Mathematical Modelingof Semiconductor Devices

Prof. Dr. Ansgar JungelFachbereich Mathematik und Statistik

Universitat Konstanz

Preliminary version

Contents

1 Introduction 3

2 Some Semiconductor Physics 112.1 Semiconductor crystals . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Mean electron velocity and effective mass approximation . . . . . 172.3 Semiconductor statistics . . . . . . . . . . . . . . . . . . . . . . . 21

3 Classical Kinetic Transport Models 273.1 The Liouville equation . . . . . . . . . . . . . . . . . . . . . . . . 273.2 The Vlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 373.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Quantum Kinetic Transport Equations 504.1 The Wigner equation . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 The quantum Vlasov and quantum Boltzmann equation . . . . . . 56

5 From Kinetic to Fluiddynamical Models 625.1 The drift-diffusion equations: first derivation . . . . . . . . . . . . 625.2 The drift-diffusion equations: second derivation . . . . . . . . . . 745.3 The hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . 765.4 The Spherical Harmonic Expansion (SHE) model . . . . . . . . . 805.5 The energy-transport equations . . . . . . . . . . . . . . . . . . . 925.6 Relaxation-time limits . . . . . . . . . . . . . . . . . . . . . . . . 1005.7 The extended hydrodynamic model . . . . . . . . . . . . . . . . . 104

1

6 The Drift-Diffusion Equations 1146.1 Thermal equilibrium state and boundary conditions . . . . . . . . 1146.2 Scaling of the equations . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Static current-voltage characteristic of a diode . . . . . . . . . . . 1186.4 Numerical discretization of the stationary equations . . . . . . . . 127

7 The Energy-Transport Equations 1317.1 Symmetrization and entropy . . . . . . . . . . . . . . . . . . . . . 1327.2 A drift-diffusion formulation . . . . . . . . . . . . . . . . . . . . . 1367.3 A non-parabolic band approximation . . . . . . . . . . . . . . . . 140

8 From Kinetic to Quantum Hydrodynamic Models 1438.1 The quantum hydrodynamic equations: first derivation . . . . . . 1438.2 The quantum hydrodynamic equations: second derivation . . . . . 1478.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

2

1 Introduction

The modern computer and telecommunication industry relies heavily on the useand development of semiconductor devices. Since the first semiconductor device(a germanium transistor) has been built by Bardeen, Brattain and Shockley in1947, a lot of different devices for special applications have been invented in thefollowing decades. A very important fact of the success of semiconductor devicesis that the device length is very small compared to previous electronic devices(like tube transistors). The first transistor of Bardeen, Brattain and Shockleyhad a characteristic length (the emitter-collector length) of 20 µm. Thanks to theprogressive miniaturization of semiconductor devices, the transistors in a modernPentium IV processor have a characteristic length of 0.18 µm. The device lengthof tunneling diodes, produced in laboratories, is only of the order of 0.075 µm.

Nowadays, semiconductor materials are contained in almost all electronic de-vices. Some examples of semiconductor devices and their use are described in thefollowing.

• Photonic devices capture light (photons) and convert it into an electronicsignal. They are used in camcorders, solar cells, and light-wave communi-cation systems as optical fibers.

• Optoelectronic emitters convert an electronic signal into light. Examplesare light-emitting diodes (LED) used in displays and indication lambs andsemiconductor lasers used in compact disk systems, laser printers, and eyesurgery.

• Flat-panel displays create an image by controlling light that either passesthrough the device or is reflected off of it. They are made, for instance, ofliquid crystals (liquid-crystal displays, LCD) or of thin semiconductor films(electroluminescent displays).

• In field-effect devices the conductivity is modulated by applying an electricfield to a gate contact on the surface of the device. The most importantfield-effect device is the MOSFET (metal-oxide semiconductor field-effecttransistor), used as a switch or an amplifier. Integrated circuits are mainlymade of MOSFETs.

• Quantum devices are based on quantum mechanical phenomena, like tun-neling of electrons through potential barriers which are impenetrable clas-sically. Examples are resonant tunneling diodes, superlattices (multi-quan-tum-well structures), quantum wires in which the motion of carriers is re-stricted to one space dimension and confined quantum mechanically in theother two directions, and quantum dots.

3

Clearly, there are many other semiconductor devices which are not mentioned(for instance, bipolar transistors, Schottky barrier diodes, thyristors). Other newdevelopments are, for instance, nanostructure devices (heterostructures) and solarcells made of amorphous silicon or organic semiconductor materials (see [14, 41]).

Usually, a semiconductor device can be considered as a device which needs aninput (an electronic signal or light) and produces an output (light or an electronicsignal). The device is connected to the outside world by contacts at which avoltage (potential difference) is applied. We are mainly interested in deviceswhich produce an electronic signal, for instance the macroscopically measurableelectric current (electron flow), generated by the applied bias. In this situation,the input parameter is the applied voltage and the output parameter is the electriccurrent through one contact. The relation between these two physical quantities iscalled current-voltage characteristic. It is a curve in the two-dimensional current-voltage space. The current-voltage characteristic does not need to be a monotonefunction and it does not need to be a function (but a relation).

The main objective of this book is to derive mathematical models which de-scribe the electron flow through a semiconductor device due to the application ofa voltage. Depending on the device structure, the main transport phenomena ofthe electrons may be very different, for instance, due to drift, diffusion, convec-tion, or quantum mechanical effects. For this reason, we have to devise differentmathematical models which are able to describe the main physical phenomenafor a particular situation or for a particular device.

This leads to a hierarchy of semiconductor models. Roughly speaking, we candivide semiconductor models in three classes: quantum models, kinetic modelsand fluiddynamical (macroscopic) models. In order to give some flavor of thesemodels and the methods used to derive them, we explain these three view-points:quantum, kinetic and fluiddynamic in a simplified situation.

The quantum view. Consider a single electron of mass m and elementarycharge q moving in a vacuum under the action of an electric field E = E(x, t).The motion of the electron in space x ∈ Rd and time t > 0 is governed by thesingle-particle Schrodinger equation

i~∂tΨ = − ~2

2m∆Ψ− qV (x, t)Ψ, x ∈ Rd, t > 0, (1.1)

with some initial condition

Ψ(x, 0) = Ψ0(x), x ∈ Rd. (1.2)

Here ~ = h/2π is the reduced Planck constant, V (x, t) the real-valued electro-static potential related to the electric field by E = −∇V, and i2 = −1. Anysolution Ψ of (1.1)-(1.2) is called wave function. Macroscopic variables are ob-

4

tained by the definitions:

electron density: n = |Ψ|2,electron current density: J = −~q

mIm(Ψ∇Ψ),

where Ψ is the complex conjugate of Ψ. More precisely, n(x, t) is the positionprobability density of the particle, i.e.

∫

A

n(x, t) dx

is the probability to find the electron in the subset A ⊂ Rd at the time t.Without too much exageration we can say that all other descriptions for the

evolution of the electron, and in particular the fluiddynamical and kinetic models,can be derived from the Schrodinger equation (1.1).

The fluiddynamical view. In order to derive fluiddynamical models, forinstance, for the evolution of the particle density n and the current density J, weassume that the wave function can be decomposed in its amplitude

√n(x, t) > 0

and phase S(x, t) ∈ R :Ψ =

√neimS/~. (1.3)

This expression makes sense as long as the electron density stays positive. We callthis change of unknowns Ψ 7→ (n, S) Madelung transform. The current densitynow reads

J = −~qm

Im

[√n

( ∇n2√n+im

~√n∇S

)]= −qn∇S,

and we can interpret S also as a velocity potential. Setting (1.3) into theSchrodinger equation (1.1) and taking the imaginary and real part of the re-sulting equation, gives, after some computations (see Chapter 8):

∂tn−1

qdiv J = 0, (1.4)

∂tJ −1

qdiv

(J ⊗ Jn

)+

~2q2m2

n∇(∆√n√n

)− q2

mnE = 0. (1.5)

Here, J ⊗ J is a tensor (or matrix) with components JjJk, where j, k = 1, . . . , d.Equation (1.4) expresses the conversation of mass. Indeed,

d

dt

∫

Rdn dx =

1

q

∫

Rddiv J dx = 0.

The equations (1.4)-(1.5) are referred to as the quantum hydrodynamic equations.They are equivalent to the Schrodinger equation (1.1) as long as n > 0. If theyare studied in the whole space, the initial conditions

n(·, 0) = |Ψ0|2, J(·, 0) = −~qm

Im(Ψ0∇Ψ0) in Rd

5

have to be prescribed.In the semi-classical limit “~ → 0” the equations (1.4)-(1.5) formally reduce

to the hydrodynamic equations

∂tn−1

qdiv J = 0, (1.6)

∂tJ −1

qdiv

(J ⊗ Jn

)− q2

mnE = 0, (1.7)

which are the Euler equations for a gas of charged particles with zero temperaturein a given electric field. Clearly, the limit “~ → 0” does not make sense since~ is a constant. After a scaling which makes the variables non-dimensional, aparameter (the so-called scaled Planck constant) appears in front of the scaledquantum term in (1.5). This parameter is small compared to one in certainphysical ranges, for instance, when the characteristic length is “large” (comparedto some reference length). Thus, the limit “~→ 0” has always to be understoodas the limit of vanishing scaled Planck constant.

The fluiddynamical formulation has some advantages:

• The equations are already formulated for macroscopic quantities.

• If the equations are considered in bounded domains (which is natural whendealing with semiconductor devices), boundary conditions can be more eas-ily prescribed compared to the Schrodinger formulation.

• In two or three space dimensions, fluiddynamical models are usually nu-merically cheaper than the Schrodinger equation.

The kinetic view. We start again with the Schrodinger equation (1.1).First, introduce the so-called density matrix

ρ(r, s, t) = Ψ(r, t)Ψ(s, t), r, s ∈ Rd, t > 0,

and make the change of coordinates

r = x+~2m

η, s = x− ~2m

η

in the density matrix defining

u(x, η, t) = ρ(x+

~2m

η, x− ~2m

η, t).

Since (~/2m)η has the dimension of length and ~/2m has the dimension of m2/s,η has the dimension of inverse velocity. We prefer to work with the variablesspace, velocity and time instead of space, inverse velocity and time and employ

6

therefore the Fourier transformation to u. We recall that the Fourier transformF is defined by

Fg(η) = g(η) = (2π)−d/2

∫

Rdg(η)e−iη·v dv

for (sufficiently smooth) functions g : Rd → C with inverse

F−1h(v) = h(v) = (2π)−d/2

∫

Rdh(v)eiη·v dη

for h : Rd → C. The inverse Fourier transform of u is called Wigner function w:

w = (2π)−d/2F−1u = (2π)−d/2u.

This function was introduced by Wigner in 1932 [43]. It can be shown (seeChapter 4) that the macroscopic electron density n and current density J definedabove in terms of the wave function can now be written in terms of the Wignerfunction as

n(x, t) =

∫

Rdw(x, v, t) dv, (1.8)

J(x, t) = −q∫

Rdv w(x, v, t) dv. (1.9)

The integrals are called the zeroth-order and first-order moments.The transport equation for w is obtained by first deriving the evolution equa-

tion for u from the Schrodinger equation (1.1) and then by taking the inverseFourier transform of the equation for u. A computation leads to the so-calledWigner equation (see Chapter 4 for details):

∂tw + v · ∇xw +q

mθ[V ]w = 0, x, v ∈ Rd, t > 0, (1.10)

where

(θ[V ]w)(x, v, t) =im

~(2π)d

∫

Rd

∫

Rd

[V

(x+

~2m

η, t

)− V

(x− ~

2mη, t

)]

×w(x, v′, t)eiη·(v−v′) dv′ dη.

The initial condition for w reads

w(x, v, 0) = (2π)−d/2u(x, v, 0)

= (2π)−d

∫

RdΨ0

(x+

~2m

η

)Ψ0

(x− ~

2mη

)eiv·η dη, x, v ∈ Rd.

7

An operator, whose Fourier transform acts as a multiplication operator on theFourier transform of the function, is called a linear pseudo-differential operator[42]. Since

θ[V ]w(x, η, t) =im

~

[V

(x+

~2m

η, t

)− V

(x− ~

2mη, t

)]w(x, η, t),

the operator θ[V ] is a pseudo-differential operator and the Wigner equation (1.10)is a linear pseudo-differential equation.

In the formal limit “~→ 0” it holds

m

~

[V

(x+

~2m

η, t

)− V

(x− ~

2mη, t

)]→ ∇xV (x, t) · η,

and using the identity i(ηu)∨ = ∇vu, we obtain, as “~→ 0”,

θ[v]w → i

(2π)d/2(∇xV · ηu)∨ =

1

(2π)d/2∇xV · ∇vu = ∇xV · ∇vw.

Hence the Wigner equation becomes in the semi-classical limit

∂tw + v · ∇xw +q

m∇xV · ∇vw = 0. (1.11)

This equation is called Liouville equation. It has to be solved in the position-velocity space (x, v) ∈ Rd × Rd for times t > 0, supplemented with the initialcondition

w(x, v, 0) = (2π)−d/2 lim~→0

u(x, v, 0)

=1

(2π)dlim~→0

∫

RdΨ0

(x+

~2m

η

)Ψ0

(x− ~

2mη

)eiv·η dη, x, v ∈ Rd.

Clearly, the above limits have to be understood in a formal way.From the solution w of the kinetic Liouville equation, the macroscopic electron

density n and current density J are computed as in the quantum case by theformulas (1.8)-(1.9).

The kinetic formulation of the motion of the electron seems much more com-plicate than the Schrodinger formulation since we pass from d + 1 variables (dfor the space and one for the time) to 2d + 1 variables (d for the space, d forthe velocity and one for the time). However, this formulation has the followingadvantages:

• In the case of an electron ensemble with many particles, an approximateformulation in the position-velocity space is possible (leading to the Vlasovequation; see Chapter 3). This means that the kinetic equation has to besolved in 2d + 1 variables instead of Md + 1 variables in the Schrodingerformulation, where M À 1 is the number of particles.

8

• Short-range interactions of the particles (collisions) can be easily includedin the classical kinetic models.

We have described how fluiddynamical and kinetic models can be derivedformally from the Schrodinger equation. Is there any relation between fluiddy-namical and kinetic models? The answer is yes, and the method to see this iscalled moment method. In the following, we describe the idea of this method.

We start with the Liouville equation (1.11). Integrating (1.11) over the veloc-ity space and using the definitions of the first two moments (1.8)-(1.9) and thedivergence theorem gives for E = −∇V

0 = ∂t

∫

Rdw dv +

d∑

k=1

∫

Rdvk∂w

∂xk

dv − q

m

d∑

k=1

Ek

∫

Rd

∂w

∂vkdv

= ∂tn−1

qdiv xJ,

which equals (1.6) expressing the conservation of mass. Now we multiply (1.11)with −qvj and integrate over the velocity space:

0 = ∂tJj − qd∑

k=1

∫

Rdvjvk

∂w

∂xk

dv +q2

m

d∑

k=1

Ek

∫

Rdvj∂w

∂vkdv

= ∂tJj − qd∑

k=1

∂

∂xk

∫

Rdvjvkw dv − q2

m

d∑

k=1

Ek

∫

Rd

∂vj∂vk

w dv.

Since ∂vj/∂vk = δjk with the Kronecker symbol δjk, this yields

∂tJ − qdiv x

∫

Rd(v ⊗ v)w dv − q2

mEn = 0. (1.12)

The term ∫

Rd(v ⊗ v)w dv

is the matrix of all second-order moments and is called energy tensor. We makenow the complete heuristic assumption that the velocity v can be approximatedby an average velocity

v =

∫vw dv∫w dv

= − J

qn,

that means, ∫

R(v ⊗ v)w dv ≈ (v ⊗ v)

∫

Rdw dv =

1

q2J ⊗ Jn

,

9

and (1.12) becomes

∂tJ −1

qdiv x

(J ⊗ Jn

)− q2

mnE = 0,

which equals (1.7). Under the above approximation, we have derived the hy-drodynamic model (1.6)-(1.7). For a more general derivation (including thermaleffects), see Chapter 5.

In a similar way, we can apply the moment method to the Wigner equa-tion (1.10) in order to derive the quantum hydrodynamic model (1.4)-(1.5) (seeChapter 8).

All the models explained in this chapter and the relations between them aresummarized in Figure 1.1.

Fluidynamical

Hydrodynamic

Quantum

Hydrodynamic

?

“~→ 0”

Kinetic

Liouville¾ momentmethod

Wigner

?

“~→ 0”

¾ momentmethod

Schrodinger©©©©©©©©¼

HHHHHHHHj

Madelungtransform

Fouriertransform

Quantum

Figure 1.1: Models for the motion of an electron in a vaccum.

The above models are all derived in a very simplified situation. For themodeling of the electron flow in semiconductors we have to take into account:

• the description of the motion of many particles;

• the influence of the semiconductor crystal lattice;

• two-particle long-range interactions (Coulomb force between the electrons);

• two-particle short-range interactions (collisions of the particles with thecrystal lattice or with other particles).

These features will be included in our models step by step in the following chap-ters.

10

2 Some Semiconductor Physics

In this chapter we present a short summary of the physics and main propertiesof semiconductors. Only these subjects relevant to the subsequent chapters areincluded here. We refer to [1, 8, 14, 28, 41] for a detailed description of solid-stateand semiconductor physics.

2.1 Semiconductor crystals

What is a semiconductor? Historically, the term semiconductor has been usedto denote solid materials with a much higher conductivity than insulators, buta much lower conductivity than metals measured at room temperature. A moreprecise definition is that a semiconductor is a solid with an energy gap larger thanzero and smaller than about 4 eV. Metals have no energy gap, whereas this gapis usually larger than 4 eV in insulators. In order to understand what an energygap is, we have to explain the crystal structure of solids.

A solid is made of an infinite three-dimensional array of atoms arranged ac-cording to a lattice

L = `~a1 +m~a2 + n~a3 : `,m, n ∈ Z ⊂ R3,

where ~a1, ~a2, ~a3 are the basis vectors of L, usually called primitive vectors of thelattice. The lattice atoms (or ions) generate a periodic electrostatic potential VL:

VL(x+ y) = VL(x) ∀x ∈ R3, y ∈ L.

The state of an electron moving in this periodic potential is described by aneigenfunction of the stationary Schrodinger equation:

− ~2

2m∆Ψ− qVL(x)Ψ = εΨ, x ∈ R3, (2.1)

where Ψ : R3 → C is called wave function. In this equation, the physical param-eters are the reduced Planck constant ~ = h/2π, the electron mass (at rest) m,the elementary charge q, and the total energy ε. In fact, (2.1) is an eigenvalueproblem, and we have to find eigenfunction-eigenvalue pairs (Ψ, ε). We considertwo examples.

Example 2.1 (Free electron motion)First, consider a free electron moving in a one-dimensional vacuum, i.e., VL(x) = 0for all x ∈ R. Then the solutions of (2.1) are given by

Ψk(x) = Aeikx +Be−ikx, x ∈ R,

and

ε =~2k2

2m,

11

for any k ∈ R.We disregard purely imaginary values of k = iγ (γ ∈ R) which alsoyield real values of the energy, since this leads to unbounded solutions of the typeexp(±ikx) = exp(∓γx) for x ∈ R. (Notice that the integral of |Ψ|2 over R canbe interpreted as the particle mass which should be finite.) Thus the eigenvalueproblem (2.1) has infinitely many solutions corresponding to different energies.The functions exp(±ikx) are called plane waves.

Example 2.2 (Infinite square-well potential)The infinite square-well potential is a one-dimensional structure in which thepotential is infinite at the boundaries and everywhere outside and zero everywherewithin. As the potential is confining an electron to the inner region, we have tosolve the Schrodinger equation (2.1) in the interval (0, L) of length L > 0 withboundary conditions

Ψ(0) = Ψ(L) = 0

and potential VL(x) = 0 for x ∈ (0, L) (see Figure 2.1). The eigenfunctions of(2.1) are given by

Ψk(x) = Ak sin(kx), x ∈ (0, L),

with discrete eigenvalues k = πn/L, n ∈ N, and energies

ε(k) =~2k2

2m, k =

πn

L.

The system only allows discrete energy states.

-x0 L

6∞

6∞

Figure 2.1: Infinite square-well potential.

Although the lattice potential VL is periodic, the wave function is defined onR3 in order to account for electron motion from one lattice cell to the next. Infact, the so-called Bloch theorem states that the Schrodinger equation (2.1) canbe reduced to a Schrodinger equation on a so-called primitive cell of the lattice.Before we state the result, we need some definitions.

Definition 2.3 (1) The reciprocal lattice L∗ corresponding to L is defined by

L∗ = ` ~a∗1 +m~a∗2 + n~a∗3 : `,m, n ∈ Z,

12

where the basis vectors (or primitive vectors) ~a∗1, ~a∗2, ~a

∗3 ∈ R3 satisfy the

relation~am · ~a∗n = 2πδmn.

(2) The connected set D ⊂ R3 is called primitive cell of L (or of L∗) if andonly if

• the volume of D equals the volume of the parallelepiped spanned by thebasis vectors of L (or of L∗);

• the whole space R3 is covered by the union of translates of D by theprimitive vectors.

(3) The (first) Brillouin zone B ⊂ R3 is that primitive cell of the reciprocallattice L∗ which consists of those points, which are closer to the origin thanto any other point of L∗ (see Figure 2.2).

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¯¯¯¯¯AAAA¯

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aaaaaaaaaaaaaaaaas

s

s

s

s

s

s

s

s - ~a1£££££°

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~a∗1

~a2

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6

Figure 2.2: The primitive vectors of a two-dimensional lattice L and its reciprocallattice L∗ and the Brillouin zone B.

We give some explainations of the above definition. As the wave planesexp(±ik · x) suggest, k can be considered as a reciprocal variable which leadsto the definitions of the reciprocal lattice and the Brillouin zone. In particular,for any x ∈ L and k ∈ L∗ with

x =3∑

m=1

αm~am and k =3∑

n=1

βn~a∗n, αm, βn ∈ Z,

13

we obtain

exp(ik · x) = exp

(i

3∑

m,n=1

αmβn · 2πδmn

)= exp

(2πi

3∑

n=1

αnβn

)= 1.

As x has the dimension of length, k has the dimension of inverse length andtherefore, k is called a wave vector. (More precisely, k is called pseudo-wavevector; see below.)

The Brillouin zone can be constructed as follows. Draw arrows from a latticepoint to its nearest neighbors and cut them in half. The planes through theseintermediate points perpendicular to the arrows form the surface of the (bounded)Brillouin zone. In two space dimensions, the Brillouin zone is always a hexagon(or a square; see Figure 2.2).

Now we are able to apply the Bloch theorem. It says that the Schrodingerequation (2.1) is equivalent to the system of Schrodinger equations

− ~2

2m∆Ψk − qVLΨk = ε(k)Ψk, x ∈ D, (2.2)

indexed by k ∈ B, with pseudo-periodic boundary conditions

Ψk(x+ y) = Ψk(x)eik·y for x, x+ y ∈ ∂D, (2.3)

where D is a primitive cell of L. (This follows from the fact that the Hamiltonoperator H = −(~2/2m)∆+ qVL(x) commutes with all the translation operatorsT` associated with the lattice, where (T`Ψ)(x) = Ψ(x+ `) for ` ∈ L, x ∈ R3.) Byfunctional analysis, the eigenvalue problem (2.2)-(2.3), for any k ∈ B, possessesa sequence of eigenfunctions Ψn

k with associated eigenvalues εn(k), n ∈ N0. Therelation between Ψk and Ψ is given by

Ψk(x) =∑

`∈L

e−ik·`Ψ(x+ `).

We can also write Ψnk as distorted waves

Ψnk(x) = un

k(x)eik·x, (2.4)

whereunk(x) =

∑

`∈L

e−ik·(x+`)Ψ(x+ `)

is clearly periodic in L. In some sense, Ψnk are plane waves which are modulated

by a periodic function unk taking into account the influence of the lattice of atoms.

The functions (2.4) are also called Bloch functions. Since the vector k appears indistorted plane waves, it is not termed wave vector but pseudo-wave vector.

The function k 7→ εn(k) is called the disperson relation or the n-th energyband. It shows how the energy of the n-th band depends on the wave vector

14

k. The union of the ranges of εn over n ∈ N is not necessarily the whole setR, i.e., there may exist energies ε∗ for which there is no n ∈ N and no k ∈ Bsuch that εn(k) = ε∗. The connected components of the set of energies with thisnon-existence property are called energy gaps. We illustrate this property by anexample.

Example 2.4 (Kronig-Penney model)We study a simple one-dimensional model for the periodic potential generated bythe lattice atoms. We use the square-well potential (see Figure 2.3)

VL(x) =

V0 : −b < x ≤ 00 : 0 < x ≤ a,

and VL is extended to R with period a+ b, i.e.

VL(x) = VL(x+ a+ b) ∀x ∈ R,

where a, b > 0 and V0 ∈ R.

-

6

−b 0

V0

a a+ b

VL(x)

x

Figure 2.3: Periodic square-well potential VL.

In order to solve the Schrodinger equation (2.1), we make the Bloch ansatz

Ψk(x) = u(x)eikx

with an (a+ b)-periodic function u. Set

α =

√2mε

~2, β =

√2m(ε− qV0)

~2.

If ε < qV0 then β can be written as β = iγ with γ > 0. A computation showsthat the general solution of (2.1) in the interval (−b, a) is given by

u(x) =

Aei(α−k)x +Be−i(α+k)x : 0 < x ≤ aCei(β−k)x +De−i(β+k)x : −b < x ≤ 0.

15

The constants A, B, C and D can be determined from the boundary conditions.More precisely, we assume that u is continuously differentiable and periodic in R:

u(a+ 0) = u(a− 0), u(a+ 0) = u(a− 0),

u(−b) = u(a), u′(−b) = u′(a),

where u(a + 0) = limxa u(x) etc. This gives four linear equations for the fourunknows A, B, C and D. They can be casted into a linear system

M(A,B,C,D)> = 0

with matrix M ∈ C4×4. (The sign “>” means the transposed of a vector.) Thenecessary conditions for getting non-trivial solutions is that detM = 0. An ele-mentary, but lenghty calculation shows that this conditon is equivalent to

cos(k(a+ b)) =

√1 +

(α2 − β22αβ

)2

sin2(βb) · cos(αa− δ), (2.5)

where

tan δ = −α2 + β2

2αβtan(βb).

The equation (2.5) is implicit in k and real-valued for all values of the energy.Indeed, if ε > qV0 we have β = iγ and sin(βb)/β = sinh(γb)/γ since sin(ix) =i sinh(x).

The right-hand side of (2.5) may be larger than +1 or smaller than −1 forcertain values of ε. Then (2.5) is solvable only for imaginary wave vectors k = iκ,κ ∈ R. But then Ψ(x) = u(x) exp(ikx) = u(x) exp(−κx) and this gives non-integrable solutions in R. We conclude that there exist values of ε for which (2.5)has no real solution k. Every connected subset of [0,∞)\R(ε), where R(ε) =ε0 ∈ [0,∞) : ∃k ∈ R : ε(k) = ε0, is an energy gap.

In Table 2.1 some values of energy gaps for some semiconductor materials arecollected. The energy gap separates two energy bands. The nearest energy band

Material Energy gap εg (eV)Silicon 1.12Germanium 0.664Gallium-Arsenide 1.424

Table 2.1: Energy gaps of selected semiconductors.

below the energy gap is called valence band; the nearest energy band above theenergy gap is termed conductor band (see Figure 2.4).

Now we are able to state the definition of a semiconductor: It is a solid withan energy gap whose value is positive and smaller than about 4 eV.

16

-

6

6?εg

ε

kvalence band

conductionband

Figure 2.4: Schematic band structure with energy gap εg.

2.2 Mean electron velocity and effective mass approxima-tion

In this section we will motivate the following two expressions:

• The mean electron velocity (or group velocity of the wave packet) in then-th band is given by

vn(k) =1

~∇kεn. (2.6)

• The effective mass tensor m∗ is defined by

(m∗)−1 =1

~2d2εndk2

. (2.7)

Motivation of (2.6). The result (2.6) can be obtained as follows. We omitin the following the index n and define the group velocity by

v(k) =

(∫

D

|Ψ|2 dx)−1 ∫

D

vk|Ψk|2 dx,

where D is a primitive cell of the lattice and vk is the particle velocity

vk = − Jkqnk

=~m

Im(Ψk∇xΨk)

|Ψk|2. (2.8)

In order to show (2.6) we have to compute the integral∫

D

Im(Ψk∇xΨk) dx.

We take the derivative of (2.2) with respect to k, use the distorted wave plane

Ψk(x) = uk(x)eik·x

and

∆x(∇kΨk) = ∆x

(eik·x∇kuk + ixΨk

)

= ∆x

(eik·x∇kuk

)+ 2i∇xΨk + ix∆xΨk

17

to obtain

(∇kε)Ψk = − ~2

2m∆x(∇kΨk)− (qVL + ε)∇kΨk

=

(− ~2

2m∆x − (qVL + ε)

)(eik·x∇kuk + ixΨk)

=

(− ~2

2m∆x − (qVL + ε)

)(eik·x∇kuk) + ix

(− ~2

2m∆x − (qVL + ε)

)Ψk

− i~2

m∇xΨk

=

(− ~2

2m∆x − (qVL + ε)

)(eik·x∇kuk)− i

~2

m∇xΨk.

Multiplying this equation by Ψk, integrating over D and integrating by partsyields

∇kε

∫

D

|Ψk|2 dx+ i~2

m

∫

D

(∇xΨk)Ψk dx

=

∫

D

eikx∇kuk

(− ~2

2m∆x − qVL − ε

)Ψk dx = 0,

using (2.2) again. Thus

∇kε = −i~2∫D(∇xΨk)Ψk dx

m∫D|Ψk|2 dx

.

Taking the real part of both sides of this equation gives

∇kε =~2

m

Im∫D(∇xΨk)Ψk dx∫D|Ψk|2 dx

= ~v(k),

which shows (2.6).The expression (2.6) has some consequences. The change of energy with

respect to time equals the product of force F and velocity vn:

∂tεn(k) = Fvn(k) = ~−1F∇kεn(k).

By the chain rule,∂tεn(k) = ∇kεn(k)∂tk

and henceF = ∂t(~k). (2.9)

Newton law’s states that the force equals the time derivative of the momentump. This motivates the definition of the crystal momentum

p = ~k.

18

Motivation of (2.7). Another consequence follows from (2.6) and (2.9):Differentiating (2.6) leads to

∂tvn =1

~d2εndk2

∂tk =1

~2d2εndk2

F.

The momentum equals m∗vn, where m∗ is the (effective) mass. Using again

Newton’s law F = ∂tp = m∗∂tvn we obtain

(m∗)−1 =1

~2d2εndk2

.

We consider this equation as a definition of the effective mass m∗. The secondderivative of εn with respect to k is a 3× 3 matrix, so the symbol (m∗)−1 is also amatrix. If we evaluate the Hessian of εn near a local minimum, i.e. ∇kεn(k0) = 0,then d2εn(k0)/dk

2 is a symmetric positive matrix which can be diagonalized andthe diagonal elements are positive:

1/m∗x 0 0

0 1/m∗y 0

0 0 1/m∗z

=

1

~2d2εndk2

(k0). (2.10)

Assume that the energy values are shifted in such a way that the energy vanishesat the local minimum k0. For wave vectors k “close” to k0, we have from Taylor’sformula and (2.10)

εn(k) = εn(k0) +∇kεn(k0) · k +1

2k>d2εndk2

(k0)k +O(|k − k0|3)

=~2

2

(k2xm∗

x

+k2ym∗

y

+k2zm∗

z

)+O(|k − k0|3),

where k = (kx, ky, kz)>. If the effective masses are equal in all directions, i.e.

m∗ = m∗x = m∗

y = m∗z, we can write, neglecting higher-order terms,

εn(k) =~2

2m∗|k|2 (2.11)

for wave vectors k “close” to a local band minimum. In this situation, m∗ is calledthe isotropic effective mass. We conclude that the energy of an electron near aband minimum equals the energy of a free electron in vacuum (see Example 2.1)where the (rest) mass of the electron is replaced by the effective mass. The effectsof the crystal potential are represented by the effective mass.

The expression (2.11) is referred to as the parabolic band approximation andusually, the range of wave vectors k is extended to the whole space, i.e., k = R3 isassumed. In order to account for non-parabolic effects, the following non-parabolicband approximation in the sense of Kane is used [41, Ch. 2.1]:

εn(1 + αεn) =~2 |k|22m∗

,

19

where α > 0 is a non-parabolicity parameter. In silicon, for instance, α =0.5 (eV)−1.

Definition of holes. When we consider the effective mass near a bandmaximum, we find that the Hessian of εn is negative definite which would leadto a negative effective mass. However, in the derivation of the mean velocityand consequently, of the effective mass definition, we have used in (2.8) that thecharge of the electrons is negative. Using a positive charge leads to a positiveeffective mass. The corresponding particles are calles holes. Physically, a holeis a vacant orbital in an otherwise filled (valence) band. Thus, the current flowin a semiconductor crystal comes from two sources: the flow of electrons in theconduction band and the flow of holes in the valence band. It is a convention toconsider rather the motion of the valence band vacancies that the motion of theelectrons moving from one vacant orbital to the next (Figure 2.5).

t = 0 :

ÁÀ¿ss

crystal atom£££±

electronJJ

ÁÀ¿scRorbital

­­À

¤¤¤º

vacancy

t > 0 :

ÁÀ¿sc

vacancy

JJ

ÁÀ¿ss¤¤¤º

electron

Figure 2.5: Motion of a valence band electron to a neighboring vacant orbital or,equivalently, motion of a hole in the inverse direction.

Semi-classical picture. In the so-called semi-classical picture, the motionof an electron in the n-th band can be approximately described by a point particlemoving with the velocity vn(k). Denoting by (x(t), vn(k, t)) the trajectory of anelectron in the position-velocity phase space, we obtain from Newton’s law (see(2.9))

∂tx = vn(k) =1

~∇kεn, ~∂tk = F, (2.12)

where F represents a driving force, for instance, F = −q∇xVL(x, t). Notice thatband transitions are excluded since the band index n is fixed in the equations.

Non-periodic potentials. If a non-periodic potential (or driving force) issuperimposed to VL, the Schrodinger equation (2.1) cannot be decomposed intothe decoupled Schrodinger equations (2.2), and the above computations are nolonger valid. In fact, the energy bands are now coupled. However, it is usuallyassumed that the non-periodic potential is so weak that the coupling of the bandscan be neglected and then, the above analysis remains approximately valid.

20

2.3 Semiconductor statistics

Number densities. We want to determine the number of electrons in theconduction band and the number of holes in the valence band per unit volume.In view of the very high number of particles (typically > 109 cm−3) it seemsappropriate to use statistical methods. We make the following assumptions:

• Electrons cannot be distinguished from one another.

• The Pauli exclusion principle holds, i.e., each level of a band can be occupiedby not more than two electrons with opposite spin.

First we compute the number of possible states within all energy bands perunit volume (2π)3. In the continuum limit, this number equals:

g(ε) =2

(2π)3

∑

ν

∫

B

δ(ε− εν(k)) dk. (2.13)

The function g(ε) is called density of states. The factor 2 comes from the twopossible states of the spin of an electron. The set B is the Brillouin zone (seeSection 2.1), and the function δ is the delta distribution defined by

∫ ∞

−∞

δ(ε0 − ε)φ(ε) dε = φ(ε0) (2.14)

for all appropriate functions φ. Mathematically, δ is a functional operating insome function space and the above integral has to be interpreted as the value ofδ(ε0 − ε) acting on φ(ε). Formally, it holds∫ ∞

−∞

δ(ε0 − ε)φ(ε) dε =∫ ∞

−∞

δ(ε− ε0)φ(ε) dε =∫ ∞

ε1

δ(ε− ε0)φ(ε) dε, (2.15)

for any ε1 < ε0.

