NC = 2MC [m1/2L mT ]

(kBT

2πh2

)3/2

(3.11)

B. Extrinsic semiconductors

In a pure (undoped) semiconductor the intrinsic carrierconcentration ni is very small, for example, for Ge at300 K it is ni = 2.4 × 1019 m−3. The atomic density ofGe is 4.4× 1028 m−3, therefore 1 ppm of impurities cangive a carrier density of 4.4 × 1022 m−3, which is muchgreater than ni. Even for light doping, the impuritieswill dominate the electrical transport properties.

In an n-type semiconductor, the pure semiconduc-tor is doped with substitutional impurities that have oneextra electron compared to the host; the donor concentra-tion is ND.12 The donor impurities create a new energylevel at ED, which lies an energy Ed = EC − ED belowthe bottom of the conduction band.

There are two equations that determine carrier densi-ties of electrons (n) and holes (p) in an n-type semicon-ductor.

From charge neutrality n = p + ND (3.12)

And from Eq. 3.7 np = n2i (3.13)

12 In general, not all donors give up their extra electron, that is,ND = N+

D + N0D, where N+

D is the ionised donor concentrationand N0

D is the neutral donor concentrations. For simplicity we

will assume that all donors are ionised, therefore ND = N+D .

FIG. 16 Extrinsic semiconductor; n-type doping introducesan energy level just below EC . Note that the donor activationenergy Ed = EC−ED has the property Ed ¿ Eg. At T = 0 K,EF lies midway between the donor level and the conductionband.

Summary: in an intrinsic semiconductor there is only oneenergy scale: the band gap Eg. In an extrinsic semicon-ductor there are two energy scales, Eg and Ed, whereEg À Ed. Therefore with temperature there are threedifferent regimes to consider in a doped semiconductor:

Intrinsic high temperature kBT > Eg n ≈ ni

Saturated room temperature kBT > Ed n ≈ ND

Freeze-out low temperature kBT < Ed n ¿ ND

1. Intrinsic ↔ saturated regions

From Eqs. 3.12 and 3.13 we obtain

n =n2

i

n+ ND. (3.14)

At high temperatures, many electrons (and holes) can becreated by thermal excitation across the band gap Eg.⇒ ni À ND, and from Eq. 3.14 we obtain n = ni, thisis the intrinsic regime. At intermediate temperatures,all the electrons are thermally excited out of the donorlevel into the conduction band. ⇒ ni ¿ ND, and fromEq. 3.14 we obtain n = ND; this is the saturated regime.

2. Saturated ↔ freeze-out regions

At intermediate temperatures, kBT ¿ Eg, the systemcan be analysed using the same model as before but withthe energy levels at the donor energy ED and at the con-duction band energy EC , as shown in Fig. 16. If thedonor activation energy Ed ¿ kBT then all the donorswill be ionised and we again obtain the expressions forthe saturation region:

n = ND Majority carriersp = n2

i /n = n2i /ND Minority carriers

At lower temperatures where kBT ¿ Ed, then thesemiconductor is in the freeze-out region. We can di-rectly use Eq. 3.7 by replacing the hole concentration

12

FIG. 17 Upper: The excitation of carriers in a semiconductoras a function of temperature (typical values for a specimen ofdoped germanium). Lower: The variation with temperatureof majority and minority carrier concentration, and mobilityµ and conductivity σ for a semiconductor.

p with the concentration of ionised donors ND (thoughthese are not mobile, of course) and by replacing Eg byEd we obtain

nND = NCND exp(−Ed/kBT )

Since n = ND we have at low temperatures

n = (NCND)1/2 exp(−Ed/2kBT ), (3.15)

which is similar to Eq. 3.8 for excitations across the band-gap in an intrinsic semiconductor. The Fermi level at lowtemperature is given by

EF =12

(EC + ED) +12

kBT log(ND/NC). (3.16)

At T = 0 K, EF lies midway between the donor level andthe conduction band.

For Si:P (3 × 1020 m−3 donors) the three differentregimes are:

Intrinsic T > 500 KSaturated 45 K < T < 500 KFreeze-out T < 45 K

In a compensated semiconductor with both donorsand acceptors, charge conservation gives n + NA = p +ND. n can be obtained by solving the following quadraticequation in n

n =n2

i

n+ ND −NA, (3.17)

and p can be obtained from np = n2i .

13

FIG. 18 Band diagram, density of states, Fermi function andcarrier concentration as functions of energy for an n-typesemiconductor.

FIG. 19 Band diagram, density of states, Fermi function andcarrier concentration as functions of energy for a p-type semi-conductor.

14

IV. CONDUCTION IN SEMICONDUCTORS

A. The Drude Model

The force on a charge q in an electric field E and amagnetic field B is

F = qE + qv×B−m∗vτ

= m∗ dvdt

, (4.1)

where τ is a relaxation or scattering time; the termm∗v/τ represents a “frictional” or resistive force. Thevelocity v can be either the velocity of an individual elec-tron or the average drift velocity of the carriers. At B = 0and in the steady state, dv/dt = 0, the mean drift ve-locity 〈v〉, the mobility µ, the conductivity σ and thecurrent density J = n q v are

〈v〉 =q〈τ〉m∗ E = µE where µ =

〈v〉E

=q 〈τ〉m∗ =

e τ

m∗

J = σ E σ = n q µ =n q2 〈τ〉

m∗ (4.2)

For electrons (charge -e) and holes (charge +e) the con-ductivities add

J = (σe + σh)E = σ E, where σ = n eµe + p e µh.(4.3)

B. The mobility µ

In a pure crystalline semiconductor (or metal) at T =0 K there is no scattering. However in a real semicon-ductor at finite temperatures there is scattering due tolattice vibrations (phonons) and defects (ionised impu-rities). For Ns scattering centres per m3 with a cross-section As, the mean free path ` of electrons with avelocity v is ` = vτ = 1/NsAs.The mean thermal velocity < vT >=

√3kBT/m ∝ T 1/2

⇒ scattering rate: 1/τ ∝ 1/µ ∝ T 1/2 Ns As

1. Charged impurities

At low temperatures scattering from charged impuri-ties gives Rutherford scattering:

Aimp ∝ 1/v4T ∝ 1/T 2 1/τ ∝ T−3/2Nimp

⇒ µimp ∝ T 3/2/Nimp.

At very low temperatures, freeze-out will occur.

FIG. 20 Temperature dependence of the mobility µ for asemiconductor in which there is scattering from phonons andcharged impurities.

TABLE VI Carrier mobilities at room temperature, incm2/Vs.

Crystal Electrons Holes Crystal Electrons Holes

Diamond 1800 1200 GaAs 8000 300

Si 1350 480 GaSb 5000 1000

Ge 3600 1800 PbS 550 600

InSb 800 450 PbSe 1020 930

InAs 30000 450 PbTe 2500 1000

InP 4500 100 AlAs 280 -

AlSb 900 400 SiC 100 10-20

2. Lattice vibrations

At high temperatures there is scattering from theacoustic phonons (lattice vibrations). For non-polarsemiconductors

Aph ∝ kBT 1/τph ∝ T 3/2

=⇒ µph ∝ T−3/2,

where µph is known as deformation potential mobility. Inpolar semiconductors (for example, GaAs) there is also astrong coupling to the optic phonons which can dominatethe mobility at room temperature. Adding scatteringrates using Matthiesen’s law gives

1τ

=1

τimp+

1τph

. (4.4)

Electrons and holes will have different mobilities, andusually µe > µh, see Table VI.

