Mathematical Modeling
of Semiconductor Devices
Prof. Dr. Ansgar Jüngel
Fachbereich Mathematik und Statistik
Universität Konstanz
Preliminary version
Contents
1 Introduction
3
2 Some Semiconductor Physics
2.1 Semiconductor crystals . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Mean electron velocity and effective mass approximation . . . . .
2.3 Semiconductor statistics . . . . . . . . . . . . . . . . . . . . . . .
11
11
17
21
3 Classical Kinetic Transport Models
3.1 The Liouville equation . . . . . . .
3.2 The Vlasov equation . . . . . . . .
3.3 The Boltzmann equation . . . . . .
3.4 Extensions . . . . . . . . . . . . . .
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27
27
31
37
44
4 Quantum Kinetic Transport Equations
4.1 The Wigner equation . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The quantum Vlasov and quantum Boltzmann equation . . . . . .
50
50
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5 From Kinetic to Fluiddynamical Models
5.1 The drift-diffusion equations: first derivation . . .
5.2 The drift-diffusion equations: second derivation .
5.3 The hydrodynamic equations . . . . . . . . . . . .
5.4 The Spherical Harmonic Expansion (SHE) model
5.5 The energy-transport equations . . . . . . . . . .
5.6 Relaxation-time limits . . . . . . . . . . . . . . .
5.7 The extended hydrodynamic model . . . . . . . .
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62
62
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100
104
6 The
6.1
6.2
6.3
6.4
Drift-Diffusion Equations
Thermal equilibrium state and boundary conditions
Scaling of the equations . . . . . . . . . . . . . . .
Static current-voltage characteristic of a diode . . .
Numerical discretization of the stationary equations
7 The
7.1
7.2
7.3
Energy-Transport Equations
131
Symmetrization and entropy . . . . . . . . . . . . . . . . . . . . . 132
A drift-diffusion formulation . . . . . . . . . . . . . . . . . . . . . 136
A non-parabolic band approximation . . . . . . . . . . . . . . . . 140
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114
114
116
118
127
8 From Kinetic to Quantum Hydrodynamic Models
143
8.1 The quantum hydrodynamic equations: first derivation . . . . . . 143
8.2 The quantum hydrodynamic equations: second derivation . . . . . 147
8.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
2
1
Introduction
The modern computer and telecommunication industry relies heavily on the use
and development of semiconductor devices. Since the first semiconductor device
(a germanium transistor) has been built by Bardeen, Brattain and Shockley in
1947, a lot of different devices for special applications have been invented in the
following decades. A very important fact of the success of semiconductor devices
is that the device length is very small compared to previous electronic devices
(like tube transistors). The first transistor of Bardeen, Brattain and Shockley
had a characteristic length (the emitter-collector length) of 20 µm. Thanks to the
progressive miniaturization of semiconductor devices, the transistors in a modern
Pentium IV processor have a characteristic length of 0.18 µm. The device length
of tunneling diodes, produced in laboratories, is only of the order of 0.075 µm.
Nowadays, semiconductor materials are contained in almost all electronic devices. Some examples of semiconductor devices and their use are described in the
following.
• Photonic devices capture light (photons) and convert it into an electronic
signal. They are used in camcorders, solar cells, and light-wave communication systems as optical fibers.
• Optoelectronic emitters convert an electronic signal into light. Examples
are light-emitting diodes (LED) used in displays and indication lambs and
semiconductor lasers used in compact disk systems, laser printers, and eye
surgery.
• Flat-panel displays create an image by controlling light that either passes
through the device or is reflected off of it. They are made, for instance, of
liquid crystals (liquid-crystal displays, LCD) or of thin semiconductor films
(electroluminescent displays).
• In field-effect devices the conductivity is modulated by applying an electric
field to a gate contact on the surface of the device. The most important
field-effect device is the MOSFET (metal-oxide semiconductor field-effect
transistor), used as a switch or an amplifier. Integrated circuits are mainly
made of MOSFETs.
• Quantum devices are based on quantum mechanical phenomena, like tunneling of electrons through potential barriers which are impenetrable classically. Examples are resonant tunneling diodes, superlattices (multi-quantum-well structures), quantum wires in which the motion of carriers is restricted to one space dimension and confined quantum mechanically in the
other 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 new
developments are, for instance, nanostructure devices (heterostructures) and solar
cells made of amorphous silicon or organic semiconductor materials (see [14, 41]).
Usually, a semiconductor device can be considered as a device which needs an
input (an electronic signal or light) and produces an output (light or an electronic
signal). The device is connected to the outside world by contacts at which a
voltage (potential difference) is applied. We are mainly interested in devices
which produce an electronic signal, for instance the macroscopically measurable
electric current (electron flow), generated by the applied bias. In this situation,
the input parameter is the applied voltage and the output parameter is the electric
current through one contact. The relation between these two physical quantities is
called current-voltage characteristic. It is a curve in the two-dimensional currentvoltage space. The current-voltage characteristic does not need to be a monotone
function and it does not need to be a function (but a relation).
The main objective of this book is to derive mathematical models which describe the electron flow through a semiconductor device due to the application of
a voltage. Depending on the device structure, the main transport phenomena of
the electrons may be very different, for instance, due to drift, diffusion, convection, or quantum mechanical effects. For this reason, we have to devise different
mathematical models which are able to describe the main physical phenomena
for a particular situation or for a particular device.
This leads to a hierarchy of semiconductor models. Roughly speaking, we can
divide semiconductor models in three classes: quantum models, kinetic models
and fluiddynamical (macroscopic) models. In order to give some flavor of these
models 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 elementary
charge 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 the
single-particle Schrödinger equation
~2
∆Ψ − qV (x, t)Ψ,
2m
i~∂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 electrostatic potential related to the electric field by E = −∇V, and i2 = −1. Any
solution Ψ of (1.1)-(1.2) is called wave function. Macroscopic variables are ob-
4
tained by the definitions:
n = |Ψ|2 ,
~q
electron current density: J = − Im(Ψ∇Ψ),
m
electron density:
where Ψ is the complex conjugate of Ψ. More precisely, n(x, t) is the position
probability density of the particle, i.e.
Z
n(x, t) dx
A
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 Schrödinger equation (1.1).
The fluiddynamical view. In order to derive fluiddynamical models, for
instance, for the evolution of the particle density n and the current density
J, we
√
assume 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 call
this change of unknowns Ψ 7→ (n, S) Madelung transform. The current density
now reads
·
µ
¶¸
√
~q
im √
∇n
J = − Im
n √ +
n∇S
= −qn∇S,
m
~
2 n
and we can interpret S also as a velocity potential. Setting (1.3) into the
Schrödinger equation (1.1) and taking the imaginary and real part of the resulting equation, gives, after some computations (see Chapter 8):
1
∂t n − div J = 0,
q
µ
¶
µ √ ¶
2
J ⊗J
∆ n
q2
1
~q
√
∂t J − div
+
n∇
− nE = 0.
q
n
2m2
m
n
(1.4)
(1.5)
Here, J ⊗ J is a tensor (or matrix) with components Jj Jk , where j, k = 1, . . . , d.
Equation (1.4) expresses the conversation of mass. Indeed,
Z
Z
d
1
n dx =
div J dx = 0.
dt Rd
q Rd
The equations (1.4)-(1.5) are referred to as the quantum hydrodynamic equations.
They are equivalent to the Schrödinger equation (1.1) as long as n > 0. If they
are studied in the whole space, the initial conditions
n(·, 0) = |Ψ0 |2 ,
J(·, 0) = −
5
~q
Im(Ψ0 ∇Ψ0 )
m
in Rd
have to be prescribed.
In the semi-classical limit “~ → 0” the equations (1.4)-(1.5) formally reduce
to the hydrodynamic equations
1
∂t n − div J = 0,
q
µ
¶
J ⊗J
1
q2
∂t J − div
− nE = 0,
q
n
m
(1.6)
(1.7)
which are the Euler equations for a gas of charged particles with zero temperature
in 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, a
parameter (the so-called scaled Planck constant) appears in front of the scaled
quantum term in (1.5). This parameter is small compared to one in certain
physical ranges, for instance, when the characteristic length is “large” (compared
to some reference length). Thus, the limit “~ → 0” has always to be understood
as 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 when
dealing with semiconductor devices), boundary conditions can be more easily prescribed compared to the Schrödinger formulation.
• In two or three space dimensions, fluiddynamical models are usually numerically cheaper than the Schrödinger equation.
The kinetic view. We start again with the Schrödinger 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+
~
η,
2m
s=x−
~
η
2m
in the density matrix defining
´
³
~
~
η, x −
η, t .
u(x, η, t) = ρ x +
2m
2m
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 variables
space, velocity and time instead of space, inverse velocity and time and employ
6
therefore the Fourier transformation to u. We recall that the Fourier transform
F is defined by
Z
−d/2
F g(η) = ĝ(η) = (2π)
g(η)e−iη·v dv
Rd
for (sufficiently smooth) functions g : Rd → C with inverse
Z
−1
−d/2
F h(v) = ȟ(v) = (2π)
h(v)eiη·v dη
Rd
for h : Rd → C. The inverse Fourier transform of u is called Wigner function w:
w = (2π)−d/2 F −1 u = (2π)−d/2 ǔ.
This function was introduced by Wigner in 1932 [43]. It can be shown (see
Chapter 4) that the macroscopic electron density n and current density J defined
above in terms of the wave function can now be written in terms of the Wigner
function as
Z
w(x, v, t) dv,
(1.8)
n(x, t) =
Rd
Z
J(x, t) = −q
v w(x, v, t) dv.
(1.9)
Rd
The integrals are called the zeroth-order and first-order moments.
The transport equation for w is obtained by first deriving the evolution equation for u from the Schrödinger equation (1.1) and then by taking the inverse
Fourier transform of the equation for u. A computation leads to the so-called
Wigner equation (see Chapter 4 for details):
q
θ[V ]w = 0,
m
∂t w + v · ∇ x w +
x, v ∈ Rd , t > 0,
(1.10)
where
im
(θ[V ]w)(x, v, t) =
~(2π)d
Z
Rd
0
Z
· µ
¶
µ
¶¸
~
~
V x+
η, t − V x −
η, t
2m
2m
Rd
0
×w(x, v , t)eiη·(v−v ) dv 0 dη.
The initial condition for w reads
w(x, v, 0) = (2π)−d/2 ǔ(x, v, 0)
µ
¶ µ
¶
Z
~
~
−d
= (2π)
Ψ0 x +
η Ψ0 x −
η eiv·η dη,
2m
2m
d
R
7
x, v ∈ Rd .
An operator, whose Fourier transform acts as a multiplication operator on the
Fourier transform of the function, is called a linear pseudo-differential operator
[42]. Since
· µ
¶
µ
¶¸
~
~
im
\
V x+
η, t − V x −
η, t ŵ(x, η, t),
θ[V ]w(x, η, t) =
~
2m
2m
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
· µ
¶
µ
¶¸
m
~
~
V x+
η, t − V x −
η, t
→ ∇x V (x, t) · η,
~
2m
2m
and using the identity i(ηu)∨ = ∇v ǔ, we obtain, as “~ → 0”,
θ[v]w →
1
i
(∇x V · ηu)∨ =
∇x V · ∇v ǔ = ∇x V · ∇v w.
d/2
(2π)
(2π)d/2
Hence the Wigner equation becomes in the semi-classical limit
∂t w + v · ∇ x w +
q
∇x V · ∇v w = 0.
m
(1.11)
This equation is called Liouville equation. It has to be solved in the positionvelocity space (x, v) ∈ Rd × Rd for times t > 0, supplemented with the initial
condition
w(x, v, 0) = (2π)−d/2 lim ǔ(x, v, 0)
~→0
µ
¶ µ
¶
Z
~
~
1
lim
Ψ0 x +
η Ψ0 x −
η eiv·η dη,
=
(2π)d ~→0 Rd
2m
2m
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 the
formulas (1.8)-(1.9).
The kinetic formulation of the motion of the electron seems much more complicate than the Schrödinger formulation since we pass from d + 1 variables (d
for the space and one for the time) to 2d + 1 variables (d for the space, d for
the velocity and one for the time). However, this formulation has the following
advantages:
• In the case of an electron ensemble with many particles, an approximate
formulation in the position-velocity space is possible (leading to the Vlasov
equation; see Chapter 3). This means that the kinetic equation has to be
solved in 2d + 1 variables instead of M d + 1 variables in the Schrödinger
formulation, where M À 1 is the number of particles.
8
• Short-range interactions of the particles (collisions) can be easily included
in the classical kinetic models.
We have described how fluiddynamical and kinetic models can be derived
formally from the Schrödinger equation. Is there any relation between fluiddynamical and kinetic models? The answer is yes, and the method to see this is
called 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 velocity space and using the definitions of the first two moments (1.8)-(1.9) and the
divergence theorem gives for E = −∇V
0 = ∂t
Z
Rd
w dv +
d Z
X
k=1
1
= ∂t n − div x J,
q
d
q X
∂w
Ek
dv −
vk
∂xk
m k=1
Rd
Z
Rd
∂w
dv
∂vk
which equals (1.6) expressing the conservation of mass. Now we multiply (1.11)
with −qvj and integrate over the velocity space:
d Z
X
Z
d
∂w
q2 X
∂w
0 = ∂ t Jj − q
vj vk
Ek
vj
dv +
dv
∂x
m
∂v
d
d
k
k
R
R
k=1
k=1
Z
Z
d
d
X ∂
∂vj
q2 X
Ek
vj vk w dv −
w dv.
= ∂ t Jj − q
∂xk Rd
m k=1
Rd ∂vk
k=1
Since ∂vj /∂vk = δjk with the Kronecker symbol δjk , this yields
Z
q2
∂t J − qdiv x
(v ⊗ v)w dv − En = 0.
m
Rd
The term
Z
Rd
(1.12)
(v ⊗ v)w dv
is the matrix of all second-order moments and is called energy tensor. We make
now the complete heuristic assumption that the velocity v can be approximated
by an average velocity
R
vw dv
J
v̄ = R
=− ,
qn
w dv
that means,
Z
R
(v ⊗ v)w dv ≈ (v̄ ⊗ v̄)
9
Z
Rd
w dv =
1 J ⊗J
,
q2 n
and (1.12) becomes
1
∂t J − div x
q
µ
J ⊗J
n
¶
−
q2
nE = 0,
m
which equals (1.7). Under the above approximation, we have derived the hydrodynamic model (1.6)-(1.7). For a more general derivation (including thermal
effects), see Chapter 5.
In a similar way, we can apply the moment method to the Wigner equation (1.10) in order to derive the quantum hydrodynamic model (1.4)-(1.5) (see
Chapter 8).
All the models explained in this chapter and the relations between them are
summarized in Figure 1.1.
Quantum
Madelung ©© Schrödinger HH Fourier
HH transform
transform ©©
©
©
¼
©©
Quantum
¾
Hydrodynamic
HH
moment
method
“~ → 0”
?
Hydrodynamic ¾
H
j
H
Wigner
“~ → 0”
moment
method
?
Liouville
Kinetic
Fluidynamical
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 the
modeling 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 the
crystal lattice or with other particles).
These features will be included in our models step by step in the following chapters.
10
2
Some Semiconductor Physics
In this chapter we present a short summary of the physics and main properties
of semiconductors. Only these subjects relevant to the subsequent chapters are
included here. We refer to [1, 8, 14, 28, 41] for a detailed description of solid-state
and semiconductor physics.
2.1
Semiconductor crystals
What is a semiconductor? Historically, the term semiconductor has been used
to denote solid materials with a much higher conductivity than insulators, but
a much lower conductivity than metals measured at room temperature. A more
precise definition is that a semiconductor is a solid with an energy gap larger than
zero and smaller than about 4 eV. Metals have no energy gap, whereas this gap
is usually larger than 4 eV in insulators. In order to understand what an energy
gap is, we have to explain the crystal structure of solids.
A solid is made of an infinite three-dimensional array of atoms arranged according to a lattice
L = {`a~1 + ma~2 + na~3 : `, m, n ∈ Z} ⊂ R3 ,
where a~1 , a~2 , a~3 are the basis vectors of L, usually called primitive vectors of the
lattice. The lattice atoms (or ions) generate a periodic electrostatic potential V L :
VL (x + y) = VL (x) ∀x ∈ R3 , y ∈ L.
The state of an electron moving in this periodic potential is described by an
eigenfunction of the stationary Schrödinger equation:
−
~2
∆Ψ − qVL (x)Ψ = εΨ,
2m
x ∈ R3 ,
(2.1)
where Ψ : R3 → C is called wave function. In this equation, the physical parameters 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 eigenvalue
problem, and we have to find eigenfunction-eigenvalue pairs (Ψ, ε). We consider
two examples.
Example 2.1 (Free electron motion)
First, consider a free electron moving in a one-dimensional vacuum, i.e., VL (x) = 0
for all x ∈ R. Then the solutions of (2.1) are given by
Ψk (x) = Aeikx + Be−ikx ,
and
ε=
~2 k 2
,
2m
11
x ∈ R,
for any k ∈ R. We disregard purely imaginary values of k = iγ (γ ∈ R) which also
yield real values of the energy, since this leads to unbounded solutions of the type
exp(±ikx) = exp(∓γx) for x ∈ R. (Notice that the integral of |Ψ| 2 over R can
be interpreted as the particle mass which should be finite.) Thus the eigenvalue
problem (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 the
potential is infinite at the boundaries and everywhere outside and zero everywhere
within. As the potential is confining an electron to the inner region, we have to
solve the Schrödinger equation (2.1) in the interval (0, L) of length L > 0 with
boundary 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) =
~2 k 2
,
2m
k=
πn
.
L
The system only allows discrete energy states.
∞
∞
6
6
-
x
0
L
Figure 2.1: Infinite square-well potential.
Although the lattice potential VL is periodic, the wave function is defined on
R3 in order to account for electron motion from one lattice cell to the next. In
fact, the so-called Bloch theorem states that the Schrödinger equation (2.1) can
be reduced to a Schrödinger 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 + ma~∗2 + na~∗3 : `, m, n ∈ Z},
12
where the basis vectors (or primitive vectors) a~∗1 , a~∗2 , a~∗3 ∈ R3 satisfy the
relation
a~m · a~∗n = 2πδmn .
(2) The connected set D ⊂ R3 is called primitive cell of L (or of L∗ ) if and
only if
• the volume of D equals the volume of the parallelepiped spanned by the
basis vectors of L (or of L∗ );
• the whole space R3 is covered by the union of translates of D by the
primitive vectors.
(3) The (first) Brillouin zone B ⊂ R3 is that primitive cell of the reciprocal
lattice L∗ which consists of those points, which are closer to the origin than
to any other point of L∗ (see Figure 2.2).
sa
aa
aa
aa
~∗
aa
s a2
a
6 aa
aa
aa
a
©s
©
sa
A ©©
aa ¯¯ B
A
©
aa
A
©
¯a
©
a
A¯
©
aa
s
©
a
¯
¯ a~1
a
©
a
a
£
¯
aa
A ©© £ 0 aa
a
¯a
aa
A
©©
aq
¯ a
A
as a~∗1
£
©
A
¯
©
s©
£
aa
°£
aa
a
~
2
aa
aa s
aa
aa
aa
aa
as
Figure 2.2: The primitive vectors of a two-dimensional lattice L and its reciprocal
lattice L∗ and the Brillouin zone B.
We give some explainations of the above definition. As the wave planes
exp(±ik · x) suggest, k can be considered as a reciprocal variable which leads
to the definitions of the reciprocal lattice and the Brillouin zone. In particular,
for any x ∈ L and k ∈ L∗ with
x=
3
X
αm~am
and
k=
3
X
n=1
m=1
13
βn~a∗n ,
αm , βn ∈ Z,
we obtain
Ã
exp(ik · x) = exp i
3
X
m,n=1
αm βn · 2πδmn
!
Ã
= exp 2πi
3
X
n=1
αn βn
!
= 1.
As x has the dimension of length, k has the dimension of inverse length and
therefore, k is called a wave vector. (More precisely, k is called pseudo-wave
vector; see below.)
The Brillouin zone can be constructed as follows. Draw arrows from a lattice
point to its nearest neighbors and cut them in half. The planes through these
intermediate 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 Schrödinger
equation (2.1) is equivalent to the system of Schrödinger equations
−
~2
∆Ψk − qVL Ψk = ε(k)Ψk ,
2m
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 Hamilton
operator H = −(~2 /2m)∆ + qVL (x) commutes with all the translation operators
T` associated with the lattice, where (T` Ψ)(x) = Ψ(x + `) for ` ∈ L, x ∈ R3 .) By
functional analysis, the eigenvalue problem (2.2)-(2.3), for any k ∈ B, possesses
a sequence of eigenfunctions Ψnk with associated eigenvalues εn (k), n ∈ N0 . The
relation between Ψk and Ψ is given by
X
Ψk (x) =
e−ik·` Ψ(x + `).
`∈L
We can also write Ψnk as distorted waves
Ψnk (x) = unk (x)eik·x ,
where
unk (x) =
X
(2.4)
e−ik·(x+`) Ψ(x + `)
`∈L
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 in
distorted 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 energy
band. 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 set
R, i.e., there may exist energies ε∗ for which there is no n ∈ N and no k ∈ B
such that εn (k) = ε∗ . The connected components of the set of energies with this
non-existence property are called energy gaps. We illustrate this property by an
example.
Example 2.4 (Kronig-Penney model)
We study a simple one-dimensional model for the periodic potential generated by
the lattice atoms. We use the square-well potential (see Figure 2.3)
½
V0 : −b < x ≤ 0
VL (x) =
0 : 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
V0
VL (x)
-x
−b
a
0
a+b
Figure 2.3: Periodic square-well potential VL .
In order to solve the Schrödinger equation (2.1), we make the Bloch ansatz
Ψk (x) = u(x)eikx
with an (a + b)-periodic function u. Set
r
r
2mε
2m(ε − qV0 )
, β=
.
α=
2
~
~2
If ε < qV0 then β can be written as β = iγ with γ > 0. A computation shows
that the general solution of (2.1) in the interval (−b, a) is given by
½
Aei(α−k)x + Be−i(α+k)x : 0 < x ≤ a
u(x) =
Cei(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),
u0 (−b) = u0 (a),
where u(a + 0) = limx&a u(x) etc. This gives four linear equations for the four
unknows 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.) The
necessary conditions for getting non-trivial solutions is that det M = 0. An elementary, but lenghty calculation shows that this conditon is equivalent to
s
¶2
µ 2
α − β2
sin2 (βb) · cos(αa − δ),
(2.5)
cos(k(a + b)) = 1 +
2αβ
where
α2 + β 2
tan(βb).
2αβ
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 for
certain 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 nonintegrable 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.
tan δ = −
In Table 2.1 some values of energy gaps for some semiconductor materials are
collected. The energy gap separates two energy bands. The nearest energy band
Material
Silicon
Germanium
Gallium-Arsenide
Energy gap εg (eV)
1.12
0.664
1.424
Table 2.1: Energy gaps of selected semiconductors.
below the energy gap is called valence band; the nearest energy band above the
energy gap is termed conductor band (see Figure 2.4).
Now we are able to state the definition of a semiconductor: It is a solid with
an energy gap whose value is positive and smaller than about 4 eV.
16
ε
6
conduction
band
εg 6
?
valence band
- k
Figure 2.4: Schematic band structure with energy gap εg .
2.2
Mean electron velocity and effective mass approximation
In this section we will motivate the following two expressions:
• The mean electron velocity (or group velocity of the wave packet) in the
n-th band is given by
1
vn (k) = ∇k εn .
(2.6)
~
• The effective mass tensor m∗ is defined by
(m∗ )−1 =
1 d 2 εn
.
~2 dk 2
(2.7)
Motivation of (2.6). The result (2.6) can be obtained as follows. We omit
in the following the index n and define the group velocity by
¶−1 Z
µZ
2
v̂k |Ψk |2 dx,
|Ψ| dx
v(k) =
D
D
where D is a primitive cell of the lattice and v̂k is the particle velocity
v̂k = −
Jk
~ Im(Ψk ∇x Ψk )
.
=
qnk
m
|Ψk |2
(2.8)
In order to show (2.6) we have to compute the integral
Z
Im(Ψk ∇x Ψk ) dx.
D
We take the derivative of (2.2) with respect to k, use the distorted wave plane
Ψk (x) = uk (x)eik·x
and
¡
¢
∆x (∇k Ψk ) = ∆x eik·x ∇k uk + ixΨk
¡
¢
= ∆x eik·x ∇k uk + 2i∇x Ψk + ix∆x Ψk
17
to obtain
~2
(∇k ε)Ψk = −
∆x (∇k Ψk ) − (qVL + ε)∇k Ψk
¶
µ 2m 2
~
∆x − (qVL + ε) (eik·x ∇k uk + ixΨk )
=
−
2m
¶
µ
¶
µ
~2
~2
ik·x
∆x − (qVL + ε) (e ∇k uk ) + ix −
∆x − (qVL + ε) Ψk
=
−
2m
2m
~2
− i ∇x Ψ k
µ m2
¶
~
~2
=
−
∆x − (qVL + ε) (eik·x ∇k uk ) − i ∇x Ψk .
2m
m
Multiplying this equation by Ψk , integrating over D and integrating by parts
yields
Z
Z
~2
2
∇k ε
|Ψk | dx + i
(∇x Ψk )Ψk dx
m D
D
¶
µ
Z
~2
ikx
∆x − qVL − ε Ψk dx = 0,
=
e ∇ k uk −
2m
D
using (2.2) again. Thus
∇k ε = −
i~2
R
(∇x Ψk )Ψk dx
.
m D |Ψk |2 dx
DR
Taking the real part of both sides of this equation gives
R
~2 Im D (∇x Ψk )Ψk dx
R
= ~v(k),
∇k ε =
2 dx
m
|Ψ
|
k
D
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 :
∂t εn (k) = F vn (k) = ~−1 F ∇k εn (k).
By the chain rule,
and hence
∂t εn (k) = ∇k εn (k)∂t k
F = ∂t (~k).
(2.9)
Newton law’s states that the force equals the time derivative of the momentum
p. 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
1 d 2 εn
1 d 2 εn
∂
k
=
F.
t
~ dk 2
~2 dk 2
The momentum equals m∗ vn , where m∗ is the (effective) mass. Using again
Newton’s law F = ∂t p = m∗ ∂t vn we obtain
∂t v n =
1 d 2 εn
.
~2 dk 2
We consider this equation as a definition of the effective mass m∗ . The second
derivative of εn with respect to k is a 3 × 3 matrix, so the symbol (m∗ )−1 is also a
matrix. If we evaluate the Hessian of εn near a local minimum, i.e. ∇k εn (k0 ) = 0,
then d2 εn (k0 )/dk 2 is a symmetric positive matrix which can be diagonalized and
the diagonal elements are positive:


1/m∗x
0
0
1 d 2 εn
 0
0  = 2 2 (k0 ).
1/m∗y
(2.10)
~ dk
0
0
1/m∗z
(m∗ )−1 =
Assume that the energy values are shifted in such a way that the energy vanishes
at the local minimum k0 . For wave vectors k “close” to k0 , we have from Taylor’s
formula and (2.10)
1 d 2 εn
εn (k) = εn (k0 ) + ∇k εn (k0 ) · k + k > 2 (k0 )k + O(|k − k0 |3 )
dk
µ 2
¶ 2
2
2
2
ky
~
k
kx
=
+ ∗ + z∗ + O(|k − k0 |3 ),
∗
2 mx my mz
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,
~2
|k|2
(2.11)
2m∗
for wave vectors k “close” to a local band minimum. In this situation, m∗ is called
the isotropic effective mass. We conclude that the energy of an electron near a
band 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 effects
of the crystal potential are represented by the effective mass.
The expression (2.11) is referred to as the parabolic band approximation and
usually, the range of wave vectors k is extended to the whole space, i.e., k = R 3 is
assumed. In order to account for non-parabolic effects, the following non-parabolic
band approximation in the sense of Kane is used [41, Ch. 2.1]:
εn (k) =
εn (1 + αεn ) =
19
~2 |k|2
,
2m∗
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 band
maximum, we find that the Hessian of εn is negative definite which would lead
to a negative effective mass. However, in the derivation of the mean velocity
and consequently, of the effective mass definition, we have used in (2.8) that the
charge of the electrons is negative. Using a positive charge leads to a positive
effective mass. The corresponding particles are calles holes. Physically, a hole
is a vacant orbital in an otherwise filled (valence) band. Thus, the current flow
in a semiconductor crystal comes from two sources: the flow of electrons in the
conduction band and the flow of holes in the valence band. It is a convention to
consider rather the motion of the valence band vacancies that the motion of the
electrons moving from one vacant orbital to the next (Figure 2.5).
t=0:
electron
orbital
vacancy
J^
­
À
­
Js
¿
¿
Rc
s
s
º¤
±
£
ÁÀ
¤ ÁÀ
£
¤
£
crystal atom
vacancy
t>0:
J^
Jc
¿
¿
s
s
s
º¤
ÁÀ
¤ ÁÀ
¤
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 motion
of an electron in the n-th band can be approximately described by a point particle
moving with the velocity vn (k). Denoting by (x(t), vn (k, t)) the trajectory of an
electron in the position-velocity phase space, we obtain from Newton’s law (see
(2.9))
1
~∂t k = F,
(2.12)
∂t x = vn (k) = ∇k εn ,
~
where F represents a driving force, for instance, F = −q∇x VL (x, t). Notice that
band transitions are excluded since the band index n is fixed in the equations.
Non-periodic potentials. If a non-periodic potential (or driving force) is
superimposed to VL , the Schrödinger equation (2.1) cannot be decomposed into
the decoupled Schrödinger equations (2.2), and the above computations are no
longer valid. In fact, the energy bands are now coupled. However, it is usually
assumed that the non-periodic potential is so weak that the coupling of the bands
can 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 the
conduction 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 seems
appropriate 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 occupied
by not more than two electrons with opposite spin.
First we compute the number of possible states within all energy bands per
unit volume (2π)3 . In the continuum limit, this number equals:
Z
2 X
g(ε) =
δ(ε − εν (k)) dk.
(2.13)
(2π)3 ν B
The function g(ε) is called density of states. The factor 2 comes from the two
possible states of the spin of an electron. The set B is the Brillouin zone (see
Section 2.1), and the function δ is the delta distribution defined by
Z ∞
δ(ε0 − ε)φ(ε) dε = φ(ε0 )
(2.14)
−∞
for all appropriate functions φ. Mathematically, δ is a functional operating in
some function space and the above integral has to be interpreted as the value of
δ(ε0 − ε) acting on φ(ε). Formally, it holds
Z ∞
Z ∞
Z ∞
δ(ε − ε0 )φ(ε) dε, (2.15)
δ(ε − ε0 )φ(ε) dε =
δ(ε0 − ε)φ(ε) dε =
−∞
ε1
−∞
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 with
isotropic effective mass. The energy is given by
ε(k) = εc +
~2
|k|2 ,
2m∗e
k ∈ R3 .
Transforming k to the spherical coordinates (ρ, θ, φ) and then substituting z =
~2 ρ2 /2m∗e , we obtain
Z
1
gc (ε) =
δ(ε − εc − ~2 |k|2 /2m∗e ) dk
4π 3 R3
Z 2π Z π Z ∞
1
=
δ(ε − εc − ~2 ρ2 /2m∗e ) ρ2 sin θ dρ dθ dφ
4π 3 0
0
0
µ ∗ ¶3/2 Z ∞
√
4π 1 2me
δ(ε − εc − z) z dz.
=
3
2
4π 2
~
0
21
Define the Heaviside function H by
H(x) =
½
0 : x<0
1 : x > 0.
Then, by (2.14),
1
gc (ε) =
2π 2
1
=
2π 2
µ
µ
2m∗e
~2
2m∗e
~2
¶3/2 Z
¶3/2
∞
−∞
√
δ(ε − εc − z)
√
zH(z) dz
ε − εc H(ε − εc ).
The electron density n is the integral of the probability density of occupancy
of an energy state per unit volume:
Z
2 X
f (εν (k)) dk,
n=
(2π)3 ν B
where f (ε) is a distribution function. Using (2.14), (2.15) and the definition
(2.13) of gc , we have
Z Z ∞
1 X
n =
δ(ε − εν (k)) f (ε) dε dk
4π 3 ν B −∞
Z ∞
=
gc (ε) f (ε) dε.
(2.16)
−∞
Fermi-Dirac distribution. We will motivate that the electron distribution
follows the Fermi-Dirac distribution
f (ε) =
1
1+
e(ε−εF )/kB T
,
(2.17)
where the number εF which is called Fermi energy or Fermi level will be interpreted later. Consider a system of non-interacting particles with discrete quantum 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 unit
time. The average number of particles that make a transition from state i to
state j is
A(T )e−εi /kB T Wij ,
where A(T ) is some function. The temperature T enters because the transition
will absorb energy from, or omit energy to, the reservoir. Conversely, the average
number of particles that go from state j to state i is
A(T )e−εj /kB T Wji .
