DISSERTATIONdie Maxwell-Boltzmann- und Fermi-Dirac-Statistik als Spezialfälle enthalten sind. Die Lagrange-Multiplikatoren, die von der Maximierung stammen, können ebenfalls als

Diese Dissertation haben begutachtet:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DISSERTATION

Diffusive higher-order moment

equations for semiconductors

ausgeführt zum Zwecke der Erlangung des akademischen Gradeseines Doktors der technischen Wissenschaften unter der Leitung von

Univ.-Prof. Dr. rer. nat. Ansgar Jüngel

E101Institut für Analysis und Scientific Computing

eingereicht an der Technischen Universität WienFakultät für Mathematik und Geoinformation

von

Dipl.-Phys. Stefan Krause

Matrikelnummer 0727846

Wien, im November 2010

Contents

Kurzfassung v

Abstract vii

1 Introduction 11.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Diffusive semiconductor models . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Semiconductor physics 52.1 Crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Derivation of the model hierarchy 133.1 The scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Intended generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Entropy maximization . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Current densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 Diffusion matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.8 Drift-diffusion formulation . . . . . . . . . . . . . . . . . . . . . . . . 323.9 Dual-entropy formulation . . . . . . . . . . . . . . . . . . . . . . . . . 353.10 The case N = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Explicit models 434.1 Simplification of the integrals . . . . . . . . . . . . . . . . . . . . . . 434.2 Drift-diffusion models . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Energy-transport models with Maxwell-Boltzmann statistics . . . . . 514.4 Energy-transport models with Fermi-Dirac statistics . . . . . . . . . . 554.5 Extended energy-transport models with Maxwell-Boltzmann statistics 62

iii

iv CONTENTS

5 Discretization of the equations 655.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Thermal equilibrium and Poisson equation . . . . . . . . . . . . . . . 665.3 Transformation and ansatz spaces . . . . . . . . . . . . . . . . . . . . 695.4 The discretized equations for g1 and J1 . . . . . . . . . . . . . . . . . 715.5 The discretized equations for g0 and J0 . . . . . . . . . . . . . . . . . 75

6 Numerical results 776.1 Numerical setting and scaling . . . . . . . . . . . . . . . . . . . . . . 776.2 Fermi integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Thermal equilibrium and iteration . . . . . . . . . . . . . . . . . . . . 816.4 Some implementation details . . . . . . . . . . . . . . . . . . . . . . . 836.5 Results for Maxwell-Boltzmann . . . . . . . . . . . . . . . . . . . . . 876.6 Results for Fermi-Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . 996.7 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Summary and outlook 117

Bibliography 121

Acknowledgments 125

Kurzfassung

Die Computer- und Telekommunikationsindustrie setzt heutzutage in großem Ma-ße Halbleiterbauelemente ein. Um die Produktionskosten zu senken, benutzt mannumerische Software zur Simulation von Prototypen und passt deren Parameter andie gewünschten Eigenschaften an. Ein wichtiger Aspekt ist die Strom-Spannungs-Kennlinie eines Bauteils. Zur Beschreibung der Elektronenbewegung zwischen zweiKontakten wird ein System von Gleichungen verwendet, das aus den Grundgleichun-gen der Festkörperphysik oder Quantenmechanik hergeleitet wird.

Ein möglicher Startpunkt ist die Boltzmann-Gleichung, die die Bahn jedes ein-zelnen Teilchens beschreibt. Allerdings muss die Anzahl der Freiheitsgrade drastischreduziert werden, damit die Lösung in für die Industrie realistischer Zeit berechnetwerden kann. Dies führt auf sogenannte makroskopische Modelle, deren einfachsterVertreter das Drift-Diffusions-Modell

∂tm0 + divx J0 = 0 , J0 = m0∇xΦ−∇xm0

ist, das aus einem Moment m0, der Elektronendichte, und einem Strom J0 besteht.Die erste Beziehung ist eine Erhaltungsgleichung für die Masse, und die zweite de-finiert den Strom mit einem Driftterm m0∇xΦ und einem Diffusionsterm ∇xm0.

Es stellt eine starke Einschränkung dar, dass in diesem Modell die Temperaturals konstant Raumtemperatur angenommen wird. Dies wird um so schwerwiegen-der, je größer die angelegte Spannung bzw. je kleiner die charakteristische Längewird. Durch die Miniaturisierung der Bauteile ist es notwendig, genauere Modellezu entwerfen, die insbesondere die Temperatur als variable Größe enthalten. Ver-schiedene Methoden führen auf verschiedene Varianten von sogenannten Energie-Transport-Modellen, die alle aus einer Elektronendichte m0, einer Energiedichte m1,einem Elektronenstrom und einem Energiestrom bestehen. Im einfachsten Fall istdie Temperatur dann durch m1 =

32m0T gegeben.

Der neuartige und wesentliche Inhalt der Arbeit ist, dass wir in der Lage sind,Modelle mit einer beliebigen Anzahl von Momenten herzuleiten. Diese haben ver-nünftige mathematische Eigenschaften, obwohl die Voraussetzungen relativ allge-mein sind, der Nachteil jedoch ist, dass die Grenzübergänge nur formal durchgeführtsind. Allerdings können wir explizit zeigen, dass die hergeleiteten Modelle einige Ei-genschaften besitzen, die man erwarten würde, z. B. dass die Diffusionsmatrizenpositiv definit sind und daher das ganze System parabolisch ist.

v

vi KURZFASSUNG

Der Zustand im thermischen Gleichgewicht ist definiert durch die Maximierungder Entropie unter gewissen Nebenbedingungen. Dies ist allgemein gehalten, wobeidie Maxwell-Boltzmann- und Fermi-Dirac-Statistik als Spezialfälle enthalten sind.Die Lagrange-Multiplikatoren, die von der Maximierung stammen, können ebenfallsals Unbekannte verwendet werden, weil sie bijektiv mit den Momenten zusammen-hängen. Ferner wird gezeigt, dass weitere Darstellungen möglich sind, nämlich dieDrift-Diffusions-Formulierung und eine Variante in dualen Entropievariablen.

Um wieder zur Motiviation der effizienten Lösbarkeit zurückzukehren, wird zumAbschluss gezeigt, dass gängige numerische Verfahren anwendbar sind. Das Fermi-Dirac-Energie-Transport-Modell wird diskretisiert mit hybridisierten gemischten fi-niten Elementen vom Marini-Pietra-Typ in einer Dimension. Die simulierte ballisti-sche n+nn+-Diode dient als einfaches Modell eines Kanals eines Feldeffekttransistors.Ein iterativer Gummel-Algorithmus wird statt des vollen Newtonverfahrens benutzt,wobei einige Schwierigkeiten bei der Implementation aufgetreten sind. Bei der Be-rechnung der Fermi-Integrale muss man einen Kompromiss aus Genauigkeit undGeschwindigkeit in Kauf nehmen. Für größere Werte des Parameters η der Fermi-Dirac-Statistik gibt es noch offene Fragen bezüglich der Konvergenz des Verfahrens.

Abstract

The modern computer and telecommunication industry relies heavily on the func-tionality of semiconductor devices. In order to reduce production costs the proto-types are simulated by a numerical software and parameters are adjusted to meetthe desired properties of the device. One of the key features is the current-voltagecharacteristic. Therefore a set of equations is needed to model the transport of elec-trons from one contact to another, and the derivation of those models is based onfundamental principles of solid state physics or quantum mechanics.

The starting point is the Boltzmann equation which contains the trajectoriesof each single electron. In order to be solvable for industrial needs, the number ofdegrees of freedom has to be reduced which leads to so-called macroscopic models.The easist one is the drift-diffusion model consisting of one moment m0 for the elec-tron density and one associated charge current density J0. One possible formulationof the model is

∂tm0 + divx J0 = 0 , J0 = m0∇xΦ−∇xm0

consisting of a balance equation which guarantees the conservation of mass and asecond equation which defines the current. The current consists of a drift termm0∇xΦ and a diffusion term ∇xm0.

The temperature is assumed to be constant at ambient temperature which isquite a strong limitation. The approximation gets worse with increasing voltagesand decreasing characteristic lengths. But due to the miniaturization of the semi-conductor devices better models had to be developed which especially comprise thetemperature as a variable. This leads to various types of energy-transport modelswhich all have in common that they contain the electron density m0, the energydensity m1, the electron current, and the energy current. The temperature is givenas a function of m0 and m1; one of the easiest cases is m1 =

32m0T .

The new and major aspect of this thesis is that we can derive models withan arbitrary number of moments, that the models have reasonable mathematicalproperties, and that the assumptions are rather general. The drawback, however, isthat the limits are not performed rigorously; we usually assume that the limits exist.Nevertheless, we can prove that the derived model equations have some propertiesone would expect, this is, for example, that the diffusion matrices are positive definiteso that the complete system is of parabolic type.

vii

viii ABSTRACT

The state of thermal equilibrium is defined by the maximization of entropy undersome constraints. Our case is general but includes the Maxwell-Boltzmann distri-bution as well as the Fermi-Dirac distribution. The Lagrange multipliers stemmingfrom the maximization can be used as the unknowns of the model since they canbe mapped one-to-one onto the moments. It is shown that even more representa-tions are possible, so the drift-diffusion formulation and a version using dual-entropyvariables.

