Quantum Kinetic Theory and Applications: Electrons, Photons, Phonons

Quantum Kinetic Theory and Applications

Electrons, Photons, Phonons

Quantum Kinetic Theoryand Applications

Electrons, Photons, Phonons

FEDIR T. VASKOOLEG E. RAICHEV

Institute of Semiconductor PhysicsNAS of Ukraine, Kiev

Fedir T. Vasko Oleg E. RaichevInstitute of Semiconductor Physics, NAS Institute of Semiconductor Physics, NAS45 Prospekt Nauki 45 Prospekt NaukiKiev 03028 Ukraine Kiev 03028 Ukraine

Library of Congress Control Number: 2005926337

ISBN-10: 0-387-26028-5 e-ISBN: 0-387-28041-3ISBN-13: 978-0387-26028-0

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Preface

Physical kinetics is the final section of the course of theoretical physicsin its standard presentation. It stays at the boundary between gen-eral theories and their applications (solid state theory, theory of gases,plasma, and so on), because the treatment of kinetic phenomena alwaysdepends on specific structural features of materials. On the other hand,the physical kinetics as a part of the quantum theory of macroscopicsystems is far from being complete. A number of its fundamental is-sues, such as the problem of irreversibility and mechanisms of chaoticresponses, are now attracting considerable attention. Other importantsections, for example, kinetic phenomena in disordered and/or stronglynon-equilibrium systems and, in particular, phase transitions in thesesystems, are currently under investigation. The quantum theory of mea-surements and quantum information processing actively developing inthe last decade are based on the quantum kinetic theory.

Because a deductive theoretical exposition of the subject is not con-venient, the authors restrict themselves to a lecture-style presentation.Now the physical kinetics seems to be at the stage of development when,according to Newton, studying examples is more instructive than learn-ing rules. In view of these circumstances, the methods of the kinetictheory are presented here not in a general form but as applications fordescription of specific systems and treatment of particular kinetic phe-nomena.

The quantum features of kinetic phenomena can arise for several rea-sons. One naturally meets them in strongly correlated systems, when itis impossible to introduce weakly interacting quasiparticles (for exam-ple, in a non-ideal plasma), or in more complicated conditions, such asin the vicinity of the phase transitions. Next, owing to complexity ofthe systems like superconductors, ferromagnets, and so on, the manifes-tations of kinetic phenomena change qualitatively. The theoretical con-

v

vi QUANTUM KINETIC THEORY

sideration of these cases can be found in the literature. Another reasonfor studying quantum features of transport and optical phenomena hasemerged in the past decades, in connection with extensive investigationof kinetic phenomena under strong external fields and in nanostructures.The quantum features of these phenomena follow from non-classical dy-namics of quasiparticles, and these are the cases the present monographtakes care of, apart from consideration of standard problems of quan-tum transport theory. Owing to intensive development of the physics ofnanostructures and wide application of strong external (both stationaryand time-dependent) fields for studying various properties of solids, thetheoretical methods presented herein are of current importance for anal-ysis and interpretation of the experimental results of modern solid statephysics.

This monograph is addressed to several categories of readers. First,it will be useful for graduate students studying theory. Second, the top-ics we cover should be interesting for postgraduate students of variousspecializations. Third, the researchers who want to understand the back-ground of modern theoretical issues in more detail can find a numberof useful results here. The phenomena we consider involve kinetics ofelectron, phonon, and photon systems in solids. The dynamical prop-erties and interactions of electrons, phonons, and photons are brieflydescribed in Chapter 1. Further, in Chapters 28, we present main the-oretical methods: linear response theory, various kinetic equations forthe quasiparticles under consideration, and diagram technique. The pre-sentation of the key approaches is always accompanied by solutions ofconcrete problems, to illustrate applications of the theory. The remain-ing chapters are devoted to various manifestations of quantum transportin solids. The choice of particular topics (their list can be found in theContents) is determined by their scientific importance and methodolog-ical value. The 268 supplementary problems presented at the end of thechapters are chosen to help the reader to study the material of the mono-graph. Focusing our attention on the methodical aspects and discussinga great diversity of kinetic phenomena in line with the guiding principlea method is more important than a result, we had to minimize bothdetailed discussion of physical mechanisms of the phenomena consideredand comparison of theoretical results to experimental data.

It should be emphasized that the kinetic properties are the impor-tant source of information about the structure of materials, and manypeculiarities of the kinetic phenomena are used for device applications.These applied aspects of physical kinetics are not covered in detail either.However, the methods presented in this monograph provide the theoret-ical background both for analysis of experimental results and for device

PREFACE vii

simulation. In the recent years, these theoretical methods were appliedfor the above-mentioned purposes so extensively that any comprehensivereview of the literature seems to be impossible in this book. For thisreason, we list below only a limited number of relevant monographs andreviews.

Fedir T. VaskoOleg E. RaichevKiev, December 2004

Monographs:1. J. M. Ziman, Electrons and Phonons, the Theory of Transport Phenomena in

Solids, Oxford University Press, 1960.2. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, W. A. Benjamin,

Inc., New York, 1962.3. A. A. Abrikosov, L. P. Gorkov and I. E. Dzialoszynski, Methods of Quantum

Field Theory in Statistical Physics, Prentice-Hall, 1963.4. S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics,

Saunders, PA, USA, 1966.5. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bu-

reau, New York, 1974.6. E. M. Lifshitz and L. P. Pitaevski, Physical Kinetics, Pergamon Press, Oxford,

1981.7. H. Bottger and V. V. Bryksin, Hopping Conduction in Solids, VCH Publishers,

Akademie-Verlag Berlin, 1985.8. V. L. Gurevich, Transport in Phonon Systems (Modern Problems in Condensed

Matter Sciences, Vol. 18), Elsevier Science Ltd., 1988.9. V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Metals and Semi-

conductors (Modern Problems in Condensed Matter Sciences, Vol. 19), Elsevier Sci-ence Ltd., 1987.

10. A. A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, 1988.11. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic

Properties of Semiconductors, World Scientific, Singapore, 1990.12. N. N. Bogolubov, Introduction to Quantum Statistical Mechanics, Gordon and

Breach, 1992.13. G. D. Mahan, Many Particle Physics, Plenum, New York, 1993.14. H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of

Semiconductors, Springer, Berlin, 1997.15. Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 1997.16. D. K. Ferry and S. M. Goodnick, Transport in Nanostructures, Cambridge

University Press, New York, 1997.17. R. P. Feynmann, Statistical Mechanics, Addison-Wesley, 1998.18. A. M. Zagoskin, Quantum Theory of Many-Body Systems: Techniques and

Applications, Springer-Verlag, New York, 1998.19. F. T. Vasko and A. V. Kuznetsov, Electron States and Optical Transitions in

Semiconductor Heterostructures, Springer, New York, 1998.

viii QUANTUM KINETIC THEORY

20. J. Rammer, Quantum Transport Theory (Frontiers in Physics, Vol. 99), West-view Press, 1998.

21. T. Dittrich, P. Hanggi, G.-L. Ingold, B. Kramer, G. Schon, and W. Zverger,Quantum Transport and Dissipation, Wiley-VCH, Weinheim, 1998.

22. B. K. Ridley, Quantum Processes in Semiconductors, Oxford University Press,1999.

23. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation, Springer, Berlin, Heidelberg, New York, 2000.

Reviews:1. D. N. Zubarev, Double-Time Greens Functions, Sov. Phys. - Uspekhi 3, 320

(1960).2. R. N. Gurzhi and A. P. Kopeliovich, Low-Temperature Electrical Conductivity

of Pure Metals, Sov. Phys. - Uspekhi 133, 33 (1981).3. T. Ando, A. B. Fowler, and F. Stern, Electronic Properties of Two-Dimensional

Systems, Rev. Mod. Phys. 54, 437 (1982).4. J. Rammer and H. Smith, Quantum Field-Theoretical Methods in Transport

Theory of Metals, Rev. Mod. Phys. 58, 323 (1986); J. Rammer, Quantum TransportTheory of Electrons in Solids: A Single-Particle Approach, Rev. Mod. Phys. 63, 781(1991).

5. G. D. Mahan, Quantum Transport Equation for Electric and Magnetic Fields,Physics Reports 145, 251 (1987).

6. W. R. Frensley, Boundary Conditions for Open Quantum Systems Driven Farfrom Equilibrium, Rev. Mod. Phys. 62, 745 (1990).

7. B. Kramer and A. MacKinnon, Localization: Theory and Experiment, Rep.Prog. Phys. 56, 1469 (1993).

8. C. H. Henry and R. F. Kazarinov, Quantum Noise in Photonics, Rev. Mod.Phys. 68, 801 (1996).

9. C. W. J. Beenakker, Random-Matrix Theory of Quantum Transport, Rev. Mod.Phys. 69, 731 (1997).

10. Ya. M. Blanter and M. Buttiker, Shot Noise in Mesoscopic Conductors,Physics Reports 336, 1 (2000).

11. P. Lipavsky, K. Morawetz, and V. Spicka, Kinetic Equation for Strongly In-teracting Dense Fermi Systems, Annales de Physique 26, 1 (2001).