22
We use the principle of detailed balance. It says that in thermal equilibrium, the
two number must coincide. Therefore
Wji
= e(εj −εi )/kB T .
Wij
(2.18)
Now let fi be the average number of electrons in the state i. By Pauli’s
exclusion 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 which
the state i is occupied. Similarly, 1 − fi is the fraction of configurations in which
the state i is empty. The number of transitions from state i to state j of a large
ensemble M of configurations is
M fi (1 − jj )Wij ,
since the transition only takes place if the state i is occupied and the state j is
empty. By the principle of detailed balance,
M fi (1 − fj )Wij = M fj (1 − fi )Wji .
This equation is equivalent to
fj Wji
fi
=
1 − fi
1 − fj Wij
or, by (2.18),
fi εi /kB T
fj εj /kB T
e
=
e
.
1 − fi
1 − fj
This relation holds for any state i and any state j. Therefore, both sides are
independent of i and j :
fi εi /kB T
e
= const. = eεF /kB T .
1 − fi
This defines the number εF . Solving the above equation for fi yields
fi =
1
1 + e(εi −εF )/kB T
.
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, we
obtain from (2.17)
½
1 : ε < εF
f (ε) →
0 : ε > εF
and f (εF ) = 1/2. This provides a physical interpretation of the Fermi energy εF
for 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, the
states are rather empty (f (ε) < 1/2 for ε > εF ).
For energies which are much larger than the Fermi energy in the scale of
kB T, i.e. ε − εF À kB T, we can approximate the Fermi-Dirac distribution by the
Maxwell-Boltzmann distribution
f (ε) = e−(ε−εF )/kB T
since 1 + exp((ε − εF )/kB T ) ∼ exp((ε − εF )/kB T ) for ε − εF À kB T. We call this
the non-degenerate case. A semiconductor in which Fermi-Dirac statistics have
to 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
Z ∞
gc (ε) dε
n=
.
(ε−εF )/kB T
−∞ 1 + e
Similar expressions hold for the hole density p for energies below the valence band
minimum εv :
¶
µ
Z ∞
1
dε
p =
gv (ε) 1 −
1 + e(ε−εF )/kB T
−∞
Z ∞
gv (ε) dε
=
.
(εF −ε)/kB T
−∞ 1 + e
The electron and hole densities can be computed more explicitely in the
parabolic band approximation with isotropic effective mass:
µ ∗ ¶3/2 Z ∞ √
µ
¶
2me
εF − ε c
ε − εc dε
1
= Nc F1/2
,
n =
(ε−εF )/kB T
2π 2
~2
kB T
εc 1 + e
µ
µ ∗ ¶3/2 Z εv √
¶
εv − ε F
2mh
εv − ε dε
1
= Nv F1/2
p =
,
(εF −ε)/kB T
2π 2
~2
kB T
−∞ 1 + e
where we have introduced the effective densities of states
¶3/2
µ ∗
¶3/2
µ ∗
mh kB T
me kB T
, Nv = 2
Nc = 2
2π~2
2π~2
and the Fermi integral
2
F1/2 (y) = √
π
Z
∞
0
√
x dx
,
1 + ex−y
y ∈ R.
In the non-degenerate case εc − εF À kB T and εF − εv À kB T we can replace
the Fermi integral by a simpler function:
F1/2 (y) ∼ exp(y)
24
as y → −∞.
Then the particle densities become
µ
¶
εF − ε c
n = Nc exp
,
kB T
p = Nv exp
µ
εv − ε F
kB T
¶
.
(2.19)
Intrinsic semiconductors. A pure semiconductor crystal with no impurities
is called an intrinsic semiconductor. In this case, the conduction band electrons
can only have come from formerly occupied valence band levels leaving holes
behind them. The number of conduction band electrons is therefore equal to the
number of valence band holes:
n = p =: ni .
The intrinsic density ni can be computed in the non-degenerate case from (2.19):
¶ p
¶
µ
µ
p
εg
εv − ε c
√
= Nc Nv exp −
(2.20)
ni = np = Nc Nv exp
kB T
2kB T
since the energy gap is εg = εc −εv . The Fermi energy of an intrinsic semiconductor
can be computed using (2.19) and (2.20):
εF = εc + kB T ln(n/Nc )
= εc + kB T ln(ni /Nc )
1
1
Nv
= εc − εg + kB T ln
2
2
Nc
1
3
m∗h
=
(εc + εv ) + kB T ln ∗ .
2
4
me
This asserts that as T → 0, the Fermi energy lies precisely in the middle of the
energy gap. Furthermore, since ln(m∗h /m∗e ) is of order one, the correction is only
of order kB T. In most semiconductors and at room temperature, the energy gap
εg is much larger than kB T (silicon: εg = 1.12 eV, kB T = 0.0259 eV at room
temperature). This shows that the non-degeneracy assumptions
1
3
m∗
ε − εF ≥ εc − εF = εg + kB T ln h∗ À kB T,
2
4
me
3
m∗
1
εF − ε ≥ εF − εv = εg + kB T ln h∗ À kB T
2
4
me
are satisfied and that the result is consistent with the assumptions (see Figure
2.6).
Doping. The intrinsic density is too small to result in a significant conductivity (for instance, silicon: ni = 6.93 · 109 cm−3 ). However, it is possible
to replace some atoms in a semiconductor crystal by atoms which provide free
electrons in the conduction band or free holes in the valence band. This process is called doping. Impurities that contribute to the carrier density are called
25
ε 6
εc
εF
6
εv + 21 εg εg À kB T
?
εv
-
k
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 acceptors
if 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. For
instance, when we dope a germanium crystal, whose atoms have each 4 valence
electrons, with arsenic, whose atoms have each 5 valence electrons, each arsenic
impurity provides one additional electron. These additional electrons are only
weakly binded to the arsenic atom. Indeed, the binding energy is about 0.013 eV
and thermal excitation provides enough energy to excite the additional electrons
to the conduction band. Generally speaking, let εd and εa be the energy level of
a donor electron and an acceptor hole, respectively. Then εc − εd and εa − εv are
small compared to kB T (Figure 2.7). This means that the additional particles
contribute at room temperature to the electron and hole density.
ε 6
εc
εd
6
εa
εv
?
εg
-
k
Figure 2.7: Illustration of the donor and acceptor energy levels εd and εa .
26
3
3.1
Classical Kinetic Transport Models
The Liouville equation
We first analyze the motion of an ensemble of M electrons in a vacuum under the
action 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, but
also one- or two-dimensional models can be considered. Since the electrons have
equal mass m, the trajectories (xi (t), vi (t)) of the ensemble satisfy the system
of ordinary differential equations in the (2d · M )-dimensional ensemble positionvelocity space
∂t x i = v i ,
∂ t vi =
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 set
F = (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 trajectory
w(t; x0 , v0 ) = (x(t), v(t)). For instance, the force can be given by an electric
field 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 ∼
5
10 ) and therefore, the (numerical) solution of (3.1) is very expensive or even
not feasible. It seems reasonable to use a statistical description. We assume
that instead of the precise initial position x0 and initial velocity v0 we are given
the joint probability density fI (x, v) of the initial position and velocity of the
electrons. This density has the properties
Z
Z
dM
fI (x, v) ≥ 0 for x, v ∈ R ,
fI (x, v) dx dv = 1.
(3.3)
RdM
Then
ZZ
RdM
fI (x, v) dx dv
B
is the probability to find the particle ensemble in the subset B of the (x, v)-space
at 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 trajectories
w(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 time
gives
d
f (x(t), v(t), t)
dt
= ∂t f + ∂ t x · ∇ x f + ∂ t v · ∇ v f
1
= ∂t f + v · ∇x f + F · ∇v f.
m
0 =
(3.5)
This equation is referred to as the classical Liouville equation for an electron
ensemble. When the force is given by the electric field,
F = −qE = q∇x V (x, t),
where V is the electrostatic potential, the Liouville equation reads
∂t f + v · ∇ x f +
q
∇x V · ∇v f = 0,
m
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 density
f (x, v, t) of an electron ensemble in the electric field −q∇x V under the assumptions 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 the
divergence theorem:
Z
Z
f (x, v, t) dx dv
∂t
RdM RdM
Z
Z
³
´
q
div x (vf ) + div v (f ∇x V ) dx dv
= −
m
RdM RdM
= 0.
By (3.3), we conclude for all t ≥ 0
Z
Z
Z
f (x, v, t) dx dv =
RdM
RdM
RdM
Z
RdM
fI (x, v) dx dv = 1.
(3.8)
• Moments: We define the zeroth- and first-order moments
Z
n(x, t) =
f (x, v, t) dv,
RdM
Z
J(x, t) = −q
vf (x, v, t) dv,
RdM
where q is the elementary charge. The functions n and J are interpreted
as the position probability density (or particle density) and as the particle current density, respectively. The conservation property (3.8) can be
restated as
Z
Z
nI (x) dx, t ≥ 0,
(3.9)
n(x, t) dx =
RdM
RdM
with the initial electron density
nI (x) =
Z
RdM
fI (x, v) dv.
This means that the total number of electrons is conserved in time. By
formally integrating (3.6) over v ∈ RdM we obtain the so-called particle
continuity equation
1
∂t n − div x J = 0,
q
x ∈ RdM , t > 0.
(3.10)
We consider an electron ensemble moving in a semiconductor crystal. As
explained in Chapter 2, the ions in the crystal lattice induce a lattice-periodic
potential which influences the motion of the charged particles. Let k 7→ εn (k) be
the 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
∇k εn (k).
~
29
Let ki ∈ Rd be the wave vector and xi ∈ Rd the position vector of the i-th electron
and set k = (k1 , . . . , kM ) ∈ B M , x = (x1 , . . . , xM ) ∈ RdM . We fix the energy band
and omit therefore the index n. The motion of the electrons in the (x, k)-space
can be described semi-classically by the equations
∂t x = v(k) =
1
∇k ε(k),
~
~∂t k = 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 index is fixed. Notice also that the equation ~∂t k = F is an expression of Newton’s
law since p = ~k is the crystal momentum vector of the electrons. An analogous computation as in (3.5) yields the semi-classical Liouville equation for the
distribution function f (x, k, t) of an electron ensemble:
1
∂t f + v(k) · ∇x f + F · ∇k f = 0,
~
where
v(k) · ∇x f =
M
X
j=1
x ∈ RdM , k ∈ B M , t > 0,
(3.11)
v(kj ) · ∇xj f.
As the Brillouin zone is a bounded subset, we have to impose boundary conditions
for 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 to
the origin, i.e. k ∈ B if and only if −k ∈ B.
The macroscopic particle and current densities are now defined by
Z
f (x, k, t) dk,
n(x, t) =
BM
Z
v(k) f (x, k, t) dk.
J(x, t) = −q
BM
The periodicity of f in ki and the point-symmetry of B imply the conservation
property (3.9) and the conservation law (3.10). Here we also need the assumption
div k F = 0 such that terms arising in the integration by parts cancel. This
assumption 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
|k|2 ,
2m∗
k ∈ RdM ,
for electrons is used, then v(k) = ~k/m and ∇k f = (~/m∗ )∇v f , and the semiclassical 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 to
be 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 formally
derive a low-dimensional equation, the so-called Vlasov equation. The idea of the
derivation 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 finally
to 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 and
denote by x = (x1 , . . . , xM ) ∈ RdM , v = (v1 , . . . , vM ) ∈ RdM the position and
velocity coordinates of the electrons, respectively. We impose the following assumptions:
• 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 and
by two-particle interaction forces.
The first assumption will be discarded later on. The second assumption means
that magnetic fields are ignored (see [34, Ch. 1.3] for an inclusion of magnetic
effects). 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 of
an electric field acting on the i-th electron and of the sum of M − 1 two-particle
interaction forces exerted on the i-th electron by the other electrons:
Fi (x, t) = −qEext (xi , t) − q
M
X
Eint (xi , xj ),
i = 1, . . . , M.
(3.12)
j=1, j6=i
The interaction force Eint is independent of the electron indices, which interprets
the fact that the electrons are indistinguishable. We assume that the force exerted
by the i-th electron on the j-th electron is equal to the negative force exerted by
the j-th electron on the i-th electron:
Eint (xi , xj ) = −Eint (xj , xi ) ∀xi , xj ∈ Rd ,
31
(3.13)
which implies Eint (x, x) = 0.
The classical Liouville equation for the density f (x, v, t) of the ensemble reads:
∂t f +
M
X
i=1
M
q X
v i · ∇ xi f −
Eext (xi , t) · ∇vi f
m i=1
M
q X
Eint (xi , xj ) · ∇vi f = 0.
−
m i,j=1
(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 }. Then
the property (3.13) implies that also f (x, v, t) is independent of the numbering
of the particles for all t > 0. We introduce the density f (a) of a subensemble
consisting of a < M electrons:
Z
(a)
f (x, v, t) dxa+1 · · · dxM dva+1 · · · dvM .
f (x1 , . . . , xa , v1 , . . . , va , t) =
R2d(M −a)
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 Z
X
i=1
R2d(M −a)
=
a
X
i=1
+
v i · ∇ xi
M Z
X
i=a+1
a
X
=
vi · ∇xi f dxa+1 · · · dxM dva+1 · · · dvM
i=1
Z
R2d(M −a)
R2d(M −a)
f dxa+1 · · · dxM dva+1 · · · dvM
div xi (vi f ) dxa+1 · · · dxM dva+1 · · · dvM
vi · ∇xi f (a) .
Similarly,
M
q X
−
m i=1
= −
Z
R2d(M −a)
div vi (Eext (xi , t)f ) dxa+1 · · · dxM dva+1 · · · dvM
a
q X
div vi (Eext (xi , t)f (a) ).
m i=1
32
The last integral becomes
M
q X
−
m i=1
Z
R2d(M −a)
div vi (Eint (xi , xj )f ) dxa+1 · · · dxM dva+1 · · · dvM
a
q X
Eint (xi , xj ) · ∇vi f (a)
= −
m i,j=1
M Z
M
q X X
div vi (Eint (xi , xj )f ) dxa+1 · · · dxM dva+1 · · · dvM
−
m i=a+1 j=1 R2d(M −a)
a
M Z
q X X
−
div vi (Eint (xi , xj )f ) dxa+1 · · · dxM dva+1 · · · dvM .
m i=1 j=a+1 R2d(M −a)
The second integral vanishes by the divergence theorem. The last integral is equal
to
Z
a
q X
(M − a)div vi
−
Eint (xi , xa+1 )
m i=1
R2d
µZ
¶
f dxa+2 · · · dxM dva+2 · · · dvM dxa+1 dva+1
R2d(M −a)
a
X
q
= −
m
×f
i=1
(a+1)
(M − a)div vi
Z
R2d
Eint (xi , x∗ )
(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 Z
X
j=2
=
R4d
Z
div v1 (Eint (x1 , xj )f (x, v, t)) dx2 dx3 dv2 dv3
R4d
+
Z
div v1 (Eint (x1 , x2 )f (x1 , x2 , x3 , v1 , v2 , v3 , t)) dx2 dx3 dv2 dv3
div v1 (Eint (x1 , x3 )f (x1 , x2 , x3 , v1 , v3 , v2 , t)) dx2 dx3 dv2 dv3
Z
= 2div v1
Eint (x1 , x2 )
R2d
¶
µZ
f (x1 , x2 , x3 , v1 , v2 , v3 , t) dx3 dv3 dx2 dv2
×
R2
Z
= 2div v1
Eint (x1 , x∗ )f (2) (x1 , x∗ , v1 , v∗ , t) dx∗ dv∗ .
R4d
R2
33
We summarize the above computations and obtain from (3.14)
∂t f (a) +
a
X
a
vi · ∇xi f (a) −
i=1
a
X
q X
Eext (xi , t) · ∇vi f (a)
m i=1
q
Eint (xi , xj ) · ∇vi f (a)
m i,j=1
Z
a
X
q
− (M − a)
div vi
Eint (xi , x∗ )f∗(a+1) dx∗ dv∗ = 0,
m
2d
R
i=1
−
(3.16)
where
f∗(a+1) = f (a+1) (x1 , . . . , xa , x∗ , v1 , . . . , va , v∗ , t).
These equations for 1 ≤ a ≤ M − 1 are called the Bogoliubov-Born-GreenKirkwood-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 the
formal limit, whereas the expression (M − a)Eint in the last term on the left-hand
side of (3.16) remains finite. For M À 1, we can substitute this expression by
M Eint . Thus, for M À 1, we obtain from (3.16), neglecting terms of order 1/M,
∂t f
(a)
+
q
−
m
a
X
i=1
a
X
i=1
a
v i · ∇ xi f
div vi
(a)
Z
R2d
q X
Eext (xi , t) · ∇vi f (a)
−
m i=1
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
Y
P (xi , vi , t).
(3.18)
i=1
The interpretation of this factorization is that we assume that the electrons of
the subensemble move independently of each other. Using this ansatz in (3.17)
for a = 1 gives
∂t P + v 1 · ∇ x 1 P −
where
Eeff (x, t) = Eext (x, t) +
Z
q
Eeff (x1 , t) · ∇v1 P = 0,
m
R2d
M P (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 satisfies
the 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 information of the evolution of the electron subensemble is thus contained in the function
P. We define
F (x, v, t) = M P (x, v, t),
Z
F (x, v, t) dv,
n(x, t) =
Rd
x, v ∈ Rd , t ≥ 0.
The quantity n(x, t) represents the density of the electrons per unit volume. The
number density F satisfies the so-called classical Vlasov equation
q
Eeff · ∇v F = 0, x, v ∈ Rd , t > 0,
m
Z
n(x∗ , t) Eint (x, x∗ ) dx∗ .
Eeff (x, t) = Eext (x, t) +
∂t F + v · ∇ x F −
Rd
(3.19)
(3.20)
This equation has the form of a single-particle Liouville equation with the force
Eeff . Many-particle physics enters only through the effective field Eeff , which in
turn depends on the density n and hence on F. This means that (3.19) is a
nonlinear equation with a nonlocal nonlinearity of quadratic type. It provides a
macroscopic description of the motion of many-particle systems with weak longrange forces. However, it does not account for strong short-range forces such as
scattering 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 of
a 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
∂t x = v,
∂t v = −
q
Eeff ,
m
x(0) = x0 ,
v(0) = v0 ,
the expression
0 = ∂t F + v · ∇ x F −
=
q
Eeff · ∇v F
m
d
F (x(t), v(t), t)
dt
and therefore
F (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 Coulomb
force:
x−y
q
, x, y ∈ R3 , x 6= y,
Eint (x, y) = −
4πεs |x − y|3
where the permittivity εs is a material constant. (The following considerations
are also valid in non-vacuum.) Then the effective field reads (see (3.20))
Z
q
x−z
Eeff (x, t) = Eext (x, t) −
dz.
n(z, t)
4πεs R3
|x − z|3
It is well known that the function
Z
1
f (z)
φ(x) =
dz,
4π R3 |x − z|
x ∈ R3 ,
satisfies the equation on ∆φ = f in R3 under some regularity assumptions on f.
Therefore
Z
Z
1
x−z
1
1
f (z)∆x
f (z)div x
dz
dz =
f (x) =
4π R3
|x − z|
4π R3
|x − z|3
and
1
0 = curl x ∇x φ(x) =
curl x
4π
We conclude that
Z
R3
f (z)
x−z
dz.
|x − z|3
q
n(x, t),
εs
curl Eeff (x, t) = curl Eext (x, t), x ∈ R3 , t > 0.
div Eeff (x, t) = div Eext (x, t) −
We assume that the external field is generated by ions of charge +q which are
present in the material (for instance, doping atoms in the semiconductor crystal).
Then, by Coulomb’s law
Z
x−z
+q
C(z, t)
dz,
Eext (x, t) =
4πεs R3
|x − z|3
where C is the number density of the background ions. Since
q
div Eext = C, curl Eext = 0,
εs
we obtain
q
(n − C), curl Eeff = 0 in R3 .
εs
Since Eeff is vortex-free, there is a potential Veff such that Eeff = −∇Veff . Then
we can rewrite the above equation as
div Eeff = −
εs ∆Veff = q(n − C),
36
x ∈ R3 .
(3.21)
This equation which also holds in Rd , d ≥ 1, is called Poisson equation. The
Vlasov equation (3.19) with the Coulomb interaction field, i.e. with E eff = −∇Veff
and (3.21), is referred to as Vlasov-Poisson system.
Instead of taking the classical Liouville equation (3.6) as basis for the derivation 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 by
proceeding as above the semi-classical Vlasov equation
q
∂t F + v(k) · ∇x F − Eeff · ∇k F = 0,
~
Z
Eeff (x, t) = Eext (x, t) +
Rd
x ∈ Rd , k ∈ B, t > 0,
n(x∗ , t) Eint (x, x∗ ) dx∗
(3.22)
(3.23)
with the particle position density
n(x, t) =
Z
F (x, k, t) dk
B
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 :
x ∈ Rd , k ∈ ∂B, t > 0.
F (x, k, t) = F (x, −k, t),
(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 in
the conduction band and the total number of quantum states in this volume in
the 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 k
in x at time t.
3.3
The Boltzmann equation
The Vlasov equation accounts for long-range particle interactions, like the Coulomb force (see Example 3.1). Short-range interactions, like collisions of the
particles with other particles or with the crystal lattice, are not included. We
wish to extend the Vlasov equation to include collision mechanisms, which will
lead 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 to
the literature in [9, Sec. 1.5.3] and [16].
The starting point of the derivation is to postulate that the rate of change
of the number density F (x, v, t) of the particle ensemble due to convection and
the effective field, dF/dt, and the rate of change of F due to collisions, Q(F ),
balance:
dF
= Q(F ).
dt
This equation has to be understood along the particle trajectories. Explicitely,
it reads (in the classical case, see (3.19))
∂t F + v · ∇ x F −
q
Eeff · ∇v F = Q(F ),
m
x, v ∈ Rd , t > 0,
(3.25)
where the effective field Eeff is given by (3.20). In order to derive an expression
for Q(F ) we assume that the rate P (x, v 0 → v, t) of a particle at (x, t) to change
its velocity v 0 into v due to a scattering event is proportional to
• the occupation probability F (x, v 0 , t) and
• the probability 1 − F (x, v, t) that the state (x, v) is not occupied at time t.
Thus
P (x, v 0 → v, t) = s(x, v 0 , v)F (x, v 0 , t)(1 − F (x, v, t)),
where s(x, v 0 , v) is the so-called scattering rate. (We made a similiar consideration
in Section 2.3.) Then the rate of change of F due to collisions is the sum of
P (x, v 0 → v, t) − P (x, v → v 0 , t)
for all velocities v 0 in the volume element dv 0 . In the continuum limit, the sum is
in fact an integral and we get
Z
(Q(F ))(x, v, t) =
[P (x, v 0 → v, t) − P (x, v → v 0 , t)] dv 0
(3.26)
d
R
Z
=
[s(x, v 0 , v)F 0 (1 − F ) − s(x, v, v 0 )F (1 − F 0 )] dv 0 ,
Rd
where F = F (x, v, t), F 0 = F (x, v 0 , t). The equation (3.25), together with the
effective field equation (3.20) and the definition of the collision operator (3.26),
is called the classical Boltzmann equation. When the Coulomb force is used
to model the long-range interactions, then (3.25), (3.20), (3.26), and (3.21) are
termed Boltzmann-Poisson system.
The Boltzmann equation has two nonlinearities: a quadratic nonlocal nonlinearity caused by the self-consistent field Eeff and another quadratic nonlocal
nonlinearity caused by the collision integral (3.26). A rigorous mathematical
38
analysis of the initial-value problem (3.25)-(3.26) (existence and uniqueness of
solutions) is very difficult. We refer to the research papers [23, 36] and to the
review paper [4] for details on the mathematical analysis of this problem.
Starting from the semi-classical Vlasov equation (3.22) we can include the
collision effects similiarly as above and obtain the semi-classical Boltzmann equation
q
(3.27)
∂t F + v(k) · ∇x F − Eeff · ∇k F = Q(F ), x ∈ Rd , k ∈ B, t > 0,
~
where the effective field Eeff is defined in (3.23). This equation models the semiclassical motion of electrons incorporating the quantum effects of the semiconductor 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 the
quantum of lattice vibrations are called phonons. We can distinguish so-called
acoustic phonons and optical phonons. These names refer to the vibration models
of the lattice (see [14, Ch. 9.2] or [28, Ch. 7]). The collision operator for electronphonon interactions is given by
Z
(Qα (F ))(x, k, t) = [sα (x, k 0 , k)F 0 (1 − F ) − sα (x, k, k 0 )F (1 − F 0 )] dk 0 ,
B
where F = F (x, k, t), F 0 (F (x, k 0 , t), and the scattering rate is
sα (x, k, k 0 ) = σα (x, k, k 0 )[(1 + Nα )δ(ε(k 0 ) − ε(k) + ~ωα )
+ Nα δ(ε(k 0 ) − ε(k) − ~ωα )],
the function σα (x, k, k 0 ) is symmetric in k and k 0 , ~ωα is the (constant) phonon
energy, Nα is the phonon occupation number given by Bose-Einstein statistics
µ
µ
¶
¶−1
~ωα
Nα = exp
−1
,
kB T
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 is
non-zero only if ε(k 0 ) − ε(k) = ±~ωα , i.e., the rate accounts for phonon emission and absorption of energy ~ωα . The above expression for sα shows that the
scattering rates can be highly non-smooth; in fact, Qα is a distribution.
When an atom different from the semiconductor material is introduced into
the semiconductor it may donate either an electron or a hole, leaving behind an
ionized charged impurity center. These ionized impurities can act as scattering
agents for free carriers propagating through the crystal. The interaction of carriers with neutral impurities is also possible, but we do not consider it here. The
collision operator reads
Z
(Qimp (F ))(x, k, t) =
σimp (x, k, k 0 ) δ(ε(k 0 ) − ε(k)) (F 0 − F ) dk 0 ,
B
where F = F (x, k, t), F 0 = F (x, k 0 , t) and σimp is symmetric in k and k 0 [11].
The electron-electron interaction arises from the long-range nature of the
Coulomb force. The corresponding collision operator
Z
(Qee (F ))(x, k, t) =
[see (x, k, k1 , k 0 , k10 )F 0 F10 (1 − F )(1 − F1 )
B3
− see (x, k 0 , k10 , k, k1 )F F1 (1 − F 0 )(1 − F10 )] dk 0 dk1 dk10 ,
where F = F (x, k, t), F 0 = F (x, k 0 , t), F1 = F (x, k1 , t), and F10 = F (x, k10 , t), is a
nonlocal nonlinearity of fourth order. The influence of electron-electron interaction on the carrier dynamics is more pronounced in degenerate semiconductors in
which Fermi-Dirac statistics instead of Maxwell-Boltzmann statistics has to be
used (see Section 2.3). Also electron-hole interactions are possible but we do not
consider these collision events here. For more details of scattering processes, we
refer 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
Z
(Q(F ))(x, k, t) = [s(x, k 0 , k)F 0 (1 − F ) − s(x, k, k 0 )F (1 − F 0 )] dk 0 ,
(3.29)
B
where we have again set F = F (x, k, t) and F 0 = F (x, k 0 , t).
We study now some properties of the collision integral (3.29). Since s(x, k, k 0 )
is the transition rate from state k to state k 0 , the principle of detailed balance
implies that
s(x, k, k 0 )
0
= e(ε(k)−ε(k ))/kB T
0
s(x, k , k)
holds (compare with (2.18)).
40
∀x ∈ Rd , k, k 0 ∈ B,
(3.30)
Lemma 3.2 (see [36]) Let (3.30) hold for some function ε(k) and let s(x, k, k 0 ) >
0 for all x ∈ Rd , k, k 0 ∈ B.
(1) It holds
Z
(Q(F ))(x, k, t) dk = 0
B
∀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-decreasing
functions χ : R → R
µ
¶
Z
F (x, k, t) ε(k)/kB T
(Q(F ))(x, k, t) χ
e
dk ≤ 0,
1 − F (x, k, t)
B
¶
µ
Z
1 − F (x, k, t) −ε(k)/kB T
e
dk ≥ 0.
(Q(F ))(x, k, t) χ
F (x, k, t)
B
(3) The property Q(F ) = 0 is equivalent to the existence of −∞ ≤ εF ≤ +∞
such that
1
, k ∈ B.
F (k) =
1 + exp((ε(k) − εF )/kB T )
Let F be a solution to the Boltzmann equation (3.27), (3.28). Then Lemma
3.2(1) implies that the total number of electrons is conserved in time:
Z
Z
n(x, t) =
F (x, k, t) dk =
FI (x, k) dk = nI (x, t), x ∈ Rd , t > 0.
B
B
Physically, this is reasonable: collisions neither destroy nor generate particles.
The statements of Lemma 3.2(2) are also called H-theorems. Finally, Lemma
3.2(3) means that precisely the Fermi-Dirac distribution functions are in the
kernel of the collision operator.
Proof of Lemma 3.2. (1) Changing k and k 0 in the second integral gives
Z
Z
Z
0
0
0
Q(F ) dk =
s(x, k , k)F (1 − F ) dk dk −
s(x, k, k 0 )F (1 − F 0 ) dk 0 dk
2
2
B
ZB
ZB
=
s(x, k 0 , k)F 0 (1 − F ) dk 0 dk −
s(x, k 0 , k)F 0 (1 − F ) dk dk 0
B2
B2
= 0,
where F = F (x, k, t) and F 0 = F (x, k 0 , t).
(2) We only show the second inequality. The proof of the first one is similar.
We set
1 − F (k)
M (k) = e−ε(k)/kB T , h(k) =
M (k).
F (k)
41
Then (3.30) is equivalent to
s(x, k 0 , k)
s(x, k, k 0 )
=
,
M (k)
M (k 0 )
(3.31)
and we obtain with the notations h = h(k), h0 = h(k 0 ), M = M (k), M 0 = M (k 0 ),
Z
Q(F )χ(h) dk
B
·
¸
Z
M 0
0
0
s(x, k, k )
F (1 − F )χ(h) − F (1 − F )χ(h) dk 0 dk
=
0
M
2
·
¸
ZB
s(x, k, k 0 )
1 − F0 0
0 1−F
FF
M χ(h) dk 0 dk
=
M χ(h) −
0
0
M
F
F
2
ZB
0
s(x, k, k )
=
F F 0 (h − h0 )χ(h) dk 0 dk,
(3.32)
0
M
B2
and, by changing k and k 0 and using (3.31),
Z
Q(F )χ(h) dk
B
·
¸
Z
1 − F0 0
1−F
s(x, k 0 , k) 0
0
0
FF
M χ(h ) −
M χ(h ) dk dk 0
=
0
M
F
F
2
·
¸
ZB
1−F
1 − F0 0
s(x, k, k 0 ) 0
0
0
FF
M χ(h ) −
M χ(h ) dk 0 dk
=
0
0
M
F
F
2
ZB
s(x, k, k 0 ) 0
=
F F (h0 − h)χ(h0 ) dk 0 dk.
(3.33)
0
M
2
B
Adding (3.32) and (3.33) yields
Z
Z
s(x, k, k 0 )
1
F F 0 (h − h0 )(χ(h) − χ(h0 )) dk dk 0
Q(F )χ(h) dk =
0
2 B
M
B
≥ 0,
since s(x, k, k 0 ), M 0 , F , and F 0 are non-negative and χ is non-decreasing.
(3) By (2), with χ(x) = x, the property Q(F ) = 0 is equivalent to
F F 0 (h − h0 )2 = 0
for almost all k, k 0 ∈ B.
This implies F = 0 or h = h0 almost everywhere. The equation h = h0 is
equivalent to
1 − F (k 0 )
1 − F (k)
M (k) =
M (k 0 )
F (k)
F (k 0 )
42
for almost all k, k 0 ∈ B.
Thus, both sides are constant and we call this constant exp(−εF /kB T ) with
εF ∈ R. Notice that the constant must be positive since 0 ≤ F ≤ 1 expect if
F ≡ 1. But then
1 − F (k)
e−εF /kB T
=
= e(ε(k)−εF )/kB T
F (k)
M (k)
and F (k) = 1 + exp((ε(k) − εF )/kB T ))−1 . Choosing εF = ±∞ yields the other
two possibilities F ≡ 0 or F ≡ 1.
¤
In the literature, two approximations of the collision operator are frequently
used:
• the low-density approximation:
Z
(Q(F ))(x, k, t) =
σ(x, k 0 , k)(M F 0 − M 0 F ) dk 0 ,
(3.34)
B
where σ(x, k 0 , k) = s(x, k, k 0 )/M (k 0 ) is called collision cross-section.