Finally, which in some sense closes the circle to the motivation given in thebeginning, the models are accessible for numerical algorithms. The Fermi-Diracenergy-transport model is discretized using hybridized mixed Marini-Pietra finiteelements in one dimension. This is applied to the n+nn+ ballistic diode serving asa simple model for the channel of a field effect transistor. An iterative Gummelmethod is used instead of the full Newton scheme, but some difficulties had tobe solved in the implementation of the algorithm. Regarding the calculation ofthe Fermi integrals, a compromise had to be made between accuracy and speed.Unfortunately there are still open questions concerning the convergence for highervalues of the parameter η in the Fermi-Dirac statistics.

Chapter 1

Introduction

1.1 General overview

The modern computer and telecommunication industry relies heavily on the func-tionality of semiconductor devices. In order to reduce production costs the proto-types are simulated by a numerical software and parameters are adjusted to meetthe desired properties of the device. One of the key features is the current-voltagecharacteristic. Therefore a set of equations is needed to model the transport of elec-trons from one contact to another, and the derivation of those models is based onfundamental principles of solid state physics or quantum mechanics.

The classical or semiclassical transport of charged carriers in semiconductors isgoverned by the Boltzmann equation. Its solution is the microscopic distributionfunction f(t, x, p) depending on the time t, the spatial variable x, and the momen-tum p. The meaning of “microscopic” is that every single electron is consideredwhich leads to a huge number of degrees of freedom in the system. This is why thenumerical simulation, usually done with Monte-Carlo methods, is extremely time-consuming and thus not appropriate for the use in the production industry. More-over, the distribution function itself carries much more information than needed,and the physically relevant and measurable macroscopic quantities like current ortemperature are obtained by integration.

It is obvious that one would like to aim for macroscopic models which directlycontain the quantities mentioned above. Whereas the Monte-Carlo algorithm re-sults in an approximation of the solution of the exact Boltzmann equation, derivinglower-dimensional model equations is an approximation procedure on itself. Thederivation is carried out by multiplying the Boltzmann equation by functions de-pending on the momentum and then integrating over the momentum space. Thesemoments are usually interpreted as electron density, energy density, and so on, andother quantities in the equations are referred to as the corresponding current den-sities. Depending on the scaling and the type of moments, diffusive models andhydrodynamic models are distinguished – see [15] for a more detailed presentation.

1

2 CHAPTER 1. INTRODUCTION

When quantum effects are not negligible the microscopic starting point has tobe replaced by the Schrödinger or the Wigner-Boltzmann equation. The classicalderivation can be mimicked in the quantum case, and the result are analogous quan-tum diffusive or quantum hydrodynamic models. For example, when the electroncurrent density J of the simplest classical diffusive model is given by

J = n∇xΦ−∇xn ,

then the quantum counterpart is

J = n∇xΦ−∇xn +ε2

6n∇x

(

∆x

√n√n

)

+ higher-order terms ,

where ε is some small parameter. Since divx J contains a fourth-order derivative, itis not surprising that the quantum model is of much higher complexity – both forthe analysis and the numerics.

1.2 Diffusive semiconductor models

The easiest classical diffusive model is the so-called drift-diffusion model which wasjust mentioned above. It consists of one moment m0 for the electron density andone associated charge current density J0. One possible formulation of the model is

∂tm0 + divx J0 = 0 , J0 = m0∇xΦ−∇xm0

consisting of a balance equation which guarantees the conservation of mass and asecond equation which defines the current. The current consists of a drift termm0∇xΦ and a diffusion term ∇xm0. Models of this type were first introduced byvan Roosbroeck in 1950, see [24].

The temperature is assumed to be constant at ambient temperature which isquite a strong limitation. The approximation gets worse with increasing voltagesand decreasing characteristic lengths. But due to the miniaturization of the semi-conductor devices better models had to be developed which especially comprise thetemperature as a variable. This leads to various types of energy-transport modelswhich all have in common that there are two moments m0 and m1 describing theelectron and energy density, respectively, and two corresponding current densitiesJ0 and J1. The balance equation for the energy contains two more terms, that is,the Joule heating term and a collision term for processes that do not conserve theenergy – for example inelastic collisions. The temperature is given as a function ofm0 and m1; one of the easiest cases is m1 =

32m0T .

The new and major aspect of this thesis is that we can derive models withan arbitrary number of moments, that the models have reasonable mathematicalproperties, and that the assumptions are rather general. The drawback, however, isthat the limits are not performed rigorously, for example we assume that fα tends

1.3. OUTLINE OF THE THESIS 3

to F for α → 0, but we do not specify neither the function spaces nor whether thisis a strong or a weak limit. Nevertheless, the models consisting of N + 1 momentsm0, . . . , mN and currents J0, . . . , JN which – yet in a special case – look like

∂tmi + divx Ji − iJi−1 · ∇xΦ =Wi , Ji = −N∑

k=0

(Dik∇xλk + kλkDi,k−1∇xΦ) .

The diffusion matrix consisting of Dik can be shown to be positive definite so thatthe complete system is of parabolic type.

The variables λ0, . . . , λN are Lagrange multipliers stemming from a constraintentropy maximization process which defines the equilibrium state. In many discus-sions the Maxwell-Boltzmann statistics is taken as a basis, sometimes the Fermi-Dirac statistics is considered as a generalization. Here we just claim the existenceof such a function defining the equilibrium state and some of its properties – thenMaxwell-Boltzmann and Fermi-Dirac are included as a special case. Furthermore, itis shown that the model equation can be transformed into a so-called drift-diffusionformulation, that is, the current definitions are written as

Ji = −∇xgi − giri(g)∇xΦ

with new variables g0, . . . , gN under weak additional assumptions.Finally, which in some sense closes the circle to the motivation given in the be-

ginning, the models are accessible for numerical algorithms. Following [7] we presentthe discretization of the Fermi-Dirac energy-transport model in drift-diffusion formu-lation with hybridized mixed Marini-Pietra finite elements in one dimension. Thisis applied to the n+nn+ ballistic diode serving as a simple model for the channelof a field effect transistor. An iterative Gummel method is used instead of the fullNewton scheme, but some difficulties had to be solved in the implementation of thealgorithm. Regarding the calculation of the Fermi integrals, a compromise had tobe made between accuracy and speed. Unfortunately there are still open questionsconcerning the convergence for higher values of the parameter η in the Fermi-Diracstatistics.

1.3 Outline of the thesis

The thesis is structured as follows: After Chapter 1 – this introduction – Chapter 2describes some basic principles of semiconductors. This seemed to be reasonabledue to the fact that the Boltzmann transport equation is the starting point of themodeling and since some terms like Fermi energy or parabolic bands repeatedlyappear therein. In order to keep in short, however, more advanced topics like directand indirect semiconductors are omitted; we refer to the corresponding books onsolid state physics.

4 CHAPTER 1. INTRODUCTION

The Chapters 3 and 4 contain the major work of the thesis, that is, the modelingprocedure in a setting as general as possible. To achieve explicit models, typicalassumptions – parabolic bands and relaxation-time operator – are made afterwardsto list a variety of models in different formulations.

The Chapters 5 and 6 use the equations of the Fermi-Dirac energy-transportmodel from the previous chapter. At first the model is discretized using finiteelements, and then an iterative Gummel solver is used for the numerics. Whereasthe first part is more or less standard in this context, the latter one needed a fewnontrivial adaptations to work in the Fermi-Dirac setting.

As usual the thesis concludes with a summary of what has been done and withan outlook of what can or has to be improved which is located in Chapter 7.

Chapter 2

Semiconductor physics

In the beginning we discuss some basic facts of solid state physics which we aregoing to need during the modeling process. More specifically, the important wordswill be lattices, band structures, distribution functions, collision operators, and theBoltzmann equation, where the latter one will be the typical starting point for amacroscopic electron transport model in a semiconductor.

2.1 Crystal lattices

The crystal structure of a solid can be seen as an infinite repetition of a group ofatoms which is typical of the solid. The repetition is performed in a regular way,that is, the crystal looks the same when viewed from an arbitrary point where thisatom group is attached. According to this periodicity we define the discrete set

L = n1c1 + n2c2 + n3c3 | n1, n2, n3 ∈ Z ⊂ R3

of points of the R3, where c1, c2, c3 is a basis of the R3, and call it a lattice. Thetranslation vectors c1, c2, c3 are chosen in a way that the crystal contains an atomgroup at each point r ∈ L. In the case that the converse is true, that means eachposition of an atom group can be expressed as a linear combination of c1, c2, c3 withinteger coefficients, then the lattice L is said to be primitive.

