ADVANCED THEORY OFSEMICONDUCTOR DEVICES

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HIGH-TEMPERATURE ELECTRONICSEdited by Randall Kirschman1998 Hardcover 912 pp ISBN 0-7803-3477-9

ADVANCED THEORY OFSEMICONDUCTOR DEVICES

Karl HessUniversity of Illinois at Urbana-Champaign

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Hess, Karl, 1945-Advanced theory of semiconductor devices I Karl Hess.

p. cm.Includes bibliographical references (p. )."IEEE Electron Devices Society, sponsor.""IEEE Solid-State Circuits Society, sponsor."ISBN 0-7803-3479-5I. Semiconductors. I. Title.

TK7871.85.H475 1999621.3815'2--dc21 99-44500

CIP

To thememory of my fatherKarl Joseph Hess

CONTENTS

Preface

Acknowledgments

xiii

xv

Chapter 1 A Brief Review of the Basic Equations 11.1 The Equationsof ClassicalMechanics, Application

to Lattice Vibrations 21.2 The Equations of QuantumMechanics 9

Chapter 2 The Symmetry of the Crystal Lattice2.1 Crystal Structures of Silicon and GaAs2.2 Elements of Group Theory 22

2.2.1 Point Group 222.2.2 TranslationalInvariance 26

2.3 Bragg Reflection 29

1919

Chapter 3 The Theory of Energy Bands in Crystals 333.1 CouplingAtoms 333.2 EnergyBands by Fourier Analysis 343.3 Equations of Motion in a Crystal 423.4 Maxima of EnergyBands-Holes 463.5 Summary of Important Band-Structure

Parameters 503.6 Band Structure of Alloys 50

Chapter 4 Imperfections of Ideal Crystal Structure 574.1 ShallowImpurity Levels-Dopants 584.2 Deep Impurity Levels 604.3 Dislocations,Surfaces, and Interfaces 62

vii

viii Contents

Chapter5 Equilibrium Statistics for Electrons andHoles 675.1 Density of States 675.2 Probability of Finding Electrons in a State 735.3 Electron Density in the Conduction Band 75

Chapter 6 Self-Consistent Potentials and Dielectric Properties 816.1 Screening and the Poisson Equation in One

Dimension 826.2 Self-Consistent Potentials and the Dielectric

Function 83

Chapter 7 ScatteringTheory7.1 General Considerations-Drude Theory 897.2 Scattering Probability from the Golden

Rule 947.2.1 Impurity Scattering 947.2.2 PhononScattering 967.2.3 Scattering by a S-ShapedPotential 102

7.3 Important Scattering Mechanisms in Silicon andGallium Arsenide 103

89

Chapter 8 The Boltzmann Transport Equation 1098.1 Derivation 1098.2 Solutions of the Boltzmann Equation in the

Relaxation Time Approximation 1148.3 Distribution Function and Current Density 1218.4 Effect of TemperatureGradients and Gradients of

the Band Gap Energy 1258.5 Ballistic and QuantumTransport 1278.6 The Monte Carlo Method 129

Chapter 9 Generation-Recombination 1359.1 Important Matrix Elements 135

9.1.1 RadiativeRecombination 1359.1.2 Auger Recombination 139

9.2 Quasi-Fermi Levels (Imrefs) 1399.3 Generation-RecombinationRates 1409.4 Rate Equations 144

Chapter 10 The HeteroJunction Barrier 14710.1 Thermionic Emission of Electrons over

Barriers 147

Contents ix

10.2 Free Carrier Depletion of SemiconductorLayers 151

10.3 Connection Rules for the Potential at anInterface 153

10.4 Solution of Poisson's Equation in the Presence ofFree Charge Carriers 15410.4.1 ClassicalCase 15410.4.2 QuantumMechanicalCase 157

10.5 Pronounced Effects of Size Quantization andHeterolayer Boundaries 162

Chapter11 The DeviceEquations of Shockleyand Stratton 16711.1 The Method of Moments 16711.2 Moment for the Average Energy and Hot

Electrons 17011.2.1 Steady-StateConsiderations 17111.2.2 VelocityTransients and Overshoot 17511.2.3 Equation of Poisson andCarrier Velocity 176

Chapter12 Numerical DeviceSimulations 18112.1 General Considerations 18112.2 Numerical Solution of the Shockley

Equations 18412.2.1 Numerical Simulation Beyond the Shockley

Equations 188

Chapter13 Diodes13.1 SchottkyBarriers-ohmic Contacts 19413.2 The p-n Junction 201

13.2.1 Introductionand Basic Physics 20113.2.2 Basic Equations for the Diode Current 20713.2.3 Steady-StateCurrent in ForwardBias 21113.2.4 ACCarrier Concentrationsand Current in Forward

Bias 21313.2.5 Short Diodes 21513.2.6 Recombinationin DepletionRegion 21613.2.7 ExtremeForwardBias 21913.2.8 AsymmetricJunctions 22113.2.9 Effects in ReverseBias 223

13.3 High-Field Effects in SemiconductorJunctions 22613.3.1 Role of Built-InFields in ElectronHeating and p-n

JunctionCurrents 226

193

x Contents

13.3.2 Impact Ionization in p-n Junctions 22913.3.3 Zener Tunneling 23613.3.4 Real Space Transfer 240

13.4 Negative Differential Resistance andSemiconductor Diodes 241

Chapter 14 Laser Diodes 24714.1 Basic Geometry and Equations for Quantum Well

Laser Diodes 24814.2 Equations for Electronic Transport 25014.3 Coupling of Carriers and Photons 25314.4 Numerical Solutions of the Equations for Laser

Diodes 257

Chapter 15 Transistors 26515.1 Simple Models 266

15.1.1 Bipolar Transistors 26615.1.2 Field Effect Transistors 272

15.2 Effects of Reduction in Size, Short Chan-nels 27815.2.1 Scaling Down Devices 27815.2.2 Short Gates and Threshold Voltage 279

15.3 HotElectron Effects 28115.3.1 Mobility in Small MOSFETs 28115.3.2 Impact Ionization, Hot Electron Degrada-

tion 284

Chapter 16 Future Semiconductor Devices 29116.1 New Types of Devices 291

16.1.1 Extensions of ConventionalDevices 29116.1.2 Future Devices for Ultrahigh Integration 293

16.2 Challenges in Nanostructure Simulation 29516.2.1 Nanostructures in Existing Semiconductor

Devices 29616.2.2 QuantumDots 29716.2.3 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 TemperatureDependenceof the Band Structure 307

Appendix 0 Hall Effect and Magnetoresistance 309

Appendix E The Power Balance Equation 311

Appendix F The Self-Consistent Potential at a Heterojunction 315

Appendix G Schottky Barrier Transport 317

Index 321

About the Author 333

PREFACE

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

My goal 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 the SolidStateTheory edited by Landsberg, the encyclopedic description ofmost devices by Sze, and the text on numerical device simulation by Selberherr.

Karl HessUniversity ofIllinois at Urbana-Champaign

xiii

ACKNOWLEDGMENTS

I would like to express my sincere thanks to Wolfgang Fichtner, who has sti-mulated the revision and invested much time to give advice for improvement.B. G. Streetman, R. Dutton, andM. Lundstrom havegivenvaluable advicedur-ing many important stages of the development of this book. Others have con-tributed to various sections: P. D.Yoder to the section on thedensityof statesandMonteCarlo simulations, J. Bude to the sections on impact ionization throughhis insight, S. Laux to the chapteron diodesas the majorcontributor to the newtreatment of p-n junctions, Alex Trellakis to the explicit solution for the evenpartof thedistribution function, M.Grupen to the insights presented in the chap-teron laserdiodes basedonhis pathbreaking workon laserdiodesimulation, andL. F.Register to the theory of collision broadening in MonteCarlo simulationsand to someof the treatment of the electron phonon interactions.

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

Special thanks go to L. R. Cooper from the Office of Naval Research, toM. Stroscio from the ArmyResearch Office, and to George Lea from the Na-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.