C. Hall effect and magnetoresistivity tensor (Drude model)

In an electric field E and a magnetic field B the equa-tion of motion, Eq. 4.1, for the steady state drift velocity

15

FIG. 21 Experimental temperature dependence of the mobil-ity µ of free electrons in n-type Ge. For samples (1) to (6), thedonor concentration ND varies between 1018 and 1013 cm−3.

becomes (put v = J/nq)

E = ρ0(J + sµB× J) = ρJ, (4.5)

where s = +1 for positive charges and s = −1 for neg-ative charges, ρ0 = 1/σ0 = 1/n|q|µ, and µ and ρ arealways taken as positive quantities. For a magnetic fieldalong the z-axis, the resistivity tensor ρ can be writtenas

ρ = ρ0

1 −sµB 0sµB 1 0

0 0 1

. (4.6)

This relates the components of the current density tothose of the electric field. In this simple model the mag-netoresistivity is independent of magnetic field, sincefor a current along the x-axis only (as in a Hall bar orwire) we have

Ex = ρxx Jx = (1/σ0)Jx, (4.7)

where σ0 = neµ is the zero-field conductivity (now usingthe electronic charge e = |e|). The electric field perpen-dicular to the current, the Hall field EH , is given by

EH = Ey = ρyxJx =sB

neJx. (4.8)

FIG. 22 Experimental conductivity σ of n-type Ge as a func-tion of temperature. For the samples (1) to (6) the donorconcentration ND varies between 1018 and 1013 cm−3.

The Hall coefficient is defined as RH = EH/JxB, hence

RH =s

ne. (4.9)

Note the Hall coefficient RH is positive for holes andnegative for electrons. The geometry of the fields andcurrents is shown in Fig. 23, where the electric field is atthe Hall angle φH to the current, which is given by

tanφH =ρyx

ρxx= µB = ωcτ, (4.10)

where ωc = eB/m∗ is the cyclotron frequency.

D. The Hall effect for two carriers

In semiconductors with electrons (concentration n, mo-bility µe) and holes (concentration p, mobility µh), theconductivity is given by Eq. 4.3. The Hall coefficient isthen given by

RH =1e

(p µ2h − nµ2

e)(p µh + nµe)2

. (4.11)

If the mobilities are equal, then the Hall coefficient equalszero for n = p. Note that a negative Hall coefficient can

16

FIG. 23 Geometry of the Hall effect.

FIG. 24 Schematic setup for a Hall effect measurement. B isthe magnetic field, i is the current through the sample, andUH is the measured Hall voltage.

be obtained even in p-type semiconductors if the holemobility is much lower than the electron mobility, whichis often the case. This is also the reason why the Hallcoefficient is usually negative in intrinsic semiconductorswhere the carrier concentrations are equal. The Hall co-efficient is also modified if the electron relaxation time isenergy dependent and for an anisotropic effective mass,as in Si and Ge.

E. Electrical measurements

Several techniques are used to measure the resistivityand Hall effect of semiconductors. Three common onesare four-terminal measurements, the four-point probeand the Corbino disk.

1. Four-terminal measurements

A four-terminal arrangement is shown in Fig. 26. A DCcurrent I is passed through the sample and the voltagesbetween contacts 1 and 2 (VR) and contacts 1 and 3 (Hallvoltage VH) are measured. The current is determinedfrom the voltage VSR across a standard resistor RS . The

FIG. 25 Temperature dependence of the Hall constant RH

for (a) p-type and (b) n-type silicon. For p-type with a boronconcentration of 2× 1017 cm−3, intrinsic conductivity sets inat ∼ 1300 K.

FIG. 26 Four-terminal measurement setup.

resistance R between contacts 1 and 2 is

R =VR

I=

a ρ

b c=

a

σ b c, (4.12)

where ρ and σ = ρ−1 are resistivity and conductivity ofthe material. a is the distance between the contacts, b isthe sample width, and c is the sample thickness.

The direction of the magnetic field B is also shownin Fig. 26, allowing measurement of the Hall coefficient,which is

RH =EH

JB=

VH b c

c I B=

VH b

I B, (4.13)

where EH is the Hall field (perpendicular to both the

17

FIG. 27 The linear four-point probe technique.

current I and the field B) and J is the current density.The Hall voltage contacts may not be exactly oppositeeach other and the measured VH will have contributionsfrom the longitudinal resistance as well as the transverseHall voltage. There will also be a thermoelectric volt-age V0 even with no current flowing. Hence, the voltagebetween the two contacts can be written

V (I,B) = V0 + I R + I B RH/b. (4.14)

The resistance in zero field is calculated from the changein VR when the current is reversed:

R = [VR(I, 0)− VR(−I, 0)]/2I. (4.15)

The Hall coefficient is calculated from the change in VH

when the field B is reversed:

RH = [VH(I, B)− VH(I,−B)]b/2IB. (4.16)

The resistance in a magnetic field R(B) is calculated fromthe change of VR, when both the current I and the fieldB are reversed:

R(B) = [VR(I, B)− VR(−I,−B)]/2 I. (4.17)

2. Four-probe measurements

The four-point probe resistance on the top surface of asemiconductor structure, see Fig. 27, having a verticallynon-uniform structure can be obtained by solving the

Laplace equation. The non-uniformity is modeled13 as aseries of distinct layers, each having a uniform resistivity.The result is complicated; for the case of a finite layerthickness x on an insulating substrate

V

I=

ρ(x)πx

ln[

sinh(x/s)sinh(x/2s)

]. (4.18)

In a thin layer of uniform resistivity ρ, where the separa-tion of the probes is very much greater than the thicknessof the layer, x ¿ s, this expression reduces to

V

I=

ρ ln 2πx

, (4.19)

and the sheet resistance is given by

ρs =ρ

x=

π

ln 2V

I. (4.20)

3. Corbino disk

A two terminal measurement of the conductivity σxx

is possible using the Corbino geometry shown in Fig. 28.

FIG. 28 Corbino disk. One electrode is at the centre of thedisc, and the other is around the circumference. Current flowsradially, so that a Hall field cannot be established.

13 P. A. Schumman, Jr. and E. E. Gardner, Journal of the Electro-chemical Society 116, 87 (1969).

18

V. OPTICAL PROPERTIES OF SEMICONDUCTORS

A. Optical properties

The interaction of light with semiconductors gives animportant and powerful technique for investigating theirproperties. Interactions occur over a wide range of fre-quencies. The main features shown in Fig. 29 are:

À The fundamental absorption region arises fromtransitions between the conduction and valencebands, with high absorption. A well defined ab-sorption edge at typically 1 eV determines the bandenergy gap.

Á Just below the band edge, structure is found in thespectrum due to bound electron-hole states knownas excitons.

 At longer wavelengths the absorption increasesagain due to free carrier absorption.

à At energies between 0.02 and 0.05 eV absorp-tion peaks occur due to interactions with thephonons. This is very strong in polar semicon-ductors (reststrahlen region).

Ä Impurity levels also give absorption lines, dueto the excited states and ionisation of donor andacceptor atoms.

Å Extra absorption lines are found, at low energies,in a magnetic field, due to cyclotron resonanceof the carriers. In magnetic semiconductors, spinwaves called magnons are also detected.

B. Optical constants

The dielectric function ε(ω, k) describes the responseof a crystal to an electromagnetic field; it depends on theelectronic band structure, and can be studied by opticalspectroscopy. In the infrared, visible and optical regimek is very small compared to the shortest reciprocal latticevector, k ¿ G, and is usually taken to be zero.

An electromagnetic wave propagating along the z -axisin a semiconductor has a perpendicular electric fieldwhich can be written as

E(z) = E0 ei (ωt−kz) = E0 eiωte−ik1ze−k2z, (5.1)

where k = k1 − i k2 is the complex wavevector of thelight. The intensity of the light varies I(z) = E0 E∗

0 =I0 exp(−az), where a = 2 k2 is the optical attenuationcoefficient. The speed of light in a non-magnetic semi-conductor is given by

v =c

n=

ω

k=

1√εε0µ0

and hence

n2 =c2

ω2k2 = ε,

(5.2)

FIG. 29 Absorption spectra of a hypothetical semiconductor.

which defines a complex dielectric constant or rela-tive permittivity ε = ε1 + j ε2, and a complex refrac-tive index n =

√ε = n1 + i n2 = ck/ω.

The reflectivity is measured when a plane wave of am-plitude Ei is normally (90) incident from vacuum (ε = 1,n = 1) onto a medium with dielectric constant ε. The re-flected wave has amplitude Er and the fraction of powerreflected is called the reflectivity

R =|Er|2|Ei|2 =

∣∣∣∣1−√ε

1 +√

ε

∣∣∣∣2

=|1− n|2|1 + n|2 =

(1− n1)2 + n22

(1 + n1)2 + n22

.