When c1, c2, c3 generate a primitive lattice, the set

C = λ1c1 + λ2c2 + λ3c3 | 0 ≤ λ1, λ2, λ3 ≤ 1 ⊂ R3

is connected, compact, and convex with volume c1 · (c2 × c3), which is nonzero sincec1, c2, c3 is linearly independent, and is called a primitive cell. We assume thatthe basis c1, c2, c3 is right-handed so that the expression for the volume is positive.More generally, every connected, compact, and convex subset C ⊂ R3 with the samevolume is called a primitive cell, when the whole R3 can be covered by translation,in other words when

⋃

r∈L

(r + C) = R3 .

5

6 CHAPTER 2. SEMICONDUCTOR PHYSICS

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Figure 2.1: Geometric construction of the Wigner-Seitz cell, for simplicity in two dimen-sions only.

The so to speak most symmetric primitive cell is given by

CWS =

r ∈ R3∣

∣ |r| ≤ |r − r′| ∀ r′ ∈ L \ r

and called the Wigner-Seitz cell; | · | denotes the Euclidean norm on R3. In otherwords, CWS consists of all points that are not further away from the origin (which liesin L) than from any other grid point in L. The Wigner-Seitz cell can be constructedgeometrically by drawing connection lines from the origin to its 26 nearest neighborsn1c1 + n2c2 + n3c3 where (n1, n2, n3) ∈ −1, 13 \ (0, 0, 0) and then erecting theperpendicular planes in the center point of these lines. The area encompassed byall these planes is the Wigner-Seitz cell, see Figure 2.1.

The grid points in L belong to the position space. When switching to the canon-ically conjugate of the spatial variable, that is, the momentum variable, we also haveto change to another lattice. We define vectors

b1 =2π

vol(C)(c2 × c3) , b2 =

2π

vol(C)(c3 × c1) , b3 =

2π

vol(C)(c1 × c2) ,

where vol(C) = c1 · (c2× c3) denotes the volume of a primitive cell. The well-knownrules for the cross product immediately result in

cj · bℓ = 2πδjℓ .

Up to the factor 2π which is convention, b1, b2, b3 is the dual basis of c1, c2, c3.The analogous discrete set

L∗ = n1b1 + n2b2 + n3b3 | n1, n2, n3 ∈ Z ⊂ R3

is called the reciprocal lattice, and for any primitive cell B of L∗ with vol(B) =b1 · (b2 × b3) holds

vol(C) vol(B) = (2π)3 .

2.2. BAND STRUCTURES 7

Whereas the vectors cj naturally are of unit meter, the vectors bj by definition areof unit 1/meter. Thus the elements of L∗ cannot be momentum vectors, rather theyare so-called wave vectors usually denoted by k. These quantities come in handywhen discussing scattering, reflection, and diffraction of waves whose wavelengthsor frequencies are characterized by wave vectors. In our transport equations forelectrons, however, the momentum p is the more suitable variable.

The Wigner-Seitz construction can be done for the reciprocal lattice in the verysame way, and the resulting cell is called the Brillouin zone BWS . As mentionedabove, it contains wave vectors k and not momentum vectors p. Thus we define

B = ~BWS

due to the relation p = ~k and call the set B, by a slight abuse of denotation, theBrillouin zone. We will use this interpretation within the modeling chapter.

2.2 Band structures

A free electron, that is an electron with no interaction to other electrons or electricpotentials, with momentum p has got the energy

E(p) =|p|22m

(2.1)

where m is the mass of the electron. In principle every value for |p| is possible, andthe energy E depends continuously on |p|. When an electron feels a positive chargegenerating a potential Φ, for example the nucleus of an atom, its motion is governedby the Schrödinger equation

Hψ =

(

− ~2

2m∆x − eΦ(x)

)

ψ = Eψ .

As long as the energy E is below a certain value – at higher energies the atom iscalled to be ionized – there are solutions for the Schrödinger equation for a finitenumber of energies only. Mathematically spoken, the Hamilton operator H has adiscrete spectrum for lower energies and a continuous spectrum for higher ones.This gives rise to the atomic shell structure and finally to the periodic table of theelements.

Every single electron in a solid sees a potential stemming from the ionized lat-tice atoms superposed with another stemming from the electron-electron interaction.Considering n electrons at the positions x1, . . . , xn and N lattice ions at fixed posi-tions X1, . . . , XN , the Hamilton operator is given by

H = − ~2

2m

n∑

i=1

∆xi− eQ

4πε0

n∑

i=1

N∑

I=1

1

|xi −XI |+

e2

8πε0

n∑

i,i′=1i 6=i′

1

|xi − xi′ |


where Q is the charge of each ion; the last term is halved because of the mutualinteraction for each pair of electrons at (xi, xi′) and (xi′ , xi). The last term is also theone that turns out to be the most troublesome since it interlinks the variables of thedifferent electrons. Thus much effort was spent to replace it by an approximationincluding an effective potential. Inserting

−e8πε0

n∑

i′=1, i′ 6=i

1

|xi − xi′ |≈ Φi(xi) ,

where Φi is such a potential, yields

H = H1 + . . .+Hn , Hi = − ~2

2m∆xi

− eQ

4πε0

N∑

I=1

1

|xi −XI |− eΦi(xi) .

Now Hi acts on xi only so that the n-electron Schrödinger equation resolves intoa sum of n one-electron Schrödinger equations. Together with the product ansatzψ(x1, . . . , xn) =

∏ni=1 ψi(xi), this is called the Hartree-Fock approximation.

For a fixed i, the complete potential term of Hi can be written as −eΦL(xi) witha compound potential ΦL that acts on the ith electron. In the limit of huge N , orin other words, in a quasi-infinite crystal, ΦL turns out to be periodic with respectto the lattice – for each lattice vector r ∈ L we have ΦL(x) = ΦL(x + r). In thefollowing we consider a single electron and therefore omit the index i.

Theorem 2.1 (Bloch decomposition)Every solution of the Schrödinger equation Hψ = Eψ with the Hamilton operator

H = − ~2

2m∆x − eΦL(x)

whose potential ΦL is periodic with respect to a crystal lattice L can be decomposedinto a product of a plane wave characterized by a k ∈ BWS and a periodic so-calledBloch function; in formulas it holds for the solution ψ

ψ(x) = eik·xu(x) , u(x) = u(x+ r) ,

for all x ∈ R3 and r ∈ L. The main advantage is that, due to its periodicity, thefunction u is completely determined by its values in a primitive cell.

To denote all solutions we add the wave vector k as a parameter for ψ, and sincethere will be many solutions with the same k, we add another countable index ν ∈ N;then we get

ψν,k(x) = eik·xuν,k(x) .

Of course there exists an associated eigenvalue Eν,k to every eigenstate ψν,k. Therange of the continuous map Eν : BWS → R with Eν(k) = Eν,k is called the νth

2.3. SEMICONDUCTORS 9

energy band. With this definition we see that every electron in a solid must have anenergy E that is contained in the set E =

⋃

ν∈N ranEν . The ground state of everysystem of electrons is governed on the one hand by the Pauli exclusion principleand on the other hand by taking up the minimal total energy. Thus there exists anenergy EF with the property that all states with lower or equal energy are occupiedand all states with higher or equal energy are unoccupied at zero temperature. Thisenergy EF is called the Fermi energy, and it is not defined as consistently as onecould expect in the various books about solid state physics, for example see [1] or[19]. The location of the energy bands and of the Fermi energy determines theelectrical conductivity characteristics of the solid.

Whenever there is a band ranEν such that EF lies in its interior, then this energyband is partially filled by electrons and partially empty. This means that an electroncan absorb a very little portion of energy in order to change to a free higher state inthe same band where it can travel across the crystal and contribute to the current.Thus the conductivity is very high which is typical of metals like silver, copper, oraluminum – with a specific resistance around ≈ 10−7 Ωm.

When EF does not belong to any energy band or, depending on the specificdefinition of EF , lies on the boundary of a band, then the solid is said to have aband gap. The band gap Eg is the biggest connected subset of R containing EF (atleast on its boundary) which is not contained in any band. We denote by εg thesize of the band gap which then is the energy distance between the highest occupiedand the lowest unoccupied state. The conductivity at zero temperature is exactlyzero whereas it increases with increasing temperature since the probability that anelectron can jump across the gap becomes strictly greater than zero. The actualvalue of this probability strongly depends on εg. When it is too large, there willbe no measurable current and the solid is called an insulator, for example quartzwith εg ≈ 9 eV and ≈ 1018 Ωm. When the band gap is much smaller so thatconductivity becomes probable, the solid is called a semiconductor, for exampleεg ≈ 1.15 eV and ≈ 10−3 Ωm for silicon.

2.3 Semiconductors

The low conductivity of an ideal pure semiconductor crystal, strongly depending onthe temperature and being relevant not until temperatures too high to be technicallyreasonable, is referred to as the intrinsic conductivity. When an electron in thehighest completely filled band, the valence band, is thermally excited to the lowestcompletely empty band, the conduction band, it leaves behind a hole in the valenceband. The electron, a particle, and the hole, a so-called quasiparticle, travel inantipodal directions due to their opposite charge and both contribute to the current.When we denote by e and h the electron and hole density, respectively, we havee = h = i and call i the intrinsic density. We want to note that the number ofelectrons is changed, usually increased, when there are crystallographic defects such


as dislocations or impurity atoms. These defects locally distort the band structureand cause unpredictable effects. This is why the semiconductor industry has to workwith high-purity crystals.