The Beckman Institute of the University of Illinois and its first directors,T.L.Brown andJ. Jonas,haveprovided an idealenvironment 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 workandcorrecting my feebleattempts in IMEX.

Very special thanks go to my wife Sylvia, my daughterUrsula S., and mysonKarlH. for their lovingsupport.

Karl HessUniversity ofIllinois at Urbana-Champaign

xv

CHAPTER 1A BRIEF REVIEW OFTHERELEVANT BASICEQUATIONS OF PHYSICS

From a mathematical viewpoint, all equations of physics (both microscopic andmacroscopic) are relevant for semiconductor devices. In an absolutely strictmathematical way, we therefore would have to proceed from the fundamentalsof quantum field theory and write down the ~ 1023 coupled 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 oneof the 1023 equations.

Any realistic approach oriented toward engineering applications has to pro-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 inclusionof a factor of 2 in some equations).

Most effects of statistics can be understood classically, and we will needonly a very limited amountof 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 densityin 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 semi-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.

1.1 THEEQUATIONS OFCLASSICAL MECHANICS,APPLICATION TOLATTICE 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)coordinates.

This function is now called the HamiltonianfunctionH(pi,Xi). The laws ofmechanics are

and

dPi _dt -

aH(Pi,Xi)aXi

(1.1)

dx, aH(Pi,Xi)dt = dPi (1.2)

where t is time and i = 1,2,3. Instead of Xi, we sometimes denote the spacecoordinates by x,y,Z.

Some simple special cases can be solved immediately. The free particle(potential energy =zero) moves according to

H= LPi2/2mi

and we have from Eq. (1.1)

~i = 0; Pi= constant,

which is Newton's first law of steady motion without forces.If we have a potential energy V (Xl) that varies in the Xl direction, we obtain

from Eq. (1.1)

dp, _ aV(x.) - F. (1.3)dt --~= 0

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

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

Sec. 1.1 The Equations of Classical Mechanics, Application to Lattice Vibrations 3

o

o

o

o

o

oFigure 1.1 Displacement u;(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 can be reduced 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 hooked 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 displacements u;(r),where i = x,y,Z and r is the number (identification) of the atom. It is importantto note that r is not equal to the continuous space coordinate r in this chapter, al-though it has similar significance because it labels the atoms. The displacementof an atom in a set of regularly arranged atoms is shown in Figure 1.1.

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

T IM~ 2( ) h u(r) __ au;(r)= 2 ~Ui r were atn

(1.4)

(1.5)

andM is 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)+...n rs

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, which

makes 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 energyU, the first derivative vanishes and

We further have

Bi=O (1.7)

(1.8)Br:~ = iPu = 1JS{'I} dUi(r)duj(s) }l

We now use the fact that the crystal is translationally invariant-that is, wecan shift the coordinate system by s atoms (start to count s atoms later), and thecrystal is transformed into itself (at least if it is infinite). Therefore,

(1.9)

(1.11)

Furthermore, a rigid displacement (all Ui equal) of the crystal does not change Uand we therefore have, from Eqs. (1.5) and (1.9),

LB?J = 0 (1.10)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 of Xi:

. ( ) dH(Pi,Ui)Pi r =- dU;and

Eq. (1.11) gives

Pi(r) = MUi(r) (1.12)

(1.13)

Sec. 1.1 The Equations of Classical Mechanics, Application to Lattice Vibrations S

Figure 1.2 A ring of atoms representing cyclic boundary conditions.

Here, also, the indices m,n are used to number the atoms (as ',S above). There-fore

p;(r) = - LBijuj(s)s.]

and, together with Eq. (1.12), one obtains

Mu;(r)+LBijuj(s) = 0s,j

(1.14)

Remember that the index S in Eq. (1.14) runs over a large number of atoms;that is, up to about 1023 in a typical crystal. The r can also assume any of thesenumbers. In other words, we have about 1023 coupled equations to solve. Thissituation is very typical for any type of solid-state problem, but by far not ashopeless as it may seem. Powerful methods have been developed to reduce thenumber of equations and the following treatment is representative. We will as-sume for simplicity that the crystal is one dimensional and avoid the complicatedgeometrical arrangement of atoms in a real crystal. (We will learn more aboutthis when we discuss the electrons and their motion in crystals.)

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

B'" =1= 0 only for s = r 1Notice that we dropped the indices i, j because this is a one-dimensional

problem. Without any loss of generality, we may assume r = O. Then we have

BOs =/=0 for s = 1and

BOs= 0 otherwise

Furthermore,

according to Eq. (1.9) and

B-10 == BO(- 1)

according to Eq. (1.8). Therefore,

B01 =BO(- l )

From Eq. (1.10) we obtain

(1.15)

(1.16)

It is now customary to denote B01 = 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 any r

Mu(r) = -2au(r) +au(r-l)+au(r+ 1) (1.17)Eq. (1.17) leaves us still with 1023 coupled differential equations. However,

these equations are now in tridiagonal form, all with coefficient e, Such a formcan be reduced to one equation by skillful substitution. The substitution canbe 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)

Sec. 1.1 The Equations of Classical Mechanics, Application to Lattice Vibrations 7

v

_1!a

1!a

q

Figure 1.3 Dispersion relation v(q) for lattice vibrations. (Remember, E =hv.)

and

.. -4au. 2 (aq )U= --sIn -M 2

This gives

(1.20)

(1.21)

with

v =2lsin a2q

l~

This means that the atoms are oscillating in time with frequency v, which is afunction of the wave vector q. The function is shown in Figure 1.3.

There areseveral important points to notice. First,atq= -1t/ a andq = 1t/ a,the energy has its highest value. For these q, the wavelength A. = 21t/q has thevalue A= 2a. As can be seen from Figure 1.4, this is the shortest wavelengththat we really need to describe the physics of the lattice vibrations. Shorterwavelengths lead only to "wiggles" between the atoms, but the displacementsare actually the same. For example, if q = 31t/a and A= 2/3a, the atoms aredisplaced in exactly the same way as for q = tt]. In other words, for any qoutside the zone -1t/a ~ q ~ 1t/a, which is called the Brillouin zone, we canfind a q inside the zone that describes the same displacement, energy, and so

(\ v=~ f\ (/\ ,\ I \\/.'V v

Figure 1.4 Illustration of the shortest possible physical wavelength of lattice vibrations.

8 Chap. 1 A Brief Review of the Basic Equations

on. Notice that in a real crystal the arrangement of atoms is different in dif-ferent directions. Therefore, the three-dimensional Brillouin zone is usually acomplicated geometrical figure (seeChapter2).

Second, q is not a continuous variable because of the boundary conditions.Consider, for example, the ringof Figure 1.2witheight atoms and

u(O) = u(8)

or, in general, forN atoms, we have

u(N)= u(O) and u(N) = u(O)eiqNa

Therefore eiqNa equalsone, andwe conclude that q = 21tl/Nawhere1is aninteger. If werestrictq to the firstBrillouin zone,wehave-N/2 ~ I ~N/2. Thismeans that q assumes only discrete (not continuous) values. However, becauseof the largenumberN, it can almostberegarded as continuous.

Third, without emphasizing it, we have developed a microscopic theory ofsoundpropagation in solids. For smallwave vectors q (Le., largeA),we have

. qa qaSln- ~-2 2

and

v = ~qa (1.22)

UsingAV = vs, whereVs is the velocity of sound, we obtain

Vs = 2TCa~ (1.23)

whichis a microscopic description of the soundvelocity.In real crystals additional complications arisefromthe fact that wecanhave

two 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 chainwith 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, and the corresponding latticevibrations are calledopticalphonons. Itis very important to note that in principle all black atomscan oscillate in phaseagainstthe whiteones. Thismeansthat we can havehigh frequencies (energies)even if the wavelength is very large or the q vector is very small (Figure 1.5).

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

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

Sec. 1.2 The Equations of QuantumMechanics

v

Optic

9

7ta q

(1.24)

Figure 1.6 Schematic v(q) diagramfor acousticandopticphononsin one dimension.