Contents

Preface v

1. ELEMENTS OF QUANTUM DYNAMICS 11. Dynamical Equations 12. S-Operator and Probability of Transitions 63. Photons in Medium 114. Many-Electron System 175. Electrons under External Fields 256. Long-Wavelength Phonons 35Problems 46

2. ELECTRON-IMPURITY SYSTEM 517. Kinetic Equation for Weak Scattering 518. Relaxation Rates and Conductivity 579. Quasi-Classical Kinetic Equation 6510. Multi-Photon Processes 7211. Balance Equations 7912. Conductance of Microcontacts 87Problems 94

3. LINEAR RESPONSE THEORY 9913. Kubo Formula 9914. Diagram Technique 10715. Bethe-Salpeter Equation 11516. Greens Function as a Path Integral 12317. Dispersion of Dielectric Permittivity 130

ix

x QUANTUM KINETIC THEORY

18. Interband Absorption under External Fields 139Problems 147

4. BOSONS INTERACTINGWITH ELECTRONS 15519. Kinetic Equation for Boson Modes 15520. Spontaneous and Stimulated Radiation 16221. Phonon Instabilities 16922. Boson Emission by 2D Electrons 176Problems 184

5. INTERACTING PHONON SYSTEMS 18923. Phonon-Phonon Collisions 18924. Thermal Conductivity of Insulators 19725. Balance Equations for Phonons 20126. Relaxation of Long-Wavelength Phonons 20727. Polaritons and Dielectric Function of Ionic Crystals 215Problems 224

6. EFFECTS OF ELECTRON-ELECTRONINTERACTION 22928. Hartree-Fock Approximation 22929. Shift of Intersubband Resonance 23530. Exciton Absorption 24331. Electron-Electron Collision Integral 25032. Coulomb Drag Between 2D Electrons 25633. Dynamical Screening 262Problems 273

7. NON-EQUILIBRIUM ELECTRONS 28134. Electron-Boson Collision Integral 28135. Quasi-Isotropic and Streaming Distributions 28936. Diffusion, Drift, and Energy Balance 29937. Heating under High-Frequency Field 30938. Relaxation of Population 321Problems 331

8. NON-EQUILIBRIUM DIAGRAMTECHNIQUE 341

Contents xi

39. Matrix Greens Function 34140. Generalized Kinetic Equation 34741. General Formulation of NDT 35542. NDT Formalism for Electron-Boson System 36443. Weak Localization under External Fields 374Problems 381

9. KINETICS OF BOUNDED SYSTEMS 39144. Boundary Conditions at Non-Ideal Surface 39145. Size-Dependent Conductivity 39846. Thermal Conductivity of Bounded Insulators 40547. Electron Relaxation by Near-Surface Phonons 411Problems 420

10. QUANTUM MAGNETOTRANSPORT 42548. Method of Iterations 42649. Greens Function Approach 43450. Quasi-Classical Conductivity 44351. Quantum Hall Effect 45252. Magnetooptics 465Problems 476

11. PHOTOEXCITATION 48353. Photogeneration Rate 48354. Response to Ultrafast Excitation 49055. Partially Inverted Electron Distribution 49756. Photoinduced Interband Hybridization 50857. Excitation of Coherent Phonons 520Problems 528

12. BALLISTIC AND HOPPING TRANSPORT 53758. Quantized Conductance 53859. One-Dimensional Conductors 54960. Tunneling Current 56261. Coulomb Blockade 57462. Polaronic Transport 585Problems 596

xii QUANTUM KINETIC THEORY

13. MULTI-CHANNEL KINETICS 60763. Spin-Flip Transitions 60764. Spin Hydrodynamics 61765. Coupled Quantum Wells 62866. Auger Processes 63667. Kondo Effect 646Problems 654

14. FLUCTUATIONS 66368. Non-Equilibrium Fluctuations 66469. Quasi-Classical Approach 67470. Light Scattering 68571. Fluctuations in Mesoscopic Conductors 69872. NDT Formalism for Fluctuations 708Problems 720

Appendices 733Harmonic Oscillator 733Many-Band KP-Approach 737Wigner Transformation of Product 743Double-Time Greens Functions 747Many-Electron Greens Functions 751Equation for Cooperon 759Greens Function in Magnetic Field 763Hamiltonian of Tunnel-Coupled Systems 771

Index779

Chapter 1

ELEMENTS OF QUANTUM DYNAMICS

The dynamical equations for quantum systems, the Schroedinger equation forpure states and the density-matrix equation for mixed states, form the theoreticalbackground for description of transport phenomena in systems with different kindsof elementary excitations (quasiparticles). Both single-particle formulation of theseequations and many-particle formalism, which is required for the cases of interactingquasiparticles, are presented below. This chapter is not a systematic introduction toquantum theory. It contains only the description of some basic equations and defini-tions (probability of transitions, second quantization, and so on). The properties ofconcrete quasiparticles (electrons, phonons, and photons in different materials) andtheir interaction are also discussed in order to use the corresponding results in thenext chapters.

1. Dynamical EquationsLet us start our consideration of the quantum dynamics with the sim-

plest case of a single particle propagating along the x direction. The evo-lution of such a particle is described by the time-dependent Schroedingerequation:

i()xtt

= H()xt , ()xt=t0 =

()x , (1)

where the initial state at t = t0 is determined by the wave function()x , which depends on the set of quantum numbers . The HamiltonianH can depend on time. A simple example of quantum evolution is aparticle moving in a one-dimensional potential. The Hamiltonian Hx forsuch a case is obtained from the classical expression for the energy afterreplacing the momentum by the operator proportional to the Planck

1

2 QUANTUM KINETIC THEORY

constant :

Hx =p2

2m+ U(x) , p = i

x, (2)

where m is the mass of the particle. The character of the dynamicsdepends essentially on the potential energy U(x). We mention, for ex-ample, formation of confined states in a potential well or tunneling pen-etration of the particle through a potential barrier. Different observablevalues (such as coordinate, velocity, and energy) of the system are de-termined by the quantum-mechanical average

Q()

t =

dx()xtQ()xt , (3)

where the operator Q corresponds to the classical expression for theobservable value. Note that Q()t is expressed through a quadratic formof the -functions. Since Q()t is real, any operator Q must be Hermitian.In particular, H = H+, because the Hamiltonian corresponds to theenergy of the system.

The operator nature of the characteristics of quantum systems makesit possible to rewrite Eqs. (1) and (3) in the integral representation. Weintroduce a kernel

H(x, x1) =[p212m

(x x1)]+ U(x1)(x x1) (4)

containing Diracs -function, and transform the Schroedinger equation(1) to the following integral form:

i()xtt

=

dx1H(x, x1)()x1t

, ()xt=t0 = ()x . (5)

The kernel for the observable value, Q(x, x1) = [Q1(x x1)], is in-troduced in the same way (here Q1 acts on the coordinate x1 of the-function), and we obtain

Q()

t =

dx

dx1

()xt

Q(x, x1)

()x1t

. (6)

In these formulations, the state with quantum numbers is described bythe wave function ()xt and by the operators of physical values appearingin Eqs. (1) and (3), or by the x-dependent kernels in Eqs. (5) and (6).Such a description is called the coordinate (or x-) representation.

In many cases, the description of quantum dynamics can be simplifiedby using the Fourier-transformed wave function introduced according to

Elements of Quantum Dynamics 3

the relations

()pt =

dxe(i/)px()xt , ()xt =

1L

p

e(i/)px()pt . (7)

In order to avoid the ambiguities due to -functions, the motion of theparticle is considered here for an interval of length L, with the use ofappropriate boundary conditions. In the limit L , the momentump in Eq. (7) is a quasi-discrete variable with values (2n/L), wheren is an integer. The substitution p (2n/L) does not depend onthe type of the boundary conditions used (hard-wall, periodic, etc.),provided that n is a large number. A Fourier transformation of Eq. (1)with the Hamiltonian (2) leads to the Schroedinger equation

i()ptt

=p1

H(p, p1)()p1t

, (8)

H(p, p1) 1L

dxe(i/)pxHxe(i/)p1x,

which is similar to Eq. (5). The kernel H(p, p) depends on a pair ofmomenta. The initial condition to Eq. (8) is determined by the Fouriertransformation of ()x . In the above example of the particle in a one-dimensional potential, the Hamiltonian kernel is transformed to

H(p, p1) =p2

2mp,p1 + U(p, p1), (9)

where p,p1 is the Kronecker symbol (below we use two equivalent no-tations a,b and ab for such symbols). The kinetic energy acquires itsclassical form, while the action of the potential is described by the kernelU(p, p1). The expression for an observable through

()pt is written as

Q()

t =1L2

p1p2

()p1tQ(p1, p2)

()p2t

, (10)

where the kernel Q(p1, p2) can be written in terms of Q in a similar wayas the Hamiltonian kernel in Eq. (8). The structure of Eqs. (8) and (10)is analogous to that of Eqs. (5) and (6). This description is called themomentum (or p-) representation of the problem under consideration.

Obviously, the nature of quantum dynamics does not depend on therepresentation used. For this reason, it is convenient to consider thewave function as a projection of the ket-vector |, t, which describes thestate defined by the quantum numbers , onto the bra-vector, k|, which


determines the representation:

()kt = k|, t k x, p, . . . . (11)

It should be noted that the above-introduced bra- and ket-vectors are notusual functions. They are Hermitian conjugate elements of the Hilbertspace satisfying the relations of orthogonality, normalization, and com-pleteness:

k| = |k+, k|k = k,k ,k

|kk| = 1, (12)

where 1 is the unit operator. Using these notations, one may formulateany dynamical problem in the operator form.

The Schroedinger equation for the state in this representation takesthe following form:

i |, tt

= H |, t , |, t = t0 = | , (13)

with the initial condition determined by the ket-vector |. A similarequation for the Hermitian conjugate vector , t| contains , t|H onthe right-hand side. Using Eq. (11) and rewriting the kernel Q(k1, k2)as k1|Q|k2, we define the observable Q

()

t as follows:

Q()

t =k1k2

, t|k1k1|Q|k2k2|, t = , t|Q|, t, (14)

so that the classical observable is expressed through the diagonal matrixelement. As a result, the dynamics of the system with a fixed initial state|, t = 0 (such a system is said to be in the pure state) is described byEqs. (13) and (14).

Transforming the double sum in Eq. (14) as

k1k2k1|Q|k2k2|, t

, t|k1, it is convenient to separate the operator |, t, t| there. Thisoperator,

()t |, t, t|, (15)

known as the density matrix or as the statistical operator, describesthe quantum dynamics of the system. The quantity k2|, t, t|k1 =k2|()t |k1 is also called the density matrix in the |k-representation.One may consider, for example, x- or p-representation, or a representa-tion based upon discrete quantum numbers (problem 1.1). The descrip-tion of the quantum dynamics based on the density matrix formalism isconvenient for the cases when the initial state |, t = t0 of the quantum


system is not defined (for example, because of the quantum-mechanicaluncertainty).