The intrinsic conductivity of pure silicon and germanium is about a factor of30,000 to 60,000 lower than the one of pure copper which makes them unsuitablefor electronic applications. For their use in electronic devices like computers it isnecessary to insert additional charge carriers into the crystal during the manufac-turing process. This is done by substituting a certain percentage of the fourth maingroup semiconductor element atoms by element atoms of the adjacent main groups.This process is called doping; when elements like gallium or indium from the thirdmain group are used, the semiconductor is called p-doped, in the case of arsenic orantimony from the fifth group n-doped. It is possible to locally control the amountand the type of doping atoms which defines a function dop : Ω → R called thedoping profile, where Ω is the crystal domain.

The arsenic atoms in an n-doped region have five outmost electrons of whichonly four are used for the chemical bindings. The fifth electron has got a higherenergy lying inside the band gap very close the conduction band. So it can be easilythermally excited to an energy level in the conduction band at room temperaturegiving rise to a much higher conductivity of the entire crystal. The other way around,indium atoms in a p-doped region lack the fourth electron and induce unoccupiedenergy levels in the gap slightly above the valence band to which electrons can easilybe excited.

When Ec(p) is the energy in the conduction band for a momentum vector p ∈ B,we can expand it into a Taylor series as

Ec(p) = Ec(0) +∇pEc(0) · p+1

2p · HesspEc(0) p+ higher-order terms ,

where HesspEc denotes the Hessian of Ec. We assume that Ec reaches its minimumat p = 0 and that the Hessian is positive definite at this point. Then the linear termvanishes, and with the definition of the so-called effective mass m∗ = (HesspEc(0))

−1

we get Ec(p) = Ec(0) +12p · (m∗)−1p + higher-order terms. Neglecting the higher-

order terms and additionally assuming that the 3× 3 matrix m∗ is a multiple of theunit matrix, we can interpret m∗ as a positive real number and get the parabolicband approximation

ε(p) = Ec(p)− Ec(0) =|p|22m∗

with the relative energy ε which is exactly the same as (2.1) – just with the electronmass m replaced by the effective one m∗. With the assumption that the electronsare not too far away from the the minimum Ec(0), we can choose B = R3.

The probability that a level with energy E is occupied is given by the Fermi-Diracfunction

f(E) =1

1 + e(E−µ)/kBT.

2.4. BOLTZMANN EQUATION 11

In the zero temperature limit T → 0 we have µ = EF , and the function f degeneratesto a step function which is 1 for E < EF and 0 for E > EF . The chemical potentialµ depends on the temperature T and must be chosen in a way that the electrondensity comes out correctly. To calculate this density one needs to know how manystates there are with a given energy E, and this is measured by the so-called densityof states g(E). It characterizes the number of states – that can be occupied by anelectron – per energy and volume. The electron density in the conduction band isthen given by

n =

∫ ∞

Ec(0)

f(E)g(E) dE .

Under some assumptions g(E) turns out to be proportional to√

E − Ec(0) whichleads to n =

∫∞

0ε1/2f(ε) dε – compare to (4.1) in a generalized but scaled setting.

2.4 Boltzmann equation

We have seen that the integral of the function ε 7→ f(ε)g(ε) over the full energyrange [0,+∞[ results in the electron density. Analogously, we can consider anothernonnegative function f depending on time t, space x, and momentum p such that

n(t, x) =

∫

B

f(t, x, p) d3p

describes the electron density at x to the time t – compare to (3.2) with κ0 = 1 andm0 = n. We denote by Ωt ⊂ R × R3 the set of all possible time variables t ∈ Rand space variables x ∈ R3. One can think of Ωt = ]0, t∗[ × Ω with a maximaltime t∗ and a bounded set Ω characterizing the semiconductor domain. Thus theconsidered function is f : Ωt × B → R.

When keeping track of a single electron along a trajectory (x(t), p(t)) in thephase space, one can show that

f(

t, x(t), p(t))

= f(

0, x(0), p(0))

. (2.2)

The proof uses Liouville’s theorem from Hamiltonian mechanics together with thefact that the vector field that describes the time evolution of (x, p) is divergence-free.This is true in our case since the driving forces do not depend on the the momentump. The most prominent counterexample are magnetic fields with the Lorentz forcethat depends on the velocity of the charge. Differentiating (2.2) with respect to tyields

∂tf(t, x, p) + ∂tx · ∇xf(t, x, p) + ∂tp · ∇pf(t, x, p) = 0

which is called the Boltzmann equation.The derivative ∂tx – a velocity – can be interpreted as the crystal momentum,

at least up to the effective mass, and thus we get ∂tx = ∇pε. Note that this leads


to the familiar relation ∂tx = p/m∗ for parabolic bands where ε = |p|2/(2m∗). Thederivative ∂tp – a force – is the electric force stemming from the voltage applied atthe semiconductor. Thus we set ∂tp = Fel = −eE = e∇xΦ, where −e is the chargeof the electron and Φ the electric potential.

Unless the applied voltage is very small, the interactions between the particlescannot be neglected. This leads to the situation that the right-hand side of theBoltzmann equation is not zero any more, but

∂tf +∇pε · ∇xf + e∇xΦ · ∇pf = Q(f)

with a usually nonlinear collision operator Q. A serious discussion, choice, and de-velopment of collision operators is an elaborate task as there are lots of possibleprocesses one can consider. Of course every type of interaction obeys the conserva-tion of mass and charge, however, the energy can be altered, so in inelastic processes.

A typical phenomenon is the scattering of an electron, for example at a latticeatom, changing its momentum vector from p to p′. The probability is given by

Pr(t, x, p→ p′) = s(x, p, p′)f(t, x, p)(

1− f(t, x, p′))

,

where s describes the scattering rate. Integrating yields

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

∫

B

(

Pr(t, x, p′ → p)− Pr(t, x, p→ p′))

d3p′ .

In a low-density approximation, that is, 0 ≤ f ≪ 1, we have 1 − f ≈ 1 whichsimplifies the expression for the probability. Under few another assumptions, amongthem for example is that f is not too far away from the equilibrium characterizedby feq , one can show

Q(f)(t, x, p) ≈ f(t, x, p)− feq(x, p)

τ(x, p), τ(x, p) =

(∫

B

s(x, p, p′) d3p′)−1

,

which is called a collision operator of relaxation-time type with characteristic timeτ .

Chapter 3

Derivation of the model hierarchy

In this chapter we want to present how to derive a large number of different models.We start with the Boltzmann equation in diffusion scaling, apply some expansionand pass on to the limit α → 0 in multiple situations. We first show the idea in avery special case and try to be as general as possible in the complete derivation.

The scaling of the Boltzmann equation is taken from [23] or [15]. The derivationof the model hierarchy follows [16] and [17] where the procedure is presented forthe Maxwell-Boltzmann and Fermi-Dirac statistics, respectively. The descriptionhere is held more general and contains both papers as a special case – anyway, theMaxwell-Boltzmann case can be seen as a limit of the Fermi-Dirac case.

3.1 The scaling

We decompose the collision operator into a sum of two parts, namely

∂tf +∇pε · ∇xf + e∇xΦ · ∇pf = Q1(f) +Q2(f) .

We assume that the average time τC between two collisions is characteristic of theoperator Q1 whereas the average time τR between two recombination processes ischaracteristic of the operator Q2. With a drift velocity v of the electrons we get

v =λCτC

=λRτR

where λC and λR are the mean free paths corresponding to τC and τR, respectively.We now use τR for the time scale and

√λCλR for the spatial scale leading to

t = τR t , x =√

λCλR x , ε = kBTamb ε , p =kBTamb

vp , Φ =

kBTamb

eΦ ,

Q1 =1

τCQ1 , Q2 =

1

τRQ2

13

14 CHAPTER 3. DERIVATION OF THE MODEL HIERARCHY

where the scaled quantities are marked by a tilde and Tamb is the ambient tem-perature. Inserting everything into the Boltzmann equation and multiplying by τCyields

τCτR∂tf +

τC v√λCλR

∇pε · ∇xf +τC v√λCλR

∇xΦ · ∇pf = Q1(f) +τCτRQ2(f) .

The idea behind the two operators with two average times and the scaling is thefollowing: We define by

α =

√

τCτR

the so-called Knudsen number α and assume that 0 < α ≪ 1. The physical inter-pretation is that the time between two consecutive collisions is much much smallerthan the one between two recombination processes, in other words, the collisionsare dominant. Inserting this definition we get the Boltzmann equation in diffusionscaling

α2∂tfα + α(∇pε · ∇xfα +∇xΦ · ∇pfα) = Q1(fα) + α2Q2(fα) (3.1)

where the tildes have been removed for simplicity. Furthermore, the distributionfunction f is parametrized by an index α. The scaling is called diffusive since thedominant collisions are the cause for diffusion.

