Non-Linear Carrier Dynamics in Semiconductor Microstructures

Non-Linear Carrier Dynamics in Semiconductor Microstructures

Karol KALNADepartment of Theory of Semiconductor Microstructures

Institute of Electrical Engineering, Slovak Academy of SciencesDubravska cesta 9, 842 39 Bratislava, Slovakia

Tutor: RNDr. Martin MOSKO, CSc.

Bratislava, December 1994

i

To Viki and Kajo— who are performing non-linear dynamics

Contents

I Formulation of Theory 1

1 Introduction 3

2 Liouville-von Neumann Quantum Equation 52.1 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Boltzmann Transport Equation 93.1 Electron-Electron Hamiltonian in Jellium Model . . . . . . . . . . . . . . . . . . . . . . . 103.2 Interaction Representation and Distribution Function . . . . . . . . . . . . . . . . . . . . 113.3 Drift Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Scattering Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.1 Born Approximation for Electron-Phonon Interaction . . . . . . . . . . . . . . . . 163.5 Screening and Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Interparticle Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6.1 Born Approximation for Electron-Electron Interaction . . . . . . . . . . . . . . . . 213.7 Linear Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.7.1 Thomas-Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7.2 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.8 Limitations of Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Methods for Solution of Boltzmann Transport Equation 294.1 Iterative Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Monte-Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Comparison of Ensemble Monte-Carlo Method with

Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Goals of the Doctoral Thesis 39

A Plasmons as Collective Excitations in Electron Gas 41

0

Part I

Formulation of Theory

1

Chapter 1

Introduction

Minituarization of electronic devices was fast progressing during the last twenty years. There are somebasic reasons for the unceasing interest in this field. Very small devices not only work faster and consumeless energy but also provide new functions and show new physical effects. In the active region of a verysmall device very high electric fields are usually present. Therefore non-linear carrier dynamics must betaken into account in the construction of proper physical models.

The non-linear dynamics is joined with the two main fields of interest: i) device research for themicroelectronic applications and ii) new physical effects in the structures of reduced dimensionality. Thefast development of both fields begun when sophisticated epitaxial growth techniques (MBE, MOCVD)and nanolitography techniques appeared. The progress of these fine technologies enables to constructartificial quantum systems e.g. quantum wells, quantum wires, quantum dots, ballistic structures andsuperlattices.

To describe non-linear transport in semiconductor devices it is necessary to incorporate such effectslike the electron heating, size effects, real-space transfer, time duration of the scattering, etc. Newphenomena and different time scales appear when the dimensionality of the device is reduced to 2-, 1- or0-dimensions in comparison with classical 3-dimensional devices.

The carrier dynamics and carrier interactions (electron-electron, electron-hole and/or electron-pho-non) are usually much more complicated than in the linear transport regime. First principle formulationsof the carrier behaviour in solids are summarized in the Table I to give the full framework of statisticalmechanics [1]. In the classical transport theory Boltzmann equation, within the framework of classicalmechanics, is derived from the Liouville-von Neumann equation by using infinite hierarchy of distributionfunctions. The derivation of the quantum dynamical equations from the basic ideas of quantum theorystarts from the Liouville-von Neumann equation for density matrix. The basic formal similarity betweenthe quantum and classical approaches can be summarized as in Table II.

The goal of this work is to review the Boltzmann transport theory including the derivation of theBoltzmann transport equation (BTE) from the first principles and the methods which enable to solveBTE exactly. The Liouville-von Neumann quantum equation and the relative Master equations arepresented in Section 2, represent starting points for the derivation of the BTE . In Section 3 the BTEis derived for the homogeneous time-dependent transport problem assuming carrier-phonon and carrier-carrier interactions. Here we mention also our original contribution to this problem (the paper is enclosedat the end of the text). Numerical techniques (iteration procedure, Monte-Carlo technique) which provideexact solution of the BTE are outlined in Section 4. The problem of equivalency of BTE and Monte Carlois also discussed in detail. Finally, in Section 5 the goals of our doctoral thesis are specified.

3

4 CHAPTER 1. INTRODUCTION

1. Equilibrium classical stat. mechanics 2. Nonequilibrium classical stat. mechanicsBasic tools: Basic tools:Distribution function, Maxwell distribution Boltzmann transport equationPartition function Master equationCanonical grand ensemble Liouville-von Neumann equation

Hierarchy of distribution function

Applications: Applications:Classical theory of specific heat, equation of state Calculation of thermal conductivityvirial development, molecular distribution Electrical conductivityfunction Viscosity, equations of hydrodynamicsPhase transitions Diffusion coefficients3. Equilibrium quantum stat. mechanics 4. Nonequilibrium quantum stat. mechanicsBasic tools: Basic tools:Distribution functions: Fermi-Dirac, Einstein-Bose Quantum forms of transport equationQuantum-mechanical partition function Liouville equation for the density matrixCanonical grand ensemble, density matrix Wigner phase-space functions

Applications: Applications:Quantum theory of specific heat of solids Quantum theory of electrical conductivitySpecific heat of an electron gas Nuclear spin relaxationMagnetic properties of an electron gas Thermal conductivity of solidsHard-sphere Bose gas MagnetoresistancePhase transitions Line broadening in solids

Table I: Subdivisions of statistical mechanics.

Classical statistical mechanics Quantum statistical mechanicsPhase space density: WN (p, r, t) Density matrix: (x, x′, t) Wigner function: WQ(k, r, t)

or: nm(t)Liouville equation:∂WN

∂t − [H,WN ] = 0 ih∂∂t − [H, ] = 0 ∂WQ

∂t +∑ pi,α

m∂WQ

∂ri,α+ ΩWQ = 0

Observed averages: Observed quantities: Observed quantities:< O >=

∫dp∫dr O(p, r)WN < O >= Tr (O) < O >=

∫dk∫drO(k, r)WQ

Reduced distribution functions: Reduced density matrices: Reduced distribution functions:fs(p1 . . . ps, r1 . . . rs, t) = (x1 . . . xs, x

′1 . . . x

′s) = fQ

s (k1 . . . ks, r1 . . . rs, t) =∫. . .∫dps+1 . . . drN× ∫

. . .∫dxs+1 . . . dxN (x1 . . .

∫. . .∫dks+1 . . . drN×

×WN (p1 . . . xN ) . . . xN , x′1 . . . x

′N ), s < N ×WQ(r1 . . . kN , t)

Liouville hierarchy Master function for probabi-Master function: P (n1 . . . nN , t) litiesMaster equation Master equation Master equationBoltzmann transport equation Uhlenbeck-Uhling equation

Table II: Comparison between classical and quantum mechanics.

Chapter 2

Liouville-von Neumann QuantumEquation

A quantum-mechanical system is described by vector

R

QS

Figure 2.1: Concept of the great reservoir (R)and the quantum system (QS) inside

|ψ〉 in the Hilbert space which provides the absolute de-scription of the state [2]. In some situations when theconsidered quantum system is a part (subsystem) of agreater system or when we can not obtain the total in-formation it is useful to introduce the density matrix [3].

If we consider the closed system in which a subsystemis inserted, we have the concept of the reservoir and thesystem (see Fig. 2.1). Dynamical variables of the reser-voir in state |ψR〉, which surrounds our system of interestare ξ. Dynamical variables of the system in the state |ψS〉are x. If we want to express the mean value (denoted by angle brackets) of some physical quantity O(x)related only to our system, the following integral has to be calculated:

< O > = 〈ψ(ξ, x)|O(x)|ψ(ξ, x)〉 (2.1)

For the calculation of mean values it is helpful to define the operator

(x, x′) =∫

dξ ψ∗(ξ, x′)ψ(ξ, x) (2.2)

called the density matrix. Then (2.1) has the form:

< O > =∫

dx [O(x)(x, x′)]x=x′ = Tr[O]. (2.3)

The density matrix can be expressed as = |ψS〉〈ψS | (2.4)

in the operator form of Dirac notation [4].¿From the statistical point of view the density matrix is introduced for a pure state as

= |n〉〈n| (2.5)

where |n〉 is the Fock state. The density matrix is hermitian in this case: † = and its trace is equal tounity: Tr = 1. For a mixed state which can be found with the normalized propability pn in the state|n〉, the definition of the density matrix reads

=∞∑

n=0

pn|n〉〈n|, (2.6)

5

6 CHAPTER 2. LIOUVILLE-VON NEUMANN QUANTUM EQUATION

where [5]∞∑

n=0

pn = 1 ; pn ≥ 0.

The time development of the quantum states |ψ〉 is governed by the general expression

|ψ(ξ′;x′, t)〉 =∑ξx

Uξξ′xx′(t) |ψ(ξ;x, 0)〉 (2.7)

where U is an arbitrary time operator and the time argument was carried out from the x variable.Therefore, the density matrix (2.2) can be written as

(t) = U(t) (0) U∗(t) (2.8)

in contrast to any dynamical quantity

O(t) = U∗ O(0) U(t)(t).

For an isolated system with states determinated by Schrodinger equation with the hamiltonian H , theequation of motion reads [6]

ih∂(t)∂t

= [H, (t)]. (2.9)

Equation (2.9) is the Liouville-von Neumann equation for the density matrix. In the next we will focusour attention to possible forms of the hamiltonian in the equation (2.9).

2.1 Master Equation

The equation,

ih∂

∂t= L (2.10)

which generates time evolution of the density matrix (2.2) is known as master equation. Above, L isan evolution operator which takes into account both the Hamiltonian of the quantum system and itsinteraction with the reservoir (see Fig. 2.1). Consequently, L is typically non-Hermitian. The masterequation (2.10) governs the evolution of the system, e.g. a harmonic oscillator or a free particle in thecollection of oscillators which we shall refer to as the heat bath.

Starting from the quantum equation (2.10) we derive the master equation for the opened quantumsystem QS. If one uses the projection operator P [7, 8] associated with the quantum system, one can find[9, 10]

ih∂P(t)∂t

= PL[P(t)] +PLe−i(1−P)Lt(1−P)(0)− i

∫ t

0

dτ PLe−iτ(1−P)L(1−P)L[P(t− τ)]. (2.11)

In the following we try to find a more practicable form of the equation (2.11) starting from the masterequation (2.10).

The calculation becomes more tractable when the density matrix is not given in the position rep-resentation (2.2). If we define d as the distance from the diagonal of an arbitrary matrix in the positionrepresentation and k as the wave number in the direction parallel to the diagonal, we have in a newrepresentation

(k, d) =∫

dx eikx (x− d/2, x+ d/2). (2.12)

Note that this is just the double Fourier transform of the usual Wigner function [11]

W (p, q) =12π

∫dz (q − z/2, q + z/2)eipz.

2.1. MASTER EQUATION 7

The density matrix operator in (2.12) can now be expressed as

(q − 12d, q +

12d) = Tr δ(Q− q)eiPd/2eiPd/2 (2.13)

by means of the shift operator exp(iPd/2). Here Q and P are the natural position and momentumoperators, respectively. From equations (2.12) and (2.13), utilizing the commutation relation [P,Q] = −i,we obtain

(k, d) = Trei(kQ+dP ). (2.14)

If we wish to find the expression for the operator L in (2.10), we have to calculate the time derivative ofthe time-dependent density matrix (2.14)

L = ih∂ (k, d)/∂t (k, d)

. (2.15)

To evaluate its time derivative, we need the formula for an arbitrary operator O:

deO(t)

dt=∫ 1

0

dλeλO(t) ∂O∂te(1−λ)O(t).

The time derivative of the density matrix is then given by

∂

∂t= Tr

∫ 1

0

dλ eiλ(kQ+dP )

[i(k

∂Q

∂t+ d

∂P

∂t)]eiλ(kQ+dP ). (2.16)

To derive a more appropriate master equation we can define the damping coefficient γ as γ = κ2/4when the evolution of the harmonic oscillator is generated by

m(d2x

dt2+ Ω0x

2) = −κdφdt.

The field φ obeysd2φ

dt2− d2φ

dr2= κ

dx

dtδ(x).

Above, Ω0 is the frequency of the undumped harmonic oscillator and κ is a constant. After the re-transformation from the (k, d) representation into the standard position representation (2.16) arrives to[12]

ih∂

∂t= [HS , ] − 2iγ(x− x′)

(∂

∂x− ∂

∂x′

)− 4iγ(x− x′)2h− 4γ(x− x′)f

(∂

∂x+∂

∂x′

). (2.17)

where HS is the free oscillator hamiltonian of the system (see Fig. 2.1). Above, functions f and h aregiven by

f(t,Γ, β) =1

Ωπ

∫ Γ

0

dω

[∫ t

0

dτ cos(ωτ) sin(Ωτ)e−γτ

]ω coth(βω/2)

g(t,Γ, β) =1π

∫ Γ

0

dω

[∫ t

0

dτ cos(ωτ) cos(Ωτ)e−γτ

]ω coth(βω/2)

h(t,Γ, β) = g − γf

where Ω is the angular frequency of the damped harmonic oscillator defined by

Ω =√

Ω20 − γ2,

β is inverse thermal energy (β = 1/kBT ) and Γ is the cutoff which is introduced to regularize logarithmicdivergencies which appear when the above integrals are calculated (see discussion in [12]).