The energy versus q relation 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 cannot be described by simple forces between neighbor-ing atoms, and one calls the phonons polar optical phonons if these long-rangeforces are important.

As mentioned, lattice vibrations are important in various ways. Electrons in-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. Itis only the sum of electons and lattice vibrations which is Hamiltonian.

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

1.2 THEEQUATIONS OFQUANTUM 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 always beviewed 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

hA= !Pi

10 Chap. 1 A Brief Review of the Basic Equations

where Ii == 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 understoodas boundary value problems. In his theory, electrons are representedby a wavefunction 'I'(r), which can have real and imaginaryparts, 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 nowcalled the Ham-iltonian operatorH. Formally this operator is obtained from the classicalHam-iltonian by replacing momentum with the operatorVn/i (i = imaginary unit),where

v= (~,~,~)The meaning of the wave function '1'(r) was not clearly understood at the

time Schrodingerderived his famous equation. It is now agreed that 1'I'(r)12 isthe probabilityof finding an electronin a volumeelementdr at r. In otherwords,we have to think of the electron as a point chargewith a statisticalinterpretationof its whereabouts (the wave-like nature). It is usually difficult to get a deeperunderstanding of this viewpoint of nature;evenEinsteinhad troublewith it. It is,however, a very successful viewpoint that describes exactly all phenomenaweare interestedin. To obtain a better feeling for the significance of W(r), we willsolve Eq. (1.25) for several special cases. As in the classical case, the simplestsolutionis obtainedfor constantpotential. Choosingan appropriate energyscale,we put V (r) =0 everywhere.

By inspectionwe can see that the functionCexp(ikr) ~ C(coskr+isinkr) (1.26)

is a solutionofEq. (1.25)with

(1.27)

andC a constant.The significance of the vector k can be understood from analogies to well-

knownwavephenomenain optics and from the classical equations. BecauseEis the kinetic energy, 1ik has to be equal to the classicalmomentump to satisfyE = p2/ 2m. On the other hand, in optics

[k] = 2rt/'"A (1.28)

which gives, togetherwithEq. (1.24),nk=p

which is consistentwith the mechanical result.

Sec. 1.2 The Equations of QuantumMechanics 11

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

1'I'(r)12 = leI2(cos2 k . r + sin2 k . r ) = ICl2This means that the probabilityof findingthe electron at any place is equal toe2Ifwe know that the electron has to be in a certain volume Vol (e.g., of a crystal),then the probabilityof finding the electron in the crystal must be one. Therefore,

{ ICl2dr =VoIICl2 = 1iVoI

and

lei = 1/v% (1.29)

(1.30)

In other words, the probability of finding an electron with momentum lik at acertain point r is the same in the whole volume and equals l/Vol. Wewill givea more detailed discussion of this somewhat peculiar result in the next section.The unfamiliar reader is referred to an introductory text (e.g., Feynman [3]).

Note that by confining the electron to a volume, we have already contra-dicted our assumption of constant potential V(r) = O. (Electrons can only beconfined in potential wells.) If, however, the volume is large, our mistake isinsignificant for many purposes.

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 potentialenergyV(r) is zero over the distance (O,L) on thex-axis and infiniteat the boundaries0 and L.

The Schrodinger equation, Eq. (1.25), reads in one dimension (x-direction,V(x) = 0)

_ r,,2 a2",(x) = E (x)2m ax2 'I'

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 finitewidth, the electron can even leak out of the well (tunnel).This is a very important quantum phenomenon the reader should be familiarwith. Wewill return to the tunneling effect below.

The wave function, Eq. (1.31), corresponds to energies E (called eigen en-ergies)

(1.32)

12 Chap. 1 A Brief Review of the Basic Equations

Because 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 states and are characterized by the quantum number n. Thewave function and corresponding energy are therefore also denoted by 'l'n, En.

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

E = hv (1.33)

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 besolved by standard discretization (transforming the differentiations into finite dif-ferences) and by solving the resulting matrix equations by standard solvers suchas found in EISPACK and LAPACK. 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 for n = 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 solverby Rutishauser [6].

Frequently, one needs to obtain an explicit solution of Schroedinger's equa-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

(HO+HI)'I' = 0 (1.34)

Sec. 1.2 The Equations of Quantum Mechanics 13

where Ho and HI are differential operators of arbitrary complication and Eis asmall positivenumber.

If we know the solution 'Vo of the equation

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

(Ho +EH1)('I'0+E'I'I) =Ho'l'o +Hl'1'O + HoE'I'l +E2Hl"'lWenowcan neglectthe termproportional to E2 (becauseE is small), and becauseHo'l'o == 0, we have

h-I"'0+HO'l'l =0 (1.35)This equation is nowconsiderablysimpler than Eq. (1.34) because '1'0 is known.Therefore, 'VI can be determined easily if Ho has a simple form no matter howcomplicated HI is. Repeated application of this principle leads to perturbationtheory including higher orders (EIE3.... ). The derivation is given in many text-bookson quantummechanics(seeBaym [1]). Here we quote only the result thatis used at severaloccasions.

Assumethat we know the solutionsof a Schrodingerequation:

HO"'n=En"'n n=1,2,3, ...and we would like to know the solutionsof

(1.36)

(1.38)

(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

~ IMmnl2Wm= Em +Mmm+ ...Jn=l=m Em-En

with

and

~ Mmn4>m == "'m+ ~ E -E 'JInn=l=m m n

(1.39)

Mmn = r 'I'~Hl'1'mdr (l.40)lVol

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

First-orderperturbation theory (to order e) amounts to setting 'I'm = cI>m andWm == Em +Mmm. The only change then is in the value of the eigen energy byMmm, which can be obtained by the integration in Eq. (1.40); the integrand isknown 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 thepotential 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 will be interested in the following type of problem: Theelectron initially is in an eigenstate ofHo, denoted, for example, by a wave vectork for the free electron. What is the probability that the electron will beobservedin a different eigenstate characterized by the wave vector k', after it interactswith a potential V(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, whichis also derived in almost every text on quantum mechanics by so-called time-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.

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(ky'i72

The function in brackets deserves special attention and is plotted in Fig-ure 1.7. Notice that as t approaches infinity, the function plotted in Fig-ure 1.7 becomes more and more peaked at its center (E(k') =E(k)). Inthe limit 1 --t 00, the so-called &-function is approached, which is definedby

lim 4sin2[(E(k' ) - E(kt/ (2/i)J = 2xo(E(k')_E(k (1.42)

t--+oo (E(k') - E(k) )21 Ii

Sec. 1.2 The Equations of QuantumMechanics 15

(1.45)

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 &-function is the following: For any continuousfunction f(E'), we have

Lco f(E')o(E -E')dE' = fee) (1.43)2. We assume that the perturbation is harmonic, which means we have a

potential of the form

V(r,t) = V(r)(e- irot +eirot )For t --+ 00, we obtain the transition probability

S(k,k') = 2: Il: '!'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 as t increases. 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"kconstant - e-' "e' "drVol Vol

The integration is over the volume Vol of the crystal. In manypractical cases, this volume will be much larger than the de Brogliewavelength Aof the electron, which is of the order of 100A in 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 oftenas 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)vollVol '

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

16 Chap. 1 A Brief Reviewof the Basic Equations

takencare of momentumconservation; the free electron in a constantpotential does not change its momentum.