• the relaxation-time approximation:
(Q(F ))(x, k, t) = −
where
τ (x, k) =
µZ
F (x, k, t) − M (k)n(x, t)
,
τ (x, k)
0
s(x, k, k ) dk
B
0
(3.35)
¶−1
is called the relaxation time describing the average time between two consecutive scattering events at (x, k).
These approximations can be derived as follows. The low-density collision
operator 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),
Z
(Q(F ))(x, k, t) =
[s(x, k 0 , k)F 0 − s(x, k, k 0 )F ] dk 0
ZB
s(x, k 0 , k)
=
[M (k)F 0 − M (k 0 )F ] dk 0
M (k)
ZB
=
σ(s, k 0 , k)(M F 0 − M 0 F ) dk 0 ,
B
where M (k) = exp(−ε(k)/kB T ) and M 0 = M (k 0 ). Notice that by (3.30), the
collision cross-section σ(x, k 0 , k) is symmetric in k and k 0 .
43
When the initial datum FI is close to a multiple of the so-called Maxwellian
M (k) it is reasonable to approximate F 0 in (3.34) by n(x, t)M (k 0 ). Then we obtain
from (3.34)
Z
(Q(F ))(x, k, t) =
σ(x, k 0 , k)(nM M 0 − M 0 F ) dk 0
ZB
=
s(x, k, k 0 )(nM − F ) dk 0
B
Z
=
s(x, k, k 0 ) dk 0 · (n(x, t)M (k) − F (x, k, t))
B
= −
F (x, k, t) − n(x, t)M (k)
.
τ (x, k)
Along the trajectories (x(t), k(t)), where
∂t x = v(k),
~∂t k = −qEeff ,
x(0) = x0 ,
k(0) = k0 ,
the Boltzmann equation with the relaxation-time approximation for constant τ
reads
1
dF
= − (F − nM ), t > 0.
dt
τ
The solution of this linear ordinary differential equation (for given n(x, t)) is
Z
1 t
−t/τ
F (x(t), k(t), t) = e
FI (x0 , k0 ) +
n(x(s), s)M (k(s))e(s−t)/τ ds.
τ 0
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 state
FI along the trajectories after a time of order τ. This explains the name relaxation
time.
A summary of the main models derived in Section 3.1-3.3 and its relations
are presented in Figure 3.1.
3.4
Extensions
We discuss two extensions of the Boltzmann equation:
• multi-valley models and
• bipolar models.
44
classical ¾ no two-particle
interactions
Liouville
classical ¾
Vlasov
6
no collisions
6
parabolic
band
parabolic
band
semi-classical ¾no two-particle
interactions
Liouville
classical
Boltzmann
parabolic
band
semi-classical ¾ no collisions
Vlasov
6
semi-classical
Boltzmann
Figure 3.1: Relations between the (semi-)classical Liouville, Vlasov and Boltzmann equations.
ε(k) 6
conduction
band
6
energy
gap
valence
band
?
-
L
Γ
X
|k|
Figure 3.2: Schematic band structure of GaAs.
In the semiconductor material GaAs, the (conduction band) energy ε(k) has
one 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- and
X-valleys; see Figure 3.2 [28, Ch. 4]. As the highest energy X-valley can be
occupied only at very high electric field strengths, we will consider only the Γand L-valleys. We assume furthermore the parabolic band approximation. Then,
after a suitable translation and rotation of the three-dimensional k-space,
!
Ã
ky2
~2
kz2
kx2
εΓ (k) = ∆Γ −
+
+
,
2 m∗Γx m∗Γy m∗Γz
Ã
!
ky2
~2
kx2
kz2
εL (k) = ∆L +
+
+
, k = (kx , ky , kz )> ,
2 m∗Lx m∗Ly m∗Lz
where m∗Γα , m∗Lα are the effective masses of an electron in the Γ-valley, L-valley,
respectively, in the different directions. Since we assume low field strengths, we
split 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 within
each valley since the transfer to a higher-energetic valley requires the presence of
high 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
q
∂t FΓ + vΓ (k) · ∇x FΓ − Eeff · ∇k FΓ = QΓ (FΓ ) + QΓ→L (FΓ , FL ),
~
q
∂t FL + vL (k) · ∇x FL − Eeff · ∇k FL = QL (FL ) + QL→Γ (FΓ , FL ),
~
where
1
1
∇k εΓ (k), vL (k) = ∇k εL (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 collision
operators
Z
Qα (Fα ) =
σα (x, k 0 , k)(Mα (k)Fα0 − Mα (k 0 )Fα ) dk 0 , α = Γ, L,
vΓ (k) =
R3
QΓ→L and QL→Γ are the intervalley low-density collision operators
Z
(σL→Γ (x, k 0 , k)MΓ (k)FL0 − σΓ→L (x, k, k 0 )ML (k 0 )FΓ ) dk 0
QΓ→L (FΓ , FL ) =
3
ZR
QL→Γ (FΓ , FL ) =
(σΓ→L (x, k 0 , k)ML (k)FΓ0 − σL→Γ (x, k, k 0 )MΓ (k 0 )FL ) dk 0
R3
sΓ→L (x, k, k 0 ) is the cross-section from the state (x, k) of the Γ-valley into a state
(x, k 0 ) of the L-valley, similarly for sL→Γ (x, k, k 0 ), and the Maxwellians MΓ and
ML are given by
µ
¶
µZ
¶ ¶−1
µ
εα (k)
εα (k)
∗
∗
Mα (k) = Nα exp −
, Nα =
dk
, α = Γ, L.
exp −
kB T
kB T
R3
The effective field Eeff is computed from (3.23), i.e.
Z
Eeff (x, t) = Eext (x, t) +
(nΓ (x∗ , t) + nL (x∗ , t)) Eint (x, x∗ ) dx∗ ,
R3
Z
nα (x, t) =
Fα (x, k, t) dk, α = Γ, L.
R3
It can be shown similarly as in Lemma 3.2 that under the assumption
σΓ→L (x, k, k 0 ) = σL→Γ (x, k 0 , k) ∀x, k, k 0 ∈ 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 conduction
band. However, also holes in the valence band contribute to the carrier flow
in semiconductors (see Section 2.2). It is possible that an electron moves from
the valence band to the conduction band, leaving a hole in the valence band
behind it. For such a process, which is termed generation of an electron-hole
pair, energy absorption is necessary. The inverse process that an electron of the
conduction band moves to the lower energetic valence band, occupying an empty
state, is called recombination of an electron-hole pair. For such an event, energy
is 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 by
a transfer of energy to a free carrier which is then excited to a higher energy
state. The inverse Auger process, in which an electron-hole pair is generated, is
called impact ionization. The energy comes from the collision of a high-energy
free carrier with the lattice.
In a radiative recombination event, an electron from the conduction band
recombines with a hole from the valence band emitting a photon. The energy
lost by the electron is equal to the energy gap of the material, and a photon of
this amount of energy is produced. Radiative generation events occur from the
absorption of a photon of energy greater than or equal to the bandgap energy.
Finally, thermal recombination or generation events arise from phonon emission or absorption, respectively (see Figure 3.3).
ε(k) 6 conduction band
ε(k) 6 conduction band
t
d
6
- energy
?
d
emission
valence band -
k
¾
t
energy
absorption
valence band -
k
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 of
the holes is given by the Boltzmann equation including an operator accounting
for the recombination-generation processes. More precisely, in the semi-classical
framework,
q
∂t Fn + vn (k) · ∇x Fn − Eeff · ∇k Fn = Qn (Fn ) + In (Fn , Fp ),
~
q
∂t Fp + vp (k) · ∇x Fp + Eeff · ∇k Fp = Qp (Fn ) + Ip (Fn , Fp ),
~
(3.36)
(3.37)
where
1
1
∇k εn , v p = ∇k εp
~
~
are the mean velocities related to the electron conduction band εn and the hole
valence band εp , respectively, Qn and Qp are the collision operators
Z
(Qα (Fα ))(x, k, t) =
[sα (x, k 0 , k)F 0 (1 − F ) − sα (x, k, k 0 )F (1 − F )] dk 0 ,
vn =
B
α = n, p, and the recombination-generation operators are given by
Z
¤
£
g(x, k 0 , k)(1 − Fn )(1 − Fp0 ) − r(x, k, k 0 )Fn Fp0 dk 0 ,
(In (Fn , Fp ))(x, k, t) =
ZB
(Ip (Fn , Fp ))(x, k, t) =
[g(x, k, k 0 )(1 − Fn0 )(1 − Fp ) − r(x, k 0 , k)Fn0 Fp ] dk 0 .
B
Here g(x, k, k 0 ) ≥ 0 is the rate of generation of an electron at the state (x, k) and
of a hole at the state (x, k 0 ), and r(x, k 0 , k) ≥ 0 is the analogous recombination
rate. Similarly as for the transition rates sn and sp (see (3.30)) the relation
µ
¶
εn (k) − εp (k 0 )
0
r(x, k, k ) = exp
g(x, k 0 , k)
kB T
is assumed to hold. Then the operators In and Ip can be written, in the lowdensity approximation, as follows:
·
µ
¶
¸
Z
εn (k) − εp (k 0 )
0
0
Fn Fp dk 0 ,
(In (Fn , Fp ))(x, k, t) =
g(x, k , k) 1 − exp
k
T
B
·
µ
¶
¸
ZB
εn (k 0 ) − εp (k)
0
0
(In (Fn , Fp ))(x, k, t) =
g(x, k, k ) 1 − exp
Fn Fp dk 0 .
k
T
B
B
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 charged
holes in the electric field Eeff .
In the case of Coulomb forces in R3 , the effective field is given by
Z
x − x∗
1
ρ(x∗ , t)
dx∗ ,
Eeff (x, t) =
4πεs R3
|x − x∗ |3
48
where ρ is the total space charge density. The charge density is the sum of the
electron density n, the hole density p and the densities ND , NA of the implanted
positively charged donor ions and the negatively charged acceptor ions, respectively, 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
Z
Z
n(x, t) =
Fn (x, k, t) dk, p(x, t) =
Fp (x, k, t) dk.
B
B
Thus the Poisson equation (3.21) for the electrostatic potential defined by E eff =
−∇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 Boltzmann equations (3.36) and (3.37) and integrating over x ∈ Rd and k ∈ B yields
Z
Z Z
(In (Fn , Fp ) − Ip (Fn , Fp )) dk dx = 0,
(n − p)(x, t) dx =
∂t
Rd
Rd
B
by Lemma 3.2(1) and by definition of In and Ip . If the doping profile C does not
depend on t, we obtain the conservation of the total charge density
Z
Z
(n(x, t) − p(x, t) − C(x)) dx = 0.
ρ(x, t) dx = ∂t
∂t
Rd
Rd
49
4
4.1
Quantum Kinetic Transport Equations
The Wigner equation
As the dimensions of a semiconductor device decrease, quantum mechanical transport 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 capable of describing quantum effects and, on the other hand, are sufficiently simple
to allow for efficient numerical simulations. In order to derive such models, we
start with the Schrödinger equation.
The motion of an electron ensemble consisting of M particles in a vacuum
under the action of an electric field E = −∇x V is described by the Schrödinger
equation
M
~2 X
i~∂t ψ = −
∆x ψ − qV (x, t)ψ,
2m j=1 j
ψ(x, 0) = ψI (x),
x ∈ RdM , t > 0,
(4.1)
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 electrostatic
potential, and ψ = ψ(x, t) is called the wave function of the electron ensemble.
Macroscopic variables are defined by
n = |ψ|2 ,
~q
electron current density: J = − Im(ψ∇x ψ),
m
electron density:
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 the
density matrix by
n(x, t) = ρ(x, x, t),
¢
i~q ¡
J(x, t) =
ψ∇x ψ − ψ∇x ψ (x, t)
2m
i~q
=
(∇s − ∇r )ρ(x, x, t).
2m
(4.3)
(4.4)
Differentiating (4.2) with respect to t and using the Schrödinger equation (4.1)
50
leads to the Heisenberg equation
³
´
i~∂t ρ(r, s, t) = i~ ∂t ψ(r, t)ψ(s, t) + ψ(r, t)∂t ψ(s, t)
µ 2
¶
~
∆r ψ + qV ψ (r, t)ψ(s, t)
=
2m
¶
µ
~2
∆s ψ − qV ψ (s, t)
(4.5)
+ ψ(r, t) −
2m
~2
(∆s ρ − ∆r ρ)(r, s, t) − q(V (s, t) − V (r, t))ρ(r, s, t),
= −
2m
with r, s ∈ RdM , t > 0.
The kinetic form of the quantum equations is obtained by a change of unknowns and a Fourier transformation. We recall that the Fourier transform F is
defined by
Z
1
(F g)(η) = ĝ(η) =
g(v)e−iη·v dv
(2π)dM/2 RdM
for (sufficiently smooth) functions g : RdM → C with inverse
Z
1
−1
(F h)(v) = ȟ(v) =
h(η)eiη·v dη
(2π)dM/2 RdM
for functions h : RdM → C. Introduce the change of coordinates
r =x+
and set
~
η,
2m
s=x−
~
η
2m
µ
¶
~
~
u(x, η, t) = ρ x +
η, x −
η, t .
2m
2m
(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 Fourier
transform of a function g(v) has the dimension of velocity cm/s. We compute
µ
¶
~
~
div η (∇x u)(x, η, t) = div η (∇r ρ + ∇s ρ) x +
η, x −
η, t
2m
2m
~
=
(∆r ρ − ∆s ρ)(r, s, t).
2m
Hence the transformed Heisenberg equation for u reads
∂t u + idiv η (∇x u) +
where
q
(δV )u = 0,
m
x, η ∈ RdM , t > 0,
· µ
¶
µ
¶¸
im
~
~
(δV )(x, η, t) =
V x+
η, t − V x −
η, t .
~
2m
2m
51
(4.7)
(4.8)
The initial condition is
µ
u(x, η, 0) = ψI
¶ µ
¶
~
~
x+
η ψI x −
η ,
2m
2m
x, η ∈ RdM .
We prefer to work with the variables velocity v and space x instead of inverse
velocity η and space x and apply therefore the Fourier transformation to u. The
function
w = (2π)−dM/2 F −1 u = (2π)−dM/2 ǔ
or, in terms of the wave function,
1
w(x, v, t) =
(2π)dM
Z
RdM
¶ µ
¶
~
~
η ψ x−
η eiη·v dη
ψ x+
2m
2m
µ
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
Z
n(x, t) =
w(x, v, t) dv,
(4.9)
RdM
Z
vw(x, v, t) dv,
(4.10)
J(x, t) = −q
RdM
since the first identity follows from (4.3) and
n(x, t) = ρ(x, x, t) = u(x, 0, t) = (2π)
dM/2
ŵ(x, 0, t) =
Z
RdM
w(x, v, t) dv,
and for the second identity we use the fact that the Fourier transform translates
differential operators into a multiplication:
Z
i
i(∇η ŵ)(x, η, t) =
w(x, v, t)∇η e−iη·v dv
(2π)dM/2 RdM
Z
1
w(x, v, t)ve−iη·v dv
=
(2π)dM/2 RdM
= vw(x,
c η, t),
and therefore, using (4.4),
i~q
(∇s − ∇r )ρ(x, x, t) = −iq(∇η u)(x, 0, t)
2m
= −iq(2π)dM/2 (∇η ŵ)(x, 0, t) = −q(2π)dM/2 vw(x,
c 0, t)
Z
vw(x, v, t) dv.
= −q
J(x, t) =
RdM
52
The integrals (4.9) and (4.10) are called the zeroth and first moments of the
Wigner function, respectively, in analogy to the classical situation (see Section
3.1).
The transport equation for w is obtained by taking the inverse Fourier transform of the equation (4.7) for u :
∂t ǔ + i(div η ∇x u)∨ +
q
[(δV ) u]∨ = 0.
m
(4.11)
Using
∨
i(div η ∇x u) (x, v, t) =
=
=
=
Z
i
(div η ∇x u(x, η, t))eiv·η dη
(2π)dM/2 RdM
Z
v
∇x u(x, η, t)eiv·η dη
·
(2π)dM/2 RdM
v · ∇x ǔ(x, v, t)
(2π)dM/2 v · ∇x w(x, v, t),
where we have employed integration by parts, and
(θ[V ]w) (x, v, t) :=
=
=
1
[(δV )u]∨ (x, v, t)
(4.12)
(2π)dM/2
Z
1
(δV )(x, η, t) u(x, η, t) eiη·v dη
(2π)dM RdM
Z
Z
1
0
(δV )(x, η, t) w(x, v 0 , t) eiη·(v−v ) dv 0 dη,
dM
(2π)
RdM RdM
it follows from (4.11)
∂t w + v · ∇ x w +
q
θ[V ]w = 0,
m
x, v ∈ RdM , t > 0.
(4.13)
This equation is called Wigner equation or quantum Liouville equation. The
initial condition for w reads
w(x, v, 0) = wI (x, v),
x, v ∈ RdM ,
(4.14)
where
1
wI (x, v) =
(2π)dM
Z
RdM
µ
¶ µ
¶
~
~
ψI x +
η ψI x −
η eiv·η dη,
2m
2m
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 transform of the function, is called a linear pseudo-differential operator [42]. Indeed,
it holds
\
θ[V
]w(x, η, t) = (δV )(x, η, t)ŵ(x, η, t),
53
and so, θ[V ] is a pseudo-differential operator and the Wigner equation (4.13) is
a 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 solution
of the Schrödinger equation, and
• the question of positivity of the solution of the Wigner equation.
It is interesting to perform the formal semi-classical limit “~ → 0”. We
compute, as “~ → 0”,
(δV )(x, η, t) → i∇x V (x, t) · η
and, using i(ηu)∨ = ∇v ǔ,
θ[V ]w → i(2π)−dM/2 [(∇x V · η) u]∨ = (2π)−dM/2 ∇x V · ∇v ǔ = ∇x V · ∇v w.
Therefore, (4.13) becomes, as “~ → 0”,
∂t w + v · ∇ x w +
q
∇x V · ∇v w = 0,
m
and we recover the classical Liouville equation (3.6). Clearly, the above limits
have to be understood in a formal way.
The solution of the initial-value problem (4.13)-(4.14) is given by
1
w(x, v, t) =
(2π)dM
Z
RdM
µ
¶ µ
¶
~
~
η, t ψ x −
η, t eiv·η dη,
ψ x+
2m
2m
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 wave
function ψ. In quantum physics this is referred to as a pure quantum state. For
more general initial data, we cannot expect that (4.16) holds true. However, we
can derive a similar expression. For this we choose the initial data
Z
n
o
2
dM
dM
dM
dM
wI ∈ L (R × R ) = w : R × R → C :
|w(x, v)|2 dx dv < ∞ .
R2dM
From functional analysis, there exists a complete orthonormal system (φ(k) )k∈N
of 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) = ŵI (x, η)
54
can be expanded in the series
ρI (r, s) =
X
λk φ(k) (r)φ(k) (s)
k∈N
with
λk =
Z
RdM
Z
RdM
ρI (r, s) φ(k) (r)φ(k) (s) dr ds.
Let ψ (k) be the solution of the Schrödinger equation
~2
∆x ψ (k) − qV ψ (k) ,
2m
ψ (k) (x, 0) = φ(k) (x), x ∈ RdM .
i~∂t ψ (k) = −
x ∈ RdM , t > 0,
Then the solution of the Wigner equation (4.13)-(4.14) is given by
µ
¶
¶
µ
Z
X
~
1
~
(k)
(k)
x−
η, t ψ
η, t eiv·η dη,
w(x, v, t) =
λk
ψ
x+
(2π)dM k∈N
2m
2m
dM
R
for t > 0. This means that the Wigner equation is capable of describing so-called
mixed quantum states.
The electron density is
Z
n(x, t) =
w(x, v, t) dv
RdM
dM/2
= (2π)
ŵ(x, 0, t)
"
¶
¶#
µ
µ
X
1
~
~
(k)
=
λk ψ (k) x +
η, t ψ
η, t
x−
(2π)dM/2 k∈N
2m
2m
η=0
X
λk
=
|ψ (k) (x, t)|2 .
dM/2
(2π)
k∈N
If the initial electron density is non-negative or, more precisely, if λk ≥ 0 for all
k ∈ N, the electron density is non-negative for all t > 0. One may ask if also
the Wigner function is non-negative for all t > 0 if this is true initially. This
would 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,
1
w(x, v, t) =
(2π)dM
Z
RdM
µ
¶ µ
¶
~
~
ψ x+
η, t ψ x −
η, t eiv·η dη
2m
2m
is non-negative if and only if either ψ ≡ 0 or
¡
¢
ψ(x, t) = exp −x> A(t)x − a(t) · x + b(t) ,
55
x ∈ RdM , t > 0,
where A(t) is a CdM ×dM matrix with symmetric positive definite real part and
a(t) ∈ CdM , b(t) ∈ C. A necessary condition for the non-negativity of w for mixed
quantum states is not known.
The Wigner equation (4.13) does not account for the effect of the crystal
lattice on the motion of the electron ensemble. The motion of one electron in the
crystal is described by the Schrödinger equation
i~∂t ψ = −
~2
∆x ψ − q(VL + V )ψ,
2m
x ∈ R3 , t > 0,
where VL is the lattice potential satisfying
VL (x + γ a~j ) = VL (x) x ∈ R3 , j = 1, 2, 3,
with the primitive vectors a~j of the lattice and the lattice length scale γ (see
Section 2.1). By using a Bloch decomposition and performing the limit γ → 0,
it is possible to derive the Wigner equation in a crystal:
q
∂t w + v(k) · ∇x w + θ[V ]w = 0,
~
(4.17)
where v(k) = (1/~)∇k ε(k) and ε(k) is the energy band in which the electrons
are moving. The pseudo-differential operator θ[V ] is defined as in (4.12), and
the function w is the inverse of the Fourier transform of the density matrix
transformed from (r, s) to (x, η) as in (4.6). We refer to [6, 38] for details on
the derivation.
4.2
The quantum Vlasov and quantum Boltzmann equation
The quantum Liouville equation has the same disadvantages as its classical analogue:
• The equation has to be solved in a very high-dimensional phase space since
M 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 of
dimension 2d. We proceed similarly as in Section 3.2. In particular, we impose
the following assumptions.
Consider an ensemble of M electrons with mass m in a vacuum under the
action of the real-valued electrostatic potential V (x, t), x ∈ RdM , t > 0. The
motion of the particle ensemble is described by the wave function ψ which is a
solution of the many-particle Schrödinger 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
M
X
M
1X
Vext (xj , t) +
V (x1 , . . . , xM , t) =
Vint (xi , xj ),
2 i,j=1
j=1
(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 12 in (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 is
conserved 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 the
fact that the electrons are indistinguishable.
It is possible to use, instead of (4.18), the condition (4.20) as a hypothesis
which 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 wave
function is anti-symmetric or symmetric. The anti-symmetry property represents
the 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 the
slightly stronger condition (4.18) as a hypothesis.
57
We wish to model the evolution of subensembles. The density matrix of a
subensemble of particles is defined by
Z
(a) (a) (a)
ρ(r(a) , ua+1 , . . . , uM , s(a) , ua+1 , . . . , uM ) dua+1 . . . duM ,
ρ (r , s , t) =
Rd(M −a)
where
r(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 the
complete electron ensemble is governed by the Heisenberg equation (see (4.5))
M
M
X
¢
~2 X ¡
i~∂t ρ = −
∆ sj − ∆ r j ρ − q
(Vext (sj , t) − Vext (rj , t))ρ
2m j=1
j=1
M
q X
(Vint (sj , s` ) − Vint (rj , r` )) ρ.
−
2 j,`=1
We set uj = sj = rj for j = a + 1, . . . , M in the above equation, integrate
over (ua+1 , . . . , uM ) ∈ Rd(M −a) and use the indistinguishability property (4.21) to
obtain, after an analogous calculation as in Section 3.2, a quantum equivalent of
the BBGKY hierarchy:
i~∂t ρ
(a)
M
M
X
¢ (a)
~2 X ¡
= −
(Vext (sj , t) − Vext (rj , t))ρ(a)
∆ sj − ∆ r j ρ − q
2m j=1
j=1
Z
a
X
−q(M − a)
du∗ ,
(Vint (sj , u∗ ) − Vint (rj , u∗ )) ρ(a+1)
∗
j=1
Rd
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 get
for fixed a ≥ 1 and M À 1, neglecting terms of order 1/M,
i~∂t ρ
(a)
M
M
X
¢ (a)
~2 X ¡
∆ sj − ∆ r j ρ − q
(Vext (sj , t) − Vext (rj , t)) ρ(a)
= −
2m j=1
j=1
Z
a
X
−q
(Vint (sj , u∗ ) − Vint (rj , u∗ )) M ρ(a+1)
du∗ .
∗
j=1
Rd
58
Similarly to the classical case, a particular solution of this equation is given by
the so-called Hartree ansatz
ρ(a) (r1 , . . . , ra , s1 , . . . , sa , t) =
a
Y
R(rj , sj , t),
j=1
where R solves the equation
i~∂t R = −
~2
(∆s − ∆r )R − q(Veff (s, t) − Veff (r, t))R,
2m
r, s ∈ Rd , t > 0, (4.22)
with the effective potential
Veff (x, t) = Vext (s, t) +
Z
Rd
M R(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+
~
η,
2m
s=x−
~
η
2m
and set U (x, η, t) = M R(r, s, t). Then U solves the equation
µ
µ
µ
¶
¶¶
~
~
q
Veff x +
η, t − Veff x −
η
U = 0.
∂t U + idiv η ∇x U + i
~
2m
2m
Finally, the inverse Fourier transform W = (2π)−d/2 Ǔ is a solution of the quantum
Vlasov equation
∂t W + v · ∇ x W +
q
θ[Veff ]W = 0,
m
x, v ∈ Rd , t > 0.
(4.23)
The pseudo-differential operator θ[Veff ] is defined as in (4.12), and the effective
potential is
Z
n(z, t)Vint (x, z) dz,
(4.24)
Veff (x, t) = Vext (x, t) +
Rd
where
n(x, t) =
Z
Rd
W (x, v, t) dv = U (x, 0, t) = M R(x, x, t)
is the quantum electron number density. As the effective potential depends on
the Wigner function W, it is a nonlinear pseudo-differential equation.
Contrary to the classical Vlasov equation, the quantum Vlasov equation does
not preserve the non-negativity of the solution W (cf. the discussion in Section
4.1). However, the number density n remains non-negative for all times, if the
initial single-particle density matrix R(r, s, 0) is positive semi-definite.
59
In the semi-classical limit “~ → 0” the quantum Vlasov equation formally
converges to the classical Vlasov equation
∂t W + v · ∇ x W +
q
∇x Veff · ∇v W = 0.
m
In semiconductor modelling, a usual choice for Vint is the Coulomb potential
Vint (x, y) = −
q
1
,
4πεs |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
Z
n(z, t)
q
dz
Veff (x, t) = Vext (x, t) −
4πεs R3 |z − x|
solves the Poisson equation
εs ∆Veff = q(n − C),
where
C(x) = −
εs
Vext (x)
q
is the doping concentration if Vext is generated by ions of charge +q in the semiconductor material.
The presented quantum Vlasov equation models the motion of an ensemble of
many 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 solved
in a bounded domain with appropriate boundary conditions at the device
contacts and insulating surfaces.
• Short-range interactions modelled by scattering events of particles have to
be included in the model.
For the first two generalizations, we refer to [34, Ch. 1] (also see (4.17)). The
quantum mechanical modelling of collisions of electrons (with phonons, for instance) is a very difficult task. One approach is to formulate the so-called quantum
Boltzmann equation
∂t W + v · ∇ x W +
q
θ[Veff ]W = Q(W ),
m
60
x, v ∈ Rd , t > 0,
by adding a heuristic collision term to the right-hand side of the quantum Vlasov
equation (4.23). In numerical studies, the relaxation-time model
¶
µ
1 n
Q(W ) =
W0 − W ,
τ n0
or the Fokker-Planck model
1
Q(W ) = div v
τ
µ
k B T0
∇v W + vW
m
is often used, where τ is the relaxation time,
Z
Z
W (x, v, t) dv, n0 (x) =
n(x, t) =
Rd
Rd
¶
,
W0 (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 energyband quantum Boltzmann equation
∂t W + v(k) · ∇x W +
q
θ[Veff ]W = Q(W ),
m
(4.25)
the above collision models read as follows: The relaxation-time collision term is
¶
µ
1 n
Q(W ) =
W0 − W ,
τ n0
where now
n(x, t) =
Z
Rd
W (x, k, t) dk,
n0 (x) =
Z
Rd
W0 (x, k) dk,
and the Fokker-Planck terms is given by
µ
¶
1
mkB T0
Q(W ) = div k
∇k W + kW .
τ
~2
(4.26)
A summary of the kinetic models derived in this and the previous section is
presented in Figure 4.1. Notice that for each model, there is an energy-band
version in the (x, k, t) variables which reduces to a model in the (x, v, t) variables
in the parabolic band approximation.
61
no two-particle
Liouville ¾
interactions
6
¾
no collisions
6
“~ → 0”
quantum
Liouville
Vlasov
“~ → 0”
¾no two-particle
interactions
quantum
Vlasov
¾ no collisions
Boltzmann
“~ → 0”
6
quantum
Boltzmann
Figure 4.1: Relations between the classical and quantum kinetic equations.
5
5.1
From Kinetic to Fluiddynamical Models
The drift-diffusion equations: first derivation
We derive the drift-diffusion equations (formally) from the bipolar Boltzmann
equations
q
∂t fn + vn (k) · ∇x fn − E · ∇k fn = Qn (fn ) + In (fn , fp ),
~
q
∂t fp + vp (k) · ∇x fp + E · ∇k fp = Qp (fp ) + Ip (fn , fp )
~
(5.1)
(5.2)
with the low-density collision operators
Z
(Qn (fn ))(x, k, t) =
φn (x, k, k 0 )
B
¶
µ
¶ ¸
·
µ
εn (k 0 )
εn (k)
0
fn − exp −
fn dk 0 , (5.3)
× exp −
kB T
kB T
Z
(Qp (fp ))(x, k, t) =
φp (x, k, k 0 )
B
·
µ
¶
µ
¶ ¸
εp (k)
εp (k 0 )
0
× exp −
fp − exp −
fp dk 0 , (5.4)
kB T
kB T
and the recombination-generation rates
Z
(In (fn , fp ))(x, k, t) = −
g(x, k, k 0 )
·B µ
¶
¸
εn (k) − εp (k 0 )
0
× exp
fn fp − 1 dk 0 ,
kB T
Z
(Ip (fn , fp ))(x, k, t) = −
g(x, k, k 0 )
·B µ
¶
¸
εn (k 0 ) − εp (k)
0
× exp
fn fp − 1 dk 0 .
kB T
62
(5.5)
(5.6)
Here, fj0 means evaluation at k 0 , i.e. fj0 = fj (x, k 0 , t), j = n, p, and B denotes the
Brillouin zone. We assume the parabolic band approximation
εn (k) = εc +
~2
|k|2 ,
2mn
εp (k) = εv −
~2
|k|2 ,
2mp
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
1
~k
∇k εn (k) =
,
~
mn
~k
1
.
vp (k) = − ∇k εp (k) =
~
mp
vn (k) =
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. In
order 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 holespare of the same order such
that we can define the reference velocity v̄ = kB T /mn . This means that
we assume that the thermal energy kB T is of the same order as the kinetic
energy mn v̄ 2 /2.
(2) Collisions occur on a much shorter time scale than recombination-generation
events. Thus, scaling
1
Qns ,
τc
1
Qps ,
=
τc
Qn =
Qp
1
Ins ,
τR
1
Ip = Ips
τR
In =
(5.7)
(5.8)
with the collision relaxation time τc and the recombination-generation relaxation time τR , we assume that
τ c ¿ τR .
This is our main assumption.
(3) The reference length ι0 is given by the geometric average of the mean free
paths ιc := τc v̄ and ιR := τR v̄:
√
ι0 = ι R ιc .
The mean free path is the average time between two successive collisions.
63
(4) We use the reference time τR , the reference wave vector mn v̄/~, and the
reference field strength kB T /qι0 :
t = τ R ts ,
k=
mn v̄
ks ,
~
E=
kB T
Es .
ι0 q
(5.9)
By assumption (2), the parameter α2 := ιc /ιR = τc /τR satisfies α2 ¿ 1. With
the scaling (5.7)-(5.9) we can rewrite (5.1) (omitting the index s):
1
v̄
kB T
1
1
∂t f n + k · ∇ x f n −
E · ∇k fn = Qn (fn ) + In (fn , fp ).