8 CHAPTER 2. LIOUVILLE-VON NEUMANN QUANTUM EQUATION

The first term on the right-hand side of master equation (2.17) is the von Neumann term similar to(2.9). The second term (2.17) is the damping term which is responsible for the dissipation of the energyand the third term describes the quantum diffusion due to the fluctuations. However, diffusion coefficienth is still time dependent. The last term of the equation (2.17) contains a correction proportional to f .Coefficient f tends to disappear for harmonic oscillator in the classical (high-temperature) limit. In thislimit the equation (2.17) has the form

ih∂

∂t= [HS , ] − 2iγ(x− x′)

(∂

∂x− ∂

∂x′

)− 4iγ(x− x′)2h (2.18)

which can also be obtained [13] with the help of Feynman-Vernon influence-functional technique [14].In this last part we will briefly model the effect of linear attenuation and amplification ([15], p 254)

by coupling the quantum system to the reservoir. The hamiltonian H from (2.9) can be taken in theform [10]

H = HS +HR +HSR (2.19)

The hamiltonian HS consists of the harmonic oscillators with the characteristic frequency ω and thearbitrary function of the annihilation operators d and creation operators d†,

HS = hω(d†d + 1/2) + f(d, d†). (2.20)

HR is the hamiltonian of the reservoir which we assume in the form

HR =∑

λ

hωλ(b†λbλ + 1/2) (2.21)

where ωλ is the characteristic phonon frequency in the λ mode and b, b† are the annihilation and creationoperators, respectively, satisfying the relation

[b†λ(t), bλ′(t′)] = δλλ′δ(t− t′).

Finally, the coupling between the oscillator and the bath is described by the hamiltonian [16]

HSR = h[dΓ†(t) + d†Γ(t)]. (2.22)

The bath operators Γ(t) are given byΓ†(t) =

∑λ

gλb†λ(t)

in the case of attenuation, andΓ(t) =

∑λ

g∗λbλ(t).

in the case of amplification. Here gλ is a coupling constant for a single mode λ.For what follows, it is important to introduce the basic postulate: our reservoir from Fig. 2.1 must

be found in the unceasingly thermodynamic equilibrium (Born-Markov approximation). Then we candispose of the reservoir variables and density matrix from (2.9) has only the system variables. Themaster equation describes time evolution of the density matrix of the system in the interaction picture

ih∂

∂t= [H ′

S , ] +i

2γ(n+ 1)(2dd† − d†d− d†d) +

i

2γn(2d†d − dd†− dd†) (2.23)

where H ′S is hamiltonian in the interaction representation (3.15) for the closed quantum system (unin-

teracting with the environment), n is the average number of particles (phonons, photons, etc.) in theλ mode ([15], p 264) and γ is the decay rate. The second term on the right hand-side of the masterequation (2.23) ensures the transition from the quantum system to the reservoir and the third term of(2.23) ensures the transition from the reservoir to the system.

Chapter 3

Boltzmann Transport Equation

To study non-equilibrium statistics of classical gas, Boltzmann [17] introduced the BTE which describesthe time evolution of the distribution function f due to binary interparticle collisions [18]. A classicaldistribution function f(r,v, t) depends on the particle position r, on the particle velocity v, and on thetime t [19]. This model may be used also in the theory of semiclassical electron transport in solids.

We will derive semiclassical BTE from the Liouville-von Neumann equation (2.9) using certain ap-proximations. Significant assumptions and logical interrelations of solid-state physics, necessary to deriveBTE, are shown in the Table III.

In representation of the second quantization one can introduce the annihilation and creation operatorsa, a†, respectively, for fermions. Similarly, the annihilation and creation operators b, b† can be definedfor bosons. The solid-state system will be described by the hamiltonian in which the particle spin isomitted (we consider the zero magnetic field and we neglect the spin-orbit interaction). Due to adiabatic(Born-Oppenheimer) approximation [20] the hamiltonian of our system can be written as

H = He +HL +HF +Hi +Hee (3.1)

where He is the kinetic energy operator

He =∑k

εka†kak, εk =hk2

2m. (3.2)

with m the effective mass (besides Born-Oppenheimer approximation the effective mass approximation[21] is used to introduce He ). HL is the hamiltonian of lattice vibrations,

HL =∑q

hωq

(b†qbq +

12

)(3.3)

with ωq as the phonon frequency of the mode q. The hamiltonian

HF =eh

m

∑k

(k ·E t+ 1/2 e2E2t2) a†kak (3.4)

describes the interaction of an electron with the applied electric field E, where e is the charge of anelectron [22]. Interaction of an electron with lattice vibrations can be expressed as

Hi =∑k,q

h (Mqb†qa†k−qak +M∗qbqa†k−qak), (3.5)

where Mq is the coupling parameter.

9

10 CHAPTER 3. BOLTZMANN TRANSPORT EQUATION

Electron problem Lattice problem Interactions– Many-electron problem – Harmonic approximation – Character of the ion-electron

interactions– Single-electron problem ina periodic potential. Band – Theories attempting to – Various assumptions aboutstructure. Effective mass calculate dispersion relation the behavior of ions (rigid,approximation deformed)

– Single-electron dynamics with – Phonon-phonon interaction – Calculation of (self-consis-an assumed dispersion law: due to anharmonic effects tent) screened interaction

ε = ε(k) potentials and frequencies

– Approximative treatment – Equilibrium between phonons. – Perturbation treatmentof inter-electron interactions Umklapp processes

Table I: Assumptions in solid-state transport theory

The hamiltonian Hee for an electron-electron interaction reads

Hee =12ν

∑k,k’,q

vq a†k+qa†k’−qak’ak. (3.6)

In the hamiltonian (3.6) vq represents the Fourier transformation of the Coulomb potential e2/(4πκr),which is given by

vq =∫d3r

e2

4πκ|r| exp(−q · r) =e2

κq2for q = 0 or vq =

∫d3r

e2

rfor q = 0 (3.7)

where ν is the normalization volume of the system and κ is the dielectric permittivity.

3.1 Electron-Electron Hamiltonian in Jellium Model

The interacting homogeneous electron gas is known as the free electron gas or the jellium model. Thisparticular system can be thought of as a model for many systems, if one assumes that the charge densityof the positive ions of the metal or semiconductors is uniformly smeared out over the volume of thesystem so that the electrons can move practically freely through the material. The valence electrons areonly weakly bound and are therefore only weakly localized in the crystal lattice.

In the jellium model the three terms (HL, HF and Hi) can be omitted in the complete hamilto-nian (3.1). The hamiltonian can be rewritten as

H = He +Hb +Heb +Hee. (3.8)

Here Hb is the electrostatic energy of the background ions. The homogeneous electron gas has no crystalstructure because the positive charge of the ions is spread uniformly about the unit cell of the crystal.Then the hamiltonian Hb can be expressed as the average density of the electrons

Hb =e2

2

N∑i,j

〈a†iai〉〈a†jaj〉|ri − rj | =

e2

2

N∑i,j

i(ri)j(rj)|ri − rj | =

N2e2

2ν

∑r

1|r| (3.9)

since the number operator may be summed to give the number of particles N . The hamiltonian Heb

is the interaction of the electrons with the background ions and corresponds to the external potential.

3.2. INTERACTION REPRESENTATION AND DISTRIBUTION FUNCTION 11

Using transformation (3.7), this hamiltonian can be written as

Heb = −e2N∑i,j

a†iai

|ri − rj | = −e2N2

ν

∑r

1|r| . (3.10)

In the hamiltonian (3.10) the term with q = 0 was canceled [23] after the Fourier transformation (3.7) ofthe Coulomb potential between the electrons and the ion background. The last hamiltonian Hee in theexpression (3.8) is self-interaction of the electrons and will be split into two terms

Hee = ∑k, k’q=0

+∑k, k’q=0

vq2ν

a†k+qa†k’−qak’ak. (3.11)

Further only first term of (3.11) will be provided

12ν

∑k,k’

vq=0 a†ka†k’ak’ak = − 12ν

∑k,k’

vq=0 a†ka†k’akak’ =

=12ν

∑k,k’

vq=0 a†kaka†k’ak’ − a†kak’δk,k’ =12ν

(N2 −N

)vq=0 (3.12)

using anticommutation relations for the fermion operators. When the results (3.9),(3.10) and (3.12) aresummarized in the equation (3.8) then the constant terms cancel mutually ([24] p 25)

Hi +Heb +H(q=0)ee =

N2

2νvq=0 − N2

νvq=0 +

12ν

(N2 −N

)vq=0.

The remaining constant expression ∼ N will be transformed to an integral over the unit cube throughthe change of the scale given by rnew = rold/L. Because ν = L3 one has

−N

2νvq=0 = −NL

2

2ν

∫ 1

−1

d3re2

|r| = − cN

2ν1/3(3.13)

where c is a constant which does not depend on the volume. The corresponding energy contribution perparticle will thus vanish as ν−1/3 in the thermodynamic limit. Consequently, the term (3.13) can beneglected.

Finally, the electron-electron interaction hamiltonian was obtained in the form

Hee =∑k, k’q=0

vq a†k+qa†k’−qak’ak, vq =e2

2νκq2. (3.14)

where the term with q = 0 is not present. If the entire interaction is regarded as a perturbation,the zeroth and the first (3.14) order terms in a perturbation theory are precisely corresponding to theHartree-Fock energy of the electron gas [25].

3.2 Interaction Representation and Distribution Function

In the following we will use the interaction representation

O′(t) = exp[i

h(He +HL)t

]O exp

[− i

h(He +HL)t

](3.15)


for an arbitrary operator O. Using the representation (3.15) and the hamiltonian (3.1), from the equation(2.9) we have

ih∂′

∂t= [H ′

F (t) +H ′i(t) +H ′

ee(t), ′(t)]. (3.16)

The formal solution of (3.16) satisfying the initial condition (−∞) = 0(He +HL + Hi + Hee), where0 is the equilibrium density matrix, can be expressed in the integral form

′(t) = ′(t0) − i

h

∫ t

t0

dt′ [H ′F (t′) +H ′

i(t′) +H ′

ee(t′), ′(t′)]. (3.17)

The choice of the initial condition 0 for the equation (3.16) causes complications. The states ofthe unpertubed free basis obtained in the representation (3.15) are not stationary even in the initialstate, because the electron-phonon interaction H ′

i can lead to spontaneous fluctuations of the thermaldistributions of phonons and electrons. If one assumes the weak coupling between both systems, sucheffects can be neglected [26].

Further we will work in the representation in which He +HL is diagonal. Let eigenvalues and eigen-functions of He are εk and |k〉, respectively. Eigenvalues and eigenfunctions of the operator HL areEN and |N〉. We assume phonons in thermal equilibrium in spite of their interaction with electrons,which gain energy from the external field. Such approximation is usually valid for not too high electronconcentrations [27].

If the operator O depends only on the electron variables, than

〈k′N ′|O|kN〉 = 〈k′|O|k〉δN ′N .

It is evident from (2.3), that we can write

< O > = Tr[Of ] =∑k,k’

〈k|O|k′〉〈k′|f |k〉.

The electron distribution function f(k, t) is given by diagonal elements of the matrix as

〈k|f |k〉 =∑N

〈kN ||kN〉. (3.18)

Non-diagonal matrix elements can be expressed as

〈kN |(t)|k′N ′〉 = 〈k|f(t)|k′〉P (N)δN,N ′ (3.19)

if one assumes that P (N) is given by the Boltzmann factor (kB is the Boltzmann constant):

P (N) =exp(−EN/kB)∑

N ′ exp(−EN ′/kBT ). (3.20)

This is a fundamental statement of the statistical mechanics introducing the existence of non-phaserelation between the probability amplitudes for different states |N〉 and the probability of finding thelattice in any one of these states [28]. Note that an identical relation as (3.19) is satisfied by ′(t) andf ′(t).

The distribution function f(k, t) does not depend on the spatial variable r, i.e. an infinite homogeneoussystem of electrons is considered. For an inhomogeneous problem, Wigner distribution function [11, 29],which serves as a bridge bettween quantum and classical transport theory, or N-body Green function[30] has to be used. Function f(k, t) can be used to obtain ensemble averages for observable macroscopicquantities.