Let us give a more general definition of the density matrix. Consideran ensemble of identical systems, which are distributed over the states with probabilities P at the initial moment of time t = t0 (such a systemis called the mixed state, or the mixture of states). We introduce theobservable quantity Qt according to

Qt =

PQ()

t . (16)

The probability for realization of -states is normalized as

P = 1.Since the operator Q does not depend on the initial conditions, thedensity matrix for the mixed state is introduced as

t

P |, t, t|, (17)

and the observable (16) is obtained from Eqs. (14) and (16) in the form

Qt =k

k|Qt|k Sp(Qt). (18)

Here and below Sp(A) (or, equivalently, SpA), where A is an arbitraryoperator, denotes the sum of the diagonal matrix elements of this oper-ator and is called the trace of the operator.

The equation of evolution for the density matrices (15) and (17) de-scribing dynamics of pure and mixed states, respectively, is obtainedin the following way. Let us take a derivative of the density matrixover time and use Eq. (13) together with the corresponding Hermitianconjugate equation. As a result,

i

t|, t, t| = H|, t, t| |, t, t|H. (19)

Now, let us multiply this equation by P and calculate the sums over of both its sides. Since H does not depend on , we obtain, accordingto Eq. (17), the operator equation

itt

= [H, t] (20)

describing the evolution of the quantum system. The right-hand side ofEq. (20) is written using the commutator defined as [A, B] = AB BA,where A and B are arbitrary operators. The initial condition for Eq.(20) in the case of a pure state may be expressed as t=t0 = ||, whilefor a mixed state one needs additional physical restrictions removing theuncertainty of the initial state.


2. S-Operator and Probability of TransitionsThe evolution of the system with time-dependent Hamiltonian Ht is

described by the Schroedinger equation (1.13). The ket-vectors |t |, t at the instants t and t are connected through the evolution operatorS (also known as S-operator or scattering matrix):

|t = S(t, t)|t. (1)

Equation (1.13) leads to the operator equation for S(t, t), with the initialcondition at t = t:

i

tS(t, t) = HtS(t, t), S(t, t)t=t = 1. (2)

For the case of a time-independent Hamiltonian, Ht = H, this equationis solved as

S(t, t) = exp[ i

H(t t)

] S(t t), (3)

and the temporal evolution is determined only by the difference t t.If the initial value of the ket-vector, |, t = 0 = |, belongs to oneof the vectors determined by the eigenstate problem H| = |, theevolution is harmonic:

|t = exp( i

t

)|. (4)

In the case of a mixed initial state, the evolution is described by a sumof oscillating factors with different energies .

In the general case of the time-dependent Hamiltonian, it is convenientto transform Eq. (2) to the integral form:

S(t, t) = 1 i

ttdH S(, t). (5)

The solution of this equation is obtained by iterations and is written as

S(t, t) = 1 +n=1

( i

)n ttdt1 . . .

tn2t

dtn1 tn1t

dtn

Ht1 . . . Htn1Htn . (6)Introducing the operator of chronological ordering, T , we rewrite Eq.(6) as follows (problem 1.2):

S(t, t) = 1+n=1

(i/)nn!

ttdt1 . . .

ttdtn1

ttdtnT

{Ht1Ht2 . . . Htn

},


T{HtHt

}=

{HtHt , t > t

HtHt, t < t . (7)

One can write S(t, t) of Eqs. (6) and (7) as a chronologically orderedexponential operator

S(t, t) = T{exp

[ i

ttdH

]}. (8)

This expression, together with Eq. (2), leads to the following propertiesof the evolution operator:

S(t, t) = S+(t, t), S+(t, t)S(t, t) = 1, S(t, t1)S(t1, t) = S(t, t), (9)

which can be checked by calculating the time derivatives (problem 1.3).Below we consider a system with time-independent Hamiltonian H

in the presence of a weak harmonic perturbation. In other words, wediscuss the evolution of the system with the Hamiltonian

H + (veit +H.c.) H + Vt, (10)

where the operator v is small. The letters H.c. in Eq. (10) indicatethe Hermitian conjugate contribution to the perturbation. A solution ofthis problem not only describes a response of the system to the harmonicperturbation, but also allows one to consider a modification of stationarystates under the time-independent perturbation v + v+ , where = 0.It is convenient to use the interaction representation by introducing anew ket-vector |t) according to |t = S(t)|t), where S(t) is the S-operator (introduced by Eq. (3)) for the Hamiltonian H. Substituting|t = S(t)|t) into Eq. (1.13), we multiply the latter by S+(t) from theleft and obtain the following Schroedinger equation in the interactionrepresentation:

i|t)t

= S+(t)VtS(t)|t), |, t = 0) = |i. (11)

To solve Eq. (11) with the accuracy of the first order in the perturbationVt, we substitute the unperturbed ket-vector |i to the right-hand sideof this equation. Since we assume that the unperturbed system is in theinitial state i, we have

|it) |i+ 1i

t0dtV (t)|i, V (t) = S+(t)VtS(t). (12)

The probability of finding the system at the instant t in the state f(described by the ket-vector S(t)|f of zero-order approximation), calcu-lated with the accuracy of the second order in the perturbation, is equal


to |f |S+(t)|it|2 = |f |it)|2. The probability of transition between thestates i and f is defined as a time derivative of this quantity:

Wif =d|f |it)|2

dt. (13)

We note that both |i and |f are the solutions of the eigenstate problemH|i = i|i. An explicit expression for Wif is determined after a simpleintegration over time:

Wif =12

d

dt

vfi ei(fi)t 1i(fi ) + vif ei(fi+)t 1i(fi + )

2

, (14)

where fi = (f i)/ is the frequency of transitions between the statesi and f of the unperturbed system and vfi = f |v |i. The probabilityof transitions has resonant behavior: at t it is not equal to zeroonly when coincides with one of the transition frequencies. We notethat, on the small-time scale, a non-zero probability exists also for non-resonant conditions, owing to the energy-time uncertainty. In the caseof time-independent perturbations ( = 0), the energy of the system isconserved, and the transitions occur between the degenerate states only.

Let us consider first the asymptotic behavior of Wif at large times un-der a time-independent perturbation V 2v (note that v is Hermitian).For this case, taking into account |vif |2 = |vfi|2, we obtain

Wif =4|vif |2

2d

dt

2 2 cosfit2f i

=8|vif |2

2d

dt

[2 sin2(fit/2)

2f itt

]. (15)

If fit 1, the function 2 sin2(fit/2)/2f it goes to (fi); see Fig.1.1 and problem 1.4, where different presentations of Diracs -functionare discussed. As a result, the probability of transition becomes

Wif =2|f |V |i|2(f i). (16)

This important result is known as Fermis golden rule. We stress againthat the energy of the system is conserved, and only the states withf = i contribute into the probability (16) under a time-independentperturbation.

The probability of resonant transitions in the case of time-dependentperturbations is calculated in a similar way. If fi = 0, only the termscontaining fi in the factor | . . . |2 of Eq. (14) are important at larget, and one obtains

Wif () =2|f |v |i|2(f i ). (17)


Figure 1.1. Function F () = sin2(t)/2t for t =3, 10, and 30 (dotted, dashed,and solid curves, respectively).

The energy conservation law

f = i + (18)

is fulfilled for interlevel transitions excited by a harmonic perturbationwith the energy of quantum .

Equations (16) and (17) can be derived in an alternative way, underthe assumption that the perturbation Vt is adiabatically turned on att = . The first-order solution of the time-dependent Schroedingerequation (1.13) with the Hamiltonian H + Vt and boundary condition|, t = = |i is written as

|it |i+ 1i

t

dtetS(t, t)VtS+(t, t)|i , (19)

where +0 describes the adiabatic turning-on. Consider, for exam-ple, a time-independent perturbation V . The integral in Eq. (19) iseasily calculated by substituting = t t. Since S(t, t) = S(), weobtain

f |it = et f |V |ii f + i

. (20)

The transition probability d|f |it|2/dt is reduced to Eq. (16) accordingto the first expression for the -function in problem 1.4. The case of


time-dependent perturbation is considered in a similar way, leading toEq. (17).

The probabilities of transitions, derived above in a pure quantum-mechanical approach, are the important characteristics determining ki-netic properties of different systems. Indeed, let us introduce the oc-cupation number njt for the state j, i.e., the average number of theparticles in the state j at the instant t, according to (see Eq. (1.17))

njt = j|t|j =

P |j|t|2. (21)

One may expect that, under proper conditions (in the subsequent chap-ters this question will be considered in detail), the temporal evolutionof the occupation numbers is determined by the balance equation

njtt

=j

Wjj(njt njt

), (22)

where Wjj is given by Eq. (16) for the case of time-independent per-turbations and Wjj = Wjj() + Wjj(), see Eq. (17), for time-dependent harmonic perturbations. The first and second terms on theright-hand side of Eq. (22) describe incoming (arrival) and outgoing (de-parture) contributions to the balance of occupation, respectively. Wenote that the arrival rate from the state j to the state j is equal toWjjnjt, while the departure rate from the state j to all other states isequal to njt

j Wjj . The balance equation (22) conserves the number

of the particles and, for the case of time-independent perturbations, theenergy of the system. In order to describe the temporal evolution of theother characteristics of the system (those which are sensitive to phasecorrelation), one has to consider quantum kinetic equations for t; seethe next chapters.

In the case of harmonic perturbations, one may express the powerabsorbed by the system through the transition probability (17). Theabsorbed power U is defined as the energy of the quantum, , multi-plied by the difference between the rate of transition from the state j tothe state j (which corresponds to absorption of the quantum) and therate of emission of the quantum associated with the transitions from jto j:

U = jj

[Wjj()nj(1 nj)Wjj()nj(1 nj)

]=

jj

Wjj()(nj nj). (23)


On the other hand, the absorbed power U for the system excited by anelectric field Et is determined by the electrodynamical expression It Et,where It is the electric current density induced by the field and the lineover the expression denotes the averaging over the period 2/. Withinthe accuracy of E2t , which corresponds to the perturbation theory ap-plied above, one may consider It in the framework of the linear-responseapproximation and describe U through the frequency-dependent con-ductivity of the system; see Chapter 3.