Finally we have to scale the Poisson equation

∆xΦ = −

ε0εr

which holds in every electrodynamic system. The charge density consists of twocomponents, the electron part −ne with the positive electron particle density n andthe doping part ndope with the doping ion density ndop . The latter one is positive inour cases since we consider n-doped devices only. ε0 is the vacuum permittivity andεr the relative permittivity of the semiconductor material. The densities are scaledwith the maximum doping value ndop,max , that is

n = ndop,max n .

Then the Poisson equation in scaled form becomes

ε0εrkBTamb

e2λCλRndop,max

∆xΦ = n− ndop .

3.2 The idea

We denote by ε : B → R the microscopic energy in the band depending on themomentum vector p ∈ B of the Brillouin zone B ⊆ R3. Here we assume parabolicbands meaning ε = 1

2|p|2 and thus ∇pε = p.

3.2. THE IDEA 15

We multiply (3.1) with α−2 and with α−2ε, respectively, and integrate withrespect to p over B which yields

∂t

∫

B

fα d3p+

1

α

∫

B

p · ∇xfα d3p+

1

α

∫

B

∇xΦ · ∇pfα d3p

=1

α2

∫

B

Q1(fα) d3p+

∫

B

Q2(fα) d3p ,

∂t

∫

B

εfα d3p+

1

α

∫

B

εp · ∇xfα d3p+

1

α

∫

B

ε∇xΦ · ∇pfα d3p

=1

α2

∫

B

εQ1(fα) d3p+

∫

B

εQ2(fα) d3p .

To simplify the notation we define

⟨

h⟩

=

∫

B

h d3p

for any term h which depends on p. Rewriting ∇xfα as a divergence and usingintegration by parts in the potential term, we get

∂t⟨

fα⟩

+1

αdivx

⟨

fαp⟩

=1

α2

⟨

Q1(fα)⟩

+⟨

Q2(fα)⟩

,

∂t⟨

εfα⟩

+1

αdivx

⟨

εfαp⟩

− 1

α

⟨

fαp⟩

· ∇xΦ =1

α2

⟨

εQ1(fα)⟩

+⟨

εQ2(fα)⟩

.

We assume that the equilibrium state is characterized by a function

Mf = exp(λ0 + λ1ε)

where λ0 and λ1 are two Lagrange multipliers stemming from an entropy maximiza-tion process. We are going to discuss this in more detail later on. We insert aChapman-Enskog expansion

fα = Mfα + αgα =⇒ gα =fα −Mfα

α

which can be, in other words, seen as a definition of gα. Since 〈hp〉 = 0 for an evenfunction h in |p|, we can pass on to the limit α → 0 leading to

∂t⟨

MF

⟩

+ divx

⟨

Gp⟩

=⟨

Q2(F )⟩

,

∂t⟨

εMF

⟩

+ divx⟨

εGp⟩

−⟨

Gp⟩

· ∇xΦ =⟨

εQ2(F )⟩

where we have assumed that fα → F and gα → G for α → 0.With the interpretation m0 = 〈MF 〉, m1 = 〈εMF 〉, J0 = 〈Gp〉, and J1 = 〈εGp〉

as the particle density, energy density, particle current density, and energy currentdensity, respectively, we get the so-called balance equations

∂tm0 + divx J0 = 0 , ∂tm1 + divx J1 − J0 · ∇xΦ = W1


with W1 = 〈εQ2(F )〉 and the assumption 〈Q2(F )〉 = 0. These equations are part ofone possible widely known time-dependent energy-transport model.

In order to get equations of J0 and J1 we need to specify G. This can be doneby assuming a very special case

Q1(fα) =Mfα − fα

τ1= − α

τ1gα

which is called a relaxation-time operator. The characteristic relaxation time τ1may depend on t, x, and p, but for simplicity we assume for the moment that it isconstant. We insert this expression into (3.1) and obtain

α∂tfα +∇pε · ∇xfα +∇xΦ · ∇pfα = − 1

τ1gα + αQ2(fα) .

In the limit α → 0 we get

G = −τ1(p · ∇xMF +∇xΦ · ∇pMF )

so that for the jth component

Jij = −τ1⟨

εipjp · ∇xMF

⟩

− τ1⟨

εipj∇pMF

⟩

· ∇xΦ .

We perform integration by parts in the second term yielding

⟨

pj∇pMF

⟩

= −⟨

MF

⟩

ej = −m0ej

for i = 0 and

⟨

εpj∇pMF

⟩

= −⟨

MF∇p(εpj)⟩

= −⟨

pjMFp⟩

−⟨

εMF

⟩

ej

= −⟨

pjpjMF

⟩

ej −m1ej = −5

3m1ej

for i = 1 since pℓ 7→ pjMFp is an odd function for ℓ 6= j and 〈pjpjMF 〉 has thesame value for each j.

We use the same reasoning in the first term of Jij and get 23〈εi+1∂xj

MF 〉. Torewrite this we define new variables

g0 =2τ13

⟨

εMF

⟩

, g1 =2τ13

⟨

ε2MF

⟩

and observe that the gradients ∇xg0 and ∇xg1 coincide with the first term of J0 andJ1, respectively, as long as τ1 does not depend on x. Thus we get

J0 = −∇xg0 + τ1m0∇xΦ , J1 = −∇xg1 +5τ13m1∇xΦ .

3.3. INTENDED GENERALIZATIONS 17

In a final step we want to replace m0 and m1 by g0 and g1. This is easy in thissimple case as the integrals can be calculated explicitly to

m0 =(2π)3/2eλ0

(−λ1)3/2, m1 =

3

2

(2π)3/2eλ0

(−λ1)5/2, g0 = τ1

(2π)3/2eλ0

(−λ1)5/2, g1 =

5τ12

(2π)3/2eλ0

(−λ1)7/2.

When we interpret −λ1, which has to be positive to make MF integrable, as theinverse electron temperature 1/T , we get the current density definition equations

J0 = −∇xg0 +g0T∇xΦ , J1 = −∇xg1 +

g1T∇xΦ .

They are called to be in drift-diffusion formulation since they consist of a drift termgi∇xΦ and a diffusion term ∇xgi. They are well-known in the scope of energy-transport models.

Generally, when deriving moment equations from some kind of a scaled Boltz-mann equation, one is confronted with the so-called closure problem. For exampleduring a Hilbert expansion, where terms are expanded into powers of a small pa-rameter α and the model equations are obtained be equating the coefficients, thenumber of moments is larger than the number of equations. Thus the model isunder-determined without further information. In our situation the closure problemis obviated by passing on to the diffusive limit α → 0 and defining the equilibriumstate by a constraint maximization process.

3.3 Intended generalizations

We want to keep the approach presented in the last section to derive a hierarchy ofdifferent models.

We do not want to stick to only two balance equations and two current densities.Before the introduction of the first energy-transport model, the drift-diffusion modelwith only one moment and current density and with constant temperature played amajor role and is still in use in electronical engineering. Thus we will be introducingN + 1 moments m0, . . . , mN and as many current densities J0, . . . , JN .

For defining more moments we must introduce so-called weight functions κi inmi = 〈κiMF 〉. We will see that κi = εi is one possible suitable choice. For the bandenergy ε we want to drop the special relation ε = 1

2|p|2, in other words we want to

allow bands different from parabolic bands.The equilibrium state function Mf from the previous section is associated with

Maxwell-Boltzmann statistics for the electrons. This function has to be generalizedsince N + 1 Lagrange multipliers are needed for models with N + 1 moments. Fur-thermore, a main focus of this thesis is to deal with equilibrium states governed byFermi-Dirac statistics which is new within this generalized setting.

The choice of Q1 as a relaxation-time operator is a strong restriction. It will comeout that letting Q1 rather arbitrary significantly increases the degree of abstraction


during the derivation process. Especially there will not be a closed expression forG any more. Certain assumptions on Q1, however, will guarantee the solvability ofthe implicit equations for G.

In the special case of the relaxation-time operator which is used in the numericalexamples at the end, we try to keep the relaxation time τ1 as general as possible.Since an unknown dependence on p prevents us from getting rid of the momentumintegrals, however, we need to specify p 7→ τ1 in a suitable way. A physicallyreasonable and analytically satisfactory choice is a certain power of ε.

3.4 Entropy maximization

We assume that N ≥ 1 which means that we have at least two moments. We aregoing to discuss the drift-diffusion model case (N = 0) later on which is comparablystraight-forward anyway. Now we define what we want to denote as weight functions:

Definition 3.1 (Weight functions)For every i = 0, . . . , N we take a scalar-valued weight function κi : B → R definedfor every momentum vector p in the Brillouin zone B. We write κ = (κ0, . . . , κN)

T

for the column vector of all weight functions. Although we want to have differentchoices for the higher weights, we fix κ0 = 1 and κ1 = ε which is the same as in theprevious section.

