Theory of semiconductor devices karl hess

Preface Acknowledgments xiiixv 1.1 The Equations of Classical Mechanics, Application to Lattice Vibrations 21.2 The Equations of Quantum Mechanics 9 Chapter 2 The Symmetry of the Crystal

ADVANCED THEORY OF

SEMICONDUCTOR DEVICES

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NONVOLATILE SEMICONDUCTOR MEMORY TECHNOLOGY:

A Comprehensive Guide to Understanding and Using NVSM Devices

Edited by William Brown and Joe E Brewer

ADVANCED THEORY OF

SEMICONDUCTOR DEVICES

Karl Hess

University of Illinois at Urbana-Champaign

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Karl Joseph Hess

Preface

Acknowledgments

xiiixv

1.1 The Equations of Classical Mechanics, Application

to Lattice Vibrations 21.2 The Equations of Quantum Mechanics 9

Chapter 2 The Symmetry of the Crystal Lattice

2.1 Crystal Structures of Silicon and GaAs

2.2 Elements of Group Theory 22

2.2.1 Point Group 22 2.2.2 TranslationalInvariance 26

3.2 Energy Bands by Fourier Analysis 34

3.3 Equations of Motion in a Crystal 42

3.4 Maxima of Energy Bands-Holes 46

3.5 Summary of Important Band-Structure

Parameters 503.6 Band Structure of Alloys 50

4.1 ShallowImpurity Levels-Dopants 58

4.2 Deep Impurity Levels 60

4.3 Dislocations,Surfaces, and Interfaces 62

vii

Chapter5 Equilibrium Statistics for Electrons and Holes 67

5.1 Density of States 67

5.2 Probability of Finding Electrons in a State 735.3 Electron Density in the Conduction Band 75

Chapter 6 Self-Consistent Potentials and Dielectric Properties 81

6.1 Screening and the Poisson Equation in One

Dimension 826.2 Self-Consistent Potentials and the Dielectric

Function 83

Chapter 7 Scattering Theory

7.1 General Considerations-DrudeTheory 89

7.2 Scattering Probability from the Golden

7.2.1 Impurity Scattering 94 7.2.2 Phonon Scattering 96 7.2.3 Scattering by a S-ShapedPotential 1027.3 Important Scattering Mechanisms in Silicon and

Gallium Arsenide 103

89

Chapter 8 The Boltzmann Transport Equation 109

8.1 Derivation 109

8.2 Solutions of the Boltzmann Equation in the

Relaxation Time Approximation 1148.3 Distribution Function and Current Density 1218.4 Effect of Temperature Gradients and Gradients ofthe Band Gap Energy 125

8.5 Ballistic and Quantum Transport 127

8.6 The Monte Carlo Method 129

Chapter 9 Generation-Recombination 135

9.1 Important Matrix Elements 135

9.1.1 RadiativeRecombination 135 9.1.2 Auger Recombination 1399.2 Quasi-Fermi Levels (Imrefs) 139

9.3 Generation-RecombinationRates 140

9.4 Rate Equations 144

Chapter 10 The HeteroJunction Barrier 147

10.1 Thermionic Emission of Electrons over

Barriers 147

Contents ix

10.2 Free Carrier Depletion of Semiconductor

Layers 15110.3 Connection Rules for the Potential at an

Interface 15310.4 Solution of Poisson's Equation in the Presence of

Free Charge Carriers 15410.4.1 Classical Case 154 10.4.2 Quantum MechanicalCase 15710.5 Pronounced Effects of Size Quantization and

Heterolayer Boundaries 162

Chapter11 The DeviceEquations of Shockleyand Stratton 167

11.1 The Method of Moments 167

11.2 Moment for the Average Energy and Hot

Electrons 17011.2.1 Steady-StateConsiderations 171 11.2.2 VelocityTransientsand Overshoot 175 11.2.3 Equation of Poisson and Carrier Velocity 176

12.1 General Considerations 181

12.2 Numerical Solution of the Shockley

Equations 18412.2.1 Numerical Simulation Beyond the Shockley

Bias 213 13.2.5 Short Diodes 215 13.2.6 Recombinationin Depletion Region 216 13.2.7 Extreme Forward Bias 219

13.2.8 AsymmetricJunctions 221 13.2.9 Effects in ReverseBias 22313.3 High-Field Effects in SemiconductorJunctions 226

13.3.1 Role of Built-InFields in Electron Heating andp-n

Junction Currents 226

193

13.3.2 Impact Ionization in p-n Junctions 229 13.3.3 Zener Tunneling 236

13.3.4 Real Space Transfer 24013.4 Negative Differential Resistance and

Semiconductor Diodes 241

14.1 Basic Geometry and Equations for Quantum WellLaser Diodes 248

14.2 Equations for Electronic Transport 250

14.3 Coupling of Carriers and Photons 253

14.4 Numerical Solutions of the Equations for Laser

Diodes 257

15.1 Simple Models 266

15.1.1 Bipolar Transistors 266 15.1.2 Field Effect Transistors 27215.2 Effects of Reduction in Size, Short Chan-

nels 27815.2.1 Scaling Down Devices 278 15.2.2 Short Gates and Threshold Voltage 27915.3 HotElectron Effects 281

15.3.1 Mobility in Small MOSFETs 281 15.3.2 Impact Ionization, Hot Electron Degrada-

tion 284

Chapter 16 Future Semiconductor Devices 291

16.1 New Types of Devices 291

16.1.1 Extensions of ConventionalDevices 291 16.1.2 Future Devices for Ultrahigh Integration 29316.2 Challenges in Nanostructure Simulation 295