(b) Second,weconsideran arbitrarypotentialhavingthe following Four-ier representation:

V(r) =I,Vqeiq.r (1.47)q

Then the matrix element, which we nowdenote byMk,k" becomes

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

=I,VqOt'-k,qq

How do we interpret this result? If we also allow the potential ofEq. (1.47) to have a time dependence(e.g., as eirot ), then the potentialcan be interpretedas that of a wave (e.g., an electromagnetic wave).In this caseEq. (1.49) simply tells us that the wavevectorsof all scat-tering agents (Le., their momenta) are conserved,because we have

k'-k=q (1.49)

It is important to notice that Eq. (1.49) is also valid for a static,time-independent potential-that is, evena staticpotential"supplies"momentumaccording to its Fourier components-in the same way awavedoes. This seemsstrangeat firstglance. Tosee the significance,consider the boundaryof a billiard table. This boundary is an impen-etrable abrupt potential step whose Fourier decomposition involvesall values of q. Indeed, the boundary can supply any momentum tothe ball to make it bounce back. The above two examples show thattheGoldenRule essentiallytakescare of energyandmomentumcon-servation. This is also the reason for its generality and importance.Remember, however, that this is true only for cases when time t atwhich we observe the scattered particle is long after the potential isswitchedon. For short times (in practice these are times of the orderof 10-14's), the function in Eq. (1.42) cannotbeapproximated by a 0function, and energyneed 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 moreexample is given.

(c) Consider the "tunneling problem" of Figure 1.8. Although an elec-tric field F is applied in the z-direction, the electron in Figure 1.8is confined in a small well. Classically, it would stay in the well.However, because the barrier is not infinite, as assumedin Eq. (1.31),the wave function is not zero at the well boundarybut penetrates the

Sec. 1.2 The Equations of Quantum Mechanics

Figure 1.8 Electrons in a potential well plus applied electric field F.

17

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'in in the well and 'l'ou out-side the well, and regard the electric field as a perturbation. Thisperturbation gives a term (the potential energy) eFz in 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 Op-

penheimer), I have swept under the rug the fact that Wou and Win arethe solutions of different Hamiltonians. 'l'in is 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

1i2V2H=----eFz

2mAn exact justification of this procedure is complicated and is dis-

cussed in great detail in Duke's treatise of tunneling [2] (see alsoAppendix A).

We emphasize that the matrix elements represent all that needs to be knownto obtain perturbation theory solutions. These matrix elements are given by threedimensional integrals. Alternatively they can be viewed as scalar products in avector space denoting 'l'n by 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 BasicEquations

PROBLEMS1.1 Solveby perturbationtheory (to first order) for y:

dysin(x) dye +-=ydX axwhere e is a small positivequantity.

1.2 Calculate the matrixelements for wave functionsof the form 'II= ~eik-r andyVol

(a) HI oc e;qr

(b) HI oc o(r)(c) HI oc Irl-2

(d) HI ee exp{ -}:' } ro >0 Polar coordinatesare helpful in parts c and d.

1.3 Consider a one-dimensional crystal latticewith two ions (atoms) repeated in a circulararrangement. The two ions (atoms) are identical, with massM, but are connected bysprings of alternatingstrength (Dl ,D2).(a) Derive the equations of motion. (Consider only nearest neighbor interactions,where the force is proportionalto the differencein displacements.)(b) Findand sketchthedispersionrelationof thepossiblevibrationalmodes. (Assumeall displacements are travelingwaveswithsinusoidaltimedependence, that is, U; (ra) ==E;ei(qra-rot) .)

(c) Discussthe formof the dispersionrelationand the natureof themodesfor q 1t/ aand q = 1t/ a, whereq is the wavevector.(d) Find the velocityof sound (ro/q for q ~ 0).(e) Showthat the group velocitydro/dq becomeszero at the Brillouinzone boundary.(This is a general result.)

REFERENCES

[1] Baym,G. Lectures on Quantum Mechanics. NewYork: Benjamin, 1969,p. 248.[2] Duke,C. B. Tunneling in Solids. NewYork: AcademicPress, 1969,p. 207.[3] Feynman, R. P. Lectures on Physics, Vol. III. Reading, MA: Addison-Wesley, 1964,

pp. 1.1-4.15.[4] Golub, G. H., and Loan, C. F.Matrix Computation. John Hopkins Univ. Press: Balti-

more, 1989.[5] Landsberg,P.T. SolidState Theory. NewYork: WileylInterscience, 1969,pp. 327-56.[6] Rutishauser, H. "Numerical eigenvaluesolver,"Numerical Mathematics, vol. 13, 1969t

p.4.

CHAPTER2THE SYMMETRY OFTHE CRYSTAL LATTICE

This chapter and the next three are a crash course in the solid-state physics un-derlying semiconductor devices. They can be read as a reminder to the readerof 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 cannot be accomplished by casual reading, but onlyby 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 objectof 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 anythingranging 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-coplanartranslations 81, 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 bya tetrahedral arrangement of the nearest neighbor atoms. Their lattice can beviewed 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

,e I ~----

// .(a) Simple cubic (b) Face-centered cubic

I'.,~----/

/

(c) Body-centered cubic

Figure 2.1 The three types of cubic Bravais lattices.

~~------~--~'-~ .. ~_. --~

--------l-----:--~

,~I I

!'

Figure 2.2 Crystal structure of silicon (or GaAs if two kinds of atoms are on the appropriatelattice sites). Notice the tetrahedral arrangement of nearest neighbor atoms andthe equivalence to a face-centered cubic lattice (if a two-atom basis is assumed).

Sec. 2.1 Crystal Structures of Silicon and GaAs 21

a

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 illustrated in Figure 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 twobasis atoms can vibrate against each other. By two sublattices, we mean thatwe can also view the silicon crystal as two interconnected face-centered cubiclattices (sublattices), each having one basis atom.

The three translation vectors 8t, 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 for lie

Figures 2.4a and 2.4b are photographs of a gallium-arsenide crystal modelin 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 crystal-lographic directions.

2.2 ELEMENTSOF GROUPTHEORY

2.2.1 Point Group

From a more mathematical viewpoint, it is important to note that crystal lat-tices-and with them all their physical properties-are transformed into them-selves by certain geometrical operations (rotations, reflections, etc.). The set ofall these operations is called the point group of the crystal lattice. Forty-eightsuch operations (translations are excluded for the moment) transform a cube in-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

Sec. 2.2 Elements of Group Theory

Table2.1 Elements of the PointGroup Td.

Ql (XIX2X3) Q2(Xli2i3) Q3(it X2i3) Q4(ili2X3)QS(X2X3X.) Q6(i2X3il) Q1(i2i3Xt} Q8(X2i3i.)Q9(X3XtX2) QlO(i3itX2) Qlt (X3Xli2) Q12(i3Xti2)Q13(itX3i2) QI4(it i3X2) Qls(.i3i2 Xl) Q16(X3i2Xl)Q17(X2XIX3) QI8(i2XI X3) QI9(XIX3X2) Q20(Xl i3 i2)Q21 (X3X2Xl) Q22(i3X2il) Q23(X2XI X3) Q24(i2ilX3)

Source: AfterMorgan, D. L, in Landsberg, P.T; 00., SolidStateTheory: Methods andApplications, TableC.7.!. Copy-right1969JohnWiley & Sons,Ltd. Reprinted bypermission.

operation Q3 (il ,X2,X3) means, for example,

23

for any function f 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 Chapter 3.)

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. Figure 2.5 shows the basicgeometry of a MOS transistor (which is described in detail in Chapter 15).

It is known that bulk silicon exhibits an isotropic conductivity 0". The in-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 transistorson various wafers of these three surface orientations. It was found that 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 can be obtained by a straightforward calcu-lation. The current density j as a function of electric field F is given by Ohm'slaw

j =oF (2.2)

In isotropic materials the conductivity 0" 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 Symmetryof 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.

sions x, y)

( ~; ) = (~: ~;) ( ~ )Denoting the conductivity matrix by 0 and the current density and electric fieldagain by their vectors j and F, we have

j=oF (2.4)

We now apply to Eq. (2.4) one of the symmetry operations Q of a regularsquare (our system is two dimensional, and the (100) surface has the symmetryof a square instead of a cube). The specific symmetry operation we choose isa rotation by 90. In the notation of Table 2.1 and in three dimensions, such arotation would be QOQ14.