It is also possible to allow other types of weight functions, for example the mo-mentum p itself. This leads to vector-valued moments which are often used forderivations starting from the Boltzmann equation in hydrodynamic scaling. We donot want to discuss this here.

Assumption 3.2We claim that the energy ε and each weight κi is even, that means ε(−p) = ε(p) andκi(−p) = κi(p) for every p ∈ B and every index i = 0, . . . , N . Thus the gradients∇pε and ∇pκi are odd functions.

In order to perform the expansion in the next section, we have to define an equilib-rium state function. The set Ωt contains all the time variables t and space variablesx of interest as usual. We specify the maximization problem we want to examine:

Definition 3.3 (Entropy maximization)Let H be a relative entropy functional which means what H maps the distributionfunction f on a function of t and x. Thus,

H(f)(t, x) =⟨

some expression in f⟩

is a real number for every (t, x) ∈ Ωt. We call a function f ∗ : Ωt × B → R whichsatisfies

H(f ∗)(t, x) = max

H(f)(t, x)∣

∣

∣

⟨

κif(t, x, p)⟩

= mi(t, x) ∀ i

(3.2)

3.4. ENTROPY MAXIMIZATION 19

for given moment functions m0, . . . , mN : Ωt → R a maximizer of H under theconstraint 〈κf〉 = m where m = (m0, . . . , mN )

T.

It is a rather elaborate task to investigate the existence and/or uniqueness of f ∗,however, it is clear that some trivial conditions are needed for m; for example whenf and κi are nonnegative, then mi has to be nonnegative. Although we are a bitsloppy and assume unique solvability in our cases, we give a sketch on what can gowrong when we discuss two possibilities for H .

The Lagrange multiplier theorem states that, if there exists such a maximizerf ∗, then we have Lagrange multipliers λ0, . . . , λN . Thus we can define:

Definition 3.4 (Equilibrium function)Let f : Ωt × B → R be a distribution function with moments mi = 〈κif〉. Whenthe maximization problem is uniquely solvable for these moments m, we denote thismaximizer f ∗ by Ef . Then, by this definition, the moments of f and Ef coincide,

⟨

κif⟩

=⟨

κiEf⟩

.

We assume that Ef only implicitly depends on the variables t, x, and p, which meansthat the dependence of Ef on t and x is completely determined by the Lagrangemultipliers λ0, . . . , λN : Ωt → R and the dependence on p by the weights κ0, . . . , κN .Hence, Ef is an even function in p, too.

Now we want to consider two possible entropies which will be used in the nextchapter for the explicit models. First we define for Maxwell-Boltzmann statistics:

Definition 3.5 (Maxwell-Boltzmann entropy)We call HM with

HM(f)(t, x) = −⟨

f(ln f + ε− 1)⟩

the relative Maxwell-Boltzmann entropy functional for f : Ωt × B → R.

Theorem 3.6If the maximization problem for HM is uniquely solvable for the moments m = 〈κf〉with a given function f , it holds

Mf(t, x, p) = Ef(t, x, p) = exp(

λ(t, x) · κ(p)− ε(p))

.

Since κ1 = ε we can omit the energy term by defining λ1 = λ1 − 1 and λi = λi forall i 6= 1 and thus

Mf(t, x, p) = exp(

λ(t, x) · κ(p))

.

Proof With the Gâteaux derivatives

DHM(f)g = limδ→0

HM(f + δg)−HM(f)

δ

= − limδ→0

1

δ

⟨

(f + δg)(

ln(f + δg) + ε− 1)

− f(ln f + ε− 1)⟩


= −⟨

limδ→0

(f + δg) ln(f + δg)− f ln f

δgg + εg − g

⟩

=⟨

(ln f + 1)g + εg − g⟩

= −⟨

(ln f + ε)g⟩

and D(f 7→ 〈κif〉 −mi)g = 〈κig〉 we get

0 = DHM(f ∗)g +

N∑

i=0

λiD(

f ∗ 7→ 〈κif ∗〉 −mi

)

g =⟨

(

−(ln f ∗ + ε) + λ · κ)

g⟩

=⇒ −(ln f ∗ + ε) + λ · κ = 0 =⇒ f ∗ = exp(λ · κ− ε) .

We have already mentioned that the unique solvability of the constrained maximiza-tion problem nontrivially depends on the choice of B and κ. Ihara shows in [14]that the problem is solvable as soon as Lagrange multipliers can be found. This isin some sense the converse of the Lagrange multipliers theorem.

Furthermore, there are cases in which the maximization problem for HM doesnot have a solution. In [9] and [18] the authors Junk et al. consider an unboundedBrillouin zone B (for example B = R3) and show that no solution exists if one of theweight functions grows stronger than |p|2 when |p| tends to infinity. This problemcould be obviated by relaxing the Nth constraint to an inequality, which means thatwe claim 〈κif〉 = mi for i = 0, . . . , N − 1 and only 〈κNf〉 ≤ mN . In our derivation,however, the relaxation is unsatisfactory since we want the moments of f to coincidewith those of the maximizer Ef .

The other entropy we want to consider is the following:

Definition 3.7 (Fermi-Dirac entropy)We call HF with

HF(f)(t, x) = −⟨

f(ln f + ε) +(1− ηf) ln(1− ηf)

η

⟩

,

where η > 0 is a parameter, the relative Fermi-Dirac entropy functional for f : Ωt ×B → R.

Theorem 3.8If the maximization problem for HF is uniquely solvable for the moments m = 〈κf〉with a given function f , it holds

Ff(t, x, p) = Ef(t, x, p) =1

η + exp(−λ(t, x) · κ(p) + ε(p)).

Just as before we can rewrite λ1 to get

Ff(t, x, p) =1

η + exp(−λ(t, x) · κ(p)) .

3.4. ENTROPY MAXIMIZATION 21

Proof With the Gâteaux derivatives

DHF(f)g = limδ→0

HF(f + δg)−HF(f)

δ

= − limδ→0

1

δ

⟨

(f + δg)(

ln(f + δg) + ε)

+(1− η(f + δg)) ln(1− η(f + δg))

η

− f(ln f + ε)− (1− ηf) ln(1− ηf)

η

⟩

= −⟨

limδ→0

(f + δg) ln(f + δg)− f ln f

δgg + εg

+ limδ→0

(1− η(f + δg)) ln(1− η(f + δg))− (1− ηf) ln(1− ηf)

ηδgg

⟩

= −⟨

(ln f + 1)g + εg +−η ln(1− ηf)− η

ηg

⟩

= −⟨(

lnf

1− ηf+ ε

)

g

⟩

and D(f 7→ 〈κif〉 −mi)g = 〈κig〉 we get

0 = DHF(f∗)g+

N∑

i=0

λiD(

f ∗ 7→ 〈κif ∗〉−mi

)

g =

⟨(

−(

lnf ∗

1− ηf ∗+ε

)

+ λ ·κ)

g

⟩

=⇒ −(

lnf ∗

1− ηf ∗+ ε

)

+ λ · κ = 0 =⇒ f ∗ =1

η + exp(−λ · κ+ ε).

The calculations for both of the entropies suggest the following:

Assumption 3.9We assume that the dependence of Ef on t, x, and p is completely determined by

the expression λ · κ =∑N

i=0 λi(t, x)κi(p) which is an extension to the assumptionson Ef in the Definition 3.4. Therefore the derivatives of Ef can be written as

∂tEf = E ′f

N∑

i=0

κi(p)∂tλi(t, x) ,

∇xEf = E ′f

N∑

i=0

κi(p)∇xλi(t, x) ,

∇pEf = E ′f

N∑

i=0

λi(t, x)∇pκi(p)

with another function E ′f acting on λ · κ. We further assume that E ′

f > 0. An easycalculation shows that

M′f = eλ·κ = Mf and F ′

f =e−λ·κ

(η + e−λ·κ)2= Ff(1− ηFf)


and that M′f > 0 and F ′

f > 0 actually hold.

The Fermi-Dirac setting actually is a generalization of the Maxwell-Boltzmann set-ting:

Theorem 3.10It holds

HM = limη→0

HF and Mf = limη→0

Ff .

Proof It is

limη→0

(1− ηf) ln(1− ηf)

η= lim

η→0

(

−f ln(1− ηf)− f)

= −f

and

limη→0

1

η + exp(−λ · κ) = exp(λ · κ) .

3.5 Balance equations

We multiply (3.1) with κiα−2 and integrate with respect to p over B which yields

∂t⟨

κifα⟩

+1

αdivx

⟨

κifα∇pε⟩

− 1

α

⟨

fα∇pκi⟩

· ∇xΦ

=1

α2

⟨

κiQ1(fα)⟩

+⟨

κiQ2(fα)⟩

. (3.3)

The boundary term∫

∂Bκifα∇xΦ · ν∂B dσp in the integration by parts vanishes: In

one case it holds B = R3 and fα decays exponentially for |p| → ∞ whereas κi onlygrows polynomially. In the other case B is a bounded set and we impose periodicboundary conditions for fα, which means together with the assumptions on κi that(κifα)(−p) = (κifα)(p) for every p ∈ ∂B.