τR
ι0
ι0 mn v̄
τc
τR
Multiplying this equation by τc = ιc /v̄ and using α = ιc /ι0 and kB T = mn v̄ 2 , we
obtain
α2 ∂t fn + α(k · ∇x fn − E · ∇k fn ) = Qn (fn ) + α2 In (fn , fp ).
(5.10)
In a similar way, we have
³m
´
n
2
k · ∇x fp − E · ∇k fp = Qp (fp ) + α2 Ip (fn , fp ).
α ∂t f p + α
mp
(5.11)
The scaled collision and recombination-generation terms have the same form as
(5.3)-(5.6) but the exponential terms
µ
¶
µ
¶
εn (k)
εp (k)
exp −
and exp −
kB T
kB T
are replaced by
µ
|k|2
εc
−
exp −
kB T
2
¶
µ
mn |k|2
εv
+
exp −
kB T
mp 2
and
¶
and the rates φn , φp and g are multiplied by (mn v̄/~)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
Z
£
¤
Qj (fj ) =
Φj (x, k, k 0 ) Mj (k)fj0 − Mj (k 0 )fj dk 0 , j = n, p,
(5.12)
Rd
where
Φj (x, k, k 0 ) = Nj φn (x, k, k 0 ),
and
µ
¶
|k|2
1
exp −
,
Mn (k) =
Nn
2
j = n, p,
µ
¶
mn |k|2
1
exp −
Mp (k) =
Np
mp 2
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 to
one.
For the following analysis define the functions
Z
λj (k) =
Φj (x, k, k 0 )Mj (k 0 ) dk 0 , k ∈ Rd , j = n, p,
Rd
for fixed x ∈ Rd , and the Banach spaces
Xj = {f : Rd → R measurable: kf kXj < ∞},
Yj = {f : Rd → R measurable: kf kYj < ∞}
with associated norms
kf kXj =
kf kYj =
µZ
Rd
µZ
Rd
2
f (k) λj (k)Mj (k)
2
−1
f (k) (λj (k)Mj (k))
dk
−1
¶1/2
dk
,
¶1/2
.
Lemma 5.1 Let j = n or j = p, let Φn , Φp > 0 be symmetric in k and k 0 and
assume that for all isometric matrices A it holds
Φ(x, Ak, Ak 0 ) = Φ(x, k, k 0 ) ∀x, k, k 0 ∈ 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
Z
g(k) dk = 0.
Rd
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
Z
f (k)λj (k)dk = 0
(5.13)
Rd
holds.
65
(4) The equations
Qn (hni ) = ki Mn (k),
Qp (hpi ) =
mn
ki Mp (k),
mp
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
Z
k ⊗ hn dk = −µn (x) Id,
(5.14)
Rd
Z
mn
k ⊗ hp dk = −µp (x) Id,
(5.15)
Rd m p
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 assumptions
on Φn and Φp (regularity conditions) which are particularly needed in the proof of
Lemma 5.1(4). We only sketch the proof of part (3) since properties of so-called
Hilbert-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 )1/2 f and
Qs (fs ) = (λM )−1/2 Q(f ). Then
µ Z
¶
−1/2
0
0
0
M
Qs (fs ) = (λM )
Φ(x, k, k )f dk − λf
=
Z
Rd
0
Rd
Φ(x, k, k )
µ
MM0
λλ0
¶1/2
fs0 dk 0 − fs ,
where M 0 = M (k 0 ) and λ0 = λ(k 0 ). Since Φ is symmetric in k and k 0 by assumption, the operator Qs : L2 (Rd ) → L2 (Rd ) is self-adjoint.
Next we analyze the kernel of Qs . By definition of λ we have
Z
Φ(x, k, k 0 )M (k 0 )/λ(k) dk 0 = 1
Rd
66
and therefore
0 ≤
=
=
=
µ
¶2
Z Z
1
fs
fs0
0
0
Φ(x, k, k )M M √
dk dk 0
−√
0
0
2 Rd Rd
λM
λM
¶
Z µZ
0
M
1
Φ(x, k, k 0 )
dk 0 fs2 dk
2 Rd
λ
Rd
¶
Z
Z µ
1
0 M
Φ(x, k, k ) 0 dk (fs0 )2 dk 0
+
2 Rd
λ
Rd
µ
¶1/2
Z Z
MM0
0
Φ(x, k, k )
−
fs0 fs dk dk 0
0
λλ
Rd Rd
!
µ
¶
Z
Z Ã
0 1/2
MM
Φ(x, k, k 0 )
fs0 fs dk 0 dk
fs2 −
0
λλ
Rd
Rd
Z
Qs (fs )fs dk.
−
Rd
Hence Qs (fs ) = 0 implies
f
f0
√ s −√ s =0
λM
λ0 M 0
for k, k 0 ∈ 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 some
σ ∈ R then the formulation (5.12) of Q(f ) immediately yields Q(f ) = 0. Thus
N (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 )⊥
2
d
⇔ ∀h ∈ L (R ), Qs (h) = 0 :
Z
Rd
gs h dk = 0.
The equivalence between the solvability of Qs (fs ) = g and g ∈ N (Q∗s )⊥ is also
known as the
√ Fredholm alternative [44, Appendix, (39)]. As the kernel of Q s is
spanned by λM , Qs (fs ) = gs has a solution if and only if
Z
√
gs λM dk = 0
Rd
67
or, in the original variables,
0=
Z
√
Rd
gs λM dk =
Z
Rd
g dk.
(3) We only give a sketch of the proof and refer to [37] for details. It is possible
to show that the operator −Q is coercive in the following sense:
Z
(−Q(f ))(k)f (k)M (k)−1 dk ≥ ckf k2X
(5.16)
Rd
for all f ∈ X satisfying (5.13). Clearly, this implies that Q is one-to-one on the
subset of functions satisfying (5.13), and the uniqueness is proved. In order to
show the coerciveness property prove that Id + Q is a Hilbert-Schmidt operator
and use general properties of those operators (see [37]).
(4) The existence of solutions hi of (Q(hi ))(k) = ki M (k) follows from part (2)
since
Z
ki Mj (k) dk = 0 for i = 1, . . . , d, j = n, p.
Rd
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)1 M (Ak) = k1 N −1 exp(−|Ak|2 /2) = k1 N −1 exp(−|k|2 /2) = k1 M (k), (5.17)
since A is isometric. We have for all k ∈ Rd , using the assumption Φ(x, Ak, Ak 0 ) =
Φ(x, k, k 0 ) and (5.17),
Z
(Q(h1 ◦ A))(k) =
Φ(x, k, k 0 )[M (k)h1 (Ak 0 ) − M (k 0 )h1 (Ak)] dk 0
d
R
Z
=
Φ(x, Ak, Ak 0 )[M (Ak)h1 (Ak 0 ) − M (Ak 0 )h1 (Ak)] dk 0
d
ZR
=
Φ(x, Ak, w)[M (Ak)h1 (w) − M (w)h1 (Ak)] dw
=
=
=
=
Rd
(Q(h1 ))(Ak)
(Ak)1 M (Ak)
k1 M (k)
(Q(h1 ))(k)
68
and thus Q(h1 ◦ A − h1 ) = 0. Another computation leads to
Z
Z Z
h1 (Ak)λ(k) dk =
Φ(x, k, k 0 )h1 (Ak)M (k 0 ) dk 0 dk
d
d
d
R
ZR ZR
=
Φ(x, Ak, Ak 0 )h1 (Ak)M (Ak 0 ) dk 0 dk
d
d
ZR ZR
Φ(x, v, w)h1 (v)M (w) dw dv
=
d
d
ZR R
h1 (v)λ(v) dv
=
Rd
or
Z
Rd
(h1 ◦ A − h1 )(k)λ(k) dk = 0.
This is the orthogonality condition (5.13) which ensures the uniqueness of the
solution of the equation Q(h1 ◦ A − h1 ) = 0. Therefore, h1 ◦ A − h1 = 0. We
conclude that h1 remains invariant under a rotation with axis k1 , and we can
write h1 as
h1 (k) = H(k1 , |k|2 − k12 ).
Now let A be the isometric matrix of the mapping k 7→ (−k1 , k2 , . . . , kd ).
Since k 7→ k1 M (k) is an odd function, a similar computation as above yields
Q(h1 ◦ A) = −Q(h1 ) and
Z
(h1 ◦ A + h1 )λ(k) dk = 0.
Rd
This implies as above that h1 ◦ A + h1 = 0 and thus, h1 is an odd function with
respect to k1 .
In a similar way, we can show that
hi (k) = Hi (ki , |k|2 − ki2 ),
i = 2, . . . , d,
and Hi are odd functions with respect to ki . In fact all the functions Hi equal H
since, for instance, exchanging k1 and k2 in
Q(H(k1 , k22 + · · · + kd2 )) = k1 M (k12 + · · · + kd2 )
(with slight abuse of notation) leads to
Q(H(k2 , k12 + k32 + · · · + kd2 )) = k2 M (k12 + · · · + kd2 ) = 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 − kj2 does not
depend on kj , we obtain for all i 6= j
Z
Z
ki H(kj , |k|2 − kj2 )dk = 0.
ki hj (k)dk =
Rd
Rd
69
Furthermore
Z
Rd
ki hi (k)dk =
=
=
Z
Z
Z
Rd
ki H(ki , |k|2 − ki2 )dk
Rd
kj H(kj , |k|2 − kj2 )dk
Rd
kj hj (k)dk,
for all i and j. Thus, the above integral is independent of i and we can set
Z
k1 h1 (k)dk.
µ := −
Rd
The constant µ depends on the parameter x since h1 depends on x through Q.
Moreover, by (5.16),
Z
(Q(h1 ))(k)h1 (k)M (k)−1 dk ≥ 0,
µ(x) = −
Rd
and this implies
The theorem is proved.
Z
Rd
k ⊗ h(k) dk = −µ(x) Id.
¤
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
Z
Z
fn0 dk = n(x, t),
fp0 dk = p(x, t),
Rd
Rd
we interpret n and p as the scaled number densities of electrons and holes, respectively. In order to obtain more informations from (5.10)-(5.11) we use the
Hilbert expansion method. The idea is to expand fn and fp in terms of powers of
α,
fn = fn0 + αfn1 + α2 fn2 + · · · ,
fp = fp0 + αfp1 + α2 fp2 + · · · ,
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 are
linear)
70
• for the terms of order 1:
Qn (fn0 ) = 0,
Qp (fp0 ) = 0;
(5.19)
• for the terms of order α:
k · ∇x fn0 − E · ∇k fn0 = Qn (fn1 ),
k · ∇x fp0 + E · ∇k fp0 = Qn (fp1 );
(5.20)
(5.21)
• and for the terms of order α2 :
∂t fn0 + k · ∇x fn1 − E · ∇k fn1 = Qn (fn2 ) + In (fn0 , fp0 ),
∂t fp0 + k · ∇x fp1 + E · ∇k fp1 = Qp (fp2 ) + Ip (fn0 , fp0 ).
(5.22)
(5.23)
We already solved (5.19). By (5.18) and ∇k Mj (k) = −kMj (k) (j = n, p) we
can write (5.20)-(5.21) as
Qn (fn1 ) = Mn k · (∇x n + nE),
mn
Qp (fp1 ) =
Mp k · (∇x p − nE).
mp
Lemma 5.1(2) shows that these equations have solutions and that any solution
can be expressed as
fn1 = (∇x n + nE) · hn + σn Mn ,
fp1 = (∇x p − pE) · hp + σp Mp
for some unspecified parameters σn (x, t), σp (x, t). It is convenient to define
Jn (x, t) = µn (∇x n + nE),
Jp (x, t) = −µp (∇x p − pE),
(5.24)
where µn µp are introduced in Lemma 5.1(4). We will see below that Jn and Jp
can be interpreted as scaled current densities.
The equation (5.22) is solvable, by Lemma 5.1(2), if and only if
Z
0 =
(∂t fn0 + k · ∇x fn1 − div k (Efn1 ) − In (fn0 , fp0 )) dk
Rd
Z
Z
= ∂t n +
k · ∇x fn1 dk −
In (fn0 , fp0 ) dk.
Rd
From (5.14) follows that
Z
Z
k · ∇x fn1 dk =
Rd
Rd
d
X
Rd
k · ∇x (µ−1
n Jn · hn + σn Mn ) dk
∂
=
∂xi
i,j=1
µ
Z
−1
µn Jnj
d
X
∂
Jnj δij
= −
∂x
i
i,j=1
= −div x Jn .
71
Rd
¶
ki hnj dk + ∇x σn ·
Z
Rd
kMn (k) dk
Furthermore
Z
In (fn0 , fp0 )
R := −
Rd
µ
¶
µ 2
¶
Z Z
h
εc − ε v
|k|
mn |k 0 |2
0
g(x, k, k ) exp
=
+
exp
kB T
2
mp 2
Rd Rd
i
× npMn (k)Mp (k 0 ) − 1 dk 0 dk
·
µ
¶
¸
Z Z
εc − ε v
np
0
g(x, k, k ) exp
=
− 1 dk 0 dk
kB T
Nn Np
Rd Rd
2
= A(x)(np − ni ),
where
1
A(x) = 2
ni
and
ni =
p
Z
Rd
Z
g(x, k, k 0 ) dk dk 0
Rd
Nn Np exp
µ
εv − ε c
2kB T
¶
is the scaled intrinsic density. We conclude that (5.22) and (by a similar computation) (5.23) are solvable if and only if
∂t n − div x Jn = −R,
∂t p + div x Jp = −R.
(5.25)
Here we can see that the quantities Jn and Jp can be indeed interpreted as current
densities. In order to scale back to the physical variables we notice that the scaled
number densities, now called ns and ps , are obtained from
¶d Z
µ
Z
~
~d n
,
ns =
fn0 dks =
fn0 dk =
mn v̄
(kB T mn )d/2
Rd
Rd
Z
~d p
,
ps =
fp0 dks =
(kB T mn )d/2
Rd
where n and p are now the unscaled variables. Thus, using
ts =
t
,
τR
Es =
ι0
E,
UT
xs =
x
,
ι0
µns =
mn
µn ,
τc q
we get from (5.24)-(5.25) after some computations:
∂t n − div x (µn UT ∇x n + µn nE) = −A(np − n2i ),
where UT = kB T /q,
~d
As = A and
(kB T mn )d/2
72
(kB T mn )d/2 2
nis = n2i .
~d
In particular, the unscaled intrinsic density reads (cf. (2.20))
¶
µ
¶
µ
√
2πkB T mn mp d
εv − ε c
ni =
.
exp
~2
2kB T
The unscaled current density and the evolution equation for n are given by
1
(5.26)
∂t n − div x Jn = −R, Jn = qµn (UT ∇x n + nE).
q
Similarly, we can compute,
1
∂t p + div x Jp = −R, Jp = −qµp (UT ∇x p − pE).
(5.27)
q
The equations (5.26)-(5.27) are referred to as the drift-diffusion equations for
given electric field. For a selfconsistent treatment of the electric field, (5.26)(5.27) have to be supplemented by the Poisson equation
−div (εs E) = 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 coefficients
Dn := µn UT , Dp := µp UT
and mobilities µn , µp . The quotient of diffusivity and mobility is constant, and
the equations
Dp
Dn
=
= UT
µn
µp
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 to
Chapter 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 following
hypotheses:
• The free mean path ιc between two consecutive scattering events is much
smaller than the free mean path ιR between two recombination-generation
events (typically, ιc ∼ 10−7 m, ιR ∼ 10−4 m).
√
• The device diameter is of the order of ι0 = ιR ιc (typically, ι0 ∼ 10−5 m).
• The electrostatic potential is of the order of UT = 0.026 V (at T = 300 K).
Thus, the drift-diffusion model is appropriate for semiconductor devices with
characteristic lengths not smaller than 1 . . . 10 µm and applied voltages much
smaller than 1 V. However, in application this model is used also for higher applied
voltages. It gives reasonable results as long as the characteristic length is not
much smaller than 1 µm.
73
5.2
The drift-diffusion equations: second derivation
Using a relaxation approximation of the collision operator in the Boltzmann equationm we can give a much shorter derivation of the drift-diffusion model than presented in Section 5.1. The derivation combines the Hilbert expansion method and
the moment method. For simplicity, we start with the unipolar classical Boltzmann equation in the diffusion scale (5.10) neglecting recombination-generation
effects:
´
³
q
2
(5.28)
α ∂t f + α v · ∇x f − E · ∇v f = Q(f ), x, v ∈ Rd , t > 0,
m
where E = E(x, t) is the electric field and α > 0 is a parameter which is small
compared to one. We assume a low-density collision operator Q(f ) with a scattering rate independent of v and v 0 (see (5.12)):
Z
(Q(f ))(x, v, t) =
φ(x) [M (v)f (x, v 0 , t) − M (v 0 )f (x, v, t)] dv 0 ,
(5.29)
Rd
where
M (v) =
µ
m
2πkB T
¶d/2
µ
¶
m|v|2
exp −
2kB T
is the Maxwellian. We introduce the average of f in the velocity space by
Z
[f ](x, t) =
f (x, v, t) dv.
Rd
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 termed
relaxation-time operator (also see (3.35)).
We use the Hilbert expansion
f = f0 + αf1 + α2 f2 + · · ·
in the Boltzmann equation (5.28) and identify terms of the same order of α:
• terms of order α0 :
• terms of order α1 :
v · ∇ x f0 −
M (v)[f0 ] − f0 = 0,
(5.31)
q
M (v)[f1 ] − f1
E · ∇ v f0 =
,
m
τ (x)
(5.32)
74
• terms of order α2 :
∂t f 0 + v · ∇ x f 1 −
M (v)[f2 ] − f2
q
E · ∇ v f1 =
.
m
τ (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 integrating
over v ∈ Rd implies
Z
J := −q
vf1 dv
RdZ
Z
´
³
q
= qτ (x)
M (v)v dv
v · ∇x f0 − E · ∇v f0 v dv + q[f1 ]
m
Rd
Rd
µZ
¶
Z
q
= qτ (x)
v ⊗ ∇v M (v) dv · En .
v ⊗ vM (v) dv · ∇x n −
m Rd
Rd
Since
Z
for i 6= j,
Z
kB T
kB T
2
2 −z 2 /2
vi M (v) dv =
z
e
dz
=
,
(2π)1/2 m R
m
Rd
Z
Z
m
v ⊗ vM (v) dv = −Id,
v ⊗ ∇v M (v) dv = −
k B T Rd
Rd
we have
Rd
vi vj M (v) dv = 0
Z
q 2 τ (x)
J=
m
µ
kB T
∇x n + nE
q
¶
= qµn (UT ∇x n + nE),
where µn = qτ /m is the electron mobility and UT = kB T /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 :
Z
Z
Z
q
1
∂t n +
v · ∇x f1 dv − E ·
∇v f1 dv =
(M (v)[f2 ] − f2 ) dv.
m
τ (x) Rd
Rd
Rd
The third term of the left-hand side vanishes, by the divergence theorem. The
integral of the right-hand side also vanishes. Finally, the second term on the lefthand side equals −(1/q)div J. Thus we have derived the drift-diffusion equations
1
∂t n − div J = 0
q
J = qµn (UT ∇x n + nE).
75
5.3
The hydrodynamic equations
We derive the hydrodynamic model from the Boltzmann equation by employing the so-called moment method. The idea of this method is to multiply the
Boltzmann equation by powers of the velocity components, to integrate over the
velocity space and to derive evolution equations for the integrals. We consider
the classical Boltzmann equation for one type of charge carriers (say, electrons):
∂t f + v · ∇ x f −
q
E · ∇v f = Q(f )
mn
(5.34)
with the low-density collision operator
Z
Q(f ) =
φ(x, v, v 0 )(M f 0 − M 0 f ) dv 0 ,
Rd
where the Maxwellian is given by
M (v) =
µ
mn
2πkB T0
¶d/2
µ
mn |v|2
exp −
2kB T0
¶
(5.35)
and f 0 and M 0 means evaluation at v 0 . We assume that the scattering rate φ
is symmetric in v and v 0 . The first moments of the distribution function f are
defined by
Z
f (x, v, t) dv,
h1i(x, t) =
Rd
Z
hvj i(x, t) =
vj f (x, v, t) dv,
d
ZR
hvi vj i(x, t) =
vi vj f (x, v, t) dv, i, j = 1, . . . , d,
Rd
and so on. We write
hvi = (hvj i)j ,
hv 2 i = hv ⊗ vi = (hvi vj i)ij ,
hv 3 i = (hvi vj vk i)ijk .
Multiply (5.34) by powers of vj and integrate over v ∈ Rd . This leads to the
moment equations
Z
Q(f ) dv,
∂t h1i + div x hvi =
Rd
Z
q
2
∂t hvi + div x hv i +
v Q(f ) dv,
h1iE =
mn
Rd
Z
2q
2
3
∂t hv i + div x hv i +
v ⊗ v Q(f ) dv,
hvi ⊗ E =
mn
Rd
76
and so on. The symmetry of φ implies
¶
Z
Z µZ
0
Q(f ) dv =
φ(x, v, v )M (v) dv f 0 dv 0
d
d
d
R
R
¶
Z µRZ
0
0
0
φ(x, v, v )M (v ) dv f dv
−
= 0.
Rd
Rd
The moments are related to physical quantities. In fact, we define the number
density n, the current density J, the energy tensor E, and the energy e by
n = h1i,
E=
mn 1
hv ⊗ vi,
2 n
J = −qhvi,
d
mn 1 X
mn 1
e=
h|v|2 i.
hvj vj i =
2 n j=1
2 n
With these definitions we obtain the conservation laws of mass, momentum and
energy:
1
∂t n − div x J = 0,
q
Z
2q
q2
∂t J −
div x (nE) −
nE = −q
v Q(f ) dv,
mn
mn
Rd
Z
mn
mn
div x hv|v|2 i − J · E =
|v|2 Q(f ) dv.
∂t (ne) +
2
2 Rd
(5.36)
(5.37)
(5.38)
The moment method has two difficulties. First, a truncation of the hierarchy
of moment equations does not give a closed system, i.e., the equation for the j-th
moment contains always a moment of order j + 1. Second, the terms originating
from the collision operator generally do not depend on the moments in a simple
way. The first difficulty can be overcome by making an ansatz for the distribution
function. In order to deal with the integrals involving the collision operator, we
choose a particular scattering rate φ.
Lemma 5.1 shows that the Maxwellian (5.35) lies in the kernel of Q(f ): The
ansatz f = nM would yield J = −qhvi = 0. More interesting equations can be
obtained from the so-called shifted Maxwellian
fe (x, v, t) = n
µ
mn
2πkB T
¶d/2
µ
mn |v − v̄|2
exp −
2kB T
¶
,
where n = n(x, t), T = T (x, t) and v̄ = v̄(x, t) are interpreted as the electron
number density, electron temperature and mean velocity, respectively. Substitut-
77
ing z =
p
mn /kB T (v − v̄) and using the identities
Z
√
2
e−x /2 dx = 2π,
ZR
2
xe−x /2 dx = 0,
ZR
Z
√
2
2 −x2 /2
xe
dx =
1 · e−x /2 dx = 2π,
R
ZR
Z
2
2
3 −x /2
xe
dx = 2 xe−x /2 dx = 0,
R
R
the moments corresponding to the above ansatz are
Z
fe dv = n,
h1i =
Rd
!
r
Z
Z Ã
kB T
n
2
hvi =
v fe dv =
v̄ +
z
e−|z| /2 dz = nv̄,
d/2
mn
(2π)
Rd
Rd
hv ⊗ vi =
Z
Rd
Ã
v̄ +
r
kB T
z
mn
!
⊗
Ã
v̄ +
r
kB T
z
mn
!
n
2
e−|z| /2 dz
d/2
(2π)
Z
kB T
n
2
= (v̄ ⊗ v̄)n +
z ⊗ z e−|z| /2 dz
d/2
mn (2π)
Rd
kB T
= (v̄ ⊗ v̄)n +
nId,
Id = identity matrix in Rd×d ,
mn
hvj |v|2 i =
d Z
X
i=1
Rd
Ã
v̄j +
r
kB T
zj
mn
!Ã
v̄i +
r
kB T
zi
mn
!2
n
2
e−|z| /2 dz
d/2
(2π)
Z
d
kB T
n X
2
v̄j
|zi |2 e−|z| /2 dz
= v̄j |v̄| n +
d/2
mn (2π)
Rd
i=1
Z
d
kB T
n X
2
+2
v̄i
zi zj e−|z| /2 dz
d/2
mn (2π)
Rd
i=1
µ
¶3/2
Z
kB T
n
2
+
zj |z|2 e−|z| /2 dz
d/2
mn
(2π)
Rd
d
kB T X
(v̄j + 2v̄i δij )
= v̄j |v̄|2 n +
n
mn i=1
µ
¶
2
mn 2 d + 2
=
v̄j n
|v̄| +
kB T .
mn
2
2
2
78
Thus, the energy is the sum of kinetic and thermal energy:
e=
mn 2 d
mn |J|2 d
|v̄| + kB T = 2 2 + kB T,
2
2
2q n
2
and the third-order term can be written as
2
v̄n(e + kB T ).
hv|v|2 i =
mn
In order to compute the terms coming from the collision operator, we assume
that the scattering rate is independent of v and v 0 , i.e. φ(x, v, v 0 ) = φ0 (x). Then
Z
Z
Z
Z
Z
0
0
0
0
fe v dv
fe dv − φ0
M dv
v Q(fe ) dv = φ0
M v dv
Rd
Rd
Rd
Rd
Rd
= −φ0 hvi
φ0
=
J,
q
Z
Z
Z
2
2
|v| Q(fe ) dv = φ0
M |v| dv · n − φ0
M 0 dv 0 · h|v|2 i
d
d
d
R
R
R
¶
µ
2
k B T0
−
e .
= φ0 n d
mn
mn
We introduce the relaxation time by τ = 1/φ0 . Then
¶
µ
Z
Z
mn
d
J
n
2
−q
e − k B T0 .
v Q(fe ) dv = − ,
|v| Q(fe ) dv = −
τ
2 Rd
τ
2
Rd
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
1
∂t n − div x J = 0,
q
µ
¶
1
J ⊗J
qkB
q2
J
∂t J − div x
+
∇(T n) −
nE = − ,
q
n
mn
mn
τ
¶
µ
n
1
d
∂t (ne) − div x [J(e + kB T )] − J · E = −
e − k B T0 .
q
τ
2
(5.39)
(5.40)
(5.41)
For vanishing right-hand sides, these equations are the Euler equations of gas dynamics for a gas of charged particles in an electric field. Sometimes an additional
term −div (κ(n, t)∇T ) is added to the left-hand side of (5.41). This term whose
presence is purely heuristic is a diffusion term and makes the equation (5.41) of
parabolic type. This is desirable since we expect the temperature to satisfy an
equation related to the parabolic heat equation. The heat conductivity κ(n, T )
is usually modeled (in R3 ) by [10]
κ(n, T ) =
2
3 kB
τ
nT.
2 mn
79
The equations (5.39)-(5.41) including the aboce heat flow (but with different
relaxation times in the momentum and energy equations) has been first derived
by Bløtekjær [13] and Baccarani and Wordeman [10].
The equations (5.39) and (5.40) are of hyperbolic type. The system of equations (5.39)-(5.41), with or without the additional heat flux term, are referred to
as 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 density
and the hole energy density can be derived from the corresponding Boltzmann
equation for holes (see (5.2)). The equations are as above with −q replaced by
q. The corresponding equations for the electrons and holes are referred to as the
bipolar hydrodynamic model.
5.4
The Spherical Harmonic Expansion (SHE) model
We start with the semi-classical Boltzmann equation for the distribution function
f (x, k, t) (see Section 3.3)
q
1
∂t f + ∇k ε · ∇x f + ∇x V · ∇k f = Q(f ),
~
~
x ∈ Rd , k ∈ B, t > 0.
(5.42)
In the following we only consider the evolution of the electrons in a single conduction band ε(k). We proceed similarly as in [11, 18]. The collision operator is
assumed to be the sum of lattice-defect collision terms (due to ionized impurities,
acoustic and optical phonons) and electron-electron collision terms. These terms
are separated in an elastic collision part and an inelastic collision part:
Q(f ) = Qel (f ) + α2 Qinel (f ).
We have assumed that the electric field is so large that the typical energy qU of
the electrons, where U is the voltage bias between the contacts, is much larger
than the energy ~ω of the phonons:
α2 =
~ω
¿ 1,
qU
and we have expanded the collision operator in terms of α2 . By Section 3.3, we
can write the elastic collision operator as
Z
(Qel (f ))(x, k) =
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))(f (k 0 ) − f (k)) dk 0
(5.43)
B
for functions f : Rd × B → R, where
φ(x, k, k 0 ) = σimp (x, k, k 0 ) + (2Nop + 1)σop (x, k, k 0 ) + (2Nac + 1)σac (x, k, k 0 ),
80
is the scattering rate, Nop , Nac are the occupation numbers of optical or acoustic
phonons, respectively, and δ is the delta distribution. Clearly, the integral (5.43)
has to be understood symbolically. The operator Qinel (f ) contains the inelastic
correction 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 0 :
φ(x, k, k 0 ) > 0,
φ(x, k, k 0 ) = φ(x, k 0 , k) ∀x, k, k 0 .
(5.44)
The Spherical Harmonic Expansion (SHE) model is derived in the so-called
diffusion scaling, i.e. by introducing the space and time scale
ts = α2 t.
xs = αx,
Then (xs , ts ) are called macroscopic variables. In these variables, the Boltzmann
equation (5.42) reads
µ
¶
1
q
2
α ∂t f + α
∇k ε · ∇x f + ∇x V · ∇k f = Qel (f ) + α2 Qinel (f ),
(5.45)
~
~
where we have written again (x, t) instead of (xs , ts ). Notice that this is exactly
the same scaling used in the derivation of the drift-diffusion equations (see Section
5.1).
We use now the Hilbert expansion method to ”solve” (5.45). Inserting the
expansion
f = f0 + αf1 + α2 f2 + · · ·
into (5.45), using the linearity of the operators Qel and Qinel and identifying terms
of the same order in α, we find:
• terms of order α0 :
Qel (f0 ) = 0,
(5.46)
• terms of order α1 :
Qel (f1 ) =
q
1
∇k ε · ∇ x f 0 + ∇x V · ∇ k f 0 ,
~
~
(5.47)
• terms of order α2 :
1
q
Qel (f2 ) = ∂t f0 + ∇k ε · ∇x f1 + ∇V · ∇k f1 − Qinel (f0 ).
~
~
81
(5.48)
First we wish to solve (5.46). For this, we need to prove some properties of
the operator Qel defined on the space
Z
n
o
2
L (B) = f : B → B measurable:
|f (k)|2 dk < ∞ .
B
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))},
Z
n
o
⊥
2
N (Qel ) =
f ∈ L (B) : ∀e ∈ R(ε) :
f (k)δ(e − ε(k)) dk = 0 .
B
(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 assumptions
on the regularity of φ and g(ε(k)). The orthogonal complement N (Qel )⊥ is the
set of all functions f ∈ L2 (B) such that
Z
hf, F i =
f (k) F (k) dk = 0 ∀F ∈ N (Qel ).
B
The range R(ε) of ε is defined by R(ε) = {ε ∈ R : ∃k ∈ B : ε(k) = e}. The
operator −Qel is called non-negative if
Z
− Qel (f )f dk ≥ 0 ∀f ∈ L2 (B).
B
Proof of Lemma 5.2. (1) Let x ∈ Rd , f, g ∈ L2 (B) and set f = f (k), f 0 = f (k 0 ),
g = g(k), g 0 = g(k 0 ), ε = ε(k) and ε0 = ε(k 0 ). Then, using the property
Z
Z
0
0
0
δ(ε − ε)ψ(k, k ) dk =
δ(ε − ε0 )ψ(k, k 0 ) dk 0
B
B
for any function ψ and the symmetry of φ, we obtain
Z
Z Z
1
Qel (f )g dk =
φ(x, k, k 0 )δ(ε0 − ε)(f 0 − f )g dk 0 dk
2
B
B B
Z Z
1
+
φ(x, k 0 , k)δ(ε − ε0 )(f − f 0 )g 0 dk dk 0
2 B B
Z Z
1
φ(x, k, k 0 )δ(ε0 − ε)(f 0 − f )(g 0 − g) dk 0 dk
= −
2 B B
Z
=
Qel (g)f dk.
B
82
This shows that Qel is symmetric and hence self-adjoint. Taking f = g gives
Z
Z Z
1
Qel (f )f dk = −
φ(x, k, k 0 )δ(ε0 − ε)(f 0 − f )2 dk 0 dk ≤ 0
(5.49)
2
B
B B
and thus, −Qel is non-negative.