3.3. DRIFT TERM 13

Since HF does not contain lattice variables, the equation (3.17) gives with the use of (3.18) thefollowing equation for the electron distribution function [31]:

〈k|f ′(t0)|k〉 − 〈k|f ′(t)|k〉 =i

h

∫ t

t0

dt′ 〈k|[H ′F (t′), f ′(t′)]|k〉 +

+i

h

∫ t

t0

dt′∑N

〈kN |[H ′i(t

′), ′(t′)]|kN〉 +i

h

∫ t

t0

dt′∑N

〈kN |[H ′ee(t

′), ′(t′)]|kN〉. (3.21)

3.3 Drift Term

Using the definition (3.15) and the eigenvalues εk and EN , the following relations are valid for the diagonalmatrix elements:

〈k|f ′(t)|k〉 = 〈k|f(t)|k〉 (3.22)

〈k|[H ′F (t), f ′(t)]|k〉 = 〈k|[HF , f(t)]|k〉. (3.23)

Relation (3.22) can be used on the left-hand side of the equation (3.21). The first term on the right-handside of (3.21) can be simplified using (3.23). After that equation (3.21) is transformed to

∫ t

t0

dt′∂f(k, t′)∂t′

+i

h〈k|[HF , f(t′)]|k〉 +

+i

h

∑N

〈kN |[H ′i(t

′), ′(t′)]|kN〉 +i

h

∑N

〈kN |[H ′ee(t

′), ′(t′)]|kN〉

= 0. (3.24)

Now we restrict ourselves to the electric field slowly varying in time (adiabatically switched at timet = −∞). In the model without electron-photon interaction and magnetic field the external vectorpotential Aext is small [32]. Then the term proportional to A2 is neglected [33] in the hamiltonian HF .

If we use this reduced expression of field hamiltonian (3.4) in the second term on the right-hand sideof the equation (3.24), we have

〈k|[HF , f(t)]|k〉 =eh

m〈k|

∑k

k ·Et a†kakf(t) − f(t)∑k

k · Et a†kak|k〉 =

=eh

m

〈k|

∑k

k · Et a†kak|k′〉〈k′|f(t)|k〉 − 〈k|f(t)|k′〉〈k′|∑k

k · Et a†kak|k〉.

Matrix element with the operators can be expressed as

eh

tE · 〈k|

∑k’

k′a†k’ak’|k′〉 = eE · 〈k|∑k’

r a†k’ak’|k′〉 = −ieE · ∇k’〈k|∑k’

a†k’ak’|k′〉δ(k − k′).

Then we can derive the drift term in the form

〈k|[HF , f(t)]|k〉 = −ieE · ∇k〈k|a†kak|k〉〈k|f(t)|k〉 − 〈k|f(t)|k〉∇k〈k|f(t)|k〉 == −ieE · 〈k|f(t)|k〉∇k〈k|a†kak|k〉 − 〈k|a†kak|k〉∇k〈k|f(t)|k〉 − 〈k|f(t)|k〉∇k〈k|a†kak|k〉 == −ieE · ∇kf(k, t) (3.25)

where f(k, t) is our electron distribution function defined by (3.18).


3.4 Scattering Term

In order to obtain the scattering term from the third term of the equation (3.24), the density matrix′(t′) will be replaced by the series expansion

′(t0) + ′1(t0, t′) + ′2(t0, t

′) + . . . (3.26)

The evaluation of series expansion after their substitution in place of the density matrix ′(t′) is thegeneral problem which the Many-body theory struggles to solve by using various methods. The mostusefull method is the diagrammatic representation of the every term ([34], p 214) arising due to theexpansion series and a consequent iteration procedure and time ordering process [35, 36].

In the expansion (3.26) it is assumed [37, 38] that the terms of higher order than 1 disappear in theapproximation of weak scattering. Then one can start the iteration process and express ′1(t0, t′) as

′1(t0, t′) = − i

h

∫ t′

0

dτ [H ′F (τ) +H ′

i(τ) +H ′ee(τ),

′(t0)]. (3.27)

Now we insert (3.27) into the expression (3.24) and we omit the contribution from the term

i

h

∑N

〈kN |[H ′ee(t

′), ′(t′)]|kN〉.

This term will be taken into account in the next section. We have

〈kN |[H ′i(t

′), ′(t′)]|kN〉 = 〈kN |[H ′i(t

′), ′(t0)]|kN〉 − i

h

∫ t′

0

dτ 〈kN |[H ′i(t

′), [H ′F (τ), ′(t0)]]|kN〉 −

− i

h

∫ t′

0


′), [H ′i(τ),

′(t0)]]|kN〉 − i

h

∫ t′

0


′), [H ′ee(τ),

′(t0)]]|kN〉. (3.28)

The terms proportional to H ′i in the first order

∑N

∫ t

t0

dt′ 〈k′N |H ′i(t

′)|kN〉 =∑N

P (N)〈k′N |Hi|kN〉∫ t

t0

dt′ expi

h(εk’ − εk)t′

vanish because they have the matrix elements which are diagonal in the lattice quantum numbers [39].Then on the right-hand side of (3.28) remains the third term only (According to the perturbation theory,the term in the first order of the arbitrary interaction hamiltonian is not contributed to the transitionprobability between energy eigenstates, the contributed term must be in the second order of the interactionhamiltonian [40]). The scattering term in the expression (3.28) can be immediately found [41] in the form(

i

h

)2 ∫ t

t0

dt′∫ t′

0

dτ∑N

〈kN | [H ′i(t

′), [H ′i(τ),

′(t0)]] |kN〉 =

= − 1h2

∫ t

t0

dt′∫ t′

0

dτ∑N

〈kN | H ′i(t

′)H ′i(τ)

′(t0) − H ′i(t

′)′(t0)H ′i(τ) −

−H ′i(τ)

′(t0)H ′i(t

′) + ′(t0)H ′i(τ)H

′i(t

′) |kN〉 (3.29)

In order to employ the properties of the fermion and the boson operators the following trick will beapplied. The density matrix may be written in the matrix element in (3.29) as the sum of the pure stateaccording to the definition (2.5) in the following way

〈kN |H ′i(t

′)H ′i(τ)|k′N ′〉〈k′N ′|kN〉 − 〈kN |H ′

i(t′)|k′N ′〉〈k′N ′|H ′

i(τ)|kN〉 −−〈kN |H ′

i(τ)|k′N ′〉〈k′N ′|H ′i(t

′)|k′N ′〉 + 〈kN ||k′N ′〉k′N ′〉H ′i(τ)H

′i(t

′)|kN〉 == 〈kN |H ′

i(t′)H ′

i(τ)|kN〉 − 〈kN |H ′i(t

′)|k′N ′〉〈k′N ′|H ′i(τ)|kN〉 −

−〈kN |H ′i(τ)|k′N ′〉〈k′N ′|H ′

i(t′)|k′N ′〉 + 〈kN |H ′

i(τ)H′i(t

′)|kN〉. (3.30)

3.4. SCATTERING TERM 15

When the interaction representation (3.15) is uncovered in the calculation then, for instance, the firstterm in (3.30) is

〈kN |H ′i(t

′)|k′N ′〉〈k′N ′|H ′i(τ)|kN〉 =

= 〈kN | exp[i

h(He +HL)t′]Hi exp(− i

h(He +HL)t′]|k′N ′〉 ×

×〈k′N ′| exp[i

h(He +HL)τ ]Hi exp[− i

h(He +HL)τ ]|kN〉 =

= exp[i

h(εk + EN )t′]〈kN |Hi|k′N ′〉 exp[

i

h(εk’ + EN ′)t′] ×

× exp[i

h(εk’ + EN ′ ]τ ]〈k′N ′|Hi|kN〉 exp[− i

h(εk + EN )τ ] =

= exp[i

h(εk − εk’ + EN − EN ′)(t′ − τ)]〈kN |Hi|k′N ′〉〈k′N ′|Hi|kN〉. (3.31)

Futher the last two matrix elements with the electron-phonon interaction hamiltonian (3.5) will becalculated as

〈kN |Hi|k′N ′〉〈k′N ′|Hi|kN〉 = h2|Mq|2(〈kN |b†qa†k−qak|k′N ′〉 + 〈kN |bqa†k−qak|k′N ′〉)×× (〈k′N ′|b†qa†k−qak|kN〉 + 〈k′N ′|bqa†k−qak|kN〉) . (3.32)

When the operator bqa†k+qak appearing in the hamiltonian (3.5) and describing transition involvingabsorption of a phonon is applied to the state |kN〉 then one can obtain for the matrix element from(3.32)

〈kN | bqa†k+qak |k′N ′〉 =√

(1 − f(k + q))f(k)Nq. (3.33)

The other matrix elements vanishe by the calculation.Correspondingly, the matrix element for transition involving emission of a phonon is

〈kN | bqa†k−qak |k′N ′〉 =√

(1 − f(k − q))f(k)(Nq + 1). (3.34)

In the above expressions (3.33) and (3.34) the all occupation states was replaced by their statistical meanvalues. If the electron and phonon systems are in equilibrium before the transition, these mean valuesare, respectively, the Fermi (3.18) and the Bose (3.20) distribution.

If the expressions (3.33) and (3.34) are performed in the calculation of the matrix elements (3.32) theterms which are appearing as the combination of the absorption and the emission hamiltonian part, e.g.

〈kN |b†qa†k−qak|k′N ′〉〈k′N ′|bqa†k−qak|kN〉is zero after integration over k,q. Therefore, only the pure terms describing the absorption and theemission are employed in the electron-phonon interaction.

Collecting results (3.31), (3.32), (3.33) and (3.34) and inserting their into the equation (3.30) the timeintegral is possible to obtain∫ t

t0

dt′∫ t′

0

dτ cos [ω(t′ − τ)] =1ω2

sin2 ω(t− t0) = 2π(t− t0)1 − cos 2ω(t− t0)

2πω2(t− t0)

Above ω is defined ashω = εk − εk’ ∓ hωq

where the minus sign is for the absorption of a phonon and the plus sign is for the emission.The last factor for large enought t− t0 is a sharply peaked function of ω at ω = 0 and in integrals of

smooth function of ω it acts as a delta function. Therefore, for large enought t− t0, the following resultis derived ∫ t

t0

dt′∫ t′

0

dτ cos [ω(t′ − τ)] = 2π∆t δ(ω).


It is important to emphasize that the above result enables us easily to incorporate the electron-phononscattering term (3.29) into equation (3.24).

Since the integrand in (3.24) must be zero (integral in (3.24) is zero for any t > t0), one gets from(3.24) and (3.29) the BTE in the known form

∂f(k, t)∂t

+e

hE · ∇kf(k, t) =

[∂f(k, t)∂t

]S

. (3.35)

with the standard electron-phonon (S = ep) scattering terms [42]:[∂f(k, t)∂t

]ep

=2πh

∑q

|Mq|2δ(εk − εk−q − hωq)[Nqf(k− q)(1 − f(k)) − (Nq + 1)f(k)(1 − f(k− q))]+

+δ(εk − εk−q + hωq)[(Nq + 1)f(k− q)(1 − f(k)) −Nqf(k)(1 − f(k − q))]. (3.36)

We note that the time variable t is omitted in f(k) in order to simplify notations. The scatteringterm (3.36) have the quasi-classical form because the Fermi exclusion principle occurs in the transitionpropabilities. This form can be obtained by the change in the classical collision phase volume [43].

Finally, it should be noted that it is possible to obtain higher-order corrections to the BTE (3.35),when the term ′2(t0, t

′) is also considered in the expansion (3.26). Such a derivation is presented inour paper [44] which is enclosed at the end of this text. The derivation results in the correction termsfor electron-phonon interaction which describe the so-called intracollisional field effect [45, 46] — theinfluence of the electric field E on the scattering process.

3.4.1 Born Approximation for Electron-Phonon Interaction

The simpler way may be used by the introduction of the electron-phonon scattering term to the BTE.The famous Born approximation [47] will be utilized by the searching of a transition probability from theinitial state |i〉 to the final state |f〉 what is also known as the Fermi golden rule. ¿From the Dirac (timedependent) perturbation theory this probability in the first-order approximation is

P ( i→ f ) =2πh

|〈 f |H | i 〉|2 δ(Ef − Ei) (3.37)

where H can be the arbitrary interaction hamiltonian which is considered as a perturbation [48].With the help of the electron-phonon hamiltonian (3.5) the probability of an electron transmition

from state |k〉 into state |k + q〉 is possible to calculate using formula (3.37). Here is important to bearin mind that neither the interaction representation (3.15) nor series expansion (3.26) is not necessary tointroduce. First of all a transition involving absorption of a phonon is considered. Application of theoperators from the hamiltonian (3.5) on the state |kN〉 was investigated by (3.33) and the argument inthe delta function is

Ef − Ei = εk+q − εk − hωq.

Analogously, the matrix element for transition involving emission of a phonon was demonstrated by(3.34) and the energy conservation argument is

Ef − Ei = εk−q − εk + hωq.