(5.3)If the refractive index is purely imaginary, n = i n2, thenR = 1, and there is 100% reflection of the incident wave.

C. A microscopic model

A simple model14 can be used to gain insight into opti-cal interactions. Consider a bound charge, mass m, witha natural resonance frequency of ω0. In an applied elec-tric field the equation of motion for the displacement rfrom equilibrium is

Force = qE −mγdr

dt−mω2

0r = md2r

dt2, (5.4)

where γ is a damping term representing an energy-lossmechanism, and mω2

0r is the restoring force. For a sinu-soidal electric field at an angular frequency ω we have

r =qE/m

(ω20 − ω2)− iγω

.

14 Jenkins, Chap. 5.

19

If there are N charges per unit volume15 then the polar-isation P (charge× displacement) produced is

P =(Nq2/m)E

(ω20 − ω2)− iγω

= χε0E = (ε− 1) ε0E.

The dielectric constant ε(ω) is frequency dependent andgiven by

ε(ω) = 1 +P

ε0E= 1 +

(Nq2/ε0m)(ω2

0 − ω2)− iγω. (5.5)

This simple model gives the following two limits for ω = 0and ω = ∞

ε(0) = 1 +Nq2

ε0mω20

and ε(∞) = 1, (5.6)

where ε(0) is the static dielectric constant. At lowfrequencies the charges follow the field, and there is aresonant response at ω = ω0. At high frequencies thecharges can no longer respond. This is a classical model,and is known as the Lorentz oscillator. The equivalentquantum result is

ε(ω) = 1 +(Nq2/ε0m)f

(ω20 − ω2)− iγω

, (5.7)

where ω0 = (Em − En)/h is the frequency for the tran-sition between two quantum states Em and En, and f isthe oscillator strength for this transition. In real ma-terials there will be many contributions to the dielectricconstant:

ε(ω) = 1 +∑

i

(Nq2/ε0mi)fi

(ω20 − ω2)− iγiω

. (5.8)

For any given mechanism this can be written as

ε(ω) = ε(∞) +(ε(0)− ε(∞))ω2

0

(ω20 − ω2)− iγω

, (5.9)

where ε(∞) expresses the contribution from all the othereffects.

Given the dielectric constant ε(ω) the refractive indexn, the wavevector k, the absorption coefficient and thereflection coefficient R can be calculated from Eqs. 5.2and 5.3. These quantities are shown in Fig. 30 for asingle oscillator with different values of ω0/γ. Figure 30shows reflectivity curves for a system of two oscillatorswith resonant frequencies ω1 and ω2.

15 In Chap. 3, N was the total number of electrons and n = N/Vwas the density. In this chapter n is the refractive index and Nis the density.

FIG. 30 Components of the complex permittivity ε = ε1+iε2

(top: left & right), and the complex refractive index n =n1+in2 (middle: left & right), as a function of frequency ω forthe case of a single oscillator: (a) ω0/γ = 25; (b) ω0/γ = 10;(c) ω0/γ = 5. The bottom figure shows reflection (top), at-tenuation (middle) and absorption (bottom) curves calculatedfor ω0/γ = 25. The absorption curve is displaced for clarity.

FIG. 31 Resultant reflectivity of system of two oscillators ofresonant frequency ω1 and ω2. (a) ω2/ω1 = 1.1; (b) ω2/ω1 =1.2; (c) ω2/ω1 = 1.5.

20

FIG. 32 Infrared reflectivity due to lattice vibrations in GaP.The continuous line is a calculated fit to a classical dampedsimple harmonic oscillator.

FIG. 33 The different phonon modes as a function of thewavevector k = 2π/λ. There are transverse acoustic (TA),longitudinal acoustic (LA), transverse optic (TO), and longi-tudinal optic (LO) phonon branches. The frequency of theoptic modes is typical 10 THz (50 meV).

D. Phonon absorption

The simplest application of this model16 is in polarsemiconductors, such as GaAs, where the bonding ispartly ionic and the atoms are effectively charged. Anelectric field applied to an ionic crystal causes a displace-ment of the positive and negative ions - thus affectingthe dielectric constant ε(ω). As a result there is a verystrong coupling between light and the transverse op-tic phonon at an angular frequency ωTO. The dielectricconstant, from Eq. 5.9, can be written as

ε(ω) = ε(∞) +(ε(0)− ε(∞)) ω2

TO

(ω2TO − ω2)− iγω

, (5.10)

where ε(∞) allows for the contributions from all the otheroscillators and interactions in the semiconductor. The

16 Jenkins p. 266, Hook and Hall p. 265

FIG. 34 Observed absorption spectrum for As impurities inGe.

damping is usually very low for this mode and henceε(ω) can be negative for ω > ωTO. The refractive indexn =

√ε is then purely imaginary, which gives 100%

reflectivity. As ω increases further, ε(ω) ≈ 0, and allelectric interactions are switched off. This corresponds tothe longitudinal optic phonon frequency at ωLO. Theregion between these two frequencies is one of very highreflectivity and is known as the reststrahlen region. Itdoes not exist in covalent semiconductors such as Si orGe.

Figure 32 shows the reflectivity for GaP - the re-strahlen region ωTO < ω < ωLO, (see Fig. 33 for thephonon dispersion curves) corresponds to a reflectivityof unity between 25-27 µm.

E. Shallow donors and acceptors

Another application is in the absorption of infrared ra-diation by shallow impurity sites. In a simple hydrogenicmodel the energy levels are given by

En = −R∗Hp2

R∗H =RHm∗

ε2RH = 13.6 eV,

(5.11)where p is the Bohr quantum number. For Ge, this wouldgive R∗H = 10 meV for all impurities, compared with14 meV for Ge:As, 9.8 meV for Ge:Sb and 12.8 meVfor Ge:P. Note that spin-orbit splitting (the interactionbetween the magnetic field produced by the orbital mo-tion and the electron spin) lifts the degeneracy of the 2pstate. The band structure of the Ge also modifies the

21

FIG. 35 Calculated (left, using hydrogenic model) and ob-served (right) energy levels for As impurities in Ge.

valence band edge

conduction band edgeωg

E

k0

valence band edgeω

E

k0

Ω

kc-kc

onset of directphoton transition

Adsorption

Transparentregion

photon energy ħωħωg

onset of indirectphoton transition

Adsorption

photon energy ħωE +g ħΩ Evert

onset ofdirect transition

FIG. 36 Direct gap and indirect gap semiconductors, andtheir associated photon absorption characteristics.

simple model.

F. Band-gap absorption

The absorption across the energy gap from the conduc-tion band to the valence band is the strongest intrinsicabsorption in most semiconductors and lies in the near-infrared through the visible to the ultraviolet. There aretwo types of absorption: direct gap and indirect gap.In each case momentum and energy must be conserved.However the momentum of the photon, hω/c is very smalland can be neglected. In direct gap materials (GaAs,InSb, etc.) a vertical transition occurs for hω ≥ Eg. An

FIG. 37 Optical absorption in pure indium antimonide, InSb.The transition is direct because both conduction and valenceband edges are at the centre of the Brillouin zone (k = 0).

FIG. 38 Variation of optical absorption coefficient with pho-ton energy. GaAs is a direct gap material.

electron-hole pair is created at a wavevector given by

hω = Eg +h2k2

2me+

h2k2

2mh= Eg +

h2k2

2mr, (5.12)

22

where mr is the reduced effective mass

1mr

=1

me+

1mh

.

The absorption coefficient αd is proportional to the den-sity of states and hence

αd = Cd (hω − Eg)1/2. (5.13)

In indirect gap materials a phonon is created or absorbedto conserve momentum:

hω = Eg + hΩ q = −kmin. (5.14)

In this case

αi = Ci (hω − Eg)1/2. (5.15)

G. Excitons

Just below the band-edge, structure is seen due to theformation of excitons, which are hydrogenic-like boundstates formed from an electron in the conduction bandand a hole in the valence band. Hence we have

hω = Eg +h2k2

2mr− R∗H

n2, (5.16)

where R∗H is now the effective Rydberg energy

R∗H =mre

4

8ε2ε20h

2.