(2) Let f ∈ N (Qel ), i.e. Qel (f ) = 0. By (5.49), this implies
δ(ε0 − ε)(f 0 − f )2 = 0
for almost all k, k 0 ∈ B.
The delta distribution has the ”property” δ(z) = 0 for all z 6= 0. (We treat the
delta distribution as a function; this is mathematically not correct but avoids
technical calculations.) Therefore
ε(k 0 ) = ε(k) and f (k 0 ) = f (k) for almost all k, k 0 ∈ B.
Hence f must be constant on each energy surface {k : ε(k) = e}. This implies
that 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
Z
0 =
f (k)F (k) dk
B
Z
=
f (k)g(ε(k)) dk
B
µZ
¶
Z
=
f (k)
g(e)δ(ε(k) − e) de dk
B
R
¶
Z µZ
=
f (k)δ(ε(k) − e) dk g(e) de.
(5.50)
R
B
Here we have used the definition of the delta distribution
Z
δ(z − e)ψ(e) de = ψ(z) for any (regular) function ψ.
R
(5.51)
Equation (5.50) holds for all F ∈ N (Qel ) and thus for any function g. We infer
Z
f (k)δ(ε(k) − e) dk = 0 for almost all e ∈ R(ε).
B
This proves (2).
(3) The first integral in
Z
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))f (k 0 )g(ε(k 0 )) dk 0
(Qel (gf ))(k) =
B
Z
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))f (k)g(ε(k)) dk 0
−
B
83
equals, after the change of unknown e = ε(k 0 ) and the definition (5.51),
¯
¯−1
Z
¯
¯
dε
0
0
0
φ(x, k, k )δ(e − ε(k))f (k )g(e) ¯¯det (k )¯¯ de
dk
R(ε)
¯
¯−1 ¯¯
¯
¯ ¯
dε
= φ(x, k, k 0 )f (k 0 )g(ε(k)) ¯¯det (k 0 )¯¯ ¯
¯
dk
e=e(k)=ε(k 0 )
¯
¯−1
Z
¯
dε 0 ¯¯
0
0 ¯
= g(ε(k))
φ(x, k, k )δ(e − ε(k))f (k ) ¯det (k )¯ de
dk
R(ε)
Z
= g(ε(k))
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))f (k 0 ) dk 0 .
B
Notice that the use of the transformation formula is rather formal since ε may not
satisfy the required technical assumptions. However, the above computation can
be made mathematically rigorous by employing the coarea formula (see [11, 18]
for details). We infer
Z
(Qel (gf ))(k) = g(ε(k))
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))f (k 0 ) dk 0
B
Z
−g(ε(k))
φ(x, k, k 0 )δ(ε(k 0 ) − ε(k))f (k) dk 0
B
= g(ε(k))(Qel (f ))(k).
This proves (3).
¤
Remark 5.3 From the proof of Lemma 5.2(3) we conclude the following general
result: For all ψ(k, k 0 ) and g(ε(k)) it holds
Z
Z
0
0
0
0
ψ(k, k )δ(ε(k ) − ε(k))g(ε(k )) dk = g(ε(k))
ψ(k, k 0 )δ(ε(k 0 ) − ε(k)) dk 0 .
B
B
(5.52)
We can now solve the equations (5.46)-(5.48). By Lemma 5.2(2), any solution
of (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
¶
µ
1
∂F
Qel (f1 ) = ∇k ε · ∇x F + q∇x V
.
~
∂ε
As ∇x F + q∇x V (∂F/∂ε) only depends on ε(k) (and on the parameters x, t), any
solution of this equation can be written, by Lemma 5.2(3), as
µ
¶
∂F
f1 (x, k, t) = −λ(x, k) · ∇x F + q∇x V
+ F1 (x, ε(k), t)
∂ε
84
where F1 ∈ N (Qel ) and λ(x, k) is a solution of
1
Qel (λ) = − ∇k ε.
~
(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 above
equation is solvable if and only if ∇k ε ∈ R(Qel ) (the range of Qel ). It is not
difficult to see that R(Qel ) is closed. From functional analysis (and the selfadjointness of Qel ) follows that R(Qel ) = N (Qel )⊥ . Thus (5.53) is solvable if and
only if ∇k ε ∈ N (Qel )⊥ . By Lemma 5.2(2) this is equivalent to
Z
∇k ε(k)δ(e − ε(k)) dk = 0 ∀e ∈ R(ε).
B
Now, using the relation
dH
(z) = δ(z),
dz
where H is the Heaviside function, we have
Z
Z
∇k ε(k)δ(e − ε(k)) dk = −
∇k H(e − ε(k)) dk = 0,
B
(5.54)
B
since ε is periodic on B. Thus, (5.53) is solvable. We choose the particular solution
of (5.47) as
¶
µ
∂F
.
f1 (x, k, t) = −λ(x, k) · ∇x F + q∇x V
∂ε
It remains to solve (5.48). Again, (5.48) is solvable if and only if the right-hand
side of (5.48) is an element of N (Qel )⊥ or, equivalently,
¶
Z µ
1
q
∂t f0 + ∇k ε · ∇x f1 + ∇x V · ∇k f1 − Qel (f0 ) δ(e − ε(k)) dk = 0 (5.55)
~
~
B
for all e ∈ R(ε). We define the density of states of energy e by
Z
N (e) =
δ(e − ε(k)) dk
B
(compare with (2.13) in Section 2.3). Then, by (5.52),
Z
Z
∂t f0 δ(e − ε(k)) dk = ∂t
F (x, ε(k), t)δ(e − ε(k)) dk
B
B
Z
= ∂t F (x, e, t)
δ(e − ε(k)) dk
B
= N (e)∂t F (x, e, t).
85
(5.56)
Furhermore, we introduce the electron current density by
Z
1
J(x, e, t) = −q
∇k ε(k)f1 (x, k, t)δ(e − ε(k)) dk.
B ~
Then we can write the second term in (5.55) as
Z
1
1
∇k ε · ∇x f1 δ(e − ε(k)) dk = − div x J(x, e, t).
q
B ~
Using the expression for f1 and the property (5.52), we have
¶
µ
Z
1
∂F
J(x, e, t) = q
(x, ε(k), t)
∇k ε(k)λ(x, k) · ∇x F + q∇x V
∂ε
B ~
× δ(e − ε(k)) dk
¶
µ
∂F
(x, e, t),
= D(x, e) ∇x F + q∇x V
∂ε
where
Z
q
D(x, e) =
∇k ε(k) ⊗ λ(x, k)δ(e − ε(k)) dk ∈ Rd×d
(5.57)
~ B
is called diffusion matrix.
The last term in (5.55) is a collision term, averaged over the energy surface
e = ε(k):
Z
(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 ψ
we obtain, using the definition of δ and integrating by parts,
Z
Z
ψ(e)
∇k f1 (x, k, t)δ(e − ε(k)) dk de
R(ε)
B
Z
=
ψ(ε(k))∇k f1 (x, k, t) dk
B
Z
dψ
= − f1 (x, k, t) (ε(k))∇k ε(k) dk
dε
Z
ZB
dψ
(e)
= −
∇k ε(k)f1 (x, k, t)δ(e − ε(k)) dk de
R(ε) dε
B
Z
~
dψ
=
(e)J(x, e, t) de
q R(ε) dε
Z
~
∂J
= −
ψ(e) (x, e, t) de.
q R(ε)
∂ε
Since ψ is arbitrary, it follows
Z
~ ∂J
∇k f1 (x, k, t)δ(e − ε(k)) dk = −
(x, e, t)
q ∂ε
B
86
and
q
~
Z
B
∇x V · ∇k f1 δ(e − ε(k)) dk = −∇x V ·
∂J
(x, e, t).
∂ε
We have shown that the Hilbert expansion (5.46)-(5.48) is solvable if and only
if f0 (x, k, t) = F (x, ε(k), t), where F is a solution of
1
∂J
N (ε)∂t F − div x J − ∇x V ·
= S(F ),
(5.59)
q
∂ε
¶
µ
∂F
, x ∈ Rd , ε ∈ R(ε), t > 0, (5.60)
J(x, ε, t) = D(x, ε) ∇x F + q∇x V
∂ε
where D(x, ε) is given by (5.57). The equations (5.59)-(5.60) are referred to
as Spherically Harmonic Expansion (SHE) model. It has been first derived in
the physical literature for spherically symmetric band diagrams and rotationally
invariant collision operators from the Boltzmann equation [26, 27]. In fact, the
above derivation of the SHE model does not require any assumption of spherical
symmetry. The equations can be written more clearly in terms of the total energy
u = 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
∇x F = ∇x f − q∇x V
∂f
,
∂u
∂t F = ∂t f − q∂t V
∂f
∂u
and therefore
n(x, u, t)∂t f − div x (d(x, u, t)∇x f ) = S(f ) + n(x, u, t)q∂t V
∂f
.
∂u
This shows, together with the following proposition, that the SHE model is of
parabolic type.
Proposition 5.4 The diffusion matrix D is a symmetric non-negative d × d
matrix. Moreover, there exists a constant K > 0 such that for all z ∈ Rd and all
x ∈ Rd and e ∈ R(ε),
Z
K
>
z D(x, e)z ≥
|∇k ε(k) · z|2 δ(e − ε(k)) dk.
N (ε) B
The fact that the matrix D(x, ε) is non-negative is a direct consequence of
the non-negativity of the operator −Qel (see Lemma 5.2(1)). The symmetry of
D(x, e) is related to the so-called Onsager reciprocity relation of non-equilibrium
thermodynamics [22].
87
Proof. We only prove that D(x, ε) = (Dij )ij is symmetric. The proof of the
second assertion is more technical and can be found in [11, Sec. III.4]. We
compute, for (smooth) functions ψ,
Z
Z Z
∂ε
q
λj δ(e − ε(k)) dk ψ(e) de
Dij ψ(e) de = −
~ R B ∂ki
R
Z
= −q
Qel (λi )λj ψ(ε(k)) dk
B
(using the definition of δ)
Z
λi Qel (λj ψ(ε)) dk
= −q
B
(since Qel is self-adjoint by Lemma 5.2(1))
Z
= −q
λi Qel (λj )ψ(ε) dk
B
(by Lemma 5.2(3))
Z Z
q
∂ε
=
λi
δ(e − ε(k)) dk ψ(e) de
~ R B ∂kj
Z
=
Dji ψ(e) de.
R
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 following
example.
Example 5.5 (Spherically symmetric energy bands)
Assume that the scattering rate only depends on ε(k), i.e.
φ(x, k, k 0 ) = φ(x, ε(k))
for all k, k 0 with ε(k 0 ) = ε(k),
that the energy band is spherically symmetric (thus, B = Rd ) and strictly monotone in |k|, i.e.
ε = ε(|k|),
and that d = 3. Notice that the first assumption makes sense since due to the
term δ(ε(k 0 ) − ε(k)) in the definition of Qel , the scattering rate only needs to
be defined on the energy surface {k 0 : ε(k 0 ) = ε(k)} of energy ε(k). The second
assumption implies that ε(|k|) is invertible.
From the first assumption and the definition (5.56) of the density of states
N (e) follows, with ε = ε(k), ε0 = ε(k 0 ),
Z
(Qel (f ))(x, k, t) = φ(x, ε)
δ(ε0 − ε)f (k 0 ) dk 0 − φ(x, ε)N (ε)f (k)
Rd
1
([f ] − f ) (x, k, t),
=
τ (x, ε)
88
(5.61)
where
τ (x, ε) =
1
φ(x, ε)N (ε)
is called relaxation time and
1
[f ](k) =
N (ε)
Z
Rd
δ(ε0 − ε)f (k 0 ) dk 0
is the average of f on the energy surface {k 0 ∈ B : ε(k 0 ) = ε(k)}. The expression
(5.61) is called relaxation-time operator.
It follows that a solution λ(x, k) of Qel (λ) = −~−1 ∇k ε (see (5.53)) is given by
λ(x, k) = τ (x, ε(k))~−1 ∇k ε(k).
Indeed, since, by (5.52) and (5.54),
Z
1
[λ] =
δ(ε0 − ε)τ (x, ε0 )~−1 ∇k ε(k 0 ) dk 0
N (ε) Rd
Z
1
−1
τ (x, ε)~
δ(ε0 − ε)∇k ε(k 0 ) dk 0
=
N (ε)
Rd
= 0,
we obtain
Qel (λ) =
λ
1
1
([λ] − λ) = −
= − ∇k ε.
τ (x, ε)
τ (x, ε)
~
Thus, again using (5.52), the diffusion matrix becomes
Z
−2
∇k ε(k) ⊗ ∇k ε(k)τ (x, ε)δ(e − ε) dk
D(x, e) = q~
Rd
Z
−2
∇k ε(k) ⊗ ∇k ε(k)δ(e − ε(k)) dk.
= q~ τ (x, e)
Rd
This expression can be further simplified under the second assumption. As ε
only depends on |k|, it is convenient to introduce spherical coordinates k = ρω,
where ρ = |k| ∈ [0, ∞) is the modulus of k and ω ∈ S d−1 are the angle variables
on the unit sphere S d−1 ⊂ Rd . Then
∇k ε(|k|) = ε0 (|k|)
k
= ε0 (ρ)ω
|k|
and the volume element dk transforms symbolically to ρd−1 F (ω) dρ dω, where
F (ω) is the modulus of the determinant of the Jacobian of the angle transform.
89
We compute
−2
Z
∞
Z
ε0 (ρ)2 ω ⊗ ωδ(e − ε(ρ))ρd−1 F (ω) dρ dω
Z0 ∞
Z
−2
0
2
d−1
ω ⊗ ωF (ω) dω
= qτ (x, e)~
ε (ρ) δ(e − ε(ρ))ρ
dρ ·
0
S d−1
Z
Z
0
d−1
−2
ω ⊗ ωF (ω) dω
ε (ρ)δ(e − η)ρ
dη ·
= qτ (x, e)~
D(x, e) = qτ (x, e)~
S d−1
S d−1
R(ε)
0
(transforming η = ε(ρ) and dη = ε (ρ) dρ)
Z
¯
−2 0
d−1 ¯
= qτ (x, e)~ ε (ρ)ρ ¯
·
ω ⊗ ωF (ω) dω.
e=ε(ρ)
S d−1
In the three dimensional case d = 3 (third assumption) this becomes
D(x, e) =
4πq
τ (x, e)ε0 (|k|)|k|2 Id,
3~2
(5.62)
where |k| is such that e = ε(|k|) and Id ∈ R3×3 is the identity matrix. Indeed,
from


sin θ cos φ
ω =  sin θ sin φ  , 0 ≤ φ < 2π, 0 ≤ θ < π,
cos θ
and F (ω) dω = sin θ dθ dφ we obtain
Z
Z
(ω ⊗ ω)ij F (ω) dω =
S2
2π
0
Z
π
ωi ωj sin θ dθ dφ.
0
For i 6= j the integral vanishes. We compute for
Z 2π
Z π
4
4π
2
i=j=1:
cos φ dφ ·
sin3 θ dθ = π · =
,
3
3
0
0
Z 2π
Z π
4π
4
2
,
i=j=2:
sin φ dφ ·
sin3 θ dθ = π · =
3
3
0
0
Z π
4π
2
,
i=j=3:
2π ·
cos2 θ sin θ dθ = 2π · =
3
3
0
and (5.62) follows.
Furthermore,
Z
Z ∞Z
N (e) =
δ(e − ε(|k|)) dk =
δ(e − ε(ρ))ρd−1 F (Ω) dρ dω
d
S d−1
Z 0
ZR
d−1
ρ
=
δ(e − η) 0 dη ·
F (ω) dω
ε (ρ)
R
S d−1
Z
S d−1 ¯¯
= 0 ¯
F (ω) dω,
ε (ρ) e=ε(ρ) S d−1
90
and for d = 3 we get
Z
F (ω) dω =
S d−1
and therefore
Z
2π
0
Z
π
sin θ dθ dφ = 4π
0
|k|2
with e = ε(|k|).
(5.63)
ε0 (|k|)
It is convenient to express D(x, e) and N (e) in terms of the function γ, where
N (e) = 4π
|k|2 = γ(ε(|k|)).
Differentiating this equation with respect to |k| yields
2|k| = γ 0 (ε(|k|))ε0 (|k|)
and thus
p
2 γ(e)
ε (|k|) =
with e = ε(|k|).
γ 0 (e)
It follows from (5.62) and (5.63) for d = 3
p
γ(e)γ 0 (e)
= 2π γ(e)γ 0 (e),
N (e) = 4π p
2 γ 0 (e)
p
2 γ(e)
γ(e)
γ(e)
4q
4πq
p
Id.
Id
=
D(x, e) =
3~2 2πφ(x, e) γ(e)γ 0 (e) γ 0 (e)
3~2 φ(x, e)γ 0 (e)2
0
Example 5.6 (Parabolic band approximation)
We assume that d = 3, φ = φ(x, ε(k)) and
~2
|k|2 .
∗
2m
∗
2
This implies γ(e) = 2m e/~ and (see Example 5.5)
µ ∗ ¶3/2
√
2m
N (e) = 2π
e,
2
~
2q
e
D(x, e) =
Id, e > 0.
∗
3m φ(x, e)
ε(k) =
It remains to determine the average collision operator S(F ) (see (5.58). The
precise structure of this term depends on the assumptions on the inelastic collision
operator. Here, we only notice that a simplified expression is given by the FokkerPlanck approximation
·
µ
¶¸
∂
∂F
S(F ) =
A(ε) F + kB TL
,
(5.64)
∂ε
∂ε
where TL is the lattice temperature and A(ε) depends on the scattering rate of
the 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 SHE
model 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 Boltzmann
equation. However it is numerically more expensive than usual macroscopic models due to the additional energy variable. In this section we derive a macroscopic
energy-transport model from the SHE model following [11].
In [12] an energy-transport model has been derived directly from the Boltzmann equation. In this approach, the electron-electron collision term is assumed
to be dominant compared to the phonon collisions which can be criticized. Here
we assume that the electron-electron collision term is dominant compared to the
inelastic part of the phonon collision term. More precisely, we start with the SHE
model
1
∂J
N (ε)∂t F − div x J − ∇x V ·
= S(F ),
q
∂ε
µ
¶
∂F
J = D(x, ε) ∇x F + q∇x V ·
, x ∈ Rd , ε ∈ R, t > 0,
∂ε
(5.65)
(5.66)
for the distribution function F (x, ε, t). The density of states N (ε), the diffusion
matrix 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 operator contains the inelastic correction to phonon collisions and electron-electron
collisions. Let β 2 be the ratio of the typical electron-electron collision time and
the phonon-collision time. We assume that β 2 ¿ 1 which corresponds to so-called
hot-electron regimes. Moreover, we suppose that
• S(F ) = β 2 Sph (F ) + Se (F ),
• xs = βx, ts = β 2 t, Js = J/β,
• Se is a linear operator.
In the first assumption, Sph and Se are the averaged inelastic phonon and electronelectron operators, respectively. The second assumption means that we are considering a macroscopic space and time scale and “small” current densities. Finally, the third assumption is only for convenience (see below). In the macroscopic
scale, the SHE equation (5.65) reads, writing again (J, x, t) instead of (J s , xs , ts ),
1
∂J
1
N (ε)∂t F − div x J − ∇x V ·
= Sph (F ) + 2 Se (F ),
q
∂ε
β
µ
¶
∂F
J = D(x, ε) ∇x F + q∇x V ·
.
∂ε
92
(5.67)
(5.68)
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 β 0 :
∂J0
1
− Sph (F0 ) = Se (F1 ).
N (ε)∂t F0 − div x J0 − ∇x V ·
q
∂ε
(5.70)
If Se is not a linear operator, we would need to replace Se (F1 ) by (DF0 Se )(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
Z
(Se (F ))(x, e, t) =
(Qe (F ))(x, k, t)δ(e − ε(k)) dk,
B
Z
(Sph (F ))(x, e, t) =
(Qac (F ) + Qop (F ))(x, k, t)δ(e − ε(k)) dk,
B
and Qe , Qac and Qop are the operators of electron-electron, acoustic phonon and
optical phonon collisions, respectively (see [11] for precise definitions of these
operators).
(1) It holds for any (smooth) function F : R → R
Z
Z
(Se (F ))(ε) dε = 0,
(Se (F ))(ε)ε dε = 0,
R
R
Z
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)/kB T )
and kB is the Boltzmann constant.
(3) The equation Se (F ) = G is solvable if and only if
Z
Z
G(ε) dε = 0 and
G(ε)ε dε = 0.
R
R
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
¶µ ¶
Z µ
1
∂J0
1
N (ε)∂t Fµ,T − div x J0 − ∇x V ·
− Sph (F0 )
dε = 0. (5.71)
ε
q
∂ε
R
We introduce the particle density and the particle energy density, respectively,
by
Z
Z
n(µ, T ) =
Fµ,T (ε)N (ε) dε, E(µ, T ) =
Fµ,T (ε)εN (ε) dε,
(5.72)
R
R
and the macroscopic particle and energy current densities, respectively, by
Z
Z
1
J0 (x, ε, t)ε dε.
J(x, t) =
J0 (x, ε, t) dε, S(x, t) =
q R
R
Then, using Lemma 5.7(1), we can write (5.71) as
Z
Z
1
∂J0
dε + Sph (Fµ,T ) dε = 0,
∂t n − div x J = ∇x V ·
q
∂ε
ZR
ZR
∂J0
ε dε + Sph (Fµ,T )ε dε
∂t E − div x S = ∇x V ·
RZ ∂ε
Z R
= −∇x V · J0 dε + Sph (Fµ,T )ε dε
R
R
= −∇x V · J + W (µ, T ),
where
W (µ, T ) =
Z
R
(Sph (Fµ,T )) (ε)ε dε
(5.73)
is called relaxation term. It can be seen that W also depends on the lattice
temperature TL , i.e. W (µ, T ) = W (µ, T, TL ) (this requires knowledge about the
precise structure of Sph ; we refer to [12]).
It remains to compute the fluxes J and S. We can interpret Fµ,T (ε) as a
function of the variables u1 := qµ/kB T , u2 := −1/kB T and ε:
µ
¶
qu −1
F (u1 , u2 , ε) = F
,
, ε := Fµ,T (ε).
kB T kB T
94
Then
F (u1 , u2 , ε) =
1
1+
e−(εu2 +u1 )
and
∂F
e−(εu2 +u1 )
=
= Fµ,T (1 − Fµ,T ),
∂u1
(1 + e−(εu2 +u1 ) )2
∂F
εe−(εu2 +u1 )
=
= εFµ,T (1 − Fµ,T ),
∂u2
(1 + e−(εu2 +u1 ) )2
u2 e−(εu2 +u1 )
1
∂F
=
=−
Fµ,T (1 − Fµ,T ).
−(εu
+u
)
2
2
1
∂ε
(1 + e
)
kB T
This yields
·
∂F
J0 = D(x, ε)
∇
∂u1
µ
qu
kB T
¶
∂F
+
∇
∂u2
µ
−1
kB T
¶
∂F
+ q∇x V ·
∂ε
¸
and
¶
µ
¶
−1
q∇x V
qu
+ D12 ∇
− D11
,
J = D11 ∇
kB T
kB T
kB T
¶
µ
¶
µ
q∇x V
−1
qu
+ D22 ∇
− D21
,
qS = D21 ∇
kB T
kB T
kB T
µ
where the coefficients
Z
Dij (x, µ, T ) =
D(x, ε)Fµ,T (1 − Fµ,T )εi+j−2 dε,
R
i, j = 1, 2,
(5.74)
(5.75)
(5.76)
are d × d matrices.
Summarizing, we can write the energy transport equations as
1
∂t n − div x J = 0,
q
∂t E − div x S + ∇x V · J = W (µ, T ),
(5.77)
(5.78)
where n(µ, T ), E(µ, T ) are defined in (5.72), W (µ, T ) is given by (5.73), and
finally, 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 the
following 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. Dij> = Dij for i, j = 1, 2.
95
(3) Assume that the 2d functions
∂ε
∂ε
∂ε
∂ε
,...,
, ε
,...,ε
∂k1
∂kd ∂k1
∂kd
are linearly independent. Then D(x, µ, T ) is symmetric positive definitive
for 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 assumption
on the band structure. It expresses for the case d = 3 that the energy band
needs to have a real three-dimensional structure, excluding bands depending only
on one or two variables, for instance. Notice that the statement of part (4)
justifies the name “relaxation term”: The temperature of the system is expected
to relax to the (constant) lattice temperature if there are no forces influencing
the temperature.
Proof. We only prove parts (1)-(3) since for the proof of (4) we need the precise
structure of Sph (see [12, Lemma 4.11] for a detailed proof). Part (1) follows from
the symmetry of the matrix, and part (2) is a consequence of the symmetry of
D. 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
¶
µ
Z
D(x, ε) εD(x, ε)
>
>
ξFµ,T (1 − Fµ,T ) dε
ξ Dξ =
ξ
εD(x, ε) ε2 D(x, ε)
R
(by the definition of D)
Z h
i
¡ ¢>
¡ ¢>
¡ (1) ¢>
Dξ (1) + 2ε ξ (1) Dξ (2) + ε2 ξ (2) Dξ (2)
ξ
=
R
=
≥
=
>
×Fµ,T (1 − Fµ,T ) dε
(since D = D(x, ε) is symmetric)
Z
¡ (1)
¢
¡
¢
>
ξ + εξ (2) D ξ (1) + εξ (2) Fµ,T (1 − Fµ,T ) dε
R
Z Z
¡
¢
K
|∇k ε(k) · ξ (1) + εξ (2) |2 δ(ε − ε(k))
N (ε) R B
×Fµ,T (1 − Fµ,T ) dk dε
(by Proposition 5.4)
¶ ¯2
Z Z ¯µ
¯
¯
K
∇
ε(k)
k
¯
· ξ ¯¯ δ(ε − ε(k)) Fµ,T (1 − Fµ,T ) dk dε
¯
ε(k)∇k ε(k)
N (ε) R B
0;
since otherwise would
µ
∇k ε(k)
ε(k)∇k ε(k)
¶
· ξ = 0,
96
ξ 6= 0,
imply that (∇k ε, ε(k)∇k ε) is linearly dependent, which is a contradiction.
¤
More explicit expressions for n, E and D can be derived under the assumptions
of 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)/kB T ).
This is possible if ε − qµ À kB T. In particular,
Fµ,T (1 − Fµ,T ) ≈ e−(ε−qµ)/kB T · 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)
Z p
Z
−(ε−qµ)/kB T
qµ/kB T
γ(ε)γ 0 (ε)e−ε/kB T dε,
n(µ, T ) =
e
N (ε) dε = 2πe
ZR p
ZR
−(ε−qµ)/kB T
qµ/kB T
E(µ, T ) =
e
εN (ε) dε = 2πe
γ(ε)γ 0 (ε)εe−ε/kB T dε,
R
R
where |k|2 = γ(ε(k)). The diffusion matrices (5.76) become
Z
4q qµ/kB T
γ(ε)εi+j−2 −ε/kB T
Dij (µ, T ) = 2 e
e
dε · Id.
0
2
3~
R φ(x, ε)γ (ε)
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
|k|2 ,
2m∗
k ∈ R3 .
• The scattering rate is given by the so-called Chen model [17, 21]
√
φ(x, ε) = φ0 (x) ε.
97
(5.79)
By Example 5.6,
N (ε) = 2π
and therefore
n(µ, T ) = 2π
= 2π
µ
µ
2m∗
~2
2m
~2
∗
¶3/2
¶3/2
e
e
µ
2m∗
~2
qµ/kB T
qµ/kB T
Z
¶3/2
∞
√
ε,
√
e−ε/kB T ε dε
0
1
√ (kB T )3/2
2
Z
∞
e−z
2 /2
z 2 dz,
0
after substituting ε/kB T = z 2 /2. From
r
Z ∞
Z
Z
1
1
1√
π
−z 2 /2 2
−z 2 /2 2
−z 2 /2
e
z dz =
e
z dz =
e
· 1 dz =
2π =
2 R
2 R
2
2
0
we conclude
n(µ, T ) = π
3/2
µ
2m∗ kB T
~2
¶3/2
where
Nc (T ) = 2
eqµ/kB T = (4π 2 )3/2 Nc (T )eqµ/kB T ,
µ
m∗ kB T
2π~2
¶3/2
is the effective density of states defined in Section 2.3. For the energy density we
compute
µ ∗ ¶3/2
Z ∞
2m
qµ/kB T
e
E(µ, T ) = 2π
e−ε/kB T ε3/2 dε
~2
0
µ ∗ ¶3/2
Z
2m
2
qµ/kB T 1
5/2 1
√ (kB T )
e
e−z /2 z 4 dz
= 2π
2
~
2 R
8
µ ∗
¶3/2
Z
π
2m kB T
2
qµ/kB T
= √
e
(kB T ) · 3 e−z /2 z 2 dz
2
~
8
R
3
(4π 2 )3/2 Nc (T )eqµ/kB T (kB T )
=
2
3
n(µ, T ) · kB T.
=
2
√
The diffusion matrices become, using γ(ε) = 2m∗ ε/~2 and φ(x, ε) = φ0 (x) ε,
Z
2q qµ/kB T ∞ εi+j−1 −ε/kB T
Dij (µ, T ) =
e
e
dε · Id
3m∗
φ(x, ε)
0
Z ∞
2q
qµ/kB T
=
e
εi+j−3/2 e−ε/kB T dε · Id,
3m∗ φ0 (x)
0
98
and from the relations
√
Z ∞
Z ∞
1
π
2 −z 2 /2
1/2 −ε/kB T
3/2
z e
dz =
ε e
dε = √ (kB T )
(kB T )3/2 ,
2
2
0
√
Z ∞
Z0 ∞
3
π
1
4 −z 2 /2
5/2
3/2 −ε/kB T
z e
dz = ·
(kB T )5/2 ,
ε e
dε = √ (kB T )
3
2 2
2
√
Z0 ∞
Z0 ∞
5 3
1
π
6 −z 2 /2
5/2 −ε/kB T
7/2
z e
dz = · ·
ε e
dε = √ (kB T )
(kB T )7/2 ,
5
2 2 2
2
0
0
we conclude
¶
µ
3
πq
(kB T )Id
Id
3/2 qµ/kB T
2
(kB T ) e
D(µ, T ) =
3
(k T )Id 15
(kB T )2 Id
3m∗ φ0 (x)
2 B
4
µ
¶
3
q~3 n(µ, T )
Id
(k
T
)Id
B
2
= √
∈ R6×6 .
3
15
2
∗
2
(k
T
)Id
(k
T
)
Id
B
3 8π(m ) φ0 (x)
2 B
4
√
Notice that det D > 0 as long as n > 0 and T > 0, and the energy-transport
equations are of parabolic type. Moreover, the matrix D(µ, T ) can be identified
by 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 using
√
the Fokker-Planck approximation (5.64) with A(ε) = φ0 εN (ε)2 and assuming
the hypotheses of Example 5.10. Then
·
µ
¶¸
Z ∞
∂Fµ,T
∂
A(ε) Fµ,T + kB TL
ε dε
W (µ, T, TL ) =
∂ε
∂ε
0
µ
·
¶¸
Z ∞
TL
∂ √
qµ/kB T
2 −ε/kB T
1−
εN (ε) e
ε dε
= φ0 e
∂ε
T
0
µ
¶Z ∞
√
TL
qµ/kB T
1−
= −φ0 e
εN (ε)2 e−ε/kB T dε
T
0
µ
¶Z ∞
µ ∗ ¶3
TL
2m
qµ/kB T
2
e
1−
ε3/2 e−ε/kB T dε
= −4π φ0
~2
T
0
µ
¶
µ ∗ ¶3
TL
2m
eqµ/kB T 1 −
(kB T )5/2
= −3π 5/2 φ0
2
~
T
µ ∗ ¶3/2
2m
= −3πφ0
n(µ, T ) · kB (T − TL )
~2
3 n(µ, T )kB (T − TL )
=
,
2
τ0
where
1
τ0 =
2πφ0
is called relaxation time.
µ
99
~2
2m∗
¶3/2
In this chapter we have derived four semiconductor models: the hydrodynamic
equations, the SHE model, the energy-transport equations, and the drift-diffusion
model. In Figure 5.1 we summarize the derivations of the models and their
relations.
dominant elastic ´ Boltzmann Q
Q
collisions (Hilbert´´
Q moment method
expansion) ´
Q
´
dominant electron-electron QQ
s
+́
collisions (Hilbert
SHE
Hydrodynamic
expansion) [12]
½
@
½
momentum
dominant @
½
electron-electron@
½ relaxation-time
?
½
limit
collision (Hilbert@
R
@
=
Energy-transport ½
expansion)
energy relaxation- time limit
?