3. Photons in MediumWe begin our consideration of the quantum dynamics of concrete phys-

ical systems with the case of electromagnetic field in the spatially inho-mogeneous medium described by the dielectric permittivity tensor r.Starting from an expression for the energy of the electromagnetic fieldin the absence of free electric charges, we derive the Hamiltonian of thefield and, after a quantization procedure, describe the field as a set ofoscillators corresponding to elementary quasiparticles known as photons.The photons are an example of bosons, the particles with a symmetricwave function corresponding to the Bose-Einstein statistics.

The energy of the field is determined by the expression

Ef =18

(V )

dr(Ert rErt +H2rt)

=18

(V )

dr{

1c2

Artt

rArtt

+ ([Art])2}, (1)

where the integrals are taken over the normalization volume V . In Eq.(1) we assume a local relation between the electrostatic induction andthe field: rErt. On the other hand, the magnetic induction is equal toHrt because the kinetic phenomena are considered in this book for non-magnetic materials only. The electric and magnetic field strengths, Ertand Hrt, which satisfy the Maxwell equations in medium, are expressedonly through the vector potentialArt, since we have chosen the Coulombgauge rArt = 0 leading to zero scalar potential in the absence offree charges.

It is convenient to represent the electromagnetic field described by thevector potential Art as

Art =

[q(t)Ar + q(t)A

r ], (2)


where the modes Ar with frequencies are determined by the waveequation following from the Maxwell equations:

[ [Ar ]](

c

)2rAr = 0. (3)

The modes satisfy the orthogonality and normalization conditions ac-cording to

(V )

drAr rA

r = 2c2 . The coefficients q(t) in the

expression (2) can be considered as the generalized coordinates of -thmode. They satisfy the oscillator equation

d2q(t)dt2

+ 2q(t) = 0 (4)

corresponding to the harmonic oscillations with eigenfrequencies . Inthe presence of external sources described by the electric current densityIrt, one must add the term c1

(V )

dr Irt Art to the right-hand sideof Eq. (1) and (4/c)Ir to the right-hand side of Eq. (3). Such acontribution describes the interaction of the modes Ar with externalcharges.

Introducing the generalized momentum p(t) dq(t)/dt, we applythe orthogonality and normalization conditions for the modes of Eq. (3)to rewrite the energy of the field given by Eq. (1) as a sum of oscillatorenergies:

Ef =12

{|p(t)|2 + 2 |q(t)|2

}. (5)

Since the solutions of Eq. (4) are proportional to exp(it), we havethe relation p = iq . It is convenient to introduce the canonicallyconjugate variables

Q(t) =q(t) + q(t)

2, P(t) = i

q(t) q(t)2

, (6)

which are used here in order to rewrite the energy of the field as Ef =

{|P(t)|2 +2 |Q(t)|2

}/2. The equations of motion acquire Hamil-

tonian form: P = Q = Ef /Q , Q = Ef /P .In order to quantize the electromagnetic field, we have to replace

the canonically conjugate variables Q(t) and P(t) by the operators ofgeneralized coordinate and momentum, Q and P , which satisfy thecommutation relation

[Q , P ] = i . (7)

Let us use the expression for the energy as a sum of the oscillatory con-tributions (5) and take into account the relation P = i/Q . Then


we write the Hamiltonian of quantized field in the Q-representation:

Hph =12

{2

2

Q2+ 2Q

2

}. (8)

A solution of the eigenstate problem Hph{n} = E{n}{n} deter-mines a set of occupation numbers, {n}, for the given modes. Thesymmetrized wave function, corresponding to the Bose-Einstein statis-tics, is a product of the eigenfunctions of different modes, n (Q), whilethe total energy, E{n}, is given by a sum of the oscillator energies:

{n} =

n (Q), E{n} =

(n +

12

). (9)

The occupation numbers n are integers (n 0). As follows from Eq.(9), the wave function {n} is symmetric with respect to permutationsof each oscillatory function n(Q) (see Appendix A) with another oscil-latory function. The matrix elements of the generalized coordinate forthe transitions between the states with quantum numbers n and n areequal to zero if n = n 1, while for the transition between adjacentlevels these matrix elements are

n |Q |n =

2

{ n + 1 ,n ,

n = n + 1n = n 1

. (10)

The matrix elements of the generalized momentum are n |P |n =in |Q |n, where the signs correspond to the transitions be-tween the states with occupation numbers n = n 1 and n . Thisequation is consistent with the relation between the Fourier componentsof coordinate and momentum used in Eq. (6). Instead of a pair of canon-ically conjugate operators Q and P , we introduce, by analogy to Eq.(A.11), two Hermitian conjugate creation and annihilation operators forthe mode :

b =Q + iP

2, b+ =

Q+ iP+2

. (11)

Representing the contribution of the state in the Hamiltonian (8) as{. . .} = (Q+ iP+ )(Q + iP) + , we rewrite Hph in the form

Hph =

(b+ b +

12

). (12)

The Hamiltonian of the field is given as a sum of the contributions hoscdetermined by Eq. (A.12), with the oscillator frequencies .


Therefore, the electromagnetic field in a medium is presented as asuperposition of quantized normal vibrations with frequencies andoccupation numbers n . It is convenient to use a representation de-scribed by the ket-vector |{n} depending on the sets of occupationnumbers {n}. Using these sets as independent variables of the problem(instead of the generalized coordinates Q), one may define the creationand annihilation operators through their matrix elements

n |b+ |n =n + 1n ,n+1,

n |b |n =nn ,n1, (13)

instead of using Eq. (11). This means that the operators b+ and b ,while acting on the ket-vector |{n}, change the occupation number ofthe photons of the mode by 1, respectively:

b+ |n1n2 . . . n . . . =n + 1|n1n2 . . . n + 1 . . .,

b |n1n2 . . . n . . . =n |n1n2 . . . n 1 . . .. (14)

The commutation rules for these operators are obtained by using eitherthe matrix elements (13) or the expressions of these operators throughQ and P , Eq. (11). For the Hermitian conjugate operators, one has

[b , b+ ] = n ,n , (15)

while the operators of the same kind (creation or annihilation) merelycommute with each other. It is the commutation rule (15) that leadsto the appearance of zero-field oscillation energy

/2 in Eq. (12);

see also Eqs. (A.11) and (A.12). By analogy to the case of a singleoscillator, see Eq. (A.18), the set of ket-vectors |{n} is presented as

|{n} =

(b)nn !

|{0}, (16)

where |{0} describes the vacuum state where only zero-field oscillationsdue to quantum-mechanical uncertainty are present. The set of ket-vectors also satisfies the completeness, orthogonality, and normalizationconditions:

|{n}{n}| = 1, {n}|{n} = {n},{n}. (17)

The generalized Kronecker symbol {n},{n} is equal to unity only whenall the occupation numbers from the sets {n} and {n} coincide. The


description of the electromagnetic field given by Eqs. (13)-(17) is calledthe occupation number representation or the second quantization. It isanalogous to the description of a single oscillator given by Eqs. (A.16)-(A.19). In this representation, the sets of independent variables describ-ing the system are the numbers of quanta of the field in each mode.These quanta are called the photons in medium, i.e., the system is de-scribed in terms of quasiparticles. The operator of the photon numberfor the mode is introduced as n = b+ b . The justification of thisdefinition is the same as for a single oscillator, and the ket-vector |{n}is the eigenvector of the operator n corresponding to the eigenvalue n ,according to n |{n} = n |{n}; see Eq. (A.20).

Using the expansion (2) and expressing the amplitudes of vibrationsaccording to Eqs. (6) and (11) through the creation and annihilationoperators as q =

/ b and q+ =

/ b

+ , we get the quantized

operator of the vector potential

Ar =

(Ar b +A

r b

+

), (18)

where the modes Ar are determined from Eq. (3). The operators ofthe second-quantized fields, Er and Hr, can be written by using Eq.(18) together with the relation q(t) = iq(t) and by expressingthese fields through the vector potential according to Ert = c1Artand Hrt = [Art]. The classical vector of the radiation flux density(Poynting vector), Srt = (c/4)[Ert Hrt], is expanded in terms of themodes as follows:

Srt = 14

[Artt

[Art]]=

i

4

[(

q (t)A

r

q (t)A

r

) (q(t)[Ar ] + q(t)[Ar ])

]. (19)

In the second quantization representation, the operator of the radiationflux density, Sr, is written as

Sr =i

4

[(b A

r b+ A

r

)

(b [Ar ] + b+ [Ar ]

)](20)

after expressing the amplitudes q(t) in Eq. (19) through the corre-sponding creation and annihilation operators.