16.2.1 Nanostructures in Existing Semiconductor

Devices 296 16.2.2 Quantum Dots 297 16.2.3 Structural, Atomistic, and Many-Body

Effects 297

Appendix A Tunneling and the Golden Rule

Appendix B The One Band Approximation

301 305

Contents xi

Appendix C Temperature Dependenceof the Band Structure 307

Appendix F The Self-Consistent Potential at a Heterojunction 315

This book evolved from my earlier book of the same title Chapters have beenadded (e.g., one on laser diodes); others have been completely rewritten (e.g.,the chapter on the Boltzmann equation)

Semiconductor devices are now the substrates of information and tation-the substrates of Internet browsers that sift with great speed through aworld of information and represent the information visually to the user, and thesubstrates of artificial intelligence They form the basis of all computer chips, ofsolar cell arrays, and of the newer red lights on cars They are essential in fibercommunications, and laser diodes are among the most sophisticated semiconduc-tor devices They are truly ubiquitous and can be found in increasing numbers

compu-in cars, kitchens and even compu-in electronic door locks Trillions of the basic conductor devices,p-njunction diodes, are fabricated daily, and Moore's law ofincreasing the integration and reducing the device size every 18 months has beenpersistently obeyed

semi-Mygoal is to present a description of the theoretical concepts underlyingdevice function and to cover device theory from the principles of condensedmatter physics and chemistry to the numerical mathematics of device simulationtools, all in a form understandable for anyone who knows advanced calculus andsome numerical algorithms important for the solution of the device equations,the Boltzmann equation, and the Schroedinger equation This goal could not beachieved Instead I have presented only an overview of some of the most impor-tant concepts of selected devices To obtain a truly broad knowledge of devicetheory, the reader will need to study additional books that are referenced, particu-larly theSolidStateTheoryedited by Landsberg, the encyclopedic description ofmost devices by Sze, and the text on numerical device simulation by Selberherr

Karl Hess

University ofIllinois at Urbana-Champaign

xiii

advicedur-treatment of p-n junctions, Alex Trellakis to the explicit solution for the even

partof the distribution function, M Grupen to the insights presented in the teronlaserdiodes basedonhis pathbreaking workonlaserdiodesimulation, and

chap-L F Register to the theory of collision broadening in MonteCarlo simulationsandto someof the treatment of the electron phonon interactions

I thankJ.P Leburton, U.Ravaioli, M.Staedele, F.Oyafuso, B Klein, ister, E Rosenbaum, and B Tuttlefor reading selected chapters and suggestingimprovements, and the students in my classes who have found and correctedmany mistakes

F.Reg-Special thanks go toL R Cooper from the Office of Naval Research, to

M Stroscio from the Army Research Office, and to George Lea from the tionalScience Foundation for their insights regarding the importance of topicalareas and for their encouragement G J Iafrate has worked with me on manytopics and has influenced my thinking from velocity overshoot to quantum ca-pacitance

Na-The Beckman Institute of the University of Illinois and its first directors,T.L Brown andJ.Jonas,haveprovided anidealenvironment to covertheoreticalexpertise of a rangeof disciplines including basic physics, chemistry, electricalengineering, and numerical mathematics

William Henstrom has performed above and beyond duty in creating layoutandcomposite workandcorrectingmyfeebleattempts inIMEX.

Very special thanks go to my wife Sylvia, my daughterUrsula S., and myson KarlH for theirlovingsupport

KarlHess

University ofIllinois at Urbana-Champaign

xv

of quantum field theory and write down the ~1023coupled equations for all theatoms in the semiconductor device Then we would have to solve these equa-tions, including the complicated geometrical boundary conditions However, theoutcome of such an attempt is clear to everyone who has tried to solve only one

of the 1023equations

Any realistic approach oriented toward engineering applications has to ceed differently Based on the experience and investigations of many excellentscientists in this field, we neglect effects that would only slightly influence the re-sults In this way many relativistic effects become irrelevant In my experience,the spin of electrons plays a minor role in the theory of most current semicon-ductor devices and can be accounted for in a simple way (the correct inclusion

pro-of a factor pro-of 2 in some equations)

Most effects of statistics can be understood classically, and we will needonly a very limitedamountof quantum statistical mechanics This leaves us es-sentially with the Hamiltonian equations (classical mechanics), the Schrodingerequation (quantum effects), the Boltzmann equation (statistics), and the Maxwellequations (electromagnetics)

It is clear that the atoms that constitute a solid are coupled, and therefore theequations for the movement of atoms and electrons in a solid are coupled Thisstill presents a major problem, a many-body problem We will see, however,that there are powerful methods to decouple the equations and therefore makesingle particle solutions possible The many interacting electrons in a solid arethen, for example, replaced by single independent electrons moving in a periodicpotential Complex many body effects, such as superconductivity, are then ex-cluded from our treatment, which is justified because of the low electron density

in typical semiconductors We also exclude in our treatment effects of extremelyhigh magnetic fields because these are unimportant for most device applications

1

2 Chap 1 A Brief Review of the Basic Equations

In this way, the fundamental laws of physics are finally reduced to laws of conductor devices that are tractable and whose limitations are clearly stated Thefollowing sections are written with the intent to remind the reader of the basicphysics underlying device operation and to review some of the physicist's toolkit in solid-state theory

semi-1.1 THE EQUATIONS OF CLASSICAL MECHANICS,

APPLICATION TO LATTICE VIBRATIONS

Hamilton was able to give the laws of mechanics a very elegant and powerfulform He found that these laws can be closely linked to the sum of kinetic andpotential energy written as a function of momentumlike (Pi)and spacelike (Xi)

which is Newton's first law of steady motion without forces

If we have a potential energyV(Xl) that varies in theXl direction, we obtainfrom Eq.(1.1)