H. Free carrier absorption

For mobile charge carriers we can find17 the opticalresponse from the dielectric constant in Eq. 5.5. In thiscase we put ω0 = 0 (the charges are not bound to anyposition) and γ = 1/τ , where τ is the relaxation or scat-tering time used in the conductivity to obtain

ε(ω) = 1 +(Nq2/ε0m)

(−ω2 − iω/τ)= 1− ω2

p

(ω2 + iω/τ). (5.17)

Here n is the carrier concentration (for n-type semicon-ductors) and ωp = (Ne2/ε0m)1/2 is the plasma fre-quency, which is the natural response frequency of thecharge carriers and is typically 1012 Hz. At room tem-perature ωpτ < 1 and free carrier effects are relativelysmall. At low temperature (and in metals), ωpτ À 1,and the dielectric constant is

ε(ω) = 1− ω2p

ω2ωpτ > 1. (5.18)

17 Jenkins p. 276

FIG. 39 An exciton is a bound electron-hole pair, usually freeto move together through the crystal. In some respects it issimilar to an atom of positronium, formed from a positronand an electron.

FIG. 40 Exciton levels in relation to the conduction bandedge, for a simple band structure at k = 0.

FIG. 41 Energy levels of an exciton created in a direct pro-cess.

If ω < ωp, then ε(ω) < 0 - which corresponds to an imag-inary refractive index and 100% reflectivity. As ω > ωp,the reflectivity falls rapidly. In semiconductors we mustadd the free electron response to the other mechanismsand the plasma frequency becomes ωp = (Ne2/εε0m)1/2,where ε can be taken as the static dielectric constant.

23

FIG. 42 Effect of an exciton level on optical absorption inGaAs at 21 K. The band gap Eg is 1.521 eV and the excitonbinding energy is 0.0034 eV.

FIG. 43 Reflectivity of a semiconductor in the plasma fre-quency regime.

I. Cyclotron resonance

In a magnetic field B, free electrons move in circularorbits in a plane at right angles to the field; the angu-lar frequency is ωc = eB/m, where ωc is the cyclotronfrequency. In semiconductors, the conduction electronsbehave similarly with the cyclotron frequency defined asω∗c = eB/m∗, where m∗ is the cyclotron effective mass.For spherical energy surfaces, appropriate for holes andelectrons in III-V semiconductors, m∗ = me or m∗ = mh.

Si is an indirect band gap semiconductor and theelectron energy surfaces are prolate spheroids18 orientedalong the < 100 > axes; these six pockets of electrons areshown in Fig. 44. The electron energy in the conductionband for the minimum along the kx axis is given

Ee(k) = EC +h2

2 mL(kx−kmin)2+

h2

2 mT(k2

x+k2y), (5.19)

where kmin is the position of the energy minimum along

18 “Pointy” like a zeppelin because mL > mT , not squashed likeoblate spheroids.

FIG. 44 Surfaces of constant energy for Si.

the kx axis. The cyclotron effective mass is anisotropic(two masses mL and mT ) and depends on the angle θ ofthe magnetic field relative to the longitudinal axis of theenergy surface

1m∗2 =

sin2 θ

mT mL+

cos2 θ

m2T

. (5.20)

For θ = 0, m∗ = mT , while for θ = π/2, m∗ =√

mLmT .Since there are six equivalent energy surfaces in Si, forany general direction of magnetic field there are threedifferent cyclotron effective masses from the electrons.

Equation 5.20 is derived assuming that the mag-netic field lies in the x-z plane and is given byB = B(sin θ, 0, cos θ). The equation of motion is ob-tained from the Lorentz force

hdkdt

= −ev×B, (5.21)

where the velocity is derived from

v =1h∇kE,

and is given by v = h(kx/mT , ky/mT , (kz − kmin)/mL).Equation 5.21 gives the following three equations

dkx

dt= − eB

mTky cos θ, (5.22)

dky

dt= − eB

mL(kz − kmin) sin θ +

eB

mTkx cos θ, (5.23)

anddkz

dt=

eB

mTky sin θ. (5.24)

These three equations can be used to derive Eq. 5.20.Figure 45 shows the absorption for silicon when θ = 30,from which the two cyclotron masses, mL and mT , canbe determined.

24

FIG. 45 Cyclotron resonance signal from silicon at 4 K usingν = 2.4 × 1010 Hz radiation. The magnetic field lies in the(100) plane and is at θ = 30 to the [001] axis. Note 10 kOe= 1 T.

25

VI. SOLID STATE ELECTRONICS

A. Inhomogeneous semiconductors

Semiconductors form the basis19 for most electronicdevices. A key feature in real devices is that the dopingis inhomogeneous - the simplest example of this is thep-n junction diode. In an inhomogeneous semiconductor,there is an extra mechanism of charge transport due tothe thermal diffusion of charges in a carrier concentra-tion gradient.

Initially when p-type and n-type semiconductors areplaced in contact the excess electrons in the n-type sidecan lower their energy by flowing into the empty statesin the p-type material, thereby charging it negatively.The charging produces an electric field that will tend tooppose the diffusion of further electrons - this electricfield is responsible for the energy levels on either sideof the junction being displaced relative to one another,Eventually a dynamic equilibrium will be reached - thisoccurs when the EF levels are coincident - a processcalled the principle of detailed balance. There will be asimilar process happening in the valence band, that is,there is a diffusion of holes from the p-type material intothe n-type. We will concentrate on the electrons, but willremember to add the hole effects.

B. Diffusion currents

Charged particles move in the gradient of a potentialV according to the equation

J = −n q µ∇V = n q µE,

where E = −∇V is the electric field. Particles also dif-fuse from high to low concentrations. For a concentra-tion gradient along the x-axis the diffusion current Jdiff

(along the x-axis) is given by

Jdiff = −qD∇n, (6.1)

where D is the diffusion constant. Hence the totalcurrent density J is

J = n q µE− q D∇n, (6.2)

where E is the electric field (and for electrons we putq = −e). In a semiconductor the electron density is givenby Eq. 3.3

n = NC exp[−(EC − EF )/kBT ],

where in an inhomogeneous semiconductor both EC(x)(known as band-bending) and EF (x) can depend on po-sition. Hence

∂n

∂x=

n

kBT

∂EF

∂x− n

kBT

∂EC

∂x.

19 Hook and Hall, Chapter 6

In an isolated semiconductor in equilibrium, the Fermilevel must be constant, therefore ∂EF /∂x = 0. ThenEq. 6.2 in 1D becomes

J = nq

µE − D

kBT

∂EC

∂x

(6.3)

But the local or internal electric field E seen by the elec-trons is given by the force from the position dependentenergy EC(x)

qE =∂EC

∂x.

In an isolated inhomogeneous semiconductor, the totalcurrent density J = 0, as the diffusion current produces acharge imbalance which gives an electric field and currentwhich balances the diffusion current. Hence from Eq. 6.3

D =kBT

qµ; (6.4)

this is the Einstein relation for diffusion. Typical val-ues for Ge at room temperature are

µe = 0.39 m2/Vs De = 1.01× 10−2 m2/sµh = 0.19 m2/Vs Dh = 0.39× 10−2 m2/s.

In an electric field and a concentration gradient the elec-tron and hole current densities will be

Je = eµe

nE +

kBT

e

∂n

∂x

and

Jh = eµh

pE − kBT

e

∂p

∂x

.

(6.5)

Note that the signs of the electron and hole diffusion cur-rents are different, as each diffuses down a concentrationgradient but have opposite charges.

For a semiconductor with an external applied voltage,the Fermi level will depend on position and the totalcurrent density is then

J = −nqD

kBT

∂EF

∂x= −nµ

∂EF

∂x= neµEext, (6.6)

where Eext is the electric field from the external voltage.Hence the effective conductivity is given by σ = neµas before. But in the semiconductor the current can bedriven by the internal electric field or by carrier diffusion.This applies to a semiconductor in equilibrium.