Thereafter the probability Pa for the absorption of a phonon q is obtained in the form

Pa (k → k + q) =2πh|Mq|2(1 − f(k + q))f(k)Nqδ(εk+q − εk − hωq) (3.38)

and for emission of a phonon −q in the form

Pe (k → k − q) =2πh|Mq|2(1 − f(k − q))f(k)(Nq + 1)δ(εk−q − εk + hωq) (3.39)

3.5. SCREENING AND PLASMONS 17

Note here that the principle of microscopic reversibility tells us that the possibility of a processbetween two states must be independent of the direction in which the process runs. Therefore, collisionterm in (3.29) is given by the sum of all collision probabilities from state k−q into state k with emissionof a phonon −q and from state k + q into state k with absorption of a phonon q, minus all scatteringprobabilities from the state k into state k +q with absorption of a phonon q and from state k into statek − q with emission of a phonon −q as[

∂f(k, t)∂t

]ep

=∑q

[Pe(k − q → k) + Pa(k + q → k) − Pa(k → k + q) − Pe(k → k − q)] . (3.40)

Putting the expressions (3.38) and (3.39) to the above equation and changing the summation index fromq to −q in the second and the third terms on the right-hand side of (3.40), thus that is∑

q

Nq (1 − f(k)) f(k + q)δ(εk − εk+q − hωq) = q → −q =⇒

=⇒∑q

Nq (1 − f(k)) f(k − q)δ(εk − εk−q − hωq)

for the second term and∑q

Nq (1 − f(k + q)) f(k)δ(εk+q − εk − hωq) = q → −q =⇒

=⇒∑q

−Nq (1 − f(k− q)) f(k)δ(εk − εk−q + hωq)

for the third term, the final result is derived in the form given by (3.36).The collision term for the phonon system have the similar structure and is given by the sum of

all collision processes involving the emission of a phonon (3.38), minus the corresponding absorptionprocesses (3.39) [

∂Nq

∂t

]scattering

=∑k

[Pe(k + q → k) − Pa(k → k + q)] .

But the thermal equilibrium of the phonons is supposed as was mentioned before, in the section 3.1.When the electron an phonon system is in a nonequilibrium state, e.g., by the applying of the external fieldthat this perturbed system is also reverted in the equilibrium state by the other scattering mechanismsthus as the scattering on the impurities, selfscattering, etc. or by the Umklapp processes.

3.5 Screening and Plasmons

The interaction among carriers in solid state systems can be analysed starting from the hamiltonian (3.14).In the previous section it was mentioned that the terms of the higher order in Hee can be neglected in theweak scattering limit. But this assumption cannot be applied in many cases of real systems. Anyway onemust consider the higher order terms in the perturbation expansion although the calculation is anythingbut simple. It turns out that every order in the perturbation theory beyond the present one diverges forzero scattering angles due to the use of bare Coulomb potential vq. In the following we show how to avoidthis problem. Only the summation of the entire perturbation series, or at least infinite sub-series, leavesa finite, sensible result. It is possible to overcome these problems with a great effort by the introductionof Feymann diagrams [49].

Fortunately, another possible way can be chosen. The main reason for the divergence of the higherorder perturbation terms originates from the observation that the long-range tail of the Coulomb potentialis mannering it. Hence, one thing that seems natural to do is to divide the total Coulomb interaction intoa short-range and a long-range part. Then, for example, the correlation energy of the short-range part


Figure 3.1: Division of the (a) full Coulomb potential into the (b) short-range and (c) long-range componentsand their comparison with the (d) screened Coulomb potential

may be calculated by the perturbation theory since this part should not give rise to divergent expessions.However, a non-perturbative approach which is the infinite order summation is required to calculate thelong-range correlation energy.

The division of the hamiltonian (3.14) can be done in the following way

Hee =∑

0<q<qc

vq∑k,k’

a†k+qa†k’−qak’ak

︸︷︷︸long-range part

+∑q>qc

vq∑k,k’

a†k+qa†k’−qak’ak

︸︷︷︸short-range part

(3.41)

where the first term of the divided hamiltonian is a long-range part and the second term is a short-rangepart of the Coulomb interaction. Fig. 3.1 shows the full Coulomb potential and the division (3.41) intothe short-range and long-range components [50].

We want to show that the short-range part of the divided Coulomb interaction can be replaced by ascreened interaction ([34], p 279). The long-range component which is thereby neglected then reemergesin the form of additional collective oscillations of the electron gas.

For a moment the electron gas is taken to be a uniformly distributed density of space charge. If anadditional negative charge is introduced at the point r, two things happen. By the Coulomb repulsion,the charge is driven away from the immediate vicinity of the negative point charge. This rearrangementmeans a positive charge-cloud around it relative to the average charge density of the electron gas. Thisin turn means a screening of the electron charge.

It is possible (Appendix A) to rewrite the hamiltonian (3.14) in the random phase approximation(RPA) as the sum of three different contributions [51]

Hee = H(sc)ee +H(pl)

ee +H(e−pl)ee (3.42)

3.6. INTERPARTICLE TERM 19

which can be explicitly written as:

H(sc)ee =

∑q>qc

vq∑k,k’

a†k+qa†k’−qak’ak (3.43)

H(pl)ee =

∑k

hωp(b†kbk +12) (3.44)

H(e−pl)ee =

∑q<qc

uq

∑k

(2k · q − q2)(bqa†k+qak − b†−qa†k+qak). (3.45)

The coefficient uq in the last hamiltonian is given as

uq =(

e2h3

8νκωpm2q2

)1/2

The form of the hamiltonian (3.42) is obtained after several transformations [52] of the originalhamiltonian (3.14) which involve a Fourier transformation of the interparticle term and an introductionof additional canonical conjugate variables (see Appendix A).

The three terms in the expression (3.42) have a precise physical meaning. The term H(sc)ee describes

a screened Coulomb interaction between two electrons k,k′ in which the momentum q is exchanged inthe transition k,k′ → k + q,k′ + q. A characteristic inverse screening length is defined by the cut-offwave vector qc which is introduced arbitrarily to separate long-range and short-range components in theFourier expansion of the term vq in (3.14). The term H

(pl)ee in (3.42) represents the energy of the gas

of free bosons, which are here identified with the plasmons, i.e., with the quantized oscillations (withthe frequency ωp) of the electron gas. The last term, given by (3.45), describes the electron-plasmoninteraction through a process in which the momentum k is transferred to an electron with absorptionof a plasmon of the wave vector q or emission of a plasmon of the wave vector −q. Processes involvingmore than one plasmon are neglected in the RPA.

According to the previous analysis, two main contributions to the carrier-carrier scattering can beidentified [53]: i) the individual carrier-carrier interaction via a screened Coulomb potential in the form

V (r) =e2

4πκrexp(−qc · r) (3.46)

which accounts for the two-body short-range interaction and ii) the electron-plasmon interaction, whichaccounts for the collective long-range behaviour of the electron gas.

In semiconductors, the plasmon energy can be of the same order of magnitude as the characteristicphonon energies. The cut-off wave vector qc can be obtained from an independent analysis of a wavevector and a frequency-dependent dielectric function. It is found that it is reasonable for the most casesto assume that qc is equal to the Debye screening length for non-degenerate conditions.

3.6 Interparticle Term

In the derivation of scattering term (3.36) it was assumed that electrons do not interact among themselves.This assumption is unessential on the way to the BTE because the scattering term can be split into twocontributions: i) an electron-phonon scattering part which was evaluated in the previous section and ii)an interparticle scattering part which will be considered in this section. Symbolically one has[

∂f(k, t)∂t

]S

=[∂f(k, t)∂t

]ep

+[∂f(k, t)∂t

]ee

.

Similarly to the derivation of the scattering term (3.36) the density matrix ′(t′) is taken as (3.26).After an iteration process, we find the following perturbation expansion [54]:

〈kN |[H ′ee(t

′), ′(t′)]|kN〉 = 〈kN |[H ′ee(t

′), ′(t0)]|kN〉 − i

h

∫ t′

0

dτ 〈kN |[H ′ee(t

′), [H ′F (τ), ′(t0)]]|kN〉−


− i

h

∫ t′

0


′), [H ′i(τ),

′(t0)]]|kN〉 − i

h

∫ t′

0


′), [H ′ee(τ),

′(t0)]]|kN〉. (3.47)

In the expansion (3.26) the terms of the order higher than two do not contribute. This is obvious inthe terms of the odd-order in H ′

ee because [36]:

〈kN |[H ′ee(t

′), ′(t0)]|kN〉 = 0. (3.48)

The evaluation of the even-order terms in H ′ee up to the fourth order is more complicated. However,

if the weak scattering approximation and short-range interaction are applied [55], the terms of the fourthand higher orders are contributing less and less so that they can be neglected. The same problem wouldappear when the particles are recalculated and considered as quasi-particles in the theory of Fermi liquid[55]. The relation between the pure particle distribution function and the quasi-particle distributionfunction was found for one electron approximation [36].

Now we can return to the expression (3.47). The first term on the left-hand side is neglected asmentioned above. The second term where the field hamiltonian (3.4) appears in first-order, [57] does notcontribute in the weak scattering limit.

The third term on left-hand side of (3.47) is zero because all the terms withH ′i in the first-order [38, 37]

do not contribute to the change of the electron distribution function. The electron-electron scatteringterm (ee) can be obtained from the last term with H ′

ee in the second-order. Taking in account therelation between the particle and the quasi-particle distribution function which was mentioned above thematrix element with Hee in the second order can be expressed as a two-quasi-particle scattering matrixplus corrections to this matrix of order higher than H2

ee [56]. This term corresponds to the thermalHartree-Fock correction and the second-order Hartree-Fock correction to the single particle energy.

We will consider the scattering of two electrons from initialization states, the first one with the wavevector k and the second one with the wave vector k′, to final states by the exchange of the wave vectorq, namely the first into the state with the wave vector k + q and the second into the state with thewave vector k′ − q, similarly to the form of the electron-electron hamiltonian (3.43). Afterwards, if theoperator a†k+qa†k’−qak’ak puts on the state |k k′〉, the investigated matrix element is

〈k k′|a†k+qa†k’−qak’ak|k + q k′ + q〉 =√

(1 − f(k + q))(1 − f(k′ − q))f(k′)f(k) (3.49)

employing the known properties of the creation and annihilation fermion operators and the fact that allthe occupation states was replaced by their statistical average values like in the case of the electron-phononmatrix element.

If the result (3.49) is used one finds [36, 57][∂f(k)∂t

]ee

=∑k’,l,l’

| M(kl → k′l′)|2δ(εk + εl − εk’ − εl’) ×

× f(k)f(l)[1 − f(k′)][1 − f(l′)] − f(k′)f(l′)[1 − f(k)][1 − f(l)] (3.50)

where the time dependence in the distribution function f is again omitted for simplicity. The twoconduction electrons with the wave vectors k, l, are scattered to states with the wave vectors k′, l′ afterbinary collision. The matrix element M in (3.50) is the transition probability of this process. Theelectron-electron exchange effect is often forgotten in the calculations of | M(kl → k′l′)|2 [60]. Thiscoefficient is given ([61], p 661) through the definition of the transition probability as

| M(kl → k′l′)|2 =12(M2(kl → k′l′) + M2(kl → l′k′) −M(kl → k′l′)M(kl → l′k′)

). (3.51)

If the electron wave functions are approximated by the plane waves or by the Bloch wave functions,what gives the same result ([58], p 173), the matrix element M can be obtained in the form

M(k, l → l′,k′) ≡ 〈kl|M|k′l′〉 = −e δ(k + l − k′ − l′) vk−k’ (3.52)

where vq is the Coulomb potential which was defined in the electron-electron hamiltonian (3.14).

3.7. LINEAR SCREENING 21

3.6.1 Born Approximation for Electron-Electron Interaction

Making use of the Born approximation (3.37) in the same way as it was shown by the uncover of theelectron-phonon term (3.36), the electron-electron term (3.43) can be acquired.

The matrix element from (3.37) of the process which was considered by the previous derivation of theelectron-electron scattering term is equal to the result obtained in (3.49). Then the probability of theelectron transition from the states k,k′ into the states k + q,k′ − q is

P (k k′ → k + q k′ − q) =2πh| M(k k′ → k + q k′ − q)|2 δ(εk+q + εk’−q − εk − εk’) ×

×(1 − f(k + q))(1 − f(k′ − q))f(k′)f(k). (3.53)

Because of the Fermi exclusion principle the probability of the transition from k+q,k−q into the statesk,k′

P (k + q k′ − q → k k′) =2πh| M(k k′ → k + q k′ − q)|2 δ(εk + εk’ − εk+q − εk’−q) ×

×(1 − f(k))(1 − f(k′))f(k′ − q)f(k + q) (3.54)

is necessary to know. The electron-electron collision term in (3.35) is composed by the sum of all collisionprobabilities from the state k and all possible states k′ into the state k + q and all possible states k′ − qminus the sum of all collision probabilities from the state k + q and all possible states k′ − q into thestate k and all possible states k′ as[

∂f(k)∂t

]ee

=∑k’,q

P (k k′ → k + q k′ − q) − P (k + q k′ − q → k k′). (3.55)

Inserting the expressions (3.53) and (3.54) in (3.55), the electron-electron scattering term (3.50) is ob-tained.

3.7 Linear Screening

Screening is one of the most important concepts not only in transport theory, but in many-body theoryin general. Free carriers are redistributed in presence of the probe charge (for example donor, acceptor,electron, hole, etc.). This redistribution will stabilize into a new distribution of charge around the probecharge. This new distribution is just the right amount of charge to cancel the electric field of the probecharge at large distance. If the electric field is not cancelled at large distances, more charge will still beattracted until it is sufficient for cancellation.