Drift-Diffusion
Figure 5.1: Hierarchy of classical semiconductor models. The drift-diffusion
model can be directly derived from the Boltzmann equation via a moment
method or from the hydrodynamic model in the combined momentum and energy
relaxation-time limit.
5.6
Relaxation-time limits
In this section we will derive the drift-diffusion and energy-transport equations
from the (full) hydrodynamic model by performing so-called relaxation-time limits. We recall the full hydrodynamic model:
1
∂t n − div J = 0,
(5.80)
q
µ
¶
J ⊗J
1
qkB
q2
J
∂t J − div
−
∇(nT ) + n∇V = − ,
(5.81)
q
n
m
m
τp
µ 2
¶
1
τp k B
κ0 nT ∇T
(5.82)
∂t (ne) − div [J(e + kB T )] + J · ∇V − div
q
m
µ
¶
n
d
=−
e − k B T0
in Ω ⊂ Rd ,
τw
2
where τp and τw are the momentum energy relaxation times, respectively, and
e=
m |J|2 d
+ kB T
q 2 2n2
2
100
is the total energy. The coupling of the electrostatic potential V to the electron
density through the Poisson equation needs not to be considered in the subsequent
considerations. Therefore, V will be treated as a given function.
We scale the equations (5.80)-(5.82) by introducing reference values for the
physical variables (see Table 5.1). Here, L > 0 is a reference length and C is a
reference value for the particle density. When the Poisson equation is added to
the above system of equations, C can be chosen, for instance, as the supremum
of the doping profile C = C(x). The reference time τ0 is given by the assumption
that the thermal energy is of the same order as the geometric average of the
kinetic energies needed to cross the domain in time τ0 , τp , respectively:
s µ ¶ s µ ¶
2
2
L
L
.
m
k B T0 = m
τ0
τp
The reference velocity u0 is equal to the velocity needed to cross the domain in
time τ0 .
variable
x
t
n(x, t)
T (x, t)
V (x, t)
J(x, T )
e(x, t)
reference value
L
τ0
C
T0
UT
qCu0
k B T0
definition
e.g. diam(Ω)
τ0 = L2 m/kB T τp
e.g. supΩ |C|
lattice temperature
UT = kB T0 /q
u0 = L/τ0
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 = τ 0 ts ,
n = Cns ,
T = T 0 Ts ,
V = U T Vs ,
J = qCu0 · Js .
Replacing the dimensional variables in (5.80)-(5.82) by the scaled ones we obtain
the scaled equations (writing x, t, n etc. instead of xs , ts , ns etc.)
∂ n − div J = 0,
(5.83)
µ t
¶
τp
τp
J ⊗J
∂t J − div
− ∇(nT ) + n∇V = −J,
(5.84)
τ0
τ0
n
µ
¶
d
τ0
e−
∂t (ne) − div [J(e + T )] + J · ∇V − div (κ0 nT ∇T ) = −
, (5.85)
τw
2
101
and the scaled energy becomes
e=
τp |J|2 d
+ T.
τ0 2n2
2
(5.86)
We consider the following physical situations:
Drift-diffusion limit in the hydrodynamic equations. The kinetic energy needed to cross the domain in time τ0 is assumed to be much smaller than
the thermal energy, and the momentum and energy relaxation times are assumed
to be of the same order. This means
1À
mu20
L2 m
τp
=
=
2
k B T0
k B T 0 τ0
τ0
The second condition implies
and
O(1) =
τp
τp τ0
=
· .
τw
τ0 τw
τ0
À 1.
τw
The drift-diffusion limit means that we perform (formally) the limits
τp
→0
τ0
and
τw
→ 0.
τ0
Then (5.85)-(5.86) become in the limit
d
d
e= T = .
2
2
This means that the limit temperature equals one. Scaling back to the unscaled
variables the physical limit temperature equals the lattice temperature. From
(5.83)-(5.84) we conclude
∂t n − div J = 0,
J = ∇n − n∇V.
In the original variables this reads
1
∂t n − div J = 0,
q
J = qµn (UT ∇n − n∇V ),
(5.87)
where
qτp
m
is called the electron mobility. The equations (5.87) together with the Poisson
equation are the drift-diffusion equations derived in Sections 5.1 and 5.2.
µn =
Energy-transport limit in the hydrodynamic equations. The kinetic
energy 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 energy
relaxation time, i.e.
τp
τ0
τ0 τp
¿ 1 and O(1) =
=
· ,
τ0
τw
τp τw
which implies
τp
¿ 1.
τw
In the energy-transport limit we only perform (formally) the limit
τp
→ 0.
τ0
Equations (5.83)-(5.86) become in the limit
∂t
µ
d
nT
2
¶
∂t n − div J = 0, J = ∇(nT ) − n∇V,
(5.88)
µ
¶
d+2
d τ0
− div
JT + κ0 nT ∇T + J · ∇V = −
n(T − 1), (5.89)
2
2 τw
or, in the original variables,
1
∂t n − div J = 0, J = qµn (UT ∇(nT ) − n∇V ) ,
q
¶
µ
d nkB (T − T0 )
d
∂t
kB T n − div S + J · ∇V = −
,
2
2
τw
2
J
τp k B
S = kB T +
κ0 nT ∇T.
q
m
In order to see that this system of equations can be written as an energy-transport
model, we introduce the chemical potential µ by
n = N (kB T )d/2 eqµ/kB T ,
where N is the effective density of states (see Example 5.10). A lengthy computation shows that the above system of equations can be written equivalently in
the variables qµ/kB T and −1/kB T as
1
∂t n − div J = 0,
q
where
and
d
E = kB T · n,
2
∂t E − div S + J · ∇V = W (µ, T ),
W (µ, T ) = −
µ
d nkB (T − T0 )
2
τw
¶
µ
¶
qµ
−1
q∇V
J = D11 ∇
,
+ D12 ∇
− D11
kB T
kB T
kB T
µ
¶
µ
¶
qµ
q∇V
−1
S = D21 ∇
+ D22 ∇
− D21
,
kB T
kB T
kB T
103
with the diffusion matrix
qτp
nkB T
(Dij )ij =
m
Ã
1
d+2
kB T
2
d+2
T
³¡ ¢ 2 kB ´
d+2 2
+ κ0 (kB T )2
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 conductivity
term div (κ0 nT ∇T ) is necessary to obtain a uniformly parabolic set of equations.
We recall, however, that this term is introduced heuristically, and no justification
is available.
Drift-diffusion limit in the energy-transport equations. It is possible to derive the drift-diffusion model from the energy-transport equations. For
this we consider the energy-transport model in the scaled form (5.88)-(5.89) and
assume that the reference time τ0 is much larger than the energy relaxation time:
τ0
À 1.
τw
This is exactly the second condition needed in the drift-diffusion limit. The
formal limit τw /τ0 → 0 in (5.88)-(5.89) yields T = 1 and, after scaling back to
the original variables, the drift-diffusion equations (5.87).
The relaxation-time limits studied in this section are summarized in Figure
5.2.
τp /τ0 → 0 ¡
¡
ª
¡
¡
¡
¡Hydrodynamic@
@
¡
Energy-transport
-
τw /τ0 → 0
@
τp /τ0 → 0
@ τw /τ0 → 0
@
@
R
@
Drift-diffusion
Figure 5.2: Relaxation-time limits in the hydrodynamic and energy-transport
models
5.7
The extended hydrodynamic model
The heuristic addition of the heat flux −div (κ∇T ) in the energy equation of the
hydrodynamic 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 socalled maximum entropy principle. This method provides a set of closed moment
equations consisting of the conservation laws of mass, momentum, energy and
energy 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 been
first used by Anile and Pennisi [2] for the derivation of charge transport equations
for semiconductors. It is based on principles of extended thermodynamics [35].
We start with the semiclassical Boltzmann equation in R3
q
∂t f + v(k) · ∇x f − E · ∇k f = Q(f ),
~
x ∈ R3 , k ∈ B, t > 0,
(5.90)
where v(k) = ~−1 ∇k ε(k). The collision operator Q(f ) is assumed to satisfy
Z
Q(f ) dk = 0.
B
In order to simplify the presentation, we assume parabolic bands
ε(k) =
~2 |k|2
,
2m∗
k ∈ B = R3 ,
such that v(k) = ~k/m∗ . The maximum entropy principle also works for general
band diagrams (see [3] for details). We define the first moments
Z
hMi i =
f (x, k, t) Mi (k) dk,
R3
where
M0 (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 Boltzmann equation (5.90) by Mi and integrating over k ∈ R3 :
~
div x hki = 0,
m∗
Z
q
~
Q(f )k dk,
∂t hki + ∗ div x hk ⊗ ki + Eh1i =
m
~
R3
Z
~
2q
2
2
∂t h|k| i + ∗ div x hk|k| i + E · hki =
Q(f )|k|2 dk,
m
~
R3
q
~
∂t hk|k|2 i + ∗ div x hk ⊗ k|k|2 i + E · h∇k ⊗ (k|k|2 )i
~
Z m
∂t h1i +
=
R3
Q(f )k|k|2 dk,
105
(5.91)
(5.92)
(5.93)
(5.94)
since, using integrating by parts
Z
Z
q
q
2
∇k f |k| dk =
f ∇k |k|2 dk
E·
− E·
~
~
3
R3
ZR
2q
=
f k dk
E·
~
R3
2q
=
E · hki.
~
We can write these moment equations in a more compact form as
∂t hMi i + div x Fi + E · Gi = Pi ,
i = 0, 1, 2, 3,
where Fi , Gi and Pi are defined in an obvious way. The moments hMi i can
be considered as the fundamental variables since they have a direct physical
interpretation. Indeed, we have:
n = hM0 i
q~
J = − ∗ hM1 i
m
~2
ne =
hM2 i
2m∗
~3
S = −
hM3 i
2(m∗ )2
electron density,
electron current density,
electron energy density,
energy flux.
The goal now is to express Fi , Gi and Pi in terms of the moments hMi i. Since
F0 = hM1 i,
F2 = hM3 i,
G0 = 0,
G1 = hM0 i,
G2 = hM1 i,
we only have to compute F1 , F3 , G3 and the production terms P0 , . . . , P3 . For this
we choose a distribution function f0 which can be used to evaluate the unknown
moments, i.e., we set
Z
F1 = hk ⊗ ki =
f0 (k ⊗ k) dk,
R3
Z
F3 = hk ⊗ k|k|2 i =
f0 (k ⊗ k)|k|2 dk,
R3
and so on. The function f0 is defined as the extremal of the entropy functional
Z
s(f ) =
f (log f − 1) dk
R3
under the constraints that it yields exactly the known moments hMi i. This means
that we have to minimize s (or, equivalently, to maximize the physical entropy
−s) under the constraints
Z
Mi (k)f0 (k) dk = hMi i, i = 0, . . . , 3.
(5.95)
R3
106
Using Lagrange multipliers, we have to analyze the functional
H(f ) =
Z
R3
f (log f − 1) dk +
3
X
λi
i=0
µZ
R3
¶
Mi f dk − hMi i .
A simple computation shows that
dH(f )
(g) =
df
Z
R3
g log f dk +
3
X
λi
i=0
Z
R3
Mi g dk.
The necessary condition
dH(f0 )
(g) = 0 ∀g
df
for the extremal f0 leads to
Z
R3
³
g log f0 +
and hence
3
X
´
λi Mi dk = 0 ∀g
3
X
´
λi Mi (k) ,
i=0
³
f0 (k) = exp −
i=0
k ∈ R3 .
(5.96)
The Lagrange multipliers are obtained by inserting this expression into (5.95):
hMi i =
Z
R3
³
Mi (k) exp −
3
X
i=0
´
λi Mi (k) dk.
(5.97)
This is an implicit equation for λ0 , . . . , λ3 . Once we have found the values of
λ0 , . . . , λ3 (which are functions of Mi and hMi i), we can define the maximum
entropy distribution function f0 from (5.96), and inserting the expression for f0
into the definitions of Fi , Gi and Pi gives finally a closed set of equations.
Unfortunately, the implicit equation (5.97) usually cannot be inverted directly
to give explicit expressions for the Lagrange multipliers. If we assume that the
distribution function f0 is in some sense not far from the thermal equilibrium
distribution function
¶
µ
ε(k)
εF
−
(5.98)
feq (k) = exp
kB T
kB T
(see Lemma 3.2(3) for Boltzmann statistics), we can develop the Lagrange multipliers in terms of a small parameter δ > 0. Comparing (5.98) with (5.96) suggests
that the moments hM1 i and hM3 i (which appear in (5.98)) are of order 1 whereas
the moments hM2 i and hM4 i (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
hMi i in the following way:
λi = ai1 + ai2 δhM1 i · δhM3 i + ai3 δhM1 i · δhM1 i
+ ai4 δhM3 i · δhM3 i + O(δ 3 ), i = 0, 2,
λj = aj1 · δhM1 i + aj2 · δhM3 i + O(δ 3 ), j = 1, 3,
where we replaced hM1 i and hM3 i by the rescaled moments δhM1 i and δhM3 i,
respectively, and where the functions aij depend on hM0 i and hM2 i. The facts
that λ0 and λ2 do not contain first-order terms in δ and that λ1 and λ3 do not
contain 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 and
δ is a small anisotropy parameter (see [3]). The above equations can be written,
up to second order in δ, as
(0)
(2)
λ0 = λ 0 + δ 2 λ0 ,
(1)
λ1 = δλ1 ,
(0)
(2)
λ2 = λ 2 + δ 2 λ2 ,
(1)
λ3 = δλ3 .
We insert this expansion into (5.96) and use the approximation
e−x = 1 − x +
x2
+ O(x3 ) (x → 0)
2
to obtain, up to second order in δ,
h
i
(0)
(2)
(1)
(0)
(2)
(1)
f0 = exp −(λ0 + δ 2 λ0 ) − δλ1 · k − (λ2 + δ 2 λ2 )|k|2 − δλ3 · k|k|2
h
i
h
i
(0)
(0)
(1)
(1)
(2)
2
2
2 (2)
2
= exp −(λ0 + λ2 |k| ) exp −δ(λ1 + λ3 |k| ) · k − δ (λ0 + λ2 |k| )
in
h
(2)
(2)
(1)
(1)
(0)
(0)
= exp −(λ0 + λ2 |k|2 ) 1 − δ(λ1 + λ3 |k|2 ) · k − δ 2 (λ0 + λ2 |k|2 )
o
δ 2 (1)
(1)
+ ((λ1 + λ3 |k|2 ) · k)2 .
(5.99)
2
Using this expression in the constraints (5.95) allows to compute the Lagrange
multipliers.
(0)
(0)
(1)
(1)
(2)
First we compute λ0 and λ2 . Setting A = λ1 + λ3 |k|2 and B = λ0 +
(2)
λ2 |k|2 , we infer
·
¸
Z
(0)
(0)
δ2
2
2
−λ2 |k|2
−λ0
1 − δA · k − δ B + (A · k) dk.
e
n = hM0 i = e
2
R3
108
Equating equal powers of δ gives
n = e
(0)
−λ0
(0)
Z
(0)
R3
e−λ2
|k|2
(0)
= (2λ2 )−3/2 e−λ0
Z
dk
R3
e−|u|
2 /2
du
(0)
(0)
= π 3/2 (λ2 )−3/2 e−λ0 .
(5.100)
Similarly, the expression
~2
hM2 i
2m∗
·
¸
Z
(0)
δ2
~2 −λ(0)
2
−λ2 |k|2
2
2
0
e
|k| 1 − δA · k − δ B + (A · k) dk
e
=
2m∗
2
R3
ne =
yields
ne =
=
=
=
Z
(0)
~2 −λ(0)
2
0
e−λ2 |k| |k|2 dk
e
∗
2m
R3
Z
2
~
2
(0) −5/2 −λ(0)
(2λ2 )
e 0
e−|u| /2 |u|2 du
∗
2m
R3
2
(0)
~
(0)
(2λ2 )−5/2 e−λ0 · 3(2π)3/2
∗
2m
3~2 π 3/2 (0) −5/2 −λ(0)
(λ2 )
e 0 .
4m∗
(5.101)
(0)
(0)
The equations (5.100) and (5.101) can be used to compute λ0 and λ2 . From
3 ~2 (0) −1
ne
e=
=
(λ )
n
4 m∗ 3
follows
(0)
λ2
3~2
=
,
4m∗ e
(0)
λ0
3
= − ln n + ln
2
µ
¶
4πm∗
e .
3~2
(1)
(1)
(2)
Now we compute the remaining Lagrangian multipliers λ1 , λ3 , λ0 , and
(2)
λ2 . Recalling that we have replaced hM1 i and hM3 i by the rescaled expressions
δhM1 i and δhM3 i we obtain
q~
δhM1 i
m∗
¸
·
Z
2
(0)
q~ −λ(0)
δ
2
−λ
|k|
2
2
e 2
k 1 − δA · k − δ B + (A · k) dk
= − ∗e 0
m
2
R3
δJ = −
109
and
~3
δhM3 i
2(m∗ )2
·
¸
Z
(0)
(0)
δ2
~3
2
−λ2 |k|2
2
−λ0
2
e
k|k| 1 − δA · k − δ B + (A · k) dk.
e
−
2(m∗ )2
2
R3
equal powers of δ leads to
Z
´
³
(0)
q~ −λ(0)
(1)
(1)
−λ2 |k|2
2
0
(5.102)
e
k λ1 + λ3 |k| · k dk,
− ∗e
m
R3
· ³
¸
Z
´2
(0)
1
(1)
(1)
(2)
(2)
−λ2 |k|2
2
e
k
(λ1 + λ3 |k| ) · k − (λ0 + λ2 |k|2 ) dk, (5.103)
2
R3
δS = −
=
Equating
J =
0 =
and
Z
³
´
(0)
(0)
~3
(1)
(1)
−λ0
−λ2 |k|2
2
2
e
S = −
λ
e
k|k|
+
λ
· k dk,
(5.104)
|k|
1
3
2(m∗ )2
R3
· ³
¸
Z
´2
(0)
(1)
(1)
(2)
(2)
2
−λ2 |k|2
2 1
2
(λ1 + λ3 |k| ) · k − (λ0 + λ2 |k| ) dk.(5.105)
e
k|k|
0 =
2
R3
(1)
(1)
The multipliers λ1 and λ3 can be computed by solving the linear system given
(2)
(2)
by (5.102) and (5.104). Then the remaining parameters λ0 and λ2 are solutions
of the linear system (5.103) and (5.105). The general structure of the multipliers
is as follows:
λ1
(1)
= a11 (n, e)J + a12 (n, e)S,
(1)
λ3
(2)
λ0
(2)
λ2
= a31 (n, e)J + a32 (n, e)S,
= a01 (n, e)J · J + a02 (n, e)J · S + a03 (n, e)S · S,
= 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
µ
¶3/2
µ
¶
3~2
3~2 |k|2
f0 = n
exp −
(5.106)
4πm∗ e
4m∗ e
n
× 1 − (a21 J + a22 S) · k − (a11 + a31 |k|2 )J · J − (a12 + a32 |k|2 )J · S
¤2 o
δ2 £
− (a13 + a33 |k|2 )S · S +
(a21 + a41 |k|2 )J · k + (a22 + a42 |k|2 )S · k
.
2
Hence, the unknown moments are given by
Z
F1 = hk ⊗ ki =
f0 (k ⊗ k) dk = F1 (n, e, J, S),
2
R3
F3 = hk ⊗ k|k| i = F3 (n, e, J, S),
G3 = h∇k ⊗ (k|k|2 )i = h|k|2 iId + 2hk ⊗ ki =
110
m∗
ne · Id + F1 (n, e, J, S).
~2
From (5.106) we see that the dependence of F1 and F3 on J, S is at most
quadratic, i.e., F1 and F3 are in fact functions of J · J, J ⊗ J, J · S, S ⊗ S
and so on. The precise formulas for F1 and F3 are given in [3, (124), (125)].
The production terms Pi are approximated by
Z
Q(f0 )k dk,
P1 =
R3
Z
Q(f0 )|k|2 dk,
P2 =
3
ZR
Q(f0 )k|k|2 dk.
P3 =
R3
In order to get explicit formulas for Pi as functions of n, J, e and S, the collision
operator 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 refer
to 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 the
deviations of the microscopic velocity v(k) from the averaged macroscopic velocity
u = −J/qn:
~k
ū(k) = v(k) + u = ∗ + u.
(5.107)
m
The electron temperature T and the heat flux σ are defined as moments of the
deviations:
Z
m∗
nkB T =
f |ū|2 dk,
3 R3
Z
m∗
nσ =
f |ū|2 ū dk.
2 R3
Then we can reformulate the energy density, using (5.107), as
Z
~
f |k|2 dk
ne =
2m∗ R3
Z
¡
¢
m∗
=
f |u|2 − 2u · ū + |ū|2 dk
(using (5.107))
2 R3
3
m∗
n|u|2 + nkB T,
=
2
2
since
Z
R3
f ū dk =
Z
R3
f
µ
~k
−u
m∗
111
¶
dk = −
J
− nu = 0.
q
(5.108)
The energy flux can be written as
Z
~3
f k|k|2 dk
S = −
∗
2
2(m ) R3
Z
m∗
f (ū − u)(|ū|2 − 2u · ū + |u|2 ) dk (using (5.107)
= −
2 R3
Z
m∗
f (ū|ū|2 − u|ū|2 + 2(u · ū)ū − u|u|2 ) dk (using (5.108)
= −
2 R3
Z
m∗
3
∗
f (u · ū)ū +
= −nσ + nkB T u − m
nu|u|2 .
2
2
3
R
Computing
Z
X Z
f (u · ū)ūj dk =
ui
¶
~kj
+ uj dk
f
m∗
R3
R3
i
"µ ¶ Z
#
Z
2
X
~
~2
f (ki uj + kj ui ) dk
f ki kj dk + nui uj + ∗
=
ui
m∗
m R3
R3
i
i
h³ ~2 ´2
2
hk
⊗
ki
·
u
+
3nu|u|
,
=
m∗
j
we obtain
S = ne − nσ +
Introducing
F¯1 = q
and
µ
µ
µ
~
m∗
~
m∗
¶2
~ki
+ ui
m∗
¶2
¶µ
hk ⊗ ki · u + 3nu|u|2 .
F1 ,
F̄3 =
~4
F3
2(m∗ )3
~2
~3
q~
P
,
P̄
=
P
,
P̄
=
−
P3 ,
1
2
2
3
m∗
2m∗
2(m∗ )3
we can write the system (5.91)-(5.94) as follows:
P̄1 = −
1
∂t n − div J = 0,
q
where
q2
∂t J − div F̄1 (n, e, S, J) − ∗ En = P̄1 ,
m
∂t (ne) − div S − E · J = P̄2 ,
´
1³ q
∂t S − div F̄3 (n, e, S, J) +
ne · Id + F̄1 · E = P¯3 ,
2 m∗
m∗ |J|2 3
+ nkB T,
2q 2 n
2
1
3J|J|2
S = ne − nσ + F̄1 (n, e, J, S) − 3 2 ,
q
q n
ne =
112
and F̄1 , F̄3 , P̄1 , P̄2 and F̄1 are given explicitly in [3]. The above system of equations
is 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) are
n, J, ne and S, the independent variables of the above equations are n, J, T and
σ. Notice that in the above derivation, there was no need to introduce heuristic
terms.
113
6
The Drift-Diffusion Equations
In this chapter the drift-diffusion model is studied in more detail. We recall the
equations for electrons and holes:
1
∂t n − div Jn
q
1
∂t p + div Jp
q
Jn
Jp
εs ∆V
= −R(n, p),
(6.1)
= −R(n, p),
(6.2)
= qµn (UT ∇n − n∇V ),
= −qµp (UT ∇p + p∇V ),
= q(n − p − C), x ∈ Ω, t > 0,
(6.3)
(6.4)
(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.
∂t n = ∂ t p = 0
and
J n = Jp = 0
in Ω.
This implies R(n, p) = 0 or np = n2i in Ω and
0 = UT ∇n − n∇V = n∇(UT ln n − V ),
0 = UT ∇p + p∇V = p∇(UT ln p + V ).
It is physically reasonable to assume that the particle densities n and p are
positive in Ω. (In fact, this can be proved using maximum principle arguments
if the boundary data is positive; see [33]). This yields
α := UT ln n − V = const.,
β := UT ln p + V = const.
(6.6)
or
n = 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 = n2i
and the fact that V is determined only up to an additive constant. Then the
sum of the two equations in (6.6) yields α + β = UT ln(np) = 2UT ln ni and,
substituting V by V + γ, where γ := UT ln ni − α, gives
n = e(α+γ)/UT eV /UT = ni eV /UT
114
(6.7)
and
p = e(β−γ)/UT e−V /UT = e(β+α−UT ln ni )/UT e−V /UT = ni e−V /UT .
(6.8)
The electrostatic potential satisfies the semilinear elliptic equation
εs ∆V
= q(n − p − C) = q(ni eV /UT − ni e−V /UT − C)
´
³
V
−C
in Ω.
= q 2ni sinh
UT
In order to get a unique solution of this differential equation, we have to impose
boundary conditions.
We assume that the boundary of the semiconductor domain consists of two
parts: one part, called the Dirichlet part, on which the particle densities and the
potential are prescribed,
n = nD , p = p D , V = V D
on ΓD ,
(6.9)
and another part, the Neumann part, which models the insulating boundary
segments on which the normal components of the current densities and the electric
field 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 :
nD pD = n2i
(6.13)
• The boundary potential is the superposition of the built-in-potential Vbi and
the applied voltage U :
VD = Vbi + U.
(6.14)
Clearly, in thermal equilibrium we have U = 0. The built-in potential is the
potential corresponding to the equilibrium densities given by (6.7) and (6.8):
nD = ni eVbi /UT ,
pD = ni e−Vbi /UT .
Substituting (6.12) in (6.13) gives n2D − CnD = n2i such that
µ
¶
µ
¶
q
q
1
1
2
2
2
2
nD =
C + C + 4ni , pD =
−C + C + 4ni
2
2
115
(6.15)
on ΓD .
Thus we get from (6.15)
nD
Vbi = UT ln
= UT ln
ni
Ã
C
+
2ni
s
C2
+1
4n2i
!
= UT arsinh
C
.
2ni
(6.16)
Therefore, the thermal equilibrium state (ne , pe , Ve ) is the (unique) solution
of
¶
Ve
−C
in Ω,
εs ∆Ve = q 2ni sinh
UT
Ve = Vbi on ΓD , ∇Ve · ν = 0 on ΓN ,
µ
where Vbi is given by (6.16), and
ne = ni eVe /UT ,
pe = ni e−Ve /UT
in Ω.
Moreover, the drift-diffusion equations (6.1)-(6.5) are complemented with the
boundary conditions (6.9)-(6.11), where (nD , pD , VD ) are defined by (6.14)-(6.16),
and the initial conditions
n(·, 0) = nI ,
6.2
p(·, 0) = pI
in Ω.
Scaling of the equations
The drift-diffusion model (6.1)-(6.5), (6.9)-(6.11) contains several physical parameters. By bringing the equations in a non-dimensional form, we can identify
the relevant scaled parameters. In fact, we show that there are only two relevant
parameters. Define
C = sup |C(x)|, L = diam(Ω).
x∈Ω
We expect that the particle densities are of the order of C and the potential is of
the order of UT . Then the scaled variables ns , ps and Vs , defined by
n = Cns ,
p = Cps ,
V = U T Vs ,
are of the order 1. We also set
x = Lxs ,
C = CCs ,
t = τ ts
with
L2 µ
, µ = max(µn , µp ).
UT
This fixes the time scale to the order τ. A simple computation shows that the
scaled quantities satisfy the equations (where we omitted the index “s”):
τ=
∂t n − µ0n div (∇n − n∇V ) = −R(n, p),
∂t p − µ0p div (∇p + p∇V ) = −R(n, p),
λ2 ∆V
116
= n − p − C,
(6.17)
(6.18)
(6.19)
where µ0α = µα /µ (α = n, p), and
λ=
s
εs U T
qL2 C
q
is called the scaled Debye length. (The Debye length is the quantity εs UT /qC =
Lλ.) The scaled recombination-generation term reads
R(n, p) =
np − δ 4
,
τp0 (n + δ 2 ) + τn0 (p + δ 2 )
where τα0 = τα /τ (α = n, p) and
ni
C
is the ratio of the intrinsic density and the maximal doping concentration. The
relevant parameters are λ and δ. Typically, they are small compared to one. For
instance, in silicon we have
δ2 =
εs = 1.05 · 10−10 As/Vm,
µn = 0.15 m2 /Vs,
ni = 1016 m−3 ,
µp = 0.045 m2 /Vs.
Choosing
L = 10−6 m,
C = 1023 m−3 ,
UT = 0.026 V (i.e. T = 300 K),
we obtain
λ2 = 1.7 · 10−4 ,
δ 2 = 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
(C + C 2 + 4δ 4 ), pD =
2
C
= Vbi + U = arsinh 2 + U
2δ
√
1
(−C + C 2 + 4δ 4 ),
2
(6.20)
VD
on ΓD ,
(6.21)
and the built-in potential can be written as
Ã
!
r
C
C2
nD
δ2
Vbi = ln
+
+
1
=
ln
=
ln
.
2δ 2
4δ 4
δ2
pD
117
(6.22)
6.3
Static current-voltage characteristic of a diode
In this subsection we derive the current-voltage characteristic of a pn-junction
diode in the case of “small” current densities. The diode is defined by
½
Cn : x ∈ Ω n
C(x) =
−Cp : x ∈ Ωp ,
where Cn > 0 is the doping concentration in the n-region Ωn , −Cp < 0 is the
doping density in the p-region Ωp , and Ω = Ωn ∪ Ωp , Ωn ∩ Ωp = ∅. The Dirichlet
boundary ΓD is supposed to consist of the two boundary parts Γ1 ⊂ Ωn and
Γ2 ⊂ Ωp (see Figure 6.1). We set Γ = Ωn ∩ Ωp . The applied voltage U is given by
½
0
: x ∈ Γ1
U (x) =
U0 : x ∈ Γ 2 ,
where U0 is a constant.
Γ1
Ωn
Γ
ΓN
Ωp
ΓN
Γ2
Figure 6.1: Geometry of a pn-junction diode.
The total current is defined by
Z
(Jn + Jp ) · ν ds,
I=
Γ1
where ν is the exterior unit normal to ∂Ωn . We assume (for simplicity) that
µ0n = µ0p = 1 (see (6.17)-(6.18)). Then, taking the difference of the stationary
versions of (6.17)-(6.18), we have div (Jn +Jp ) = 0 and, by the divergence theorem
and Jn · ν = Jp · ν = 0 on ΓN (see (6.10)),
Z
Z
Z
(Jn + Jp ) · ν dx + (Jn + Jp ) · ν dx.
div (Jn + Jp ) dx =
0=
Γ
Γ1
Ωn
Thus
I=
Z
Γ1
(Jn + Jp ) · ν ds = −
Z
Γ
(Jn + Jp ) · ν ds.
Our main goal is to describe the function I = I(U0 ) in more detail.
118
(6.23)
Our first main assumption is to consider “small” current densities. More
precisely, we assume that the current densities are of the order of δ 4 . Then we
can rescale Jn and Jp by
J n = δ 4 jn ,
J p = δ 4 jp .
(6.24)
We see later in this section why we use the exponent 4 in the rescaling. The
new variables jn and jp are of the order O(1). It is convenient to introduce new
variables u, w by
n = δ 2 eV u, p = δ 2 e−V w.
The functions u and w are called Slotboom variables. The advantage is that they
symmetrize the current relations in the following sense:
δ 2 eV ∇u = eV ∇(e−V n) = ∇n − n∇V = Jn ,
−δ 2 e−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)):
δ 2 div (eV ∇u) = R(δ 2 eV u, δ 2 e−V w),
δ 2 div (e−V ∇w) = R(δ 2 eV u, δ 2 e−V w),
λ2 ∆V = δ 2 eV u − δ 2 e−V w − C
in Ω.
(6.25)
(6.26)
(6.27)
The boundary conditions for u and w follow from (6.20)-(6.22):
2
u = δ −2 e−V n = δ −2 e−Vbi nD = δ −2 e− ln(nD /δ ) nD = 1,
2
w = δ −2 eV p = δ −2 eVbi pD = δ −2 eln(δ /pD ) pD = 1,
V = Vbi + U = Vbi
on Γ1 ,
and
u = δ −2 e−Vbi −U nD = e−U0 ,
w = δ −2 eVbi +U pD = 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 still
equivalent 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 the
parameters λ and δ are small compared to one. Hence the solution of the reduced
problem 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 rigorously
by means of asymptotic analysis; see [33]).