C. Diffusion length

An important parameter is the diffusion length Ldiff.Consider a region of semiconductor where excess carriers(say electrons) are injected (by photoionisation or in ap-n junction) to give a non-equilibrium carrier concen-tration ∆n(x) = n(x) − n0, where n0 is the equilibrium

26

concentration. These will diffuse away until they recom-bine with holes in a recombination time τr. For diffusionalong the x-axis

Jdiff = −qD∂∆n

∂x, (6.7)

where the diffusion current decreases as electrons recom-bine, to return the concentration back to the equilibriumvalue n0. In the region between x → x+dx the equationof continuity for the excess charge density is

Jdiff(x + dx) = Jdiff(x)− q∆n(x)dx

τr

rate out rate in recombination rate

which can written as

−∂Jdiff

∂x=

q∆n(x)τr

charge/unit area/sec

Hence

qD∂2∆n(x)

∂x2=

q∆n(x)τr

,

which has a solution

∆n(x) = ∆n(0) exp(−x/Ldiff), (6.8)

where Ldiff =√

Dτr =√

µkBTτr/e is the diffusionlength. For τr = 15 µs and for electrons in Ge at roomtemperature, Ldiff = 0.1 mm.The diffusion current for electrons, Eq. 6.7, can then bewritten as

Jdiff,e = −eD∂∆n

∂x=

eDe

Le∆n(x)

=eDe

Le∆n(0)e−x/Ldiff .

(6.9)

D. The p-n junction in equilibrium

A p-n junction is an example of an inhomogeneoussemiconductor.

Electrons leaving the n-type region leave behind staticpositively charged positive ions, and similarly on the p-type side there will be a region of negatively charged ac-ceptor ions. Therefore in the junction region there will betwo depletion layers containing fixed equal and oppositecharges - these can be considered to behave like a chargedcapacitor. If the function is at x = 0, the depletion layerextends to −lp and +ln into the p and n-type materialsrespectively. With no applied external voltage, the car-riers move so that the Fermi level is constant across thejunction. In the depletion region, the current from theelectric field balances the diffusion current at all points,

FIG. 46 Comparison of band profiles for a p-n junction whichis (a) unbiased; (b) forward based; and (c) reverse biased.

as in Section VI.B, with no net current flow in a dynamicequilibrium.Band bending occurs. Define Ei as the Fermi level inan intrinsic semiconductor. Then in the n-type region,from Eq. 3.3 we have

n = NC exp[−(EC − EF )/kBT ] = ni exp[(EF − Ei)/kBT ]= ni exp(e Φn/kBT ) = ND,

where ND is the donor concentration and Φn is measuredin volts. Hence

Φn =EF − Ei

e=

kBT

eln

(ND

ni

).

A similar expression is obtained for the p-type region,therefore the total internal potential Φ of the junctionis

Φ = Φn + Φp =kBT

e

ln

(ND

ni

)+ ln

(NA

ni

). (6.10)

The internal potential is less than the energy gap; in Si,Egap = 1.12 eV, for NA = 1022 m−3, ND = 4×1021 m−3,ni = 1.6× 1016 m−3, and Φ = 0.67 eV.The ionised impurity charge concentrations (immobile)in the depletion layer can be taken as ND and NA over

27

FIG. 47 Left: (a) Net charge density ρ(x), and (b) free chargedensity, in a p-n junction. Right: Electric field across anequilibrium p-n junction.

the depletion layer widths dn and dp. Charge neutralitygives

ND dn = NA dp = Qd. (6.11)

These charges act like a capacitor Cd with charge Qd =ND dn per unit area with a mean charge separation of(dn + dp)/2. The voltage across this capacitor is theinternal voltage

Φ =Qd

Cd=

eND dn (dn + dp)2εε0

=eNDd2

n

2εε0+

eNAd2p

2εε0= Φn + Φp.

Hence

d2n =

2εε0Φn

eNDand d2

p =2εε0Φp

eNA.

If ND = NA, then Φn = Φp = Φ/2, and the width of thedepletion layer is

W = dn + dp =√

4εε0ΦeND

. (6.12)

For a typical dopant level in Si, W ≈ 2µm. In the spacecharge region the Fermi level is far from the conductionand valence bands. Consequently the carrier concentra-tions are small (≈ ni).

E. A biased p-n junction

Consider a p-n device, across which a voltage V is ap-plied. Almost all the applied voltage will be droppedoccur across the junction itself, as the bulk material hasa very low resistance. In forward bias (V > 0) the p-type region is positive and the n-type region is negative;current then flows from p-type to n-type. The oppositeoccurs in reverse bias, V < 0, see Fig.46.

For many situations, a p-n junction can be treated asan abrupt junction in which the energy bands changediscontinuously at x = 0. The electron concentration at

FIG. 48 Energy band diagram at zero bias showing the num-ber of conduction band electrons able to cross from N → P .

FIG. 49 Energy band diagram at forward bias, showing anincreased number of conduction band electrons able to crossN → P .

x > 0 (in the n-type) is fixed at ND (majority carriers)so that the electron concentration at x = 0− (just inthe p-type) from the electrons which diffuse across thejunction is

np(x = 0−) = ND exp[−e(Φ− V )/kBT ]

= n0p exp(eV/kBT ),

(6.13)

where V is the bias voltage and n0p is the equilibrium

electron concentration (the minority carrier concentra-tion) in the p-type region. Hence there is injection ofminority carriers (for V > 0, forward bias) into thep-type region. This produces a diffusion current into thep-region where recombination will occur. This leads toan excess electron density ∆n(x) for x < 0 given by

∆n(x) = ∆n(0) exp(−|x|/Le), (6.14)

where Le is the diffusion length for electrons and

∆n(0) = n0p [exp(eV/kBT )− 1]

with n0p = n2

i /NA. The diffusion length is typically50 µm, much greater than the depletion layer thickness- hence the abrupt junction approximation is normallysatisfactory. The diffusion current of electrons is givenby

Jdiff = −qDdn

dx=

eDe

Le∆n(0)

=eDe

Len0

p [exp(eV/kBT )− 1],(6.15)

28

TABLE VII Carrier concentrations in a p-n junction (diffusion model)

Zero bias p-type n-type

Electron Minority carriers Majority carriers

concentrations n0p = n2

i /NA = ND e−eΦ/kBT nn = ND

Hole Majority carriers Minority carriers

concentrations pp = NA p0n = n2

i /ND = NA e−eΦ/kBT

Bias V p-type n-type

Electron Diffusing minority carriers Majority carriers

concentrations at x = 0 np(0) = n0p eeV/kBT nn = ND

Hole Majority carriers Diffusing minority carriers

concentrations at x = 0 pp = NA pn(0) = p0n eeV/kBT

where De = µekBT/e is the diffusion constant. A similarexpression will apply to the hole current which will addto the electron current. Hence the total current can bewritten as

I = I0 (exp(eV/kBT )− 1), (6.16)

which is the famous Shockley’s law (1949) which de-scribes the I-V characteristics of a p-n junction diode. I0

is the saturation current, which for a cross-section areaA is

I0 = eA

(De

Len0

P +Dh

Lhp0

n

)

= eAn2i

(De

Le NA+

Dh

Lh ND

).

(6.17)

I0 is strongly temperature dependent because of the n2i

factor. In a biased p-n junction the diffusion currentis carried by the excess minority carriers in a diffusionlength each side of the space charge region, known asthe diffusion region. These carriers recombine with themajority carriers which are supplied by the voltage sourceand hence a majority carrier current flows in the bulk n-type and p-type material.

For Ge, this simple theory works well. But in Si, the in-trinsic carrier concentration ni is much smaller and thereis also a contribution to the current from carrier gener-ation or recombination in the space-charge region. Inreverse bias, any electron-hole pairs which are thermallygenerated in the space-charge region are rapidly sepa-rated by the strong electric field there and give a gen-eration current. In forward bias, the recombination ofelectron and holes in the space-charge region produces arecombination current.

In real diodes the experimental Shockley’s law maybecome

I = I0(exp(eV/αkBT )− 1), (6.18)

where the “diode ideality factor” α lies between 1 and 2.In real diodes the doping is not usually the same in both

regions (a single-sided junction). A more heavily dopedp-type region would give a p+n junction. The width ofthe depletion layer is determined by the doping level onthe lower doped side.