Because we have given the conductivity in matrix form, we also would liketo express the rotation in matrix form. From calculus, we know that a rotationby an angle 4> can be represented by

(coso sinq

QcI> = -sin cosowhich gives for q> = 90

Q90= (_~ ~)

Applying the operation Q90 to Eq. (2.4) from the left, we obtain

Q9ni= Q90crQ901Q90F (2.5)Here we have inserted before the field vector the operation Q901Q90. Q9"ol is theinverse operation of Q90 and therefore Q901Q90 is the identity matrix

Sec. 2.2 Elementsof GroupTheory 2S

From a physical point of view, it is now important to note that for

26 Chap. 2 The Symmetry of the Crystal Lattice

like are all matrices of rank two and therefore can be treated in cubic crystalsas 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 evenmorepowerfulsymmetry, the trans-lational invariance is treatednext.

2.2.2 Translational Invarlance

Wehavenot yet discussedin detail the other typeof symmetry: the symmetry oftranslations. If we apply a translation Rl [see Eq. (2.1)] to a crystal, the crystalis transformed into itself (exceptat the boundaries, whichwe disregard). Wecannowargue (as we did for the point group) that anyphysicalproperty, denotedbyf( r), of the crystal has to be the samebefore and after this translation:

f(r+Rl)=f(r) (2.11)where r is the space coordinateand Rl is a lattice vector. In other words f( r) isa function that is periodicwith respect to all lattice vectorsR1 The periodicityis a hint that Fourier expansion will bea powerful mathematical tool to treat allthose functions f(r). Therefore, it is customary to introduce physical proper-ties of crystals in terms of Fourier series. Becausewe are dealingwith a three-dimensional entitywithgivenperiodicity, anyphysicalpropertyor function f(r)is expandedas

with

f(r) = LAKheiKhorKh

(2.12)

AK = -!. r!(r)e-iK"ordr (2.13)h 010

n is the basic volumethat generates the crystal when repeatedover and over byR,. The subscripth ofKh is an integerand labelsthe vectorsK. Thechoiceof thebasic volumeis not unique. Wecould, for example, choose the cell that has thethree vectorsaI, a2, a3 as boundaries. Wecan also choose the 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 choose the cell, the volume is the same and is given by thefollowing productof the three lattice generating vectors

n = 81 (82 x 83) (2.14)UsingEq. (2.11),we have

f(r+Rl) = LAKhei~o(r+Rl) =f(r)Kh

(2.15)

Sec. 2.2 Elementsof GroupTheory

and therefore,

FromEq. (2.16), it follows that

KhR, = 21t (integer)It can be shown (by inspection) that any vector

Kh =hlbl +h2b2+h3b3with

21tbl = n 82 x 83,

21tb2= na3xal,

21tb3 = n al x a2

27

(2.16)

(2.17)

(2.18)

satisfies Eq. (2.17).The vectors Kh are called reciprocal lattice vectors (their unit is em-1 ).

Thesevectors alsogenerate a lattice, the reciprocal crystal lattice, whichis com-plementary to the crystal lattice. Cubic lattices have reciprocal cubic lattices.However, the reciprocal latticeof a face-centered cubic crystal is body-centeredcubic, and viceversa.

Welearnedin Chapter 1 about the importance of a reciprocal latticevector,the vector27t/a (in one dimension) or anymultiple of it. Wehave seen that thewave vectorq of phonons is basically restricted to a zone -1t/ a ::;q ~ 1t/a andassumes the discrete values 21CI/Na, where -N/2 s 1s N /2.

For a three-dimensional crystal, the possible values that the wave vector qcan assume are

q=Kh/N with O::;lhll,lh21,lh31::;N/2 (2.19)Herewehaveassumed that the crystalcontains N repetitions of the basis in eachof themaindirections 81, 82, 83.

As mentioned, the largest physical values of Iql define an area called theBrillouin zone (Figure 2.6). This zone is theWigner-Seitz cell of the reciprocallattice, as canbeseen fromEq. (2.19). The conceptof the Brillouin zone is notonly important for phonons, but also for electrons. The relevance and signifi-canceof the zone conceptfor electrons can be seen fromBloch's theorem, andthe following discussions of Braggreflection.

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 but 1'1'12, whichgives the probability density of finding the electron. This means that 'II canchange by a phase factor whose square equals unity as then 1'1'12 is invariant.

28 Chap.2 The Symmetry of theCrystal Lattice

Figure 2.6 Brillouinzone of the face-centered cubic lattice (body-centered cubic reciprocallattice). Noticethat the zoneextendsto 21t/a in one direction (X) in contrasttothe one-dimensional zone. The labels I', X andK denotesymmetry points. T isthe zonecenter [0,0,0]; X theendpointin [1,0,0] direction; andL andK, the zoneendpoints in [1,1,1] and [1,1,0] directions, respectively.

Bloch found that the wave function 'V of an electron in a crystal can be labeledby a wavevector k (analogousto q for the phonons)and fulfills the relation

'I'(k, r + R[) = eik.R/'I'(k,r)

for any lattice vectorR/.It can be shownthat Eq. (2.20) is equivalentto [seeProblem 2.1]

whereUk (r) is a function periodicwith respect to R/. That is,

uk(R/ + r) = uk(r)

(2.20)

(2.21)

(2.22)

It is important to note that the wave function is unchanged if we replace k byk +Kh, whereKh is a reciprocallattice vector. This can be seen fromEq. (2.21);Uk(r) canbeexpanded intotheFourier seriesliven byEq. (2.15)because it ispe-riodic. The multiplication of this series bye; hr just leads to an identical series,which is only reordered in the sequence of reciprocal lattice vectors. Thereforeit is clear that the wave vector k, which for a free electron is proportional to themomentum,must have a differentmeaning in the crystal. To explore this mean-ing we will use perturbation theory by regarding the crystal as a perturbationofthe free electronbehavior. Detailedderivations of the aboveequationshavebeengiven in the literature [1].

Sec. 2.3 BraggReflection 29

o o o o o

o

o

oFigure 2.7 Bragg reflectionof electrons (waves)by crystal planes.

2.3 BRAGG REFLECTION

To calculate the matrix elements, we Fourier decompose the periodic crystalpotential V (r) :

V(r) = LBKheiKhorh

Thematrix element is then

(k'tV(r)lk)= LBKhOk-k',-Khh

andvanishes exceptfor

k+Kh =k'

(2.23)

(2.24)

(2.25)

The collision of the "light" electron with the "huge" crystal latticewill be elas-tic (we ignore lattice vibrations), and therefore the energy before and after thecollision willbe the same,whichis equivalent to

Ik( = fk'fSquaring Eq. (2.25) and usingEq. (2.26), we have

-2kKh= IKhl2

(2.26)

(2.27)

This represents the well-known condition for Bragg reflection. It is alsoclear that Bragg reflection occurs for k at the Brillouin zone boundary (solveEq. (2.27) in one dimension). Bragg reflection simply means that the electronsare reflected by crystalplanes, so Eq. (2.25)is valid(Figure 2.7).

Wereturnnowto the Blochtheorem [Eq. (2.21)] and regard uk(r) as a con-stant. One then recovers the wave function of the free electron, and lik is themomentum of the free electron. For a free electron, momentum is conserved. Ina crystal, k is not conserved because the crystalitself can contribute vectors Kh'as can be seen fromEq. (2.25). We now understand that there is not only onek attributed to the wave function, but all k +Kh-or, as statedbefore, all wavefunctions withwave vectors k+Kh are equivalent. In otherwords, wecan againrestrictourselves to use k within the Brillouin zone. If k lies outside the zone,we substract Kh untilweobtaina valueinsidethe zone.

30 Chap. 2 The Symmetry of the Crystal Lattice

What happens to the energy of the electrons? Is there a maximum energyat the Brillouin zone boundary as in the case of lattice vibrations? There is not;the electron energy as a function of k is multiple valued and only limited withincertain bands. A helpful analogy is the following.