Let the probe charge is distributed according to the charge distribution ρi(r). The name screeningcharge is applied to the mobile charge attracted by the probe electric field. It will also have its owndistribution in space, ρs(r). The screened potential from the probe charge and the screening charge isgiven as

φ(r) =∫

d3r′ρi(r′) + ρs(r′)

|r− r′| . (3.56)

The screening charge is from the unbound conduction electrons of the metal or semiconductor. Intheir motion through the crystal, they spend a little more time near the probe charge, if it is attractive,than they do elsewhere in the solid. When these motions are averaged, there is more electron densitynear the probe charge than elsewhere, and this is the screening charge. If the probe charge is repulsive forelectrons, they tend to spend less time near the probe charge, so the average charge is depleted near theprobe charge. Here the screening charge is positive, since it signifies a reduction in the average densityof electrons, which have negative charge.


If the displacement field is indicated D then the dielectric response function is defined as the ratio oflongitudinal components in the limit

κ(q) = limρi→0

Dl(q)El(q)

= limρi→0

[ρi(q)

ρi(q) + ρs(q)

]. (3.57)

In this limit κ(q) becomes a property of the material and is independent of the charge distribution.The linear screening model assumes this definition is true for finite ρi(q), which gives for the potential

φ′ =1q2ρi(q)κ(q)

φ′(r) =∫

d3q

(2π)31q2

ρi(q)κ(q)

eiq·r. (3.58)

The potential φ′ is the total potential from the screening charge and probe charge, expressed throughκ(q). The linear screening approximation is to calculate φ′(r) in place of φ(r). Another feature of linearscreening model is that the screening charge density ρs(q) is proportional, in q space, to the impuritycharge density ρi(q). Linear screening model assumes that

κ(q) =ρi(q)

ρs(q) + ρi(q)ρs(q)ρi(q)

=1

κ(q)− 1

are valid for finite values of ρi, rather than for infinitesimally small ones.The exact dielectric function for the homogeneous electron gas was not derived up to now. Instead,

approximate solutions have been obtained.

3.7.1 Thomas-Fermi Theory

In this section we present a simple model for the screening of immobile probe charge. The derivation willbe begun with the exact Poisson equation for the screened potential energy V

V (r) =e

κ[ρi(r) + ρs(r)]. (3.59)

In the Thomas-Fermi theory [59], the electron particle density n(r) is represented locally as a free-particle system. Thus we write the screening charge as the difference between n(r) and the equilibriumcharge density n0,

ρs(r) = −e[n(r) − n0]. (3.60)

Assuming strongly degenerate carrier gas, the local density for a free-particle system is n(r) = k3F (r)/3π2,

where the Fermi wave vector kF is now a local quantity. Vector kF is determined by the condition thatthe chemical potential εF is independent of position,

k2F (r)2m

= εF (r) = εF − V (r). (3.61)

Assume the potential V (r) is slowly varying in space. If the absolute Fermi level is εF , then theeffective Fermi level εF (r) is reduced or raised by the value of local potential V . If the relations (3.60)and (3.61) are utilized in the equation (3.59), this yields

V = +e

κ

ρi(r) + en0 − en0

[1 − V (r)

εF

]3/2. (3.62)


This equation can be solved exactly only for electrons εF = 0 and ρi = Zeδ3(r) to give description of theprobe potentials and charge distributions ([61], pp 667–671).

To get a linear screening model one needs assume that V/εF 1. Then one may expand the term[1 − V (r)/κF ]3/2 to obtain the equation

V =e

κρi(r) +

3e2n0

2κεFV (r). (3.63)

To define the Thomas-Fermi wave vector qTF , (3.63) can be rewritten as

( − q2TF )V (r) =e

κρi(r) (3.64)

q2TF =3e2n0

2κεF.

Equation (3.64) may be solved in Fourier transformed space to obtain

V (r) = − e

κ

∫d3q

(2π)3ρi(q)

q2 + q2TF

eiq·r. (3.65)

Comparing (3.65) with (3.58), the Thomas-Fermi dielectric function is

κ(q) = 1 +q2TF

q2. (3.66)

Employing the expression for Fermi energy by the zero temperature

εF (0) =h2

2m(3π2n0

)2/3

to the definition of the Thomas-Fermi wave vector, following, more practical, formula can be found

q2TF =e2m

h2κ

(3n0

π4

)1/3

.

For example, an analytical result can be obtained when the probe charge is a point charge ρi(q) = Qi.Then

V (r) = − eQi

4π2κ

∫ ∞

0

dqq2

q2 + q2TF

∫ 1

−1

dz eiqrz = −e Qi

i4πκr

∫ ∞

−∞dq

qeiqr

q2 + q2TF

= − eQi

4πκre−qT F r.

The last integral is calculated by closing the integration contour in the upper half of the complex planearound the pole iqTF . The screened interaction has the form of Yukawa potential. The interactiondeclines rapidly at large distances because of the exponential dependence exp(−qTF r). Finally, one canget the formula [62]

Vq = −eQi

κν

1q2 + q2TF

for the using in the relation (3.52) or

V (r) = −4πeQi

ν

∑k

eik·r

k2 + q2TF

. (3.67)

in the position representation. Thomas-Fermi theory provides only a static model for κ(q) because theseis no time dependence in the ρi(r) distribution in (3.59). It is not usually used to describe the dynamicresponse κ(q, ω).


3.7.2 Random Phase Approximation

The RPA approach, presented in this section, enables to derive dynamic dielectric function κ(q, ω), whichis commonly called the Lindhard dielectric function [63].

One introduces, as in the Thomas-Fermi theory, a probe charge density ρi(r, t) or its equivalent Fouriertransform ρi(q, ω). The equivalent probe potential is Vi(q, ω). Similarly, the potential of the screeningcharge ρs(r, t) is Vs(q, ω). Then the total potential reads

V (q, ω) = Vi(q, ω) + Vs(q, ω). (3.68)

Poisson’s equation for each potential can be written as

Vα(q, ω) = −4πeq2

ρα(q, ω), α = s, i. (3.69)

When one solves the equations of motion of the electron, the potential V (r, t) or V (q, ω) is used.Once V (q, ω) is known, the dielectric function in the linear screening model is

ε(q, ω) =Vi(q, ω)V (q, ω)

and the calculation is completed. In the method of self-consistent field, we assume that electrons respondto V [64]. Thus we write the hamiltonian of free particles (3.2) plus the interaction hamiltonian oftight binding model ([58], p 82) with the self-consistent potential V (q, t) (unlike to V (q, ω), V (q, t) isFourier-trasformed only through r variable) as an effective hamiltonian

Heff = He +1ν

∑k,q

V (q, t) a†k+qak (3.70)

where ν is the volume of the system. Note that there are no explicit electron-electron interactions here.They are indirectly included in the interaction term: The part of V (q, t) from screening Vs(q, t) is causedby electron-electron interactions. This is a rather crude way to include these interactions, since it neglectseffects of correlation and exchange. This is the major defect of the RPA. The main objective of othermodels was to improve the RPA by including these effects [65, 66, 67]. Here we restrict ourselves to thehamiltonian (3.70).

The probe charge is regarded as a classical oscillating system and we try to find the quantum responseof electron gas to this classical oscillation. Furthermore, the probe is assumed to oscillate with a singlefrequency: ρi(r, t) = ρi(r) exp(−iωt) and Vi(r, t) = Vi(r) exp(−iωt). The time average response of aquantity O(q, ω) is defined as

< O(q, t) > = O(q, ω) exp(−iωt).

When the probe charge is present, its time response reads

< s(q, t) > = −e < (q, t) > = −e∑k

< a†k+qak > = −e ρ(q, ω) exp(−iωt). (3.71)

To simplify notations, we use −e < > instead of < s > in the following derivations. Then equation(3.69) is

Vs(q, ω) =e2

κq2ρ(q, ω). (3.72)

To obtain the screened potential Vs, we need to derive the expression for the screening charge den-sity s(q, t). This is obtained by writing an equation of motion for this operator and then solving itapproximately. In the homogeneous electron gas, the particle density operator n(q, t) has an expectation


value of zero unless there is a perturbation of the system. A perturbation V (q, t) will cause polariza-tion of electron system, so that the average < (q, t) > will now have a finite value. In the linearscreening model, we assume that this average is proportional to the potential causing the perturbation< (q, t) > ∼ < V (q, t) >. Our goal is to determine the constant of proportionality.

The density matrix (2.3) is expressed in the form a†k+qak. The equation of motion (2.9) with hamil-tonian (3.70) can be then written as

ih∂

∂ta†k+qak = [Heff , a

†k+qak]. (3.73)

The impurity potential V (q, t) is assumed to oscillate at a single frequency as exp(−iωt). The timederivative on the left-hand side of (3.73) gives −iωa†k+qak [see (3.71)]. The commutator on the right-handside of (3.73) gives ∑

l

εl[a†lal, a†k+qak] = (εk+q − εk)a†k+qak

1ν

∑q’l

V (q′, t)[a†l+q’al, a†k+qak] =

1ν

∑q’

V (q′, t)[a†k+q+q’ak − a†k+qak−q’].

Thus (3.73) arrives to

(εk − εk+q + ω)a†k+qak =1ν

∑q’

V (q′, t)[a†k+q+q’ak − a†k+qak−q’] V (q, t)ν

(a†kak − a†k+qak+q)

where the sum on the right-hand side is approximated by considering only the term with q′ = −q. Ifone wishes to evaluate the average values in a state ψ then the complex numbers with a phase factorexp(iφ) are obtained. These phase factors are determined by indexes of operators a†qaq’. In the RPAthe phase factors are uncorrelated for q = −q′ and the sum of the terms with q = −q′ gives zero [51].The approximate equation can be solved and summed over k to obtain

(q, t) =V (q, t)ν

∑k

a†kak − a†k+qak+q

εk − εk+q + ω. (3.74)

Multiplying (3.74) by exp(−iωt), performing the quantum-mechanical averaging and replacing the eigen-values of the operators a†kak and a†k+qak+q by their statistical averages f(k) and f(k + q) one obtains

ρ(q, ω) =V (q, ω)

ν

∑k

f(k) − f(k + q)εk − εk+q + ω

≡ V (q, ω)P (1)(q, ω). (3.75)

This result can now be used in (3.72):

Vs(q, ω) = vq V (q, ω) P (1)(q, ω), vq =e2

κq2. (3.76)

In (3.76) the screening particle density ρ(q, ω) is proportional to the self-consistent potential V (q, ω).Substituting Vs(q, ω) from equation (3.76) into the equation (3.68) one gets

V (q, ω) = Vi(q, ω) + vqP(1)(q, ω)V (q, ω) =

Vi(q, ω)1 − vqP (1)(q, ω)

. (3.77)

The ratio Vi(q, ω)/V (q, ω) is the RPA dielectric function with analytic continuation (δ is a infinitesimalquantity),

κRPA(q, ω) = 1 − vqP(1)(q, ω)

P (1)(q, ω) =1ν

∑k

f(k) − f(k + q)εk − εk+q + ω − iδ

. (3.78)


This derivation is equivalent [64] to the so-called Sawada-Brout scheme [68, 69] in which the sameapproximations the RPA and the replacement of diagonal components (k) by their expectation valuesfor free-electron gas, are adopted, and to the so-called collective approach [70].

3.8 Limitations of Boltzmann Transport Equation

Some problems with the introduction of electron distribution function f(k, t) were mentioned in thesection 3.1. Briefly, the BTE provides semi-classical approach and suffers from the limitations of thatapproach.

One limitations of the BTE arises from the fact that the field and the scattering events are treatedas non-iteracting perturbations. This approximation does not to hold in the very high field regime [71]where the particle dynamics is affected by field during the collision. Only a little progress has been madeto overcome this restriction (only several attempts appeared in the literature [46, 72] to incorporate thisintracollisional field effect).

The collision term on the right-hand side of (3.35) represents a summation over individual scatteringevents. It implies that the collision duration τc must be much smaller than any other time scale in theproblem [39]. For instance, the collisions are regarded as instantaneous processes which interrupt freeflight. That means weak electron-phonon and electron-electron coupling. Validity of the BTE requires

τc τL, (3.79)

where the transit time τL is defined as the ratio l/v, where l is the carrier mean free path and v is typicalcarrier velocity. Under the condition (3.79) one expects the response of the distribution function to fieldsand scattering to be instantaneous and distribution should then evolve in a Markovian sense to localequation of motion. But the model fails when the energy absorbed by the electron from an external fieldbecomes comparable with the width of the energy band.

The electron distribution function f(k, t) defined by relation (3.18) represents essentially the diagonalelements of the electron density matrix (k, t) in momentum space. This construction makes practicalsense only if: i) the momentum states |k〉 are good approximations to the true electron states, ii) thedistribution function satisfies a closed equation of motion and iii) correct boundary conditions on f canbe constructed. The first condition is actually not restrictive, since we can always choose a better setof states. Nevertheless, the wave vector of the electron is defined by the centre of gravity of the wavepacket |k〉 what implies that the extension of the wave packet in k-space must be small compared withthe dimension of the Brillouin zone. The second condition is more difficult to fulfil. If the Liouville-vonNeumann equation (2.9) is formally projected out into a closed equation for f , it involves necessarily thedynamics of phonon subsystem. If it is solved for the off-diagonal elements in terms of f , the equation forf can be closed, but has the form of an integro-differential equation. If the electrons are not independentthis is no longer true.