Example 2.5 (Density of states in the parabolic band approximation)In the case of the parabolic band approximation we can compute the integral(2.13). Consider a single band near the bottom of the conduction band withisotropic effective mass. The energy is given by

ε(k) = εc +~2

2m∗e

|k|2, k ∈ R3.

Transforming k to the spherical coordinates (ρ, θ, φ) and then substituting z =~2ρ2/2m∗

e, we obtain

gc(ε) =1

4π3

∫

R3

δ(ε− εc − ~2|k|2/2m∗e) dk

=1

4π3

∫ 2π

0

∫ π

0

∫ ∞

0

δ(ε− εc − ~2ρ2/2m∗e) ρ

2 sin θ dρ dθ dφ

=4π

4π31

2

(2m∗

e

~2

)3/2 ∫ ∞

0

δ(ε− εc − z)√z dz.

21

Define the Heaviside function H by

H(x) =

0 : x < 01 : x > 0.

Then, by (2.14),

gc(ε) =1

2π2

(2m∗

e

~2

)3/2 ∫ ∞

−∞

δ(ε− εc − z)√zH(z) dz

=1

2π2

(2m∗

e

~2

)3/2 √ε− εcH(ε− εc).

The electron density n is the integral of the probability density of occupancyof an energy state per unit volume:

n =2

(2π)3

∑

ν

∫

B

f(εν(k)) dk,

where f(ε) is a distribution function. Using (2.14), (2.15) and the definition(2.13) of gc, we have

n =1

4π3

∑

ν

∫

B

∫ ∞

−∞

δ(ε− εν(k)) f(ε) dε dk

=

∫ ∞

−∞

gc(ε) f(ε) dε. (2.16)

Fermi-Dirac distribution. We will motivate that the electron distributionfollows the Fermi-Dirac distribution

f(ε) =1

1 + e(ε−εF )/kBT, (2.17)

where the number εF which is called Fermi energy or Fermi level will be inter-preted later. Consider a system of non-interacting particles with discrete quan-tum states. The system is put into contact with a reservoir at temperature T.Let εi and εj be the total energies of a particle in the state i and j, respectively.Let Wij be the probability that an electron of state i will go to state j in a unittime. The average number of particles that make a transition from state i tostate j is

A(T )e−εi/kBTWij,

where A(T ) is some function. The temperature T enters because the transitionwill absorb energy from, or omit energy to, the reservoir. Conversely, the averagenumber of particles that go from state j to state i is

A(T )e−εj/kBTWji.

22

We use the principle of detailed balance. It says that in thermal equilibrium, thetwo number must coincide. Therefore

Wji

Wij

= e(εj−εi)/kBT . (2.18)

Now let fi be the average number of electrons in the state i. By Pauli’sexclusion principle, each state can only have 0 or 1 electron in it at any time.Hence, 0 ≤ fi ≤ 1, and fi can be seen as the fraction of configurations in whichthe state i is occupied. Similarly, 1− fi is the fraction of configurations in whichthe state i is empty. The number of transitions from state i to state j of a largeensemble M of configurations is

Mfi(1− jj)Wij,

since the transition only takes place if the state i is occupied and the state j isempty. By the principle of detailed balance,

Mfi(1− fj)Wij =Mfj(1− fi)Wji.

This equation is equivalent to

fi1− fi

=fj

1− fjWji

Wij

or, by (2.18),fi

1− fieεi/kBT =

fj1− fj

eεj/kBT .

This relation holds for any state i and any state j. Therefore, both sides areindependent of i and j :

fi1− fi

eεi/kBT = const. = eεF /kBT .

This defines the number εF . Solving the above equation for fi yields

fi =1

1 + e(εi−εF )/kBT.

Hence, (2.17) is motivated.At zero temperature, we expect that all states below a certain energy are

occupied and all states above that energy are empty. Indeed, as T → 0, weobtain from (2.17)

f(ε)→

1 : ε < εF0 : ε > εF

and f(εF ) = 1/2. This provides a physical interpretation of the Fermi energy εFfor any temperature: For energies below the Fermi level, the states are rather

23

occupied (f(ε) > 1/2 for ε < εF ), and for energies above the Fermi level, thestates are rather empty (f(ε) < 1/2 for ε > εF ).

For energies which are much larger than the Fermi energy in the scale ofkBT, i.e. ε− εF À kBT, we can approximate the Fermi-Dirac distribution by theMaxwell-Boltzmann distribution

f(ε) = e−(ε−εF )/kBT

since 1+exp((ε− εF )/kBT ) ∼ exp((ε− εF )/kBT ) for ε− εF À kBT. We call thisthe non-degenerate case. A semiconductor in which Fermi-Dirac statistics haveto be used is called degenerate.

Formulas for the number densities. We use the Fermi-Dirac distribution(2.17) in the formula (2.16) for the electron density

n =

∫ ∞

−∞

gc(ε) dε

1 + e(ε−εF )/kBT.

Similar expressions hold for the hole density p for energies below the valence bandminimum εv:

p =

∫ ∞

−∞

gv(ε)

(1− 1

1 + e(ε−εF )/kBT

)dε

=

∫ ∞

−∞

gv(ε) dε

1 + e(εF−ε)/kBT.

The electron and hole densities can be computed more explicitely in theparabolic band approximation with isotropic effective mass:

n =1

2π2

(2m∗

e

~2

)3/2 ∫ ∞

εc

√ε− εc dε

1 + e(ε−εF )/kBT= NcF1/2

(εF − εckBT

),

p =1

2π2

(2m∗

h

~2

)3/2 ∫ εv

−∞

√εv − ε dε

1 + e(εF−ε)/kBT= NvF1/2

(εv − εFkBT

),

where we have introduced the effective densities of states

Nc = 2

(m∗

ekBT

2π~2

)3/2

, Nv = 2

(m∗

hkBT

2π~2

)3/2

and the Fermi integral

F1/2(y) =2√π

∫ ∞

0

√x dx

1 + ex−y, y ∈ R.

In the non-degenerate case εc− εF À kBT and εF − εv À kBT we can replacethe Fermi integral by a simpler function:

F1/2(y) ∼ exp(y) as y → −∞.

24

Then the particle densities become

n = Nc exp

(εF − εckBT

), p = Nv exp

(εv − εFkBT

). (2.19)

Intrinsic semiconductors. A pure semiconductor crystal with no impuritiesis called an intrinsic semiconductor. In this case, the conduction band electronscan only have come from formerly occupied valence band levels leaving holesbehind them. The number of conduction band electrons is therefore equal to thenumber of valence band holes:

n = p =: ni.

The intrinsic density ni can be computed in the non-degenerate case from (2.19):

ni =√np =

√NcNv exp

(εv − εckBT

)=√NcNv exp

(− εg2kBT

)(2.20)

since the energy gap is εg = εc−εv. The Fermi energy of an intrinsic semiconductorcan be computed using (2.19) and (2.20):

εF = εc + kBT ln(n/Nc)

= εc + kBT ln(ni/Nc)

= εc −1

2εg +

1

2kBT ln

Nv

Nc

=1

2(εc + εv) +

3

4kBT ln

m∗h

m∗e

.

This asserts that as T → 0, the Fermi energy lies precisely in the middle of theenergy gap. Furthermore, since ln(m∗

h/m∗e) is of order one, the correction is only

of order kBT. In most semiconductors and at room temperature, the energy gapεg is much larger than kBT (silicon: εg = 1.12 eV, kBT = 0.0259 eV at roomtemperature). This shows that the non-degeneracy assumptions

ε− εF ≥ εc − εF =1

2εg +

3

4kBT ln

m∗h

m∗e

À kBT,

εF − ε ≥ εF − εv =1

2εg +

3

4kBT ln

m∗h

m∗e

À kBT

are satisfied and that the result is consistent with the assumptions (see Figure2.6).

Doping. The intrinsic density is too small to result in a significant con-ductivity (for instance, silicon: ni = 6.93 · 109 cm−3). However, it is possibleto replace some atoms in a semiconductor crystal by atoms which provide freeelectrons in the conduction band or free holes in the valence band. This pro-cess is called doping. Impurities that contribute to the carrier density are called

25

-

6

k

εv

εc

ε

εFεv +

12εg

6

?εg À kBT

Figure 2.6: Illustration of the energy gap εg in relation to εc, εv and εF .

donors if they supply additional electrons to the conduction band, and acceptorsif they supply additional holes to (i.e., capture electrons from) the valence band.A semiconductor which is doped with donors is termed n-type semiconductor,and a semiconductor doped with acceptors is called p-type semiconductor. Forinstance, when we dope a germanium crystal, whose atoms have each 4 valenceelectrons, with arsenic, whose atoms have each 5 valence electrons, each arsenicimpurity provides one additional electron. These additional electrons are onlyweakly binded to the arsenic atom. Indeed, the binding energy is about 0.013 eVand thermal excitation provides enough energy to excite the additional electronsto the conduction band. Generally speaking, let εd and εa be the energy level ofa donor electron and an acceptor hole, respectively. Then εc− εd and εa− εv aresmall compared to kBT (Figure 2.7). This means that the additional particlescontribute at room temperature to the electron and hole density.

-

6

k

εv

εc

ε

εa

εd 6

?

εg

Figure 2.7: Illustration of the donor and acceptor energy levels εd and εa.

26

3 Classical Kinetic Transport Models

3.1 The Liouville equation

We first analyze the motion of an ensemble ofM electrons in a vacuum under theaction of an electric field E. The electrons will be discribed as classical particles,i.e., we associate the position vector xi ∈ Rd and the velocity vector vi ∈ Rd

with the i-th particle of the ensemble. The space dimension is usually three, butalso one- or two-dimensional models can be considered. Since the electrons haveequal mass m, the trajectories (xi(t), vi(t)) of the ensemble satisfy the systemof ordinary differential equations in the (2d ·M)-dimensional ensemble position-velocity space

∂txi = vi, ∂tvi =Fi

m, t > 0, i = 1, . . . ,M, (3.1)

where Fi = Fi(x, v, t) are forces and x = (x1, . . . , xM), v = (v1, . . . , vM). We setF = (F1, . . . , FM). The initial conditions are given by

x(0) = x0, v(0) = v0. (3.2)

The system (3.1)-(3.2) constitutes an initial-value problem for the trajectoryw(t;x0, v0) = (x(t), v(t)). For instance, the force can be given by an electricfield acting on the electron ensemble:

Fi = −qE(x, t).

In this case the forces are independent of the velocity.In semiconductor applications, M is a very large number (typically, M ∼

105) and therefore, the (numerical) solution of (3.1) is very expensive or evennot feasible. It seems reasonable to use a statistical description. We assumethat instead of the precise initial position x0 and initial velocity v0 we are giventhe joint probability density fI(x, v) of the initial position and velocity of theelectrons. This density has the properties

fI(x, v) ≥ 0 for x, v ∈ RdM ,

∫

RdM

∫

RdMfI(x, v) dx dv = 1. (3.3)

Then ∫∫

B

fI(x, v) dx dv

is the probability to find the particle ensemble in the subset B of the (x, v)-spaceat time t = 0.

Let f(x, v, t) be the probability density of the electron ensemble at time t.We wish to derive a differential equation for f. For this we use the Lionville

27

theorem [16]. It states that the function f does not change along the trajectoriesw(t;x0, v0) = (x(t), v(t)), i.e.

f(x(t), v(t), t) = f(w(t;x0, v0), t) = fI(x0, v0) (3.4)

for all x0, v0 ∈ RdM and t ≥ 0. Differentiating this equation with respect to timegives

0 =d

dtf(x(t), v(t), t)

= ∂tf + ∂tx · ∇xf + ∂tv · ∇vf

= ∂tf + v · ∇xf +1

mF · ∇vf. (3.5)

This equation is referred to as the classical Liouville equation for an electronensemble. When the force is given by the electric field,

F = −qE = q∇xV (x, t),

where V is the electrostatic potential, the Liouville equation reads

∂tf + v · ∇xf +q

m∇xV · ∇vf = 0, x, v ∈ RdM , t > 0, (3.6)

with initial condition

f(x, v, 0) = fI(x, v), x, v ∈ RdM . (3.7)

This equation governs the evolution of the position-velocity probability densityf(x, v, t) of an electron ensemble in the electric field −q∇xV under the assump-tions that

• the electrons move according to the laws of classical mechanics and

• the electrons move in a vacuum.

In the following we describe some properities of the solution of (3.6)-(3.7).We assume that fI satisfies (3.3).

• Non-negativity: From (3.4) we conclude that

f(x, v, t) ≥ 0 ∀x, v ∈ RdM

and for all t ≥ 0 for which a solution exists.

28

• Conservation property: We integrate (3.6) over RdM × RdM and use thedivergence theorem:

∂t

∫

RdM

∫

RdMf(x, v, t) dx dv

= −∫

RdM

∫

RdM

(div x(vf) +

q

mdiv v(f∇xV )

)dx dv

= 0.

By (3.3), we conclude for all t ≥ 0∫

RdM

∫

RdMf(x, v, t) dx dv =

∫

RdM

∫

RdMfI(x, v) dx dv = 1. (3.8)

• Moments: We define the zeroth- and first-order moments

n(x, t) =

∫

RdMf(x, v, t) dv,

J(x, t) = −q∫

RdMvf(x, v, t) dv,

where q is the elementary charge. The functions n and J are interpretedas the position probability density (or particle density) and as the parti-cle current density, respectively. The conservation property (3.8) can berestated as ∫

RdMn(x, t) dx =

∫

RdMnI(x) dx, t ≥ 0, (3.9)

with the initial electron density

nI(x) =

∫

RdMfI(x, v) dv.

This means that the total number of electrons is conserved in time. Byformally integrating (3.6) over v ∈ RdM we obtain the so-called particlecontinuity equation

∂tn−1

qdiv xJ = 0, x ∈ RdM , t > 0. (3.10)

We consider an electron ensemble moving in a semiconductor crystal. Asexplained in Chapter 2, the ions in the crystal lattice induce a lattice-periodicpotential which influences the motion of the charged particles. Let k 7→ εn(k) bethe n-th energy band of the crystal for wave vectors k of the Brillouin zone B.The corresponding mean velocity is given by (see Section 2.2)

vn(k) =1

~∇kεn(k).

29

Let ki ∈ Rd be the wave vector and xi ∈ Rd the position vector of the i-th electronand set k = (k1, . . . , kM) ∈ BM , x = (x1, . . . , xM) ∈ RdM .We fix the energy bandand omit therefore the index n. The motion of the electrons in the (x, k)-spacecan be described semi-classically by the equations

∂tx = v(k) =1

~∇kε(k), ~∂tk = F,

where v(k) = (v(k1), . . . , v(kM)), and F = (F1, . . . , FM) are some driving forces.Notice that band transitions are excluded in this formulation since the band in-dex is fixed. Notice also that the equation ~∂tk = F is an expression of Newton’slaw since p = ~k is the crystal momentum vector of the electrons. An analo-gous computation as in (3.5) yields the semi-classical Liouville equation for thedistribution function f(x, k, t) of an electron ensemble:

∂tf + v(k) · ∇xf +1

~F · ∇kf = 0, x ∈ RdM , k ∈ BM , t > 0, (3.11)

where

v(k) · ∇xf =M∑

j=1

v(kj) · ∇xjf.

As the Brillouin zone is a bounded subset, we have to impose boundary conditionsfor k. We choose the periodic boundary conditions

f(x, k1, . . . , kj, . . . , kM , t) = f(x, k1, . . . ,−kj, . . . , kM , t), kj ∈ ∂B,

for all j = 1, . . . ,M. This formulation makes sense since B is point symmetric tothe origin, i.e. k ∈ B if and only if −k ∈ B.

The macroscopic particle and current densities are now defined by

n(x, t) =

∫

BMf(x, k, t) dk,

J(x, t) = −q∫

BMv(k) f(x, k, t) dk.

The periodicity of f in ki and the point-symmetry of B imply the conservationproperty (3.9) and the conservation law (3.10). Here we also need the assumptiondiv kF = 0 such that terms arising in the integration by parts cancel. Thisassumption is satisfied if F is given by an electric field

F = −qE(x, t)

or by an electro-magnetic field

F = −q(E(x, t) + v ×Bind(x, t)),

30

where Bind is the induction vector.When the parabolic band approximation (see Section 2.2)

ε(k) =~2

2m∗|k|2, k ∈ RdM ,

for electrons is used, then v(k) = ~k/m and ∇kf = (~/m∗)∇vf , and the semi-classical Liouville (3.11) reduces to its classical counterpart (3.6).

3.2 The Vlasov equation

The main disadvantage of the (semi-)classical Liouville equation is that it has tobe solved in a very high-dimensional phase space: typically, M ∼ 105, d = 3,and the dimension of the (x, k)-space is 3 · 105! In the following we will formallyderive a low-dimensional equation, the so-called Vlasov equation. The idea of thederivation is first to assume a certain structure of the interaction (force) field,then to integrate the Liouville equation in a certain sub-phase space and finallyto carry out the formal limit “M →∞”, where M is the number of particles.

More precisely, we consider an ensemble of M electrons with equal mass anddenote by x = (x1, . . . , xM) ∈ RdM , v = (v1, . . . , vM) ∈ RdM the position andvelocity coordinates of the electrons, respectively. We impose the following as-sumptions:

• Assumption 1: The electrons move in a vacuum.

• Assumption 2: The force field F only depends on position and time.

• Assumption 3: The motion is governed by an external electric field andby two-particle interaction forces.

The first assumption will be discarded later on. The second assumption meansthat magnetic fields are ignored (see [34, Ch. 1.3] for an inclusion of magneticeffects). The last assumption is crucial for the derivation of the Vlasov equation.It means that the force field Fi exerted on the i-th electron is given by the sum ofan electric field acting on the i-th electron and of the sum of M − 1 two-particleinteraction forces exerted on the i-th electron by the other electrons:

Fi(x, t) = −qEext(xi, t)− qM∑

j=1, j 6=i

Eint(xi, xj), i = 1, . . . ,M. (3.12)

The interaction force Eint is independent of the electron indices, which interpretsthe fact that the electrons are indistinguishable. We assume that the force exertedby the i-th electron on the j-th electron is equal to the negative force exerted bythe j-th electron on the i-th electron:

Eint(xi, xj) = −Eint(xj, xi) ∀xi, xj ∈ Rd, (3.13)

31

which implies Eint(x, x) = 0.The classical Liouville equation for the density f(x, v, t) of the ensemble reads:

∂tf +M∑

i=1

vi · ∇xif −q

m

M∑

i=1

Eext(xi, t) · ∇vif

− q

m

M∑

i,j=1

Eint(xi, xj) · ∇vif = 0. (3.14)

We assume that the initial density is independent of the numbering of the particles(Assumption 4):

fI(x1, . . . , xM , v1, . . . , vM) = fI(xπ(1), . . . , xπ(M), vπ(1), . . . , vπ(M)) (3.15)

for all xi, vi ∈ Rd, i = 1, . . . ,M, and for all permutations π of 1, . . . ,M. Thenthe property (3.13) implies that also f(x, v, t) is independent of the numberingof the particles for all t > 0. We introduce the density f (a) of a subensembleconsisting of a < M electrons:

f (a)(x1, . . . , xa, v1, . . . , va, t) =

∫

R2d(M−a)

f(x, v, t) dxa+1 · · · dxM dva+1 · · · dvM .

We integrate the above Liouville equation with respect to xa+1, . . . , xM , va+1, . . . ,vM in order to obtain an equation for f (a). By the divergence theorem,

M∑

i=1

∫

R2d(M−a)

vi · ∇xif dxa+1 · · · dxM dva+1 · · · dvM

=a∑

i=1

vi · ∇xi

∫

R2d(M−a)

f dxa+1 · · · dxM dva+1 · · · dvM

+M∑

i=a+1

∫

R2d(M−a)

div xi(vif) dxa+1 · · · dxM dva+1 · · · dvM

=a∑

i=1

vi · ∇xif(a).

Similarly,

− q

m

M∑

i=1

∫

R2d(M−a)

div vi(Eext(xi, t)f) dxa+1 · · · dxM dva+1 · · · dvM

= − q

m

a∑

i=1

div vi(Eext(xi, t)f(a)).

32

The last integral becomes

− q

m

M∑

i=1

∫

R2d(M−a)

div vi(Eint(xi, xj)f) dxa+1 · · · dxM dva+1 · · · dvM

= − q

m

a∑

i,j=1

Eint(xi, xj) · ∇vif(a)

− q

m

M∑

i=a+1

M∑

j=1

∫

R2d(M−a)

div vi(Eint(xi, xj)f) dxa+1 · · · dxM dva+1 · · · dvM

− q

m

a∑

i=1

M∑

j=a+1

∫

R2d(M−a)

div vi(Eint(xi, xj)f) dxa+1 · · · dxM dva+1 · · · dvM .

The second integral vanishes by the divergence theorem. The last integral is equalto

− q

m

a∑

i=1

(M − a)div vi

∫

R2d

Eint(xi, xa+1)

(∫

R2d(M−a)

f dxa+2 · · · dxM dva+2 · · · dvM)dxa+1 dva+1

= − q

m

a∑

i=1

(M − a)div vi

∫

R2d

Eint(xi, x∗)

×f (a+1)(x1, . . . , xa, x∗, v1, . . . , vd, v∗, t) dx∗ dv∗,

using the indepency of f on the numbering of the particles.We illustrate the last argument by the example M = 3, a = 1:

3∑

j=2

∫

R4d

div v1(Eint(x1, xj)f(x, v, t)) dx2 dx3 dv2 dv3

=

∫

R4d

div v1(Eint(x1, x2)f(x1, x2, x3, v1, v2, v3, t)) dx2 dx3 dv2 dv3

+

∫

R4d

div v1(Eint(x1, x3)f(x1, x2, x3, v1, v3, v2, t)) dx2 dx3 dv2 dv3

= 2div v1

∫

R2d

Eint(x1, x2)

×(∫

R2

f(x1, x2, x3, v1, v2, v3, t) dx3 dv3

)dx2 dv2

= 2div v1

∫

R2

Eint(x1, x∗)f(2)(x1, x∗, v1, v∗, t) dx∗dv∗.

33

We summarize the above computations and obtain from (3.14)

∂tf(a) +

a∑

i=1

vi · ∇xif(a) − q

m

a∑

i=1

Eext(xi, t) · ∇vif(a)

− q

m

a∑

i,j=1

Eint(xi, xj) · ∇vif(a) (3.16)

− q

m(M − a)

a∑

i=1

div vi

∫

R2d

Eint(xi, x∗)f(a+1)∗ dx∗ dv∗ = 0,

wheref (a+1)∗ = f (a+1)(x1, . . . , xa, x∗, v1, . . . , va, v∗, t).

These equations for 1 ≤ a ≤ M − 1 are called the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We wish to perform the formal limit “M →∞”. For this, we assume that |Eint| is of order 1/M (Assumption 5) such that|Fi| is of order one as M → ∞. Then the third term in (3.16) vanishes in theformal limit, whereas the expression (M−a)Eint in the last term on the left-handside of (3.16) remains finite. For M À 1, we can substitute this expression byMEint. Thus, for M À 1, we obtain from (3.16), neglecting terms of order 1/M,

∂tf(a) +

a∑

i=1

vi · ∇xif(a) − q

m

a∑

i=1

Eext(xi, t) · ∇vif(a)

− q

m

a∑

i=1

div vi

∫

R2d

M f (a+1)∗ Eint(xi, x∗) dx∗ dv∗ = 0. (3.17)

In order to solve this equation we make the ansatz

f (a)(x1, . . . , xa, v1, . . . , va, t) =a∏

i=1

P (xi, vi, t). (3.18)

The interpretation of this factorization is that we assume that the electrons ofthe subensemble move independently of each other. Using this ansatz in (3.17)for a = 1 gives

∂tP + v1 · ∇x1P −q

mEeff(x1, t) · ∇v1P = 0,

where

Eeff(x, t) = Eext(x, t) +

∫

R2d

MP (x2, v2, t)Eint(x, x2) dx2 dv2.

It can be shown that (3.18) is indeed a particular solution of (3.16) if P satisfiesthe above equation. Clearly, we need to assume that the initial density f (a)(·, ·, 0)

34

admits such a factorization for all a ∈ N (Assumption 6). The whole informa-tion of the evolution of the electron subensemble is thus contained in the functionP. We define

F (x, v, t) = MP (x, v, t),

n(x, t) =

∫

RdF (x, v, t) dv, x, v ∈ Rd, t ≥ 0.

The quantity n(x, t) represents the density of the electrons per unit volume. Thenumber density F satisfies the so-called classical Vlasov equation

∂tF + v · ∇xF −q

mEeff · ∇vF = 0, x, v ∈ Rd, t > 0, (3.19)

Eeff(x, t) = Eext(x, t) +

∫

Rdn(x∗, t)Eint(x, x∗) dx∗. (3.20)

This equation has the form of a single-particle Liouville equation with the forceEeff . Many-particle physics enters only through the effective field Eeff , which inturn depends on the density n and hence on F. This means that (3.19) is anonlinear equation with a nonlocal nonlinearity of quadratic type. It provides amacroscopic description of the motion of many-particle systems with weak long-range forces. However, it does not account for strong short-range forces such asscattering of particles (see Section 3.3).

The Vlasov equation (3.19)-(3.20) is supplemented by the initial condition

F (x, v, 0) = FI(x, v), x, v ∈ Rd.

Since the solution F of (3.19)-(3.20) can be interpreted as the probability ofa particle to be in the state (x, v) at time t, we expect that

0 ≤ F (x, k, t) ≤ 1, x, v ∈ Rd, t > 0.

Indeed, assuming that 0 ≤ FI(x, k) ≤ 1, we obtain from the trajectory equations

∂tx = v, ∂tv = − q

mEeff , x(0) = x0, v(0) = v0,

the expression

0 = ∂tF + v · ∇xF −q

mEeff · ∇vF

=d

dtF (x(t), v(t), t)

and thereforeF (x(t), v(t), t) = FI(x0, v0) ∈ [0, 1].

35

Example 3.1 (Coulomb force in R3)The most important long-range force acting between two electrons is the Coulombforce:

Eint(x, y) = −q

4πεs

x− y|x− y|3 , x, y ∈ R3, x 6= y,

where the permittivity εs is a material constant. (The following considerationsare also valid in non-vacuum.) Then the effective field reads (see (3.20))

Eeff(x, t) = Eext(x, t)−q

4πεs

∫

R3

n(z, t)x− z|x− z|3 dz.

It is well known that the function

φ(x) =1

4π

∫

R3

f(z)

|x− z| dz, x ∈ R3,

satisfies the equation on ∆φ = f in R3 under some regularity assumptions on f.Therefore

f(x) =1

4π

∫

R3

f(z)∆x1

|x− z| dz =1

4π

∫

R3

f(z)div xx− z|x− z|3 dz

and

0 = curl x∇xφ(x) =1

4πcurl x

∫

R3

f(z)x− z|x− z|3 dz.

We conclude that

divEeff(x, t) = divEext(x, t)−q

εsn(x, t),

curlEeff(x, t) = curlEext(x, t), x ∈ R3, t > 0.

We assume that the external field is generated by ions of charge +q which arepresent in the material (for instance, doping atoms in the semiconductor crystal).Then, by Coulomb’s law

Eext(x, t) =+q

4πεs

∫

R3

C(z, t)x− z|x− z|3 dz,

where C is the number density of the background ions. Since

divEext =q

εsC, curlEext = 0,

we obtaindivEeff = − q

εs(n− C), curlEeff = 0 in R3.

Since Eeff is vortex-free, there is a potential Veff such that Eeff = −∇Veff . Thenwe can rewrite the above equation as

εs∆Veff = q(n− C), x ∈ R3. (3.21)

36

This equation which also holds in Rd, d ≥ 1, is called Poisson equation. TheVlasov equation (3.19) with the Coulomb interaction field, i.e. with Eeff = −∇Veffand (3.21), is referred to as Vlasov-Poisson system.

Instead of taking the classical Liouville equation (3.6) as basis for the deriva-tion of the Vlasov equation, we can also start from the semi-classical formulation(3.11). Under the assumptions (3.12) and (3.13) on the force fields, we obtain byproceeding as above the semi-classical Vlasov equation

∂tF + v(k) · ∇xF −q

~Eeff · ∇kF = 0, x ∈ Rd, k ∈ B, t > 0, (3.22)

Eeff(x, t) = Eext(x, t) +

∫

Rdn(x∗, t)Eint(x, x∗) dx∗ (3.23)

with the particle position density

n(x, t) =

∫

B

F (x, k, t) dk

and the initial condition

F (x, k, 0) = FI(x, k), x ∈ Rd, k ∈ B.

The number density is assumed to satisfy periodic boundary equations in k :

F (x, k, t) = F (x,−k, t), x ∈ Rd, k ∈ ∂B, t > 0. (3.24)

We notice that the precise meaning of the number density F is the following:F (x, k, t) is the ratio of the number of occupied states in the “volume” dx dk inthe conduction band and the total number of quantum states in this volume inthe conduction band. Then (see also Section 2.3)

0 ≤ F (x, k, t) ≤ 1, x ∈ Rd, k ∈ B, t > 0,

if 0 ≤ FI(x, k) ≤ 1, and F (x, k, t) is also called the occupation number of state kin x at time t.

3.3 The Boltzmann equation

The Vlasov equation accounts for long-range particle interactions, like the Cou-lomb force (see Example 3.1). Short-range interactions, like collisions of theparticles with other particles or with the crystal lattice, are not included. Wewish to extend the Vlasov equation to include collision mechanisms, which willlead to the Boltzmann equation. We present only a phenomenological derivation,formulated first by Boltzmann in 1872 for the description of non-equilibrium

37

phenomena in dilute gases. For details on a more rigorous derivation, we refer tothe literature in [9, Sec. 1.5.3] and [16].

The starting point of the derivation is to postulate that the rate of changeof the number density F (x, v, t) of the particle ensemble due to convection andthe effective field, dF/dt, and the rate of change of F due to collisions, Q(F ),balance:

dF

dt= Q(F ).

This equation has to be understood along the particle trajectories. Explicitely,it reads (in the classical case, see (3.19))

∂tF + v · ∇xF −q

mEeff · ∇vF = Q(F ), x, v ∈ Rd, t > 0, (3.25)

where the effective field Eeff is given by (3.20). In order to derive an expressionfor Q(F ) we assume that the rate P (x, v′ → v, t) of a particle at (x, t) to changeits velocity v′ into v due to a scattering event is proportional to

• the occupation probability F (x, v′, t) and

• the probability 1−F (x, v, t) that the state (x, v) is not occupied at time t.

ThusP (x, v′ → v, t) = s(x, v′, v)F (x, v′, t)(1− F (x, v, t)),

where s(x, v′, v) is the so-called scattering rate. (We made a similiar considerationin Section 2.3.) Then the rate of change of F due to collisions is the sum of

P (x, v′ → v, t)− P (x, v → v′, t)

for all velocities v′ in the volume element dv′. In the continuum limit, the sum isin fact an integral and we get

(Q(F ))(x, v, t) =

∫

Rd[P (x, v′ → v, t)− P (x, v → v′, t)] dv′ (3.26)

=

∫

Rd[s(x, v′, v)F ′(1− F )− s(x, v, v′)F (1− F ′)] dv′,

where F = F (x, v, t), F ′ = F (x, v′, t). The equation (3.25), together with theeffective field equation (3.20) and the definition of the collision operator (3.26),is called the classical Boltzmann equation. When the Coulomb force is usedto model the long-range interactions, then (3.25), (3.20), (3.26), and (3.21) aretermed Boltzmann-Poisson system.

The Boltzmann equation has two nonlinearities: a quadratic nonlocal non-linearity caused by the self-consistent field Eeff and another quadratic nonlocalnonlinearity caused by the collision integral (3.26). A rigorous mathematical

38

analysis of the initial-value problem (3.25)-(3.26) (existence and uniqueness ofsolutions) is very difficult. We refer to the research papers [23, 36] and to thereview paper [4] for details on the mathematical analysis of this problem.

Starting from the semi-classical Vlasov equation (3.22) we can include thecollision effects similiarly as above and obtain the semi-classical Boltzmann equa-tion

∂tF + v(k) · ∇xF −q

~Eeff · ∇kF = Q(F ), x ∈ Rd, k ∈ B, t > 0, (3.27)

where the effective field Eeff is defined in (3.23). This equation models the semi-classical motion of electrons incorporating the quantum effects of the semicon-ductor crystal lattice as explained in Section 2.2. The Boltzmann equation (3.27)is supplemented by the initial condition

F (x, k, 0) = FI(x, k), x ∈ Rd, k ∈ B, (3.28)

and by the periodic boundary conditions (3.24).In semiconductor crystals, there are three main classes of scattering events:

• electron-phonon scattering,

• ionized impurity scattering, and

• electron-electron scattering.

We explain these collision mechanisms now in detail.At finite temperature, the atoms in a crystal lattice undergo variations about

their fixed equilibrium positions. These lattice vibrations are quantized and thequantum of lattice vibrations are called phonons. We can distinguish so-calledacoustic phonons and optical phonons. These names refer to the vibration modelsof the lattice (see [14, Ch. 9.2] or [28, Ch. 7]). The collision operator for electron-phonon interactions is given by

(Qα(F ))(x, k, t) =

∫

B

[sα(x, k′, k)F ′(1− F )− sα(x, k, k′)F (1− F ′)] dk′,

where F = F (x, k, t), F ′(F (x, k′, t), and the scattering rate is

sα(x, k, k′) = σα(x, k, k

′)[(1 +Nα)δ(ε(k′)− ε(k) + ~ωα)

+Nαδ(ε(k′)− ε(k)− ~ωα)],

the function σα(x, k, k′) is symmetric in k and k′, ~ωα is the (constant) phonon

energy, Nα is the phonon occupation number given by Bose-Einstein statistics

Nα =

(exp

(~ωα

kBT

)− 1

)−1

,

39

the index α refers to either “opt” for optical phonons or “ac” to acoustic phonons,and δ is the delta distribution used in Section 2.3. The above transition rate isnon-zero only if ε(k′) − ε(k) = ±~ωα, i.e., the rate accounts for phonon emis-sion and absorption of energy ~ωα. The above expression for sα shows that thescattering rates can be highly non-smooth; in fact, Qα is a distribution.

When an atom different from the semiconductor material is introduced intothe semiconductor it may donate either an electron or a hole, leaving behind anionized charged impurity center. These ionized impurities can act as scatteringagents for free carriers propagating through the crystal. The interaction of car-riers with neutral impurities is also possible, but we do not consider it here. Thecollision operator reads

(Qimp(F ))(x, k, t) =

∫

B

σimp(x, k, k′) δ(ε(k′)− ε(k)) (F ′ − F ) dk′,

where F = F (x, k, t), F ′ = F (x, k′, t) and σimp is symmetric in k and k′ [11].The electron-electron interaction arises from the long-range nature of the

Coulomb force. The corresponding collision operator

(Qee(F ))(x, k, t) =

∫

B3

[see(x, k, k1, k′, k′1)F

′F ′1(1− F )(1− F1)

− see(x, k′, k′1, k, k1)FF1(1− F ′)(1− F ′1)] dk

′ dk1 dk′1,

where F = F (x, k, t), F ′ = F (x, k′, t), F1 = F (x, k1, t), and F′1 = F (x, k′1, t), is a

nonlocal nonlinearity of fourth order. The influence of electron-electron interac-tion on the carrier dynamics is more pronounced in degenerate semiconductors inwhich Fermi-Dirac statistics instead of Maxwell-Boltzmann statistics has to beused (see Section 2.3). Also electron-hole interactions are possible but we do notconsider these collision events here. For more details of scattering processes, werefer to [41, Ch. 6] or [14, Ch. 9].

The collision operator Q(F ) in (3.27) can generally be written as

Q(F ) = Qopt(F ) +Qac(F ) +Qimp(F ) +Qee(F ).