We compare energy and wave vector with the true kinetic energy and rota-tional frequency of a spinning wheel in a movie. As the wheel spins faster andfaster, the picture of the wheel seems to stop as soon as the frequency of the mov-ing pictures Vo is the same as the spinning frequency of the wheel v. Increasingv leads to a picture in which the wheel seems to spin in the opposite directionuntil v = 2vo and then in forward direction again for 2vo~ v ::; 3vQ and so on.If we see only the movie, we do not know which kinetic energy or frequency vthe wheel really unless somebody tells us in which frequency range

nvO ~ v ~ (n+ l)vo (2.28)

the wheel is spinning.This situation is analogous in crystals with respect to the energy. For any

given k vector, the energy can be viewed as multiple valued and we need tospecify a range, or band, in order to find the energy E from k. The theory ofE(k), band-structure theory, is treated in Chapter 3.

PROBLEMS

2.1 Show the equivalence of Eqs. (2.20) and (2.21)

w(r+Rt} = e''kR/w(r)

and

where Uk(r) has the period of the lattice.

2.2 Find the surface atomic densities of (1(0), (110),and (111)planes in a silicon structurewith a lattice constant of Q. Which plane has the highest density?

2.3 (a) Find the reciprocal lattice of the three-dimensional face-centered cubic lattice. Useas lattice vectors (i, y, i are the unit vectors in the respective direction)

Q

81 = 2(i+y)82 = ~(~+z)83 = ~(z+i)

(b) Sketch the first Brillouin zone.

(c) Find the volume of the first Brillouin zone.

Reference 31

(d) Repeat parts a~ for the body-centered cubic lattice with primitive lattice vectorsa

81 = 2(X+Y- Z)82 = ~(-i+y+z)

a (A A A)83 =2 x-y+z

REFERENCE

[1] Madelung, O.Introduction to Solid State Theory. New York: Springer-Verlag, 1978, pp.36-55.

CHAPTER 3

THE THEORY OF ENERGYBANDS IN CRYSTALS

3.1 COUPLING ATOMS

In Chapter 2 we hinted at a band structure for the E(k) relation from ratherformal arguments. We now introduce the bands from phenomenological consid-erations.

Consider a series of quantum wells, as shown in Figures 3.1a through 3.1c.The wells in Figure 3.1a are separated and essentially independent. Each wellhas, therefore, a series of discrete levels. In Figure 3.1b, the wells are closertogether and coupled by the possibility of tunneling. This coupling causes asplitting of the energy levels into N closely spaced levels if we have N coupledquantum wells. The effect is much the same as the phenomenon associated withcoupled oscillators (the frequency of the oscillators then splits into a series offrequency maxima) or coupled pendulums in mechanics. We can see that thesingle levels are, therefore, replaced by bands ofenergies.

If the wells are coupled very closely together, a new phenomenon can hap-pen: It is possible that the bands spread more and overlap. Such an overlap istypical for metals. There is, however, an effect that can split up the bands, even ifwe put the wells closer and closer together. This happens, in fact, in semiconduc-tors such as diamond and silicon. The effect is known as bonding-antibondingsplitting, which is schematically explained in Figures 3.2a and 3.2b. The wavefunctions plotted in the figures give approximately the same probability for find-ing an electron in either well. However, the probability of finding an electronbetween the wells vanishes in Figure 3.2a and is finite in Figure 3.2b. The sit-uation is known from molecules where the electron holds together the positivenuclei of the ions by being in between them. Therefore, a state resembling thatin Figure 3.2b is called a bonding state, whereas Figure 3.2a (with no probabilityof finding an electron in between) represents an antibonding state.

Under certain circumstances, this separation into bonding and antibondingstates can lead to an additional splitting of the bands and therefore to the appear-ance of additional "energy gaps"-regions without states for the electrons. (In

33

34 Chap. 3 The Theory of Energy Bands in Crystals

(8)

(b)

(c)

Figure 3.1 Quantum wells coupled together with increasing coupling strength.

(a) (b)

Figure 3.2 Possible forms of wave function for two coupled wells.

the case of diamond and silicon, the splitting follows the formation of so-calledsp3 hybrids, whichis discussed in Harrison [5].)

3.2 ENERGY BANDS BY FOURIER ANALYSIS

Although theabove discussion establishes thebandsbymoving thewells(atoms)closer to each other, we can also go about this in another way. We can startwith a free electronand introduce the crystaljust by usingour knowledge aboutBrillouin zonesand restricting the energy function to this zone.Wethenhavetoplot the parabola of the free electron E(k) relation, E = li,2k2 / 2m, as shown inFigure3.3.

At firstglance, it seemsthatwehavedonenothing morethanreplota parabo-la in a verycomplicated way. However, wewill seein the following that theE(k)relation in a crystal is similar to Figure 3.3 except that at the intersections andzoneboundaries the function splits, as indicated by thedashedlines,whichleads

Sec. 3.2 Energy Bands by Fourier Analysis 3S

_2!a i k

Figure 3.3 The free electron parabola plotted in the Brillouin zone. Notice that the energynow becomes a multiple valued function of k and we have to label the differentbranches (numbers 1,2, 3, ... ) to distinguish among them.

to the formation of bands and energy gaps. A rather rigorous theory of the E(k)relation that contains most of these features automatically is described in thefollowing discussion.

This theory is based on direct Fourier analysis of the Schrodinger equation.Bloch's theorem tells us that the wave function can bewritten in the form

(3.1)

where Uk(r) is periodic. Therefore, wecanFourier expand Uk in terms of recip-rocallattice vectors as discussed inEq. (2.12) to obtain

W(k,r) = eJ'krLAKheiKhrh

(3.2)

Inserting Eq. (3.2) into the Schrodinger equation [Eq. (1.25)] with a periodic

36 Chap. 3 The Theory of Energy Bands in Crystals

crystal-potential V(r), we obtain

;,: L,lk+KhI2AKhei(k+~}.r+V(r) L,AKhei(k+Kh}rh h

=E(k) L,AKhei(k+Kh)r (3.3)h

The method of solving differential equations by Fourier transformation pro-ceeds now by multiplying Eq. (3.3) by a term, e-i(k+Kdor, and integrating overthe volume Vol of the crystal. Using Eqs. (1.46) and (1.51), we then obtain

(E? - E(k))AKI +LAKh(K1IV(r)IKh) = 0 (3.4)h

where we have used the definition

1i2 2 02mlk+Ktl =Et (3.5)

Because Eq. (3.4) is homogeneous, a nonzero solution for the AKh existsonly if the determinant of the coefficients vanishes; that is,

(3.6)

where 1and h are integers (I, h = 0,1,2, ... ).Equation (3.6) is called the secular equation and gives us the possible val-

ues of the energy E(k) as a function of wave vector k. It can only be solvednumerically; a large number of Fourier components (Kh) are usually necessaryto give a reasonable description of energy bands in semiconductors. However,we can see some significant features of the solution if we restrict ourselves to aone-dimensional model with two reciprocal lattice vectors 0 (h = 0) and -21t/a(h = -1), which corresponds to one incident and one reflected wave. Then theonly matrix element that matters is (OIV(r)l- 1), which we denote by M. Thesecular equation then reads (putting (OIV(r)10) = (-lIV(r)l- 1) = 0 by properchoice of the energy scale):

which givesIEg-E(k) M

M* E~l -E(k) 1=0 (3.7)

E(k) = ~(E8 +E~l) ~J(E8 -E~1)2 +41M12 (3.8)withM* being the complex conjugate ofM. Plotting this E(k) relation gives ex-actly the first two bands (0 and 1) of Figure 3.3 with the splitting at the Brillouinzone boundary of magnitude 21MI. This splitting is called the energy gap EG.

The student of E(k) relations may wish to pause here and consider the spe-cial case V(r) = O. As can be seen from Eq. (3.6), this leads to the empty lattice

Sec. 3.2 Energy Bands by Fourier Analysis 37

(3.9)

bands E(k) =E? Note, however, that owing to the complicated reciprocal latticevectors of the cubic face centered semiconductors, the bands are more compli-cated than in Figure 3.3. Nice examples are given in [7].