In a homogeneous and isotropic medium with dielectric permittivity, the modes Ar are the plane waves Aq exp(iq r) with wave vector qand polarization . The amplitudes Aq are determined from the vectorequation

[q [qAq]] +(q

c

)2Aq = 0 (21)

following from Eq. (3) and from the gauge condition (q Aq) = 0.Equation (21) is equivalent to a system of three algebraic equations forthe components of the vector Aq. The requirement of orthogonalityand normalization for the amplitudes is written as

V

2c2(Aq Aq) = , (22)

so that one can introduce the unit vectors of polarization, eq, accordingto Aq =

2c2/V eq. These vectors have the properties of transver-

sity (following from the gauge conditions), orthogonality, and normal-ization (following from Eq. (22)), while Eq. (21) leads to a polarization-independent dispersion relation for the photon of frequency q :

(q eq) = 0, (eq eq) = , q = cq, c =c. (23)

These relations describe propagation of the photons whose unit vectorsof polarization, eq=1 and eq=2, are directed in the plane perpendicularto the wave vector q. The dispersion of the photons is linear in q, and theproportionality coefficient c is the velocity of light in the medium. TheHamiltonian and the radiation flux density operator for the homogeneousand isotropic medium are expressed, according to Eqs. (12) and (20),through the photonic creation and annihilation operators for the states = (q, ). The polarization vectors and the frequency of these statesare given by Eq. (23). The operator of electric field can be obtainedfrom Eq. (18):

Er = iq

2qV

eqeiqr(bq b+q

), (24)

where we assumed that eq = eq. The magnetic-field operator is givenby a similar expression, which is obtained according to Hr = [ Ar]and contains the polarization factor [qeq] under the sum. The matrixelement of Sr for the case of plane waves is determined according toEqs. (20), (22), and (13) as n |Sr|n = (q/q)cV 1q(n + 1/2). Inthis form, the Poynting vector has direct meaning of the flux of photonenergy density with velocity c in the direction of q. In non-homogeneousmedia, the description of the modes based upon Eqs. (3) and (18) is


more sophisticated, though relatively simple results exist for the case ofone-dimensional inhomogeneities (problems 1.5 and 1.6).

Finally, let us calculate the averaged occupation number of the mode for the equilibrium distribution of photons with temperature Tph. Thisdistribution is described by the density matrix

eq = Z1 exp(Hph/Tph), Z = Sp exp(Hph/Tph). (25)

The partition function Z is expressed through the photon energy (9)written as E{n} =

n + E0, where E0 is the energy of zero

vibrations, according to Z = exp(E0/Tph)Z and

Z ={n}

en/Tph =

n

en/Tph

=

(1 e/Tph

)1. (26)

The mean value of the occupation number of the mode is defined asn = Spn eq . It is expressed through Z as

n = Z1{n1}

neE{n1}/Tph = Tph

lnZ()

. (27)

Calculating the derivative in Eq. (27), we obtain the equilibrium Planckdistribution

n =[e/Tph 1

]1. (28)

This distribution allows one to describe various equilibrium propertiesof the boson gas (problems 1.7 and 1.8). It is valid for all kinds of thebosons whose number is not fixed.

4. Many-Electron SystemIn contrast to the case of photons, the dynamics of a system of elec-

trically charged particles depends on their interactions with externalelectric fields (created by different, with respect to the system underconsideration, charges) and externally applied magnetic fields (note thatwe consider non-magnetic materials only), as well as on the interactionof these particles with each other. The existence of the spin variableleads to a further sophistication of such dynamics. Below we discuss thequantum dynamics for electrons, charged particles with two differentspin states. The electrons are an example of fermions, the particles withan antisymmetric, with respect to particle permutation, wave functioncorresponding to the Fermi-Dirac statistics.


The Hamiltonian of the electron system in external fields is writtenas

He =j

hj + Hf . (1)

Here hj is the one-electron (the index j numbers the electrons) operatorof the kinetic energy. It is given by the equation

hj =(pj eAxjt/c)2

2m, (2)

where pj eAxjt/c is the operator of kinematic momentum expressedthrough the canonical momentum p satisfying the ordinary commutationrelations [p, x ] = i and through the vector potential Axjt. Thesecond term of Eq. (1), Hf , is the operator of the field energy Ef , thelatter is given by the first part of Eq. (3.1). Using the expressions

Ert = 1c

Artt

rt , Hrt = [Art] (3)

relating the electric and magnetic fields to the vector potential Art andscalar potential rt, we rewrite Ef as

Ef =

(V )

dr8

{1c2

Artt

rArtt

+ ([Art])2}

+

(V )

dr4c

rtrArtt

+

(V )

dr8

rtrrt . (4)

This equation generalizes Eq. (3.1) to the case of non-zero gradient ofthe scalar potential. The tensor r is assumed to be symmetric. Belowwe again employ the Coulomb gauge (rArt) = 0 and assume thatthe fields go to zero at the boundaries of the region V (one may alsouse the periodic boundary conditions). The first term of the expression(4) corresponds to the energy of transverse vibrations of the field anddescribes the photons in medium. After the quantization of the fielddone in the previous section, we can denote this term as Hph. Thesecond term on the right-hand side of Eq. (4) is equal to zero because

rtrArtt

= (rtr

Artt

)(5)

in the gauge used, and the integral over the volume V is reduced to asurface integral over an infinitely remote boundary where the fields areequal to zero. The third term of the expression (4) can be rewritten


according to = ()(), and only the last termhere remains finite after integrating over the volume. Next, by using thePoisson equation (rrt) = 4rt, where rt is the charge density,we obtain the following expression for this term:

(V )

dr8

rt (rrt) =12

(V )

drrtrt. (6)

Now we see that the third term on the right-hand side of Eq. (4) de-scribes the interaction of electric charges with the longitudinal part ofthe electric field. We denote it below as Eint. In a homogeneous andisotropic medium with constant dielectric permittivity , one can easilysolve the Poisson equation as rt = 1

(V )

drrt/|r r| so that Eintis expressed through the charge densities only:

Eint =12

(V )

(V )

drdrrtrt|r r| . (7)

One should remember that both rt and rt include the contribu-tions of the external fields and charges. To extract these contributionsfrom Eint, it is convenient to separate the contributions coming fromthe internal (i) and external (e) charges under the integrals of Eq. (7).Then,

Eint =12

(V )

drdr(i)rt

(i)rt

|r r| +12

(V )

drdr(e)rt

(e)rt

|r r|

+

(V )

drdr(e)rt

(i)rt

|r r| . (8)

The first term on the right-hand side of Eq. (8) is the energy of Coulombinteraction between the electrons of the system (the electrostatic en-ergy). The second term is the energy of interaction between the ex-ternal charges. It should be omitted in the following, because such acontribution is not relevant to the dynamics of the system under consid-eration. Finally, the last term is the energy of interaction of electronswith the longitudinal part of the external field. It can be rewritten ase1

(V )

drUrt(i)rt , where Urt is the potential energy of an electron in the

external field. One may introduce the potential of the external field asUrt/e. Below we omit the index i in

(i)rt .

To transform Eint into the operator of the interaction, Hint, one shoulduse the charge density operator r = e

j (r xj) instead of rt. As a

result, we obtain

Hint =12

jj

e2

|xj xj |+

j

Uxjt. (9)


Because the interaction of an electron with itself should not be consid-ered, the prime sign at the sum denotes the exclusion of the terms whoseindices coincide (in other words, j = j is assumed).

Combining Hint with the kinetic-energy part

j hj , we find that thetotal Hamiltonian of the system of interacting electrons in the presenceof external fields is given as

He =j

hj + Hee,

hj = hj + Uxjt, Hee =12

jj

e2

|xj xj |, (10)

where hj is the single-particle Hamiltonian comprising both kinetic andpotential energy operators, and Hee is the Hamiltonian of Coulomb inter-action between the electrons in the medium with dielectric permittivity. It is represented as a binary sum over all particles. We stress that thevector potential Axjt standing in hj includes a contribution of the ex-ternal fields. This contribution, in particular, describes the interactionof electrons with a stationary magnetic field and with electromagneticwaves (photon field). In the above consideration, we have omitted thecontribution corresponding to the Pauli interaction of the electron spinwith the magnetic field. We have also neglected relativistic corrections,which are small if the energy of the particle is small in comparison tothe energy 2mc2.

The evolution of a many-electron system in external fields is describedby the Schroedinger equation analogous to Eq. (1.1):

i{xj}t

t= He{xj}t. (11)

It determines the wave function of x-representation, which depends onthe set of coordinates {xj}. The charge density at the point (r, t) isexpressed through r according to the general rule (1.3) for observablevalues:

rt =

d{xj}{xj}tr{xj}t, (12)

where the charge density operator r is introduced above as a sum ofthe -functions multiplied by the electron charge. Calculating the timederivative of Eq. (12) with the use of Eq. (11), we take into accountthat He is Hermitian and find

rtt

=i

d{xj}{xj}t

[He, r

]{xj}t. (13)


One can see that both the potential energy Uxjt and Coulomb interactionenergy Hee commute with r and do not contribute to the right-handside of Eq. (13). The calculation of the commutators [hj , (r xj)](problem 1.9) gives us the continuity relation

rtt

+ Irt = 0, Irt =

d{xj}{xj}tIr{xj}t. (14)

The current density Irt at the point (r, t) is introduced by analogy toEq. (12), and the current density operator Ir is given by

Ir =e

2m

j

[(pj

e

cAxjt)(r xj) + (r xj)(pj

e

cAxjt)

]. (15)

This equation allows us to represent the Hamiltonian He introduced byEq. (10) in the form

He = He 1c

dr Ir Art (16)

so that He accounts for the interaction with longitudinal fields only,while the second term gives us the interaction with transverse fieldsentering through the vector potential Art. These fields are described bythe photon Hamiltonian Hph defined in Sec. 3 after the quantization ofthe field.

Because the many-electron wave function {xj}t is antisymmetric withrespect to permutation of electrons (Fermi-Dirac statistics), one can in-troduce the occupation number representation with the aid of the fol-lowing antisymmetric function of N particles:

{k}({xj}) = det(k)xj /

N ! , (17)

where ()x is a complete set of one-electron wave functions numberedby the quantum numbers , and det . . . denotes the determinant ofthe matrix . . . . The set {k} in Eq. (17) determines the state of Nelectrons with coordinates {xj} so that one can write N2 functions of thekind of Eq. (17). A complete system of linearly independent functionsis obtained from Eq. (17) with the aid of a set of quantum numbers,{k}, ordered according to 1 < 2 < . . . . The expansion of {xj}t interms of the functions (17),

{xj}t =n{k}

{nk}t{k}({xj}), (18)


contains the numbers n which can be either 1 or 0. This propertydirectly follows from the antisymmetry of the wave functions (17) andis known as the Pauli principle. The numbers are chosen in such away that n = 1 if the state belongs to {k}({xj}) and n = 0for other , provided

n = N . Therefore, each state of the N -

electron system corresponds to a set of occupation numbers {n}. Itis not necessary to specify the functions ()x standing in Eq. (17) inorder to introduce the ket-vector |{n}t instead of {nk}t. Such ket-vectors form a complete, orthogonal, and normalized set: {n}|{n} ={n},{n} (here and below the argument t is omitted). The creationoperator for the state is introduced by the relation

a+ |{n} =

1 n(1)p()| . . . , n + 1, . . ., (19)

where p() =

k


For the binary operators of the form H2 = (1/2)

jj vjj , where vjj =v(xj ,xj), we have

H2 =12

1234

12|v|43a+1 a+2 a3 a4 ,

12|v|43 =

dx

dx(1)x(2)x

v(x,x)(4)x (3)

x . (24)

The operator Hee given by Eq. (10) is an example of such operators.Equations (23) and (24) are justified in view of the equivalence of theirmatrix elements calculated with arbitrary ket-vectors |{n} to the usualexpressions for the matrix elements of H and H2 calculated in the anti-symmetric basis (17) (problem 1.10).