In the following we assume that the collision operator is given by

(Q(F ))(x, k, t) =

∫

B

[s(x, k′, k)F ′(1− F )− s(x, k, k′)F (1− F ′)] dk′, (3.29)

where we have again set F = F (x, k, t) and F ′ = F (x, k′, t).We study now some properties of the collision integral (3.29). Since s(x, k, k ′)

is the transition rate from state k to state k′, the principle of detailed balanceimplies that

s(x, k, k′)

s(x, k′, k)= e(ε(k)−ε(k′))/kBT ∀x ∈ Rd, k, k′ ∈ B, (3.30)

holds (compare with (2.18)).

40

Lemma 3.2 (see [36]) Let (3.30) hold for some function ε(k) and let s(x, k, k ′) >0 for all x ∈ Rd, k, k′ ∈ B.

(1) It holds ∫

B

(Q(F ))(x, k, t) dk = 0 ∀x ∈ Rd, t > 0,

for all (regular) functions F : R3 ×B × [0,∞)→ [0, 1].

(2) It holds for all functions F : R3×B× [0,∞)→ [0, 1] and all non-decreasingfunctions χ : R→ R

∫

B

(Q(F ))(x, k, t)χ

(F (x, k, t)

1− F (x, k, t)eε(k)/kBT

)dk ≤ 0,

∫

B

(Q(F ))(x, k, t)χ

(1− F (x, k, t)F (x, k, t)

e−ε(k)/kBT

)dk ≥ 0.

(3) The property Q(F ) = 0 is equivalent to the existence of −∞ ≤ εF ≤ +∞such that

F (k) =1

1 + exp((ε(k)− εF )/kBT ), k ∈ B.

Let F be a solution to the Boltzmann equation (3.27), (3.28). Then Lemma3.2(1) implies that the total number of electrons is conserved in time:

n(x, t) =

∫

B

F (x, k, t) dk =

∫

B

FI(x, k) dk = nI(x, t), x ∈ Rd, t > 0.

Physically, this is reasonable: collisions neither destroy nor generate particles.The statements of Lemma 3.2(2) are also called H-theorems. Finally, Lemma3.2(3) means that precisely the Fermi-Dirac distribution functions are in thekernel of the collision operator.

Proof of Lemma 3.2. (1) Changing k and k′ in the second integral gives

∫

B

Q(F ) dk =

∫

B2

s(x, k′, k)F ′(1− F ) dk′ dk −∫

B2

s(x, k, k′)F (1− F ′) dk′ dk

=

∫

B2

s(x, k′, k)F ′(1− F ) dk′ dk −∫

B2

s(x, k′, k)F ′(1− F ) dk dk′

= 0,

where F = F (x, k, t) and F ′ = F (x, k′, t).(2) We only show the second inequality. The proof of the first one is similar.

We set

M(k) = e−ε(k)/kBT , h(k) =1− F (k)F (k)

M(k).

41

Then (3.30) is equivalent to

s(x, k′, k)

M(k)=s(x, k, k′)

M(k′), (3.31)

and we obtain with the notations h = h(k), h′ = h(k′), M =M(k), M ′ =M(k′),

∫

B

Q(F )χ(h) dk

=

∫

B2

s(x, k, k′)

[M

M ′F ′(1− F )χ(h)− F (1− F ′)χ(h)

]dk′ dk

=

∫

B2

s(x, k, k′)

M ′FF ′

[1− FF

Mχ(h)− 1− F ′

F ′M ′χ(h)

]dk′ dk

=

∫

B2

s(x, k, k′)

M ′FF ′(h− h′)χ(h) dk′ dk, (3.32)

and, by changing k and k′ and using (3.31),

∫

B

Q(F )χ(h) dk

=

∫

B2

s(x, k′, k)

MF ′F

[1− F ′

F ′M ′χ(h′)− 1− F

FMχ(h′)

]dk dk′

=

∫

B2

s(x, k, k′)

M ′F ′F

[1− F ′

F ′M ′χ(h′)− 1− F

FMχ(h′)

]dk′ dk

=

∫

B2

s(x, k, k′)

M ′F ′F (h′ − h)χ(h′) dk′ dk. (3.33)

Adding (3.32) and (3.33) yields

∫

B

Q(F )χ(h) dk =1

2

∫

B

s(x, k, k′)

M ′FF ′(h− h′)(χ(h)− χ(h′)) dk dk′

≥ 0,

since s(x, k, k′), M ′, F , and F ′ are non-negative and χ is non-decreasing.(3) By (2), with χ(x) = x, the property Q(F ) = 0 is equivalent to

FF ′(h− h′)2 = 0 for almost all k, k′ ∈ B.

This implies F = 0 or h = h′ almost everywhere. The equation h = h′ isequivalent to

1− F (k)F (k)

M(k) =1− F (k′)F (k′)

M(k′) for almost all k, k′ ∈ B.

42

Thus, both sides are constant and we call this constant exp(−εF/kBT ) withεF ∈ R. Notice that the constant must be positive since 0 ≤ F ≤ 1 expect ifF ≡ 1. But then

1− F (k)F (k)

=e−εF /kBT

M(k)= e(ε(k)−εF )/kBT

and F (k) = 1 + exp((ε(k) − εF )/kBT ))−1. Choosing εF = ±∞ yields the othertwo possibilities F ≡ 0 or F ≡ 1. ¤

In the literature, two approximations of the collision operator are frequentlyused:

• the low-density approximation:

(Q(F ))(x, k, t) =

∫

B

σ(x, k′, k)(MF ′ −M ′F ) dk′, (3.34)

where σ(x, k′, k) = s(x, k, k′)/M(k′) is called collision cross-section.

• the relaxation-time approximation:

(Q(F ))(x, k, t) = −F (x, k, t)−M(k)n(x, t)

τ(x, k), (3.35)

where

τ(x, k) =

(∫

B

s(x, k, k′) dk′)−1

is called the relaxation time describing the average time between two con-secutive scattering events at (x, k).

These approximations can be derived as follows. The low-density collisionoperator is obtained by assuming that the distribution function F is small:

0 ≤ F (x, k, t)¿ 1.

Then 1− F (k) is approximated by one in (3.28) and, employing (3.30),

(Q(F ))(x, k, t) =

∫

B

[s(x, k′, k)F ′ − s(x, k, k′)F ] dk′

=

∫

B

s(x, k′, k)

M(k)[M(k)F ′ −M(k′)F ] dk′

=

∫

B

σ(s, k′, k)(MF ′ −M ′F ) dk′,

where M(k) = exp(−ε(k)/kBT ) and M ′ = M(k′). Notice that by (3.30), thecollision cross-section σ(x, k′, k) is symmetric in k and k′.

43

When the initial datum FI is close to a multiple of the so-called MaxwellianM(k) it is reasonable to approximate F ′ in (3.34) by n(x, t)M(k′). Then we obtainfrom (3.34)

(Q(F ))(x, k, t) =

∫

B

σ(x, k′, k)(nMM ′ −M ′F ) dk′

=

∫

B

s(x, k, k′)(nM − F ) dk′

=

∫

B

s(x, k, k′) dk′ · (n(x, t)M(k)− F (x, k, t))

= −F (x, k, t)− n(x, t)M(k)

τ(x, k).

Along the trajectories (x(t), k(t)), where

∂tx = v(k), ~∂tk = −qEeff , x(0) = x0, k(0) = k0,

the Boltzmann equation with the relaxation-time approximation for constant τreads

dF

dt= −1

τ(F − nM), t > 0.

The solution of this linear ordinary differential equation (for given n(x, t)) is

F (x(t), k(t), t) = e−t/τFI(x0, k0) +1

τ

∫ t

0

n(x(s), s)M(k(s))e(s−t)/τ ds.

It is possible to show that

et/τ [F (x(t), k(t), t)− n(x(t), t)M(k(t))]→ 1 as t→∞,

i.e., the function relaxes to the equilibrium density nM from the perturbed stateFI along the trajectories after a time of order τ. This explains the name relaxationtime.

A summary of the main models derived in Section 3.1-3.3 and its relationsare presented in Figure 3.1.

3.4 Extensions

We discuss two extensions of the Boltzmann equation:

• multi-valley models and

• bipolar models.

44

semi-classical

Liouville

semi-classical

Vlasov¾no two-particle

interactions

semi-classical

Boltzmann¾ no collisions

6parabolicband

classical

Liouville

6parabolicband

classical

Vlasov¾ no two-particle

interactions

6parabolicband

classical

Boltzmann¾ no collisions

Figure 3.1: Relations between the (semi-)classical Liouville, Vlasov and Boltz-mann equations.

6

-

L Γ X

ε(k)

|k|

energygap

6

?valenceband

conductionband

Figure 3.2: Schematic band structure of GaAs.

In the semiconductor material GaAs, the (conduction band) energy ε(k) hasone maximum and several minima which are called energy valleys. More precisely,GaAs has three valleys: the low-energy Γ-valley and the higher energetic L- andX-valleys; see Figure 3.2 [28, Ch. 4]. As the highest energy X-valley can beoccupied only at very high electric field strengths, we will consider only the Γ-and L-valleys. We assume furthermore the parabolic band approximation. Then,after a suitable translation and rotation of the three-dimensional k-space,

εΓ(k) = ∆Γ −~2

2

(k2xm∗

Γx

+k2ym∗

Γy

+k2zm∗

Γz

),

εL(k) = ∆L +~2

2

(k2xm∗

Lx

+k2ym∗

Ly

+k2zm∗

Lz

), k = (kx, ky, kz)

>,

where m∗Γα, m∗

Lαare the effective masses of an electron in the Γ-valley, L-valley,

respectively, in the different directions. Since we assume low field strengths, wesplit the distribution function F into a part FΓ corresponding to the Γ-valley and

45

a part FL corresponding to the L-valley. Indeed, the electrons move only withineach valley since the transfer to a higher-energetic valley requires the presence ofhigh electric fields. Thus, for a single L-valley,

F (x, k, t) = FΓ(x, k, t) + FL(x, k, t),

and the functions FΓ and FL are satisfying the semi-classical Boltzmann equations

∂tFΓ + vΓ(k) · ∇xFΓ −q

~Eeff · ∇kFΓ = QΓ(FΓ) +QΓ→L(FΓ, FL),

∂tFL + vL(k) · ∇xFL −q

~Eeff · ∇kFL = QL(FL) +QL→Γ(FΓ, FL),

where

vΓ(k) =1

~∇kεΓ(k), vL(k) =

1

~∇kεL(k)

denote the mean velocities of the electrons in the Γ- and L-valleys, respectively(see Section 2.2), QΓ(FΓ) and QL(FL) are the intravalley low-density collisionoperators

Qα(Fα) =

∫

R3

σα(x, k′, k)(Mα(k)F

′α −Mα(k

′)Fα) dk′, α = Γ, L,

QΓ→L and QL→Γ are the intervalley low-density collision operators

QΓ→L(FΓ, FL) =

∫

R3

(σL→Γ(x, k′, k)MΓ(k)F

′L − σΓ→L(x, k, k

′)ML(k′)FΓ) dk

′

QL→Γ(FΓ, FL) =

∫

R3

(σΓ→L(x, k′, k)ML(k)F

′Γ − σL→Γ(x, k, k

′)MΓ(k′)FL) dk

′

sΓ→L(x, k, k′) is the cross-section from the state (x, k) of the Γ-valley into a state

(x, k′) of the L-valley, similarly for sL→Γ(x, k, k′), and the Maxwellians MΓ and

ML are given by

Mα(k) = N ∗α exp

(−εα(k)kBT

), N∗

α =

(∫

R3

exp

(−εα(k)kBT

)dk

)−1

, α = Γ, L.

The effective field Eeff is computed from (3.23), i.e.

Eeff(x, t) = Eext(x, t) +

∫

R3

(nΓ(x∗, t) + nL(x∗, t))Eint(x, x∗) dx∗,

nα(x, t) =

∫

R3

Fα(x, k, t) dk, α = Γ, L.

It can be shown similarly as in Lemma 3.2 that under the assumption

σΓ→L(x, k, k′) = σL→Γ(x, k

′, k) ∀x, k, k′ ∈ Rd,

46

the property QΓ→L(FΓ, FL) = QL→Γ(FΓ, FL) = 0 is equivalent to

(FΓ, FL) = const.(MΓ,ML).

So far we have only considered the transport of electrons in the conductionband. However, also holes in the valence band contribute to the carrier flowin semiconductors (see Section 2.2). It is possible that an electron moves fromthe valence band to the conduction band, leaving a hole in the valence bandbehind it. For such a process, which is termed generation of an electron-holepair, energy absorption is necessary. The inverse process that an electron of theconduction band moves to the lower energetic valence band, occupying an emptystate, is called recombination of an electron-hole pair. For such an event, energyis emitted. The basic mechanisms for generation-recombination processes are

• Auger/impact ionization generation-recombination,

• radiative generation-recombination, and

• thermal generation-recombination.

An Auger process is defined as an electron-hole recombination followed bya transfer of energy to a free carrier which is then excited to a higher energystate. The inverse Auger process, in which an electron-hole pair is generated, iscalled impact ionization. The energy comes from the collision of a high-energyfree carrier with the lattice.

In a radiative recombination event, an electron from the conduction bandrecombines with a hole from the valence band emitting a photon. The energylost by the electron is equal to the energy gap of the material, and a photon ofthis amount of energy is produced. Radiative generation events occur from theabsorption of a photon of energy greater than or equal to the bandgap energy.

Finally, thermal recombination or generation events arise from phonon emis-sion or absorption, respectively (see Figure 3.3).

-

6ε(k)

k

d

t

?

valence band

conduction band

energyemission

-

-

6ε(k)

k

t

6d

valence band

conduction band

energyabsorption

¾

Figure 3.3: Recombination (left) and generation (right) of an electron-hole pair.

47

The evolution of the distribution functions Fn of the electrons and Fp ofthe holes is given by the Boltzmann equation including an operator accountingfor the recombination-generation processes. More precisely, in the semi-classicalframework,

∂tFn + vn(k) · ∇xFn −q

~Eeff · ∇kFn = Qn(Fn) + In(Fn, Fp), (3.36)

∂tFp + vp(k) · ∇xFp +q

~Eeff · ∇kFp = Qp(Fn) + Ip(Fn, Fp), (3.37)

where

vn =1

~∇kεn, vp =

1

~∇kεp

are the mean velocities related to the electron conduction band εn and the holevalence band εp, respectively, Qn and Qp are the collision operators

(Qα(Fα))(x, k, t) =

∫

B

[sα(x, k′, k)F ′(1− F )− sα(x, k, k′)F (1− F )] dk′,

α = n, p, and the recombination-generation operators are given by

(In(Fn, Fp))(x, k, t) =

∫

B

[g(x, k′, k)(1− Fn)(1− F ′

p)− r(x, k, k′)FnF′p

]dk′,

(Ip(Fn, Fp))(x, k, t) =

∫

B

[g(x, k, k′)(1− F ′n)(1− Fp)− r(x, k′, k)F ′

nFp] dk′.

Here g(x, k, k′) ≥ 0 is the rate of generation of an electron at the state (x, k) andof a hole at the state (x, k′), and r(x, k′, k) ≥ 0 is the analogous recombinationrate. Similarly as for the transition rates sn and sp (see (3.30)) the relation

r(x, k, k′) = exp

(εn(k)− εp(k′)

kBT

)g(x, k′, k)

is assumed to hold. Then the operators In and Ip can be written, in the low-density approximation, as follows:

(In(Fn, Fp))(x, k, t) =

∫

B

g(x, k′, k)

[1− exp

(εn(k)− εp(k′)

kBT

)FnF

′p

]dk′,

(In(Fn, Fp))(x, k, t) =

∫

B

g(x, k, k′)

[1− exp

(εn(k

′)− εp(k)kBT

)F ′

nFp

]dk′.

The positive sign for the term involving Eeff in the Botzmann equation (3.37)for the holes comes from the opposite flow direction of the positively chargedholes in the electric field Eeff .

In the case of Coulomb forces in R3, the effective field is given by

Eeff(x, t) =1

4πεs

∫

R3

ρ(x∗, t)x− x∗|x− x∗|3

dx∗,

48

where ρ is the total space charge density. The charge density is the sum of theelectron density n, the hole density p and the densities ND, NA of the implantedpositively charged donor ions and the negatively charged acceptor ions, respec-tively, with which the semiconductor crystal is doped (see Section 2.3):

ρ = q(−n+ p−NA +ND),

with corresponding charges +q or −q. Here, we have set

n(x, t) =

∫

B

Fn(x, k, t) dk, p(x, t) =

∫

B

Fp(x, k, t) dk.

Thus the Poisson equation (3.21) for the electrostatic potential defined by Eeff =−∇Veff writes

εs∆Veff = q(n− p− C),

where C = ND −NA is the doping profile.The total number of each type of particles is not conserved anymore, due to

recombination-generation effects. However, taking the difference of the Boltz-mann equations (3.36) and (3.37) and integrating over x ∈ Rd and k ∈ B yields

∂t

∫

Rd(n− p)(x, t) dx =

∫

Rd

∫

B

(In(Fn, Fp)− Ip(Fn, Fp)) dk dx = 0,

by Lemma 3.2(1) and by definition of In and Ip. If the doping profile C does notdepend on t, we obtain the conservation of the total charge density

∂t

∫

Rdρ(x, t) dx = ∂t

∫

Rd(n(x, t)− p(x, t)− C(x)) dx = 0.

49

4 Quantum Kinetic Transport Equations

4.1 The Wigner equation

As the dimensions of a semiconductor device decrease, quantum mechanical trans-port phenomena play an important rule in the function of the device. Therefore,it is of great importance to devise transport models which, on one hand, are capa-ble of describing quantum effects and, on the other hand, are sufficiently simpleto allow for efficient numerical simulations. In order to derive such models, westart with the Schrodinger equation.

The motion of an electron ensemble consisting of M particles in a vacuumunder the action of an electric field E = −∇xV is described by the Schrodingerequation

i~∂tψ = − ~2

2m

M∑

j=1

∆xjψ − qV (x, t)ψ, x ∈ RdM , t > 0, (4.1)

ψ(x, 0) = ψI(x), x ∈ RdM ,

where x = (x1, . . . , xM ), i2 = −1, ~ = h/2π is the reduced Planck constant,m the electron mass, q the elementary charge, V the real-valued electrostaticpotential, and ψ = ψ(x, t) is called the wave function of the electron ensemble.Macroscopic variables are defined by

electron density: n = |ψ|2,electron current density: J = −~q

mIm(ψ∇xψ),

where ψ is the complex conjugate of ψ.Another formulation of the motion of the electron ensemble can be obtained

using the so-called density matrix

ρ(r, s, t) = ψ(r, t)ψ(s, t), r, s ∈ RdM , t > 0. (4.2)

The electron density and current density, respectively, are given in terms of thedensity matrix by

n(x, t) = ρ(x, x, t), (4.3)

J(x, t) =i~q2m

(ψ∇xψ − ψ∇xψ

)(x, t)

=i~q2m

(∇s −∇r)ρ(x, x, t). (4.4)

Differentiating (4.2) with respect to t and using the Schrodinger equation (4.1)

50

leads to the Heisenberg equation

i~∂tρ(r, s, t) = i~(∂tψ(r, t)ψ(s, t) + ψ(r, t)∂tψ(s, t)

)

=

(~2

2m∆rψ + qV ψ

)(r, t)ψ(s, t)

+ ψ(r, t)

(− ~2

2m∆sψ − qV ψ

)(s, t) (4.5)

= − ~2

2m(∆sρ−∆rρ)(r, s, t)− q(V (s, t)− V (r, t))ρ(r, s, t),

with r, s ∈ RdM , t > 0.The kinetic form of the quantum equations is obtained by a change of un-

knowns and a Fourier transformation. We recall that the Fourier transform F isdefined by

(Fg)(η) = g(η) =1

(2π)dM/2

∫

RdMg(v)e−iη·v dv

for (sufficiently smooth) functions g : RdM → C with inverse

(F−1h)(v) = h(v) =1

(2π)dM/2

∫

RdMh(η)eiη·v dη

for functions h : RdM → C. Introduce the change of coordinates

r = x+~2m

η, s = x− ~2m

η

and set

u(x, η, t) = ρ

(x+

~2m

η, x− ~2m

η, t

). (4.6)

Since (~/2m)η has the dimension of length and ~/2m has the dimension of cm2/s,η has the dimension of inverse velocity s/cm. Thus the variable v in the Fouriertransform of a function g(v) has the dimension of velocity cm/s. We compute

div η(∇xu)(x, η, t) = div η(∇rρ+∇sρ)

(x+

~2m

η, x− ~2m

η, t

)

=~2m

(∆rρ−∆sρ)(r, s, t).

Hence the transformed Heisenberg equation for u reads

∂tu+ idiv η(∇xu) +q

m(δV )u = 0, x, η ∈ RdM , t > 0, (4.7)

where

(δV )(x, η, t) =im

~

[V

(x+

~2m

η, t

)− V

(x− ~

2mη, t

)]. (4.8)

51

The initial condition is

u(x, η, 0) = ψI

(x+

~2m

η

)ψI

(x− ~

2mη

), x, η ∈ RdM .

We prefer to work with the variables velocity v and space x instead of inversevelocity η and space x and apply therefore the Fourier transformation to u. Thefunction

w = (2π)−dM/2F−1u = (2π)−dM/2u

or, in terms of the wave function,

w(x, v, t) =1

(2π)dM

∫

RdMψ

(x+

~2m

η

)ψ

(x− ~

2mη

)eiη·v dη

is called Wigner function. It was introduced by Wigner in 1932 [43].The macroscopic electron density n and the current density J can now be

written as

n(x, t) =

∫

RdMw(x, v, t) dv, (4.9)

J(x, t) = −q∫

RdMvw(x, v, t) dv, (4.10)

since the first identity follows from (4.3) and

n(x, t) = ρ(x, x, t) = u(x, 0, t) = (2π)dM/2w(x, 0, t) =

∫

RdMw(x, v, t) dv,

and for the second identity we use the fact that the Fourier transform translatesdifferential operators into a multiplication:

i(∇ηw)(x, η, t) =i

(2π)dM/2

∫

RdMw(x, v, t)∇ηe

−iη·v dv

=1

(2π)dM/2

∫

RdMw(x, v, t)ve−iη·v dv

= vw(x, η, t),

and therefore, using (4.4),

J(x, t) =i~q2m

(∇s −∇r)ρ(x, x, t) = −iq(∇ηu)(x, 0, t)

= −iq(2π)dM/2(∇ηw)(x, 0, t) = −q(2π)dM/2vw(x, 0, t)

= −q∫

RdMvw(x, v, t) dv.

52

The integrals (4.9) and (4.10) are called the zeroth and first moments of theWigner function, respectively, in analogy to the classical situation (see Section3.1).

The transport equation for w is obtained by taking the inverse Fourier trans-form of the equation (4.7) for u :

∂tu+ i(div η∇xu)∨ +

q

m[(δV )u]∨ = 0. (4.11)

Using

i(div η∇xu)∨(x, v, t) =

i

(2π)dM/2

∫

RdM(div η∇xu(x, η, t))e

iv·η dη

=v

(2π)dM/2·∫

RdM∇xu(x, η, t)e

iv·η dη

= v · ∇xu(x, v, t)

= (2π)dM/2v · ∇xw(x, v, t),

where we have employed integration by parts, and

(θ[V ]w) (x, v, t) :=1

(2π)dM/2[(δV )u]∨ (x, v, t) (4.12)

=1

(2π)dM

∫

RdM(δV )(x, η, t)u(x, η, t) eiη·v dη

=1

(2π)dM

∫

RdM

∫

RdM(δV )(x, η, t)w(x, v′, t) eiη·(v−v′) dv′ dη,

it follows from (4.11)

∂tw + v · ∇xw +q

mθ[V ]w = 0, x, v ∈ RdM , t > 0. (4.13)

This equation is called Wigner equation or quantum Liouville equation. Theinitial condition for w reads

w(x, v, 0) = wI(x, v), x, v ∈ RdM , (4.14)

where

wI(x, v) =1

(2π)dM

∫

RdMψI

(x+

~2m

η

)ψI

(x− ~

2mη

)eiv·η dη, x, v ∈ RdM .

(4.15)The operator (4.12) is a pseudo-differential operator. Generally, an operator,

whose Fourier transform acts as a multiplication operator on the Fourier trans-form of the function, is called a linear pseudo-differential operator [42]. Indeed,it holds

θ[V ]w(x, η, t) = (δV )(x, η, t)w(x, η, t),

53

and so, θ[V ] is a pseudo-differential operator and the Wigner equation (4.13) isa linear pseudo-differential equation.

We discuss now some properties of the Wigner equation, namely

• the semi-classical limit,

• the relation between the solution of the Wigner equation and the solutionof the Schrodinger equation, and

• the question of positivity of the solution of the Wigner equation.

It is interesting to perform the formal semi-classical limit “~ → 0”. Wecompute, as “~→ 0”,

(δV )(x, η, t)→ i∇xV (x, t) · η

and, using i(ηu)∨ = ∇vu,

θ[V ]w → i(2π)−dM/2 [(∇xV · η)u]∨ = (2π)−dM/2∇xV · ∇vu = ∇xV · ∇vw.

Therefore, (4.13) becomes, as “~→ 0”,

∂tw + v · ∇xw +q

m∇xV · ∇vw = 0,

and we recover the classical Liouville equation (3.6). Clearly, the above limitshave to be understood in a formal way.

The solution of the initial-value problem (4.13)-(4.14) is given by

w(x, v, t) =1

(2π)dM

∫

RdMψ

(x+

~2m

η, t

)ψ

(x− ~

2mη, t

)eiv·η dη, t > 0,

(4.16)where ψ solves (4.1) if and only if the initial datum wI satisfies (4.15). If (4.16)holds the quantum state of the electron is fully described by the single wavefunction ψ. In quantum physics this is referred to as a pure quantum state. Formore general initial data, we cannot expect that (4.16) holds true. However, wecan derive a similar expression. For this we choose the initial data

wI ∈ L2(RdM × RdM) =w : RdM × RdM → C :

∫

R2dM

|w(x, v)|2 dx dv <∞.

From functional analysis, there exists a complete orthonormal system (φ(k))k∈Nof functions φ(k) in L2(RdM) (defined similarly as above). It can be shown that

(φ(j)φ(k))j,k∈N is a complete orthonormal system in L2(RdM × RdM) and that

ρI(r, s) = wI(x, η)

54

can be expanded in the series

ρI(r, s) =∑

k∈Nλkφ(k)(r)φ

(k)(s)

with

λk =

∫

RdM

∫

RdMρI(r, s)φ

(k)(r)φ(k)(s) dr ds.

Let ψ(k) be the solution of the Schrodinger equation

i~∂tψ(k) = − ~2

2m∆xψ

(k) − qV ψ(k), x ∈ RdM , t > 0,

ψ(k)(x, 0) = φ(k)(x), x ∈ RdM .

Then the solution of the Wigner equation (4.13)-(4.14) is given by

w(x, v, t) =1

(2π)dM

∑

k∈Nλk

∫

RdMψ(k)

(x+

~2m

η, t

)ψ(k)

(x− ~

2mη, t

)eiv·η dη,

for t > 0. This means that the Wigner equation is capable of describing so-calledmixed quantum states.

The electron density is

n(x, t) =

∫

RdMw(x, v, t) dv

= (2π)dM/2w(x, 0, t)

=1

(2π)dM/2

∑

k∈Nλk

[ψ(k)

(x+

~2m

η, t

)ψ(k)

(x− ~

2mη, t

)]

η=0

=∑

k∈N

λk

(2π)dM/2|ψ(k)(x, t)|2.

If the initial electron density is non-negative or, more precisely, if λk ≥ 0 for allk ∈ N, the electron density is non-negative for all t > 0. One may ask if alsothe Wigner function is non-negative for all t > 0 if this is true initially. Thiswould allow for a probabilistic interpretation of the Wigner function. However,generally this is not true. It is shown in [30] that for a pure quantum state,

w(x, v, t) =1

(2π)dM

∫

RdMψ

(x+

~2m

η, t

)ψ

(x− ~

2mη, t

)eiv·η dη

is non-negative if and only if either ψ ≡ 0 or

ψ(x, t) = exp(−x>A(t)x− a(t) · x+ b(t)

), x ∈ RdM , t > 0,

55

where A(t) is a CdM×dM matrix with symmetric positive definite real part anda(t) ∈ CdM , b(t) ∈ C. A necessary condition for the non-negativity of w for mixedquantum states is not known.

The Wigner equation (4.13) does not account for the effect of the crystallattice on the motion of the electron ensemble. The motion of one electron in thecrystal is described by the Schrodinger equation

i~∂tψ = − ~2

2m∆xψ − q(VL + V )ψ, x ∈ R3, t > 0,

where VL is the lattice potential satisfying

VL(x+ γ ~aj) = VL(x) x ∈ R3, j = 1, 2, 3,

with the primitive vectors ~aj of the lattice and the lattice length scale γ (seeSection 2.1). By using a Bloch decomposition and performing the limit γ → 0,it is possible to derive the Wigner equation in a crystal:

∂tw + v(k) · ∇xw +q

~θ[V ]w = 0, (4.17)

where v(k) = (1/~)∇kε(k) and ε(k) is the energy band in which the electronsare moving. The pseudo-differential operator θ[V ] is defined as in (4.12), andthe function w is the inverse of the Fourier transform of the density matrixtransformed from (r, s) to (x, η) as in (4.6). We refer to [6, 38] for details onthe derivation.

4.2 The quantum Vlasov and quantum Boltzmann equa-tion

The quantum Liouville equation has the same disadvantages as its classical ana-logue:

• The equation has to be solved in a very high-dimensional phase space sinceM À 1, and its numerical solution is almost unfeasible.

• Short-range or long-range interactions are not included.

In this section we derive the quantum analogue of the classical Vlasov equation,the quantum Vlasov equation which acts on a low-dimensional (x, v)-space ofdimension 2d. We proceed similarly as in Section 3.2. In particular, we imposethe following assumptions.

Consider an ensemble of M electrons with mass m in a vacuum under theaction of the real-valued electrostatic potential V (x, t), x ∈ RdM , t > 0. Themotion of the particle ensemble is described by the wave function ψ which is asolution of the many-particle Schrodinger equation (4.1). We assume:

56

• The initial wave function is anti-symmetric:

ψ(x1, . . . , xM , 0) = sign(π)ψ(xπ(1), . . . , xπ(M), 0) (4.18)

for all permutations π of the set 1, . . . ,M and for all x = (x1, . . . , xM ) ∈RdM .

• The potential can be decomposed as

V (x1, . . . , xM , t) =M∑

j=1

Vext(xj, t) +1

2

M∑

i,j=1

Vint(xi, xj), (4.19)

where Vint is symmetric, i.e. Vint(xi, xj) = Vint(xj, xi) for all i, j = 1, . . . ,M,and of the order of magnitude 1/M as M →∞.

We discuss the assumptions. The factor 12in (4.19) is necessary since each

electron-electron pair in the sum of two-particle interactions is counted twice.Notice that the symmetry of Vint implies

V (x1, . . . , xM , t) = V (xπ(1), . . . , xπ(M), t)

for all permutations. It is possible to show that then, the anti-symmetry of ψ isconserved for all time if ψ is anti-symmetric initially.

The first hypothesis implies that the ensemble density matrix

ρ(r, s, t) = ψ(r, t)ψ(s, t), r = (r1, . . . , rM), s = (s1, . . . , sM) ∈ RdM , t ≥ 0,

remains invariant under any permutations of the r- and s-arguments:

ρ(r1, . . . , rM , s1, . . . , sM , t) = ρ(rπ(1), . . . , rπ(M), sπ(1), . . . , sπ(M), t) (4.20)

for all permutations π of 1, . . . ,M and all ri, sj ∈ Rd, t ≥ 0. This expresses thefact that the electrons are indistinguishable.

It is possible to use, instead of (4.18), the condition (4.20) as a hypothesiswhich has the advantage that the condition can be easily interpreted physically.However, (4.20) does not imply (4.18). In fact, (4.20) is satisfied if either the wavefunction is anti-symmetric or symmetric. The anti-symmetry property representsthe Pauli exclusion principle since it implies that

ψ(x1, . . . , xM , t) = 0 if xi = xj for i 6= j,

meaning that double occupancy of states is prohibited. We would need to assume(4.20) and the Pauli exclusion principle as the hypotheses. We prefer to use theslightly stronger condition (4.18) as a hypothesis.

57

We wish to model the evolution of subensembles. The density matrix of asubensemble of particles is defined by

ρ(a)(r(a), s(a), t) =

∫

Rd(M−a)ρ(r(a), ua+1, . . . , uM , s

(a), ua+1, . . . , uM) dua+1 . . . duM ,

wherer(a) = (r1, . . . , ra), s(a) = (s1, . . . , sa) ∈ Rda.

Due to (4.20), the subensemble density matrices satisfy

ρ(a)(r1, . . . , ra, s1, . . . , sa, t) = ρ(a)(rπ(1), . . . , rπ(a), sπ(1), . . . , sπ(a), t) (4.21)

for all permutations π of 1, . . . , a and all ri, sj ∈ Rd, t ≥ 0. The evolution of thecomplete electron ensemble is governed by the Heisenberg equation (see (4.5))

i~∂tρ = − ~2

2m

M∑

j=1

(∆sj −∆rj

)ρ− q

M∑

j=1

(Vext(sj, t)− Vext(rj, t))ρ

−q2

M∑

j,`=1

(Vint(sj, s`)− Vint(rj, r`)) ρ.

We set uj = sj = rj for j = a + 1, . . . ,M in the above equation, integrateover (ua+1, . . . , uM) ∈ Rd(M−a) and use the indistinguishability property (4.21) toobtain, after an analogous calculation as in Section 3.2, a quantum equivalent ofthe BBGKY hierarchy:

i~∂tρ(a) = − ~2

2m

M∑

j=1

(∆sj −∆rj

)ρ(a) − q

M∑

j=1

(Vext(sj, t)− Vext(rj, t))ρ(a)

−q(M − a)a∑

j=1

∫

Rd(Vint(sj, u∗)− Vint(rj, u∗)) ρ(a+1)∗ du∗,

for 1 ≤ a ≤M − 1, where

ρ(a+1)∗ = ρ(a+1)(r(a), u∗, s(a), u∗, t).

Since we assume that Vint is of the order of magnitude 1/M as M → ∞, we getfor fixed a ≥ 1 and M À 1, neglecting terms of order 1/M,

i~∂tρ(a) = − ~2

2m

M∑

j=1

(∆sj −∆rj

)ρ(a) − q

M∑

j=1

(Vext(sj, t)− Vext(rj, t)) ρ(a)

−qa∑

j=1

∫

Rd(Vint(sj, u∗)− Vint(rj, u∗))Mρ(a+1)∗ du∗.

58

Similarly to the classical case, a particular solution of this equation is given bythe so-called Hartree ansatz

ρ(a)(r1, . . . , ra, s1, . . . , sa, t) =a∏

j=1

R(rj, sj, t),

where R solves the equation

i~∂tR = − ~2

2m(∆s −∆r)R− q(Veff(s, t)− Veff(r, t))R, r, s ∈ Rd, t > 0, (4.22)

with the effective potential

Veff(x, t) = Vext(s, t) +

∫

RdMR(z, z, t)Vint(x, z) dz.

The kinetic formulation of (4.22) is derived as in Section 4.1. We multiply(4.22) by M, introduce the change of coordinates

r = x+~2m

η, s = x− ~2m

η

and set U(x, η, t) =MR(r, s, t). Then U solves the equation

∂tU + idiv η∇xU + iq

~

(Veff

(x+

~2m

η, t

)− Veff

(x− ~

2mη

))U = 0.