F. Applications of p-n junctions

These junctions have many applications, a few of whichare described below:

1. Varactor diode

A reverse biased (V < 0) p-n junction acts as a ca-pacitor C = εε0A/d, where A is the cross-sectional areaand d is the width of the depletion layer. Equation 6.11shows that d ∝ (Φ − V )1/2. Hence C ∝ (Φ − V )−1/2,and the capacitance can be voltage controlled. A typicalvalue might be 100 pF for an area of 1 mm2.

2. Photodiodes (or p-i-n diodes)

Optical fibre communications require fast optical de-tectors. A thin region of lightly doped (or intrinsic) Sican be placed between p-type and n-type material. A re-verse bias then depletes the whole central region to givea thick (50 µm) depletion layer. Light from a fibre pro-duces electron-hole pairs which are rapidly separated bythe internal electric field. The transit time can be veryfast (1 ns) so a rapid response is obtained.

3. Solar cells

Solar cells, see Figs. 52-54, are large area p-n junc-tions. Light absorbed in the depletion region give a light-dependent current source IL which is added to the diode

29

FIG. 50 Example of a current-voltage characteristic for a sil-icon p-n junction. Reverse voltages and currents are given asnegative values.

FIG. 51 The large reverse bias on a p-i-n (p-v-n) photodiodeproduces depletion through the whole of the v layer. Photonsabsorbed there produce carriers that are swept away to theterminals.

FIG. 52 A solar cell: (a) note the top contact design, and thepenetration of photons to various depths; (b) the energy-banddiagram for a solar cell. Only those electron-hole pairs whichseparate across the depletion layer contribute to the outputcurrent.

FIG. 53 Solar cell: (a) equivalent circuit; (b) the current-voltage characteristics show a downward shift as the light be-comes stronger.

characteristic

I = I0 [exp(eV/kBT )− 1]− IL. (6.19)

The short-circuit current is −IL and the open-circuitvoltage Voc(I = 0) is

Voc =kBT

eln

(IL + I0

I0

). (6.20)

The maximum power out IV occurs for ∂(IV )/∂V = 0and is 0.82ILVoc.

4. Light-emitting diodes (LEDs)

A p-n junction can be constructed so that a forward-biased current produces light. The injected minority car-riers recombine to emit light. Recombination can occur

30

FIG. 54 Current-voltage characteristics of a solar cell underillumination.

FIG. 55 Transition processes for electrons. The photon isshown leaving the initial state of the radiative transition.

directly across the band-gap (for the direct gap semicon-ductors usually used) or via impurity levels.Some examples are:(i) GaAs diodes at 1.4 eV (band gap) in the infra-red.(ii) GaP doped with Zn and O which form a neutralimpurity centre. This produces red light at 1.7 eV.(iii) GaP doped with N which gives colours from red togreen as the N concentration varies. (iv) GaAsxP1−x

doped with N. This is a direct gap semiconductor withcolours from red to green as x varies. There are alsosemiconductor diode lasers.

G. Appendix: Calculation of capacitance of pn-junction

The electrostatics of the depletion layer can be deter-mined by solving Poisson’s equation ∇2V = −ρ/εε0; onthe p-type side of the junction this can be written as

d2V

dx2= − ρ

εε0.

Integrating it becomes

εε0dV

dx= −ρax + constant = −ρax− ρadp.

where the constant of integration is determined knowingthat at the edge of the depletion layer the electric field iszero, −dV/dx|x=−dp

= 0.On the n-type side of the junction we have

εε0dV

dx= −ρdx + ρddn.

Continuity of dV/dX at x = 0 gives −ρadp = ρddn; ρa

and ρp have opposite signs and this latter equation canbe rewritten as NAdp = NDdn. Integrating again gives

εε0

∫ V0

Vp

dV =∫ 0

−dp

(−ρpx−ρadp)dx =[−ρa

2x2 − ρadpx

]0

−dp

⇒ εε0(V0 − Vp) = ρad2p/2− ρad2

p = −ρad2p/2.

Similar arguments on the n-type side gives

εε0(Vn − V0) = ρdd2n/2.

Therefore the potential across the junction is

V = Vp−Vn =1

2εε0(ρad2

p−ρdd2n) =

−e

2εε0(NAd2

p+NDd2n)

⇒ V =−e

2εε0NDd2

n(1 + ND/NA). (6.21)

Considering the deletion layers to act like a chargedcapacitor, then the charge on one plate will be Q =−eNDdn and Eq. 6.21 can be rewritten as

V =Q2

2eεε0

NA + ND

NAND. (6.22)

Therefore

C =Q

V=

12V − 1

2

[2eεε0

(NAND

NA + ND

)] 12

. (6.23)

31

VII. GROWTH AND DOPING OF SEMICONDUCTORS

A. Growth of bulk crystals

(a) Pure Si is obtained20 from mined quartz (SiO2),which is melted at 1700 C in a carbon-arc furnace. Thisreduces the quartz to give molten metallurgical grade Siwith many impurities, at the level > 100 ppm (parts permillion).(b) The Si interacts with HCl to give trichlorosilane,SiHCl3, leaving most impurities behind. This is reducedwith H2 gas to give polycrystalline Si with < 1 ppb (partsper billion) impurities.(c) Further purification is done from the melt. For a liq-uid and solid phase in contact in equilibrium, the ratio ofthe concentrations of a given impurity in the liquid (CL)to that in the solid, (CS) is the distribution coefficientkd = CS/CL, which is usually < 1, see Fig. 56. Hencethe semiconductor can purified by solidification from themelt. In practice, a molten region is passed through animpure ingot (zone refining), see Fig. 57; this transfersthe impurities to one end of the ingot.(d) Bulk semiconductors are normally grown from themelt, using the following:

• In the Czochralski technique, see Fig. 58, a singlecrystal is “pulled” from the melt, starting with anoriented seed crystal.

• In the Bridgman technique, see Fig. 59, the semi-conductor is melted in a crucible and crystallisationis induced by lowering the temperature.

• In the floating zone technique, see Fig. 60, a poly-crystalline rod is recrystallised as a single-crystal.

B. Epitaxial growth

In epitaxial growth, the semiconductor is grown on asinge-crystal oriented substrate by vapour-phase epi-taxy (VPE) or chemical vapour deposition (CVD),liquid-phase epitaxy (LPE) or solid-phase epitaxy(SPE). These processes occur at lower temperatures thanbulk melt growth and gives higher purity and perfectionand greater compositional uniformity. In VPE, gases arepassed over the substrate. Chemical or thermal reactionsoccur, catalysed by the substrate and the crystal grows,see Fig. 61.Typical reactions to produce Si are:

SiCl4 + 2 H2 → Si(s) + 4HClChemical reaction of adsorbed gases

SiH4(silane) → Si(s) + 2 H2

Thermal decomposition on hot substrate

20 see Jenkins Chap. 2.

FIG. 56 Phase diagram for impurities with distribution coef-ficient k = CS/CL < 1.

FIG. 57 The distribution of solute in an ingot of material:(a) single zone pass; (b) the effect of multiple zone passes.

A more recent technique is molecular beam epi-taxy (MBE). This is a thermally controlled evaporationprocess in an ultra-high vacuum (UHV) apparatus, seeFig. 62. Compounds or elements are heated in crucibles(Knudsen cells) to generate neutral atomic or molecu-lar beams directed onto a heated substrate. The beamstravel without interaction (Knudsen or molecular flow)to the substrate. For GaAs, the Ga atoms have a stick-ing probability of 1. As2 or As4 molecules do not stickunless Ga is present, when they dissociate and react withthe Ga adatoms to form GaAs. The crystal growth rateis very slow, about 1 atomic layer per second. But thisenables precise layers and abrupt changes in compositionto be incorporated. This has led to heterostructures andquantum wells (see Sec. VIII.C).

32

FIG. 58 Schematic diagram of the Czochralski (verticalpulling) of single-crystal production.