The quantity defined asF ois the force, and Eq.(1.3) is Newton's second law ofmechanics

A more involved example of the power of Hamilton's equations is given

by the derivation of the equations for the vibrations of the atoms (or ions) ofthe crystal lattice As we will see, these vibrations are of utmost importance

o

o o

o

o

Figure1.1 Displacementu;(r)of atoms in a crystal lattice.

in describing electrical resistance They also give a fine example of how themany-body problem of atomic motion canbereduced to the solution of a singledifferential equation by using the crystal symmetry (group theory from a math-ematics point of view) by a cut-off procedure Here we cut off the interatomicforces beyond the nearest neighbor interaction We also introduce below cyclicboundary conditions, which are of great importance and convenience in solid-state problems

It suffices for this section to define a crystal, and we will be mostly

interest-ed in crystalline solids, as a regular array of atoms hookinterest-ed together by atomicforces "Regular" means that the distance between the atoms is the same through-out the structure Many problems involving lattice vibrations can be solved byclassical means (i.e., using the Hamiltonian equations) because the atoms thatvibrate are very heavy Then we only have to derive the kinetic and potentialenergy Because we would like to describe vibrations (i.e.,the displacement ofthe atoms), we express all quantities in terms of the atomic displacementsu;(r),

wherei = x,y,Z and ris the number (identification) of the atom It is important

to note thatris not equal to the continuous space coordinaterin this chapter, though it has similar significance because it labels the atoms The displacement

al-of an atom in a set al-of regularly arranged atoms is shown in Figure 1.1

We follow the derivations in Landsberg [5]and express the kinetic energyT

by

T= 2 IM~ 2( )~ Ui r n wereh u·(r) au;(r) at (1.4)

(1.5)

andMis the mass of the atoms (ions)

We assume now that the total potential energy U of the atoms can be

ex-pressed in terms of a power series in the displacements,

U= Uo+ LBiu;(r) + ~ LBiju;(r)uj(s) +

ij

4 Chap 1 A Brief Review of the Basic Equations

(1.6)

where s also numbers the atoms as r does.

The following results rest on this series expansion and truncation, whichmakes a first principle derivation (involving many body effects) unnecessary.Equation (1.5) is, of course, a Taylor expansion with

-:»:I dUi(r)

Because the crystal is in equilibrium, that is, at a minimum of the potential energy

U, the first derivative vanishes and

(1.9)

(1.11)

Furthermore, a rigid displacement (allUiequal) of the crystal does not change U

and we therefore have, from Eqs (1.5) and (1.9),

r

To derive Eq (1.9), we have assumed an infinite crystal We also could haveintroduced so-called periodic or cyclic boundary conditions; that is, continue thecrystal by repeating it over and over In one dimension, this means we consideronly rings of atoms (Figure 1.2) This approach amounts to neglecting any sur-face effects or other effects that are sensitive to the finite extension of crystals

We can now derive the equations of motion by using Eqs (1.1) and (1.2)with coordinates ui(r)instead ofXi:

Figure 1.2 A ring of atoms representing cyclic boundary conditions.

Here, also, the indicesm,n are used to number the atoms (as',Sabove) fore

In the three-dimensional case, Eqs (1.7) through (1.10) are very helpful;they reduce the numbers of parameters Without going into details, we mentionthat this reduction of parameters is generally accomplished by group theoreticalarguments, and Eq (1.9) is a direct consequence of the translational invariance(group of translations)

6 Chap 1 A Brief Review of the Basic Equations

To proceed explicitly with our one-dimensional model, we need to make thedrastic assumption that each atom interacts only with its nearest neighbor (Wecan use the same method also for second, third nearest neighbor interaction,if

we proceed numerically and use a high-speed computer.) Our assumption means

B'"=1=0 only for s=r± 1

Notice that we dropped the indices i,j because this is a one-dimensionalproblem Without any loss of generality, we may assumer= O.Then we have

BOs=/=0 for s= ±1and

BOs= 0 otherwiseFurthermore,

according to Eq (1.9) and

It is now customary to denote B 01=BO(-l) by -a (a is the constant of the

"spring" forces that hold the crystal together) and therefore dXl by 2a. Theequation of motion, Eq (1.14), then becomes for anyr

Mu(r)=-2au(r) +au(r-l)+au(r+1) (1.17)

Eq (1.17) leaves us still with 1023coupled differential equations However,these equations are now intridiagonal form,all with coefficiente, Such a formcan be reduced to one equation by skillful substitution The substitution can

be derived from Bloch's theorem, which we will discuss later It also can beguessed:

u(r) =ueiqra (1.18)

Note that the amplitude u is still a function of time Here a is the distance

be-tween atoms (i.e., the lattice constant) Eq (1.17) becomes

which gives

Mueiqra= -2aueiqra+aueiqrae-iqa+aueiqraeiqa

Mil= au(2cosqa - 2)

(1.19)

are actually the same For example, if q= 31t/aand A= 2/3a,the atoms are

displaced in exactly the same way as for q=tt]« In other words, for any q

outside the zone -1t/a ~q~1t/a,which is called theBrillouin zone,we can

find a q inside the zone that describes the same displacement, energy, and so

8 Chap 1 A Brief Review of the Basic Equations

on Notice that in a real crystal the arrangement of atoms is different in ferent directions Therefore, the three-dimensional Brillouin zone is usually acomplicated geometrical figure (see Chapter2)

dif-Second, q is not a continuous variable because of the boundary conditions.