Finally, the inverse Fourier transformW = (2π)−d/2U is a solution of the quantumVlasov equation

∂tW + v · ∇xW +q

mθ[Veff ]W = 0, x, v ∈ Rd, t > 0. (4.23)

The pseudo-differential operator θ[Veff ] is defined as in (4.12), and the effectivepotential is

Veff(x, t) = Vext(x, t) +

∫

Rdn(z, t)Vint(x, z) dz, (4.24)

where

n(x, t) =

∫

RdW (x, v, t) dv = U(x, 0, t) =MR(x, x, t)

is the quantum electron number density. As the effective potential depends onthe Wigner function W, it is a nonlinear pseudo-differential equation.

Contrary to the classical Vlasov equation, the quantum Vlasov equation doesnot preserve the non-negativity of the solution W (cf. the discussion in Section4.1). However, the number density n remains non-negative for all times, if theinitial single-particle density matrix R(r, s, 0) is positive semi-definite.

59

In the semi-classical limit “~ → 0” the quantum Vlasov equation formallyconverges to the classical Vlasov equation

∂tW + v · ∇xW +q

m∇xVeff · ∇vW = 0.

In semiconductor modelling, a usual choice for Vint is the Coulomb potential

Vint(x, y) = −q

4πεs

1

|x− y| , x, y ∈ R3, x 6= y,

where εs denotes the permittivity of the semiconductor crystal (see Example 3.1).In Section 3.2 has been shown that the effective potential

Veff(x, t) = Vext(x, t)−q

4πεs

∫

R3

n(z, t)

|z − x| dz

solves the Poisson equation

εs∆Veff = q(n− C),

whereC(x) = −εs

qVext(x)

is the doping concentration if Vext is generated by ions of charge +q in the semi-conductor material.

The presented quantum Vlasov equation models the motion of an ensemble ofmany particles in a vacuum taking into account long-range particle interactions.For more realistic models, the following issues should be modeled too:

• The electrons are moving in a crystal and not in a vacuum.

• As the semiconductor device is bounded, the equations have to be solvedin a bounded domain with appropriate boundary conditions at the devicecontacts and insulating surfaces.

• Short-range interactions modelled by scattering events of particles have tobe included in the model.

For the first two generalizations, we refer to [34, Ch. 1] (also see (4.17)). Thequantum mechanical modelling of collisions of electrons (with phonons, for in-stance) is a very difficult task. One approach is to formulate the so-called quantumBoltzmann equation

∂tW + v · ∇xW +q

mθ[Veff ]W = Q(W ), x, v ∈ Rd, t > 0,

60

by adding a heuristic collision term to the right-hand side of the quantum Vlasovequation (4.23). In numerical studies, the relaxation-time model

Q(W ) =1

τ

(n

n0W0 −W

),

or the Fokker-Planck model

Q(W ) =1

τdiv v

(kBT0m∇vW + vW

),

is often used, where τ is the relaxation time,

n(x, t) =

∫

RdW (x, v, t) dv, n0(x) =

∫

RdW0(x, v) dv,

W0 is the density of the quantum mechanical thermal equilibrium (see [39, Sec.2] for its definition), and T0 is the lattice temperature. In the case of the energy-band quantum Boltzmann equation

∂tW + v(k) · ∇xW +q

mθ[Veff ]W = Q(W ), (4.25)

the above collision models read as follows: The relaxation-time collision term is

Q(W ) =1

τ

(n

n0W0 −W

),

where now

n(x, t) =

∫

RdW (x, k, t) dk, n0(x) =

∫

RdW0(x, k) dk,

and the Fokker-Planck terms is given by

Q(W ) =1

τdiv k

(mkBT0

~2∇kW + kW

). (4.26)

A summary of the kinetic models derived in this and the previous section ispresented in Figure 4.1. Notice that for each model, there is an energy-bandversion in the (x, k, t) variables which reduces to a model in the (x, v, t) variablesin the parabolic band approximation.

61

quantum

Liouville

quantum

Vlasov¾no two-particle

interactions

quantum

Boltzmann¾ no collisions

6“~ → 0”

Liouville

6“~ → 0”

Vlasov¾ no two-particleinteractions

6“~ → 0”

Boltzmann¾ no collisions

Figure 4.1: Relations between the classical and quantum kinetic equations.

5 From Kinetic to Fluiddynamical Models

5.1 The drift-diffusion equations: first derivation

We derive the drift-diffusion equations (formally) from the bipolar Boltzmannequations

∂tfn + vn(k) · ∇xfn −q

~E · ∇kfn = Qn(fn) + In(fn, fp), (5.1)

∂tfp + vp(k) · ∇xfp +q

~E · ∇kfp = Qp(fp) + Ip(fn, fp) (5.2)

with the low-density collision operators

(Qn(fn))(x, k, t) =

∫

B

φn(x, k, k′)

×[exp

(−εn(k)kBT

)f ′n − exp

(−εn(k

′)

kBT

)fn

]dk′, (5.3)

(Qp(fp))(x, k, t) =

∫

B

φp(x, k, k′)

×[exp

(−εp(k)kBT

)f ′p − exp

(−εp(k

′)

kBT

)fp

]dk′, (5.4)

and the recombination-generation rates

(In(fn, fp))(x, k, t) = −∫

B

g(x, k, k′)

×[exp

(εn(k)− εp(k′)

kBT

)fnf

′p − 1

]dk′, (5.5)

(Ip(fn, fp))(x, k, t) = −∫

B

g(x, k, k′)

×[exp

(εn(k

′)− εp(k)kBT

)f ′nfp − 1

]dk′. (5.6)

62

Here, f ′j means evaluation at k′, i.e. f ′j = fj(x, k′, t), j = n, p, and B denotes the

Brillouin zone. We assume the parabolic band approximation

εn(k) = εc +~2

2mn

|k|2, εp(k) = εv −~2

2mp

|k|2,

where εc denotes the conduction band minimum, εv the valence band maximum,and mn and mp are the effective masses of the electrons and holes, respectively.Then the velocities are given by

vn(k) =1

~∇kεn(k) =

~kmn

,

vp(k) = −1

~∇kεp(k) =

~kmp

.

In particular, we can replace B by Rd in the above integrals.The drift-diffusion equations are derived under the assumption that collisions

occur on a much shorter time scale than recombination-generation events. Inorder to make this statement more precise, we scale the Boltzmann equations(5.1) and (5.2). For this, we make the following assumptions:

(1) The effective masses of the electrons and holes are of the same order suchthat we can define the reference velocity v =

√kBT/mn. This means that

we assume that the thermal energy kBT is of the same order as the kineticenergy mnv

2/2.

(2) Collisions occur on a much shorter time scale than recombination-generationevents. Thus, scaling

Qn =1

τcQns, In =

1

τRIns, (5.7)

Qp =1

τcQps, Ip =

1

τRIps (5.8)

with the collision relaxation time τc and the recombination-generation re-laxation time τR, we assume that

τc ¿ τR.

This is our main assumption.

(3) The reference length ι0 is given by the geometric average of the mean freepaths ιc := τcv and ιR := τRv:

ι0 =√ιRιc.

The mean free path is the average time between two successive collisions.

63

(4) We use the reference time τR, the reference wave vector mnv/~, and thereference field strength kBT/qι0:

t = τRts, k =mnv

~ks, E =

kBT

ι0qEs. (5.9)

By assumption (2), the parameter α2 := ιc/ιR = τc/τR satisfies α2 ¿ 1. Withthe scaling (5.7)-(5.9) we can rewrite (5.1) (omitting the index s):

1

τR∂tfn +

v

ι0k · ∇xfn −

kBT

ι0mnvE · ∇kfn =

1

τcQn(fn) +

1

τRIn(fn, fp).

Multiplying this equation by τc = ιc/v and using α = ιc/ι0 and kBT = mnv2, we

obtain

α2∂tfn + α(k · ∇xfn − E · ∇kfn) = Qn(fn) + α2In(fn, fp). (5.10)

In a similar way, we have

α2∂tfp + α(mn

mp

k · ∇xfp − E · ∇kfp

)= Qp(fp) + α2Ip(fn, fp). (5.11)

The scaled collision and recombination-generation terms have the same form as(5.3)-(5.6) but the exponential terms

exp

(−εn(k)kBT

)and exp

(−εp(k)kBT

)

are replaced by

exp

(− εckBT

− |k|2

2

)and exp

(− εvkBT

+mn

mp

|k|22

)

and the rates φn, φp and g are multiplied by (mnv/~)d.We want to study the scaled Boltzmann equations for “small” α. First we

analyze the collision operators Qn and Qp. We rewrite them in the form

Qj(fj) =

∫

RdΦj(x, k, k

′)[Mj(k)f

′j −Mj(k

′)fj]dk′, j = n, p, (5.12)

whereΦj(x, k, k

′) = Njφn(x, k, k′), j = n, p,

and

Mn(k) =1

Nn

exp

(−|k|

2

2

), Mp(k) =

1

Np

exp

(−mn

mp

|k|22

)

64

are the scaled Maxwellians. The constants

Nn = (2π)d/2, Np = (2πmp/mn)d/2

are chosen such that the integrals of the Maxwellians over k ∈ Rd are equal toone.

For the following analysis define the functions

λj(k) =

∫

RdΦj(x, k, k

′)Mj(k′) dk′, k ∈ Rd, j = n, p,

for fixed x ∈ Rd, and the Banach spaces

Xj = f : Rd → R measurable: ‖f‖Xj<∞,

Yj = f : Rd → R measurable: ‖f‖Yj <∞

with associated norms

‖f‖Xj=

(∫

Rdf(k)2λj(k)Mj(k)

−1 dk

)1/2

,

‖f‖Yj =

(∫

Rdf(k)2(λj(k)Mj(k))

−1 dk

)1/2

.

Lemma 5.1 Let j = n or j = p, let Φn, Φp > 0 be symmetric in k and k′ andassume that for all isometric matrices A it holds

Φ(x,Ak,Ak′) = Φ(x, k, k′) ∀x, k, k′ ∈ Rd.

Then

(1) The kernel N(Qj) = f ∈ X : Qj(f) = 0 is given by

N(Qj) = σM : σ = σ(x) ∈ R.

(2) The equation Qj(f) = g with g ∈ Yj has a solution f ∈ Xj if and only if∫

Rdg(k) dk = 0.

In this situation, any solution of Qj(f) = g can be written as f + σMj,where σ = σ(x) is a parameter.

(3) The solution f ∈ Xj of Qj(fj) = g is unique if the orthogonality relation∫

Rdf(k)λj(k)dk = 0 (5.13)

holds.

65

(4) The equations

Qn(hni) = kiMn(k), Qp(hpi) =mn

mp

kiMp(k), i = 1, . . . , d,

have solutions hn(x, k) = (hni(x, k))di=1 and hp(x, k) = (hpi(x, k))

di=1 with

the property that there exist µn(x), µp(x) ≥ 0 such that

∫

Rdk ⊗ hn dk = −µn(x) Id, (5.14)

∫

Rd

mn

mp

k ⊗ hp dk = −µp(x) Id, (5.15)

where Id is the unit matrix in Rd×d and a⊗ b = a>b for a, b ∈ Rd.

In the statement of the lemma we have omitted some technical assumptionson Φn and Φp (regularity conditions) which are particularly needed in the proof ofLemma 5.1(4). We only sketch the proof of part (3) since properties of so-calledHilbert-Schmidt operators are needed, and refer to [37] for a complete proof.

Proof. In the following we omit the index j. We proceed as in [37, Prop. 1].(1) First we symmetrize the collision operator by setting fs = (λ/M)1/2f and

Qs(fs) = (λM)−1/2Q(f). Then

Qs(fs) = (λM)−1/2(M

∫

RdΦ(x, k, k′)f ′ dk′ − λf

)

=

∫

RdΦ(x, k, k′)

(MM ′

λλ′

)1/2

f ′s dk′ − fs,

where M ′ =M(k′) and λ′ = λ(k′). Since Φ is symmetric in k and k′ by assump-tion, the operator Qs : L

2(Rd)→ L2(Rd) is self-adjoint.Next we analyze the kernel of Qs. By definition of λ we have

∫

RdΦ(x, k, k′)M(k′)/λ(k) dk′ = 1

66

and therefore

0 ≤ 1

2

∫

Rd

∫

RdΦ(x, k, k′)MM ′

(fs√λM− f ′s√

λ′M ′

)2

dk dk′

=1

2

∫

Rd

(∫

RdΦ(x, k, k′)

M ′

λdk′)f 2s dk

+1

2

∫

Rd

(∫

RdΦ(x, k, k′)

M

λ′dk

)(f ′s)

2 dk′

−∫

Rd

∫

RdΦ(x, k, k′)

(MM ′

λλ′

)1/2

f ′sfs dk dk′

=

∫

Rd

(f 2s −

∫

RdΦ(x, k, k′)

(MM ′

λλ′

)1/2

f ′sfs dk′

)dk

= −∫

RdQs(fs)fs dk.

Hence Qs(fs) = 0 implies

fs√λM− f ′s√

λ′M ′= 0 for k, k′ ∈ Rd

and fs/√λM = σ = const., where σ = σ(x) is a parameter. In original variables,

Q(f) = 0 implies f = σM for all σ ∈ R. Conversely, if f = σM for someσ ∈ R then the formulation (5.12) of Q(f) immediately yields Q(f) = 0. ThusN(Q) = f ∈ X : Q(f) = 0 = σM : σ ∈ R.

(2) It is not difficult to see that the operator Qs : L2(Rd)→ L2(Rd) is linear,

continuous and has closed range R(Qs) where

R(Qs) = gs ∈ L2(Rd) : ∃fs ∈ L2(Rd) : Qs(fs) = gs.

From functional analysis follows that R(Qs) = N(Q∗s)

⊥, where Q∗s is the adjoint

of Qs and N(Q∗s) is the kernel of Q

∗s. In fact, since Qs is selfadjoint, Q

∗s = Qs. We

obtain

Qs(fs) = gs has a solution ⇔ gs ∈ R(Qs)

⇔ gs ∈ N(Q∗s)

⊥ = N(Qs)⊥

⇔ ∀h ∈ L2(Rd), Qs(h) = 0 :

∫

Rdgsh dk = 0.

The equivalence between the solvability of Qs(fs) = g and g ∈ N(Q∗s)

⊥ is alsoknown as the Fredholm alternative [44, Appendix, (39)]. As the kernel of Qs isspanned by

√λM , Qs(fs) = gs has a solution if and only if

∫

Rdgs√λM dk = 0

67

or, in the original variables,

0 =

∫

Rdgs√λM dk =

∫

Rdg dk.

(3) We only give a sketch of the proof and refer to [37] for details. It is possibleto show that the operator −Q is coercive in the following sense:

∫

Rd(−Q(f))(k)f(k)M(k)−1 dk ≥ c‖f‖2X (5.16)

for all f ∈ X satisfying (5.13). Clearly, this implies that Q is one-to-one on thesubset of functions satisfying (5.13), and the uniqueness is proved. In order toshow the coerciveness property prove that Id +Q is a Hilbert-Schmidt operatorand use general properties of those operators (see [37]).

(4) The existence of solutions hi of (Q(hi))(k) = kiM(k) follows from part (2)since ∫

RdkiMj(k) dk = 0 for i = 1, . . . , d, j = n, p.

We only show (5.14) since the proof of (5.15) is similar.Let A be the matrix of a rotation with axis k1. Then (Ak)1 = k1 for all k ∈ Rd

and

(Ak)1M(Ak) = k1N−1 exp(−|Ak|2/2) = k1N

−1 exp(−|k|2/2) = k1M(k), (5.17)

since A is isometric. We have for all k ∈ Rd, using the assumption Φ(x,Ak,Ak′) =Φ(x, k, k′) and (5.17),

(Q(h1 A))(k) =

∫

RdΦ(x, k, k′)[M(k)h1(Ak

′)−M(k′)h1(Ak)] dk′

=

∫

RdΦ(x,Ak,Ak′)[M(Ak)h1(Ak

′)−M(Ak′)h1(Ak)] dk′

=

∫

RdΦ(x,Ak, w)[M(Ak)h1(w)−M(w)h1(Ak)] dw

= (Q(h1))(Ak)

= (Ak)1M(Ak)

= k1M(k)

= (Q(h1))(k)

68

and thus Q(h1 A− h1) = 0. Another computation leads to∫

Rdh1(Ak)λ(k) dk =

∫

Rd

∫

RdΦ(x, k, k′)h1(Ak)M(k′) dk′ dk

=

∫

Rd

∫

RdΦ(x,Ak,Ak′)h1(Ak)M(Ak′) dk′ dk

=

∫

Rd

∫

RdΦ(x, v, w)h1(v)M(w) dw dv

=

∫

Rdh1(v)λ(v) dv

or ∫

Rd(h1 A− h1)(k)λ(k) dk = 0.

This is the orthogonality condition (5.13) which ensures the uniqueness of thesolution of the equation Q(h1 A − h1) = 0. Therefore, h1 A − h1 = 0. Weconclude that h1 remains invariant under a rotation with axis k1, and we canwrite h1 as

h1(k) = H(k1, |k|2 − k21).Now let A be the isometric matrix of the mapping k 7→ (−k1, k2, . . . , kd).

Since k 7→ k1M(k) is an odd function, a similar computation as above yieldsQ(h1 A) = −Q(h1) and

∫

Rd(h1 A+ h1)λ(k) dk = 0.

This implies as above that h1 A+ h1 = 0 and thus, h1 is an odd function withrespect to k1.

In a similar way, we can show that

hi(k) = Hi(ki, |k|2 − k2i ), i = 2, . . . , d,

and Hi are odd functions with respect to ki. In fact all the functions Hi equal Hsince, for instance, exchanging k1 and k2 in

Q(H(k1, k22 + · · ·+ k2d)) = k1M(k21 + · · ·+ k2d)

(with slight abuse of notation) leads to

Q(H(k2, k21 + k23 + · · ·+ k2d)) = k2M(k21 + · · ·+ k2d) = Q(H2(k2, |k|2 − k22))

and hence, by a similar argument as above, H = H2.Since H is odd with respect to the first argument and |k|2 − k2j does not

depend on kj, we obtain for all i 6= j∫

Rdkihj(k)dk =

∫

RdkiH(kj, |k|2 − k2j )dk = 0.

69

Furthermore∫

Rdkihi(k)dk =

∫

RdkiH(ki, |k|2 − k2i )dk

=

∫

RdkjH(kj, |k|2 − k2j )dk

=

∫

Rdkjhj(k)dk,

for all i and j. Thus, the above integral is independent of i and we can set

µ := −∫

Rdk1h1(k)dk.

The constant µ depends on the parameter x since h1 depends on x through Q.Moreover, by (5.16),

µ(x) = −∫

Rd(Q(h1))(k)h1(k)M(k)−1 dk ≥ 0,

and this implies ∫

Rdk ⊗ h(k) dk = −µ(x) Id.

The theorem is proved. ¤

Setting α = 0 in the scaled Boltzmann equations (5.10)-(5.11) gives

Qn(fn) = 0, Qp(fp) = 0.

By Lemma 5.1(1) these equations possess the solutions

fn0 = n(x, t)Mn, fp0 = p(x, t)Mp, (5.18)

respectively, where n(x, t) and p(x, t) are some parameters. Since∫

Rdfn0 dk = n(x, t),

∫

Rdfp0 dk = p(x, t),

we interpret n and p as the scaled number densities of electrons and holes, re-spectively. In order to obtain more informations from (5.10)-(5.11) we use theHilbert expansion method. The idea is to expand fn and fp in terms of powers ofα,

fn = fn0 + αfn1 + α2fn2 + · · · ,fp = fp0 + αfp1 + α2fp2 + · · · ,

and to derive equations for fnj and fpj. Substituting this ansatz into (5.10)-(5.11)and equating coefficients of equal powers of α yields (notice that Qn and Qp arelinear)

70

• for the terms of order 1:

Qn(fn0) = 0, Qp(fp0) = 0; (5.19)

• for the terms of order α:

k · ∇xfn0 − E · ∇kfn0 = Qn(fn1), (5.20)

k · ∇xfp0 + E · ∇kfp0 = Qn(fp1); (5.21)

• and for the terms of order α2:

∂tfn0 + k · ∇xfn1 − E · ∇kfn1 = Qn(fn2) + In(fn0, fp0), (5.22)

∂tfp0 + k · ∇xfp1 + E · ∇kfp1 = Qp(fp2) + Ip(fn0, fp0). (5.23)

We already solved (5.19). By (5.18) and ∇kMj(k) = −kMj(k) (j = n, p) wecan write (5.20)-(5.21) as

Qn(fn1) = Mnk · (∇xn+ nE),

Qp(fp1) =mn

mp

Mpk · (∇xp− nE).

Lemma 5.1(2) shows that these equations have solutions and that any solutioncan be expressed as

fn1 = (∇xn+ nE) · hn + σnMn,

fp1 = (∇xp− pE) · hp + σpMp

for some unspecified parameters σn(x, t), σp(x, t). It is convenient to define

Jn(x, t) = µn(∇xn+ nE), Jp(x, t) = −µp(∇xp− pE), (5.24)

where µn µp are introduced in Lemma 5.1(4). We will see below that Jn and Jpcan be interpreted as scaled current densities.

The equation (5.22) is solvable, by Lemma 5.1(2), if and only if

0 =

∫

Rd(∂tfn0 + k · ∇xfn1 − div k(Efn1)− In(fn0, fp0)) dk

= ∂tn+

∫

Rdk · ∇xfn1 dk −

∫

RdIn(fn0, fp0) dk.

From (5.14) follows that∫

Rdk · ∇xfn1 dk =

∫

Rdk · ∇x(µ

−1n Jn · hn + σnMn) dk

=d∑

i,j=1

∂

∂xi

(µ−1n Jnj

∫

Rdkihnj dk

)+∇xσn ·

∫

RdkMn(k) dk

= −d∑

i,j=1

∂

∂xi

Jnjδij

= −div xJn.

71

Furthermore

R := −∫

RdIn(fn0, fp0)

=

∫

Rd

∫

Rdg(x, k, k′)

[exp

(εc − εvkBT

)exp

( |k|22

+mn

mp

|k′|22

)

× npMn(k)Mp(k′)− 1

]dk′ dk

=

∫

Rd

∫

Rdg(x, k, k′)

[exp

(εc − εvkBT

)np

NnNp

− 1

]dk′ dk

= A(x)(np− n2i ),

where

A(x) =1

n2i

∫

Rd

∫

Rdg(x, k, k′) dk dk′

and

ni =√NnNp exp

(εv − εc2kBT

)

is the scaled intrinsic density. We conclude that (5.22) and (by a similar com-putation) (5.23) are solvable if and only if

∂tn− div xJn = −R, ∂tp+ div xJp = −R. (5.25)

Here we can see that the quantities Jn and Jp can be indeed interpreted as currentdensities. In order to scale back to the physical variables we notice that the scalednumber densities, now called ns and ps, are obtained from

ns =

∫

Rdfn0 dks =

(~

mnv

)d ∫

Rdfn0 dk =

~dn

(kBTmn)d/2,

ps =

∫

Rdfp0 dks =

~dp

(kBTmn)d/2,

where n and p are now the unscaled variables. Thus, using

ts =t

τR, Es =

ι0UT

E, xs =x

ι0, µns =

mn

τcqµn,

we get from (5.24)-(5.25) after some computations:

∂tn− div x(µnUT∇xn+ µnnE) = −A(np− n2i ),

where UT = kBT/q,

~d

(kBTmn)d/2As = A and

(kBTmn)d/2

~dn2is = n2i .

72

In particular, the unscaled intrinsic density reads (cf. (2.20))

ni =

(2πkBT

√mnmp

~2

)d

exp

(εv − εc2kBT

).

The unscaled current density and the evolution equation for n are given by

∂tn−1

qdiv xJn = −R, Jn = qµn(UT∇xn+ nE). (5.26)

Similarly, we can compute,

∂tp+1

qdiv xJp = −R, Jp = −qµp(UT∇xp− pE). (5.27)

The equations (5.26)-(5.27) are referred to as the drift-diffusion equations forgiven electric field. For a selfconsistent treatment of the electric field, (5.26)-(5.27) have to be supplemented by the Poisson equation

−div (εsE) = q(n− p− C),

where εs is the semiconductor permittivity and C the doping concentration.Mathematically, (5.26)-(5.27) are parabolic convection-diffusion equations with

diffusion coefficientsDn := µnUT , Dp := µpUT

and mobilities µn, µp. The quotient of diffusivity and mobility is constant, andthe equations

Dn

µn

=Dp

µp

= UT

are called Einstein relations.Usually, the drift-diffusion equations are considered in a bounded domain

such that appropriate boundary conditions have to be prescribed. We refer toChapter 6 for the choice of the boundary and initial conditions which complete(5.26)-(5.27).

The derivation of the drift-diffusion model is mainly based on the followinghypotheses:

• The free mean path ιc between two consecutive scattering events is muchsmaller than the free mean path ιR between two recombination-generationevents (typically, ιc ∼ 10−7m, ιR ∼ 10−4m).

• The device diameter is of the order of ι0 =√ιRιc (typically, ι0 ∼ 10−5m).

• The electrostatic potential is of the order of UT = 0.026V (at T = 300K).

Thus, the drift-diffusion model is appropriate for semiconductor devices withcharacteristic lengths not smaller than 1 . . . 10µm and applied voltages muchsmaller than 1V. However, in application this model is used also for higher appliedvoltages. It gives reasonable results as long as the characteristic length is notmuch smaller than 1µm.

73

5.2 The drift-diffusion equations: second derivation

Using a relaxation approximation of the collision operator in the Boltzmann equa-tionm we can give a much shorter derivation of the drift-diffusion model than pre-sented in Section 5.1. The derivation combines the Hilbert expansion method andthe moment method. For simplicity, we start with the unipolar classical Boltz-mann equation in the diffusion scale (5.10) neglecting recombination-generationeffects:

α2∂tf + α(v · ∇xf −

q

mE · ∇vf

)= Q(f), x, v ∈ Rd, t > 0, (5.28)

where E = E(x, t) is the electric field and α > 0 is a parameter which is smallcompared to one. We assume a low-density collision operator Q(f) with a scat-tering rate independent of v and v′ (see (5.12)):

(Q(f))(x, v, t) =

∫

Rdφ(x) [M(v)f(x, v′, t)−M(v′)f(x, v, t)] dv′, (5.29)

where

M(v) =

(m

2πkBT

)d/2

exp

(−m|v|

2

2kBT

)

is the Maxwellian. We introduce the average of f in the velocity space by

[f ](x, t) =

∫

Rdf(x, v, t) dv.

Then the collision operator (5.29) can be written as

(Q(f))(x, v, t) = φ(x)(M(v)[f ]− f(v))

=M(v)[f ]− f(v)

τ(x), (5.30)

where τ(x) := 1/φ(x) is called relaxation time, and the operator (5.30) is termedrelaxation-time operator (also see (3.35)).

We use the Hilbert expansion

f = f0 + αf1 + α2f2 + · · ·

in the Boltzmann equation (5.28) and identify terms of the same order of α:

• terms of order α0:M(v)[f0]− f0 = 0, (5.31)

• terms of order α1:

v · ∇xf0 −q

mE · ∇vf0 =

M(v)[f1]− f1τ(x)

, (5.32)

74

• terms of order α2:

∂tf0 + v · ∇xf1 −q

mE · ∇vf1 =

M(v)[f2]− f2τ(x)

. (5.33)

Equation (5.31) implies

f0(x, v, t) =M(v)[f0](x, t) =M(v)n(x, t),

where n := [f0] is the particle density. Multiplying (5.32) by −qv and integratingover v ∈ Rd implies

J := −q∫

Rdvf1 dv

= qτ(x)

∫

Rd

(v · ∇xf0 −

q

mE · ∇vf0

)v dv + q[f1]

∫

RdM(v)v dv

= qτ(x)

(∫

Rdv ⊗ vM(v) dv · ∇xn−

q

m

∫

Rdv ⊗∇vM(v) dv · En

).

Since∫

RdvivjM(v) dv = 0 for i 6= j,

∫

Rdv2iM(v) dv =

kBT

(2π)1/2m

∫

Rz2e−z2/2 dz =

kBT

m,

∫

Rdv ⊗∇vM(v) dv = − m

kBT

∫

Rdv ⊗ vM(v) dv = −Id,

we have

J =q2τ(x)

m

(kBT

q∇xn+ nE

)= qµn(UT∇xn+ nE),

where µn = qτ/m is the electron mobility and UT = kBT/q the thermal voltage.This shows that J(x, t) can be interpreted as the electron current density. Finally,we integrate (5.33) over v ∈ Rd:

∂tn+

∫

Rdv · ∇xf1 dv −

q

mE ·∫

Rd∇vf1 dv =

1

τ(x)

∫

Rd(M(v)[f2]− f2) dv.

The third term of the left-hand side vanishes, by the divergence theorem. Theintegral of the right-hand side also vanishes. Finally, the second term on the left-hand side equals −(1/q)div J. Thus we have derived the drift-diffusion equations

∂tn−1

qdiv J = 0

J = qµn(UT∇xn+ nE).

75

5.3 The hydrodynamic equations

We derive the hydrodynamic model from the Boltzmann equation by employ-ing the so-called moment method. The idea of this method is to multiply theBoltzmann equation by powers of the velocity components, to integrate over thevelocity space and to derive evolution equations for the integrals. We considerthe classical Boltzmann equation for one type of charge carriers (say, electrons):

∂tf + v · ∇xf −q

mn

E · ∇vf = Q(f) (5.34)

with the low-density collision operator

Q(f) =

∫

Rdφ(x, v, v′)(Mf ′ −M ′f) dv′,

where the Maxwellian is given by

M(v) =

(mn

2πkBT0

)d/2

exp

(−mn|v|2

2kBT0

)(5.35)

and f ′ and M ′ means evaluation at v′. We assume that the scattering rate φis symmetric in v and v′. The first moments of the distribution function f aredefined by

〈1〉(x, t) =

∫

Rdf(x, v, t) dv,

〈vj〉(x, t) =

∫

Rdvj f(x, v, t) dv,

〈vivj〉(x, t) =

∫

Rdvivj f(x, v, t) dv, i, j = 1, . . . , d,

and so on. We write

〈v〉 = (〈vj〉)j, 〈v2〉 = 〈v ⊗ v〉 = (〈vivj〉)ij,

〈v3〉 = (〈vivjvk〉)ijk.Multiply (5.34) by powers of vj and integrate over v ∈ Rd. This leads to themoment equations

∂t〈1〉+ div x〈v〉 =∫

RdQ(f) dv,

∂t〈v〉+ div x〈v2〉+q

mn

〈1〉E =

∫

Rdv Q(f) dv,

∂t〈v2〉+ div x〈v3〉+2q

mn

〈v〉 ⊗ E =

∫

Rdv ⊗ v Q(f) dv,

76

and so on. The symmetry of φ implies

∫

RdQ(f) dv =

∫

Rd

(∫

Rdφ(x, v, v′)M(v) dv

)f ′ dv′

−∫

Rd

(∫

Rdφ(x, v, v′)M(v′) dv′

)f dv

= 0.

The moments are related to physical quantities. In fact, we define the numberdensity n, the current density J, the energy tensor E , and the energy e by

n = 〈1〉, J = −q〈v〉,

E =mn

2

1

n〈v ⊗ v〉, e =

mn

2

1

n

d∑

j=1

〈vjvj〉 =mn

2

1

n〈|v|2〉.

With these definitions we obtain the conservation laws of mass, momentum andenergy:

∂tn−1

qdiv xJ = 0, (5.36)

∂tJ −2q

mn

div x(nE)−q2

mn

nE = −q∫

Rdv Q(f) dv, (5.37)

∂t(ne) +mn

2div x〈v|v|2〉 − J · E =

mn

2

∫

Rd|v|2Q(f) dv. (5.38)

The moment method has two difficulties. First, a truncation of the hierarchyof moment equations does not give a closed system, i.e., the equation for the j-thmoment contains always a moment of order j + 1. Second, the terms originatingfrom the collision operator generally do not depend on the moments in a simpleway. The first difficulty can be overcome by making an ansatz for the distributionfunction. In order to deal with the integrals involving the collision operator, wechoose a particular scattering rate φ.

Lemma 5.1 shows that the Maxwellian (5.35) lies in the kernel of Q(f): Theansatz f = nM would yield J = −q〈v〉 = 0. More interesting equations can beobtained from the so-called shifted Maxwellian

fe(x, v, t) = n

(mn

2πkBT

)d/2

exp

(−mn|v − v|2

2kBT

),

where n = n(x, t), T = T (x, t) and v = v(x, t) are interpreted as the electronnumber density, electron temperature and mean velocity, respectively. Substitut-

77

ing z =√mn/kBT (v − v) and using the identities

∫

Re−x2/2 dx =

√2π,

∫

Rxe−x2/2 dx = 0,

∫

Rx2e−x2/2 dx =

∫

R1 · e−x2/2 dx =

√2π,

∫

Rx3e−x2/2 dx = 2

∫

Rxe−x2/2 dx = 0,

the moments corresponding to the above ansatz are

〈1〉 =

∫

Rdfe dv = n,

〈v〉 =

∫

Rdv fe dv =

∫

Rd

(v +

√kBT

mn

z

)n

(2π)d/2e−|z|2/2 dz = nv,

〈v ⊗ v〉 =

∫

Rd

(v +

√kBT

mn

z

)⊗(v +

√kBT

mn

z

)n

(2π)d/2e−|z|2/2 dz

= (v ⊗ v)n+kBT

mn

n

(2π)d/2

∫

Rdz ⊗ z e−|z|2/2 dz

= (v ⊗ v)n+kBT

mn

nId, Id = identity matrix in Rd×d,

〈vj|v|2〉 =d∑

i=1

∫

Rd

(vj +

√kBT

mn

zj

)(vi +

√kBT

mn

zi

)2n

(2π)d/2e−|z|2/2 dz

= vj|v|2n+kBT

mn

n

(2π)d/2

d∑

i=1

vj

∫

Rd|zi|2 e−|z|2/2 dz

+ 2kBT

mn

n

(2π)d/2

d∑

i=1

vi

∫

Rdzizj e

−|z|2/2 dz

+

(kBT

mn

)3/2n

(2π)d/2

∫

Rdzj|z|2 e−|z|2/2 dz

= vj|v|2n+kBT

mn

nd∑

i=1

(vj + 2viδij)

=2

mn

vjn

(mn

2|v|2 + d+ 2

2kBT

).

78

Thus, the energy is the sum of kinetic and thermal energy:

e =mn

2|v|2 + d

2kBT =

mn

2q2|J |2n2

+d

2kBT,

and the third-order term can be written as

〈v|v|2〉 = 2

mn

vn(e+ kBT ).

In order to compute the terms coming from the collision operator, we assumethat the scattering rate is independent of v and v′, i.e. φ(x, v, v′) = φ0(x). Then

∫

Rdv Q(fe) dv = φ0

∫

RdMv dv

∫

Rdf ′e dv

′ − φ0∫

RdM ′ dv′

∫

Rdfe v dv

= −φ0〈v〉

=φ0qJ,

∫

Rd|v|2Q(fe) dv = φ0

∫

RdM |v|2 dv · n− φ0

∫

RdM ′ dv′ · 〈|v|2〉

= φ0n

(dkBT0mn

− 2

mn

e

).

We introduce the relaxation time by τ = 1/φ0. Then

−q∫

Rdv Q(fe) dv = −J

τ,

mn

2

∫

Rd|v|2Q(fe) dv = −n

τ

(e− d

2kBT0

).