The calculation, as we have presented it, looks rather simple, and the ques-tion arises why the band structures of a broad class of semiconductors have beenaccurately calculated only after 1965. One of the reasons for the problems inband-structure calculations is the fact that it is difficult to calculate the potentialV (r). It is clear that V (r), the potential an electron "feels," is generated not onlyby the atomic nuclei but also by all the other electrons in the crystal. Therefore,it is not even true that the potential can bewritten as a locally defined functionV(r). However, experience has shown that this is not too bad an approximationfor most semiconductors. But how do we determine V(r)?

From Eq. (3.6) we can see that it is not even necessary to know V(r). Allone needs are the matrix elements-that is, its Fourier components with respectto reciprocal lattice vectors. Cohen and Bergstresser [3] have demonstrated thatonly a few Fourier components are necessary to obtain relatively accurate E(k)relations if they are chosen wisely. Below we discuss how this choice should bemade. We start from the fact that the potential V (r) must be a sum of all thecontributions of the atoms that constitute the crystal.

If the atoms are located at Rl' by introducing a suitable function to, we canwrite

V (r) = L ro(r - R~)I

R~ is not necessarily a lattice vector. In the case of silicon, we have two atomsin the Wigner-Seitz cell and R~ has to reach both. The matrix element can bewritten as a Fourier coefficient obtained from

Mmn=(KnIVIKm)=: r Lro(r-RDe-iKordr01 JVOI I

(3.10)

(3.11)

where we have properly normalized the wave function to the crystal volume, theintegration extends over this volume, and K = Km - K n (m and n are integerindices as I and h above).

We now rewrite Eq. (3.10) as

u.; = _1 Le-iKoR; r e-iKo(r-RDro(r-RDd(r-RDVol 1 lVol

Notice that we can replace r - R~, by r' in the integral, which then does notdepend on the index I. Remember also that we have two atoms (ions) in theWigner-Seitz cell. To account for this, we label the position of the second atomby R1+t and position the first atom at Rl. R1 is now a lattice vector. (It isalso common to position R, in between the two atoms and to add and subtract avector t/2 to reach the two atomic positions.) Therefore, using Eq. (2.17), we

38 Chap, 3 The Theory of Energy Bands in Crystals

can rewriteEq. (3.11) as

Mmn = ~(1 +e-iKo'C)_l r e-iKor'ro(r')dr' (3.12)I Vol lv:

where the labell now runs over all Wigner-Seitz cells (not atoms). The functionro{t) is the same in eachWigner-Seitz cell.

We assume also that ro{r') vanishes rapidly if we go outside the cell andtherefore

r e-iKor' ro(r')dr' = re-iKor'ro(r') dr' (3.13)lv; io

wheren is the volumeof theWigner-Seitz cell givenby Eq. (2.14). Because thevolumeof the crystal Vol =NO, we finally obtain fromEq. (3.12)

Mmn = (1 +e-iKo't)~ re-iKor'ro(r') dr' (3.14)0.10The first term in Eq. (3.14) is the structure factorS{K)

S(K) = (I +e-iKo'C) (3.15)whereas the second term is the form factor. The structure factor can easily becalculated from the reciprocal lattice vectorsKm - Kn = K.

The reciprocal lattice vectors are the vectors of the cubic body-centeredre-ciprocallattice (e.g., for silicon)givenby

Kh = hlbl +h2b2 +h3b321t

= -(-hI +h2 +h3,hl -h2 +h3,hl +h2 - h3) (3.16)awhere hI, h2, h3 are integers. It is important to realize that there is a particularform of the combinationof integers that enters the vector componentsof Kh andnot all integer combinations are possible. For example, Kn = 2;(1,0,0) is notallowed because it would correspond to values of b: = h3 = 1/2, which is notintegral and thereforenot permitted. This is the reason for the complicatedformof empty latticebands for three-dimensional cubic face-centered lattices. Amoreextensiveillustrationcan be found in the book by Landsberg [7] on page 222.

It remains to determine the form factors. Cohen and Bergstresser [3] as-sumed that only reciprocal lattice vectorsK of squaredmagnitude0, 3, 4, 8, and11 (in units of 21t/a) contribute and all the other Fourier components vanish.Therefore, the form factor is replacedby a few unknowns. These fiveunknownsare adjusted to obtain an E(k) relation that fits best the existing experiments(optical absorption,etc.). Band-structure calculations are then reduced to the so-lution of the systemofEqs. (3.4) and (3.6), whichcan beachievedwith standardnumerical routines.

In this way,Cohen and Bergstresserfound theE(k) relation of many semi-conductors. Their results for Si, GaAs, and other materials are shown in Fig-ure 3.4. The k vectors are plotted along some of the major directions in the

Sec. 3.2 Energy Bandsby FourierAnalysis 39

Brillouin zone (see Figure 2.6) from r to X, r to L, and so on. Not all bandsare shown in the figure---()nly the most important ones that contribute to elec-tronic conduction. These are the highest bands that are still filledwith electrons,the valance band, and the next higher band separatedby an energy gap Eo, theconductionbands.

The indices of T, L, and X in Figure 3.4 denote the symmetry of the wavefunctions and their behavior with respect to transformations of the coordinatesx, y, z. r 1 means, for example, that the wave function at this point has s sym-metry (i.e., is spherically symmetric); r15 means that the wave function has psymmetry (i.e., is cylindrically symmetric around the x, y, z axes); and so on.The methoddescribedabove is called the empirical pseudo-potential method.

Toobtain a more completeviewof the band structure,it is customary to plotlinesor surfacesof constantenergyin k space. Sucha plot is shownin Figure3.5.The label on the curves is the energyof the particularcurve in electronvolts. It isimportantto note that for free electronsthe lines of equal energywouldbe circles(disregarding relativistic effects). The Bragg reflection from crystal planes givesrise to the very complicatedpattern of Figure 3.5. Only at very low energies incertainbands do the lines approachcircles (for silicon, even this is not true).

In actual calculations, it is costly to compute theE (k) relation for all pointsof the Brillouin zone. However, we know from Chapter 2 that 48 operationstransformthe cube into itself and the same48 operations transformthe Brillouinzone of a cubic lattice into itself. We, therefore, need to calculate theE(k) rela-tion only in 1/48 of the Brillouin zone and then apply the symmetryoperationsof Table2.1 (replacing XlX2X3with kxkykz) to obtainE (k) everywhere, because

QE(k)=E(k') =E(k) (3.17)The 1/48 part of the zone that can be turned into the full Brillouin zone by

the operationsQ is givenby the conditions

and

k* +k* +k* < 3/2x y z -

(3.18)

(3.19)

Here the starred k components are normalized by 21t/a; for example, k; ==kz{21t/a). The volumedefinedby Eqs. (3.21) and (3.22) is called the irreduciblewedge, and is shown in Figure 3.6. It should be noted that the translational in-variance leads also to an important law forE(k):

(3.20)

for the very same reasonswhich lead to Eq. (3.17).

40 Chap. 3 The Theory of Energy Bands in Crystals

r-4 .........-----.....- ......- ....- .....- ...L

L \- I 6 .; .~1~ GaP I '-'e~..,~.....\ K"'~

"1154 r / /.,

;~....V'...~/ \>' 2 K1 r1d~ X,lU ~5o L3 fJ.. _.-a-h....... K ,--:/-,......... e-, 2 .../. //).

f 1s \ 'a ,.' I-2 .\ 'e.....~........ / \ XS'.K1/ /

r k X KBandStructure of GaP

L r k x K rBandStructure of Si

6 L \r2.- j l~r24 ....-t~\ K1 _... ', ,~ aL, ...... r,S'. \ / r,5

C" 2 ~.... '.'. 'eX1~ /~ ... '-.!.,.K3..:. 0 r2S' ...... r25,lU ,..it'~.\ /if.......:t......... ,-. ,. r;-2 L3, ,' \' ,./ (/

'-~....K / I-4 l \ X4"' 2 lI \ , ......". K,

r- 4 .........- .....- ......--'--.-.&.-......"-----'L

6 ~r / GaAsf\~1~

[3......~. /K .'.::~4 ...".... 1 'e / ..........,.... '. X3 ~/ '.>' 2 L1 ....,f1,. ..--....~~....,.K \Q) ....., 1 r- '. X1 1~ ~So 4 -;

........... ,/~ \ ...., K2",," ' /if-2

L 15 \ I3. \ __ I \ X~I.

r k X KBandStructure of GaAs

rr k X KBandStructure of Ge

L

r

,../-4 "-....._-'-_--..l........... ~_......