Instead of a+ and a , one can use the field operators +x and xdefined by the relations

+x =

()xa+ , x =

()x a . (25)

The operator +x should be considered as the creation operator for theparticle with coordinate x, while x is the corresponding annihilationoperator. The anticommutation rules for these operators are obtainedfrom Eq. (21) according to the condition of completeness for the sets ofone-electron wave functions ()x :

[+x , +x ]+ = [x, x ]+ = 0,

[+x , x ]+ =

()x()x = (x x

). (26)

The additive and binary operators introduced by Eqs. (23) and (24) arewritten through the field operators in the following way:

H =12

dx (1)x

h(2)x a+1 a2 =

dx+x hx,

H2 =12

dx

dx+x

+xv(x,x

)xx. (27)

These expressions are analogous to the matrix elements of Eqs. (23) and(24), where the one-electron functions are replaced by the field operators,as though the -function is quantized again. This explains the origin ofthe term second quantization. If the many-electron Hamiltonian Hedepends on spin variable or/and band indices (see the next section and


Appendix B), one should consider x in Eqs. (25)(27) as a combinationof the coordinate and discrete indices so that the integral

dx must

include the sums over these indices.To calculate the observable Qt, we substitute Q in the form Q = |q|a+ a into Eq. (1.18):

Qt = SpQt =

|q|Spa+ a t spqnt. (28)

Here Sp and sp denote the traces over many-particle and single-particle states, respectively. We have introduced the average of theadditive operator q through the one-particle density matrix nt definedas

|nt| = Spa+ a t. (29)The dynamical equation for nt follows from the general many-particleequation (1.20):

i

t|nt| = Spa+ a [H, t] = Spt[a

+ a , H]. (30)

In the case of a system of non-interacting electrons described by theHamiltonian H =

j hj , one can write a closed equation for the single-

particle operator nt. We use the commutator

[a+ a , a+ a ] = a

+ a a

+ a (31)

calculated according to the anticommutation relations (21) (problem1.11). As a result, the dynamical equation for nt has the same form asEq. (1.20):

intt

= [h, nt]. (32)

Equation (32), however, contains one-electron variables only. Accordingto Eq. (28), the observable values are found by averaging the operatorq with the density matrix nt.

Let us find the averaged occupation number n of the electron state for the equilibrium distribution of electrons with temperature Te. Thisdistribution is described by the density matrix

eq = Z1 exp[(He n)/Te], Z = Sp exp[(He n)/Te]. (33)

This definition, in contrast to Eq. (3.25), accounts for the conservationof the total number N of electrons described by the particle numberoperator n =

n . The coefficient , called the chemical potential, is

determined from the condition

n = N . Under the assumption of


ideal Fermi gas, i.e., neglecting the contribution of the electron-electroninteraction to the equilibrium properties of the system, we use the Hamil-tonian He =

a

+ a . Then, the total energy is E{n} =

n ,

where n is equal to 1 (0) for the occupied (empty) states. As a result,the partition function takes the following form:

Z ={n}

exp[( )n/Te]

=

n=0,1

exp[( )n/Te] =

[1 + e()/Te

]. (34)

The mean value of the occupation number of the state is introducedas the average of the operator n with the equilibrium density matrix(33), n = Spn eq :

n = Z1{n1}

n exp

[1Te

(1

n1 E{n1}

)]= Te

lnZ

. (35)

Substituting Z from Eq. (34) into Eq. (35), we find the equilibriumFermi distribution

n =[e()/Te + 1

]1. (36)

If the temperature goes to zero, the electrons occupy only the stateswhose energies are below the chemical potential . This is the caseof a degenerate electron gas. Another limiting case, the non-degenerateelectron gas, takes place when the average occupation numbers are small,n 1, and Eq. (36) is reduced to the Boltzmann distribution n =e()/Te .

5. Electrons under External FieldsAfter the general description of the system of interacting electrons

given in Sec. 4, we are going to discuss the solutions of one-electronSchroedinger equations: h = E for stationary and i/t = h fortime-dependent problems. The Hamiltonian h is given by Eqs. (4.10)and (4.2) with particle index j omitted. We will consider the wave func-tions and energy spectra of electrons for different kinds of external fieldsentering this Hamiltonian. These particular problems form a part of thequantum mechanics and are discussed in detail in the literature. Forthis reason, in this section we cover only the problems whose solutionswill be used in the next chapters.

Free motion. Let us consider the electron states in the absence of anyexternal fields. Formally, it is convenient to assume that the electrons


are confined into a cubic volume L3 with appropriate (zero or periodic)boundary conditions, and the length L is greater than any characteristiclength of the problem. The dispersion law p is obtained as a usualkinetic energy with quasi-discrete (for L3 ) i-th component of themomentum, pi, and the -function is the plane wave:

(p)r = L3/2 exp

(i

p r

),

pi = ni2L

, ni = 1, 2, . . . , p =p2

2m. (1)

The density of states (the number of electron states with energy E perunit volume) is defined as

(E) =2L3

(E ), (2)

where the factor 2 takes into account double degeneracy of electron stateswith respect to spin (it is assumed that the quantum state indices donot include spin quantum numbers). Replacing the sum by the integralover the momentum (problem 1.12), one can obtain (E) in the form

3D(E) =m2mE

23. (3)

Therefore, the density of states in the bulk (three-dimensional) media isproportional to

E.

Electrons in crystals. To describe the electron states in crystals, onehas to introduce a periodic potential energy Ucr(r) in the Hamiltonian.Besides, it is necessary to take into account the relativistic corrections de-scribing the spin-orbit interaction (the other relativistic terms are smallin comparison to this one). The Hamiltonian has the following form:

hcr =p2

2me+ Ucr(r) +

(2mec)2 [Ucr(r) p] , (4)

where p = i is the momentum operator, me is the free electronmass, and is the vector of Pauli matrices. Owing to periodicity ofthe potential in Eq. (4), Ucr(r+Ri) = Ucr(r), where Ri is an arbitrarylattice vector, the solutions of the eigenstate problem are Bloch functions

np(r) = L3/2eipr/unp(r), unp(r+Ri) = unp(r),

En(p+G) = En(p), (5)


where p is defined inside the first Brillouin zone and referred to as quasi-momentum, G is the reciprocal lattice vector, and the electron spectrumEn(p) depends on the band index n and spin quantum number .

The spectrum En(p) and the Bloch amplitudes unp(r) are deter-mined by the geometry of the lattice and by the type of interatomicbond of the crystal. Usually, one can separate the following types: themetallic bond, the hetero- and homeopolar bonds (also known as ionicand covalent bonds, respectively), and the molecular bond, when thecrystal is formed due to van der Waals attraction between the moleculesor atoms. Simple metals can be considered as ensembles of positive ionsoscillating near their equilibrium positions in the crystal lattice and sur-rounded by a gas of nearly free electrons. The total energy of the bondis determined by a negative contribution of the electron-ion interaction( a1, where a is the lattice period) and positive kinetic energy ofstrongly degenerate electrons ( n2/3 a2, where n is the electrondensity). The value of a is determined by the condition of minimum en-ergy. In the limiting cases considered above, the crystal is formed eitherdue to long-range interaction between the molecules, or can be consid-ered as a single macroscopic molecule with periodically placed ions (themetallic case). In the case of an ionic bond in biatomic crystals, thelattice is formed by periodically placed positive and negative ions ap-pearing as a result of electron transfer between the neighboring atoms.Because each ion in such a lattice is surrounded by the ions of the oppo-site sign, the Coulomb interaction leads to the attraction that providesstability of the lattice. The covalent attractive bond appears betweenneutral atoms because of the formation of the pairs of collectivized elec-trons with antiparallel spins. This mechanism is completely analogous tothe valence bond in a single molecule. We point out that a large groupof insulators and semiconductors is characterized by the mixed ionic-covalent bond. Although the eigenstate problem for electrons in crystalsis extremely complicated, a description of these particular materials canbe considerably simplified because their energy bands are either almostempty or almost filled by electrons so that the electronic properties aredetermined by the states near the band extrema. The dynamics of thesestates is discussed below.

To describe the electron states near the band extrema (let us supposethat the latter are at p = 0), we write the Hamiltonian (4) in the basislp(r) L3/2 exp(ip r/)un(r), where l = (n, ) and the Blochamplitude in the center of the Brillouin zone, un(r) un,p=0(r), isdetermined by the equation hcrun(r) = nun(r). Here n is the energyof n-th band extremum at p = 0, and each band is doubly degeneratewith respect to the spin . The matrix elements of the Hamiltonian (4)


are

lp|hcr |lp = ppll(n +

p2

2me

)+ ppp vll ppHll(p), (6)

where the matrix elements of the velocity operator are non-diagonal inthe band index:

vll =l0 pme + (2mec)2 [ Ucr(r)]

l0 . (7)The diagonal matrix elements of the velocity operator are equal to zeroin the point of extrema. Expanding the wave functions (r) over thecomplete set lp(r), we write

(r) =lp

lplp(r), (8)

and the Schroedinger equation is transformed into a set of equations forthe envelope functions lp:

lHll(p)lp = Elp. (9)

Equation (9) defines the many-band spectrum near the extremum. Theexpressions for the matrix elements of the Hamiltonian and velocity op-erator for the two-band model and the set of equations for envelopefunctions lp in the presence of externally applied electric and magneticfields are written in Appendix B.