We consider a decomposition of fα into a sum of an equilibrium function and aremainder term:

Definition 3.11 (Chapman-Enskog expansion)Given an entropy function H and an associated equilibrium function Ef , we definefor fα the Chapman-Enskog expansion

fα = Efα + αgα (3.4)

as a decomposition in an equilibrium portion and a remainder term. Here, again, Efαhas the same moments as fα by definition, and thus 〈gα〉 = 0. During the followingderivation process we claim that fα → F and gα → G as α→ 0.

3.5. BALANCE EQUATIONS 23

We insert (3.4) into (3.3) and get

∂t⟨

κiEfα⟩

+ α∂t⟨

κigα⟩

+1

αdivx

⟨

κiEfα∇pε⟩

+ divx⟨

κigα∇pε⟩

− 1

α

⟨

Efα∇pκi⟩

· ∇xΦ−⟨

gα∇pκi⟩

· ∇xΦ =1

α2

⟨

κiQ1(fα)⟩

+⟨

κiQ2(fα)⟩

.

Since the two functions p 7→ κiEfα∇pε and p 7→ Efα∇pκi are odd, the correspondingintegrals vanish. For Q1 we claim:

Assumption 3.12For the collision operator Q1 and the weight functions κ0, . . . , κN we assume

⟨

κiQ1(f)⟩

= 0

for all distribution functions f . The physical interpretation is that Q1 conservesthe quantities related to the weights, for example the number of particles, the totalcharge, the total energy, and so on. Since κ0 = 1 the assumption especially states〈Q1(Ef)〉 = 0, but we tighten this property to Q1(Ef) = 0 which emphasizes that Efdescribes an equilibrium state.

It is not actually needed during the derivation but nonetheless physically reasonablethat the converse of the last assumption is true, that is, Q1(f) = 0 implies f = Ef .Written as an equivalence, Q1 applied to a function f vanishes if and only if f is anequilibrium function.

With these pieces of information, we can pass on to the limit α→ 0 and obtain

∂t⟨

κiEF⟩

+ divx⟨

κiG∇pε⟩

−⟨

G∇pκi⟩

· ∇xΦ =⟨

κiQ2(F )⟩

.

The first term can be immediately recognized as ∂tmi. The right-hand side is acollision expression, and, guided by the structure of well-known continuity equations,we interpret the second term as a divergence of a current density:

Theorem 3.13 (Balance equations)When we define the current densities, the “modified current densities”, and theaveraged collision operator expressions of Q2 by

Ji =⟨

κiG∇pε⟩

, Ji =⟨

G∇pκi⟩

, and Wi =⟨

κiQ2(F )⟩

,

respectively, the N + 1 derived balance equations are

∂tmi + divx Ji − Ji · ∇xΦ = Wi .

We will see later on that we get another N +1 (vector-valued) equations which canbe seen as definitions for J0, . . . , JN . Hence, we will have 2(N +1) equations for the2(N + 1) unknowns m0, . . . , mN , J0, . . . , JN . The other way around, we have to get


rid of the “modified current densities” J0, . . . , JN , however, J0 is zero anyway sinceκ0 = 1 is constant. Furthermore, as κ1 = ε, we have J1 = J0, and thus the first twobalance equation always are

∂tm0 + divx J0 = 0 and ∂tm1 + divx J1 − J0 · ∇xΦ = W1 .

The assumption W0 = 〈Q2(F )〉 = 0 is physically reasonable since it expresses massconservation for Q2.

Example 3.14In the case we set κi = εi for all i, we get

Ji =⟨

G∇pεi⟩

= i⟨

εi−1G∇pε⟩

= iJi−1 .

The balance equations then simplify to

∂tmi + divx Ji − iJi−1 · ∇xΦ =Wi .

This is a nice result since we only need κi = εi which will turn out to be a veryimportant choice of the weight functions.

On the one hand the previous example gets rid of the troublesome terms Ji, on theother hand it replaces them by expressions of Ji in a way that the N + 1 balanceequations are decoupled in some sense. This means that the first equation can besolved for m0 and J0, and for the second one which is solved for m1 and J1 theexpression J0 is already known and thus a term on the right-hand side. Of course,for eventually solving the equations numerically, we need another equation for eachpair (mi, Ji) which will be derived in the following sections.

When dropping κi = εi we want at least stick to the following case:

Theorem 3.15When the weight functions κ0, . . . , κN are chosen in a way that ∇pκi is a linearcombination of κ0∇pε, . . . , κi−1∇pε

∇pκi =

i−1∑

k=0

cikκk∇pε ,

then Ji is a linear combination of J0, . . . , Ji−1 and the balance equations are

∂tmi + divx Ji −(

i−1∑

k=0

cikJk

)

· ∇xΦ =Wi .

Thus, the potential term in the ith balance equation only contains J0, . . . , Ji−1 whichalso leads to a decoupling of the equations.

3.6. CURRENT DENSITIES 25

3.6 Current densities

The probably most elaborate task in the derivation is the specification of G =limα→0 gα which is needed to get more or less explicit formulas for the current den-sities Ji = 〈κiG∇pε〉. To get an idea on how the results can look like we assume atfirst the relaxation-time operator

Q1(fα) =Efα − fα

τ1= − α

τ1gα

with a characteristic time τ1 which may be depend on t, x, and p. Inserting Q1 and(3.4) into (3.1) and dividing by α yields

α∂tfα +∇pε · ∇x(Efα + αgα) +∇xΦ · ∇p(Efα + αgα) = − 1

τ1gα + αQ2(fα)

and finally in the limit α → 0

G = −τ1(∇pε · ∇xEF +∇xΦ · ∇pEF ) (3.5)

and thusJi = −

⟨

τ1κi(∇pε · ∇xEF +∇xΦ · ∇pEF )∇pε⟩

. (3.6)

Using the derivatives for EF we get

Ji = −⟨

τ1κi

(

∇pε · E ′F

N∑

k=0

κk∇xλk +∇xΦ · E ′F

N∑

k=0

λk∇pκk

)

∇pε

⟩

= −N∑

k=0

⟨

τ1κiκkE ′F (∇pε · ∇xλk)∇pε

⟩

−N∑

k=0

λk⟨

τ1κiE ′F (∇pκk · ∇xΦ)∇pε

⟩

and thus for the jth component

Jij = −N∑

k=0

⟨

τ1κiκkE ′F∂pjε∇pε

⟩

· ∇xλk −N∑

k=0

λk⟨

τ1κiE ′F∂pjε∇pκk

⟩

· ∇xΦ .

For each i, k = 0, . . . , N we introduce matrices Dik, Dik ∈ R3×3 by

(Dik)jℓ =⟨

τ1κiκkE ′F∂pjε ∂pℓε

⟩

and (Dik)jℓ =⟨

τ1κiE ′F∂pjε ∂pℓκk

⟩

and therefore

Ji = −N∑

k=0

Dik∇xλk −N∑

k=0

λkDik∇xΦ .

We do not consider Dik and Dik as a definition here; we are rather going to give amore general definition in the context of a general collision operator Q1 later on and


see that the definition and the special choice go well together. In order to interpretDik as diffusion matrices we want them to turn out to be positive definite. Withinthis special choice the symmetry at least is trivial.

We return to the case where Q1 is not given as a relaxation-time operator anymore. Then G cannot be explicitly calculated – hence we have to guarantee existenceand uniqueness in some suitable space. To do so, we linearize the collision operatorQ1 meaning

Q1(fα) = Q1(Efα + αgα) = Q1(Efα) + αDQ1(Efα)gα +O(α2)

with the Fréchet derivative DQ1(Efα) of Q1 at Efα. The first term in the expansion,however, is zero due to assumption 3.12. Inserting this and (3.4) into (3.1) anddividing by α yields

α∂tfα+∇pε ·∇x(Efα +αgα)+∇xΦ ·∇p(Efα +αgα) = DQ1(Efα)gα+O(α)+αQ2(fα)

and finally in the limit α → 0

DQ1(EF )G = ∇pε · ∇xEF +∇xΦ · ∇pEF . (3.7)

This is an operator equation for G.We introduce the Hilbert space L2

E(B) with the scalar product

[

u, v]

E=

⟨

uv

E ′F

⟩

for two functions u, v : B → R. Then we impose the following:

Assumption 3.16The operator DQ1(EF ) : L2

E(B) → L2E(B) – the Fréchet derivative of Q1 at EF – is

assumed to have the following properties:

• It is linear and continuous (with constant CDQ1).

• It is self-adjoint with respect to [·, ·]E .

• Its kernel kerDQ1(EF ) is spanned by EF .

• Its range ranDQ1(EF ) is closed in L2E(B).

An example of an operatorQ1 satisfying these assumptions is given in [3, Section 3.2].Under these assumptions we can show:

Theorem 3.17For any v ∈ L2

E(B) we have

DQ1(EF )u = v is solvable for u ⇐⇒[

EF , v]

E= 0 ,

3.6. CURRENT DENSITIES 27

and in case of solvability the solution u is unique under the constraint [u, EF ]E = 0.