Notice that these terms vanish in the thermal equilibrium state J = 0, T = T0.We conclude that the moment equations (5.36)-(5.38) can be written as

∂tn−1

qdiv xJ = 0, (5.39)

∂tJ −1

qdiv x

(J ⊗ Jn

)+qkBmn

∇(Tn)− q2

mn

nE = −Jτ, (5.40)

∂t(ne)−1

qdiv x[J(e+ kBT )]− J · E = −n

τ

(e− d

2kBT0

). (5.41)

For vanishing right-hand sides, these equations are the Euler equations of gas dy-namics for a gas of charged particles in an electric field. Sometimes an additionalterm −div (κ(n, t)∇T ) is added to the left-hand side of (5.41). This term whosepresence is purely heuristic is a diffusion term and makes the equation (5.41) ofparabolic type. This is desirable since we expect the temperature to satisfy anequation related to the parabolic heat equation. The heat conductivity κ(n, T )is usually modeled (in R3) by [10]

κ(n, T ) =3

2

k2Bτ

mn

nT.

79

The equations (5.39)-(5.41) including the aboce heat flow (but with differentrelaxation times in the momentum and energy equations) has been first derivedby Bløtekjær [13] and Baccarani and Wordeman [10].

The equations (5.39) and (5.40) are of hyperbolic type. The system of equa-tions (5.39)-(5.41), with or without the additional heat flux term, are referred toas the hydrodynamic equations. For constant temperature T = T0, (5.39)-(5.40)are called the isothermal hydrodynamic equations.

In a similar way, equations for the hole density, the hole current densityand the hole energy density can be derived from the corresponding Boltzmannequation for holes (see (5.2)). The equations are as above with −q replaced byq. The corresponding equations for the electrons and holes are referred to as thebipolar hydrodynamic model.

5.4 The Spherical Harmonic Expansion (SHE) model

We start with the semi-classical Boltzmann equation for the distribution functionf(x, k, t) (see Section 3.3)

∂tf +1

~∇kε · ∇xf +

q

~∇xV · ∇kf = Q(f), x ∈ Rd, k ∈ B, t > 0. (5.42)

In the following we only consider the evolution of the electrons in a single con-duction band ε(k). We proceed similarly as in [11, 18]. The collision operator isassumed to be the sum of lattice-defect collision terms (due to ionized impurities,acoustic and optical phonons) and electron-electron collision terms. These termsare separated in an elastic collision part and an inelastic collision part:

Q(f) = Qel(f) + α2Qinel(f).

We have assumed that the electric field is so large that the typical energy qU ofthe electrons, where U is the voltage bias between the contacts, is much largerthan the energy ~ω of the phonons:

α2 =~ωqU¿ 1,

and we have expanded the collision operator in terms of α2. By Section 3.3, wecan write the elastic collision operator as

(Qel(f))(x, k) =

∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))(f(k′)− f(k)) dk′ (5.43)

for functions f : Rd ×B → R, where

φ(x, k, k′) = σimp(x, k, k′) + (2Nop + 1)σop(x, k, k

′) + (2Nac + 1)σac(x, k, k′),

80

is the scattering rate, Nop, Nac are the occupation numbers of optical or acousticphonons, respectively, and δ is the delta distribution. Clearly, the integral (5.43)has to be understood symbolically. The operator Qinel(f) contains the inelasticcorrection to phonon and electron-electron collisions.

We also impose the following assumptions:

• The inelastic collision operator Qinel is linear.

• The scattering rate φ is positive and symmetric in k, k′:

φ(x, k, k′) > 0, φ(x, k, k′) = φ(x, k′, k) ∀x, k, k′. (5.44)

The Spherical Harmonic Expansion (SHE) model is derived in the so-calleddiffusion scaling, i.e. by introducing the space and time scale

xs = αx, ts = α2t.

Then (xs, ts) are called macroscopic variables. In these variables, the Boltzmannequation (5.42) reads

α2∂tf + α

(1

~∇kε · ∇xf +

q

~∇xV · ∇kf

)= Qel(f) + α2Qinel(f), (5.45)

where we have written again (x, t) instead of (xs, ts). Notice that this is exactlythe same scaling used in the derivation of the drift-diffusion equations (see Section5.1).

We use now the Hilbert expansion method to ”solve” (5.45). Inserting theexpansion

f = f0 + αf1 + α2f2 + · · ·into (5.45), using the linearity of the operators Qel and Qinel and identifying termsof the same order in α, we find:

• terms of order α0:Qel(f0) = 0, (5.46)

• terms of order α1:

Qel(f1) =1

~∇kε · ∇xf0 +

q

~∇xV · ∇kf0, (5.47)

• terms of order α2:

Qel(f2) = ∂tf0 +1

~∇kε · ∇xf1 +

q

~∇V · ∇kf1 −Qinel(f0). (5.48)

81

First we wish to solve (5.46). For this, we need to prove some properties ofthe operator Qel defined on the space

L2(B) =f : B → B measurable:

∫

B

|f(k)|2 dk <∞.

Lemma 5.2 Assume that (5.44) hold. Then

(1) −Qel is a self-adjoint and non-negative operator on L2(B).

(2) The kernel of Qel and its orthogonal complement are given by

N(Qel) = f ∈ L2(B) : ∃g : R→ R : ∀k ∈ B : f(k) = g(ε(k)),

N(Qel)⊥ =

f ∈ L2(B) : ∀e ∈ R(ε) :

∫

B

f(k)δ(e− ε(k)) dk = 0.

(3) For all functions g = g(ε(k)) and f ∈ L2(B) it holds

Qel(gf) = g Qel(f).

In the statement of the lemma we have omitted some technical assumptionson the regularity of φ and g(ε(k)). The orthogonal complement N(Qel)

⊥ is theset of all functions f ∈ L2(B) such that

〈f, F 〉 =∫

B

f(k)F (k) dk = 0 ∀F ∈ N(Qel).

The range R(ε) of ε is defined by R(ε) = ε ∈ R : ∃k ∈ B : ε(k) = e. Theoperator −Qel is called non-negative if

−∫

B

Qel(f)f dk ≥ 0 ∀f ∈ L2(B).

Proof of Lemma 5.2. (1) Let x ∈ Rd, f, g ∈ L2(B) and set f = f(k), f ′ = f(k′),g = g(k), g′ = g(k′), ε = ε(k) and ε′ = ε(k′). Then, using the property

∫

B

δ(ε′ − ε)ψ(k, k′) dk′ =∫

B

δ(ε− ε′)ψ(k, k′) dk′

for any function ψ and the symmetry of φ, we obtain∫

B

Qel(f)g dk =1

2

∫

B

∫

B

φ(x, k, k′)δ(ε′ − ε)(f ′ − f)g dk′ dk

+1

2

∫

B

∫

B

φ(x, k′, k)δ(ε− ε′)(f − f ′)g′ dk dk′

= −1

2

∫

B

∫

B

φ(x, k, k′)δ(ε′ − ε)(f ′ − f)(g′ − g) dk′ dk

=

∫

B

Qel(g)f dk.

82

This shows that Qel is symmetric and hence self-adjoint. Taking f = g gives∫

B

Qel(f)f dk = −1

2

∫

B

∫

B

φ(x, k, k′)δ(ε′ − ε)(f ′ − f)2 dk′ dk ≤ 0 (5.49)

and thus, −Qel is non-negative.(2) Let f ∈ N(Qel), i.e. Qel(f) = 0. By (5.49), this implies

δ(ε′ − ε)(f ′ − f)2 = 0 for almost all k, k′ ∈ B.

The delta distribution has the ”property” δ(z) = 0 for all z 6= 0. (We treat thedelta distribution as a function; this is mathematically not correct but avoidstechnical calculations.) Therefore

ε(k′) = ε(k) and f(k′) = f(k) for almost all k, k′ ∈ B.

Hence f must be constant on each energy surface k : ε(k) = e. This impliesthat f is a function of ε(k) and proves the first assertion.

Let f ∈ N(Qel)⊥ and F ∈ N(Qel). Then F (k) = g(ε(k)) for some function g

and

0 =

∫

B

f(k)F (k) dk

=

∫

B

f(k)g(ε(k)) dk

=

∫

B

f(k)

(∫

Rg(e)δ(ε(k)− e) de

)dk

=

∫

R

(∫

B

f(k)δ(ε(k)− e) dk)g(e) de. (5.50)

Here we have used the definition of the delta distribution∫

Rδ(z − e)ψ(e) de = ψ(z) for any (regular) function ψ. (5.51)

Equation (5.50) holds for all F ∈ N(Qel) and thus for any function g. We infer∫

B

f(k)δ(ε(k)− e) dk = 0 for almost all e ∈ R(ε).

This proves (2).(3) The first integral in

(Qel(gf))(k) =

∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))f(k′)g(ε(k′)) dk′

−∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))f(k)g(ε(k)) dk′

83

equals, after the change of unknown e = ε(k′) and the definition (5.51),

∫

R(ε)

φ(x, k, k′)δ(e− ε(k))f(k′)g(e)∣∣∣∣det

dε

dk(k′)

∣∣∣∣−1

de

= φ(x, k, k′)f(k′)g(ε(k))

∣∣∣∣detdε

dk(k′)

∣∣∣∣−1∣∣∣∣∣e=e(k)=ε(k′)

= g(ε(k))

∫

R(ε)

φ(x, k, k′)δ(e− ε(k))f(k′)∣∣∣∣det

dε

dk(k′)

∣∣∣∣−1

de

= g(ε(k))

∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))f(k′) dk′.

Notice that the use of the transformation formula is rather formal since ε may notsatisfy the required technical assumptions. However, the above computation canbe made mathematically rigorous by employing the coarea formula (see [11, 18]for details). We infer

(Qel(gf))(k) = g(ε(k))

∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))f(k′) dk′

−g(ε(k))∫

B

φ(x, k, k′)δ(ε(k′)− ε(k))f(k) dk′

= g(ε(k))(Qel(f))(k).

This proves (3). ¤

Remark 5.3 From the proof of Lemma 5.2(3) we conclude the following generalresult: For all ψ(k, k′) and g(ε(k)) it holds∫

B

ψ(k, k′)δ(ε(k′)− ε(k))g(ε(k′)) dk′ = g(ε(k))

∫

B

ψ(k, k′)δ(ε(k′)− ε(k)) dk′.(5.52)

We can now solve the equations (5.46)-(5.48). By Lemma 5.2(2), any solutionof (5.46) can be written as

f0(x, k, t) = F (x, ε(k), t)

for some function F. Thus we can reformulate (5.47) as

Qel(f1) =1

~∇kε ·

(∇xF + q∇xV

∂F

∂ε

).

As ∇xF + q∇xV (∂F/∂ε) only depends on ε(k) (and on the parameters x, t), anysolution of this equation can be written, by Lemma 5.2(3), as

f1(x, k, t) = −λ(x, k) ·(∇xF + q∇xV

∂F

∂ε

)+ F1(x, ε(k), t)

84

where F1 ∈ N(Qel) and λ(x, k) is a solution of

Qel(λ) = −1

~∇kε. (5.53)

More precisely, this equation has to be considered componentwise, i.e.

Qel(λi) = −~−1 ∂ε

∂ki, i = 1, . . . , d.

Notice that λ depends on the parameter x since φ depends on x. The aboveequation is solvable if and only if ∇kε ∈ R(Qel) (the range of Qel). It is notdifficult to see that R(Qel) is closed. From functional analysis (and the self-adjointness of Qel) follows that R(Qel) = N(Qel)

⊥. Thus (5.53) is solvable if andonly if ∇kε ∈ N(Qel)

⊥. By Lemma 5.2(2) this is equivalent to∫

B

∇kε(k)δ(e− ε(k)) dk = 0 ∀e ∈ R(ε).

Now, using the relationdH

dz(z) = δ(z),

where H is the Heaviside function, we have∫

B

∇kε(k)δ(e− ε(k)) dk = −∫

B

∇kH(e− ε(k)) dk = 0, (5.54)

since ε is periodic on B. Thus, (5.53) is solvable. We choose the particular solutionof (5.47) as

f1(x, k, t) = −λ(x, k) ·(∇xF + q∇xV

∂F

∂ε

).

It remains to solve (5.48). Again, (5.48) is solvable if and only if the right-handside of (5.48) is an element of N(Qel)

⊥ or, equivalently,∫

B

(∂tf0 +

1

~∇kε · ∇xf1 +

q

~∇xV · ∇kf1 −Qel(f0)

)δ(e− ε(k)) dk = 0 (5.55)

for all e ∈ R(ε). We define the density of states of energy e by

N(e) =

∫

B

δ(e− ε(k)) dk (5.56)

(compare with (2.13) in Section 2.3). Then, by (5.52),∫

B

∂tf0δ(e− ε(k)) dk = ∂t

∫

B

F (x, ε(k), t)δ(e− ε(k)) dk

= ∂tF (x, e, t)

∫

B

δ(e− ε(k)) dk

= N(e)∂tF (x, e, t).

85

Furhermore, we introduce the electron current density by

J(x, e, t) = −q∫

B

1

~∇kε(k)f1(x, k, t)δ(e− ε(k)) dk.

Then we can write the second term in (5.55) as∫

B

1

~∇kε · ∇xf1δ(e− ε(k)) dk = −1

qdiv xJ(x, e, t).

Using the expression for f1 and the property (5.52), we have

J(x, e, t) = q

∫

B

1

~∇kε(k)λ(x, k) ·

(∇xF + q∇xV

∂F

∂ε

)(x, ε(k), t)

× δ(e− ε(k)) dk

= D(x, e)

(∇xF + q∇xV

∂F

∂ε

)(x, e, t),

where

D(x, e) =q

~

∫

B

∇kε(k)⊗ λ(x, k)δ(e− ε(k)) dk ∈ Rd×d (5.57)

is called diffusion matrix.The last term in (5.55) is a collision term, averaged over the energy surface

e = ε(k):

(S(F ))(x, e, t) =

∫

B

(Qinel(F ))(x, k, t)δ(e− ε(k)) dk. (5.58)

It remains to compute the third term in (5.55). For any (smooth) function ψwe obtain, using the definition of δ and integrating by parts,

∫

R(ε)

ψ(e)

∫

B

∇kf1(x, k, t)δ(e− ε(k)) dk de

=

∫

B

ψ(ε(k))∇kf1(x, k, t) dk

= −∫

B

f1(x, k, t)dψ

dε(ε(k))∇kε(k) dk

= −∫

R(ε)

dψ

dε(e)

∫

B

∇kε(k)f1(x, k, t)δ(e− ε(k)) dk de

=~q

∫

R(ε)

dψ

dε(e)J(x, e, t) de

= −~q

∫

R(ε)

ψ(e)∂J

∂ε(x, e, t) de.

Since ψ is arbitrary, it follows∫

B

∇kf1(x, k, t)δ(e− ε(k)) dk = −~q

∂J

∂ε(x, e, t)

86

andq

~

∫

B

∇xV · ∇kf1δ(e− ε(k)) dk = −∇xV ·∂J

∂ε(x, e, t).

We have shown that the Hilbert expansion (5.46)-(5.48) is solvable if and onlyif f0(x, k, t) = F (x, ε(k), t), where F is a solution of

N(ε)∂tF −1

qdiv xJ −∇xV ·

∂J

∂ε= S(F ), (5.59)

J(x, ε, t) = D(x, ε)

(∇xF + q∇xV

∂F

∂ε

), x ∈ Rd, ε ∈ R(ε), t > 0, (5.60)

where D(x, ε) is given by (5.57). The equations (5.59)-(5.60) are referred toas Spherically Harmonic Expansion (SHE) model. It has been first derived inthe physical literature for spherically symmetric band diagrams and rotationallyinvariant collision operators from the Boltzmann equation [26, 27]. In fact, theabove derivation of the SHE model does not require any assumption of sphericalsymmetry. The equations can be written more clearly in terms of the total energyu = ε− qV (x, t). Introducing the change of unknowns

F (x, ε, t) = f(x, u, t) = f(x, ε− qV (x, t), t),

D(x, ε) = d(x, u, t) = d(x, ε− qV (x, t), t),

N(ε) = n(x, u, t) = N(u+ qV (x, t)),

we obtain

∇xF = ∇xf − q∇xV∂f

∂u, ∂tF = ∂tf − q∂tV

∂f

∂u

and therefore

n(x, u, t)∂tf − div x(d(x, u, t)∇xf) = S(f) + n(x, u, t)q∂tV∂f

∂u.

This shows, together with the following proposition, that the SHE model is ofparabolic type.

Proposition 5.4 The diffusion matrix D is a symmetric non-negative d × dmatrix. Moreover, there exists a constant K > 0 such that for all z ∈ Rd and allx ∈ Rd and e ∈ R(ε),

z>D(x, e)z ≥ K

N(ε)

∫

B

|∇kε(k) · z|2δ(e− ε(k)) dk.

The fact that the matrix D(x, ε) is non-negative is a direct consequence ofthe non-negativity of the operator −Qel (see Lemma 5.2(1)). The symmetry ofD(x, e) is related to the so-called Onsager reciprocity relation of non-equilibriumthermodynamics [22].

87

Proof. We only prove that D(x, ε) = (Dij)ij is symmetric. The proof of thesecond assertion is more technical and can be found in [11, Sec. III.4]. Wecompute, for (smooth) functions ψ,

∫

RDijψ(e) de = − q

~

∫

R

∫

B

∂ε

∂kiλjδ(e− ε(k)) dk ψ(e) de

= −q∫

B

Qel(λi)λjψ(ε(k)) dk

(using the definition of δ)

= −q∫

B

λiQel(λjψ(ε)) dk

(since Qel is self-adjoint by Lemma 5.2(1))

= −q∫

B

λiQel(λj)ψ(ε) dk

(by Lemma 5.2(3))

=q

~

∫

R

∫

B

λi∂ε

∂kjδ(e− ε(k)) dk ψ(e) de

=

∫

RDjiψ(e) de.

Since ψ is arbitrary, Dij(x, e) = Dji(x, e) for x ∈ Rd, e ∈ R(ε). ¤

For more explicit expressions for N(e) and D(x, e) we consider the followingexample.

Example 5.5 (Spherically symmetric energy bands)Assume that the scattering rate only depends on ε(k), i.e.

φ(x, k, k′) = φ(x, ε(k)) for all k, k′ with ε(k′) = ε(k),

that the energy band is spherically symmetric (thus, B = Rd) and strictly mono-tone in |k|, i.e.

ε = ε(|k|),and that d = 3. Notice that the first assumption makes sense since due to theterm δ(ε(k′) − ε(k)) in the definition of Qel, the scattering rate only needs tobe defined on the energy surface k′ : ε(k′) = ε(k) of energy ε(k). The secondassumption implies that ε(|k|) is invertible.

From the first assumption and the definition (5.56) of the density of statesN(e) follows, with ε = ε(k), ε′ = ε(k′),

(Qel(f))(x, k, t) = φ(x, ε)

∫

Rdδ(ε′ − ε)f(k′) dk′ − φ(x, ε)N(ε)f(k)

=1

τ(x, ε)([f ]− f) (x, k, t), (5.61)

88

where

τ(x, ε) =1

φ(x, ε)N(ε)

is called relaxation time and

[f ](k) =1

N(ε)

∫

Rdδ(ε′ − ε)f(k′) dk′

is the average of f on the energy surface k′ ∈ B : ε(k′) = ε(k). The expression(5.61) is called relaxation-time operator.

It follows that a solution λ(x, k) of Qel(λ) = −~−1∇kε (see (5.53)) is given by

λ(x, k) = τ(x, ε(k))~−1∇kε(k).

Indeed, since, by (5.52) and (5.54),

[λ] =1

N(ε)

∫

Rdδ(ε′ − ε)τ(x, ε′)~−1∇kε(k

′) dk′

=1

N(ε)τ(x, ε)~−1

∫

Rdδ(ε′ − ε)∇kε(k

′) dk′

= 0,

we obtain

Qel(λ) =1

τ(x, ε)([λ]− λ) = − λ

τ(x, ε)= −1

~∇kε.

Thus, again using (5.52), the diffusion matrix becomes

D(x, e) = q~−2

∫

Rd∇kε(k)⊗∇kε(k)τ(x, ε)δ(e− ε) dk

= q~−2τ(x, e)

∫

Rd∇kε(k)⊗∇kε(k)δ(e− ε(k)) dk.

This expression can be further simplified under the second assumption. As εonly depends on |k|, it is convenient to introduce spherical coordinates k = ρω,where ρ = |k| ∈ [0,∞) is the modulus of k and ω ∈ Sd−1 are the angle variableson the unit sphere Sd−1 ⊂ Rd. Then

∇kε(|k|) = ε′(|k|) k|k| = ε′(ρ)ω

and the volume element dk transforms symbolically to ρd−1F (ω) dρ dω, whereF (ω) is the modulus of the determinant of the Jacobian of the angle transform.

89

We compute

D(x, e) = qτ(x, e)~−2

∫ ∞

0

∫

Sd−1

ε′(ρ)2ω ⊗ ωδ(e− ε(ρ))ρd−1F (ω) dρ dω

= qτ(x, e)~−2

∫ ∞

0

ε′(ρ)2δ(e− ε(ρ))ρd−1 dρ ·∫

Sd−1

ω ⊗ ωF (ω) dω

= qτ(x, e)~−2

∫

R(ε)

ε′(ρ)δ(e− η)ρd−1 dη ·∫

Sd−1

ω ⊗ ωF (ω) dω

(transforming η = ε(ρ) and dη = ε′(ρ) dρ)

= qτ(x, e)~−2ε′(ρ)ρd−1∣∣∣e=ε(ρ)

·∫

Sd−1

ω ⊗ ωF (ω) dω.

In the three dimensional case d = 3 (third assumption) this becomes

D(x, e) =4πq

3~2τ(x, e)ε′(|k|)|k|2 Id, (5.62)

where |k| is such that e = ε(|k|) and Id ∈ R3×3 is the identity matrix. Indeed,from

ω =

sin θ cosφsin θ sinφ

cos θ

, 0 ≤ φ < 2π, 0 ≤ θ < π,

and F (ω) dω = sin θ dθ dφ we obtain

∫

S2

(ω ⊗ ω)ijF (ω) dω =

∫ 2π

0

∫ π

0

ωiωj sin θ dθ dφ.

For i 6= j the integral vanishes. We compute for

i = j = 1 :

∫ 2π

0

cos2 φ dφ ·∫ π

0

sin3 θ dθ = π · 43=

4π

3,

i = j = 2 :

∫ 2π

0

sin2 φ dφ ·∫ π

0

sin3 θ dθ = π · 43=

4π

3,

i = j = 3 : 2π ·∫ π

0

cos2 θ sin θ dθ = 2π · 23=

4π

3,

and (5.62) follows.Furthermore,

N(e) =

∫

Rdδ(e− ε(|k|)) dk =

∫ ∞

0

∫

Sd−1

δ(e− ε(ρ))ρd−1F (Ω) dρ dω

=

∫

Rδ(e− η) ρ

d−1

ε′(ρ)dη ·

∫

Sd−1

F (ω) dω

=Sd−1

ε′(ρ)

∣∣∣e=ε(ρ)

∫

Sd−1

F (ω) dω,

90

and for d = 3 we get∫

Sd−1

F (ω) dω =

∫ 2π

0

∫ π

0

sin θ dθ dφ = 4π

and therefore

N(e) = 4π|k|2ε′(|k|) with e = ε(|k|). (5.63)

It is convenient to express D(x, e) and N(e) in terms of the function γ, where

|k|2 = γ(ε(|k|)).Differentiating this equation with respect to |k| yields

2|k| = γ ′(ε(|k|))ε′(|k|)and thus

ε′(|k|) = 2√γ(e)

γ′(e)with e = ε(|k|).

It follows from (5.62) and (5.63) for d = 3

N(e) = 4πγ(e)γ ′(e)

2√γ′(e)

= 2π√γ(e)γ ′(e),

D(x, e) =4πq

3~2γ(e)

2πφ(x, e)√γ(e)γ ′(e)

2√γ(e)

γ′(e)Id =

4q

3~2γ(e)

φ(x, e)γ ′(e)2Id.

Example 5.6 (Parabolic band approximation)We assume that d = 3, φ = φ(x, ε(k)) and

ε(k) =~2

2m∗|k|2.

This implies γ(e) = 2m∗e/~2 and (see Example 5.5)

N(e) = 2π

(2m∗

~2

)3/2√e,

D(x, e) =2q

3m∗

e

φ(x, e)Id, e > 0.

It remains to determine the average collision operator S(F ) (see (5.58). Theprecise structure of this term depends on the assumptions on the inelastic collisionoperator. Here, we only notice that a simplified expression is given by the Fokker-Planck approximation

S(F ) =∂

∂ε

[A(ε)

(F + kBTL

∂F

∂ε

)], (5.64)

where TL is the lattice temperature and A(ε) depends on the scattering rate ofthe inelastic collisions, averaged on the energy surface (see [40]).

91

5.5 The energy-transport equations

The SHE model is much simpler than the Boltzmann equation since the SHEmodel has a parabolic structure and it has to be solved in a (d+ 2)-dimensional(x, e, t)-space instead of the (2d+1)-dimensional (x, k, t)-space for the Boltzmannequation. However it is numerically more expensive than usual macroscopic mod-els due to the additional energy variable. In this section we derive a macroscopicenergy-transport model from the SHE model following [11].

In [12] an energy-transport model has been derived directly from the Boltz-mann equation. In this approach, the electron-electron collision term is assumedto be dominant compared to the phonon collisions which can be criticized. Herewe assume that the electron-electron collision term is dominant compared to theinelastic part of the phonon collision term. More precisely, we start with the SHEmodel

N(ε)∂tF −1

qdiv xJ −∇xV ·

∂J

∂ε= S(F ), (5.65)

J = D(x, ε)

(∇xF + q∇xV ·

∂F

∂ε

), x ∈ Rd, ε ∈ R, t > 0, (5.66)

for the distribution function F (x, ε, t). The density of states N(ε), the diffusionmatrix D(x, ε), and the averaged inelastic collision operator S(F ) are defined in(5.56), (5.57), and (5.58), respectively. We recall that the inelastic collision op-erator contains the inelastic correction to phonon collisions and electron-electroncollisions. Let β2 be the ratio of the typical electron-electron collision time andthe phonon-collision time. We assume that β2 ¿ 1 which corresponds to so-calledhot-electron regimes. Moreover, we suppose that

• S(F ) = β2Sph(F ) + Se(F ),

• xs = βx, ts = β2t, Js = J/β,

• Se is a linear operator.

In the first assumption, Sph and Se are the averaged inelastic phonon and electron-electron operators, respectively. The second assumption means that we are con-sidering a macroscopic space and time scale and “small” current densities. Fi-nally, the third assumption is only for convenience (see below). In the macroscopicscale, the SHE equation (5.65) reads, writing again (J, x, t) instead of (Js, xs, ts),

N(ε)∂tF −1

qdiv xJ −∇xV ·

∂J

∂ε= Sph(F ) +

1

β2Se(F ), (5.67)

J = D(x, ε)

(∇xF + q∇xV ·

∂F

∂ε

). (5.68)

92

We use the Hilbert expansion

F = F0 + βF1 + · · · , J = J0 + βJ1 + · · ·

in (5.67) and identify equal powers of β. This leads to

• terms of order β−2:Se(F0) = 0, (5.69)

• terms of order β0:

N(ε)∂tF0 −1

qdiv xJ0 −∇xV ·

∂J0∂ε− Sph(F0) = Se(F1). (5.70)

If Se is not a linear operator, we would need to replace Se(F1) by (DF0Se)(F1),the derivative of Se at F0 applied to F1.

We need the following properties to the collision operator Se:

Lemma 5.7 Assume that the operators Se, Sph are given by (5.58), where

(Se(F ))(x, e, t) =

∫

B

(Qe(F ))(x, k, t)δ(e− ε(k)) dk,

(Sph(F ))(x, e, t) =

∫

B

(Qac(F ) +Qop(F ))(x, k, t)δ(e− ε(k)) dk,

and Qe, Qac and Qop are the operators of electron-electron, acoustic phonon andoptical phonon collisions, respectively (see [11] for precise definitions of theseoperators).

(1) It holds for any (smooth) function F : R→ R∫

R(Se(F ))(ε) dε = 0,

∫

R(Se(F ))(ε)ε dε = 0,

∫

R(Sph(F ))(ε) dε = 0.

(2) The kernel of Se is given by

N(Se) = F : R→ R : ∃µ ∈ R, T > 0 : F = Fµ,T,

where

Fµ,T (ε) =1

1 + exp((ε− qµ)/kBT ),

and kB is the Boltzmann constant.

(3) The equation Se(F ) = G is solvable if and only if∫

RG(ε) dε = 0 and

∫

RG(ε)ε dε = 0.

93

Proof. See [12] and [11]. The proof is similar to that of Lemma 5.2. ¤

The variable µ is called chemical potential and T is the electron temperature.The quantity εF = qµ is termed Fermi energy (see Section 2.3).

From Lemma 5.7(2) follows that the solutions of (5.69) are given by

F0(x, ε, t) = Fµ(x,t),τ(x,t)(ε),

for some parameters µ(x, t) ∈ R, T (x, t) > 0. Lemma 5.7(3) implies that (5.70)is solvable if and only if

∫

R

(N(ε)∂tFµ,T −

1

qdiv xJ0 −∇xV ·

∂J0∂ε− Sph(F0)

)(1ε

)dε = 0. (5.71)

We introduce the particle density and the particle energy density, respectively,by

n(µ, T ) =

∫

RFµ,T (ε)N(ε) dε, E(µ, T ) =

∫

RFµ,T (ε)εN(ε) dε, (5.72)

and the macroscopic particle and energy current densities, respectively, by

J(x, t) =

∫

RJ0(x, ε, t) dε, S(x, t) =

1

q

∫

RJ0(x, ε, t)ε dε.

Then, using Lemma 5.7(1), we can write (5.71) as

∂tn−1

qdiv xJ = ∇xV ·

∫

R

∂J0∂ε

dε+

∫

RSph(Fµ,T ) dε = 0,

∂tE − div xS = ∇xV ·∫

R

∂J0∂ε

ε dε+

∫

RSph(Fµ,T )ε dε

= −∇xV ·∫

RJ0 dε+

∫

RSph(Fµ,T )ε dε

= −∇xV · J +W (µ, T ),

where

W (µ, T ) =

∫

R(Sph(Fµ,T )) (ε)ε dε (5.73)

is called relaxation term. It can be seen that W also depends on the latticetemperature TL, i.e. W (µ, T ) = W (µ, T, TL) (this requires knowledge about theprecise structure of Sph; we refer to [12]).

It remains to compute the fluxes J and S. We can interpret Fµ,T (ε) as afunction of the variables u1 := qµ/kBT , u2 := −1/kBT and ε:

F (u1, u2, ε) = F

(qu

kBT,−1kBT

, ε

):= Fµ,T (ε).

94

Then

F (u1, u2, ε) =1

1 + e−(εu2+u1)

and

∂F

∂u1=

e−(εu2+u1)

(1 + e−(εu2+u1))2= Fµ,T (1− Fµ,T ),

∂F

∂u2=

εe−(εu2+u1)

(1 + e−(εu2+u1))2= εFµ,T (1− Fµ,T ),

∂F

∂ε=

u2 e−(εu2+u1)

(1 + e−(εu2+u1))2= − 1

kBTFµ,T (1− Fµ,T ).

This yields

J0 = D(x, ε)

[∂F

∂u1∇(qu

kBT

)+∂F

∂u2∇( −1kBT

)+ q∇xV ·

∂F

∂ε

]

and

J = D11∇(qu

kBT

)+D12∇

( −1kBT

)−D11

q∇xV

kBT, (5.74)

qS = D21∇(qu

kBT

)+D22∇

( −1kBT

)−D21

q∇xV

kBT, (5.75)

where the coefficients

Dij(x, µ, T ) =

∫

RD(x, ε)Fµ,T (1− Fµ,T )ε

i+j−2 dε, i, j = 1, 2, (5.76)

are d× d matrices.Summarizing, we can write the energy transport equations as

∂tn−1

qdiv xJ = 0, (5.77)

∂tE − div xS +∇xV · J = W (µ, T ), (5.78)

where n(µ, T ), E(µ, T ) are defined in (5.72), W (µ, T ) is given by (5.73), andfinally, J(µ, T ) and S(µ, T ) are given by (5.74) and (5.75), respectively.

The (2d × 2d)-matrix D = (Dij(x, µ, T ))ij and the relaxation term have thefollowing properties.

Proposition 5.8 It holds

(1) The (2d× 2d)-matrix D = (Dij)ij is symmetric, i.e. D12 = D21.

(2) The (d× d)-matrices Dij are symmetric, i.e. D>ij = Dij for i, j = 1, 2.

95

(3) Assume that the 2d functions

∂ε

∂k1, . . . ,

∂ε

∂kd, ε

∂ε

∂k1, . . . , ε

∂ε

∂kd

are linearly independent. Then D(x, µ, T ) is symmetric positive definitivefor any µ ∈ R, T > 0.

(4) The relaxation term W (µ, T, TL) is monotone in the sense of operators, i.e.

W (µ, T, TL) · (T − TL) ≤ 0.

The hypothesis of part (3) of the above proposition is a geometric assumptionon the band structure. It expresses for the case d = 3 that the energy bandneeds to have a real three-dimensional structure, excluding bands depending onlyon one or two variables, for instance. Notice that the statement of part (4)justifies the name “relaxation term”: The temperature of the system is expectedto relax to the (constant) lattice temperature if there are no forces influencingthe temperature.

Proof. We only prove parts (1)-(3) since for the proof of (4) we need the precisestructure of Sph (see [12, Lemma 4.11] for a detailed proof). Part (1) follows fromthe symmetry of the matrix, and part (2) is a consequence of the symmetry ofD. It remains to prove part (3).

Let ξ = (ξ(1), ξ(2))> ∈ R2d with ξ(i) ∈ Rd, i = 1, 2, such that ξ 6= 0. Then

ξ>Dξ =

∫

Rξ>(

D(x, ε) εD(x, ε)εD(x, ε) ε2D(x, ε)

)ξFµ,T (1− Fµ,T ) dε

(by the definition of D)

=

∫

R

[(ξ(1))>Dξ(1) + 2ε

(ξ(1))>Dξ(2) + ε2

(ξ(2))>Dξ(2)

]

×Fµ,T (1− Fµ,T ) dε (since D = D(x, ε) is symmetric)

=

∫

R

(ξ(1) + εξ(2)

)>D(ξ(1) + εξ(2)

)Fµ,T (1− Fµ,T ) dε

≥ K

N(ε)

∫

R

∫

B

|∇kε(k) ·(ξ(1) + εξ(2)

)|2 δ(ε− ε(k))

×Fµ,T (1− Fµ,T ) dk dε (by Proposition 5.4)

=K

N(ε)

∫

R

∫

B

∣∣∣∣(

∇kε(k)ε(k)∇kε(k)

)· ξ∣∣∣∣2

δ(ε− ε(k))Fµ,T (1− Fµ,T ) dk dε

> 0;

since otherwise would(

∇kε(k)ε(k)∇kε(k)

)· ξ = 0, ξ 6= 0,

96

imply that (∇kε, ε(k)∇kε) is linearly dependent, which is a contradiction. ¤

More explicit expressions for n, E and D can be derived under the assumptionsof Example 5.5 and using Maxwell-Boltzmann statistics.

Example 5.9 (Spherically symmetric energy bands)We impose the following hypotheses:

• The space dimension is d = 3 (to simplify the computations).

• The distribution function Fµ,T is approximated by exp(−(ε − qµ)/kBT ).This is possible if ε− qµÀ kBT. In particular,

Fµ,T (1− Fµ,T ) ≈ e−(ε−qµ)/kBT · 1.

• The scattering rate φ depends only on x and ε(k).

• The energy band ε is spherically symmetric and strictly monotone in |k|.