To obtain the energy spectrum of electrons in the vicinity of the n-thband extremum, one can calculate the diagonal contribution to Elp inthe second order of the perturbation theory:

Elp = n +p2

2me+

l(l =l)

(p vll)(p vll)n n

n +12

pm1p . (10)

The spectrum appears to be quadratic in p. The right equation in Eq.(10) defines the inverse effective mass tensor of the band l = (n, ). Thistensor is expressed through the velocity matrix elements (7) as

m1 =me

+

l(l =l)

vllvll + v

llv

ll

n n. (11)

In the general case, the surfaces of equal energy for Elp of Eq. (10)are the ellipsoids characterized by three principal values of the effective


masses along the main axes. In the uniaxial materials, there are two(longitudinal and transverse) effective masses, while in the cubic mate-rials the tensor (11) becomes a scalar /m. Depending on the signof m, the dispersion relation (10) describes electron (at m > 0) or hole(at m < 0) states. This consideration is valid only in the vicinity of theextremum, where the kinetic energy p2/2m is small in comparison to theinterband energies.

Therefore, the electrons moving in the periodic potential of a crystalcan be considered as free electrons with scalar effective mass m (positiveor negative near the band extrema) under certain conditions describedabove. The result (3) for the density of states of free electrons can bedirectly applied to this case by assuming that m denotes the effectivemass. More complicated cases are realized when: i) the effective massis a tensor; ii) several bands are close in energy and corresponding sev-eral branches of the spectrum with different effective masses have tobe considered; iii) the kinetic energy is not small in comparison to theinterband energies and the non-parabolicity effects (in particular, thedeviation of the energy dispersion law from the quadratic form p2/2m)have to be taken into account. Nevertheless, below we will concentrateon the simplest case of scalar effective mass.

Landau quantization. The electron states in the presence of a magneticfield H are described by the Schroedinger equation

(p eA/c)22m

r = Er, A = (0, Hx, 0), (12)

where the vector potential A is written for H||OZ and the magneticfield is supposed to be homogeneous and time-independent. Owing tothe translational invariance of the problem (12) along OY and OZ, themotion along these directions is described by the plane waves with wavenumbers py/ and pz/. The -function for -state is written as aproduct

()r = (LyLz)1/2 exp

[i

(pyy + pzz)

]

(Npy)x , (13)

and the quantum numbers include both the momenta (py and pz) andthe number of discrete level, N . The quantization of electron states isproduced by the parabolic potential energy which appears from A2contribution in Eq. (12). The equation for (Npy)x standing in Eq. (13)is [

p2x2m

+m2c2

(xXpy

)2]

(Npy)x = N

(Npy)x . (14)


The coordinate Xpy = pyc/|e|H determines the position of the centerof harmonic-oscillator wave function. The electron energy E does notdepend on py (i.e., the electron states under consideration are degeneratewith respect to py). The momentum pz along the direction of H deter-mines the longitudinal kinetic energy. The dispersion law ENpz is writtenas a sum N +p2z/2m, where N is the energy of the N -th level of the har-monic oscillator (Appendix A) with frequency c = |e|H/mc called thecyclotron frequency. The level numbers are the integers: N = 0, 1, . . . .The eigenvalues of Eq. (14) and the corresponding wave functions are

N = c(N + 1/2) ,

(Npy)x =

1

1/4l1/2H

2NN !

exp

[12

(xXpy

lH

)2]HN

(xXpy

lH

), (15)

where HN(x) is the Hermite polynomial and lH =

c/|e|H is the mag-netic length corresponding to the radius of cyclotron orbit in the classicaldynamics. The electron levels with the energies N given by Eq. (15)are called the Landau levels.

According to Eq. (2), the density of states is obtained by integrating(EENpz) over py and pz . Because of the degeneracy of the states withrespect to py , we calculate

py

. . . under the condition |Xpy | < Lx/2corresponding to the requirement that the centers of the oscillator wavefunctions are inside the normalization volume. The result contains thesum over N :

(E) =2Lx

N=0

|Xpy | d/2

(17)

corresponding to the square well (U0 is the depth of the potential welland d is the width). For the case E < U0, the underbarrier part of the


wave function is written as

z {

e(zd/2), z > d/2e(d/2+z), z < d/2 , (18)

where the underbarrier penetration length 1 is determined by =2m(U0 E). Eliminating the underbarrier part of the -function

with the aid of the boundary conditions pzz |d/20d/2+0 = 0, we obtainthe boundary condition of the third kind:[

pz i

2m(U0 E)]z |z=d/2 = 0, (19)

and the eigenstate problem should be solved in the well region |z| < d/2only. Owing to the symmetry of the system, the solution takes the form

z =

2d

{cos(pnz/), n = 1, 3 . . .sin(pnz/), n = 2, 4 . . .

, n =p2n2m

, (20)

where pn are determined by the dispersion relation following from theboundary conditions (19). For the case U0 n, one can use zeroboundary conditions for the -function at z = d/2 and obtain pn =n/d (the difference between this pn and pi in Eq. (1) should not beconfusing because pn in Eq. (20) is positive, and one obtains the sameresult for the density of states if d approaches ). Thus, the dependencen n2 is realized for the lowest energy levels. The approximation whenone uses zero boundary conditions is called the hard-wall model of thequantum well.

Another important situation takes place for narrow quantum wells,when (/d)2/2m U0. In such a case, there is a single confined statewith energy 0. The corresponding level is shallow, U00 U0, and zis a weakly varying function inside the well, z=d/2 z=0. Becauseof this property, the Schroedinger equation is transformed to

p2z2m

z = U0z=0, |z| d/2, we use the wave function (18)with =

2m(U0 0), and the energy of the level becomes

0 = U0 mU20

2(/d)2. (23)


This expression verifies the shallow level condition. The wave functionis formed mostly by the tails outside the well; see Eq. (18).

Low-dimensional states. The dynamics of electrons is modified essen-tially due to the above-described confinement effect. Apart from thewidely known case of attractive three-dimensional potentials describingthe states in atoms or the localized states on the impurity centers insolids, there exist low-dimensional systems realized in solid-state nano-structures. In the two-dimensional (2D) systems realized in quantumwells, the electrons can move in a plane (say XOY ), while the potentialenergy U(r) provides their confinement along the direction perpendicularto this plane. In the one-dimensional (1D) systems realized in quantumwires, the electrons can move only in one dimension (say along OX) andare confined in two remaining dimensions. Accordingly, the Hamiltoniancan be written as

p2

2m+ U(r), U(r) =

{U(z), (2D)U(y, z), (1D) , (24)

where m is the effective mass. The wave functions are written as prod-ucts of the plane waves, which describe free motion of electrons with 2Dmomenta p or 1D momenta p, by the localized wave functions describingthe confinement. The corresponding dispersion laws are written as sumsof the kinetic energy p2/2m and the energies of the levels which dependon the discrete quantum numbers:

(np)r = (p)x,y

(n)z , np = n +

p2

2m,

(n1n2p)r = (p)x

(n1n2)y,z , n1n2p = n1n2 +

p2

2m. (25)

The corresponding densities of states per unit square (L2 for the 2Dcase) or per unit length (L for the 1D case) are obtained in line with thegeneral definition (2), after integrating over 2D or 1D momentum:

2D(E) =2L2

np

(E np) = 2Dn

(E n),

1D(E) =2L

n1n2p

(E n1n2p) (26)

=1

n1n2

(E n1n2)

2m|E n1n2 |

,


where 2D = m/2. The energy dependence of the density of states inthe 2D case is step-like, while in the 1D case it has the inverse-square-root divergences, as in Eq. (16). Considering only the single-subbandcontributions for the 2D and 1D cases, and counting the energy E fromthe bottom of the corresponding subband, one may formally representthe ratios D(E)/2D, where D = 1D, 2D, 3D, as functions of a singleparameter, Em/2; see Fig. 1.2.

Figure 1.2. Density of states for 3D, 2D, and 1D electrons. Only the single-subbandcontributions are shown for the 2D and 1D cases, and the energy is counted from thebottom of the corresponding subband.

So far we have considered stationary problems. Below we analyze theSchroedinger equation i/t = Ht with time-dependent Hamilto-nian Ht.

Homogeneous electric field. To describe the temporal evolution ofelectron states in the electric field Et, one can express the field in theHamiltonian Ht either through the vector potential At = c

tdtEt or

through the scalar potential (Et x). For each of these variants, theSchroedinger equation in the momentum representation is written as

iptt

= (t)pt, t = p+ e t

dE ,

i

(

t+ eEt

)t = ()t, (27)


where p and denote the canonical and kinematic momenta, respec-tively, and (p) = p2/2m. The initial conditions to Eq. (27) at t0 aregiven by the Kronecker symbols pp0 and 0 , where p0 and 0 arethe quantum numbers of the canonical and kinematic momenta. Thesolution of the first equation of Eq. (27) is

(p0)pt = exp

[ i

tt0

dt(t)]pp0 , (28)

where t satisfies the classical equation of motion. The second equa-tion of Eq. (27) is a differential equation of the first order with partialderivatives, and its solution is

(0)t = exp

[ i

tt0

dt( + e

tt dE )

2

2m

]0 . (29)

The wave function given by Eq. (29) coincides with the one of Eq. (28)after a formal replacement of the kinematic momentum by the canonicalone. Therefore, the use of either vector or scalar potentials correspondsto the formulation of the problem in terms of either p or , respectively.Under a time-independent electric field E, the momentum is linear intime, p + eEt, while in the harmonic field E cost it is convenient toexpand (p0)pt into Fourier series. Such an expansion (problem 1.14)demonstrates that a shift of time by 2/ changes the wave function(28) by the phase factor exp[2i(t)/], where the line over the ex-pression denotes the averaging over the period 2/. Some consequencesof this transformation are discussed below for a more general case.