C. Diffusion doping of semiconductors

The controlled doping of semiconductors is vital for theproduction of devices. These can be incorporated duringthe growth of the crystal (chemical doping). This can alsobe done by the diffusion of impurities into the surface ofa semiconductor by thermal diffusion into the solid. Firstthe dopant atoms are deposited on the surface, usuallyby chemical reaction from gases passed over the samplein an oven. For example, passing PH3 and O2 over Sigives

4PH3 + 5 O2 → 2 P2O5 + 6H2

and

2P2O5 + 5 Si → 4P + 5 SiO2,

which delivers P atoms to the Si surface. A drive-instep is used, which heats up the Si for a time tin, so thatthe dopants diffuse into the semiconductor. The depth towhich the dopants diffuse is given by the diffusion length√

Dtin. D is the atomic diffusion coefficient, which isthermally activated

D = D0 exp(−∆E/kBT ), (7.1)

where ∆E is the activation energy for the creation ofthe defect. A typical diffusion constant at 1200C is

FIG. 59 The horizontal Bridgman technique for the growthof GaAs from the melt.

FIG. 60 The floating-zone technique for the growth of singlecrystals.

FIG. 61 Cold-wall reaction vessel for pyrolytic growth of Si.

33

FIG. 62 Schematic diagram of an MBE chamber.

FIG. 63 Schematic diagram of an ion implanter.

D = 10−16 m2s−1. Hence a drive-in time of 4000 swould give a diffusion depth of about 1 µm. Note thatthe dopant concentration will depend on the depth belowthe surface.

D. Ion implantation

Another technique is ion implantation (see Fig. 63) inwhich the semiconductor is bombarded with high-speedpositively charged ions, such as B or Sb for silicon and Si,Se or S for GaAs. The energy of the beams is typically 30-200 keV, though energies up to 2 MeV are used. The ionspenetrate for a specific range up to several microns. Notethat the ions can travel along relatively open channelsin a given crystal structure. A specific dopant profilecan be produced by using several implants with differentenergies.

There are two problems with ion implantation. First,considerable structural damage is caused to the crystal.Second, many of the dopants will initially end up on in-terstitial sites where they will not be electrically active(for a dopant to work it must be on a substitutional site).Hence the sample must be annealed after implantation torestore the crystallinity of the semiconductor and provideenough thermal energy for the implanted ions to moveonto lattice sites where they become electrically active.

FIG. 64 Sequence of operations in the production of a chip.

E. Chip production

A modern silicon fabrication plant uses a wide varietyof techniques to make devices. A silicon wafer will beprocessed on which many integrated circuits or chips willbe produced, see Fig. 64.

34

VIII. HETEROSTRUCTURES AND NANOELECTRONICS

We describe a few modern developments.

A. Introduction to low-dimensional systems

So far we have been concerned with essentially bulkproperties of semiconductors. These properties are de-scribed by coefficients that are independent of the sizeand shape of the specimens. If one or more of the dimen-sions of the sample are reduced significantly then thesample is described as a low-dimensional system.21 Wecan use modern lithographic techniques and molecularbeam epitaxy (MBE) to produce systems which behaveas films, lines and dots. Departures from bulk behaviouroccur when the size of the sample becomes comparablewith the wavelength of the relevant excitations in thesolid. This phenomenon is called a quantum size effect.A less fundamental effect is when the specimen size iscomparable with the mean free path of the excitations.This is known as a size effect (e.g., the size dependenceof the thermal conductivity of a sapphire rod).22

B. The two-dimensional electron gas (2DEG)

Consider electrons confined to a film of thickness d byinfinite potential barriers at z = 0 and z = d. Motionin the xy-plane is unconstrained. As in the 3D case wesolve Schrodinger’s equation for the band structure withboundary conditions ψ = 0 at z = 0 and z = d. Thesolution

ψ(x, y, z) = eikxxeikyy sin(kzz)

with kx = 2πp/L, ky = 2πq/L, and kz = nπ/d. L isthe macroscopic dimension of the sample in the x and ydirections. The energy levels are

E =h2

2me(k2

x + k2y + k2

z) =h2

2meL2(p2 + q2) +

n2h2

8med2;

if d is small, the final term can be large. The loweststate with n = 2 (with p = q = 0) is higher in energythan n = 1 by 3h2/8med

2, which exceeds kBT at roomtemperature for d < 60 A. The motion of electrons in thez -direction is frozen, and they behave as free particles intwo dimensions.

21 Hook and Hall, Chap. 14.22 Hook and Hall, Sec. 2.8.4.

FIG. 65 The allowed wavevectors for 2D motion in the xy-plane. The allowed states form a simple 2D lattice of spacing2π/L in both the kx and ky directions. To calculate the 2Ddensity of states the number of states in the ring betweenradii k → k + dk must be determined.

1. Density of states (DOS) in 2D

The 2D density of electron states in k-space is givenby

g(k) dk =2πk dk

(2π/L)2=

L2 k dk

2π; (8.1)

and therefore g2D(E) is given by

g2D(E) =me

πh2 (8.2)

which is independent of energy. If there are three 2Dsubbands that are occupied, the step-like density of statesshown in Fig. 66 will be obtained.

In practice quantum wells can be produced in semi-conductor heterojunctions prepared by MBE or by usingMetal oxide semiconductor (MOS) junctions.

C. Quantum wells

The advent of molecular beam epitaxy (MBE) has ledto new types of devices. In heterostructures, alternatelayers of GaAs and AlxGa1−xAs are grown on a GaAssubstrate with atomic precision, and because epitaxialgrowth is much slower than melt growth there is lower

35

FIG. 66 Density of states of electrons in a thin film. E1, E2

and E3 are the bound state energies. The dashed line showsthe energy dependence of the DOS of 3D electrons in a thickfilm, where g(E) ∼ E1/2.

disorder and higher purity. Layer widths can be con-trolled from a few nm up to µm. Conduction and valenceband offsets, ∆EC and ∆EV , shown in Fig. 67 exist be-tween the GaAs and AlxGa1−xAs so that quantum wellsare formed for both electrons and holes in the GaAs lay-ers. The free carriers are now confined along the z -axisand form standing waves in the potential well; the elec-trons are still free to move in the xy-plane. For an infinitepotential well, width a, the energies of the electron statesare

E(k) = Ep +h2k2

2m∗ =h2p2

8m∗a2+

h2k2

2m∗ , (8.3)

where p is a quantum number giving the number of half-wavelengths of the electron standing wave in the well.Because of the different effective masses, different levelswill be formed for heavy and light holes. The energystates for a given quantum number p are known as sub-bands.

D. High electron mobility transistors

If a vertical electric field is superimposed on the bandstructure, then a narrow quantum well can be formedat an AlGaAs/GaAs interface. Electrons (or holes) inthis well populate the lowest sub-band and form a two-dimensional electron gas (2DEG). These electrons can

EC

EV

Al Ga As0.3 0.7GaAsAl Ga As0.3 0.7

E (GaAs)g

ΔEV

ΔEC

E (Al Ga As)g 0.3 0.7

a

| |ψ1

2

ECEe1

(a)

(b)

FIG. 67 (a) A sample consisting of a thin layer of GaAs ofwidth a sandwiched between layers of Al0.3Ga0.7As. (b) Theelectronic structure of a GaAs quantum well. Eg(GaAs) isthe forbidden gap of bulk GaAs. The lowest confined level inthe conduction band lies Ee1 above the conduction band edgeof bulk GaAs and its wavefunction is ψ1.

++++++

doped undoped

E1

E2

AlGaAs GaAs

E1

E2

FIG. 68 The subband structure at the interface in a GaAs ac-tive channel in a HEMT structure. E1 and E2 are the confinedlevels. The approximate positions of E1 and E2 are shown inthe lower part of the figure, as well as the shape of the wave-functions. In the upper part is the approximate potential,including contributions of the conduction band offset.

come from donors in a doped layer which is separatedby a spacer layer. This is called modulation dopingand can give very high mobilities as the charged impuri-ties are well removed from the carriers. Such structuresare used in high electron mobility transistors, morecommonly known as HEMTs. Mobilities of 10 m2/Vs canbe achieved, compared to 0.06 m2/Vs in Si-MOSFETs,

36

which gives the possibility of much faster devices.