In the many-body theory the many-particle functions become functionals of the single-particle distri-butions, i.e., the system loses memory, for time

t > τs

where τs is the duration of a binary collision. BTE with intercarrier interaction can be used only on thistime scale, while quantum formulation is necessary for t < τs [73]. The time τs may be estimate as

τs ≈ 1qD v

.

If κ∞ is the high frequency permittivity and η is the electron density then the condition becomes

τs ≈ 1

(q2D v

2)1/2=[mκ∞12ηe2

]1/2

3.8. LIMITATIONS OF BOLTZMANN TRANSPORT EQUATION 27

in the case of the nondegenerate regime close to thermalization and

τs ≈ 1

(q2Dv

2)1/2=[mκ∞4πηe2

]1/2

in the case of the monokinetic distribution when we assume that all electrons have the same kineticenergy [74].

An improved description of semiconductors devices needs to include additional terms to the BTE,reflecting the memory and retardation effects [75]. The forms of these terms are yet not well understoodin the context of BTE approach. Another way is to use the two equivalent approaches [76]: i) theQuantum Boltzmann equation for linear transport [77] or ii) the Kubo formula.

Chapter 4

Methods for Solution of BoltzmannTransport Equation

As we can see from the BTE (3.35) with scattering terms (3.36) and/or (3.50), the BTE is integro-differential equation. The three main techniques are usually used to solve this equation: i) analyticalmethods which are able to find the solution after additional approximations, ii) iterative technique andiii) Monte Carlo method. The comparison of these methods is given in the Table IV.

Analytic methods may be successfully used to solve the linearized BTE [78], but the obtained resultsare valid only in the linear (Ohmic) transport regime. Analytic methods were also developed for thesolution of the nonlinearized BTE [79] with effects of acoustic and intervalley phonon scattering and withnonparabolic band structure [80], but the obtained results are not exact. Exact solution may be achievedby fully numerical techniques.

4.1 Iterative Technique

The iterative method yields the solution of the BTE by means of an iterative procedure. Since thismethod is still often used [81], we shall outline it briefly in the following.

If the integro-differential BTE for spatially uniform system (3.35) is considered, the scattering termfor the case of nondegenerate statistics can be written as[

∂f(k, t)∂t

]S

= −f(k, t)λ(k) +ν

(2π)3

∫dk′ f(k′, t) W(k,k′, t) (4.1)

where W(k,k′, t) is the transition propability rate from |k〉 to |k′〉 due to all scattering mechanisms, and

λ(k) =ν

(2π)3

∫dk′ W(k,k′, t).

By introducing the path variables

k = k− eEht,

t = t, (4.2)

which represent the collisionless trajectory in k space of electrons. The total differential of f is derivedfrom equations (3.35) and (4.1) in the form

d

dtf

(k +

eEht, t

)+ λ

(k +

eEht

)f

(k +

eEht, t

)=

ν

(2π)3

∫dk′ f(k′, t) W

(k +

eEht,k′, t

). (4.3)

29

30 CHAPTER 4. METHODS FOR SOLUTION OF BOLTZMANN TRANSPORT EQUATION

Analytical methods Iterative technique Monte Carlo methods¿From simple parametrized Numerical procedure at Stochastic calculation atdistribution function to medium level of difficulty minimum level ofcomplex series expansion difficulty

To fully exploit the potentiality of the technique,very fast computers are necessary

Approximate solutions of the An exact solution of the Boltzmann equation is obtained.Boltzmann equation are obtained.

Extension to time and space Extension to time dependent Extension to time and spacedependent phenomena is not phenomena can be easily achiev- dependent phenomena caneasily available. ed. Extension to space depen- be easily achieved.

dent phenomena is not easy.

Microscopic interpretation of phenomena in terms Microscopic interpretation ofof band structure and scattering processes is somewhat hidden. phenomena is quite

transparent.

High level of difficulty in includ- Realistic band structure and scattering modelsing realistic band structure can be easily included.and scattering models.

No direct evaluation of fluctuation phenomena can be obtained. The analysis can be imme-diately extended tofluctuation phenomena.

Electron-electron interaction is Electron-electron interaction can be easily includeddifficult to include, since it makes in method.Boltzmann equation nonlinear.It can suggest particular formsfor the distribution function.

Table I: Comparison of different methods for the solution of the BTE

Multiplying (4.3) by the factor

exp

[∫ t

0

dθ λ

(k +

eEhθ

)]

and integrating per partes from t1 to t2, one gets

f

(k +

eEht2, t2

)exp

[∫ t2

0

dθ λ

(k +

eEhθ

)]= f

(k +

eEht1, t1

)exp

[∫ t1

0

dθ λ

(k +

eEhθ

)]+

+ν

(2π)3

∫ t2

t1

dt exp

[∫ t

0

dθ λ

(k +

eEhθ

)]∫dk′ f(k′, t) W

(k +

eEht,k′, t

). (4.4)

Coming back to the variables k, t by putting

k = k +eEht2

t = t2

4.2. MONTE-CARLO METHOD 31

and writing t′ instead of t1, the equation (4.4) is

f(k, t) = f

(k− eE

h(t− t′), t′

)exp

[−∫ t

t′dθ λ

(k − eE

h(t− θ)

)]+

+ν

(2π)3

∫ t

t′dt exp

[−∫ t

t

dθ λ

(k − eE

h(t− θ)

)]∫dk′ f(k′, t) W

(k− eE

h(t− t),k′, t− t

).(4.5)

Equation (4.5), which is an integral form of the Boltzmann equation, has two different contributions: i)the first term on right-hand side is the contribution of the electrons that were in the state |k−eE(t−t′)/h〉at the time t′ and have drifted to the state |k〉 in the interval (t − t′) under the influence of the field Ewithout being scattered and ii) the second term on right-hand side is the contribution of electrons thatwere scattered from any state k′ to the new state |k − eE(t− t)/h〉 at any time t between t′ and t andthat arrive at the state |k〉 at the time t without being scattered again.

If the time t′ is arbitrary so that t′ < t and the rearrangement of variables t′′ = t − t, t′′′ = t − θ isused, the limiting form of the equation (4.5) can be considered with t′ → −∞

f(k, t) =ν

(2π)3

∫ ∞

0

dt′′ exp

[−∫ t′′

0

dt′′′ λ(k − eE

ht′′′)]

×

×∫dk′ f(k′, t− t′′) W

(k − eE

ht′′,k′, t′′

). (4.6)

The iterative technique for the solution of equation (4.6) consists in substituting an arbitrary (butreasonably chosen) function f0(k, t) on the right-hand side of the (4.5) and calculating f(k, t) as theright-hand side of the (4.5) itself. This function is resubstituted on the right-hand side and the procedureis repeated until convergence is achieved.

Numerical iterative solutions have been discussed for stationary conditions in which the dependenceof f upon t vanishes [82]. The stability of solution is very important fact in this technique. It has beenshown that the introduction of a self-scattering [83] such as that introduced for the Monte Carlo methodgreatly simplifies the evaluation of the integral in equation (4.5).

4.2 Monte-Carlo Method

Monte Carlo methods [84] are widely used in various fields of physics including nuclear physics, solid-state physics and statistical physics in general. When applied to carrier transport in solids, Monte Carloprovides exact numerical solution of the BTE without necessity to solve the BTE directly. In other words,distribution function f(k, t) obtained by the Monte Carlo satisfies the BTE (the proof for this will bepresented in the next section).

Monte Carlo method does not really solve the BTE but the obtained distribution function f(k, t)is identical with the distribution function satisfying the BTE as will be discussed in the subsequentsection. This method is by far the more popular simulation technique because it enables us to obtainexact numerical solutions of the BTE using relatively simple and very effective program algorithms,when compared with direct numerical techniques. Simultaneously, Monte Carlo provides satisfactorymicroscopic interpretation of simulated processes. The essence of Monte Carlo method is to simulatemotion of the ensemble of carrier in the k space as well as in the r space. The motion of each carrieris governed by semiclassical equations of motion and by stochastic collisions with various perturbations(phonons, ions, other carriers). These collisions cause instantaneous transitions between unperturbedBloch states, with transition probabilities given by Fermi golden rule. Using these probabilities, thecarrier free-flight time, the scattering channel and final states after scattering can be generated by randomnumbers. During the simulation any k and r dependent physical quantity can be calculated.

Historically, the stationary carrier transport in homogeneous bulk semiconductors was first inves-tigated by (single-particle) Monte Carlo simulation [85]. Development in this field started at Kyoto


Definition of physical systemInput of physical and simulation parameters

Initial conditions of motion

Stochastic determination of flight duration

Determination of electron state just before scattering

Collection of data for estimators

Is simulation sufficiently long

for desired precision???

No YesStochastic

determinationof scatteringmechanism

Finalevaluation

of estimators

Stochastic

determination ofelectronic state

just afterscattering

Print results

STOP

Figure 4.1: Flowchart of a typical Monte Carlo program (from [91])

semiconductor conference [86]. During several years, nonparabolicity effects [87], diffusion [88], high-fieldtransport with Pauli exclusion principle [89], transient and inhomogeneous transport [90] were compre-hensively included into the Monte Carlo description.

More complicated effects as the space-charge, full band-structure and carrier-carrier interactions arestudied by many-particle (ensemble) Monte Carlo method [91, 92, 93]. Introduction of the electron-electron scattering into the ensemble Monte Carlo method is formidable task. Simulations including thetwo-particle carrier-carrier scattering due to the screened Coulomb interaction [see hamiltonian (3.43)][94, 95, 96] as well as the carrier-plasmon interaction [see hamiltonian (3.45)] were developed to examinemany-body carrier dynamics [94, 98]. Thus ensemble Monte Carlo enables us to simulate single-particlescattering processes (carrier-phonon, carrier-plasmon and carrier-ion scattering) together with two-bodycollision processes (carrier-carrier scattering). Another way to include intercarrier Coulomb interactionsis the molecular dynamics technique [97]. Although this technique is fully classical, it involves both short-range and long-range Coulomb interactions and no assumptions on the screening (RPA, static screening,etc.) are necessary. The technique has to be coupled with Monte Carlo simulation of ”single-particlescattering” processes [97].

4.2. MONTE-CARLO METHOD 33

The starting point of the Monte Carlo program (see Fig. 4.1) is the definition of the physical systemof interest including the parameters of the material and the values of physical quantities, such as latticetemperature T0 and electric field E [99]. The choice of the dispersion relation ε(k) usually depends onthe simulated transport problem. For weak fields the parabolic dispersion law is used, the term αε2 isjoined to the linear term ε for stronger fields and for the strong electric fields ε(k) is calculated by thepseudopotential methods (For example, in undoped GaAs the values 3kV/cm and about 10kV/cm canbe considered as weak and very strong field limits, and between them the nonparabolic law hk2/(2m) =ε(1 + αε) has to be used. At this level we also define the parameters that control the simulation, suchas the duration of each subhistory, the desired precision of the result, and so on. The next step in theprogram is a preliminary calculation of each scattering rate as a function of electron energy. This stepwill provide information on the maximum value of these functions, which will be useful for optimalizingthe efficiency of the simulation.

In the case under consideration, in which a steady-state transport is simulated by single-particlesimulation, the time of simulation must be long enough that the initial conditions of electron motiondo not influence the final results. The choice of a good time of simulation is a compromise between theneed for ergodicity (t → ∞) and the request to save computer time. The longer the simulation time,the less influence the initial conditions will have on the average results. However, in order to avoid theundesirable effects of an inappropriate initial choice and to obtain a better convergence, the eliminationof the first part of the simulation from the statistics may be advantageous. When a simulation is madeto study a transient phenomenon and/or transport processes in an inhomogeneous system (for example,when the electron transport in a very small device is analyzed), then it is necessary to simulate manyelectrons separately; in this case the distribution of the initial electron states for the particular physicalsituation under investigation must be taken into account, and the initial transient becomes an essentialpart of the result aimed at.

The subsequent step is the generation of the flight duration (Fig. 4.1). The electron wave vector kchanges continuously during a free flight because of the applied field. Thus if λ[k(t)] is the probabilitythat an electron in the state k suffers during the time dt, the probability that an electron which suffereda collision at time t = 0 has not yet suffered another collision after time t is

exp[−∫ t

0

dt′ λ[k(t′)]]

which, generally, gives the probability that the interval (0, t) does not contain a scattering. Consequently,the probability P (t) that the electron will suffer its next collision during dt around t is

P (t)dt = λ[k(t)] exp[−∫ t

0

dt′ λ[k(t′)]]dt

Free-flight time t can be generated from the equation

r =∫ t

0

dt′ P (t′),

where r is a random number between 0 and 1.Once the electron free flight is terminated, the scattering mechanism has to be selected. The weight

of the i-th scattering mechanism (when n scattering mechanisms are present) is given by

Pi(k) =λi(k)λ(k)

, λ(k) =n∑

i=1

λi(k)

Generating random number r between 0 and 1 and testing the inequalities

j−1∑i=1

λi(k)λ(k)

< r <

j∑i=1

λi(k)λ(k)

, j = 1, . . . , n


we accept the j-th mechanism if the j-th inequality is fulfilled. It should be noted that the discussedselection of the free flight time and the scattering channel can be substantially simplified by introducingthe self-scattering λ0 [83, 100]. Since this concept does not change the physics we will not go into thedetails.