Quasienergy. If the potential energy is a periodic function of time,and the period is 2/, one can write the wave function as

(t) = exp(iEt/)uE(t), uE(t+ 2/) = uE(t), (30)

where E is referred to as quasienergy, since it is defined in the region[0, ] (we note the analogy of Eq. (30) with the Bloch function in aspatially periodic potential considered above). The states with differ-ent E are orthogonal to each other. To find uE(t), it is convenient torepresent the Hamiltonian containing a periodic potential as

Ht = H +s

Wseist, (31)

where H is the part of the Hamiltonian averaged over the period 2/,while Ws describes the oscillating part and can be associated with the


perturbation due to an external field. Introducing this field throughthe vector potential, it is easy to show that s = 1,2, because theHamiltonian is quadratic in the momentum. Further, with the Fourierexpansion uE(t) =

s exp(ist)uE(s), the Schroedinger equation is

rewritten as a set of coupled equations

(E + s H)uE(s) =s

WsuE(s s). (32)

In many cases, the time-dependent part of the Hamiltonian (31) can beconsidered as a perturbation (of the first and of the second order in thefield for s = 1 and s = 2, respectively). Within the accuracy of thesecond order, the perturbation theory gives us the following equation foruE(t) = uE(s = 0) uE:

(E H)uE =s=1

Ws(E s H)1WsuE. (33)

This is the eigenstate problem determining the quasienergy spectrum ofthe system. For an electron in a homogeneous harmonic electric field,the quasienergy can be found from Eq. (28) by averaging (t) over theperiod. As a result, the parabolic spectrum is simply shifted in energyby (eE/)2/4m.

6. Long-Wavelength PhononsThe small-amplitude vibrations of the crystal lattice are described by

a set of atomic displacement vectors us(Rnt), where Rn is the radius-vector of the elementary crystal cell numbered by the integer vectorn and the index s numbers the atoms in the cell. The expansion ofthe potential energy in the vicinity of the equilibrium positions of theatoms begins with the second-order terms, quadratic in the displace-ments. Accounting also for the anharmonic corrections described by thecubic terms, we write the total energy as

E = 12

nsk

Ms

[uks (Rn)

]2+

12

n1n2

s1s2k1k2

Gk1k2s1s2 (|Rn1 Rn2 |)

uk1s1 (Rn1)uk2s2 (Rn2) +

13!

n13

s13k13

Ak1k2k3s1s2s3 (|Rn1 Rn2 |, |Rn1 Rn3 |)

uk1s1 (Rn1)uk2s2 (Rn2)u

k3s3 (Rn3). (1)

The index k numbers the Cartesian coordinates, us(Rn) is the velocityof the atom s in the cell n, and Ms is its mass. The matrices Gk1k2s1s2


and Ak1k2k3s1s2s3 are determined by the second and third derivatives of thepotential energy in the equilibrium position. Owing to periodicity of thelattice, they depend only on the distances between the cells, |RnRn |.

Taking into account only the second-order contributions in Eq. (1),we obtain the classical equations of motion (problem 1.15)

Msuks (Rn) +

nsk

Gkk

ss (|Rn Rn |)uk

s (Rn) = 0 (2)

describing harmonic vibrations determined by the force constants Gkk

ss .The anharmonic contribution to the energy is described by the last termin the expression (1) and can be treated as a weak interaction betweenthe vibrational modes. The translational invariance allows us to repre-sent us(Rnt) as a plane wave, Us exp(iq Rn it). This substitutionreduces the number of variables in Eq. (2) from 3Ns to 3s, where N isthe number of elementary cells and s is the number of atoms in the cell.The amplitudes Us obey the following set of linear algebraic equations:

Ms2Uks

ks

[n

Gkk

ss (|Rn|)eiqRn]Uk

s = 0, (3)

where Rn Rn Rn . There are 3s solutions of this set, each corre-sponds to a branch (mode) of the vibrational spectrum.

It is convenient to express the displacements in terms of the polariza-tion vectors es(ql)/

Ms (the index l = 1, . . . , 3s numbers the vibrational

modes) found from the equationssk

[2sskk Gkk

ss (q)

]ek

s (ql) = 0,

Gkkss (q) =n

Gkk

ss (|Rn|)MsMs

eiqRn , (4)

which directly follow from Eq. (3). Because the matrix Gkkss (q) is peri-odic in q, one should consider q inside the first Brillouin zone so that q isa quasi-wave vector. The set of solutions es(ql) is normalized accordingto

s e

s(ql)es(ql

) = ll , and es(ql) = es(ql). Employing the nor-mal coordinates Qql(t), which include the exponential time-dependentfactors, we write uns(t) us(Rnt) as

uns(t) = (NMs)1/2ql

Qql(t)es(ql)eiqRn , (5)


where Qql(t) = Qql(t) because the displacements are real. In orderto express the quadratic contributions of Eq. (1) through the nor-mal coordinates, we take a sum over n with the aid of the relationN1

n exp[i(q q) Rn] = qq . The function qq is equal to 1

when q and q either coincide or differ by a reciprocal lattice vector andis equal to zero otherwise (problem 1.16). The energy of small-amplitudevibrations takes the form

12

ql

[Qql(t)Qql(t) +

2qlQ

ql(t)Qql(t)

], (6)

which is quadratic in the normal coordinates and momenta Pql(t) Qql(t). This is the energy of a set of harmonic oscillators with frequenciesql.

The quantization of the lattice vibrations can be done by analogywith the case of photons described in Sec. 3. The normal coordinatesand normal momenta are replaced by the operators Qql and Pql, whichsatisfy the commutation relations similar to Eq. (3.7):

[Qql, Pql ] = iqqll . (7)

The energy (6), after such a substitution, becomes a Hamiltonian of thekind (3.8), where the index is replaced by the quantum numbers qand l. The quasiparticles with these quantum numbers are called thephonons. Their creation and annihilation operators, b+ql and bql, satisfythe commutation relations

[bql, b+ql ] = qqll . (8)

The Hamiltonian of phonons is written as a sum of the boson modecontributions according to Eq. (3.12), and the relation between thebosonic operators and normal coordinates and momenta is the same asin Eq. (3.11). It can be rewritten as

Qql = Q+ql =

2ql(bql + b+ql),

Pql = P+ql = i

ql2

(bql b+ql). (9)

Substituting the expression for Qql into Eq. (5), we obtain the second-quantized displacement operator

uns = (NMs)1/2ql

2qles(ql)eiqRn(bql + b+ql) (10)


written through the linear combination of creation and annihilation op-erators. Therefore, the displacements us satisfy the oscillatory equations(2) analogous to the equations for the electromagnetic field in the ab-sence of free charges. To quantize the lattice vibrations, one shouldexpress the displacements in terms of the generalized normal coordi-nates and momenta and, further, introduce the elementary excitations,phonons.

Below we consider the limit of long wavelengths (the region of small q),when the lattice vibrations are described by a number of macroscopicparameters. This approach is convenient for description of electron-phonon and phonon-photon interactions. First we notice that under ashift of the lattice as a whole, when us(Rnt) do not depend on Rn, theforce in Eq. (2) must be zero. This leads to the identity

nsGkk

ss (|Rn|) = 0. (11)

Using Eq. (11) together with the symmetry property of the force matri-ces, Gkk

ss = G

kkss , we sum Eq. (3) over s. If q 0, it gives us

2s

Msus = 0. (12)

Equation (12) describes two kinds of vibrations pertinent to the long-wavelength limit: the acoustic phonons with 0 at q 0, and theoptical phonons with = 0. Under the optical vibrations, the center ofmass of the cell remains at rest, and the atoms of the cell oscillate inantiphase,

s Msus = 0. There are 3 acoustic modes and 3s 3 optical

modes.Let us consider the long-wavelength optical vibrations in a biatomic

(s = 2) crystal with ionic bond, when the oppositely charged sublat-tices oscillate as a whole with respect to each other. In this approxi-mation, the displacements uns do not depend on n, and the vibrationsare described by two variables u corresponding to two sublattices witheffective charges e and atomic masses M. According to Eq. (12),M+u+ + Mu = 0, and there is only one independent variable, therelative ionic displacement u = u+ u. Under such vibrations, eachelementary cell has a dipole moment eu, and a significant contributionto the interatomic forces comes from the long-range dipole-dipole inter-action. One has to subdivide the atomic force constant matrices Gkkss bya short-range part, proportional to the relative shift u, and a long-rangepart, given through the electric field with local strength EL. In this way,


the equations of motion become

M+u+ = k(u+ u) + eELMu = k(u+ u) eEL

, (13)

where the coefficient k describes the short-range forces. The longitudinalfield EL in Eq. (13) induces the polarization (N/V )EL, where is thepolarizability of the cell and N/V is the number of the cells per unitvolume. The total polarization P is the sum of this induced contributionand dipole moment eu. Thus, for u, EL, and P, we have

u = 2TOu+e

MEL,

P =N

V(eu+ EL). (14)

The first equation, which follows from Eq. (13), contains the reducedmass M = M+M/(M+ + M). The contribution of the short-rangeforces to this equation is expressed through the transverse mode fre-quency TO, the latter is introduced as TO = (k/M)1/2. Since thelong-range electric fields are not generated by the transverse displace-ments (see the picture of ionic crystal vibrations in Fig. 1.3), TO is thefrequency of transverse vibrations.

Figure 1.3. Pictures of the longitudinal (a) and transverse (b) vibrations of the ioniccrystal sublattices.

To consider the longitudinal vibration, one has to use an additional,with respect to Eq. (14), relation between EL and P. Owing to electricneutrality of the lattice, the Poisson equation for the

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Quantum Kinetic Theory and Applications: Electrons, Photons, Phonons

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