L

6 \ I AISb ~... r r I ! .,. ~1~...., 15., ':...':~4 L3 '.1:........ IK, ....:::.J

'- X ...........~ ,".. 3. . ">' ..,11 ;/. ?K- Q) 2 .. ...-. 1 r- L, X1 1-1lU 0 ~51...i..r...,\....--. K2 ..........;;7...,_.e

L rf '. ./.-2 3 , 15. ...... __...,.# /\ IlIii;_ ~" /

.\ X3'-......."e I\ K,

r k X KBandStructure of AISb

rr k X KBandStructure of Sn

-4 ....---.....- ....._ ......- ......--'--....L

6 ...-......-...,.---.....- .....----.

Figure 3.4 Band structure of important semiconductors. [After Cohen and Bergstresser [3].]

Sec. 3.2 Energy Bands by Fourier Analysis

~1.5x

41

Figure 3.5 Lines of equal energy in k space for a certain cut of the Brillouin zone and theconduction band of GaAs. [After Shichijo and Hess [9], Figure 4.]

Figure 3.6 Sampling region for the calculation of the band structure. The region is a 1/48part of the Brillouin zone.

42 Chap. 3 The Theory of Energy Bands in Crystals

3.3 EQUATIONS OF MOTION IN A CRYSTAL

How does an electron move in such a complicated energy band? The equations ofmotion are simpler than one would expect. We sketch only the derivation; the fullderivation takes considerable space. It is shown in Appendix B that if we restrictourselves to the consideration of one particular band (one band approximation),the Schrodinger equation can bewritten as

(E( -iV) - eVext)W = EW (3.21)

(3.22)

if a weak external potential Vext is applied to the crystal.Here E(-iV) simply means that we take the function E(k) and Taylor-

expand it. Then we replace all k by -iV. To give a one-dimensional example,we have

( .a) aE I a 1a2E I a2E -1- =(0)- - 1--- - -ax ak k=O ax 2 ak2 k=Oax2

This equation, viewed as a differential equation, is, of course, more difficultto solve than the original Schrodinger equation. The given form, however, isuseful for the general considerations that are discussed below because it does notcontain the unknown crystal potential.

To proceed, we need to invoke Ehrenfest's theorem, which gives the law ofmotion for the mean values of coordinates and momenta of a quantum system. Itsays that Eq. (1.2) is replaced by the following equation for the average velocitycomponent Vi:

. _ d('I'lxil'l') _ ~( laB, ) (3.23)V, - dt - n 'I' ak; 'I'

If we take the Hamiltonian from Eq. (3.21) and a Bloch wave function for 'I' asin Appendix B, we obtain from Eq. (3.23)

a aiJkjHI",) = iJkjE(k) I",) (3.24)

which gives for the vector of the average velocity:

dr 1v= dt = h,VkE(k) (3.25)

where

( a a a)Vk= iJkx ' iJky ' iJkzThe momentum changes only in time if we apply an external electric (or mag-netic field), and from the Ehrenfest's theorem one obtains as in Eq. (1.1) withF= -VVext

dkn- = -eFdt

(3.26)

Sec. 3.3 Equations of Motion in a Crystal 43

(3.27)

whereF is the electric field.Equation (3.26) is identical to the equation for a free electron and is true

only becausewe restrictourselves to one band. Equation(3.25) tells us that thevelocity v of the electron points always perpendicular to the curves of constantenergy in k space(because it is proportional to the gradientofE(k)O. Therefore,the motion of an electron in a crystal can be much more complicated than themotion of a free electron.

Whycanwedescribe the conductivity of a metaland semiconductor in sim-ple terms? After understanding the complicated relation between energy andmomentum as a consequence of Bragg reflection in the crystal, it is surprisingthat simplemodels that regard the electrons as free (and assumea quadratic re-lationbetween energyandmomentum) haveworked so well in the past. In fact,elementary introductions to semiconductor physicsand electronics can be givenwithoutmuchknowledge of thebandstructure at all. The reasonis that electronsreside, at least when close to equilibrium (small current densities), close to theminima of theE(k) relation. Then we can expand theE(k) relation in a Taylorseries (in onedimension):

, dEI ' 1 (j2E I ' 2E(k) =E("o) + i)k k=1Q) (k - "0) +2 i)k2 k=1Q) (k - "0) +...where ko designates the location of the minimum (maximum) of the E(k) re-lation. We now can choose the energy scale so that E(ko) =O. Furthermore,because the first derivative vanishes at an extremum, we may rename (IC - '(0)by k and the number

to obtain

bym*

(3.28)

E(k) == 1i2kl/2m* (3.29)whichis the equation for thefreeelectronwiththemassreplacedby an "effectivemass"m",

If the surfaces of equal energy in k spaces are ellipsoidal (as they are insiliconandgermanium), oneobtainsby the samereasoning, usingthe coordinatesystemof themainellipsoidal axes

,.,,2E(k) =~ 2m~ kJ (3.30)

} }

In silicon, the minimaof the conduction band do havean ellipsoidal shapewiththe twomasses mi =O.91mo, andm: = O.19mo (I standsfor the longitudinal andt for the transverse axis of the ellipsoidof revolution). This seemspuzzling; weknowfromgroup theorythat the conductivity of silicon is isotropic. The expla-nation is that silicon has six equivalent ellipsoids (minima) in the [100], [-100],

44 Chap. 3 The Theory of Energy Bands in Crystals

(3.31)

(3.32)

[010], [0-10], [001], and [OO-I]directions, andthe average overall ellipsoids givean isotropic conductivity massma of

~. = ~ (~. + ~.)a 1 tThis effective mass treatment close to a minimum can be put in a verygen-

eral form, because not only functions of numbers but also functions of operatorscan bewritten as a Taylor series [Eq. (3.22)]. This general form is the effec-tive mass theorem. The effective mass theorem is derived by expanding the oneband approximation formula [Eq.(3.21)] into theTaylor seriesofEq. (3.22) andtruncating after the third term (the term proportional to a2 jail). Weak externalpotentials Vext can be simplyaddedin as they are in Eq. (3.21). Of course, theymust notbe strongenoughto perturbthe band structure itself,which is the caseif all theirFouriercomponents (withrespectto the reciprocal latticevectors) aresmallerthan the Fouriercomponents of the crystalpotential energy V (r) .

The effective mass theorem can then be stated in the following form. As-sumewe havea periodic crystalpotential energy V(r) and additional (althoughweaker in all Fourier components) external potentials Vext. If one is interestedonly in the properties near an extremum E(ko) of the E(k) relation, then theSchrodinger equation

(1i,2 2 )- 2mV +V(r) - eVext 'I' ='1'

whichis equivalent to

can be replaced by(E(-iV) - eVext)'I' = E'I' (3.33)

(3.34)( - ~;:~ :~ -eVext) is the so-called envelope wave function, which contains all the ef-fects of the external potential (as long as it varies slowly) while all the effects ofthe crystal potential are absorbed in the effective massmI: The effective masstheorem then provides a newequation for the envelope wave function in whichthe crystal potential has disappeared, and all we had to do for this is to replacethe mass by an effective mass and count the energy from the minimum E(ko).The ellipsoidal formof Eq. (3.34) can be transformed into the familiar isotropicspherical formby suitable coordinate transformtions.

Equation (3.34) is basicfor the theory of electron devices. It forms the basisfor the theory and definition of holes, donor and acceptor states, and a basis toseparatethe electron energy in the familiar form of kinetic energy above E