Consider, for example, the ring of Figure 1.2witheightatoms and

u(O)= u(8)

or, in general, for N atoms, we have

u(N)= u(O) and u(N)= u(O)e iqNa

Therefore e iqNa equalsone, and we conclude that q= 21tl/Na where1is an integer If werestrictq to thefirstBrillouin zone,wehave -N/2~I~N/2 This

means that q assumes only discrete (not continuous) values However, because

of the largenumberN,it can almostberegarded as continuous

Third, without emphasizing it, we have developed a microscopic ·theory of

soundpropagation in solids For smallwave vectors q (Le., largeA),we have

whichis a microscopic description of the soundvelocity

In real crystals additional complications arisefromthe factthat we can havetwo or even more different kinds of atoms These atoms may oscillate as theidentical atoms in the above example There are, however, different modes ofoscillation possible Ifwe think of a chain with two different kinds of atoms, itcanhappenthatonekindof atom(black) oscillates againsttheotherkind(white).Such an oscillation can take place, and indeed does, at a veryhigh (optical)

frequency, andthe corresponding latticevibrations arecalledopticalphonons It

is very important to note that in principle all black atomscan oscillate in phaseagainstthe whiteones This meansthat we can havehigh frequencies (energies)

even if the wavelength is very large or the q vector is very small (Figure 1.5).

Figure 1.5 Two different kinds of atoms oscillating against each other This represents a

wave with high energy (frequency) and small wave vector.

Figure1.6 Schematicv(q) diagramfor acousticandopticphononsin one dimension.

The energy versus qrelation can then have two branches, the acoustic and theoptic, as shown in Figure 1.6

The presence of two different atoms can also cause long-range coulombicforces owing to the different charge on the two atom types (ionic component).The long-range forces cannotbedescribed by simple forces between neighbor-ing atoms, and one calls the phononspolar optical phonons if these long-rangeforces are important

As mentioned, lattice vibrations are important in various ways Electrons teract with the crystal lattice exciting (emitting) and absorbing lattice vibrations(the net lost energy is known as Joules heat) The system of electrons by itself istherefore not a Hamiltonian system; that is, one in which energy is conserved It

in-is only the sum of electons and lattice vibrations which in-is Hamiltonian

The interested reader is encouraged to obtain knowledge of a detailed tum picture of lattice vibrations (also phonons) and their interactions with elec-trons as described, for example, by Landsberg [5]

quan-1.2 THE EQUATIONS OF QUANTUM MECHANICS

At the beginning of the twentieth century, scientists realized that nature cannot bestrictly divided into waves and particles They found that light has particle-likeproperties and cannot alwaysbeviewed as a wave, and particles such as electronsrevealed definite wave-like behavior under certain circumstances They are, forexample, diffracted by gratings as if they had a wavelength

h

A= !Pi

10 Chap 1 A Brief Review of the Basic Equations

whereIi==h/2rc~6.58 x 10-16 eVs is Planck's constant and p is the electronmomentum

Schrodingerdemonstrated that the mechanics of atoms can be understood

as boundary value problems In his theory, electrons are representedby a wavefunction'I'(r),which can have real and imaginary parts, and follows an eigen-valuedifferential equation:

( - ::V2+ v(r)) 'Jf(r) = E'Jf(r) (1.25)

The part of the left side of Eq (1.25) that operateson 'l'is now called the iltonian operatorH. Formallythis operatoris obtainedfrom the classicalHam-iltonian by replacing momentum with the operatorVn/i (i = imaginary unit),where

Ham-v= (~,~,~)

The meaning of the wave function '1'(r) was not clearly understood at thetime Schrodingerderived his famous equation It is now agreed that1'I'(r)12 isthe probabilityof finding an electronin a volumeelementdr at r In otherwords,

we have to think of the electron as a point charge with a statisticalinterpretation

of its whereabouts (the wave-like nature) It is usually difficult to get a deeperunderstanding of this viewpoint of nature;evenEinsteinhad troublewithit It is,however, a very successful viewpoint that describes exactly all phenomena weare interestedin To obtain a better feeling for the significance of W(r), we willsolve Eq (1.25) for severalspecial cases As in the classical case, the simplestsolutionis obtainedfor constantpotential Choosingan appropriate energyscale,

The significance of the vector k can be understood from analogies to known wave phenomenain optics and from the classical equations BecauseE

well-is the kinetic energy,1ikhas to be equal to the classical momentump to satisfy

E =p 2/ 2m.On the other hand, in optics

How can the result of Eq (1.26) be understood in terms of the statisticalinterpretationof'1'(r)? Apparently

1'I'(r)12= leI2(cos2k r + sin2k r )= ICl2This means that the probabilityof findingthe electron at any place is equal toe2•

Ifwe know that the electron has tobein a certain volume Vol (e.g., of a crystal),then the probabilityof finding the electron in the crystal must be one Therefore,

Let us now consider the confinement of an electron in a one-dimensionalpotential well (although such a thing does not exist in nature) We assume thatthe potentialenergy V(r) is zero overthe distance(O,L) on the x-axis and infinite

at the boundaries0 andL.

The Schrodinger equation, Eq (1.25), reads in one dimension (x-direction,

V(x)=0)

_ r,,2 a2

",(x)= E (x)

Inspection shows that the function

",(X)=[iSin";x with n=I,2,3, (1.31)

satisfiesEq.(1.30) as well as the boundary conditions The boundary conditionsare, of course, that 'I' vanishes outside the walls, since we assumed an infiniteimpenetrable potential barrier In the case of a finite potential well, the wavefunction penetrates into the boundary and the solution is more complicated Ifthe barrier has a finite width, the electron can even leak out of the well (tunnel).This is a very important quantum phenomenon the reader should be familiarwith We will return to the tunneling effect below

The wave function, Eq (1.31), corresponds to energiesE (called eigen ergies)

en-(1.32)

12 Chap 1 A Brief Review of the Basic EquationsBecause n is an integer, the electron can assume only certain discrete energieswhile other energies are not allowed These discrete energies that can be assumedare called quantum statesand are characterized by thequantum number n. Thewave function and corresponding energy are therefore also denoted by'l'n, En.