The next step is the choice of the state after scattering. Once the scattering mechanism that causedthe end of the electron free flight has been determined, the new state after scattering of the electron, ka

must be chosen as the final state of the scattering event. If the free flight ended with a self-scattering, ka

must be taken as equal to kb, the state before scattering. When, in contrast, a true scattering occurred,then ka must be generated, stochastically, according to the differential cross section of that particularmechanism.

The last step of the simulation is the collection of statistical averages. In the single-particle simulationof steady-state transport, time averages is performed as follows. We may obtain the average value of aquantity Q(k) (e.g., the drift velocity, the mean energy, etc.) during a single history of duration t as

< Q(k) > =1t

∫ t

0

dt′ Q[k(t′)] =1t

∑i

∫ ti

0

dt′ Q[k(t′)] (4.7)

where the integral over the whole simulation time t has been separated into the sum of integrals over allfree flights of duration ti. When a steady state is investigated, t should be taken as sufficiently long that< Q > in (4.7) represents an unbiased estimator of the average of the quantity Q over the electron gas.

In a similar way we may obtain the electron distribution function: a mesh of k space is set up atthe beginning of the computer run; during the simulation the time spent by the sample electron in eachcell of the mesh is recorded and, for large t, this time conveniently normalized will represent the electrondistribution function, that is, the solution of the BTE [85]. This evaluation of the distribution functioncan be consider a special case of equation (4.7) in which we choose for Q the functions nj(k) with value 1if k lies inside the j-th cell of mesh and zero otherwise.

In ensemble Monte Carlo simulation of transient effects it is possible to compute instantaneous meanvalue < Q(t) > as

< Q(t) > =1N

N∑i=1

Q[ki(t)],

where i is the particle index and N is the whole number of simulated particles. For instance, Q(k) canbe electron energy εk, group velocity v(k) = 1/h∇kεk, etc.

4.3 Comparison of Ensemble Monte-Carlo Method with

Boltzmann Transport Equation

Comparison of the ensemble Monte Carlo method with the BTE usually includes considerations on theequivalence or nonequivalence of both transport formulations. We wish to address this point in thissection.

The ensemble Monte Carlo method calculates the time-dependent electron distribution function andcorresponding averages. It will be shown that the ensemble Monte Carlo method leads to the distributionfunction which satisfies the time-dependent Boltzmann equation [101].

Let Sn(k0,k, t) is the probability that the Monte Carlo electron started at time zero from the statek0, after n scattering events, including self-scattering and accelerated by the electric field E, at the timet will get into the state k. Then the probability that the electron which started from the state k0 at timet = 0 after an arbitrary number of scattering events will appear in the state k at time t is given as

S(k0,k, t) =∞∑

n=0

Sn(k0,k, t). (4.8)

4.3. Comparison of Ensemble Monte-Carlo Method with BTE 35

The distribution function calculated by Monte Carlo method can be expressed as

f(k, t) =∫dk0 f0(k0)S(k0,k, t) (4.9)

where f0(k0) is the initial distribution function at time t = 0. If the sum of all transition rates from k′

to k′′ for any time t including the self-scattering is defined as W (k′,k′′, t), the equation for probabilitiesSn(k0,k, t) and Sn−1(k0,k, t) has the form [85]:

Sn(k0,k, t) =∫dk′

∫dk′′

∫ t

0

dt′ Sn−1(k0,k, t′) W (k′,k′′, t′) ×

× exp

[∫ t−t′

0

dτ Γ(k′′ +

eEhτ

)]δ

(k − k′′ − eE

h(t− t′)

). (4.10)

Note that the time-dependent scattering rate in the sum (4.10) is considered. In Ref. [101] this timedependence is omitted. It is useful to know that the proof from Ref. [101] remains essentially the samealso with time dependent W (k′,k′′, t).

Differentiating (4.10) with respect to time, one can obtain

∂

∂tSn =

∫dk′ Sn−1(k0,k′, t) +

∫dk′

∫ t

0

dt Sn−1(k0,k′, t′) ×

× ∂

∂t

[W

(k′,k− eE

h(t− t′), t′

)exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)]. (4.11)

Now the time derivative in the square brackets of (4.11) will be evaluated

∂

∂t

[W

(k′,k − eE

h(t− t′), t′

)exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)]=

= W

(k′,k− eE

h(t− t′), t′

)∂

∂t

[exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)]−

− eEh

exp

[−∫ t−t′

0

dτ Γ(k − eEhτ)

]∂

∂kW

(k′,k − eE

h(t− t′), t′

)(4.12)

and, finally, the time derivative of the exponential function with gamma is

∂

∂t

[exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)]= exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)×

× ∂

∂t

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)= exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)×

×∫ t−t′

0

dτ∂

∂tΓ(k − eE

hτ) + Γ(k − eE

t− t′

h)

= −Γ(k− eE

t− t′

h) ×

× exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)− eE

h

∂

∂k

[exp

(−∫ t−t′

0

dτ Γ(k − eEhτ)

)]. (4.13)

From (4.11), using (4.10), (4.12) and (4.13), we obtain

∂

∂tSn = −eE

h

∂

∂kSn(k0,k, t) +

∫dk′ Sn−1(k0,k, t) W (k′,k, t). (4.14)


It is useful to define the probability S0(k0,k, t) that the electron which started at time t = 0 fromthe state k0 will appear at time t in state k without scattering,

S0(k0,k, t) = δ(k − k0 − eEh

) exp[−∫ t

0

dτ Γ(k0 +eEhτ)]

(4.15)

Summing (4.14) over n from n = 1 to ∞ and taking into account (4.8) and (4.15), one can find

∂S

∂t= −eE

h

∂S

∂k−Γ(k)S(k0,k, t)+

∫dk′ S(k0,k′, t) W (k′,k, t)+

∂S0

∂t+eEh

∂S0

∂k+Γ(k)S0(k0,k, t). (4.16)

The total scattering rate Γ(k) and the total transition rate W (k′,k, t) are expressed in the followingway

Γ(k) = Γ0(k) + Γr(k), W (k′k, t) = Γ0(k) δ(k′ − k) + Sr(k′,k, t) (4.17)

where Sr is the sum of the transition rates in real processes. Substituting (4.15) and (4.17) into (4.16)the equation for S is obtained

∂S

∂t= −eE

h

∂S

∂k− Γ0(k)S(k0,k, t) − Γr(k)S(k0,k, t) +

∫dk′ S(k0,k′, t) Γ0(k) δ(k′ − k) +

+∫dk′ S(k0,k, t)Sr(k′,k, t) − eE

hexp

[−∫ t

0

dτ Γ(k0 +eEhτ)]∂

∂tδ(k′ − k0 − eE

ht) −

− Γ(k0 +eEhτ)S0(k0,k, t) +

eEh

exp[−∫ t

0

dτ Γ(k0 +eEhτ)]∂

∂kδ(k′ − k0 − eE

ht) +

+ Γ(k)S0(k0,k, t)

and subsequently∂S

∂t= −eE

h

∂S

∂k− ΓrS +

∫dk′ S(k0,k′, t) W (k′,k, t). (4.18)

Multiplying both sides of (4.18) by f0(k0), integrating over k0 and taking into account (4.9) one cansee that the distribution function calculated by the ensemble Monte Carlo method satisfies the BTE

∂f

∂t= −eE

h

∂f

∂k− Γrf +

∫dk′ f(k′, t) W (k′,k, t). (4.19)

In the equation (4.19) only the real scattering processes appear. Thus the ensemble Monte Carlo methodallows one to recover the solution of the time-dependent BTE . One important fact must be emphasized,this derivation is valid merely in the limit when the amount of particles in the Monte Carlo simulation isinfinity.

¿From the form of iterative relationship (4.10) one can formally define the operator W [102] by

WSn =∑k’

∫ t

0

dt′ Sn−1(k0,k′, t) W (k′,k− eEt− t′

h). (4.20)

Then relation (4.10) is the n-th term in the series

S = S0 + WS0 + WWS0 + WWWS0 + . . . . (4.21)

There is only one k0 for which S0(k0,k, t) is not zero, as there is only one ballistic free flight pathfrom initial state to the final state. To convert these conditional probabilities to distribution function,one must weight S0 by the initial distribution function f(k0, t = 0). When this is done, one must becareful to set k0 and k appropriately in the first term of the expansion (4.21). Each following term inthe expansion represents the probability of moving from k0 to k in time t with a number of scatteringevents equal to the number of applications of the operator W, which automatically places the particle on

4.3. Comparison of Ensemble Monte-Carlo Method with BTE 37

the suitable free path. The every scattering event must be instantaneous so that the time of scatteringis not possible to define and the phenomena such as the intracollisional field effect or the memory effectcan not be included.

The equation (4.18) can be rewritten using operator (4.20):

∂S

∂t= (W − Γ)S. (4.22)

In general, we have

S(t) = exp(Ct)S(0) ≡ G(t)S(0),dS(0)dt

= C S(0) = 0 (4.23)

in equilibrium. In the equation (4.23) G(t) is termed the propagator for S(t) and define how states inS(0) evolve forward in time to give S(t). If the projection operators P and Q = 1 − P [7] which wasintroduced in the first section by the equation (2.10) is applied on the equation (4.22) a system of coupletequations is obtained [103]:

d

dt(PG) = PCPG + PCQG

d

dt(QG) = QCPG + QCQG. (4.24)

The solution of the first equation in (126) is substituted to the second to yield

d

dt(PG) = PCPG +

∫ t

0

dt′ PCQ exp[QC(t− t′)]QCPG(t′) + PCQ exp(QCt)QS(0). (4.25)

The form of the equation (4.25) is standard in transport theory [104, 9]. At this point it shouldbe mentioned that there are two contradictory opinions about the equivalency or non-equivalency ofthe BTE and ensemble Monte Carlo. Firstly, the derivation of the BTE from the ensemble MonteCarlo distribution (4.9) is performed without any approximation, i.e., both formulations are equivalent.Secondly, equation (4.25) is equivalent to the ensemble Monte Carlo, but more precise than the BTE(it contains the dependence on the prehistory t′ < t of ensemble). These two claims are presented withauthority [101, 102, 103] but (surprisingly) nobody attempted to solve this conflict ”experimentally” bycomparing numerical results from both approaches in the frame of a specific transport problem.

Chapter 5

Goals of the Doctoral Thesis

In summary, we have reviewed the derivation of the BTE from the first principles. Numerical methods forthe solution of the BTE — iterative technique and ensemble Monte Carlo — were discussed as well. TheBTE has been formulated for both carrier-phonon and carrier-carrier interaction. Screening of carrier-carrier interaction can be treated through the linear RPA screening theory which has been also reviewed.Ensemble Monte Carlo has been compared with the BTE formulation of carrier transport and the problemof the equivalency and non-equivalency of both techniques has been pointed out. On the basis of thisreview the following goals are chosen for our doctoral thesis:

1. The problem of the equivalency or non-equivalency of the BTE and ensemble Monte Carlo has beendiscussed in the previous section using analytical arguments and conclusions are contradictory.The first goal of our doctoral thesis could be to examine this problem by direct comparison of bothformulations when applied to the same transport problem. Such a numerical comparison (also notquite general) could support the equivalency (or non-equivalency) of both approaches.

2. The second goal follows the previous one. We would like to examine the following transport prob-lems:

• Ultrafast relaxation of photoexcited carrier in GaAs crystals (3D dynamics) and inGaAs/A/GaAs quantum wells (2D dynamics).

• Time response of the 3D and 2D carrier to high electric fields.

The above problems will be solved using both direct numerical solution of the BTE as well asensemble Monte Carlo simulation. The final step would be the comparison with experiment.

3. Finally, we would like to examine the reliability of the linear RPA screening theory. Since analyticalderivations of non-linear corrections would be extremely complicated, it seems natural to use a directnumerical approach to find non-linearly screened potential of the probe charge. After that sucha potential could be used in the ensemble Monte Carlo simulation of carrier-carrier scattering inorder to compare with simulations based on linear screening theory.