1. Nanostructures

Given the 2DEG in a heterostructure the electrons canbe further confined in the xy-plane by using electrodes(or gates) above the 2DEG. If the lateral dimensions of adevice, L, can be reduced to be comparable with the deBroglie wavelength λ then electron interference effectscan occur. It can also be less than the electron meanfree path. Two types of scattering are distinguished.In elastic scattering the electron wave remains coherent(i.e., no random phase shift) after scattering whereas ininelastic scattering there is a random phase shift andthe electron wave after scattering is incoherent with thewave before scattering, thus destroying any interferenceeffects. We define an elastic mean free path `e and aninelastic mean free path `i.

Several regimes are distinguished:

Diffusive regime: L > `e. This is the “normal” situ-ation considered so far. Kinetic equations can beused, using average velocities and currents.

Quantum transport regime: `e < L < `i. Interfer-ence effects must be considered. Scattering at theboundaries may be important.

Ballistic transport regime: L < `e < `i. The elec-tron may pass through the device without beingscattered.

Many new effects then arise. Mesoscopic physics is thephysics of systems or structures where average statisticalproperties cannot be used. The detailed response of thesesystems will depend on the specific structure of the indi-vidual device. For example, a single impurity atom maydetermine the response. If it moves, the explicit responsewill change.

2. Quantum dots and the single-electron transistor

A very interesting example of a mesoscopic system is aquantum dot in which a small isolated region is weaklycoupled to two reservoirs. The electrons in the dot forman almost isolated system with a series of energy levelsEp where p is a quantum number. An integral numberof electrons reside on the dot (typically 1000). Electronscan tunnel between the dot and the two reservoirs andhence the states in the dot will be occupied up to theFermi level. The energy of the dot and hence the num-ber of electrons can also be controlled by a gate electrodewhich is capacitatively coupled to the dot. As the gatepotential is swept, the conductance (in the low bias volt-age limit) through the dot exhibits a series of resonancesas the energy levels of the electrons in the dot match theFermi level EF in the reservoirs. At each resonance, one

more electron is added to the dot. This is known as aCoulomb blockade as adding electrons helps or hin-ders the conductance through the dot. The quantum dotsystem can be used to make a single-electron transistorsince the current through the device is controlled by thegate potential.

3. Low-dimensional physics

A very important consequence of the production of het-erostructures has been the study of the physics of thethe two-dimensional electron gas (2DEG). At room tem-perature the properties of a a 2DEG are similar to 3Delectrons. But at low temperatures, novel effects occur,such as the Quantum Hall Effect (QHE), discovered byvon Klitzing in l980 (Nobel prize 1985).

One-dimensional conductors have also been studied. Astriking result is that the conductance G = I/V of a one-dimensional ballistic conductor in is quantised in units of2e2/h.

E. Density of free electron states in 1D, 2D, and 3D

The density of free electron states g(E) = dn/dE in1D, 2D and 3D are derived in this section. g(E) is thenumber of available states per unit energy per unit vol-ume; this could also be called the density of availablestates per unit energy. Whatever the dimension d thedensity of states (DOS) can be written as

g(E) =dn

dE=

dn

dk× dk

dE= g(k)× dk

dE. (8.4)

The kinetic energy of free electrons is given by

E =h2k2

2mso

dk

dE=

(dE

dk

)−1

=m

h2k,

so Eq. 8.4 becomes

g(E) =dn

dE=

dn

dk× dk

dE= g(k)

m

h2k. (8.5)

The number of states in a cubic box of side L is foundby imposing periodic boundary conditions, for example,in the x-direction this condition is

eikx(x+L) = eikxx.

Therefore eikxL = 1, which means that kx = mx2πL ,

where mx is an integer. Similar boundary conditionsapply in all three directions, so the wavevector is:

~k = (kx, ky, kz) =2π

L(mx, my,mz), (8.6)

where the quantum numbers mx, my and mz are inte-gers. Hence Eq. 8.6 tells us that the electron states areuniformly distributed in k-space, with spacing 2π

L .

37

(a)

kx

ky

kf

(b)

kx

ky

kf-kf

(c)

FIG. 69 (a) Visualization of 3-dimensional k-space. The num-ber of available states is determined by the number of pointscontained in the shell of radius k and thickness dk. (b) Theground state of 2DEG at T = 0 can be represented by a Fermicircle of radius kF in k-space, centred on k = 0. States satis-fying k < kF are occupied, and those k > kF are unoccupied;the boundary between the two is called the Fermi surface.(c) A Fermi segment for a 1DEG. The confinement along they direction increases the spacing ∆ky between the adjacentstates. When ∆ky > kF , the condition Ly < λF is satisfied.The second subband is too high in energy, and only the firstsubband is populated.

1. 3D DOS: g3D(E)

In 3D the available states between k → k + dk arecontained in a shell of radius k and thickness dk, seeFig. 69(a), which has a volume 4πk2dk.

The electron states are uniformly distributed in k-space, with spacing 2π

L . Therefore the number dN ofenergy states contained in the shell is

dN = 24πk2dk

(2π/L)3,

where the extra factor of 2 is due to the spin degeneracyof spin 1/2 electrons. The volume of the sample is L3,so the number density is defined as n = N/V = N/L3.Therefore the density of states per unit volume in k-spaceis

dn

dk= g(k) =

8πk2

(2π)3=

k2

π2

Using Eq. 8.5 the 3D density of free states is

g3D =k2

π2× m

h2k=

mk

π2h2 =m

π2h3

√2mE. (8.7)

2. 2D DOS: g2D(E)

In 2D the available states between k → k + dk arecontained in a ring of radius k and thickness dk witharea 2πkdk, see Fig. 69(b).

The electron states are uniformly distributed in k-space, with spacing 2π

L ; therefore the number dN of en-ergy states contained in the ring is

dN = 22πkdk

(2π/L)2,

where the extra factor of 2 is due to the spin degeneracyof spin 1/2 electrons. The area of the sample is L2, sothe number density is defined as n = N/L2. Thereforedensity of states per unit volume in k-space is

dn

dk= g(k) =

4πk

(2π)2=

k

π

Using Eq. 8.5 the 2D density of free states is

g2D =k

π× m

h2k=

m

πh2 . (8.8)

3. 1D DOS: g1D(E)

In 1D the available states between k → k + dk arecontained at the two points at ±k with thickness dk, seeFig. 69(k), which has length 2dk.

38

The electron states are uniformly distributed in k-space, with spacing 2π

L . Therefore the number dN ofenergy states contained in the ring is

dN = 22dk

(2π/L),

where the extra factor of 2 is due to the spin degeneracyof spin 1/2 electrons. The length of the sample is L, sothe number density is defined as n = N/L. Thereforedensity of states per unit volume in k-space is

dn

dk= g(k) =

42π

=2π

Using Eq. 8.5 the 1D density of free states is

g1D =2π× m

h2k=

2m

πh2k=

1hπ

√2m

E. (8.9)

4. Summary

Table VIII summarises the results derived. Missingfrom the table is the result for zero-dimensions, whereg0D(E) = δE. In general (i) the lower the mass, thelower the density of states, and (ii) as the dimension d isreduced, g(E) becomes more “peaky” at the bottom ofthe subband.

For multiple subbands the following expressions areobtained:

g3D =m

π2h3

∑

i

√2(E − Ei) Θ(E − Ei)

g2D =m

πh2

∑

i

Θ(E − Ei)

g1D =√

2m

πh

∑

i

Θ(E − Ei)√E − Ei

g0D =∑

i

δ(E − Ei),

where Ei is the bottom of the ith subband. Θ(x) is astep function, which is equal to 1 when x ≥ 1, and equalto 0 when x < 0. It is introduced to cut off the functionsbelow the subband edges where the DOS should be zero,for example, in g1D it will avoid problems with squareroots of negative energies appearing in the denominator.

Note: to obtain the density of electrons n (the num-ber of electrons per m3) in 3D, the density of states isintegrated

n =∫ EF

0

g3D(E)dE =√

2m

π2h3

∫ EF

0

√E dE =

√2m

π2h3

23E

3/2F ,

a result obtained in PH2710.

39

TABLE VIII Density of states in different dimensions

1D 2D 3D

incremental “volume” in k-space dVk 2 dk 2πk dk 4πk2 dk

“volume” L L2 L3

g(k) 2π

kπ

k2

π

g(E) 1πh

√2mE

mπh2

√2mE

π2h3