Think ofa violin string vibratinginvarious modes at higher and lower tones(frequency v), depending on the length L, and consider Einstein's law:

If we compare the modes of vibration of the string with the form of the wavefunction for various n,then we can appreciate the title of Schrodinger's paper,

"Quantization as a Boundary Value Problem."

Devices that contain a well and feature quantized energy levels similar to theones given in Eq (1.32) do exist Quantum well lasers typically contain one ormore small wells and the well size controls the electron energy However, in mostdevices, the wells are not rectangular In silicon metal oxide semiconductor fieldeffect transistors (MOSFETs), the well is closer to triangular and its shape de-pends on the electron density (Le., the charge in the well) We will deal with thischarge-dependent well shape in Chapter 10 Here we discuss only well-definedpotential problems-c-eases where the potential is given and fixed With currenthigh-end workstations, the Schrodinger equation can then be solved numericallyfor an arbitrary (but given) potential shape One-dimensional problems can besolvedbystandard discretization (transforming the differentiations into finite dif-ferences) and by solving the resulting matrix equations by standard solvers such

as found in EISPACKandLAPACK.For two- and three-dimensional problems,this procedure still leads to a prohibitively large number of equations (whichgrows with the third power of the number of discretization points) Thereforethe discretized mesh must be coarsened even when using the fastest supercom-puters Often, however, one is interested only in relatively small sets of eigenvalues, for example, the first three [as forn= 1,2,3 in Eq (1.31)]

One then can use so-called subspace interaction techniques that only resolvecertain intervals of eigenvalues These techniques are well established for sym-metric real matrices as they occur in well-defined potential problems (see, e.g.,Golub and Loan [4]) A useful computer code is the RITZED eigenvalue solver

by Rutishauser[6]

Frequently, one needs to obtain an explicit solution of Schroedinger's tion for an arbitrary complicated form of the potential, provided only that it issmall and represents just a small perturbation to a problem for which the solu-tion is known This scenario is typical for scattering problems such as an electronpropagating in a perfect solid and then encountering a small imperfection and be-ing scattered Fortunately, for this type of problem there is a powerful method ofapproximation, perturbation theory, that gives us the solution for arbitrary weakpotentials The method is very general and applies to any kind of equation.Consider an equation of the form

whereHo and HI are differential operators of arbitrary complication andEis asmall positivenumber.

If we know the solution'Voof the equation

Ho"'o==0

then we can assume that the solution of Eq (1.34) has the form"'0 + E'"1. serting this form into Eq (1.34), we obtain

In-(Ho+EH1)('I'0+E'I'I)=Ho'l'o +£Hl'1'O + HoE'I'l+E2H l"'l

Wenowcan neglectthe term proportional toE2(becauseEis small), and because

(Ho+HI)4>m=Wm4>m with Hi e.H« (1.37)

Then it is shown in elementary texts on quantum mechanics (Baym [1]), by

re-peatedly using the method of perturbation theory as outlined, that

lVol

wheredr standsfordxdy d: (integration over volumeVol)and"'~ is the complexconjugateof'lin.

First-orderperturbation theory (to ordere)amountsto setting'I'm= cI>m and

W m ==Em+Mmm. The only change then is in the value of the eigen energy by

M mm, which can be obtained by the integration in Eq (1.40); the integrand is

known from the solution of Eq (1.36) This means that the numerical problem

14 Chap 1 A Brief Review of the Basic Equations

so,k')t

E(k')

E(k) E(k)+21t hit

Figure 1.7 Probability of a transition from k to k' according to the Golden Rule after the

potential has been on for time t.

is reduced to a volume integration (in three dimensions) To obtain solutions tohigher order, one also needs to perform the summations in Eqs (1.38) and (1.39).The formalism outlined above and the examples given are independent oftime, and the electrons are perpetually in appropriate (eigen) states In manyinstances, however, we willbeinterested in the following type of problem: Theelectron initially is in an eigenstate ofHo,denoted, for example, by a wave vector

k for the free electron What is the probability that the electron will beobserved

in adifferent eigenstate characterized by the wave vector k', after it interactswith a potentialV(r,t)? In other words, what is the probability S(k,k')per unittime that the interaction causes the system to make a transition from k to k'?The answer to this question is the famous Golden Rule of Fermi, which

is also derived in almost every text on quantum mechanics by so-called dependent perturbation theory The unfamiliar reader is urged to acquire a de-tailed understanding of the Golden Rule as derived, for example, in the text ofBaym [1] Here we only illustrate its generality and discuss results for importantspecial cases

time-1 Assume that a potential Y(r) is switched on at time 1= 0 but is timeindependent otherwise One then obtains

S(kk,)=\r *,Y(r) drI2.[sin(E(k')-E(k))1/21i]2 (1.41)

, iVol'Vk 'Vk (E(k') - E(k»y'i72

The function in brackets deserves special attention and is plotted in ure 1.7 Notice that ast approaches infinity, the function plotted in Fig-