¿From methodical point of view it will be necessary to develop:

1. the program for direct numerical solution of the BTE with carrier-carrier and carrier-phonon inter-action for both 2D and 3D systems (Monte Carlo programs for these problems are already partlydeveloped in the Department of the Theory of Semiconductor Microstructures, Institute of ElectricalEngineering, Bratislava, where our doctoral thesis should be performed)

2. the program for numerical solution of non-linear screened carrier-carrier interaction

3. the program for computation of carrier-carrier scattering rates with non-linearly screened Coulombinteraction

39

40 CHAPTER 5. GOALS OF THE DOCTORAL THESIS

Appendix A

Plasmons as Collective Excitationsin Electron Gas

A gas of charged particles moving against the background of oppositely charged particles is known asa neutral plasma. These systems can exhibit collective density oscillations which are called plasmaoscillations in the classical physics. The fundamental frequency of such oscillations can easily be derivedin its framework. Let us assume that a part of the total charge is displaced by the distance x and ρ isthe particle density of these charge. Afterwards a polarization

P = ρ e x

will arise. The polarization will cause an electric field

E = −Pκ

which acts as a restoring force eE on the displaced charges. Their dynamic is described by the Newton’sequation of the harmonic oscillator

md2xdt2

= −e2

κρx

with the frequency

ωp =

√e2ρ

κm(A.1)

which is generally called the plasma frequency.When the plasma oscillations are occuring in the electron gas, one could attempt to rewrite the

hamiltonian (3.14) to a form which contains an explicit harmonic-oscillator part of the form

12

(Π†

kΠk + ω2p Ξ†

kΞk

)(A.2)

with the suitable collective coordinates Ξk and the corresponding momenta Πk [30].The transformation is started from the appropriate collective coordinates

Ξ = Qk(r1, r2, . . . , rn)

which are written as a function of the electron coordinates. The introduction of the independent variablesQk is the crucial point because we impose that

Qk corresponds to Ξ = Ξk(ri).

41

42 APPENDIX A. PLASMONS AS COLLECTIVE EXCITATIONS IN ELECTRON GAS

The conditions under which the operators Ξ can be replaced with the operators Ξk will be given later.The solution of the Schrodinger equation

H(ri,pi)Ψ(ri) = EΨ(ri) with∂

∂QkΨ = 0, (A.3)

i.e., where Ψ(ri) is independent of Qk, is required. Defining the canonical momentum Πk = −ih ∂∂Qk

,the condition (A.3) becomes

ΠkΨ = 0 (A.4)

Using this condition the Schrodinger equation (A.3) can be transformed by adding zero to[H(ri,pi) + H(Πk)

]Ψ(ri) = EΨ(ri). (A.5)

Acting on (A.5) with a unitary transformation U the desired form[U† (H + H) U

] (U† Ψ

)= E

(U† Ψ

)H (Ξk,Πk, ri,pi) Ψ(Ξk, ri) = Ψ(Ξk, ri) (A.6)

is obtained with a harmonic-oscillator part. The transformed wave function depends explicitly on thecollective coordinates Ξk. Nevertheless, a solution of (A.6) should satisfy the transformed conditionequation (A.4) [

U† Πk U]U† (Ψ(ri)) ≡

[U† Πk U

]Ψ(ri,Ξk).

The transformed equation (A.6) with this transformed condition will be solved by the using perturbatintheory. The oscillator part and the remaining kinetic energy of the electrons can be taken ss the unper-turbed hamiltonian H0. Thus, the zeroth order wave function is useful for construction of the collectivecoordinates from a product of the Slater determinat of plane waves and oscillator wave functions.

To define the collective coordinates, the Fourier transformation of the density matrix (2.2) may bewritten as

ρ(r) =1ν

∑k

ρk exp(ik · r), ρk =∫d3r ρ(r) exp(−ik · r)

and this expansion may be interpreted as an operator equation (the hat is used to attend the operatorcharacter of the quantity)

(r) =1ν

∑k

k exp(ik · r). (A.7)

The Fourier components are

k =∫d3r(r)e−ik·r =

∫d3r

⎡⎣ N∑

j=1

δ(r − rj)

⎤⎦ e−ik·r =

N∑j=1

e−ik·rj .

These momentum components correspond to small density packages which execute oscilations with thefrequency (A.1) according to classical considerations. Then the collective coordinates are naturally chosenby the definition

Ξk(ri) = iMkk, Mk =

√e2

κk2ν. (A.8)

The complex conjugate of momentum componets is

(k)† =N∑

j=1

eik·r†j =

N∑j=1

e−i (−k)·rj = −k

43

which implies thatΞ†

k = −Ξ(−k). (A.9)

Now the entire Coulomb interaction in the position representation

H =N∑

i=1

p2i

2m+

12

∑i,j=1i=j

e2

4π|ri − rj |

can be formally rewritten as

H =N∑

i=1

p2i

2m+

12

∑i,j=1i=j

∑k =0

e2

κνk2eik·(ri−rj)

=N∑

i=1

p2i

2m+

12

∑i,j=1

∑k =0

e2

κνk2eik·(ri−rj) − N

2

∑k =0

e2

κνk2

=N∑

i=1

p2i

2m+

12

∑k =0

Ξ†kΞk − N

2

∑k =0

e2

κνk2(A.10)

Since the plasma oscillations represent the motion of many electrons, the oscillations are mostly due tothe long-range part of the Coulomb interaction. Therefore, a transformation to the collective coordinatesis performed only for this part of the interaction and leaves the short-range part as a function of theelectron coordinates. The hamiltonian (A.10) after the partition of the Coulomb interaction into thelong-range and the short-range part (3.41) can be split into following

N∑i=1

p2i

2m+

12

∑0=k<kc

(Ξ†

kΞk −NM2k

)+H(sc)

ee (A.11)

where the screening wave vector kc was defined by the equation (3.41) and the short-range part of theCoulomb potential interation is the screening hamiltonian (3.43).

The independent coordinates Ξk will be introduced in the next step (in the end, these will of coursereplace the Ξk). Similarly to (A.9) it is demanded for Ξk that

Ξ†k = −Ξ(−k). (A.12)

For the canonically conjugated operators Πk the subsequent equallings would be valid[Ξ†

k,Π†k’

]= −[Ξk,Πk’]

† = −(ihδk,k’)∗ = ihδk,k’.

These commutation relations are consistent only if

Π†k = −Π(−k) (A.13)

since then [Ξk’,Π

†k

]=[−Ξ†

(−k’),Π†k

]= −ihδk,(−k’) = −ihδk’,(−k) = − [Ξk’,Π(−k)

].

Note here that the collective operators Ξk must, of course, commute with one another just as the Ξk.It is possible to add the uncontributional term to the hamiltonian (A.10) in the operator form

H =12

∑k<kc

(Π†

k − 2Ξ†k

)Πk =

∑k<kc

(12Π†

kΠk + i Mk†kΠk

)(A.14)


which describes the free field Πk and the interaction of this field with the electron field k. The operatorH is a self-adjoint, which one readily confirms from the remarks above and by renaming the summationindices k → −k (Ξk and Πk commute, since they act on different sets of coordinates). Apparently Hgives no contribution when acting on the solution function Ψ because the condition (A.4) was demanded.

The transformation U can be taken as

U = exp

[+ih

∑k<kc

Ξ†kΞk

]= exp

[− 1h

∑k<kc

Ξ†kMkk

]. (A.15)

The operator defined in this way is unitary, since

U† = exp

[−ih

∑k<kc

ΞkΞk

]= exp

[− 1h

∑k<kc

(−Ξ†

(−k)

)(−Ξ(−k)

)]= U−1. (A.16)

By using the indentity for the two arbitrary operators O, S

exp(−S) O exp(+S) = O + [O, S]

provided that[[O, S], S] = 0

one can quickly derive the following relations:

U†ΞkU = Ξk

U†ΞkU = Ξk

U†riU = ri

U†Π†kU = Π†

k +

[Π†

k,i

h

∑l<kc

ΞlΞl

]= Π†

k +i

h(−ih)

∑l<kc

δk,lΞl = Π†k + Ξ

U†piU = pi +

⎡⎣pi,

i

h

∑k<kc

Ξ†kiMk

∑j

e−ik·rj

⎤⎦ = pi +

∑k<kc

i

hΞ†

kiMk

∑j

hδi,j(−k)e−ik·rj =

= pi +∑k<kc

kMk Ξ†ke

−ik·ri = pi +∑k<kc

kMk Ξke+ik·ri . (A.17)

Introduction of new coordinates and momenta enables us to write the relations (A.17) in an abbrevi-ated form

Ξ(new)k = Ξ(old)

k Ξ(new)k = Ξ(old)

k r(new)i = r(old)

i

Π(new)k = Π(old)

k + Ξ(old/new)k pi(new) = p

(old)i +

[∑k<kc

ΞkkMkeik·ri

](old/new)

. (A.18)

The transformed hamiltonian (A.6) can be immediately calculated with the help of the relations(A.18) as

H =

[N∑

i=1

p2i

2m+H(sc)

ee

]+H(pl)

ee +H(e−pl)I +H

(e−pl)II , (A.19)

where

H(sc)ee =

12

∑0<k<kc

[Π†

kΠk + ω2p Ξ†

kΞk −Ne2

κνk2

](A.20)

45

H(e−pl)I =

N∑i=0

∑0<k<kc

Mk

2mk · (2pi − hk) Ξke

ik·ri (A.21)

H(e−pl)II =

N∑i=1

∑0<k, k’<kc−k = k’

MkMk’

2mk · k′Ξk’Ξke

i(k+k′)·ri . (A.22)

The first part of the hamiltonian (A.19) only depends on the electron coordinates pi and ri and describesan electron gas with the short-range interaction. Futhermore, the hamiltonian (A.20) is an harmonic-oscillator part in the collective coordinates, which reproduces plasma oscillations and, finally, the twohamiltonians (A.21) and (A.22) which contain both particle and collective coordinates are describing anelectron-plasmon interaction.

The second quantization will be used in order to understand better the individual contributions (A.19).Thus, the usual oscillator creation and annihilation operators can be defined (the non-hermicity of theoperators Πk and Ξk can easily be taken into account and do not lead to any complications) as

b†k =1√2hωp

(ωpΞ

†k − iΠk

)bk =

1√2hωp

(ωpΞk + iΠ†

k

). (A.23)

These operators yield

b†kbk =1

2hωp

(ω2

p Ξ†kΞk + ΠkΠ†

k + iωp Ξ†kΠ†

k − iωp ΠkΞk

)and utilizing the relation [Ξk,Πk] = ih the more proper form is acquired

b†kbk =1

2hωp

(ω2


k + iωp

(Ξ†

kΠ†k − ΞkΠk

)− hωp

). (A.24)

Analogously, the relation

bkb†k =1

2hωp

(ω2


k + iωp

(Ξ†

kΠ†k − ΞkΠk

)+ hωp

)(A.25)

is obtained. From the equations (A.24) and (A.25) the commutation relation is as follows

[bk, b†k] = 1

Thus one has got the proof that plasmons are bosons.The harmonic-oscillator is instanted as∑

0<k<kc

hωp

(b†kbk

)=

∑0<k<kc

12

[ω2


k + iωp

(Ξ†

kΠ†k − ΞkΠk

)]. (A.26)

The expression in the round brakets vanishes if the relations (A.12) and (A.13) are taken in such a waythat for each vector k in the sphere with radius kc, the vector −k is also contained in the sphere∑

0<k<kc

Ξ†kΠ†

k =∑

0<k<kc

(Ξ−k) (Π−k) =∑

0<k<kc

ΞkΠk.

Then the hamiltonian (A.20) can be written so that it contains only plasmons

H(pl)ee =

∑0<k<kc

[hωp

(b†kbk +

12

)− Ne2

2κνk2

]. (A.27)


and the correct form of the hamiltonian (3.44) can be obtained.And now the operator (A.21) will be transformed to the second quantization with respect to electron

coordinates. By using〈p′| exp(ik · r)|p〉 = δp’,k+p

one obtains

H(e−pl)I =

∑i

∑0<k<kc

Mk

2m(2k · pi − hk2

)Ξke

ik·ri =

=∑p

∑0<k<kc

Mk

2m(2k · p − hk2

)Ξka†k+pap. (A.28)

It is appropariate to invert the definition of the plasmon creation and annihilation operators (A.23)to

Ξk =

√h

2ωp

(bk − b†−k

)what can be further used in (A.28) to give the result

H(e−pl)ee =

∑p

∑0<k<kc

Mk

2m

√h

2ωp

(2k · p− hk2

)(bka†k+pap − b†−ka†k+pap. (A.29)

This is the hamiltonian (3.45) of the electron-plasmon interaction.The second electron-plasmon interaction operator is usually omitted for the reason that it has the

sum over the phase factorsN∑

i=1

ei(k′+k)·ri

which vanishe because of a random distribution of the electron coordinates ri. This is the random phaseapproximation as it is mentioned in the Section 3.5. In the second quantization the hamiltonian (A.24)is devoted as

H(e−pl)II =

∑0<k, k’<kc−k = k’

∑p

hωp

4N

(kk

)(k′

k′

)[bkbk’ + b†−kb†−k’ − b†−kbk’ − bkb†−k’]a†k+k’+pap. (A.30)

If this part is neglected all processes are omitted where the momentum of an electron with the amountk + k′ either by absorbing two plasmons with momenta k,k′ or by emitting two plasmons with themomenta −k,−k′, or by absorbing and emitting one plasmon of each momentum ±k and ±k′.

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