First, we study the reduced problem λ = 0. From (6.27) we obtain
0 = δ 2 eV u − δ 2 e−V w − C
in Ω
and therefore
δ 2 eV =
´
√
1 ³
C + C 2 + 4δ 4 uw ,
2u
δ 2 e−V =
´
√
1 ³
−C + C 2 + 4δ 4 uw .
2w
We can discard one solution of the above quadratic equation, since we require
u, w ≥ 0. Thus (6.25) and (6.26) become
· ³
¸
´
√
1
div
C + C 2 + 4δ 4 uw ∇u = R,
(6.28)
2u
·
¸
´
√
1 ³
−C + C 2 + 4δ 4 uw ∇w = R,
div
(6.29)
2w
with the boundary conditions
u = w = 1 on Γ1 ,
u = e−U0 , w = eU0
∇u · ν = ∇w · ν = 0 on ΓN .
on Γ2 ,
(6.30)
(6.31)
Transforming back to the original variables, the particle densities n and p are
functions of u and w :
√
1
n = δ 2 eV u = (C + C 2 + 4δ 4 uw),
2
√
1
2 −V
p = δ e w = (−C + C 2 + 4δ 4 uw).
2
Notice that the doping profil C is discontinuous across Γ. This means that also the
diffusion 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 generalized (weak) sense. The particle densities are also discontinuous across Γ. Thus,
we expect that the particle densities solving the full problem have an internal
layer (i.e. sharp gradients) near Γ.
Next we consider the reduced problem δ = 0. Recall that
√
1
(C + C 2 + 4δ 4 uw)∇u,
2u
√
1
δ 4 jp = Jp = −δ 2 e−V ∇w = − (−C + C 2 + 4δ 4 uw)∇w,
2w
δ 4 jn = Jn = δ 2 eV ∇u =
and
R = δ4
τp0 (n
uw − 1
.
+ δ 2 ) + τn0 (p + δ 2 )
120
(6.32)
(6.33)
Hence we can write (6.28)-(6.29) as
uw − 1
uw − 1
, div jp = − 0
. (6.34)
2
0
2
+ δ ) + τn (p + δ )
τp (n + δ 2 ) + τn0 (p + δ 2 )
√
The Taylor expansion 1 + x = 1 + x/2 + O(x2 ) around x = 0 yields in Ωp
q
√
C + C 2 + 4δ 4 uw = −Cp + Cp2 + 4δ 4 uw
¶
µ
2δ 4 uw
8
= −Cp + Cp 1 +
+ O(δ )
Cp2
2uw 4
δ + O(δ 8 ),
=
Cp
√
2uw 4
δ + O(δ 8 ) as δ → 0
−C + C 2 + 4δ 4 uw = 2Cp +
Cp
div jn =
τp0 (n
and in Ωn
√
2uw 4
C 2 + 4δ 4 uw = 2Cn +
δ + O(δ 8 )
Cn
√
√
−C + C 2 + 4δ 4 uw = −Cn + C 2 + 4δ 4 uw
2uw 4
δ + O(δ 8 ) as δ → 0.
=
Cn
C+
Therefore, (6.32) and (6.33) give
Cp ∇w
w∇u
+ O(δ 4 ),
δ 4 jp =
+ O(δ 4 )
Cp
w
Cn ∇u
u∇w
δ 4 jn =
+ O(δ 4 ), jp = −
+ O(δ 4 )
u
Cn
in Ωp ,
jn =
in Ωn .
Here we see why we have used the scaling factor δ 4 in (6.24). Exponents larger
or smaller than 4 lead to useless relations. Moreover,
n = Cn , p = 0
n = 0,
p = Cp
in Ωn ,
in Ωp .
Now we can perform the limit δ → 0 in (6.34) and in the above equations:
uw − 1
u∇w
, div jp = − 0
Cn
τp C n
w∇u
uw − 1
∇w = 0, jn =
,
div jn = 0
Cp
τn C p
∇u = 0,
jp = −
in Ωn ,
in Ωp .
From the boundary conditions (6.30) and (6.31) we conclude
u = 1 in Ωn ,
w = e U0
121
in Ωp .
Thus
jp = −
∇w
Cn
in Ωn ,
jn =
e U0
∇u in Ωp
Cp
(6.35)
and
∆w = −Cn div jp =
w−1
τp0
∆u = Cp e−U0 div jn =
in Ωn ,
u − e−U0
τn0
in Ωp .
The boundary conditions for w and u read
w = 1 on Γ1 ,
w = e U0
on Γ,
∇w · ν = 0 on ∂Ωn ∩ ΓN
u = 1 on Γ,
∇u · ν = 0 on ∂Ωp ∩ ΓN .
and
u = e−U0
on Γ2 ,
We wish to separate the influence of the applied potential. We claim that the
solutions 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 = 0 on Γ2 ,
f = 1 on Γ,
and
∆g =
g = 0 on Γ1 ,
f
τn0
g
τp0
g = 1 on Γ,
in Ωp ,
(6.36)
∇f · ν = 0 on ∂Ωp ∩ ΓN
in Ωn ,
(6.37)
∇g · ν = 0 on ∂Ωn ∩ ΓN .
Indeed, it holds
¡ U
¢
0
e
−
1
g
w−1
∆w = eU0 − 1 ∆g =
=
in Ωn ,
0
τp
τp0
¡
¢
−U0
¡
¢
1
−
e
f
u − e−U0
=
in Ωp ,
∆u = 1 − e−U0 ∆f =
τn0
τn0
¡
¢
and it is easy to verify the boundary conditions for u and w. The advantage of
the introduction of the functions f and g is that they depend on the geometry of
the 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 continuity
of the normal components of jn in Ωp and jp in Ωn , respectively, we get from
e U0
e U0 − 1
∇u =
∇f
Cp
Cp
e U0 − 1
∇w
=−
∇g
= −
Cn
Cn
jn =
jp
in Ωp ,
in Ωn
and from (6.23)
I=−
Z
Γ
¡
(jn + jp ) · ν ds = − e
U0
−1
¢
Z µ
Γ
∇g
∇f
−
Cp
Cn
¶
· ν ds.
Notice that also the total current is scaled by the factor δ 4 . Introducing the
leakage current,
¶
Z µ
∇g ∇f
Is :=
· ν ds,
−
Cn
Cp
Γ
we obtain the scaled Shockley equation (see Figure 6.2)
¡
¢
I = I s e U0 − 1 .
(6.38)
I 6
-
−Is
U0
Figure 6.2: The Shockley equation for a pn-junction diode.
The leakage current is positive. Indeed, we use integration by parts and the
differential equations for f to infer
µ
¶
Z
Z
Z
f2
1
f
1
−f ∆f + 0
∇f · νds =
∇f · (−ν)ds +
dx
−
τn
Γ Cp
Ωp C p
Γ Cp
(using f = 1 on Γ and the differential equation for f )
¶
µ
Z
f2
1
2
dx
=
|∇f | + 0
τn
Ωp C p
(using integration by parts)
> 0.
123
Notice that −ν is the exterior unit normal to ∂Ωp . Similarly,
¶
µ
Z
Z
1
1 2
1
2
dx =
|∇g| + 0 g
∇g · ν ds.
τp
Γ Cn
Ωn C n
Therefore
Is =
Z
Ωp
1
Cp
µ
1
|∇f | + 0 f 2
τn
2
¶
dx +
Z
Ωn
1
Cn
µ
1
|∇g| + 0 g 2
τp
2
¶
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 sufficiently small applied potentials but not for “large” values of |U0 |. There exists a
threshold voltage Uth < 0 such that I(U0 ) → −∞ as U0 & Uth . This break-down
effect is used in so-called Zener diodes.
We study the high-injection case U0 À 1 for a simple one-dimensional symmetric diode modeled by the reduced problem λ = 0. For this we define Ω =
(−1, 1),
½
U0
−1 : x ∈ (−1, 0)
C(x) =
,
U (x) = − x, x ∈ Ω.
1 : x ∈ (0, 1)
2
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))
1
U0
V (1) = Vbi (1) + U (1) = arsinh 2 − ,
2δ
2
√
√
1
1
p(1) = (−1 + 1 + 4δ 4 ), n(1) = (1 + 1 + 4δ 4 ).
2
2
V (0) = 0,
Taking the difference of the first two equations in (6.39) gives
0 = (n + p)x − (n − p)Vx = (n + p)x − Vx
and
Vx = (n + p)x
124
in (0, 1).
(6.40)
Thus, adding the first two equations in (6.39) yields
2I = −(n + p)Vx = −
1 d
[(n + p)2 ].
2 dx
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
p
1 + 4δ 4 + 4I(1 − x) = (n + p)(x) = 2n(x) − 1.
We conclude
p
1
(1 + 1 + 4δ 4 + 4I(1 − x)),
2
p
1
(−1 + 1 + 4δ 4 + 4I(1 − x)).
p(x) =
2
n(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)
√
U0 p
1
+ 1 + 4δ 4 + 4I(1 − x) − 1 + 4δ 4 .
= arsinh 2 −
2δ
2
At x = 0 we obtain
p
√
1
U0
= arsinh 2 + 1 + 4δ 4 + 4I(1 − x) − 1 + 4δ 4 .
2
2δ
If U0 → ∞ or, equivalently, I → ∞,
I ∼ U02 .
(6.41)
This result means that the growth of the voltage-current characteristic slows
down in high injection. The relation (6.39) is known as Mott’s law.
For sufficiently large voltages we have Ohm’s law
I
→ const. (U0 → ∞).
U0
Whereas the Zener effect is not modeled in our drift-diffusion system (see [34] for
a more general system), we can derive the Ohm law for U0 À 1. To see this, we
rescale the current densities and the potential,
Jn = U0α jn ,
Jp = U0β jp ,
125
V = U0γ φ,
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 = U0−γ (n − p − C) → 0 in Ω, as U0 → ∞,
φ = 0 on Γ1 , φ = U01−γ on Γ2 , ∇φ · ν = 0 on ΓN .
When γ > 1, φ solves the Laplace equation with homogenous boundary conditions
in the limit U0 → ∞ and therefore, φ = 0 in Ω. This solution is useless. On the
other hand, when γ < 1, φ → ∞ on Γ2 (U0 → ∞), which is not the right scaling.
Hence γ = 1. The scaled current densities are
jp = −U0−β ∇p − U01−β p∇φ.
jn = U0−α ∇n − U01−α n∇φ,
When α > 1, we get jn = 0 in the limit which is useless. When α < 1, we have
n∇φ = 0 which does not allow to compute jn . Therefore, α = 1. Similarly, we
conclude β = 1. This yields, as U0 → ∞,
jn = −n∇φ,
jp = −p∇φ.
Notice that φ solves the problem
φ = 0 on Γ1 ,
∆φ = 0 in Ω,
φ = 1 on Γ2 , ∇φ · ν = 0 on ΓN ,
and does not depend on U0 . We obtain (see (6.20))
Z
I
1
(U0 jn + U0 jp ) · ν ds
=
U0
U0 Γ1
Z
= −
(n + p)∇φ · ν ds
Γ1
Z
p
4
2
∇φ · ν ds
= − Cn + 4δ
Γ1
= const.
We remark that the assumption that n and p are of order O(1) as U0 → ∞ is
not very realistic. However, since we evaluate n and p only on Γ1 and they are
fixed by the boundary conditions, we expect to obtain the same formula as above
without this assumption.
Whereas the Mott law I ∼ U02 holds for “large” U0 , we needed to perform
the limit U0 → ∞ to derive the Ohm law I ∼ U0 . Therefore, we expect that the
Mott law holds for “large” applied voltages and the Ohm law for “very large”
applied voltages. This can be confirmed by physical experiments. Conditions on
the size of the applied voltage which allow to distinguish the different regimes for
“small”, “large” and “very large” applied bias, unfortunately, do not follow from
the above asymptotic analysis.
126
6.4
Numerical discretization of the stationary equations
The discretization of the drift-diffusion equations using standard methods may
lead to unsatisfactory results. Indeed, for “large” electric fields −∇V, the corresponding elliptic problem is convection-dominant, and it is well known that the
numerical discretization of such problems requires special care. In a simplified
situation, we can see this directly from the equations.
Consider the one-dimensional stationary equations for the electron density in
Ω = (0, 1) with µ0n = 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 ni+1 + ni Vi+1 − Vi
−
,
h
2
h
1
(Jn,i+1/2 − Jn,i−1/2 ) = 0,
h
(6.42)
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,
n0 = nD0 , nN = nD1 .
i = 1, . . . , N − 1,
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 the
above difference equation has the explicit solution
ni = (nD1 − nD0 )
ri − 1
+ nD0 ,
rN − 1
i = 0, . . . , N,
with r =
2+c
.
2−c
The discrete electron density ni should be positive for all i. However, with the
discretization (6.42)-(6.43) this may be not true. Indeed, let (for simplicity)
nD0 = 2, nD1 = 1, N even and let
c>2
21/N + 1
.
21/N − 1
Then c > 2, −21/N < r < −1 and 1 < r N < 2. If i is odd, then r i < 0, ri +rN < 2
and
ri + rN − 2
ri − 1
ni = N
+1=
< 0.
r −1
rN − 1
127
In fact, ni < 0 holds for all odd indices i and ni > 0 for all even indices i. In
order to avoid the oscillatory behaviour, one has to choose the grid in such a way
that c = Vi − Vi−1 < 2. In Section 6.3, we have found that the potential can be
written as
·
¸
√
1
2
4
(C + C + 4δ uw)
V = ln
in Ω
2δ 2 u
for the reduced problem λ = 0, and u, w solve the problem (6.28)-(6.31). This
means that if C is discontinuous in Ω, we expect large gradients for V. To get
a non-oscillating numerical scheme, we need to choose a very fine grid at least
near the points where the field is “large” (i.e. Vi − Vi−1 < 2). Particularly for
multi-dimensional problems the restriction that the potential only varies weakly
on each of the subintervals (xi , xi+1 ) may require extremely many grid points
and 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 is
to use the Scharfetter-Gummel discretization. The idea is to symmetrize first the
current densities and then to discretize the equations. We illustrate the idea for
the one-dimensional stationary equations
Jn,x = 0, Jn = µn (nx − nVx ),
Jp,x = 0, Jp = −µp (px + pVx ),
2
λ Vxx = n − p − C in Ω = (0, 1)
subject to the boundary conditions
n(0) = nD0 ,
n(1) = nD1 ,
V (0) = 0,
p(0) = pD0 ,
p(1) = pD1 ,
V (1) = U0 .
In the following we consider only the equation for n since the treatment of
the equation for p is analogous. Define the Slotboom variable z = µn ne−V (see
Section 6.3). Then
eV zx = µn (nx − nVx ) = Jn
and we have to discretize
Jn,x = 0,
zx = Jn e−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:
µ
¶
λ2
Vi+1 − Vi Vi − Vi−1
ni − pi − C(xi ) =
−
hi−1
hi
hi−1
·
µ
¶
¸
2
hi
hi
λ
Vi+1 − 1 +
Vi +
Vi−1 ,
=
hi−1 hi
hi−1
hi−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:
Z xi+1
Z xi+1
e−Vh (x) dx.
zh,x (x) dx = Jn
zi+1 − zi =
xi
xi
Here, zh (xi ) = zi are approximations of z(xi ). The last integral can be computed
explicitely since Vh is piecewise linear:
zi+1 − zi = Jn e−Vi
¢
¡
hi
1 − e−(Vi+1 −Vi ) .
Vi+1 − Vi
This gives a definition for Jn (after transforming back to ni = zi eVi /µn ):
Jn = µ n
=
¡ −Vi+1
¢
Vi+1 − Vi
e Vi
−Vi
e
n
−
e
n
i+1
i
hi
1 − eVi −Vi+1
µn
(B(Vi+1 − Vi )ni+1 − B(Vi − Vi+1 )ni ) ,
hi
(6.45)
with the Bernoulli function
B(x) =
ex
x
,
−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
¶
µ
¶ ¸
·µ
1
Vi+1 − Vi
eVi −Vi+1
1
1
−
Jn = µ n
+
ni+1 −
ni
hi
2 1 − eVi −Vi+1
1 − eVi −Vi+1
2
ni + ni+1 Vi+1 − Vi
− µn
2
hi
µ
¶
Vi+1 − Vi ni+1 − ni
ni + ni+1 Vi+1 − Vi
= µn (Vi+1 − Vi ) coth
− µn
.
2
hi
2
hi
The function
Vi+1 − Vi
2
plays the role of an upwind factor. Indeed, for slowly varying potentials |V i+1 −
Vi | ¿ 1 we have
f (x) = 1 as x → 0,
f (Vi+1 − Vi ) = (Vi+1 − Vi ) coth
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 standard
discretization (6.42), and for convection-dominated problems |Vi+1 − Vi | À 1, the
upwind factor provides a correction such that the diffusion term
µ
¶
Vi+1 − Vi ni+1 − ni
ni+1 − ni
(Vi+1 − Vi ) coth
∼ (Vi+1 − Vi )
2
hi
hi
is of the same order as the convection term
ni+1 + ni Vi+1 − Vi
.
2
hi
The current density for the holes is discretized analogously:
¶
µ
Vi+1 − Vi pi+1 − pi
pi+1 + pi Vi+1 − Vi
Jp = −µp (Vi+1 − Vi ) coth
− µp
.
2
hi
2
hi
We obtain three discrete nonlinear systems
an,i ni+1 + bn,i ni + cn,i ni−1 = 0,
ap,i pi+1 + bp,i pi + cp,i pi−1 = 0,
αi Vi+1 + βi Vi + γi Vi−1 = λ−2 hi−1 hi (ni − pi − C(xi )),
where we have defined
an/p,i
bn/p,i
cn/p,i
µ
¶
1
f (Vi+1 − Vi ) ∓ (Vi+1 − Vi ) ,
2
µ
¶
1
1
f (Vi+1 − Vi ) ∓ (Vi+1 − Vi )
= −
hi
2
¶
µ
1
1
−
f (Vi − Vi−1 ) ∓ (Vi − Vi−1 ) ,
hi−1
2
µ
¶
1
1
=
f (Vi − Vi−1 ) ± (Vi − Vi−1 ) ,
hi−1
2
1
=
hi
and
αi = 1,
βi = −1 −
hi
,
hi−1
130
γi = −
hi
.
hi−1
7
The Energy-Transport Equations
In this chapter, we analyze the energy-transport equations for spherically symmetric and strictly monotone energy bands ε(|k|) in R3 , as derived in Section 5.5
(see Example 5.9):
1
∂t n(µ, T ) − div J1 = 0,
q
∂t E(µ, T ) − div J2 + ∇V · J1 = W (µ, T ),
εs ∆V = q(n(µ, T ) − C(x)),
(7.1)
(7.2)
x ∈ Ω, t > 0, (7.3)
where the electron and energy current densities are given by (see (5.74)-(5.75))
¶
µ
¶
µ
q∇V
−1
qµ
+ D12 ∇
− D11
J1 = D11 ∇
,
(7.4)
kB T
kB T
kB T
µ
¶
µ
¶
qµ
−1
q∇V
J2 = D21 ∇
,
(7.5)
+ D22 ∇
− D21
kB T
kB T
kB T
the electron and energy density are defined, respectively, by
Z ∞p
qµ/kB T
n(µ, T ) = 2πe
γ(ε)γ 0 (ε)e−ε/kB T dε,
Z0 ∞ p
E(µ, T ) = 2πeqµ/kB T
γ(ε)γ 0 (ε)e−ε/kB T ε dε,
(7.6)
(7.7)
0
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 temperature, and C(x) the doping profile. We assume Maxwell-Boltzmann statistics
and space-independent scattering rates, so that the diffusion matrices Dij can be
written as (see (5.76))
Z ∞
qµ/kB T
Dij (µ, T ) = e
D(ε)e−ε/kB T εi+j−2 dε, i, j = 1, 2.
(7.9)
0
The function γ is defined by |k|2 = γ(ε(|k|)). The equations (7.1)-(7.3) are considered in the bounded domain Ω ⊂ Rd (d ≥ 1), and we impose the following
initial and boundary conditions
µ(·, 0) = µI , T (·, 0) = TI
µ = µD , T = T D , V = V D
J1 · ν = J2 · ν = ∇V · ν = 0
in Ω,
on ΓD × (0, ∞),
on ΓN × (0, ∞).
We have assumed that ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅, ΓD is open in ∂Ω, and ν is
the exterior unit normal of ∂Ω.
131
7.1
Symmetrization and entropy
In this section we introduce a change of unknowns which symmetrizes the energytransport model (7.1)-(7.7) in the sense that the lower-order terms −∇V · J 1 and
−Di1 q∇V /kB T are absent in the new formulation. Moreover, we define a socalled entropy functional which is non-increasing in time and which relates the
old and the new variables.
Definition 7.1 The variables
u1 =
qµ
,
kB T
u2 = −
1
kB T
are called (primal) entropy variables; the variables
w1 =
q(µ − V )
= u1 + qV u2 ,
kB T
w2 = −
1
= u2
kB T
are called dual entropy variables.
We set u = (u1 , u2 ) and w = (w1 , w2 ). In the entropy variables the electron
and energy density can be formulated as
Z ∞p
u1
γ(ε)γ 0 (ε)eεu2 dε,
ρ1 (u) := n = 2πe
Z0 ∞ p
γ(ε)γ 0 (ε)eεu2 ε dε.
ρ2 (u) := E = 2πeu1
0
In the dual entropy variables, the energy-transport model (7.1)-(7.5) can be
written in a symmetric form.
Lemma 7.2 The equations (7.1)-(7.5) are formally equivalent to
∂t b1 (u) − div I1 = 0,
∂t b2 (u) − div I2 = W − qn∂t V,
εs ∆V = q(n − C(x)),
(7.10)
(7.11)
where
b1 (u) = n,
and
b2 (u) = E − qV n
1
J1 = L11 ∇w1 + L12 ∇w2 ,
q
= J2 − V J1 = L21 ∇w1 + L22 ∇w2 .
I1 =
I2
The diffusion coefficients Lij are given by
L11 = D11 ,
L12 = L21 = D12 − D11 qV,
132
L22 = D22 − 2D21 qV + D11 (qV )2 .
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
∂ε
∂ε
∂ε ∂ε ∂ε
,
,
, ε
, ε
, and ε
∂k1 ∂k2 ∂k3 ∂k1 ∂k2
∂k3
(7.12)
are linearly independent.
Proof. A simple computation gives, by (7.4)-(7.5),
D11 ∇u1 + D12 ∇u2 + D11 u2 q∇V
D11 ∇(u1 + u2 qV ) + (D12 − D11 qV )∇u2
L11 ∇w1 + L12 ∇w2 ,
D21 ∇u1 + D22 ∇u2 + D21 u2 q∇V − V J1
(D21 − D11 qV )∇(u1 + u2 qV ) + (D22 − 2D21 qV + D11 (qV )2 )∇u2
+ D11 qV ∇(u1 + u2 qV ) + qV (D21 − D11 qV )∇u2 − qV I1
= L21 ∇w1 + L22 ∇w2 ,
qI1 =
=
=
I2 =
=
since Proposition 5.8 implies D12 = D21 and hence L12 = L21 . This also proves
(7.10) and (7.11) since
∂t b2 (u) − div I2 = ∂t (E − qV n) − div (J2 − V J1 )
= W − ∇V · J1 − qn∂t V − qV ∂t n + ∇V · J1 + V div J1
= W − qn∂t V.
It remains to prove the positive definiteness of L. We compute
µ
¶µ
¶> µ
¶
Id −qV Id
Id −qV Id
D11 D12
L=
.
0
Id
D21 D22
0
Id
(7.13)
Here, Id is the unit matrix in R3×3 . The assumptions on the linear independency
of (∂ε/∂ki , ε∂ε/∂ki )i ensure, by Proposition 5.8, that D is positive definite.
Thus, L is positive definite since the matrix
µ
¶
Id −qV Id
0
Id
is invertible.
¤
The symmetrization property is related to the existence of an entropy functional. In fact, both properties are equivalent (see [20, 32] for details). Before we
introduce the entropy, we need to discuss the thermal equilibrium state defined
by J1 = J2 = 0. This implies
µ ¶
µ ¶
I1
w1
0=
= L∇
.
I2
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 equation yields
w1 = const. and w2 = const.
in Ω.
Therefore, T = const. and µ − V = const. in Ω. As V is only determined up to
an additive constant, we can choose this constant such that µ − V = 0 or w1 = 0
in Ω. Since T = TL on a part of the boundary, T = TL in Ω. By (7.8), this implies
W (µ, T ) = 0 in Ω for all µ. Then we obtain from (7.1), (7.2) that ∂t n = 0 and
∂t E = 0.
We notice that this and the assumption of vanishing total space charge on
the Dirichlet boundary part (i.e. n = C on ΓD ) determines the thermal state
uniquely. Indeed, from (7.6) or
Z ∞p
0
qµ/kB TL
n = Ne
with N = 2π
γ(ε)γ (ε)e−ε/kB TL dε > 0
0
in thermal equilibrium we obtain
ne−qµV /kB TL = N
⇐⇒
V =
n
k B TL
ln .
q
N
On the boundary ΓD it holds
V = UT ln
n
C(x)
= UT ln
,
N
N
where UT = kB TL /q and thus, the thermal equilibrium state (µe , Te , Ve ) is the
unique solution of
εs ∆Ve = q(N eVe /UT − C(x)) in Ω,
C(x)
on ΓD ,
∇Ve · ν = 0 on ΓN
Ve = UT ln
N
(7.14)
(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) =
³n´
E
,
ue =
µ
ue1
ue2
¶
=
µ
qµe /kB TL
−1/kB TL
¶
and χ: R2 → R is a function satisfying χ0 (u) = ρ(u). More precisely, −η is the
physical entropy density relative to the thermal equilibrium state ue . The entropy
is the integral over the entropy density:
Z
S0 (t) =
η(x, t) dx.
Ω
134
Example 7.3 (Parabolic band approximation)
Under the assumptions of Example 5.10 we can express the energy density in
terms of n and T. Since, by Example 5.10,
n = Nc (kB T )3/2 eqµ/kB T ,
3
E = kB T n,
2
where Nc = 2(2πm∗ /~2 )3/2 , we have
u1 = ln
n
3
− ln(kB T ).
Nc 2
The equilibrium entropy variables are
ue1 = ln
ne
3
− ln(kB TL ),
Nc 2
ue2 = −
1
,
k B TL
where
ne = Nc (kB TL )3/2 eqµe /kB TL
and µe = Ve is the (unique) solution of (7.14)-(7.15). A simple computation
verifies that χ(u) = n. Therefore
¶
¶ µ
µ
ln(n/ne ) − 32 ln(T /TL )
n
− n + ne
·
η =
3
k Tn
(T − TL )/kB TL T
2 B
µ
¶
n
3 nT
3
T
5
= n ln
+
− ln
− n + ne .
ne 2 T L
2 TL
2
Notice that the entropy vanishes in thermal equilibrium:
¯
η ¯n=ne , T =T = 0.
L
The free energy of the energy-transport system is the sum of the entropy and
the electric energy:
¶
Z µ
1
2
|∇(V − Ve )| (t) dx.
S(t) =
η(u) +
2εs TL
Ω
Sometimes, also the function S(t) is called entropy. It satisfies the so-called
entropy inequality:
Proposition 7.4 Assume that the boundary data is in thermal equilibrium, i.e.
µD = µ e ,
TD = TL ,
VD = Ve
on ΓD × (0, ∞),
that (7.8) holds and that the functions in (7.12) are linearly independent. Then
Z tZ
S(t) +
(∇w)> L(∇w) dx dt ≤ S(0).
0
Ω
135
The assumptions of the proposition imply that L is symmetric, positive definite. Hence the above integral which is called entropy dissipation term is nonnegative. The entropy inequality shows that the entropy is non-increasing. Under
suitable assumptions on L, ρ(u) and W it is possible to show that S(t) converges
exponentially fast to zero as t → ∞ [19]. As we have not specified our notation
of solution, computations have to be understood in a formal way. The arguments
are made rigorous in [19].
Proof. We compute the time derivative of S(t). Since χ0 (u) = ρ(u) and therefore
∂t η(u) = ∂t ρ(u) · (u − ue ) − ρ(u) · ∂t u − χ0 (u) · ∂t u
= ∂t ρ(u) · (u − ue ),
we obtain, by integrating by parts,
¸
Z ·
1
∂t ∇V · ∇(V − Ve ) dx
∂t S =
∂t n(u1 − ue1 ) + ∂t E(u2 − ue2 ) +
εs T L
Ω
Z
=
[∂t n(u1 + qV u2 − (ue1 + qVe ue2 )) + ∂t (E − qV n)(u2 − ee2 )
Ω
¸
1
− ∂t nq(V u2 − Ve ue2 ) + q∂t (V n)(u2 − ue2 ) −
∂t n(V − Ve ) dx
TL
Z
=
[div I1 (w1 − we1 ) + div I2 (w2 − we2 ) + W (u2 − ue2 )
Ω
− qn∂t V (u2 − ue2 ) − q∂t n((V − Ve )u2 + Ve (u2 − ue2 ))
+ qn∂t V (u2 − ue2 ) + q∂t nV (u2 − ue2 ) + q∂t n(V − Ve )ue2 ] dx
(using (7.10) and (7.11))
¸
Z ·
T − TL
=
−I1 · ∇w1 − I2 · ∇w2 + W
dx
kB T T L
Ω
(since we1 = 0, we2 = const.)
Z
≤ − (∇w)> L(∇w) dx.
Ω
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-transport
model using the dual entropy variables w1 and w2 . The advantage of this formulation is that first-order terms like −∇V · J1 vanish. This is also an advantage
from a numerical point of view. Indeed, the symmetric operator −div (L∇w) can
be discretized using standard methods and the discrete nonlinear system can be
solved 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 which
makes the system less flexible (for instance, when changing the energy band)
and computationally quite costly. One way to overcome this disadvantage is to
look for another formulation of the system which allows for a decoupling of the
equations like in the drift-diffusion model (see Section 6.4).
In this section we show that the energy-transport model can be formulated
as a drift-diffusion system which allows for the use of the numerical methods
developed for the classical drift-diffusion model. For this, we turn back to the
definition of the particle current density J1 and energy current density J2 of
Section 5.5:
¶
µ
Z
∂F
εi−1 dε, i = 1, 2,
Ji =
D(ε) ∇x F + q∇x V
∂ε
R
where the diffusion matrix is assumed to be independent of x and F is given by
the Maxwell-Boltzmann statistics:
F (ε) = eεu2 +u1
with u1 = qµ/kB T and u2 = −1/kB T. Then ∂F/∂ε = (−1/kB T )F and
Z
Z
q∇x V
i−1
Ji = ∇x D(ε)F (ε)ε dε −
D(ε)F (ε)εi−1 dε.
k
T
B
R
R
Setting
g1 = D11 =
g2 = D12 =
we obtain
Z
ZR
R
D(ε)F (ε) dε,
D(ε)F (ε)ε dε,
q∇V
q∇V
g1 , J2 = ∇g2 −
g2 .
kB T
kB T
In this formulation the energy-transport model simplifies since it is no longer
strongly nonlinear in the sense that the equation for n (for E, respectively) only
depends on second-order derivatives of g1 (of g2 ) and not also of g2 (of g1 ). Notice
that the equation for n in the (w1 , w2 )-formulation contains second-order derivatives 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 have
to 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
R∞
D(ε)e−ε/kB T ε dε
g2
0
R
= ∞
=: f (kB T ).
g1
D(ε)e−ε/kB T dε
0
J1 = ∇g1 −
The following lemma states under which assumptions the function f can be inverted.
137
Lemma 7.5 Assume that the functions (7.12) are linearly independent. Then
f 0 (kB T ) =
det L
>0
(kB T g1 )2
∀ T > 0.
Proof. Since
Z ∞
∂g1
1
D12
,
=
D(ε)e−ε/kB T ε dε =
2
∂(kB T )
(kB T ) 0
(kB T )2
Z ∞
∂g2
1
D22
,
=
D(ε)e−ε/kB T ε2 dε =
2
∂(kB T )
(kB T ) 0
(kB T )2
and g1 = D11 , g2 = D12 , we obtain
1
f (kB T ) = 2
g1
0
=
=
=
>
µ
∂g1
∂g2
− g2
g1
∂(kB T )
∂(kB T )
1
2
(D11 D22 − D12
)
(kB T g1 )2
det D
(kB T g1 )2
det L
(kB T g1 )2
0,
¶
using (7.13) and Proposition 5.8(3).
¤
The above lemma shows that for given g1 and g2 , the temperature
kB T (g1 , g2 ) = f −1 (g2 /g1 )
is well defined. Notice that the value g1 = 0 is not possible since this would imply
2
D11 = 0 and det D = −D12
≤ 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. For
notational convenience we set
Z ∞p
0
Qi (kB T ) = 2π
γ(ε)γ (ε)e−ε/kB T εi dε, i = 0, 1,
Z ∞0
Pi (kB T ) =
D(ε)e−ε/kB T εi dε, i = 0, 1.