Fig-ure 1.7 becomes more and more peaked at its center (E(k') =E(k)). Inthe limit1 t 00,the so-called &-function is approached, which is definedby

lim 4sin 2[(E(k') - E(k»t/ (2/i)J = 2xo(E(k')_ E(k» (1.42)

and can always be understood as a limit of ordinary functions It doeshave some remarkable properties, however, and the unfamiliar readershould consult some of the references at the end of this section A mostimportant property of the &-functionis the following: For any continuousfunction f(E'),we have

Lco

2 We assume that the perturbation is harmonic, which means we have apotential of the form

V(r,t)= V(r)(e-irot+ eirot )

Fort +00,we obtain the transition probability

S(k,k') = 2: I l: '!'k,v(r)'I'kdr \2

[O(E(k) - E(k') -lim) + O(E(k) - E(k') + lim)] (1.44)

It is clear that the o-function simply takes care of energy conservation.For a constant potential, we have to conserve energy astincreases For aharmonic perturbation, the system can gain or loose energy correspond-ing, for example, to the absorption or emission of light

3 We now turn our attention to the first term in Eq (1.41), the matrixelement, which also plays a vital role in time-independent perturbationtheory The significance of the matrix element is best illustrated by thefollowing special cases of well-defined potential problems (problems inwhich the potential is given by a certain function of coordinates):(a) V(r)= constant The matrix element is then

1 i 'k"k

constant - e-' "e' "dr

Vol VolThe integration is over the volume Volof the crystal In manypractical cases, this volume will be much larger than the de BrogliewavelengthAof the electron, which is of the order of 100Ain typicalsemiconductor problems This means that the integral of Eq (1.45)will be very close to zero, because the cosine and sine functions towhich the exponents in Eq (1.45) are equivalent are positive as often

as they are negative in the big volume There is only one exception:

In the case k'=k, the integral is equal to the volume and the matrixelement is equal to constant Therefore, we can write

constant ~ l e-ik'ore''kordr = constant Ok' k (1.46)

where ~, k, = 1 for k = k' and is zero otherwise This is known asthe Kronecker delta symbol Consequently, the matrix element has

16 Chap 1 A Brief Review of the Basic Equations

takencare of momentumconservation; the free electronin a constantpotential does not change its momentum

(b) Second,weconsideran arbitrarypotentialhavingthe following ier representation:

qThen the matrix element, which we now denote byMk,k" becomes

Mkk' = I,Vq ri(k-k'+q).r (1.48), q Vol JVOI

= I,VqOt'-k,q

qHow do we interpret this result? If we also allow the potential of

Eq (1.47)to have a time dependence(e.g., aseirot),then the potentialcan be interpreted as that of a wave (e.g., an electromagnetic wave)

In this caseEq (1.49) simplytells us that the wavevectorsof all tering agents (Le., their momenta)are conserved,because we have

It is important to notice that Eq (1.49) is also valid for a static,time-independent potential-that is, evena staticpotential"supplies"momentumaccordingto its Fouriercomponents-in the same way awavedoes This seemsstrangeat firstglance Tosee the significance,considerthe boundaryof a billiard table This boundaryis an impen-etrable abrupt potential step whose Fourier decomposition involvesall values ofq Indeed, the boundary can supply any momentum tothe ball to make it bounce back The above two examples show thatthe GoldenRule essentiallytakescare of energyand momentumcon-servation This is also the reason for its generality and importance.Remember, however, that this is true only for cases when timet atwhich we observe the scattered particle is long after the potential isswitchedon For short times (in practice these are times of the order

of 10-14's),the functionin Eq (1.42)cannotbeapproximated by a0

function, and energy need notbeconservedin processeson this shorttime scale This is at the heart of the energy time uncertainty rela-tion To illustrate the great generality of the Golden Rule, one moreexampleis given

(c) Consider the "tunneling problem" of Figure 1.8 Although an tric fieldF is applied in the z-direction, the electron in Figure 1.8

elec-is confined in a small well Classically, it would stay in the well.However, becausethe barrier is not infinite, as assumedin Eq (1.31),the wavefunction is not zero at the well boundary but penetratesthe

Figure1.8 Electrons in a potential well plus applied electric field F.

boundary In other words, there is a finite probability of finding theelectron outside the well

We can calculate the probability per unit time that the electronleaks out if we know the wave function 'l'inin the well and'l'ouout-side the well, and regard the electric field as a perturbation Thisperturbation gives a term (the potential energy) eFzin the Hamilton-ian The Golden Rule tells us that

S(w,ou) = 1[01 "'~ueFZ"'indrl' 2;O(Eou-Ein) (1.50)

In writing down this equation (which was first derived by penheimer), I have swept under the rug the fact thatWouandWinarethe solutions of different Hamiltonians 'l'inis obtained from the so-lution of the Schrodinger equation of the quantum well and'l'ou is thesolution of a free electron in an electric field with

Op-1i 2 V 2

H= eFz 2m

An exact justification of this procedure is complicated and is cussed in great detail in Duke's treatise of tunneling [2] (see alsoAppendix A)

dis-We emphasize that the matrix elements represent all that needs to be known

to obtain perturbation theory solutions These matrix elements are given by threedimensional integrals Alternatively they canbe viewed as scalar products in avector space denoting 'l'nby a vector In). For those unfamiliar with Dirac'snotation the following definition can just be used as a shorthand way of writingthe integral:

(1.51)

18 Chap 1 A Brief Reviewof the Basic Equations

PROBLEMS

1.1 Solve by perturbationtheory (to first order) fory:

dysin(x) dy

e dX +-=y axwhereeis a small positivequantity.

1.2 Calculatethe matrixelements for wavefunctionsof the form'II= ~eik-rand

(c) Discussthe formof the dispersionrelationandthe natureof the modesforq« 1t/a

andq = 1t/a, where qis the wavevector.