From Example 5.5 and (5.72) follows (if R(ε) = R)

n(µ, T ) =

∫

Re−(ε−qµ)/kBTN(ε) dε = 2πeqµ/kBT

∫

R

√γ(ε)γ ′(ε)e−ε/kBT dε,

E(µ, T ) =

∫

Re−(ε−qµ)/kBT εN(ε) dε = 2πeqµ/kBT

∫

R

√γ(ε)γ ′(ε)εe−ε/kBT dε,

where |k|2 = γ(ε(k)). The diffusion matrices (5.76) become

Dij(µ, T ) =4q

3~2eqµ/kBT

∫

R

γ(ε)εi+j−2

φ(x, ε)γ ′(ε)2e−ε/kBT dε · Id. (5.79)

Example 5.10 (Parabolic band approximation)We impose the same assumptions as in Example 5.9 and additionally,

• The energy band is given by the parabolic approximation

ε(|k|) = ~2

2m∗|k|2, k ∈ R3.

• The scattering rate is given by the so-called Chen model [17, 21]

φ(x, ε) = φ0(x)√ε.

97

By Example 5.6,

N(ε) = 2π

(2m∗

~2

)3/2√ε,

and therefore

n(µ, T ) = 2π

(2m∗

~2

)3/2

eqµ/kBT

∫ ∞

0

e−ε/kBT√ε dε

= 2π

(2m∗

~2

)3/2

eqµ/kBT 1√2(kBT )

3/2

∫ ∞

0

e−z2/2z2 dz,

after substituting ε/kBT = z2/2. From

∫ ∞

0

e−z2/2z2 dz =1

2

∫

Re−z2/2z2 dz =

1

2

∫

Re−z2/2 · 1 dz =

1

2

√2π =

√π

2

we conclude

n(µ, T ) = π3/2(2m∗kBT

~2

)3/2

eqµ/kBT = (4π2)3/2Nc(T )eqµ/kBT ,

where

Nc(T ) = 2

(m∗kBT

2π~2

)3/2

is the effective density of states defined in Section 2.3. For the energy density wecompute

E(µ, T ) = 2π

(2m∗

~2

)3/2

eqµ/kBT

∫ ∞

0

e−ε/kBT ε3/2 dε

= 2π

(2m∗

~2

)3/2

eqµ/kBT 1√8(kBT )

5/2 1

2

∫

Re−z2/2z4 dz

=π√8

(2m∗kBT

~2

)3/2

eqµ/kBT (kBT ) · 3∫

Re−z2/2z2 dz

=3

2(4π2)3/2Nc(T )e

qµ/kBT (kBT )

=3

2n(µ, T ) · kBT.

The diffusion matrices become, using γ(ε) = 2m∗ε/~2 and φ(x, ε) = φ0(x)√ε,

Dij(µ, T ) =2q

3m∗eqµ/kBT

∫ ∞

0

εi+j−1

φ(x, ε)e−ε/kBT dε · Id

=2q

3m∗φ0(x)eqµ/kBT

∫ ∞

0

εi+j−3/2e−ε/kBT dε · Id,

98

and from the relations∫ ∞

0

ε1/2e−ε/kBT dε =1√2(kBT )

3/2

∫ ∞

0

z2e−z2/2 dz =

√π

2(kBT )

3/2,

∫ ∞

0

ε3/2e−ε/kBT dε =1√2 3

(kBT )5/2

∫ ∞

0

z4e−z2/2 dz =3

2·√π

2(kBT )

5/2,

∫ ∞

0

ε5/2e−ε/kBT dε =1√2 5

(kBT )7/2

∫ ∞

0

z6e−z2/2 dz =5

2· 32·√π

2(kBT )

7/2,

we conclude

D(µ, T ) =

√πq

3m∗φ0(x)(kBT )

3/2eqµ/kBT

(Id 3

2(kBT )Id

32(kBT )Id

154(kBT )

2Id

)

=q~3n(µ, T )

3√8π(m∗)2φ0(x)

(Id 3

2(kBT )Id

32(kBT )Id

154(kBT )

2Id

)∈ R6×6.

Notice that detD > 0 as long as n > 0 and T > 0, and the energy-transportequations are of parabolic type. Moreover, the matrix D(µ, T ) can be identifiedby a R2 × R2 matrix.

Example 5.11 (Fokker-Planck relaxation term)The relaxation term (5.73) can be written explicitly in terms of µ and T usingthe Fokker-Planck approximation (5.64) with A(ε) = φ0

√εN(ε)2 and assuming

the hypotheses of Example 5.10. Then

W (µ, T, TL) =

∫ ∞

0

∂

∂ε

[A(ε)

(Fµ,T + kBTL

∂Fµ,T

∂ε

)]ε dε

= φ0eqµ/kBT

∫ ∞

0

∂

∂ε

[√εN(ε)2e−ε/kBT

(1− TL

T

)]ε dε

= −φ0eqµ/kBT

(1− TL

T

)∫ ∞

0

√εN(ε)2e−ε/kBT dε

= −4π2φ0(2m∗

~2

)3

eqµ/kBT

(1− TL

T

)∫ ∞

0

ε3/2e−ε/kBT dε

= −3π5/2φ0(2m∗

~2

)3

eqµ/kBT

(1− TL

T

)(kBT )

5/2

= −3πφ0(2m∗

~2

)3/2

n(µ, T ) · kB(T − TL)

=3

2

n(µ, T )kB(T − TL)

τ0,

where

τ0 =1

2πφ0

(~2

2m∗

)3/2

is called relaxation time.

99

In this chapter we have derived four semiconductor models: the hydrodynamicequations, the SHE model, the energy-transport equations, and the drift-diffusionmodel. In Figure 5.1 we summarize the derivations of the models and theirrelations.

Drift-Diffusion

?

Energy-transport

?

Boltzmann´´

´´

´+

QQQQQQs

Hydrodynamic

½½

½½

½½=

SHE

@@@@@R

energy relaxation- time limit

dominantcollisions

expansion)

electron-electron(Hilbert[12]

momentumrelaxation-time

limit

dominantelectron-electron

collision (Hilbertexpansion)

moment method

dominant elasticcollisions (Hilbert

expansion)

Figure 5.1: Hierarchy of classical semiconductor models. The drift-diffusionmodel can be directly derived from the Boltzmann equation via a momentmethod or from the hydrodynamic model in the combined momentum and energyrelaxation-time limit.

5.6 Relaxation-time limits

In this section we will derive the drift-diffusion and energy-transport equationsfrom the (full) hydrodynamic model by performing so-called relaxation-time lim-its. We recall the full hydrodynamic model:

∂tn−1

qdiv J = 0, (5.80)

∂tJ −1

qdiv

(J ⊗ Jn

)− qkB

m∇(nT ) + q2

mn∇V = − J

τp, (5.81)

∂t(ne)−1

qdiv [J(e+ kBT )] + J · ∇V − div

(τpk

2B

mκ0nT∇T

)(5.82)

= − n

τw

(e− d

2kBT0

)in Ω ⊂ Rd,

where τp and τw are the momentum energy relaxation times, respectively, and

e =m

q2|J |22n2

+d

2kBT

100

is the total energy. The coupling of the electrostatic potential V to the electrondensity through the Poisson equation needs not to be considered in the subsequentconsiderations. Therefore, V will be treated as a given function.

We scale the equations (5.80)-(5.82) by introducing reference values for thephysical variables (see Table 5.1). Here, L > 0 is a reference length and C is areference value for the particle density. When the Poisson equation is added tothe above system of equations, C can be chosen, for instance, as the supremumof the doping profile C = C(x). The reference time τ0 is given by the assumptionthat the thermal energy is of the same order as the geometric average of thekinetic energies needed to cross the domain in time τ0, τp, respectively:

kBT0 =

√m

(L

τ0

)2√m

(L

τp

)2

.

The reference velocity u0 is equal to the velocity needed to cross the domain intime τ0.

variable reference value definitionx L e.g. diam(Ω)t τ0 τ0 = L2m/kBTτp

n(x, t) C e.g. supΩ |C|T (x, t) T0 lattice temperatureV (x, t) UT UT = kBT0/qJ(x, T ) qCu0 u0 = L/τ0e(x, t) kBT0 thermal energy

Table 5.1: Reference values for the physical variables.

With these reference values we can define the non-dimensional variables

x = Lxs, T = T0Ts,t = τ0ts, V = UTVs,n = Cns, J = qCu0 · Js.

Replacing the dimensional variables in (5.80)-(5.82) by the scaled ones we obtainthe scaled equations (writing x, t, n etc. instead of xs, ts, ns etc.)

∂tn− div J = 0, (5.83)

τpτ0∂tJ −

τpτ0div

(J ⊗ Jn

)−∇(nT ) + n∇V = −J, (5.84)

∂t(ne)− div [J(e+ T )] + J · ∇V − div (κ0nT∇T ) = −τ0τw

(e− d

2

), (5.85)

101

and the scaled energy becomes

e =τpτ0

|J |22n2

+d

2T. (5.86)

We consider the following physical situations:

Drift-diffusion limit in the hydrodynamic equations. The kinetic en-ergy needed to cross the domain in time τ0 is assumed to be much smaller thanthe thermal energy, and the momentum and energy relaxation times are assumedto be of the same order. This means

1À mu20kBT0

=L2m

kBT0τ 20=τpτ0

and O(1) =τpτw

=τpτ0· τ0τw.

The second condition impliesτ0τwÀ 1.

The drift-diffusion limit means that we perform (formally) the limits

τpτ0→ 0 and

τwτ0→ 0.

Then (5.85)-(5.86) become in the limit

e =d

2T =

d

2.

This means that the limit temperature equals one. Scaling back to the unscaledvariables the physical limit temperature equals the lattice temperature. From(5.83)-(5.84) we conclude

∂tn− div J = 0, J = ∇n− n∇V.

In the original variables this reads

∂tn−1

qdiv J = 0, J = qµn(UT∇n− n∇V ), (5.87)

whereµn =

qτpm

is called the electron mobility. The equations (5.87) together with the Poissonequation are the drift-diffusion equations derived in Sections 5.1 and 5.2.

Energy-transport limit in the hydrodynamic equations. The kineticenergy needed to cross the domain in time τ0 is assumed to be much smaller than

102

the thermal energy, and the reference time is of the same order as the energyrelaxation time, i.e.

τpτ0¿ 1 and O(1) =

τ0τw

=τ0τp· τpτw,

which impliesτpτw¿ 1.

In the energy-transport limit we only perform (formally) the limit

τpτ0→ 0.

Equations (5.83)-(5.86) become in the limit

∂tn− div J = 0, J = ∇(nT )− n∇V, (5.88)

∂t

(d

2nT

)− div

(d+ 2

2JT + κ0nT∇T

)+ J · ∇V = −d

2

τ0τwn(T − 1), (5.89)

or, in the original variables,

∂tn−1

qdiv J = 0, J = qµn (UT∇(nT )− n∇V ) ,

∂t

(d

2kBTn

)− divS + J · ∇V = −d

2

nkB(T − T0)τw

,

S =J

qkBT +

τpk2B

mκ0nT∇T.

In order to see that this system of equations can be written as an energy-transportmodel, we introduce the chemical potential µ by

n = N(kBT )d/2 eqµ/kBT ,

where N is the effective density of states (see Example 5.10). A lengthy compu-tation shows that the above system of equations can be written equivalently inthe variables qµ/kBT and −1/kBT as

∂tn−1

qdiv J = 0, ∂tE − divS + J · ∇V = W (µ, T ),

where

E =d

2kBT · n, W (µ, T ) = −d

2

nkB(T − T0)τw

and

J = D11∇(qµ

kBT

)+D12∇

( −1kBT

)−D11

q∇VkBT

,

S = D21∇(qµ

kBT

)+D22∇

( −1kBT

)−D21

q∇VkBT

,

103

with the diffusion matrix

(Dij)ij =qτpmnkBT

(1 d+2

2kBT

d+22kBT

((d+22

)2+ κ0

)(kBT )

2

).

The above equations form an energy-transport model as discussed in Section 5.5.Notice that the determinant of this matrix is positive for positive particle

densities and temperatures if and only if κ0 > 0. Hence, the heat conductivityterm div (κ0nT∇T ) is necessary to obtain a uniformly parabolic set of equations.We recall, however, that this term is introduced heuristically, and no justificationis available.

Drift-diffusion limit in the energy-transport equations. It is possi-ble to derive the drift-diffusion model from the energy-transport equations. Forthis we consider the energy-transport model in the scaled form (5.88)-(5.89) andassume that the reference time τ0 is much larger than the energy relaxation time:

τ0τwÀ 1.

This is exactly the second condition needed in the drift-diffusion limit. Theformal limit τw/τ0 → 0 in (5.88)-(5.89) yields T = 1 and, after scaling back tothe original variables, the drift-diffusion equations (5.87).

The relaxation-time limits studied in this section are summarized in Figure5.2.

Energy-transport -τw/τ0 → 0

Drift-diffusion

Hydrodynamic¡¡

¡¡

¡¡¡ª

@@@@@@@R

τp/τ0 → 0τw/τ0 → 0

τp/τ0 → 0

Figure 5.2: Relaxation-time limits in the hydrodynamic and energy-transportmodels

5.7 The extended hydrodynamic model

The heuristic addition of the heat flux −div (κ∇T ) in the energy equation of thehydrodynamic model derived in Section 5.4 is not very satisfactory. In this section

104

we present an alternative way of deriving hydrodynamic models based on the so-called maximum entropy principle. This method provides a set of closed momentequations consisting of the conservation laws of mass, momentum, energy andenergy flux, which we refer to as the extended hydrodynamic model. In this model,no heuristic terms need to be included. The maximum entropy method has beenfirst used by Anile and Pennisi [2] for the derivation of charge transport equationsfor semiconductors. It is based on principles of extended thermodynamics [35].

We start with the semiclassical Boltzmann equation in R3

∂tf + v(k) · ∇xf −q

~E · ∇kf = Q(f), x ∈ R3, k ∈ B, t > 0, (5.90)

where v(k) = ~−1∇kε(k). The collision operator Q(f) is assumed to satisfy

∫

B

Q(f) dk = 0.

In order to simplify the presentation, we assume parabolic bands

ε(k) =~2|k|22m∗

, k ∈ B = R3,

such that v(k) = ~k/m∗. The maximum entropy principle also works for generalband diagrams (see [3] for details). We define the first moments

〈Mi〉 =∫

R3

f(x, k, t)Mi(k) dk,

whereM0(k) = 1, M1(k) = k, M2(k) = |k|2, M3(k) = k|k|2.

As in Section 5.4 we obtain the moment equations by multiplying the Boltz-mann equation (5.90) by Mi and integrating over k ∈ R3:

∂t〈1〉+~m∗

div x〈k〉 = 0, (5.91)

∂t〈k〉+~m∗

div x〈k ⊗ k〉+q

~E〈1〉 =

∫

R3

Q(f)k dk, (5.92)

∂t〈|k|2〉+~m∗

div x〈k|k|2〉+2q

~E · 〈k〉 =

∫

R3

Q(f)|k|2 dk, (5.93)

∂t〈k|k|2〉+~m∗

div x〈k ⊗ k|k|2〉+q

~E · 〈∇k ⊗ (k|k|2)〉 (5.94)

=

∫

R3

Q(f)k|k|2 dk,

105

since, using integrating by parts

− q~E ·∫

R3

∇kf |k|2 dk =q

~E ·∫

R3

f∇k|k|2 dk

=2q

~E ·∫

R3

fk dk

=2q

~E · 〈k〉.

We can write these moment equations in a more compact form as

∂t〈Mi〉+ div xFi + E ·Gi = Pi, i = 0, 1, 2, 3,

where Fi, Gi and Pi are defined in an obvious way. The moments 〈Mi〉 canbe considered as the fundamental variables since they have a direct physicalinterpretation. Indeed, we have:

n = 〈M0〉 electron density,

J = − q~m∗〈M1〉 electron current density,

ne =~2

2m∗〈M2〉 electron energy density,

S = − ~3

2(m∗)2〈M3〉 energy flux.

The goal now is to express Fi, Gi and Pi in terms of the moments 〈Mi〉. SinceF0 = 〈M1〉, F2 = 〈M3〉, G0 = 0, G1 = 〈M0〉, G2 = 〈M1〉,

we only have to compute F1, F3, G3 and the production terms P0, . . . , P3. For thiswe choose a distribution function f0 which can be used to evaluate the unknownmoments, i.e., we set

F1 = 〈k ⊗ k〉 =∫

R3

f0(k ⊗ k) dk,

F3 = 〈k ⊗ k|k|2〉 =∫

R3

f0(k ⊗ k)|k|2 dk,

and so on. The function f0 is defined as the extremal of the entropy functional

s(f) =

∫

R3

f(log f − 1) dk

under the constraints that it yields exactly the known moments 〈Mi〉. This meansthat we have to minimize s (or, equivalently, to maximize the physical entropy−s) under the constraints

∫

R3

Mi(k)f0(k) dk = 〈Mi〉, i = 0, . . . , 3. (5.95)

106

Using Lagrange multipliers, we have to analyze the functional

H(f) =

∫

R3

f(log f − 1) dk +3∑

i=0

λi

(∫

R3

Mi f dk − 〈Mi〉).

A simple computation shows that

dH(f)

df(g) =

∫

R3

g log f dk +3∑

i=0

λi

∫

R3

Mi g dk.

The necessary conditiondH(f0)

df(g) = 0 ∀g

for the extremal f0 leads to

∫

R3

g(log f0 +

3∑

i=0

λiMi

)dk = 0 ∀g

and hence

f0(k) = exp(−

3∑

i=0

λiMi(k)), k ∈ R3. (5.96)

The Lagrange multipliers are obtained by inserting this expression into (5.95):

〈Mi〉 =∫

R3

Mi(k) exp(−

3∑

i=0

λiMi(k))dk. (5.97)

This is an implicit equation for λ0, . . . , λ3. Once we have found the values ofλ0, . . . , λ3 (which are functions of Mi and 〈Mi〉), we can define the maximumentropy distribution function f0 from (5.96), and inserting the expression for f0into the definitions of Fi, Gi and Pi gives finally a closed set of equations.

Unfortunately, the implicit equation (5.97) usually cannot be inverted directlyto give explicit expressions for the Lagrange multipliers. If we assume that thedistribution function f0 is in some sense not far from the thermal equilibriumdistribution function

feq(k) = exp

(εFkBT

− ε(k)

kBT

)(5.98)

(see Lemma 3.2(3) for Boltzmann statistics), we can develop the Lagrange multi-pliers in terms of a small parameter δ > 0. Comparing (5.98) with (5.96) suggeststhat the moments 〈M1〉 and 〈M3〉 (which appear in (5.98)) are of order 1 whereasthe moments 〈M2〉 and 〈M4〉 (which do not appear in (5.98)) are of order δ ¿ 1.

107

Our mean assumption is now that we can develop λi in terms of the moments〈Mi〉 in the following way:

λi = ai1 + ai2 δ〈M1〉 · δ〈M3〉+ ai3 δ〈M1〉 · δ〈M1〉+ ai4 δ〈M3〉 · δ〈M3〉+O(δ3), i = 0, 2,

λj = aj1 · δ〈M1〉+ aj2 · δ〈M3〉+O(δ3), j = 1, 3,

where we replaced 〈M1〉 and 〈M3〉 by the rescaled moments δ〈M1〉 and δ〈M3〉,respectively, and where the functions aij depend on 〈M0〉 and 〈M2〉. The factsthat λ0 and λ2 do not contain first-order terms in δ and that λ1 and λ3 do notcontain zero- and second-order terms in δ can be regarded here as an assumption.The argument can be made more rigorous by assuming that f0 is anisotrop andδ is a small anisotropy parameter (see [3]). The above equations can be written,up to second order in δ, as

λ0 = λ(0)0 + δ2λ

(2)0 ,

λ1 = δλ(1)1 ,

λ2 = λ(0)2 + δ2λ

(2)2 ,

λ3 = δλ(1)3 .

We insert this expansion into (5.96) and use the approximation

e−x = 1− x+x2

2+O(x3) (x→ 0)

to obtain, up to second order in δ,

f0 = exp[−(λ(0)0 + δ2λ

(2)0 )− δλ(1)1 · k − (λ

(0)2 + δ2λ

(2)2 )|k|2 − δλ(1)3 · k|k|2

]

= exp[−(λ(0)0 + λ

(0)2 |k|2)

]exp

[−δ(λ(1)1 + λ

(1)3 |k|2) · k − δ2(λ(2)0 + λ

(2)2 |k|2)

]

= exp[−(λ(0)0 + λ

(0)2 |k|2)

] 1− δ(λ(1)1 + λ

(1)3 |k|2) · k − δ2(λ(2)0 + λ

(2)2 |k|2)

+δ2

2((λ

(1)1 + λ

(1)3 |k|2) · k)2

. (5.99)

Using this expression in the constraints (5.95) allows to compute the Lagrangemultipliers.

First we compute λ(0)0 and λ

(0)2 . Setting A = λ

(1)1 + λ

(1)3 |k|2 and B = λ

(2)0 +

λ(2)2 |k|2, we infer

n = 〈M0〉 = e−λ(0)0

∫

R3

e−λ(0)2 |k|2

[1− δA · k − δ2B +

δ2

2(A · k)2

]dk.

108

Equating equal powers of δ gives

n = e−λ(0)0

∫

R3

e−λ(0)2 |k|2 dk

= (2λ(0)2 )−3/2 e−λ

(0)0

∫

R3

e−|u|2/2 du

= π3/2(λ(0)2 )−3/2 e−λ

(0)0 . (5.100)

Similarly, the expression

ne =~2

2m∗〈M2〉

=~2

2m∗e−λ

(0)0

∫

R3

e−λ(0)2 |k|2 |k|2

[1− δA · k − δ2B +

δ2

2(A · k)2

]dk

yields

ne =~2

2m∗e−λ

(0)0

∫

R3

e−λ(0)2 |k|2 |k|2 dk

=~2

2m∗(2λ

(0)2 )−5/2 e−λ

(0)0

∫

R3

e−|u|2/2|u|2 du

=~2

2m∗(2λ

(0)2 )−5/2 e−λ

(0)0 · 3(2π)3/2

=3~2π3/2

4m∗(λ

(0)2 )−5/2e−λ

(0)0 . (5.101)

The equations (5.100) and (5.101) can be used to compute λ(0)0 and λ

(0)2 . From

e =ne

n=

3

4

~2

m∗(λ

(0)3 )−1

follows

λ(0)2 =

3~2

4m∗e, λ

(0)0 = − lnn+

3

2ln

(4πm∗

3~2e

).

Now we compute the remaining Lagrangian multipliers λ(1)1 , λ

(1)3 , λ

(2)0 , and

λ(2)2 . Recalling that we have replaced 〈M1〉 and 〈M3〉 by the rescaled expressionsδ〈M1〉 and δ〈M3〉 we obtain

δJ = − q~m∗

δ〈M1〉

= − q~m∗

e−λ(0)0

∫

R3

e−λ(0)2 |k|2k

[1− δA · k − δ2B +

δ2

2(A · k)2

]dk

109

and

δS = − ~3

2(m∗)2δ〈M3〉

= − ~3

2(m∗)2e−λ

(0)0

∫

R3

e−λ(0)2 |k|2k|k|2

[1− δA · k − δ2B +

δ2

2(A · k)2

]dk.

Equating equal powers of δ leads to

J = − q~m∗

e−λ(0)0

∫

R3

e−λ(0)2 |k|2k

(λ(1)1 + λ

(1)3 |k|2

)· k dk, (5.102)

0 =

∫

R3

e−λ(0)2 |k|2k

[1

2

((λ

(1)1 + λ

(1)3 |k|2) · k

)2− (λ

(2)0 + λ

(2)2 |k|2)

]dk, (5.103)

and

S = − ~3

2(m∗)2e−λ

(0)0

∫

R3

e−λ(0)2 |k|2k|k|2

(λ(1)1 + λ

(1)3 |k|2

)· k dk, (5.104)

0 =

∫

R3

e−λ(0)2 |k|2k|k|2

[1

2

((λ

(1)1 + λ

(1)3 |k|2) · k

)2− (λ

(2)0 + λ

(2)2 |k|2)

]dk.(5.105)

The multipliers λ(1)1 and λ

(1)3 can be computed by solving the linear system given

by (5.102) and (5.104). Then the remaining parameters λ(2)0 and λ

(2)2 are solutions

of the linear system (5.103) and (5.105). The general structure of the multipliersis as follows:

λ(1)1 = a11(n, e)J + a12(n, e)S,

λ(1)3 = a31(n, e)J + a32(n, e)S,

λ(2)0 = a01(n, e)J · J + a02(n, e)J · S + a03(n, e)S · S,λ(2)2 = a21(n, e)J · J + a22(n, e)J · S + a23(n, e)S · S.

Explicit expressions for the coefficients are given in [3, Section 3.4].With the above relations for the Lagrange multipliers, the maximum entropy

distribution function f0 in (5.99) becomes

f0 = n

(3~2

4πm∗e

)3/2

exp

(−3~2|k|2

4m∗e

)(5.106)

×1− (a21J + a22S) · k − (a11 + a31|k|2)J · J − (a12 + a32|k|2)J · S

− (a13 + a33|k|2)S · S +δ2

2

[(a21 + a41|k|2)J · k + (a22 + a42|k|2)S · k

]2 .

Hence, the unknown moments are given by

F1 = 〈k ⊗ k〉 =∫

R3

f0(k ⊗ k) dk = F1(n, e, J, S),

F3 = 〈k ⊗ k|k|2〉 = F3(n, e, J, S),

G3 = 〈∇k ⊗ (k|k|2)〉 = 〈|k|2〉Id + 2〈k ⊗ k〉 = m∗

~2ne · Id + F1(n, e, J, S).

110

From (5.106) we see that the dependence of F1 and F3 on J , S is at mostquadratic, i.e., F1 and F3 are in fact functions of J · J , J ⊗ J , J · S, S ⊗ Sand so on. The precise formulas for F1 and F3 are given in [3, (124), (125)].

The production terms Pi are approximated by

P1 =

∫

R3

Q(f0)k dk,

P2 =

∫

R3

Q(f0)|k|2 dk,

P3 =

∫

R3

Q(f0)k|k|2 dk.

In order to get explicit formulas for Pi as functions of n, J , e and S, the collisionoperator Q(f) has to be specified. In [3] the production terms are computed for(acoustic or optical) phonon scattering or for scattering with impurities; we referto this work for the precise expressions.

Finally, we want to compare the extended hydrodynamic model (5.91)-(5.94)with the hydrodynamic equations of Section 5.4. We need to rewrite the equations(5.91)-(5.94) by introducing the electron temperature. For this, introduce thedeviations of the microscopic velocity v(k) from the averaged macroscopic velocityu = −J/qn:

u(k) = v(k) + u =~km∗

+ u. (5.107)

The electron temperature T and the heat flux σ are defined as moments of thedeviations:

nkBT =m∗

3

∫

R3

f |u|2 dk,

nσ =m∗

2

∫

R3

f |u|2u dk.

Then we can reformulate the energy density, using (5.107), as

ne =~

2m∗

∫

R3

f |k|2 dk

=m∗

2

∫

R3

f(|u|2 − 2u · u+ |u|2

)dk (using (5.107))

=m∗

2n|u|2 + 3

2nkBT,

since ∫

R3

f u dk =

∫

R3

f

(~km∗− u)dk = −J

q− nu = 0. (5.108)

111

The energy flux can be written as

S = − ~3

2(m∗)2

∫

R3

fk|k|2 dk

= −m∗

2

∫

R3

f (u− u)(|u|2 − 2u · u+ |u|2) dk (using (5.107)

= −m∗

2

∫

R3

f (u|u|2 − u|u|2 + 2(u · u)u− u|u|2) dk (using (5.108)

= −nσ +3

2nkBTu−m∗

∫

R3

f (u · u)u+m∗

2nu|u|2.

Computing∫

R3

f (u · u)uj dk =∑

i

ui

∫

R3

f

(~kim∗

+ ui

)(~kjm∗

+ uj

)dk

=∑

i

ui

[(~2

m∗

)2 ∫

R3

fkikj dk + nuiuj +~m∗

∫

R3

f(kiuj + kjui) dk

]

=[( ~2

m∗

)2〈k ⊗ k〉 · u+ 3nu|u|2

]j,

we obtain

S = ne− nσ +

(~m∗

)2

〈k ⊗ k〉 · u+ 3nu|u|2.

Introducing

F1 = q

(~m∗

)2

F1, F3 =~4

2(m∗)3F3

and

P1 = −q~m∗

P1, P2 =~2

2m∗P2, P3 = −

~3

2(m∗)3P3,

we can write the system (5.91)-(5.94) as follows:

∂tn−1

qdiv J = 0,

∂tJ − div F1(n, e, S, J)−q2

m∗En = P1,

∂t(ne)− divS − E · J = P2,

∂tS − div F3(n, e, S, J) +1

2

( q

m∗ne · Id + F1

)· E = P3,

where

ne =m∗|J |22q2n

+3

2nkBT,

S = ne− nσ +1

qF1(n, e, J, S)−

3J |J |2q3n2

,

112

and F1, F3, P1, P2 and F1 are given explicitly in [3]. The above system of equationsis referred to as the extended hydrodynamic model for parabolic band diagrams.For non-parabolic band diagrams, similar equations can be derived; we refer to[3] for details. Whereas the independent variables in the system (5.91)-(5.94) aren, J , ne and S, the independent variables of the above equations are n, J , T andσ. Notice that in the above derivation, there was no need to introduce heuristicterms.

113

6 The Drift-Diffusion Equations

In this chapter the drift-diffusion model is studied in more detail. We recall theequations for electrons and holes:

∂tn−1

qdiv Jn = −R(n, p), (6.1)

∂tp+1

qdiv Jp = −R(n, p), (6.2)

Jn = qµn(UT∇n− n∇V ), (6.3)

Jp = −qµp(UT∇p+ p∇V ), (6.4)

εs∆V = q(n− p− C), x ∈ Ω, t > 0, (6.5)

with the recombination-generation term

R(n, p) =np− n2i

τp(n+ ni) + τn(p+ ni).

Here, Ω ⊂ Rd (d ≥ 1) is a bounded domain.

6.1 Thermal equilibrium state and boundary conditions

The thermal equilibrium state is a steady state with no current flow, i.e.

∂tn = ∂tp = 0 and Jn = Jp = 0 in Ω.

This implies R(n, p) = 0 or np = n2i in Ω and

0 = UT∇n− n∇V = n∇(UT lnn− V ),

0 = UT∇p+ p∇V = p∇(UT ln p+ V ).

It is physically reasonable to assume that the particle densities n and p arepositive in Ω. (In fact, this can be proved using maximum principle argumentsif the boundary data is positive; see [33]). This yields

α := UT lnn− V = const., β := UT ln p+ V = const. (6.6)

orn = eα/UT eV/UT , p = eβ/UT e−V/UT in Ω.

We wish to determine the constants α and β. For this we use the equation np = n2iand the fact that V is determined only up to an additive constant. Then thesum of the two equations in (6.6) yields α + β = UT ln(np) = 2UT lnni and,substituting V by V + γ, where γ := UT lnni − α, gives

n = e(α+γ)/UT eV/UT = nieV/UT (6.7)

114

andp = e(β−γ)/UT e−V/UT = e(β+α−UT lnni)/UT e−V/UT = nie

−V/UT . (6.8)

The electrostatic potential satisfies the semilinear elliptic equation

εs∆V = q(n− p− C) = q(nieV/UT − nie

−V/UT − C)

= q(2ni sinh

V

UT

− C)

in Ω.

In order to get a unique solution of this differential equation, we have to imposeboundary conditions.

We assume that the boundary of the semiconductor domain consists of twoparts: one part, called the Dirichlet part, on which the particle densities and thepotential are prescribed,

n = nD, p = pD, V = VD on ΓD, (6.9)

and another part, the Neumann part, which models the insulating boundarysegments on which the normal components of the current densities and the electricfield vanish,

Jn · ν = Jp · ν = ∇V · ν = 0 on ΓN . (6.10)

In view of the expressions (6.3)-(6.4) for Jn and Jp this is equivalent to

∇n · ν = ∇p · ν = ∇V · ν = 0 on ΓN . (6.11)

It remains to determine the boundary functions nD, pD and VD. We assume:

• The total space charge vanishes on ΓD :

nD − pD − C = 0. (6.12)

• The densities are in equilibrium on ΓD :

nDpD = n2i (6.13)

• The boundary potential is the superposition of the built-in-potential Vbi andthe applied voltage U :

VD = Vbi + U. (6.14)

Clearly, in thermal equilibrium we have U = 0. The built-in potential is thepotential corresponding to the equilibrium densities given by (6.7) and (6.8):

nD = nieVbi/UT , pD = nie

−Vbi/UT . (6.15)

Substituting (6.12) in (6.13) gives n2D − CnD = n2i such that

nD =1

2

(C +

√C2 + 4n2i

), pD =

1

2

(−C +

√C2 + 4n2i

)on ΓD.

115

Thus we get from (6.15)

Vbi = UT lnnD

ni

= UT ln

(C

2ni

+

√C2

4n2i+ 1

)= UTarsinh

C

2ni

. (6.16)

Therefore, the thermal equilibrium state (ne, pe, Ve) is the (unique) solutionof

εs∆Ve = q

(2ni sinh

Ve

UT

− C)

in Ω,

Ve = Vbi on ΓD, ∇Ve · ν = 0 on ΓN ,

where Vbi is given by (6.16), and

ne = nieVe/UT , pe = nie

−Ve/UT in Ω.

Moreover, the drift-diffusion equations (6.1)-(6.5) are complemented with theboundary conditions (6.9)-(6.11), where (nD, pD, VD) are defined by (6.14)-(6.16),and the initial conditions

n(·, 0) = nI , p(·, 0) = pI in Ω.

6.2 Scaling of the equations

The drift-diffusion model (6.1)-(6.5), (6.9)-(6.11) contains several physical pa-rameters. By bringing the equations in a non-dimensional form, we can identifythe relevant scaled parameters. In fact, we show that there are only two relevantparameters. Define

C = supx∈Ω|C(x)|, L = diam(Ω).

We expect that the particle densities are of the order of C and the potential is ofthe order of UT . Then the scaled variables ns, ps and Vs, defined by

n = Cns, p = Cps, V = UTVs,

are of the order 1. We also set

x = Lxs, C = CCs, t = τts

with

τ =L2µ

UT

, µ = max(µn, µp).

This fixes the time scale to the order τ. A simple computation shows that thescaled quantities satisfy the equations (where we omitted the index “s”):

∂tn− µ′ndiv (∇n− n∇V ) = −R(n, p), (6.17)

∂tp− µ′pdiv (∇p+ p∇V ) = −R(n, p), (6.18)

λ2∆V = n− p− C, (6.19)

116

where µ′α = µα/µ (α = n, p), and

λ =

√εsUT

qL2C

is called the scaled Debye length. (The Debye length is the quantity√εsUT/qC =

Lλ.) The scaled recombination-generation term reads

R(n, p) =np− δ4

τ ′p(n+ δ2) + τ ′n(p+ δ2),

where τ ′α = τα/τ (α = n, p) and

δ2 =ni

C

is the ratio of the intrinsic density and the maximal doping concentration. Therelevant parameters are λ and δ. Typically, they are small compared to one. Forinstance, in silicon we have

εs = 1.05 · 10−10As/Vm, ni = 1016m−3,µn = 0.15m2/Vs, µp = 0.045m2/Vs.

Choosing

L = 10−6m, C = 1023m−3, UT = 0.026V (i.e. T = 300K),

we obtainλ2 = 1.7 · 10−4, δ2 = 10−7.

The scaled current densities can now be defined by

Jn = ∇n− n∇V, Jp = −(∇p+ p∇V ).