0
Then, by (7.6)-(7.7),
n = eqµ/kB T Q0 (kB T ),
E = eqµ/kB T Q1 (kB T )
g1 = eqµ/kB T P0 (kB T ),
g2 = eqµ/kB T P1 (kB T ).
and
138
Observing the relations
n
Q0 (kB T )
=
,
g1
P0 (kB T )
E
Q1 (kB T )
=
,
g2
P1 (kB T )
we can compute, for given g1 , g2 ,
n(g1 , g2 ) = g1
Q0 (kB T (g1 , g2 ))
,
P0 (kB T (g1 , g2 ))
E = g2
Q1 (kB T (g1 , g2 ))
.
P1 (kB T (g1 , g2 ))
The chemical potential µ can be expressed in terms of g1 , g2 by using the expression n = exp(qµ/kB T )Q0 (kB T ):
kB T Q0 (kB T )P0 (kB T )
kB T Q0 (kB T )
ln
=
ln
q
n(g1 , g2 )
q
g1 Q0 (kB T )
kB T (g1 , g2 ) P0 (kB T (g1 , g2 ))
=
ln
.
q
g1
µ(g1 , g2 ) =
Thus we can summarize the energy -transport model in the (g1 , g2 )-formulation:
1
∂t n(g1 , g2 ) − div J1 = 0,
q
∂t E(g1 , g2 ) − div J2 = −∇V · J1 + W (µ(g1 , g2 ), T (g1 , g2 )),
q∇V
g1 ,
J1 = ∇g1 −
kB T (g1 , g2 )
q∇V
J2 = ∇g2 −
g2 ,
kB T (g1 , g2 )
εs ∆V = q(n(g1 , g2 ) − C(x)).
We have derived three different formulations of the energy-transport equations: 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 variables (g1 , g2 ). Table 7.1 summarizes the advantages and disadvantages of these
formulations.
Formulation
(primal)
entropy
variables (u1 , u2 )
dual entropy variables (w1 , w2 )
variables (g1 , g2 )
Advantages
Disadvantages
thermodynamical vari- drift terms, full diffusion
ables
matrix
thermodynamical vari- full diffusion matrix
ables, no drift terms
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 diffusion
matrices Dij (µ, T ) and the relaxation term W (µ, T ) in the case of Kane’s nonparabolic band approximation [31]. We impose the following hypotheses (see
[21]):
• The energy-band diagram is given by
ε(1 + αε) =
~2 |k|2
,
2m∗
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) = φ0 εβ with φ0 > 0, β > −2.
• The relaxation term is given by the Fokker-Planck approximation (see Section 5.5):
·
µ
¶¸
Z ∞
∂ (qµ−ε)/kB T
∂
(qµ−ε)/kB T
A(ε) e
+ k B TL e
ε dε,
W (µ, T ) =
∂ε
∂ε
0
where A(ε) = φ(ε)N (ε)2 .
1
0.9
0.8
0.7
ε(|k|)
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
−0.5
0
|k|
0.5
1
Figure 7.2: Illustration of the energy band ε(|k|) in the parabolic case (broken
line) and the non-parabolic case (solid line) with parameters α = 0.25, ~/2m ∗ =
1.
The parameter α is called non-parabolicity parameter. The difference to
parabolic bands is shown in Figure 7.2. The function γ defined by |k|2 = γ(ε(|k|))
reads as follows:
2m∗
γ(ε) = 2 ε(1 + αε), ε ≥ 0.
~
140
p
Recall that the density of states is given by N (ε) = 2π γ(ε)γ 0 (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 = 2πe
qµ/kB T
µ
µ
2m∗
~2
∗
¶3/2 Z
¶3/2
∞
0
p
ε(1 + αε) (1 + 2αε)e−ε/kB T dε
2m
(kB T )3/2
= 2πeqµ/kB T
~2
Z ∞p
×
1 + αkB T u (1 + 2αkB T u)u1/2 e−u du
0
= q1/2 (αkB T )(kB T )3/2 eqµ/kB T ,
and
E = 2πe
qµ/kB T
µ
2m∗
~2
¶3/2 Z
∞
√
1 + αε (1 + 2αε)ε3/2 e−ε/kB T dε
0
5/2 qµ/kB T
= q3/2 (αkB T )(kB T ) e
q3/2 (αkB T )
=
kB T n,
q1/2 (αkB T )
where
qβ (αkB T ) = 2π
µ
2m∗
~2
¶3/2 Z
∞
0
p
1 + αkB T u (1 + 2αkB T u)uβ e−u du.
For the computation of Dij we start from the formula (5.79) derived in Example
5.9 (identifying Dij and Dij · Id)
Z
4q qµ/kB T ∞ γ(ε)εi+j−2 −ε/kB T
Dij =
e
e
dε
3~2
φ(ε)γ 0 (ε)2
0
Z
4q ~2 qµ/kB T ∞ 1 + αε
=
e
εi+j−β−1 e−ε/kB T dε
2
3~2 2m∗ φ0
(1
+
2αε)
0
qµ/kB T
i+j−β
= e
(kB T )
pi+j (αkB T )
pi+j (αkB T )
(kB T )i+j−β−3/2 n.
=
q1,2 (αkB T )
where
2q
pi+j (αkB T ) =
3m∗ φ0
Z
∞
0
1 + αkB T u
ui+j−β−1 du.
(1 + 2αkB T u)2
141
Finally, we calculate the relaxation term, using the results of Example 5.11,
µ ∗ ¶3
µ
¶Z ∞
TL
2m
0
qµ/kB T
2
β
1
−
e
W = −4π φ0
ε
γ(ε)γ
(ε)2 e−ε/kB T dε
~2
T
0
¶
µ
µ ∗ ¶3
TL
2m
(kB T )β+2
= −4π 2 φ0
eqµ/kB T 1 −
2
~
T
Z ∞
(1 + αkB T u)(1 + 2αkB T u)2 u1+β e−u du
×
0
µ ∗ ¶3
2m
2
= −4π φ0
eqµ/kB T (kB T )β+1 kB (T − TL )[Γ(β + 2)
2
~
+ 5αkB T Γ(β + 3) + 8(αkB T )2 Γ(β + 4) + 4(αkB T )3 Γ(β + 5)]
kB (T − TL )n
= −
,
τ (T )
where Γ denotes the Gamma function defined by
Z ∞
us−1 e−u du, s > 0,
Γ(s) =
0
and τ (T ) is a temperature-dependent relaxation time:
µ
2m∗
~2
¶3/2
(kB T )β−1/2
[Γ(β + 2) + 5αkB T Γ(β + 3)
q1/2 (αkB T )
¤
+ 8(αkB T )2 Γ(β + 4) + 4(αkB T )3 Γ(β + 5) .
1
= 2πφ0
τ (T )
In the parabolic band case α = 0 and the Chen model β = 1/2 we recover the
expressions for n, E, D and W from Examples 5.10 and 5.11.
142
8
8.1
From Kinetic to Quantum Hydrodynamic
Models
The quantum hydrodynamic equations: first derivation
In Chapter 1 we have sketched the (formal) equivalence of the single-electron
Schrödinger equation with electrostatic potential V = V (x, t)
i~∂t ψ = −
~2
∆ψ − qV ψ,
2m
x ∈ Rd , t > 0,
and the zero-temperature quantum hydrodynamic equations
1
∂t n − div J = 0,
q
µ √ ¶
µ
¶
2
2
∆ n
J ⊗J
1
~q
q
n∇ √
∂t J − div
= 0,
+ n∇V +
2
q
n
m
2m
n
√
via the Madelung transform ψ = n exp(imS/~), where
n = |ψ|2 ,
J =−
x ∈ Rd , t > 0,
~q
Im(ψ∇ψ)
m
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 a
sequence of single states with occupation probabilities λk ≥ 0 (k ∈ N) for the
k-th state described by the Schrödinger equations
~2
∆ψk − qV ψk , x ∈ Rd , t > 0,
(8.1)
2m
ψk (x, 0) = ψI,k (x), x ∈ Rd .
P
The occupation probabilities satisfy ∞
k=0 λk = 1. We write the initial datum of
the k-th state in the form
√
ψI,k = nI,k exp(imSI,k /~).
(8.2)
i~∂t ψk = −
The electron density nk and the electron current density Jk of the k-th single
state are given by
nk = |ψk |2 ,
Jk = −
~q
Im(ψ k ∇ψk ).
m
For the fluiddynamical formulation we use the Madelung transform
√
ψk = nk exp(imSk /~).
143
(8.3)
Clearly, we have to assume that nk > 0 for all k ≥ 0. With this transform the
current 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 /~),
µ
√
√
~2
2im √
m2 √
i~ ∂t nk
∆ nk +
∇ nk · ∇Sk − 2 nk |∇Sk |2
√ − m n k ∂t S k = −
2 nk
2m
~
~
¶
√
im √
+
nk ∆Sk − qV nk .
(8.5)
~
The imaginary part of this equation is
√
√
∂t nk = −2 nk ∇ nk · ∇Sk − nk ∆Sk = −div (nk ∇Sk )
or, using (8.4),
1
∂t nk − div Jk = 0.
q
(8.6)
The real part of (8.5) reads
√
q
~ 2 ∆ nk 1
− |∇Sk |2 + V.
∂t S k =
√
2
2m
nk
2
m
In order to find an evolution equation for Jk = −qnk ∇Sk , we take the gradient
of the above equation and multiply the resulting equation by −qnk :
µ √ ¶
∆ nk
~2 q
q2
q
−qnk ∂t ∇Sk = − 2 nk ∇ √
+ nk ∇|∇Sk |2 − nk ∇V.
2m
nk
2
m
Since
−qnk ∂t ∇Sk = ∂t Jk + q∂t nk ∇Sk = ∂t Jk + (div Jk )∇Sk
and
q
nk ∇|∇Sk |2 = qdiv (nk ∇Sk ⊗ ∇Sk ) − qdiv (nk ∇Sk )∇Sk
2
¶
µ
1
Jk ⊗ J k
=
+ (div Jk )∇Sk ,
div
q
nk
the above equation can be written as
¶
µ
µ √ ¶
∆ nk
1
q2
~2 q
Jk ⊗ J k
∂t Jk − div
+ nk ∇V +
n
∇
= 0.
√
k
q
nk
2
2m2
nk
(8.7)
Here, Jk ⊗ Jk is a tensor (or matrix) with components Jk,i Jk,j for i, j = 1, . . . , d.
144
The total carrier density n and current density J of the mixed state are given
by
n=
∞
X
λ k nk ,
J=
∞
X
λ k Jk .
k=0
k=0
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
1
∂t n − div J = 0
q
and
∞
1X
λk div
∂t J −
q k=0
µ
Jk ⊗ J k
nk
¶
µ √ ¶
∞
∆ nk
~2 q X
λ k nk ∇ √
+ qn∇V +
= 0. (8.8)
2
2m k=0
nk
In order to rewrite the second and fourth term on the left-hand side of this
equation, we introduce the “current” velocities
uc = −
J
,
qn
uc,k = −
Jk
qnk
and the “osmotic” velocities
~
~
∇ log n, uos,k =
∇ log nk .
uos =
2m
2m
The notion “osmotic” comes from the fact that
~2 q
Pk =
nk (∇ ⊗ ∇) log nk
4m2
can be interpreted as a non-diagonal pressure tensor since
µ √ ¶
∆ nk
~2 q
n
∇
= div Pk .
√
k
2m2
nk
Then
∞
1X
λk div
−
q k=0
= −q
∞
X
k=0
µ
Jk ⊗ J k
nk
¶
= −q
∞
X
k=0
div (λk nk uc,k ⊗ uc,k )
λk div (nk (uc,k − uc ) ⊗ (uc,k − uc ) + 2nk uc,k ⊗ uc )
+qdiv (nu ⊗ u)
µ
¶
∞
X
2λk Jk ⊗ J
div λk nk (uc,k − uc ) ⊗ (uc,k − uc ) + 2
= −q
q
n
k=0
µ
¶
1
J ⊗J
+ div
q
n
µ
¶
1
J ⊗J
= − div
− div (nθc ),
q
n
145
where
θc = q
∞
X
λk
k=0
nk
(uc,k − uc ) ⊗ (uc,k − uc )
n
is called the current temperature. Furthermore,
µ √ ¶
∞
∆ nk
~2 q X
λ
n
∇
√
k
k
2m2 k=0
nk
·
¸
∞
∇nk ⊗ ∇nk
~2 q X
λk div (∇ ⊗ ∇)nk −
=
4m2 k=0
nk
·
∞
∇n ⊗ ∇n
~2 q X
λk div (∇ ⊗ ∇)nk + nk
=
2
4m k=0
n2
µ
¶ µ
¶
¸
∇nk ∇n
∇n ⊗ ∇nk
∇nk ∇n
− nk
⊗
−2
−
−
nk
n
nk
n
n
¶
µ
2
~q
∇n ⊗ ∇n
− div (nθos )
=
div (∇ ⊗ ∇)n −
2
4m
n
µ √ ¶
~2 q
∆ n
√
=
n∇
− div (nθos ),
2m2
n
where
θos = q
∞
X
k=0
λk
nk
(uos,k − uos ) ⊗ (uos,k − uos )
n
is termed osmotic temperature.
Hence, the momentum equation (8.8) becomes
¶
µ
µ √ ¶
~2 q
1
J ⊗J
∆ n
√
∂t J − div
− div (nθ) + qn∇V +
= 0,
n∇
q
n
2m2
n
(8.9)
where θ = θc + θos is called temperature tensor. Equation (8.9) and the mass
conservation equation
1
∂t n − div J = 0
q
are referred to as the quantum hydrodynamic equations.
The temperature tensor cannot be expressed in terms of n and J without further assumptions, and as in classical fluid dynamics, we need a closure condition
to 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 the
identity 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) = T0 nα−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) = T0 nα · Id
and div P = div (T0 nα ), with α = 1 in the isothermal case and α > 1 in the
isentropic case.
8.2
The quantum hydrodynamic equations: second derivation
In this section, the quantum hydrodynamic equations are derived as a set of nonlinear conservation laws by a moment expansion of a Wigner-Boltzmann equation
and an expansion of the thermal equilibrium Wigner distribution function up to
order O(~2 ). We proceed similarly as in [24].
We start with a Wigner-Boltzmann equation of the form (see (4.25), (4.26))
∂t w +
q
1
k B T0 m
~
∆k w,
k · ∇x w + θ[V ]w =
divk (kw) +
m
~
τ
τ ~2
x, k ∈ Rd , t > 0,
w(x, k, 0) = wI (x, k),
x, k ∈ Rd ,
(8.10)
(8.11)
where d ≥ 1 is the space dimension and w(x, k, t) is the Wigner distribution
function depending on the space variable x, the wave vector k and the time t.
Notice that we have implicitely assumed a parabolic band approximation. The
operator θ[V ] is defined in the sense of pseudo-differential operators [42] as
· ³
Z Z
´
³
´¸
i
~
m
~
(θ[V ]w)(x, k, t) =
η, t − V x −
η, t
V x+
(2π)d Rd Rd ~
2m
2m
0
× w(x, k, t)e−i(k−k )·η dk 0 dη,
where V (x, t) is the electrostatic potential. The right-hand side of (8.10) models
the interaction of the electrons with the phonons of the crystal lattice. The
physical parameters are the reduced Planck constant ~ = h/2π, the effective
mass of the electrons m, the elementary charge q, the lattice temperature T0 , and
the momentum relaxation time τ whose inverse is a measure of the strength of
the coupling between the electrons and phonons.
The existence and uniqueness of local-in-time (so-called mild) solutions to the
whole-space problem (8.10)-(8.11) has been shown in [7]. Existence of global-intime solutions of the one-dimensional problem with periodic boundary conditions
has been proved in [5].
147
The particle density n and current density J are related to the Wigner function
by the formulas
Z
Z
q~
w(x, k, t) dk and J(x, t) = −
w(x, k, t)k dk.
(8.12)
n(x, t) =
m Rd
Rd
These formulas follow from Section 3.1 observing that in the parabolic band
approximation, the Brillouin zone is extended to Rd and the mean velocity equals
v(k) = ~k/m. Usually, the electrostatic potential is self-consistently produced by
the electrons moving in the semiconductor crystal with fixed charged background
ions 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
Z
h1i = n =
w(x, k, t) dk,
Rd
Z
w(x, k, t)kj dk,
hkj i =
d
ZR
w(x, k, t)kj k` dk,
hkj k` i =
Rd
with 1 ≤ j, ` ≤ d. Integrating (8.10) over Rd with respect to k, we obtain
Z
Z ³
´
~
k B T0 m
1
q
∂t h1i + divx hki +
divk (kw) +
θ[V ]w dk =
∆
w
dk
k
m
~ Rd
τ Rd
τ ~2
= 0.
For the evaluation of the integral on the left-hand side we recall the Fourier
transform
Z
1
φ̂(x, η, t) =
φ(x, k, t)eik·η dk
(2π)d/2 Rd
with inverse
1
φ(x, k, t) =
(2π)d/2
Z
Rd
φ̂(x, η, t)e−ik·η dη.
Integrating the last equation over Rd with respect to k gives the formula
Z Z
1
φ̂(x, 0, t) =
φ̂(x, η, t)e−ik·η dη dk.
(8.13)
(2π)d Rd Rd
148
With this expression we get
· ³
Z
Z Z
´
³
´¸
i
m
~
~
θ[V ]w dk =
η, t − V x −
η, t
V x+
(2π)d/2 Rd Rd ~
2m
2m
Rd
× ŵ(x, η, t)e−ik·η dη dk
· ³
´
³
´¸
~
~
d/2 m
= i(2π)
V x+
ŵ(x, 0, t)
η, t − V x −
η, t
~
2m
2m
η=0
= 0,
and therefore
∂t h1i +
~
divx hki = 0
m
or, with (8.12),
1
divx J = 0.
(8.14)
q
This equation expresses the conservation of mass.
Now we multiply (8.10) with kj and integrate over Rd with respect to k. Since
Z ³
´
k B T0 m
∆
w
kj dk
divk (kw) +
k
τ ~2
Rd
Z ³
´
k B T0 m
= −
(∇
k
)
·
(∇
w)
dk
(k · ∇k kj )w +
k j
k
τ ~2
Rd
Z ³
kB T0 m ∂w ´
= −
kj w +
dk
τ ~2 ∂kj
Rd
= −hkj i
∂t n −
and, using integration by parts and (8.13),
Z
θ[V ]wkj dk
Rd
Z Z · ³
´
³
´¸
~
im
~
V x+
=
η, t − V x −
η, t
(2π)d/2 ~ Rd Rd
2m
2m
∂ −ik·η
e
dη dk
× ŵ(x, η, t)i
∂ηj
· ³
´
³
´¸
~
~
d/2 m ∂
V x+
= −(2π)
η, t − V x −
η, t
ŵ(x, 0, t)
~ ∂ηj
2m
2m
η=0
·
³
´¸
~
~
∂ ŵ
d/2 m
V (x +
η, t) − V x −
η, t
− (2π)
(x, 0, t)
~
2m
2m
η=0 ∂ηj
Z
∂V
= −
(x, t)
w(x, k, t) dk
∂xj
Rd
∂V
h1i,
= −
∂xj
149
this yields
d
~ X ∂
q ∂V
1
∂t hkj i +
hkj k` i −
h1i = − hkj i
m `=1 ∂x`
~ ∂xj
τ
or
d
q~2 X ∂
q 2 ∂V
Jj
∂t J j − 2
hkj k` i +
n=− .
m `=1 ∂x`
m ∂xj
τ
(8.15)
The system (8.14) and (8.15) has to be closed by expressing the term hkj k` i
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 w
is close to a wave vector displaced equilibrium density such that
³
´
m
w(x, k, t) = we x, k − u(x, t), t ,
~
where u(x, t) is some group velocity and
we (x, k, t)
µ
qV
~
|k|2 +
2mkB T
kB T
2
¶·
½
(8.16)
q
∆x V
(8.17)
8m(kB T )2
)
#
d
2
2
X
q
q2
∂
V
|∇x V |2 −
+
+ O(~4 ) ,
vj (k)v` (k)
24m(kB T )3
24(kB T )3 j,`=1
∂xj ∂x`
= A(x, t) exp −
1 + ~2
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]. The
electron temperature T = T0 is here assumed to be constant (see the following
section for non-constant temperature).
The symmetry of we with respect to k implies that the odd order moments of
we vanish. Therefore
Z
Z
´
³
m
we (x, η, t) dη,
we x, k − u(x, t), t dk =
h1i =
~
Rd
Rd
Z ³
m ´
m
hkj i =
ηj + uj we (x, η, t) dη = nuj ,
~
~
Rd
Z ³
´³
´
m
m
~2 hkj k` i = ~2
ηj + uj η` + u` we (x, η, t) dη
~
~
Rd
Z
= m2 nuj u` + ~2
ηj η` we (x, η, t) dη.
(8.18)
Rd
This implies n = h1i and J = −qnu.
We now derive an O(~2 ) approximation for n. Since
Z
Z
√
2
2 −x2 /2
xe
dx =
e−x /2 · 1 dx = 2π,
R
ZR
Z
√
2
2
4 −x /2
xe
dx = 3 e−x /2 x2 dx = 3 2π,
R
R
150
we obtain
Z
Z
−~2 |η|2 /2mkB T
d/2 −d
e
dη = (mkB T ) ~
Rd
Rd
e−|z|
2 /2
dz = (2πmkB T )d/2 ~−d
and
Z
Rd
ηj η` e
−~2 |η|2 /2mkB T
dη = (mkB T )
~ (mkB T ~ )
d/2 −d
−2
Z
Rd
zj z` e−|z|
2 /2
dz
= (2πmkB T )d/2 mkB T ~−d−2 δj` ,
and hence, with C = A(x, t)(2πmkB T )d/2 ~−d ,
Z
n =
we (x, η, t) dη
Rd
µ
q 2 ~2
q~2
qV /kB T
∆x V +
|∇x V |2
1+
= Ce
2
3
8m(kB T )
24m(kB T )
!
d
q 2 ~4
mkB T X ∂ 2 V
(8.19)
−
δj` + O(~4 )
24m2 (kB T )3 ~2 j,`=1 ∂xj ∂x`
µ
¶
q~2
q 2 ~2
qV /kB T
2
1+
= Ce
∆x V +
|∇x V | + O(~4 ).
12m(kB T )2
24m(kB T )3
In order to compute the last integral in (8.18), we have to evaluate
Z
Z
2
−~2 |η|/2mkB T
d/2+2 −d−4
zj z` zm zn e−|z| /2 dz
ηj η` ηm ηn e
dη = (mkB T )
~
Rd
= (mkB T )
where

3 :




 1 :
1 :
α(j, `, m, n) = (2π)d/2


 1 :


0 :
This yields
~
2
Z
~
d/2+2 −d−4
Rd
α(j, `, m, n),
j=`=m=n
j = ` 6= m = n
j = m 6= ` = n
j = n 6= ` = m
else.
ηj η` we (x, η, t) dη
·³
´
q~2
q 2 ~2
qV /kB T
2
= Ce
mkB T +
∆x V +
|∇
V
|
δj`
x
8kB T
24(kB T )2
#
d
q~2 X ∂ 2 V
−
α(j, `, m, n) + O(~4 ).
24kB T m,n=1 ∂xm ∂xn
Rd
151
A simple computation shows that
d
X
∂2V
∂2V
α(j, `, m, n) = 2
+ δj` ∆x V.
∂xm ∂xn
∂xj ∂x`
m,n=1
Hence
·³
Z
2
qV /kB T
~
ηj η` we (x, η, t) dη = Ce
mkB T +
q~2
∆x V
(8.20)
12kB T
¸
´
q~2
∂2V
q 2 ~2
2
+ O(~4 ).
|∇x V | δj` −
+
24(kB T )2
12kB T ∂xj ∂x`
Rd
We substitute (see (8.19))
³
CeqV /kB T mkB T +
´
q~2
q 2 ~2
2
= mnkB T + O(~4 )
∆x V +
|∇
V
|
x
12kB T
24(kB T )2
and
CeqV /kB T = n + O(~2 )
(8.21)
into (8.20) to conclude from (8.18)
~2 hkj k` i = m2 nuj u` + mnkB T δj` −
q~2
∂2V
n
+ O(~4 ).
12kB T ∂xj x`
(8.22)
The potential V is usually the sum of the electrostatic potential and an external potential modeling heterostructures (i.e. sandwiches of different materials).
As the external potential can be discontinuous at the interfaces between two
materials, also V may be discontinuous. Therefore, we wish to substitute the
derivative ∂ 2 V /∂xj ∂x` by an expression which only depends on n and its derivatives. From Taylor’s formula log(x + ε) = log x + x/ε + O(ε2 ) for x, ε > 0 we
obtain from (8.21) (with a slight abuse of notation)
log n = log(eqV /kB T + O(~2 )) + log C = log eqV /kB T + log C + O(~2 )
and therefore,
¡
¢
∂ 2 log n
∂2 ³
qV ´
d/2
=
log A(2πmkB T )
+
+ O(~2 ).
∂xj ∂x`
∂xj ∂x`
kB T
We assume that A(x, t) is slowly varying with respect to x such that we can
approximate
∂ 2 log n
∂2V
q
=
+ O(~2 ).
∂xj ∂x`
kB T ∂xj ∂x`
152
For “~ → 0”, this corresponds to the classical thermal equilibrium expression
n = const. eqV /kB T . We obtain finally from (8.22)
~2 hkj k` i = m2 nuj u` + mnkB T δj` −
~2 ∂ 2 log n
n
+ O(~4 )
12 ∂xj ∂x`
(8.23)
and
d
q~2 X ∂
hkj k` i
m2 `=1 ∂x`
µ 2
¶
d
d
X
∂ log n
qkB ∂
~2 q X ∂
∂
n
= q
(nuj u` ) +
(nT ) −
∂x`
m ∂xj
12m2 `=1 ∂x`
∂xj ∂x`
`=1
¶
µ
µ √ ¶
d
qkB T ∂n
~2 q ∂
Jj J`
∆x n
1X ∂
√
+
−
n
,
=
2
q `=1 ∂x`
n
m ∂xj
6m ∂xj
n
since J = −qnu.
With (8.14), (8.15) and the Poisson equation, the (isothermal) quantum hydrodynamic equations become
1
∂t n − divx J = 0,
q
µ
µ √ ¶
¶
1
J ⊗J
qkB T
~2 q
∆x n
√
∂t J − divx
−
∇x n +
n∇x
2
q
n
m
6m
n
2
J
q
+ n∇x V = − ,
m
τ
div(εs ∇x V ) = q(n − C(x)).
(8.24)
(8.25)
(8.26)
Notice that the coefficient of the quantum term is ~2 q/6m2 compared to the
coefficient ~2 q/2m2 in (8.9) obtained from the mixed-state approach of Section
8.1. The factor 1/3 can be interpreted as a statistical factor coming from thermal
averaging (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 particle energy and a quantum hydrodynamic model taking into account viscous (or
dissipative) effects.
Quantum hydrodynamic equations with energy. In the previous subsection we have assumed that the temperatue, which appears in the approximation 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). Then
the quantum hydrodynamic equations (8.24)-(8.26) are still valid but the pressure
term in (8.25) has now to be written as
q
∇x (nT ).
m
However, we also need an equation for the variable T. We derive this equation by
calculating the next moment equation. For this we introduce the moments
Z
2
w(x, k, t)|k|2 dk,
h|k| i =
d
ZR
w(x, k, t)k|k|2 dk.
hk|k|2 i =
Rd
Multiply (8.10) with |k|2 and integrate over Rd with respect to k to obtain
Z
~
q
2
2
∂t h|k| i + div x hk|k| i +
θ[V ]w|k|2 dk
(8.27)
m
~ Rd
¶
Z µ
1
k B T0 m
=
∆k w |k|2 dk.
divk (kw) +
2
τ Rd
~
We first compute the integrals. For the integral on the right-hand side we get,
after integration by parts,
¶
Z µ
1
k B T0 m
∆k w |k|2 dk
divk (kw) +
τ Rd
~2
¶
Z µ
dkB T0 m
2
2
−w|k| +
=
w dk
τ Rd
~2
µ
¶
2
dkB T0 m
2
=
−h|k| i +
h1i .
τ
~2
Similarly as in the previous subsection, a computation gives
Z
q
2m
2q
θ[V ]w|k|2 dk = ∇x V · hki = − 2 ∇x V · J.
~ Rd
~
~
The two moments are calculated using the wave vector displaced equilibrium
density (8.27). From (8.23) follows
~2 h|k|2 i = m2 n|u|2 + dmnkB T −
~2
n∆ log n + O(~4 ).
12
We introduce now the energy
e=
m 2 d
~2
|u| + kB T −
∆ log n,
2
2
24m
154
(8.28)
consisting of the kinetic, thermal and quantum energy parts, and the stress tensor
Pj` = −nkB T δj` +
~2 ∂ 2 log n
n
,
12m ∂xj ∂x`
j, ` = 1, . . . , d.
(8.29)
Then ~2 h|k|2 i = 2me + O(~4 ) and tedious computations lead to
Z
2
hk|k| i =
we (x, k − mu/~, t)k|k|2 dk
Rd
2m2
(une − P · u) + O(~2 ),
~2
where P = (Pj` )j` . Thus we can write (8.27), up to order O(~2 ), as
=
1
∂t (ne) + div x (une − P · u) − ∇x V · J = − (2ne − dkB T0 n).
τ
Using u = −J/qn, we obtain the full quantum hydrodynamic equations
1
∂t n − divx J = 0,
q
¶
µ
µ √ ¶
1
qkB
q2
~2 q
J
J ⊗J
∆x n
√
∂t J − divx
+
+
∇x (nT ) −
n∇
n∇x V = − ,
x
2
q
n
m
6m
m
τ
n
¶
µ
³
´
2n
P ·J
d
1
− ∇x V · J = −
e − k B T0 ,
∂t (ne) − divx Je −
q
n
τ
2
div x (εs ∇x V ) = 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 the
quantum hydrodynamic model is based on the Fokker-Planck operator
µ
¶
1
k B T0 m
LF P w =
divk (kw) +
∆k w
τ
~2
which models interactions of electrons with a heat bath consisting of an ensemble
of harmonic oscillators [15]. In [7] a more general Fokker-Planck operator has
been proposed:
¶
µ
~2
k B T0 m
1
∆k w +
∆x w .
(8.30)
divk (kw) +
LF P w =
τ
~2
k B T0 m
We wish to derive the corresponding quantum hydrodynamic model in the case
of constant temperature. For this we only need to compute the first moments of
the last term in (8.30). It holds
Z
∆x w dk = ∆x h1i = ∆x n,
Rd
Z
m
∆x wkj dk = ~∆x hkj i = − ∆x Jj , j = 1, . . . , d.
q
Rd
155
We interpret these terms as viscous terms and call the corresponding equations
the viscous quantum hydrodynamic model:
~2
1
∆x n,
∂t n − divx J =
q
τ k B T0 m
µ √ ¶
µ
¶
1
~2 q
∆x n
J ⊗J
qT
q2
√
n∇x
∂t J − divx
∇x n −
+ n∇x V
+
q
n
m
6m2
m
n
2
~
J
=
∆x J − ,
τ k B T0 m
τ
div(εs ∇x V ) = q(n − C(x)), x ∈ Ω, t > 0.
This viscous terms have the important property that they regularize the equations in the following sense. Define the total energy
¶
Z µ 2
εs
|J|2
~ q √ 2 qkB T0
2
dx,
E(t) =
|∇ n| +
n(log n − 1) + |∇V | +
6m2
m
2q
2n
Ω
consisting of the quantum energy, entropy, electric energy, and kinetic energy. If
Ω ⊂ Rd is bounded and periodic boundary conditions are used or if Ω = Rd , a
tedious (formal) computation shows that
¶
Z µ 2
√ 2
√ 2 1 |J|2
dE
~2
4~2
~q
+
(∆ n) +
|∇ n| +
dx
dt
3m2 τ kB T0 m
τ k B T0 m
τ n
Ω
Ã
!2
Z
d
X
~2
1
∇n · J − n
dx ≤ 0.
(8.31)
+
|∇Ji |
3
Ω τ k B T0 m n
i=1
Without viscous terms the variation of the energy reads
¶
Z µ 2
√ 2 1 |J|2
dE
~q
+
|∇ n| +
dx = 0.
dt
τ 3m2
τ n
Ω
Hence, the viscous terms
√ 2yield an additional term in the energy production,
essentially given by (∆ n) .
156
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