(d) Find the velocityof sound (ro/q forq~ 0).

(e) Showthat the group velocitydro/dqbecomeszero at the Brillouinzone boundary (This is a generalresult.)

CHAPTER 2

THE SYMMETRY OF THE CRYSTAL LATTICE

This chapter and the next three are a crash course in the solid-state physics derlying semiconductor devices They can be read as a reminder to the reader

un-of the most important solid-state physics principles that we will need later Theyare also written in a way to enable any novice to gain the necessary solid-stateknowledge However, this cannotbe accomplished by casual reading, but only

by going over the material with a pen

2.1 CRYSTAL STRUCTURES OF SILICON AND GaAs

In Chapter 1 we discussed the lattice vibrations without defining exactly what acrystal lattice is Here we give this definition and we see that a crystal is an object

of high symmetry This symmetry can be used to obtain general informationabout the properties of crystals and also to abbreviate complicated algebra Fulluse of the symmetry requires knowledge of group theory This knowledge is notrequired for the reader of this book Nevertheless, an attempt is made here tointroduce group theoretical techniques via practical examples

A crystal consists of a basis and a Bravais lattice The basis can be anything

ranging from atoms to giant molecules, such as deoxyribonucleic acid (DNA).The Bravais lattice is a set of points[Rr]that is generated by three non-coplanartranslations81, 82, 83,which are vectors of three-dimensional space

(2.1)and the 1; are integers

According to the properties of this lattice, under reflection, rotation, and

so on, one can distinguish 14 types of Bravais lattices For us, only the cubictypes matter (Figure 2.1) The important semiconductors are characterized by

a tetrahedral arrangement of the nearest neighbor atoms Their lattice can be

viewed as a face-centered cubic lattice with a basis of two atoms For silicon and

19

20 Chap 2 The Symmetry of the Crystal Lattice

Figure 2.2 Crystal structure of silicon (or GaAs if two kinds of atoms are on the appropriate

lattice sites) Notice the tetrahedral arrangement of nearest neighbor atoms and the equivalence to a face-centered cubic lattice (if a two-atom basis is assumed).

Figure 2.3 Vectors 81, 82, 83 generating the face-centered cubic lattice.

germanium, these two atoms are equal; for GaAs and the 111-V compounds, thetwo atoms are different This is illustratedinFigure 2.2

Canone view the lattice of silicon under all circumstances as a face-centeredcubic crystal with a basis of two atoms? In principle, the symmetry of one face-centered cubic crystal is present For some effects, however, the existence of thetwo basis atoms is vital Consider, for example, lattice vibrations It is clear thatthe two basis atoms are connected by different force (spring) constants a thantwo atoms on the side of the cube (see Figure 2.2) Therefore, opticalphononswill exist (see Problem 1.3) Instead of two different kinds of atoms vibratingagainst each other as in the problem, the two sublattices associated with the two

basis atoms can vibrate against each other By two sublattices, we mean that

we can also view the silicon crystal as two interconnected face-centered cubiclattices (sublattices), each having one basis atom

The three translation vectors8t, 82, 83 that generate a cubic face-centeredlattice are shown in Figure 2.3 It is important to note that these vectors aredifferent from the vectors that generate the simple cubic lattice If these vectors(along the sides of the cube) were chosen to generate the lattice points R[ of theface-centered lattice, some points could not be reached with integer values forlie

Figures 2.4a and 2.4b are photographs of a gallium-arsenide crystal model

in the [110] and [100] directions They illustrate the anisotropy of a crystal Inother words, lattice waves or electrons traveling in different directions encounter,

in general, different patterns (and therefore a different Brillouin zone boundary)

22 Chap 2 The Symmetry of the Crystal Lattice

Figure 2.4 Illustrations of a GaAs crystal model in the (a) [111] and (b) less regular

lat-to itself The group of these 48 operations is called Oi, Twenty-four of the 48operations that form the subgroup Td are shown in Table 2.1 In this table the

Table2.1 Elements of the PointGroup Td.

Ql (XIX2X3) Q2(Xli2i3) Q3(itX2i3) Q4(ili2X3) QS(X2X3X.) Q6(i2X3il) Q1(i2i3Xt} Q8(X2i3i.) Q9(X3XtX2) QlO(i3itX2) Qlt(X3 Xl i2) Q12(i3Xti2) Q13(itX3i2) QI4(iti3X2) Qls(.i3i2 Xl) Q16(X3i2 Xl)

Q17(X2XIX3) QI8(i2XI X3) QI9(XIX3X2) Q20(Xli3i2) Q21 (X3X2Xl) Q22(i3X2il) Q23(X2XI X3) Q24(i2ilX3) Source: After Morgan, D L, in Landsberg, P.T; 00., Solid StateTheory: Methods andApplications, TableC.7.! Copy-

right1969JohnWiley & Sons,Ltd Reprinted bypermission.

operationQ3(il,X2,X3)means, for example,

for any functionf of the coordinates

Operating on these 24 transformations with the inversion Qof(il,i2,i3)

gives another 24 symmetry operations, which, together with the operations inTable 2.1, form the 48 operations ofOi;

If f(Xl ,X2,X3) is a physical property of the crystal, and the crystal has thesymmetry Oi; then we obtain the value of f at many other points simply byapplying the symmetry operations This can save much computation time (Weexplain the use of this for band-structure calculations in Chapter3.)