The scaled Dirichlet boundary conditions read

nD =1

2(C +

√C2 + 4δ4), pD =

1

2(−C +

√C2 + 4δ4), (6.20)

VD = Vbi + U = arsinhC

2δ2+ U on ΓD, (6.21)

and the built-in potential can be written as

Vbi = ln

(C

2δ2+

√C2

4δ4+ 1

)= ln

nD

δ2= ln

δ2

pD. (6.22)

117

6.3 Static current-voltage characteristic of a diode

In this subsection we derive the current-voltage characteristic of a pn-junctiondiode in the case of “small” current densities. The diode is defined by

C(x) =

Cn : x ∈ Ωn

−Cp : x ∈ Ωp,

where Cn > 0 is the doping concentration in the n-region Ωn, −Cp < 0 is thedoping density in the p-region Ωp, and Ω = Ωn ∪Ωp, Ωn ∩Ωp = ∅. The Dirichletboundary ΓD is supposed to consist of the two boundary parts Γ1 ⊂ Ωn andΓ2 ⊂ Ωp (see Figure 6.1). We set Γ = Ωn ∩Ωp. The applied voltage U is given by

U(x) =

0 : x ∈ Γ1U0 : x ∈ Γ2,

where U0 is a constant.

Ωp

Ωn

ΓN

Γ2

ΓN

Γ1

Γ

Figure 6.1: Geometry of a pn-junction diode.

The total current is defined by

I =

∫

Γ1

(Jn + Jp) · ν ds,

where ν is the exterior unit normal to ∂Ωn. We assume (for simplicity) thatµ′n = µ′p = 1 (see (6.17)-(6.18)). Then, taking the difference of the stationaryversions of (6.17)-(6.18), we have div (Jn+Jp) = 0 and, by the divergence theoremand Jn · ν = Jp · ν = 0 on ΓN (see (6.10)),

0 =

∫

Ωn

div (Jn + Jp) dx =

∫

Γ1

(Jn + Jp) · ν dx+

∫

Γ

(Jn + Jp) · ν dx.

Thus

I =

∫

Γ1

(Jn + Jp) · ν ds = −∫

Γ

(Jn + Jp) · ν ds. (6.23)

Our main goal is to describe the function I = I(U0) in more detail.

118

Our first main assumption is to consider “small” current densities. Moreprecisely, we assume that the current densities are of the order of δ4. Then wecan rescale Jn and Jp by

Jn = δ4jn, Jp = δ4jp. (6.24)

We see later in this section why we use the exponent 4 in the rescaling. Thenew variables jn and jp are of the order O(1). It is convenient to introduce newvariables u,w by

n = δ2eV u, p = δ2e−Vw.

The functions u and w are called Slotboom variables. The advantage is that theysymmetrize the current relations in the following sense:

δ2eV∇u = eV∇(e−V n) = ∇n− n∇V = Jn,

−δ2e−V∇w = −e−V∇(eV p) = −(∇p+ p∇V ) = Jp.

We consider the stationary equations for u,w and V (see (6.17)-(6.19)):

δ2div (eV∇u) = R(δ2eV u, δ2e−Vw), (6.25)

δ2div (e−V∇w) = R(δ2eV u, δ2e−Vw), (6.26)

λ2∆V = δ2eV u− δ2e−Vw − C in Ω. (6.27)

The boundary conditions for u and w follow from (6.20)-(6.22):

u = δ−2e−V n = δ−2e−VbinD = δ−2e− ln(nD/δ2)nD = 1,

w = δ−2eV p = δ−2eVbipD = δ−2eln(δ2/pD)pD = 1,

V = Vbi + U = Vbi on Γ1,

and

u = δ−2e−Vbi−UnD = e−U0 ,

w = δ−2eVbi+UpD = eU0 ,

V = Vbi + U0 on Γ2.

On the Neumann boundary we get

∇u · ν = ∇w · ν = ∇V · ν = 0 on ΓN .

The equations (6.25)-(6.27) together with the above boundary conditions are stillequivalent to the full problem.

In order to get some informations on the current-voltage characteristic I =I(U0), we have to simplify the full problem. In Section 6.2 we have seen that theparameters λ and δ are small compared to one. Hence the solution of the reducedproblem λ = 0 and δ = 0 may be close to the solution of the full problem. In

119

fact, this is our second main assumption. (It is possible to prove this rigorouslyby means of asymptotic analysis; see [33]).

First, we study the reduced problem λ = 0. From (6.27) we obtain

0 = δ2eV u− δ2e−Vw − C in Ω

and therefore

δ2eV =1

2u

(C +√C2 + 4δ4uw

), δ2e−V =

1

2w

(−C +

√C2 + 4δ4uw

).

We can discard one solution of the above quadratic equation, since we requireu,w ≥ 0. Thus (6.25) and (6.26) become

div

[1

2u

(C +√C2 + 4δ4uw

)∇u]= R, (6.28)

div

[1

2w

(−C +

√C2 + 4δ4uw

)∇w]= R, (6.29)

with the boundary conditions

u = w = 1 on Γ1, u = e−U0 , w = eU0 on Γ2, (6.30)

∇u · ν = ∇w · ν = 0 on ΓN . (6.31)

Transforming back to the original variables, the particle densities n and p arefunctions of u and w :

n = δ2eV u =1

2(C +

√C2 + 4δ4uw),

p = δ2e−Vw =1

2(−C +

√C2 + 4δ4uw).

Notice that the doping profil C is discontinuous across Γ. This means that also thediffusion coefficients of the elliptic equations (6.28) and (6.29) are discontinuous,and we cannot expect to obtain classical (i.e. twice continuously differentiable)solutions. In fact, the elliptic problem (6.28)-(6.31) has to be solved in a gener-alized (weak) sense. The particle densities are also discontinuous across Γ. Thus,we expect that the particle densities solving the full problem have an internallayer (i.e. sharp gradients) near Γ.

Next we consider the reduced problem δ = 0. Recall that

δ4jn = Jn = δ2eV∇u =1

2u(C +

√C2 + 4δ4uw)∇u, (6.32)

δ4jp = Jp = −δ2e−V∇w = − 1

2w(−C +

√C2 + 4δ4uw)∇w, (6.33)

and

R = δ4uw − 1

τ ′p(n+ δ2) + τ ′n(p+ δ2).

120

Hence we can write (6.28)-(6.29) as

div jn =uw − 1

τ ′p(n+ δ2) + τ ′n(p+ δ2), div jp = −

uw − 1

τ ′p(n+ δ2) + τ ′n(p+ δ2). (6.34)

The Taylor expansion√1 + x = 1 + x/2 +O(x2) around x = 0 yields in Ωp

C +√C2 + 4δ4uw = −Cp +

√C2

p + 4δ4uw

= −Cp + Cp

(1 +

2δ4uw

C2p

+O(δ8)

)

=2uw

Cp

δ4 +O(δ8),

−C +√C2 + 4δ4uw = 2Cp +

2uw

Cp

δ4 +O(δ8) as δ → 0

and in Ωn

C +√C2 + 4δ4uw = 2Cn +

2uw

Cn

δ4 +O(δ8)

−C +√C2 + 4δ4uw = −Cn +

√C2 + 4δ4uw

=2uw

Cn

δ4 +O(δ8) as δ → 0.

Therefore, (6.32) and (6.33) give

jn =w∇uCp

+O(δ4), δ4jp =Cp∇ww

+O(δ4) in Ωp,

δ4jn =Cn∇uu

+O(δ4), jp = −u∇wCn

+O(δ4) in Ωn.

Here we see why we have used the scaling factor δ4 in (6.24). Exponents largeror smaller than 4 lead to useless relations. Moreover,

n = Cn, p = 0 in Ωn,n = 0, p = Cp in Ωp.

Now we can perform the limit δ → 0 in (6.34) and in the above equations:

∇u = 0, jp = −u∇wCn

, div jp = −uw − 1

τ ′pCn

in Ωn,

∇w = 0, jn =w∇uCp

, div jn =uw − 1

τ ′nCp

in Ωp.

From the boundary conditions (6.30) and (6.31) we conclude

u = 1 in Ωn, w = eU0 in Ωp.

121

Thus

jp = −∇wCn

in Ωn, jn =eU0

Cp

∇u in Ωp (6.35)

and

∆w = −Cndiv jp =w − 1

τ ′pin Ωn,

∆u = Cpe−U0div jn =

u− e−U0

τ ′nin Ωp.

The boundary conditions for w and u read

w = 1 on Γ1, w = eU0 on Γ, ∇w · ν = 0 on ∂Ωn ∩ ΓN

andu = e−U0 on Γ2, u = 1 on Γ, ∇u · ν = 0 on ∂Ωp ∩ ΓN .

We wish to separate the influence of the applied potential. We claim that thesolutions w and u of the above boundary-value problems can be written as

w = 1 +(eU0 − 1

)g in Ωn, u = e−U0 +

(1− e−U0

)f in Ωp,

where f and g are the (unique) solutions of

∆f =f

τ ′nin Ωp, (6.36)

f = 0 on Γ2, f = 1 on Γ, ∇f · ν = 0 on ∂Ωp ∩ ΓN

and∆g =

g

τ ′pin Ωn, (6.37)

g = 0 on Γ1, g = 1 on Γ, ∇g · ν = 0 on ∂Ωn ∩ ΓN .

Indeed, it holds

∆w =(eU0 − 1

)∆g =

(eU0 − 1

)g

τ ′p=w − 1

τ ′pin Ωn,

∆u =(1− e−U0

)∆f =

(1− e−U0

)f

τ ′n=u− e−U0

τ ′nin Ωp,

and it is easy to verify the boundary conditions for u and w. The advantage ofthe introduction of the functions f and g is that they depend on the geometry ofthe diode but not on the applied potential.

122

We can now compute the total current (6.23). The current densities (6.35)are only known in Ωn or Ωp; therefore we calculate I on Γ. Assuming continuityof the normal components of jn in Ωp and jp in Ωn, respectively, we get from

jn =eU0

Cp

∇u =eU0 − 1

Cp

∇f in Ωp,

jp = −∇wCn

= −eU0 − 1

Cn

∇g in Ωn

and from (6.23)

I = −∫

Γ

(jn + jp) · ν ds = −(eU0 − 1

) ∫

Γ

(∇fCp

− ∇gCn

)· ν ds.

Notice that also the total current is scaled by the factor δ4. Introducing theleakage current,

Is :=

∫

Γ

(∇gCn

− ∇fCp

)· ν ds,

we obtain the scaled Shockley equation (see Figure 6.2)

I = Is(eU0 − 1

). (6.38)

-

6

U0

I

−Is

Figure 6.2: The Shockley equation for a pn-junction diode.

The leakage current is positive. Indeed, we use integration by parts and thedifferential equations for f to infer

−∫

Γ

1

Cp

∇f · νds =

∫

Γ

f

Cp

∇f · (−ν)ds+∫

Ωp

1

Cp

(−f∆f +

f 2

τ ′n

)dx

(using f = 1 on Γ and the differential equation for f)

=

∫

Ωp

1

Cp

(|∇f |2 + f 2

τ ′n

)dx

(using integration by parts)

> 0.

123

Notice that −ν is the exterior unit normal to ∂Ωp. Similarly,

∫

Ωn

1

Cn

(|∇g|2 + 1

τ ′pg2)dx =

∫

Γ

1

Cn

∇g · ν ds.

Therefore

Is =

∫

Ωp

1

Cp

(|∇f |2 + 1

τ ′nf 2)dx+

∫

Ωn

1

Cn

(|∇g|2 + 1

τ ′pg2)dx > 0.

Notice that the unscaled Shockley equation reads

I = Is(eU0/UT − 1

).

Shockley’s equation is valid for “small” current densities or, equivalently, for“small” applied voltages U0. Experiments show that (6.38) indeed holds for suffi-ciently small applied potentials but not for “large” values of |U0|. There exists athreshold voltage Uth < 0 such that I(U0)→ −∞ as U0 Uth. This break-downeffect is used in so-called Zener diodes.

We study the high-injection case U0 À 1 for a simple one-dimensional sym-metric diode modeled by the reduced problem λ = 0. For this we define Ω =(−1, 1),

C(x) =

−1 : x ∈ (−1, 0)1 : x ∈ (0, 1)

, U(x) = −U02x, x ∈ Ω.

We assume that recombination-generation effects are neglegible. Then Jn = Jp =:I is constant. We are looking for solutions that satisfy

n(−x) = p(x), V (−x) = −V (x), x ∈ Ω.

Then V (0) = 0, and it is sufficient to solve

I = nx − nVx, I = −px − pVx, 0 = n− p− 1 in (0, 1) (6.39)

subject to the boundary conditions (see (6.20)-(6.21))

V (0) = 0, V (1) = Vbi(1) + U(1) = arsinh1

2δ2− U0

2,

p(1) =1

2(−1 +

√1 + 4δ4), n(1) =

1

2(1 +

√1 + 4δ4).

Taking the difference of the first two equations in (6.39) gives

0 = (n+ p)x − (n− p)Vx = (n+ p)x − Vx

andVx = (n+ p)x in (0, 1). (6.40)

124

Thus, adding the first two equations in (6.39) yields

2I = −(n+ p)Vx = −1

2

d

dx[(n+ p)2].

This equation can be integrated from x to 1:

(n+ p)2(x) = (n+ p)2(1) + 4I(1− x),

and the boundary conditions at x = 1 and the last equation in (6.39) give

√1 + 4δ4 + 4I(1− x) = (n+ p)(x) = 2n(x)− 1.

We conclude

n(x) =1

2(1 +

√1 + 4δ4 + 4I(1− x)),

p(x) =1

2(−1 +

√1 + 4δ4 + 4I(1− x)).

The electrostatic potential is computed from (6.40) by integrating from x to 1:

V (x) = V (1) + (n+ p)(x)− (n+ p)(1)

= arsinh1

2δ2− U0

2+√

1 + 4δ4 + 4I(1− x)−√1 + 4δ4.

At x = 0 we obtain

U02

= arsinh1

2δ2+√

1 + 4δ4 + 4I(1− x)−√1 + 4δ4.

If U0 →∞ or, equivalently, I →∞,

I ∼ U 20 . (6.41)

This result means that the growth of the voltage-current characteristic slowsdown in high injection. The relation (6.39) is known as Mott’s law.

For sufficiently large voltages we have Ohm’s law

I

U0→ const. (U0 →∞).

Whereas the Zener effect is not modeled in our drift-diffusion system (see [34] fora more general system), we can derive the Ohm law for U0 À 1. To see this, werescale the current densities and the potential,

Jn = Uα0 jn, Jp = Uβ

0 jp, V = Uγ0 φ,

125

where α, β, γ > 0, and we assume that the particle densities are of order O(1)when U0 → ∞. Also jn, jp and φ are supposed to be of order O(1) as U0 → ∞.The Poisson equation then becomes

λ2∆φ = U−γ0 (n− p− C)→ 0 in Ω, as U0 →∞,

φ = 0 on Γ1, φ = U 1−γ0 on Γ2, ∇φ · ν = 0 on ΓN .

When γ > 1, φ solves the Laplace equation with homogenous boundary conditionsin the limit U0 → ∞ and therefore, φ = 0 in Ω. This solution is useless. On theother hand, when γ < 1, φ→∞ on Γ2 (U0 →∞), which is not the right scaling.Hence γ = 1. The scaled current densities are

jn = U−α0 ∇n− U 1−α

0 n∇φ, jp = −U−β0 ∇p− U 1−β

0 p∇φ.

When α > 1, we get jn = 0 in the limit which is useless. When α < 1, we haven∇φ = 0 which does not allow to compute jn. Therefore, α = 1. Similarly, weconclude β = 1. This yields, as U0 →∞,

jn = −n∇φ, jp = −p∇φ.

Notice that φ solves the problem

∆φ = 0 in Ω,

φ = 0 on Γ1, φ = 1 on Γ2, ∇φ · ν = 0 on ΓN ,

and does not depend on U0. We obtain (see (6.20))

I

U0=

1

U0

∫

Γ1

(U0jn + U0jp) · ν ds

= −∫

Γ1

(n+ p)∇φ · ν ds

= −√C2

n + 4δ4∫

Γ1

∇φ · ν ds

= const.

We remark that the assumption that n and p are of order O(1) as U0 →∞ isnot very realistic. However, since we evaluate n and p only on Γ1 and they arefixed by the boundary conditions, we expect to obtain the same formula as abovewithout this assumption.

Whereas the Mott law I ∼ U 20 holds for “large” U0, we needed to perform

the limit U0 →∞ to derive the Ohm law I ∼ U0. Therefore, we expect that theMott law holds for “large” applied voltages and the Ohm law for “very large”applied voltages. This can be confirmed by physical experiments. Conditions onthe size of the applied voltage which allow to distinguish the different regimes for“small”, “large” and “very large” applied bias, unfortunately, do not follow fromthe above asymptotic analysis.

126

6.4 Numerical discretization of the stationary equations

The discretization of the drift-diffusion equations using standard methods maylead to unsatisfactory results. Indeed, for “large” electric fields −∇V, the corre-sponding elliptic problem is convection-dominant, and it is well known that thenumerical discretization of such problems requires special care. In a simplifiedsituation, we can see this directly from the equations.

Consider the one-dimensional stationary equations for the electron density inΩ = (0, 1) with µ′n = 1 and R = 0 (see (6.17)):

Jn = nx − nVx, Jn,x = 0 in (0, 1),

n(0) = nD0, n(1) = nD1,

for given electrostatic potential. We introduce the grid xi = ih, i = 0, . . . , N,with h > 0 and N = 1/h ∈ N. With the approximations ni, Vi of n(xi), V (xi)respectively, a standard finite-difference scheme is given by

Jn,i+1/2 =ni+1 − ni

h− ni+1 + ni

2

Vi+1 − Vi

h, (6.42)

1

h(Jn,i+1/2 − Jn,i−1/2) = 0, i = 1, . . . , N − 1. (6.43)

Then Jn,i+1/2 can be regarded as an approximation of J(xi+1/2) = J(xi + h/2).Setting ci := Vi − Vi−1, we can rewrite (6.42)-(6.43) as a difference equation

(2− ci+1)ni+1 + (ci − ci+1 − 4)ni + (2 + ci)ni−1 = 0, i = 1, . . . , N − 1,

n0 = nD0, nN = nD1.

Suppose that the electric field is almost constant in Ω such that c := ci =const. > 0 for all i = 1, . . . , N and that the grid is such that c 6= 2. Then theabove difference equation has the explicit solution

ni = (nD1 − nD0)ri − 1

rN − 1+ nD0, i = 0, . . . , N, with r =

2 + c

2− c.

The discrete electron density ni should be positive for all i. However, with thediscretization (6.42)-(6.43) this may be not true. Indeed, let (for simplicity)nD0 = 2, nD1 = 1, N even and let

c > 221/N + 1

21/N − 1.

Then c > 2, −21/N < r < −1 and 1 < rN < 2. If i is odd, then ri < 0, ri+rN < 2and

ni =ri − 1

rN − 1+ 1 =

ri + rN − 2

rN − 1< 0.

127

In fact, ni < 0 holds for all odd indices i and ni > 0 for all even indices i. Inorder to avoid the oscillatory behaviour, one has to choose the grid in such a waythat c = Vi − Vi−1 < 2. In Section 6.3, we have found that the potential can bewritten as

V = ln

[1

2δ2u(C +

√C2 + 4δ4uw)

]in Ω

for the reduced problem λ = 0, and u,w solve the problem (6.28)-(6.31). Thismeans that if C is discontinuous in Ω, we expect large gradients for V. To geta non-oscillating numerical scheme, we need to choose a very fine grid at leastnear the points where the field is “large” (i.e. Vi − Vi−1 < 2). Particularly formulti-dimensional problems the restriction that the potential only varies weaklyon each of the subintervals (xi, xi+1) may require extremely many grid pointsand is very expensive and sometimes even impossible to guarantee. The scheme(6.42)-(6.43) is therefore not practical.

One possibility to solve the drift-diffusion equations numerically accurately isto use the Scharfetter-Gummel discretization. The idea is to symmetrize first thecurrent densities and then to discretize the equations. We illustrate the idea forthe one-dimensional stationary equations

Jn,x = 0, Jn = µn(nx − nVx),

Jp,x = 0, Jp = −µp(px + pVx),

λ2Vxx = n− p− C in Ω = (0, 1)

subject to the boundary conditions

n(0) = nD0, n(1) = nD1, p(0) = pD0, p(1) = pD1,

V (0) = 0, V (1) = U0.

In the following we consider only the equation for n since the treatment ofthe equation for p is analogous. Define the Slotboom variable z = µnne

−V (seeSection 6.3). Then

eV zx = µn(nx − nVx) = Jn

and we have to discretize

Jn,x = 0, zx = Jne−V in (0, 1). (6.44)

Introduce grid points 0 = x0 < x1 < . . . < xN = 1 with hi := xi+1 − xi, i =0, . . . , N − 1. We use central finite differences to discretize the Poisson equation:

ni − pi − C(xi) =λ2

hi−1

(Vi+1 − Vi

hi

− Vi − Vi−1

hi−1

)

=λ2

hi−1hi

[Vi+1 −

(1 +

hi

hi−1

)Vi +

hi

hi−1

Vi−1

],

128

where ni, pi, and Vi are approximations of n(xi), p(xi), and V (xi), respectively.We define an approximate potential Vh as the linear interpolation of the Vi:

Vh(x) = Vi + (x− xi)Vi+1 − Vi

hi

, x ∈ [xi, xi+1].

We use the approximate potential in the last equation of (6.44), integrate over(xi, xi+1) and use the fact that Jn is constant on Ω:

zi+1 − zi =∫ xi+1

xi

zh,x(x) dx = Jn

∫ xi+1

xi

e−Vh(x) dx.

Here, zh(xi) = zi are approximations of z(xi). The last integral can be computedexplicitely since Vh is piecewise linear:

zi+1 − zi = Jne−Vi

hi

Vi+1 − Vi

(1− e−(Vi+1−Vi)

).

This gives a definition for Jn (after transforming back to ni = zieVi/µn):

Jn = µnVi+1 − Vi

hi

eVi

1− eVi−Vi+1

(e−Vi+1ni+1 − e−Vini

)

=µn

hi

(B(Vi+1 − Vi)ni+1 −B(Vi − Vi+1)ni) , (6.45)

with the Bernoulli function

B(x) =x

ex − 1, x ∈ R.

In order to understand the difference of the Scharfetter-Gummel discretization(6.45) and the standard finite-difference scheme (6.42), we rewrite (6.45) as

Jn = µnVi+1 − Vi

hi

[(1

2+

eVi−Vi+1

1− eVi−Vi+1

)ni+1 −

(1

1− eVi−Vi+1− 1

2

)ni

]

− µnni + ni+1

2

Vi+1 − Vi

hi

= µn

((Vi+1 − Vi) coth

Vi+1 − Vi

2

)ni+1 − ni

hi

− µnni + ni+1

2

Vi+1 − Vi

hi

.

The function

f(Vi+1 − Vi) = (Vi+1 − Vi) cothVi+1 − Vi

2

plays the role of an upwind factor. Indeed, for slowly varying potentials |Vi+1 −Vi| ¿ 1 we have

f(x) = 1 as x→ 0,

129

whereas for large gradients |Vi+1 − Vi| À 1 it holds

f(x) ∼ x as x→ ±∞.

Thus, for diffusion-dominated problems |Vi+1 − Vi| ¿ 1, we obtain the standarddiscretization (6.42), and for convection-dominated problems |Vi+1−Vi| À 1, theupwind factor provides a correction such that the diffusion term

((Vi+1 − Vi) coth

Vi+1 − Vi

2

)ni+1 − ni

hi

∼ (Vi+1 − Vi)ni+1 − ni

hi

is of the same order as the convection term

ni+1 + ni

2

Vi+1 − Vi

hi

.

The current density for the holes is discretized analogously:

Jp = −µp

((Vi+1 − Vi) coth

Vi+1 − Vi

2

)pi+1 − pi

hi

− µppi+1 + pi

2

Vi+1 − Vi

hi

.

We obtain three discrete nonlinear systems

an,ini+1 + bn,ini + cn,ini−1 = 0,

ap,ipi+1 + bp,ipi + cp,ipi−1 = 0,

αiVi+1 + βiVi + γiVi−1 = λ−2hi−1hi(ni − pi − C(xi)),

where we have defined

an/p,i =1

hi

(f(Vi+1 − Vi)∓

1

2(Vi+1 − Vi)

),

bn/p,i = − 1

hi

(f(Vi+1 − Vi)∓

1

2(Vi+1 − Vi)

)

− 1

hi−1

(f(Vi − Vi−1)∓

1

2(Vi − Vi−1)

),

cn/p,i =1

hi−1

(f(Vi − Vi−1)±

1

2(Vi − Vi−1)

),

and

αi = 1, βi = −1−hi

hi−1

, γi = −hi

hi−1

.

130

7 The Energy-Transport Equations

In this chapter, we analyze the energy-transport equations for spherically sym-metric and strictly monotone energy bands ε(|k|) in R3, as derived in Section 5.5(see Example 5.9):

∂tn(µ, T )−1

qdiv J1 = 0, (7.1)

∂tE(µ, T )− div J2 +∇V · J1 = W (µ, T ), (7.2)

εs∆V = q(n(µ, T )− C(x)), x ∈ Ω, t > 0, (7.3)

where the electron and energy current densities are given by (see (5.74)-(5.75))

J1 = D11∇(qµ

kBT

)+D12∇

( −1kBT

)−D11

q∇VkBT

, (7.4)

J2 = D21∇(qµ

kBT

)+D22∇

( −1kBT

)−D21

q∇VkBT

, (7.5)

the electron and energy density are defined, respectively, by

n(µ, T ) = 2πeqµ/kBT

∫ ∞

0

√γ(ε)γ ′(ε)e−ε/kBT dε, (7.6)

E(µ, T ) = 2πeqµ/kBT

∫ ∞

0

√γ(ε)γ ′(ε)e−ε/kBT ε dε, (7.7)

V is the electrostatic potential, W (µ, T ) a relaxation term satisfying

W (µ, T )(T − TL) ≤ 0 for all µ ∈ R, T > 0, (7.8)

µ is the chemical potential, T the electron temperature, TL the lattice temper-ature, and C(x) the doping profile. We assume Maxwell-Boltzmann statisticsand space-independent scattering rates, so that the diffusion matrices Dij can bewritten as (see (5.76))

Dij(µ, T ) = eqµ/kBT

∫ ∞

0

D(ε)e−ε/kBT εi+j−2 dε, i, j = 1, 2. (7.9)

The function γ is defined by |k|2 = γ(ε(|k|)). The equations (7.1)-(7.3) are con-sidered in the bounded domain Ω ⊂ Rd (d ≥ 1), and we impose the followinginitial and boundary conditions

µ(·, 0) = µI , T (·, 0) = TI in Ω,

µ = µD, T = TD, V = VD on ΓD × (0,∞),

J1 · ν = J2 · ν = ∇V · ν = 0 on ΓN × (0,∞).

We have assumed that ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅, ΓD is open in ∂Ω, and ν isthe exterior unit normal of ∂Ω.

131

7.1 Symmetrization and entropy

In this section we introduce a change of unknowns which symmetrizes the energy-transport model (7.1)-(7.7) in the sense that the lower-order terms −∇V ·J1 and−Di1q∇V/kBT are absent in the new formulation. Moreover, we define a so-called entropy functional which is non-increasing in time and which relates theold and the new variables.

Definition 7.1 The variables

u1 =qµ

kBT, u2 = −

1

kBT

are called (primal) entropy variables; the variables

w1 =q(µ− V )

kBT= u1 + qV u2, w2 = −

1

kBT= u2

are called dual entropy variables.

We set u = (u1, u2) and w = (w1, w2). In the entropy variables the electronand energy density can be formulated as

ρ1(u) := n = 2πeu1

∫ ∞

0

√γ(ε)γ ′(ε)eεu2 dε,

ρ2(u) := E = 2πeu1

∫ ∞

0

√γ(ε)γ ′(ε)eεu2ε dε.

In the dual entropy variables, the energy-transport model (7.1)-(7.5) can bewritten in a symmetric form.

Lemma 7.2 The equations (7.1)-(7.5) are formally equivalent to

∂tb1(u)− div I1 = 0, (7.10)

∂tb2(u)− div I2 = W − qn∂tV, (7.11)

εs∆V = q(n− C(x)),

whereb1(u) = n, b2(u) = E − qV n

and

I1 =1

qJ1 = L11∇w1 + L12∇w2,

I2 = J2 − V J1 = L21∇w1 + L22∇w2.

The diffusion coefficients Lij are given by

L11 = D11, L12 = L21 = D12 −D11qV, L22 = D22 − 2D21qV +D11(qV )2.

132

Moreover, the new diffusion matrix L = L(µ, T ) = (Lij)2ij=1 ∈ R6×6 is symmetric

in the sense L12 = L21 and positive definite if the functions

∂ε

∂k1,∂ε

∂k2,∂ε

∂k3, ε

∂ε

∂k1, ε

∂ε

∂k2, and ε

∂ε

∂k3(7.12)

are linearly independent.

Proof. A simple computation gives, by (7.4)-(7.5),

qI1 = D11∇u1 +D12∇u2 +D11u2q∇V= D11∇(u1 + u2qV ) + (D12 −D11qV )∇u2= L11∇w1 + L12∇w2,

I2 = D21∇u1 +D22∇u2 +D21u2q∇V − V J1= (D21 −D11qV )∇(u1 + u2qV ) + (D22 − 2D21qV +D11(qV )2)∇u2

+D11qV∇(u1 + u2qV ) + qV (D21 −D11qV )∇u2 − qV I1= L21∇w1 + L22∇w2,

since Proposition 5.8 implies D12 = D21 and hence L12 = L21. This also proves(7.10) and (7.11) since

∂tb2(u)− div I2 = ∂t(E − qV n)− div (J2 − V J1)= W −∇V · J1 − qn∂tV − qV ∂tn+∇V · J1 + V div J1

= W − qn∂tV.

It remains to prove the positive definiteness of L. We compute

L =

(Id −qV Id0 Id

)>( D11 D12

D21 D22

)(Id −qV Id0 Id

). (7.13)

Here, Id is the unit matrix in R3×3. The assumptions on the linear independencyof (∂ε/∂ki, ε∂ε/∂ki)i ensure, by Proposition 5.8, that D is positive definite.Thus, L is positive definite since the matrix

(Id −qV Id0 Id

)

is invertible. ¤

The symmetrization property is related to the existence of an entropy func-tional. In fact, both properties are equivalent (see [20, 32] for details). Before weintroduce the entropy, we need to discuss the thermal equilibrium state definedby J1 = J2 = 0. This implies

0 =

(I1I2

)= L∇

(w1

w2

).

133

From now on we assume that the functions in (7.12) are linearly independent.Then L is symmetric and positive definite, hence invertible, and the above equa-tion yields

w1 = const. and w2 = const. in Ω.

Therefore, T = const. and µ − V = const. in Ω. As V is only determined up toan additive constant, we can choose this constant such that µ− V = 0 or w1 = 0in Ω. Since T = TL on a part of the boundary, T = TL in Ω. By (7.8), this impliesW (µ, T ) = 0 in Ω for all µ. Then we obtain from (7.1), (7.2) that ∂tn = 0 and∂tE = 0.

We notice that this and the assumption of vanishing total space charge onthe Dirichlet boundary part (i.e. n = C on ΓD) determines the thermal stateuniquely. Indeed, from (7.6) or

n = Neqµ/kBTL with N = 2π

∫ ∞

0

√γ(ε)γ

′

(ε)e−ε/kBTL dε > 0

in thermal equilibrium we obtain

ne−qµV/kBTL = N ⇐⇒ V =kBTL

qlnn

N.

On the boundary ΓD it holds

V = UT lnn

N= UT ln

C(x)

N,

where UT = kBTL/q and thus, the thermal equilibrium state (µe, Te, Ve) is theunique solution of

εs∆Ve = q(NeVe/UT − C(x)) in Ω, (7.14)

Ve = UT lnC(x)

Non ΓD, ∇Ve · ν = 0 on ΓN (7.15)

and µe = Ve, Te = LL in Ω.The entropy density is now defined by

η(u) = ρ(u) · (u− ue)− χ(u) + χ(ue),

where

ρ(u) =(nE), ue =

(ue1

ue2

)=

(qµe/kBTL

−1/kBTL

)

and χ: R2 → R is a function satisfying χ′(u) = ρ(u). More precisely, −η is thephysical entropy density relative to the thermal equilibrium state ue. The entropyis the integral over the entropy density:

S0(t) =

∫

Ω

η(x, t) dx.

134

Example 7.3 (Parabolic band approximation)Under the assumptions of Example 5.10 we can express the energy density interms of n and T. Since, by Example 5.10,

n = Nc(kBT )3/2eqµ/kBT , E =

3

2kBTn,

where Nc = 2(2πm∗/~2)3/2, we have

u1 = lnn

Nc

− 3

2ln(kBT ).

The equilibrium entropy variables are

ue1 = lnne

Nc

− 3

2ln(kBTL), ue2 = −

1

kBTL

,

wherene = Nc(kBTL)

3/2eqµe/kBTL

and µe = Ve is the (unique) solution of (7.14)-(7.15). A simple computationverifies that χ(u) = n. Therefore

η =

(n

32kBTn

)·(ln(n/ne)− 3

2ln(T/TL)

(T − TL)/kBTLT

)− n+ ne

= n

(lnn

ne

− 3

2ln

T

TL

)+

3

2

nT

TL

− 5

2n+ ne.

Notice that the entropy vanishes in thermal equilibrium:

η∣∣n=ne, T=TL

= 0.

The free energy of the energy-transport system is the sum of the entropy andthe electric energy:

S(t) =

∫

Ω

(η(u) +

1

2εsTL

|∇(V − Ve)|2)(t) dx.

Sometimes, also the function S(t) is called entropy. It satisfies the so-calledentropy inequality:

Proposition 7.4 Assume that the boundary data is in thermal equilibrium, i.e.

µD = µe, TD = TL, VD = Ve on ΓD × (0,∞),

that (7.8) holds and that the functions in (7.12) are linearly independent. Then

S(t) +

∫ t

0

∫

Ω

(∇w)>L(∇w) dx dt ≤ S(0).

135

The assumptions of the proposition imply that L is symmetric, positive def-inite. Hence the above integral which is called entropy dissipation term is non-negative. The entropy inequality shows that the entropy is non-increasing. Undersuitable assumptions on L, ρ(u) and W it is possible to show that S(t) convergesexponentially fast to zero as t → ∞ [19]. As we have not specified our notationof solution, computations have to be understood in a formal way. The argumentsare made rigorous in [19].

Proof. We compute the time derivative of S(t). Since χ′(u) = ρ(u) and therefore

∂tη(u) = ∂tρ(u) · (u− ue)− ρ(u) · ∂tu− χ′(u) · ∂tu= ∂tρ(u) · (u− ue),

we obtain, by integrating by parts,

∂tS =

∫

Ω

[∂tn(u1 − ue1) + ∂tE(u2 − ue2) +

1

εsTL

∂t∇V · ∇(V − Ve)

]dx

=

∫

Ω

[∂tn(u1 + qV u2 − (ue1 + qVeue2)) + ∂t(E − qV n)(u2 − ee2)

− ∂tnq(V u2 − Veue2) + q∂t(V n)(u2 − ue2)−1

TL

∂tn(V − Ve)

]dx

=

∫

Ω

[div I1(w1 − we1) + div I2(w2 − we2) +W (u2 − ue2)

− qn∂tV (u2 − ue2)− q∂tn((V − Ve)u2 + Ve(u2 − ue2))

+ qn∂tV (u2 − ue2) + q∂tnV (u2 − ue2) + q∂tn(V − Ve)ue2] dx

(using (7.10) and (7.11))

=

∫

Ω

[−I1 · ∇w1 − I2 · ∇w2 +W

T − TL

kBTTL

]dx

(since we1 = 0, we2 = const.)

≤ −∫

Ω

(∇w)>L(∇w) dx.