The following example also gives a clear illustration of the advantageoususe of the symmetry operations Bardeen, Schrieffer, and Stem realized thatelectrons can form a two-dimensional gas at the interface between Si and Si02,

in a metal-oxide semiconductor(MOS) transistor Figure2.5 shows the basicgeometry of aMOStransistor (which is described in detail in Chapter 15)

It is known that bulk silicon exhibits an isotropic conductivity0". The teresting question arose, then, whether the conductivity of the two-dimensionalelectron sheet is also isotropic in the interface plane and whether the conductiv-ity depends on the crystallographic surface orientation of silicon To settle thesequestions, experiments were performed on(100), (110), and (111)surfaces (thereader should be familiar with the Miller indices) by fabricating many transistors

in-on various wafers of these three surface orientatiin-ons It was found that in-on (1(0)and (Ill) surfaces the conductivity is isotropic The (110) surface, however,shows an anisotropic electrical conductivity

Below we show that this result canbeobtained by astraightforward lation The current densityj as a function of electric fieldFis given by Ohm'slaw

In isotropic materials the conductivity0"is a scalar quantity If we allow foranisotropy, 0"becomes a matrix and Eq (2.2) assumes the form (in two dimen-

24 Chap 2 The Symmetry of the Crystal Lattice

Si02(insulator)

(2.3)

- Two-dimensional electron gas

Figure 2.5 MOS transistor with a two-dimensional sheet of electrons at the Si-Si02 interface.

of a square instead of a cube) The specific symmetry operation we choose is

a rotation by 90° In the notation of Table 2.1 and in three dimensions, such arotation would beQOQ14.

Because we have given the conductivity in matrix form, we also would like

to express the rotation in matrix form From calculus, we know that a rotation

by an angle4>can be represented by

( coso sinq»

QcI>= -sin<l> coso

which gives forq>=90°

Applying the operationQ90 to Eq (2.4) from the left, we obtain

Q9ni= Q90crQ90 1Q90F (2.5)Here we have inserted before the field vector the operation Q901 Q90. Q9"ol is theinverse operation ofQ90and thereforeQ901Q90 is the identity matrix

From a physical point of view, it is now important to note that for<p=90°, wehave done nothing but turned the (100) surface into itself Consequently, thecurrent density has to be related to the fieldin the same way before and after therotation Therefore,the conductivity matrix must be the same and

Eq (2.10) together with Eq (2.9) givesOxy = Oyx = 0, which together with

Eq (2.8) proves thatj and F point in the same direction on a (100) surface;that is, the surfaceis isotropic Wecan do the same proof for a(Ill)surfacethat

is turnedinto itself by a rotationof<P = 1200

• However, the (110) surfacehas

on-ly a<1> = 1800 symmetryand this is not enough to provethat this surfaceexhibitsisotropicbehavior Indeed,experiments show that the (110) surfaceconductivity

is anisotropic

Theseresultsare a specialcaseof a moregeneralrule: Anyphysicalpropertythatcan be represented as a matrixof rankrais scalarin crystallographic systemsthat can be transformed into themselves by a number of rotations (around allmain axes) largerthanr a.

The rankof a matrixis givenby the numberof indices Thus the conductivity

is of rank two; a matrix with elementsa;kem wouldbe of rank 4 In our example,the numberof rotationstransforming the surfaceinto itself was four for the (1(0)

surface (90° rotation), three for the (111) surface (1200 rotation), but only twofor the (110) surface (1800 rotation), which is not larger than the rank of theconductivity matrix

We can use this rule to determine in what form (scalar, vector, matrix) wedeal with certain physical properties of semiconductors of interest The con-ductivity and similarly the optical absorption, microwave conductivity, and the

26 Chap 2 The Symmetry of the Crystal Lattice

like are all matrices of rank two and therefore can be treated in cubic crystals

as scalar We can achieve this simplication without knowing the theory of theconductivity, which is actually developed in Chapter 8, just on the basis of thesymmetry properties of the crystal An even morepowerfulsymmetry, the trans-lationalinvariance is treated next

whereris the space coordinateandRlis a lattice vector In other wordsf( r)is

a function that is periodic with respect to all lattice vectorsR1•The periodicity

is a hint that Fourierexpansion willbea powerful mathematical tool to treat allthose functions f(r). Therefore, itis customary to introduce physical proper-ties of crystals in terms of Fourier series Because we are dealing with a three-dimensional entity withgivenperiodicity, any physicalpropertyor functionf(r)

nis the basic volumethat generatesthe crystal when repeatedover and over by

R, The subscripth ofKhis an integerand labelsthe vectorsK Thechoiceof thebasic volumeis not unique We could, for example, choose the cell that has thethree vectorsaI, a2, a3 asboundaries Wecan also choosethe so-calledWigner-Seitz cell, which is obtained as follows Connect all nearest neighboratoms bylines(aI, -81,82, -82,83, -83)and cut the connections in half by planes Thegeometrical figure enclosed by all these planes is the Wigner-Seitz cell Notethat this cell looks very different for face-centered cubic, body-centered cubic,and simplecubic lattices

However we choosethe cell, the volume is the sameand is givenby the

following productofthe three lattice generating vectors

21tb3= nal x a2

Welearnedin Chapter1aboutthe importance of a reciprocal latticevector,the vector27t/a(in one dimension) or any multiple of it Wehave seen that thewave vectorqof phonons is basically restricted to a zone-1t/a ::; q~1t/aandassumes the discrete values21CI/Na,where-N/2 s 1sN /2.

For a three-dimensional crystal, the possible values that the wave vectorq

can assume are

Because we like to discuss the consequences of translational invariance forelectrons, we need to know the consequences for the wavefunction '1'. If wetranslate the crystal into itself then it is not 'I' that is invariant but1'1'12, whichgives the probability density of finding the electron This means that'II canchange by a phase factor whose square equals unity as then1'1'12 is invariant