Integrating over (0, t) gives the result. ¤

7.2 A drift-diffusion formulation

In the previous section we have derived a formulation of the energy-transportmodel using the dual entropy variables w1 and w2. The advantage of this formu-lation is that first-order terms like −∇V · J1 vanish. This is also an advantagefrom a numerical point of view. Indeed, the symmetric operator −div (L∇w) canbe discretized using standard methods and the discrete nonlinear system can besolved via the Newton method. However, the use of the Newton method has the

136

disadvantage that the Jacobian of a (2 × 2)-matrix has to be computed whichmakes the system less flexible (for instance, when changing the energy band)and computationally quite costly. One way to overcome this disadvantage is tolook for another formulation of the system which allows for a decoupling of theequations like in the drift-diffusion model (see Section 6.4).

In this section we show that the energy-transport model can be formulatedas a drift-diffusion system which allows for the use of the numerical methodsdeveloped for the classical drift-diffusion model. For this, we turn back to thedefinition of the particle current density J1 and energy current density J2 ofSection 5.5:

Ji =

∫

RD(ε)

(∇xF + q∇xV

∂F

∂ε

)εi−1 dε, i = 1, 2,

where the diffusion matrix is assumed to be independent of x and F is given bythe Maxwell-Boltzmann statistics:

F (ε) = eεu2+u1

with u1 = qµ/kBT and u2 = −1/kBT. Then ∂F/∂ε = (−1/kBT )F and

Ji = ∇x

∫

RD(ε)F (ε)εi−1 dε− q∇xV

kBT

∫

RD(ε)F (ε)εi−1 dε.

Setting

g1 = D11 =

∫

RD(ε)F (ε) dε,

g2 = D12 =

∫

RD(ε)F (ε)ε dε,

we obtain

J1 = ∇g1 −q∇VkBT

g1, J2 = ∇g2 −q∇VkBT

g2.

In this formulation the energy-transport model simplifies since it is no longerstrongly nonlinear in the sense that the equation for n (for E , respectively) onlydepends on second-order derivatives of g1 (of g2) and not also of g2 (of g1). Noticethat the equation for n in the (w1, w2)-formulation contains second-order deriva-tives of w1 and w2. Clearly, the (g1, g2)-formulation contains first-order terms like−∇V · J1 but they can be handled numerically (see [21, 29]). Moreover, we haveto ensure that the variables T , µ, n and E can be determined uniquely from g1,g2. This will be proved in the following.

The temperature can be computed from the nonlinear equation

g2g1

=

∫∞

0D(ε)e−ε/kBT ε dε∫∞

0D(ε)e−ε/kBT dε

=: f(kBT ).

The following lemma states under which assumptions the function f can be in-verted.

137

Lemma 7.5 Assume that the functions (7.12) are linearly independent. Then

f ′(kBT ) =detL

(kBTg1)2> 0 ∀T > 0.

Proof. Since

∂g1∂(kBT )

=1

(kBT )2

∫ ∞

0

D(ε)e−ε/kBT ε dε =D12

(kBT )2,

∂g2∂(kBT )

=1

(kBT )2

∫ ∞

0

D(ε)e−ε/kBT ε2 dε =D22

(kBT )2,

and g1 = D11, g2 = D12, we obtain

f ′(kBT ) =1

g21

(g1

∂g2∂(kBT )

− g2∂g1

∂(kBT )

)

=1

(kBTg1)2(D11D22 −D2

12)

=detD

(kBTg1)2

=detL

(kBTg1)2

> 0,

using (7.13) and Proposition 5.8(3). ¤

The above lemma shows that for given g1 and g2, the temperature

kBT (g1, g2) = f−1(g2/g1)

is well defined. Notice that the value g1 = 0 is not possible since this would implyD11 = 0 and detD = −D2

12 ≤ 0—contradiction.We still have to derive formulas for n, E and µ as functions of g1 and g2. From

now on, we assume spherically symmetric stricty monotone energy bands. Fornotational convenience we set

Qi(kBT ) = 2π

∫ ∞

0

√γ(ε)γ

′

(ε)e−ε/kBT εi dε, i = 0, 1,

Pi(kBT ) =

∫ ∞

0

D(ε)e−ε/kBT εi dε, i = 0, 1.

Then, by (7.6)-(7.7),

n = eqµ/kBTQ0(kBT ), E = eqµ/kBTQ1(kBT )

andg1 = eqµ/kBTP0(kBT ), g2 = eqµ/kBTP1(kBT ).

138

Observing the relations

n

g1=Q0(kBT )

P0(kBT ),Eg2

=Q1(kBT )

P1(kBT ),

we can compute, for given g1, g2,

n(g1, g2) = g1Q0(kBT (g1, g2))

P0(kBT (g1, g2)), E = g2

Q1(kBT (g1, g2))

P1(kBT (g1, g2)).

The chemical potential µ can be expressed in terms of g1, g2 by using the expres-sion n = exp(qµ/kBT )Q0(kBT ):

µ(g1, g2) =kBT

qlnQ0(kBT )

n(g1, g2)=kBT

qlnQ0(kBT )P0(kBT )

g1Q0(kBT )

=kBT (g1, g2)

qlnP0(kBT (g1, g2))

g1.

Thus we can summarize the energy -transport model in the (g1, g2)-formulation:

∂tn(g1, g2)−1

qdiv J1 = 0,

∂tE(g1, g2)− div J2 = −∇V · J1 +W (µ(g1, g2), T (g1, g2)),

J1 = ∇g1 −q∇V

kBT (g1, g2)g1,

J2 = ∇g2 −q∇V

kBT (g1, g2)g2,

εs∆V = q(n(g1, g2)− C(x)).

We have derived three different formulations of the energy-transport equa-tions: the formulation in the (primal) entropy variables (u1, u2) (or, equivalently,µ, T ); the formulation in the dual entropy variables (w1, w2); and in the vari-ables (g1, g2). Table 7.1 summarizes the advantages and disadvantages of theseformulations.

Formulation Advantages Disadvantages(primal) entropyvariables (u1, u2)

thermodynamical vari-ables

drift terms, full diffusionmatrix

dual entropy vari-ables (w1, w2)

thermodynamical vari-ables, no drift terms

full diffusion matrix

variables (g1, g2) drift terms diagonal diffusion matrix

Figure 7.1: Comparison of different formulations of the energy-transport model.

139

7.3 A non-parabolic band approximation

We compute the electron density n(µ, T ), the energy density E(µ, T ), the diffusionmatrices Dij(µ, T ) and the relaxation term W (µ, T ) in the case of Kane’s non-parabolic band approximation [31]. We impose the following hypotheses (see[21]):

• The energy-band diagram is given by

ε(1 + αε) =~2|k|22m∗

, k ∈ Rd, α > 0.

• The relaxation time is given as in Example 5.5 by

τ(ε) =1

φ(ε)N(ε),

where we choose φ(ε) = φ0εβ with φ0 > 0, β > −2.

• The relaxation term is given by the Fokker-Planck approximation (see Sec-tion 5.5):

W (µ, T ) =

∫ ∞

0

∂

∂ε

[A(ε)

(e(qµ−ε)/kBT + kBTL

∂

∂εe(qµ−ε)/kBT

)]ε dε,

where A(ε) = φ(ε)N(ε)2.

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|k|

ε(|k|

)

Figure 7.2: Illustration of the energy band ε(|k|) in the parabolic case (brokenline) and the non-parabolic case (solid line) with parameters α = 0.25, ~/2m∗ =1.

The parameter α is called non-parabolicity parameter. The difference toparabolic bands is shown in Figure 7.2. The function γ defined by |k|2 = γ(ε(|k|))reads as follows:

γ(ε) =2m∗

~2ε(1 + αε), ε ≥ 0.

140

Recall that the density of states is given by N(ε) = 2π√γ(ε)γ ′(ε). With the

above hypotheses we are able to compute n, E , D and W.Direct computations yield (see the definitions (7.6), (7.7) and (7.9))

n = 2πeqµ/kBT

(2m∗

~2

)3/2 ∫ ∞

0

√ε(1 + αε) (1 + 2αε)e−ε/kBT dε

= 2πeqµ/kBT

(2m∗

~2

)3/2

(kBT )3/2

×∫ ∞

0

√1 + αkBTu (1 + 2αkBTu)u

1/2e−u du

= q1/2(αkBT )(kBT )3/2eqµ/kBT ,

and

E = 2πeqµ/kBT

(2m∗

~2

)3/2 ∫ ∞

0

√1 + αε (1 + 2αε)ε3/2e−ε/kBT dε

= q3/2(αkBT )(kBT )5/2eqµ/kBT

=q3/2(αkBT )

q1/2(αkBT )kBTn,

where

qβ(αkBT ) = 2π

(2m∗

~2

)3/2 ∫ ∞

0

√1 + αkBTu (1 + 2αkBTu)u

βe−u du.

For the computation of Dij we start from the formula (5.79) derived in Example5.9 (identifying Dij and Dij · Id)

Dij =4q

3~2eqµ/kBT

∫ ∞

0

γ(ε)εi+j−2

φ(ε)γ ′(ε)2e−ε/kBT dε

=4q

3~2~2

2m∗φ0eqµ/kBT

∫ ∞

0

1 + αε

(1 + 2αε)2εi+j−β−1e−ε/kBT dε

= eqµ/kBT (kBT )i+j−β pi+j(αkBT )

=pi+j(αkBT )

q1,2(αkBT )(kBT )

i+j−β−3/2 n.

where

pi+j(αkBT ) =2q

3m∗φ0

∫ ∞

0

1 + αkBTu

(1 + 2αkBTu)2ui+j−β−1 du.

141

Finally, we calculate the relaxation term, using the results of Example 5.11,

W = −4π2φ0(2m∗

~2

)3

eqµ/kBT

(1− TL

T

)∫ ∞

0

εβγ(ε)γ′

(ε)2e−ε/kBT dε

= −4π2φ0(2m∗

~2

)3

eqµ/kBT

(1− TL

T

)(kBT )

β+2

×∫ ∞

0

(1 + αkBTu)(1 + 2αkBTu)2u1+βe−u du

= −4π2φ0(2m∗

~2

)3

eqµ/kBT (kBT )β+1kB(T − TL)[Γ(β + 2)

+ 5αkBT Γ(β + 3) + 8(αkBT )2 Γ(β + 4) + 4(αkBT )

3 Γ(β + 5)]

= −kB(T − TL)n

τ(T ),

where Γ denotes the Gamma function defined by

Γ(s) =

∫ ∞

0

us−1e−u du, s > 0,

and τ(T ) is a temperature-dependent relaxation time:

1

τ(T )= 2πφ0

(2m∗

~2

)3/2(kBT )

β−1/2

q1/2(αkBT )[Γ(β + 2) + 5αkBT Γ(β + 3)

+ 8(αkBT )2 Γ(β + 4) + 4(αkBT )

3 Γ(β + 5)].

In the parabolic band case α = 0 and the Chen model β = 1/2 we recover theexpressions for n, E , D and W from Examples 5.10 and 5.11.

142

8 From Kinetic to Quantum Hydrodynamic

Models

8.1 The quantum hydrodynamic equations: first deriva-tion

In Chapter 1 we have sketched the (formal) equivalence of the single-electronSchrodinger equation with electrostatic potential V = V (x, t)

i~∂tψ = − ~2

2m∆ψ − qV ψ, x ∈ Rd, t > 0,

and the zero-temperature quantum hydrodynamic equations

∂tn−1

qdiv J = 0,

∂tJ −1

qdiv

(J ⊗ Jn

)+q2

mn∇V +

~2q2m2

n∇(∆√n√n

)= 0, x ∈ Rd, t > 0,

via the Madelung transform ψ =√n exp(imS/~), where

n = |ψ|2, J = −~qm

Im(ψ∇ψ)

are the electron density and electron current density, respectively.In this section we consider an electron ensemble which is represented by a

mixed state (see Section 4.1). A mixed quantum mechanical state consists of asequence of single states with occupation probabilities λk ≥ 0 (k ∈ N) for thek-th state described by the Schrodinger equations

i~∂tψk = − ~2

2m∆ψk − qV ψk, x ∈ Rd, t > 0, (8.1)

ψk(x, 0) = ψI,k(x), x ∈ Rd.

The occupation probabilities satisfy∑∞

k=0 λk = 1. We write the initial datum ofthe k-th state in the form

ψI,k =√nI,k exp(imSI,k/~). (8.2)

The electron density nk and the electron current density Jk of the k-th singlestate are given by

nk = |ψk|2, Jk = −~qm

Im(ψk∇ψk).

For the fluiddynamical formulation we use the Madelung transform

ψk =√nk exp(imSk/~). (8.3)

143

Clearly, we have to assume that nk > 0 for all k ≥ 0. With this transform thecurrent density now reads

Jk = −qnk∇Sk. (8.4)

We wish to find evolution equations for nk and Jk. Setting (8.3) into (8.1)gives, after division of exp(imSk/~),

i~2

∂tnk√nk

−m√nk∂tSk = − ~2

2m

(∆√nk +

2im

~∇√nk · ∇Sk −

m2

~2√nk|∇Sk|2

+im

~√nk∆Sk

)− qV√nk. (8.5)

The imaginary part of this equation is

∂tnk = −2√nk∇√nk · ∇Sk − nk∆Sk = −div (nk∇Sk)

or, using (8.4),

∂tnk −1

qdiv Jk = 0. (8.6)

The real part of (8.5) reads

∂tSk =~2

2m2

∆√nk√nk

− 1

2|∇Sk|2 +

q

mV.

In order to find an evolution equation for Jk = −qnk∇Sk, we take the gradientof the above equation and multiply the resulting equation by −qnk:

−qnk∂t∇Sk = − ~2q2m2

nk∇(∆√nk√nk

)+q

2nk∇|∇Sk|2 −

q2

mnk∇V.

Since−qnk∂t∇Sk = ∂tJk + q∂tnk∇Sk = ∂tJk + (div Jk)∇Sk

and

q

2nk∇|∇Sk|2 = qdiv (nk∇Sk ⊗∇Sk)− qdiv (nk∇Sk)∇Sk

=1

qdiv

(Jk ⊗ Jknk

)+ (div Jk)∇Sk,

the above equation can be written as

∂tJk −1

qdiv

(Jk ⊗ Jknk

)+q2

2nk∇V +

~2q2m2

nk∇(∆√nk√nk

)= 0. (8.7)

Here, Jk ⊗ Jk is a tensor (or matrix) with components Jk,iJk,j for i, j = 1, . . . , d.

144

The total carrier density n and current density J of the mixed state are givenby

n =∞∑

k=0

λknk, J =∞∑

k=0

λkJk.

In the following, we will derive evolution equations for n and J from (8.6) and(8.7). Multiplication of (8.6) and (8.7) by λk and summation over k yields

∂tn−1

qdiv J = 0

and

∂tJ −1

q

∞∑

k=0

λkdiv

(Jk ⊗ Jknk

)+ qn∇V +

~2q2m2

∞∑

k=0

λknk∇(∆√nk√nk

)= 0. (8.8)

In order to rewrite the second and fourth term on the left-hand side of thisequation, we introduce the “current” velocities

uc = −J

qn, uc,k = − Jk

qnk

and the “osmotic” velocities

uos =~2m∇ log n, uos,k =

~2m∇ log nk.

The notion “osmotic” comes from the fact that

Pk =~2q4m2

nk(∇⊗∇) log nk

can be interpreted as a non-diagonal pressure tensor since

~2q2m2

nk∇(∆√nk√nk

)= divPk.

Then

−1

q

∞∑

k=0

λkdiv

(Jk ⊗ Jknk

)= −q

∞∑

k=0

div (λknkuc,k ⊗ uc,k)

= −q∞∑

k=0

λkdiv (nk(uc,k − uc)⊗ (uc,k − uc) + 2nkuc,k ⊗ uc)

+qdiv (nu⊗ u)

= −q∞∑

k=0

div

(λknk(uc,k − uc)⊗ (uc,k − uc) +

2λk

q2Jk ⊗ Jn

)

+1

qdiv

(J ⊗ Jn

)

= −1

qdiv

(J ⊗ Jn

)− div (nθc),

145

where

θc = q∞∑

k=0

λknk

n(uc,k − uc)⊗ (uc,k − uc)

is called the current temperature. Furthermore,

~2q2m2

∞∑

k=0

λknk∇(∆√nk√nk

)

=~2q4m2

∞∑

k=0

λkdiv

[(∇⊗∇)nk −

∇nk ⊗∇nk

nk

]

=~2q4m2

∞∑

k=0

λkdiv

[(∇⊗∇)nk + nk

∇n⊗∇nn2

− nk

(∇nk

nk

− ∇nn

)⊗(∇nk

nk

− ∇nn

)− 2∇n⊗∇nk

n

]

=~2q4m2

div

((∇⊗∇)n− ∇n⊗∇n

n

)− div (nθos)

=~2q2m2

n∇(∆√n√n

)− div (nθos),

where

θos = q∞∑

k=0

λknk

n(uos,k − uos)⊗ (uos,k − uos)

is termed osmotic temperature.Hence, the momentum equation (8.8) becomes

∂tJ −1

qdiv

(J ⊗ Jn

)− div (nθ) + qn∇V +

~2q2m2

n∇(∆√n√n

)= 0, (8.9)

where θ = θc + θos is called temperature tensor. Equation (8.9) and the massconservation equation

∂tn−1

qdiv J = 0

are referred to as the quantum hydrodynamic equations.The temperature tensor cannot be expressed in terms of n and J without fur-

ther assumptions, and as in classical fluid dynamics, we need a closure conditionto obtain a closed set of equations (similarly as in Section 4.2). In the literature,the following condition has been used: The temperature is a scalar (times theidentity matrix),

θ = T · Id,and the temperature scalar T satisfies either

146

• T = T0, where T0 > 0 is a constant (the so-called isothermal case), or

• T = T (n) = T0nα−1 depends on the carrier density, where T0 > 0 and α > 1

are constants (the so-called isentropic case).

In these cases the (classical) pressure P = nθ becomes

P (n) = T0nα · Id

and divP = div (T0nα), with α = 1 in the isothermal case and α > 1 in the

isentropic case.

8.2 The quantum hydrodynamic equations: second deri-vation

In this section, the quantum hydrodynamic equations are derived as a set of non-linear conservation laws by a moment expansion of a Wigner-Boltzmann equationand an expansion of the thermal equilibrium Wigner distribution function up toorder O(~2). We proceed similarly as in [24].

We start with a Wigner-Boltzmann equation of the form (see (4.25), (4.26))

∂tw +~mk · ∇xw +

q

~θ[V ]w =

1

τdivk(kw) +

kBT0m

τ~2∆kw, (8.10)

x, k ∈ Rd, t > 0,

w(x, k, 0) = wI(x, k), x, k ∈ Rd, (8.11)

where d ≥ 1 is the space dimension and w(x, k, t) is the Wigner distributionfunction depending on the space variable x, the wave vector k and the time t.Notice that we have implicitely assumed a parabolic band approximation. Theoperator θ[V ] is defined in the sense of pseudo-differential operators [42] as

(θ[V ]w)(x, k, t) =i

(2π)d

∫

Rd

∫

Rd

m

~

[V(x+

~2m

η, t)− V

(x− ~

2mη, t)]

× w(x, k, t)e−i(k−k′)·η dk′ dη,

where V (x, t) is the electrostatic potential. The right-hand side of (8.10) modelsthe interaction of the electrons with the phonons of the crystal lattice. Thephysical parameters are the reduced Planck constant ~ = h/2π, the effectivemass of the electrons m, the elementary charge q, the lattice temperature T0, andthe momentum relaxation time τ whose inverse is a measure of the strength ofthe coupling between the electrons and phonons.

The existence and uniqueness of local-in-time (so-called mild) solutions to thewhole-space problem (8.10)-(8.11) has been shown in [7]. Existence of global-in-time solutions of the one-dimensional problem with periodic boundary conditionshas been proved in [5].

147

The particle density n and current density J are related to the Wigner functionby the formulas

n(x, t) =

∫

Rdw(x, k, t) dk and J(x, t) = −q~

m

∫

Rdw(x, k, t)k dk. (8.12)

These formulas follow from Section 3.1 observing that in the parabolic bandapproximation, the Brillouin zone is extended to Rd and the mean velocity equalsv(k) = ~k/m. Usually, the electrostatic potential is self-consistently produced bythe electrons moving in the semiconductor crystal with fixed charged backgroundions of density C(x):

div(εs∇V ) = q(n− C(x)), x ∈ Rd,

where εs denotes the permittivity of the material.In order to derive macroscopic equations from (8.10) we apply a moment

method. The first moments are defined by

〈1〉 = n =

∫

Rdw(x, k, t) dk,

〈kj〉 =

∫

Rdw(x, k, t)kj dk,

〈kjk`〉 =

∫

Rdw(x, k, t)kjk` dk,

with 1 ≤ j, ` ≤ d. Integrating (8.10) over Rd with respect to k, we obtain

∂t〈1〉+~mdivx〈k〉+

q

~

∫

Rdθ[V ]w dk =

1

τ

∫

Rd

(divk(kw) +

kBT0m

τ~2∆kw

)dk

= 0.

For the evaluation of the integral on the left-hand side we recall the Fouriertransform

φ(x, η, t) =1

(2π)d/2

∫

Rdφ(x, k, t)eik·η dk

with inverse

φ(x, k, t) =1

(2π)d/2

∫

Rdφ(x, η, t)e−ik·η dη.

Integrating the last equation over Rd with respect to k gives the formula

φ(x, 0, t) =1

(2π)d

∫

Rd

∫

Rdφ(x, η, t)e−ik·η dη dk. (8.13)

148

With this expression we get∫

Rdθ[V ]w dk =

i

(2π)d/2

∫

Rd

∫

Rd

m

~

[V(x+

~2m

η, t)− V

(x− ~

2mη, t)]

× w(x, η, t)e−ik·η dη dk

= i(2π)d/2m

~

[V(x+

~2m

η, t)− V

(x− ~

2mη, t)]

η=0

w(x, 0, t)

= 0,

and therefore

∂t〈1〉+~m

divx〈k〉 = 0

or, with (8.12),

∂tn−1

qdivxJ = 0. (8.14)

This equation expresses the conservation of mass.Now we multiply (8.10) with kj and integrate over Rd with respect to k. Since

∫

Rd

(divk(kw) +

kBT0m

τ~2∆kw

)kj dk

= −∫

Rd

((k · ∇kkj)w +

kBT0m

τ~2(∇kkj) · (∇kw)

)dk

= −∫

Rd

(kjw +

kBT0m

τ~2∂w

∂kj

)dk

= −〈kj〉

and, using integration by parts and (8.13),∫

Rdθ[V ]wkj dk

=im

(2π)d/2~

∫

Rd

∫

Rd

[V(x+

~2m

η, t)− V

(x− ~

2mη, t)]

× w(x, η, t)i ∂∂ηj

e−ik·η dη dk

= −(2π)d/2m~

∂

∂ηj

[V(x+

~2m

η, t)− V

(x− ~

2mη, t)]

η=0

w(x, 0, t)

− (2π)d/2m

~

[V (x+

~2m

η, t)− V(x− ~

2mη, t)]

η=0

∂w

∂ηj(x, 0, t)

= − ∂V∂xj

(x, t)

∫

Rdw(x, k, t) dk

= − ∂V∂xj

〈1〉,

149

this yields

∂t〈kj〉+~m

d∑

`=1

∂

∂x`

〈kjk`〉 −q

~∂V

∂xj

〈1〉 = −1

τ〈kj〉

or

∂tJj −q~2

m2

d∑

`=1

∂

∂x`

〈kjk`〉+q2

m

∂V

∂xj

n = −Jjτ. (8.15)

The system (8.14) and (8.15) has to be closed by expressing the term 〈kjk`〉in terms of the primal variables n and J. As in the case of classical kinetic theory(see Chapter 5), we achieve the closure by assuming that the Wigner function wis close to a wave vector displaced equilibrium density such that

w(x, k, t) = we

(x, k − m

~u(x, t), t

),

where u(x, t) is some group velocity and

we(x, k, t) (8.16)

= A(x, t) exp

(− ~2

2mkBT|k|2 + qV

kBT

)[1 + ~2

q

8m(kBT )2∆xV (8.17)

+q2

24m(kBT )3|∇xV |2 −

q2

24(kBT )3

d∑

j,`=1

vj(k)v`(k)∂2V

∂xj∂x`

+O(~4)

],

where we recall that v(k) = ~k/m. The form (8.16) is derived from an O(~2)approximation of the thermal equilibrium density first given by Wigner [43]. Theelectron temperature T = T0 is here assumed to be constant (see the followingsection for non-constant temperature).

The symmetry of we with respect to k implies that the odd order moments ofwe vanish. Therefore

〈1〉 =

∫

Rdwe

(x, k − m

~u(x, t), t

)dk =

∫

Rdwe(x, η, t) dη,

〈kj〉 =

∫

Rd

(ηj +

m

~uj

)we(x, η, t) dη =

m

~nuj,

~2〈kjk`〉 = ~2∫

Rd

(ηj +

m

~uj

)(η` +

m

~u`

)we(x, η, t) dη

= m2nuju` + ~2∫

Rdηjη`we(x, η, t) dη. (8.18)

This implies n = 〈1〉 and J = −qnu.We now derive an O(~2) approximation for n. Since

∫

Rx2e−x2/2 dx =

∫

Re−x2/2 · 1 dx =

√2π,

∫

Rx4e−x2/2 dx = 3

∫

Re−x2/2x2 dx = 3

√2π,

150

we obtain∫

Rde−~2|η|2/2mkBT dη = (mkBT )

d/2~−d

∫

Rde−|z|2/2 dz = (2πmkBT )

d/2~−d

and∫

Rdηjη`e

−~2|η|2/2mkBT dη = (mkBT )d/2~−d(mkBT~−2)

∫

Rdzjz`e

−|z|2/2 dz

= (2πmkBT )d/2mkBT~−d−2δj`,

and hence, with C = A(x, t)(2πmkBT )d/2~−d,

n =

∫

Rdwe(x, η, t) dη

= CeqV/kBT

(1 +

q~2

8m(kBT )2∆xV +

q2~2

24m(kBT )3|∇xV |2

− q2~4

24m2(kBT )3mkBT

~2d∑

j,`=1

∂2V

∂xj∂x`

δj`

)+O(~4) (8.19)

= CeqV/kBT

(1 +

q~2

12m(kBT )2∆xV +

q2~2

24m(kBT )3|∇xV |2

)+O(~4).

In order to compute the last integral in (8.18), we have to evaluate

∫

Rdηjη`ηmηne

−~2|η|/2mkBT dη = (mkBT )d/2+2~−d−4

∫

Rdzjz`zmzne

−|z|2/2 dz

= (mkBT )d/2+2~−d−4α(j, `,m, n),

where

α(j, `,m, n) = (2π)d/2

3 : j = ` = m = n1 : j = ` 6= m = n1 : j = m 6= ` = n1 : j = n 6= ` = m0 : else.

This yields

~2∫

Rdηjη`we(x, η, t) dη

= CeqV/kBT

[(mkBT +

q~2

8kBT∆xV +

q2~2

24(kBT )2|∇xV |2

)δj`

− q~2

24kBT

d∑

m,n=1

∂2V

∂xm∂xn

α(j, `,m, n)

]+O(~4).

151

A simple computation shows that

d∑

m,n=1

∂2V

∂xm∂xn

α(j, `,m, n) = 2∂2V

∂xj∂x`

+ δj`∆xV.

Hence

~2∫

Rdηjη`we(x, η, t) dη = CeqV/kBT

[(mkBT +

q~2

12kBT∆xV (8.20)

+q2~2

24(kBT )2|∇xV |2

)δj` −

q~2

12kBT

∂2V

∂xj∂x`

]+O(~4).

We substitute (see (8.19))

CeqV/kBT(mkBT +

q~2

12kBT∆xV +

q2~2

24(kBT )2|∇xV |2

)= mnkBT +O(~4)

andCeqV/kBT = n+O(~2) (8.21)

into (8.20) to conclude from (8.18)

~2〈kjk`〉 = m2nuju` +mnkBTδj` −q~2

12kBTn∂2V

∂xjx`

+O(~4). (8.22)

The potential V is usually the sum of the electrostatic potential and an exter-nal potential modeling heterostructures (i.e. sandwiches of different materials).As the external potential can be discontinuous at the interfaces between twomaterials, also V may be discontinuous. Therefore, we wish to substitute thederivative ∂2V/∂xj∂x` by an expression which only depends on n and its deriva-tives. From Taylor’s formula log(x + ε) = log x + x/ε + O(ε2) for x, ε > 0 weobtain from (8.21) (with a slight abuse of notation)

log n = log(eqV/kBT +O(~2)) + logC = log eqV/kBT + logC +O(~2)

and therefore,

∂2 log n

∂xj∂x`

=∂2

∂xj∂x`

(log(A(2πmkBT )

d/2)+

qV

kBT

)+O(~2).

We assume that A(x, t) is slowly varying with respect to x such that we canapproximate

∂2 log n

∂xj∂x`

=q

kBT

∂2V

∂xj∂x`

+O(~2).

152

For “~ → 0”, this corresponds to the classical thermal equilibrium expressionn = const. eqV/kBT . We obtain finally from (8.22)

~2〈kjk`〉 = m2nuju` +mnkBTδj` −~2

12n∂2 log n

∂xj∂x`

+O(~4) (8.23)

and

q~2

m2

d∑

`=1

∂

∂x`

〈kjk`〉

= qd∑

`=1

∂

∂x`

(nuju`) +qkBm

∂

∂xj

(nT )− ~2q12m2

d∑

`=1

∂

∂x`

(n∂2 log n

∂xj∂x`

)

=1

q

d∑

`=1

∂

∂x`

(JjJ`n

)+qkBT

m

∂n

∂xj

− ~2q6m2

n∂

∂xj

(∆x

√n√n

),

since J = −qnu.With (8.14), (8.15) and the Poisson equation, the (isothermal) quantum hy-

drodynamic equations become

∂tn−1

qdivxJ = 0, (8.24)

∂tJ −1

qdivx

(J ⊗ Jn

)− qkBT

m∇xn+

~2q6m2

n∇x

(∆x

√n√n

)

+q2

mn∇xV = −J

τ, (8.25)

div(εs∇xV ) = q(n− C(x)). (8.26)

Notice that the coefficient of the quantum term is ~2q/6m2 compared to thecoefficient ~2q/2m2 in (8.9) obtained from the mixed-state approach of Section8.1. The factor 1/3 can be interpreted as a statistical factor coming from thermalaveraging (see [25]).

8.3 Extensions

In this section we derive two extensions of the quantum hydrodynamic equations:a quantum hydrodynamic model including an evolution equation for the parti-cle energy and a quantum hydrodynamic model taking into account viscous (ordissipative) effects.

Quantum hydrodynamic equations with energy. In the previous sub-section we have assumed that the temperatue, which appears in the approxima-tion of the thermal equilibrium Wigner function (8.16), is constant. We suppose

153

now that the temperature is time and space dependent, i.e. T = T (x, t). Thenthe quantum hydrodynamic equations (8.24)-(8.26) are still valid but the pressureterm in (8.25) has now to be written as

q

m∇x(nT ).

However, we also need an equation for the variable T. We derive this equation bycalculating the next moment equation. For this we introduce the moments

〈|k|2〉 =

∫

Rdw(x, k, t)|k|2 dk,

〈k|k|2〉 =

∫

Rdw(x, k, t)k|k|2 dk.

Multiply (8.10) with |k|2 and integrate over Rd with respect to k to obtain

∂t〈|k|2〉+~mdiv x〈k|k|2〉+

q

~

∫

Rdθ[V ]w|k|2 dk (8.27)

=1

τ

∫

Rd

(divk(kw) +

kBT0m

~2∆kw

)|k|2 dk.

We first compute the integrals. For the integral on the right-hand side we get,after integration by parts,

1

τ

∫

Rd

(divk(kw) +

kBT0m

~2∆kw

)|k|2 dk

=2

τ

∫

Rd

(−w|k|2 + dkBT0m

~2w

)dk

=2

τ

(−〈|k|2〉+ dkBT0m

~2〈1〉).

Similarly as in the previous subsection, a computation gives

q

~

∫

Rdθ[V ]w|k|2 dk =

2q

~∇xV · 〈k〉 = −

2m

~2∇xV · J.

The two moments are calculated using the wave vector displaced equilibriumdensity (8.27). From (8.23) follows

~2〈|k|2〉 = m2n|u|2 + dmnkBT −~2

12n∆ log n+O(~4).

We introduce now the energy

e =m

2|u|2 + d

2kBT −

~2

24m∆ log n, (8.28)

154

consisting of the kinetic, thermal and quantum energy parts, and the stress tensor

Pj` = −nkBTδj` +~2

12mn∂2 log n

∂xj∂x`

, j, ` = 1, . . . , d. (8.29)

Then ~2〈|k|2〉 = 2me+O(~4) and tedious computations lead to

〈k|k|2〉 =

∫

Rdwe(x, k −mu/~, t)k|k|2 dk

=2m2

~2(une− P · u) +O(~2),

where P = (Pj`)j`. Thus we can write (8.27), up to order O(~2), as

∂t(ne) + div x (une− P · u)−∇xV · J = −1

τ(2ne− dkBT0n).

Using u = −J/qn, we obtain the full quantum hydrodynamic equations

∂tn−1

qdivxJ = 0,

∂tJ −1

qdivx

(J ⊗ Jn

)+qkBm∇x(nT )−

~2q6m2

n∇x

(∆x

√n√n

)+q2

mn∇xV = −J

τ,

∂t(ne)−1

qdivx

(Je− P · J

n

)−∇xV · J = −2n

τ

(e− d

2kBT0

),

div x(εs∇xV ) = q(n− C(x)),

where e and P = (Pj`) are defined by (8.28) and (8.29), respectively.

The viscous quantum hydrodynamic equations. The derivation of thequantum hydrodynamic model is based on the Fokker-Planck operator

LFPw =1

τ

(divk(kw) +

kBT0m

~2∆kw

)

which models interactions of electrons with a heat bath consisting of an ensembleof harmonic oscillators [15]. In [7] a more general Fokker-Planck operator hasbeen proposed:

LFPw =1

τ

(divk(kw) +

kBT0m

~2∆kw +

~2

kBT0m∆xw

). (8.30)

We wish to derive the corresponding quantum hydrodynamic model in the caseof constant temperature. For this we only need to compute the first moments ofthe last term in (8.30). It holds

∫

Rd∆xw dk = ∆x〈1〉 = ∆xn,

∫

Rd∆xwkj dk = ~∆x〈kj〉 = −

m

q∆xJj, j = 1, . . . , d.

155

We interpret these terms as viscous terms and call the corresponding equationsthe viscous quantum hydrodynamic model:

∂tn−1

qdivxJ =

~2

τkBT0m∆xn,

∂tJ −1

qdivx

(J ⊗ Jn

)+qT

m∇xn−

~2q6m2

n∇x

(∆x

√n√n

)+q2

mn∇xV

=~2

τkBT0m∆xJ −

J

τ,

div(εs∇xV ) = q(n− C(x)), x ∈ Ω, t > 0.

This viscous terms have the important property that they regularize the equa-tions in the following sense. Define the total energy

E(t) =

∫

Ω

(~2q6m2|∇√n|2 + qkBT0

mn(log n− 1) +

εs2q|∇V |2 + |J |

2

2n

)dx,

consisting of the quantum energy, entropy, electric energy, and kinetic energy. IfΩ ⊂ Rd is bounded and periodic boundary conditions are used or if Ω = Rd, atedious (formal) computation shows that

dE

dt+

∫

Ω

(~2q3m2

~2

τkBT0m(∆√n)2 +

4~2

τkBT0m|∇√n|2 + 1

τ

|J |2n

)dx

+

∫

Ω

~2

τkBT0m

1

n3

(∇n · J − n

d∑

i=1

|∇Ji|)2

dx ≤ 0. (8.31)

Without viscous terms the variation of the energy reads

dE

dt+

∫

Ω

(~2qτ3m2

|∇√n|2 + 1

τ

|J |2n

)dx = 0.

Hence, the viscous terms yield an additional term in the energy production,essentially given by (∆

√n)2.

156

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