Proof For any linear and densely defined operator L we have

(ranL)⊥ = kerL∗

where L∗ is the adjoint operator of L in (L2E(B), [·, ·]E) – for details see [6, Proposi-

tion 1.13]. For L = DQ1(EF ) we have L = L∗, and thus passing on to the orthogonalcomplement yields

ranL = ranL = (ranL)⊥⊥ = (kerL∗)⊥ = (kerL)⊥

since ranL is closed. Together with the kernel characterization of L we concludethat v ∈ ranL whenever EF ⊥ v which is exactly the proposition.

This is an adequate criterion to check solvability:

Theorem 3.18The operator equations

DQ1(EF )ϕ = κiE ′F∂pjε and DQ1(EF )ϕ = E ′

F∂pjκi

are solvable for ϕ and ϕ, respectively, for each i = 0, . . . , N and j = 1, 2, 3.We denote the solutions by ϕij and ϕij, respectively, and to get uniqueness, weclaim [ϕij , EF ]E = [ϕij, EF ]E = 0. Furthermore, we use the column vectors ϕi =(ϕi1, ϕi2, ϕi3)

T and ϕi = (ϕi1, ϕi2, ϕi3)T.

Proof The only thing we have to check is

[

EF , κiE ′F∂pjε

]

E=⟨

EFκi∂pjε⟩

= 0 ,[

EF , E ′F∂pjκi

]

E=⟨

EF∂pjκi⟩

= 0

which is true since both integrands are odd in pj .

With the help of ϕij and ϕij we can construct an expression for G:

Theorem 3.19It holds

G =

N∑

k=0

(ϕk · ∇xλk + λkϕk · ∇xΦ) .

Proof We calculate

DQ1(EF )G =

N∑

k=0

3∑

j=1

(

DQ1(EF )ϕkj ∂xjλk + λkDQ1(EF )ϕkj ∂xj

Φ)

=

N∑

k=0

3∑

j=1

(

κkE ′F∂pjε ∂xj

λk + λkE ′F∂pjκk∂xj

Φ)


=

N∑

k=0

(

κkE ′F∇pε · ∇xλk + λkE ′

F∇pκk · ∇xΦ)

which coincides with the right hand side of (3.7) due to the derivatives in assump-tion 3.9.

Comparing this general term for G with the expression G = −τ1(∇pε ·∇xEF +∇xΦ ·∇pEF ) from equation (3.5) for the relaxation-time operator, we get

ϕk = −τ1κkE ′F∇pε and ϕk = −τ1E ′

F∇pκk ,

and obviously it holds [ϕkj, EF ]E = [ϕkj, EF ]E = 0 since again both integrands areodd in pj.

Inserting the expression for G into the definition for the current densities yields

Ji =⟨

κiG∇pε⟩

=

N∑

k=0

(

⟨

κi(ϕk · ∇xλk)∇pε⟩

+⟨

κiλk(ϕk · ∇xΦ)∇pε⟩

)

.

In the following we are going to define the moments in the sum as diffusion expres-sions and prove some important properties.

3.7 Diffusion matrices

Definition 3.20 (Diffusion matrices)For each i, k = 0, . . . , N we define matrices Dik, Dik ∈ R3×3 by

(Dik)jℓ = −⟨

κiϕkℓ∂pjε⟩

and (Dik)jℓ = −⟨

κiϕkℓ∂pjε⟩

,

and we call Dik and Dik a diffusion matrix and a “modified diffusion matrix”, re-spectively. Then the current densities can we written as

Ji = −N∑

k=0

(

Dik∇xλk + λkDik∇xΦ)

.

Furthermore, we define matrices D, D ∈ R3(N+1)×3(N+1) by the block structure

D =

D00 · · · D0N...

. . ....

DN0 · · · DNN

and D =

D00 · · · D0N...

. . ....

DN0 · · · DNN

.

Theorem 3.21 (Symmetry)The matrices D and D are symmetric.

3.7. DIFFUSION MATRICES 29

Proof We show that (Dik)T = Dki and (Dik)

T = Dki which is the same as theproposition. With DQ1(EF ) being self-adjoint we get

−(Dik)jℓ =⟨

κiϕkℓ∂pjε⟩

=[

κiE ′F∂pjε, ϕkℓ

]

E=[

DQ1(EF )ϕij, ϕkℓ

]

E

=[

ϕij , DQ1(EF )ϕkℓ

]

E=[

ϕij, κkE ′F∂pℓε

]

E=⟨

κkϕij∂pℓε⟩

= −(Dki)ℓj

and

−(Dik)jℓ =⟨

ϕkℓ∂pjκi⟩

=[

E ′F∂pjκi, ϕkℓ

]

E=[

DQ1(EF )ϕij , ϕkℓ

]

E

=[

ϕij , DQ1(EF )ϕkℓ

]

E=[

ϕij, E ′F∂pℓκk

]

E=⟨

ϕij∂pℓκk⟩

= −(Dki)ℓj .

As a summary to this point we have

∂tmi + divx Ji = Wi + Ji · ∇xΦ , Ji = −N∑

k=0

(

Dik∇xλk + λkDik∇xΦ)

.

With the derivatives of EF we calculate

∂tmi = ∂t⟨

κiEF⟩

=⟨

κi∂tEF⟩

=

⟨

κiE ′F

N∑

k=0

κk∂tλk

⟩

=

N∑

k=0

⟨

κiκkE ′F

⟩

∂tλk .

With a matrix K ∈ R(N+1)×(N+1) with entries (K)ik = 〈κiκkE ′F 〉 we obtain

(K∂tλ)i − divx

N∑

k=0

Dik∇xλk = Wi + Ji · ∇xΦ+ divx

N∑

k=0

λkDik∇xΦ .

In order to get a well-posed problem, in our case a system of N + 1 parabolicpartial differential equations for the N + 1 unknowns λ0, . . . , λN , we need K andD to be positive definite. K is at least positive semi-definite since it is the Grammatrix of the functions κ0E ′

F , . . . , κNE ′F with respect to the scalar product [·, ·]E ;

definiteness comes with the following assumption:

Assumption 3.22

• The set

κi∂pjε∣

∣ i = 0, . . . , N, j = 1, 2, 3

of functions in p is linearly independent. Then, of course, κ0, . . . , κN andκ0E ′

F , . . . , κNE ′F are linearly independent, too.

• The operator −DQ1(EF ) is coercive on (kerDQ1(EF ))⊥ which means there isa constant cDQ1

> 0 such that[

−DQ1(EF )ϕ, ϕ]

E≥ cDQ1

‖ϕ‖2Efor all ϕ ∈ (kerDQ1(EF ))⊥.


This assumption is sufficient for:

Theorem 3.23 (Positivity)The diffusion matrix D is positive definite.

Proof Let ξ ∈ R3(N+1). Then, with the definitions for Dik and ϕij we calculate

ξTDξ =N∑

i,k=0

ξ3i+1

ξ3i+2

ξ3i+3

T

Dik

ξ3k+1

ξ3k+2

ξ3k+3

=N∑

i,k=0

3∑

j,ℓ=1

ξ3i+j(Dik)jℓξ3k+ℓ

= −N∑

i,k=0

3∑

j,ℓ=1

⟨

ξ3i+jξ3k+ℓϕkℓκi∂pjε⟩

= −N∑

i,k=0

3∑

j,ℓ=1

⟨

ξ3i+jξ3k+ℓϕkℓDQ1(EF )ϕij

E ′F

⟩

= −N∑

i,k=0

3∑

j,ℓ=1

[

ξ3i+jξ3k+ℓϕkℓ, DQ1(EF )ϕij

]

E

= −N∑

i,k=0

3∑

j,ℓ=1

[

DQ1(EF )(ξ3i+jϕij), ξ3k+ℓϕkℓ

]

E

=

[

−DQ1(EF )(

N∑

i=0

3∑

j=1

ξ3i+jϕij

)

,N∑

k=0

3∑

ℓ=1

ξ3k+ℓϕkℓ

]

E

.

The expression∑

i

∑

j ξ3i+jϕij occurs twice and lies in (kerDQ1(EF ))⊥. With thecoercivity and boundedness of −DQ1(EF ) we continue

ξTDξ ≥ cDQ1

∥

∥

∥

∥

∥

N∑

i=0

3∑

j=1

ξ3i+jϕij

∥

∥

∥

∥

∥

2

E

≥ cDQ1

C2DQ1

∥

∥

∥

∥

∥

DQ1(EF )(

N∑

i=0

3∑

j=1

ξ3i+jϕij

)∥

∥

∥

∥

∥

2

E

=cDQ1

C2DQ1

∥

∥

∥

∥

∥

N∑

i=0

3∑

j=1

ξ3i+jκiE ′F∂pjε

∥

∥

∥

∥

∥

2

E

=cDQ1

C2DQ1

⟨∣

∣

∣

∣

∣

N∑

i=0

3∑

j=1

ξ3i+jκi∂pjε

∣

∣

∣

∣

∣

2

E ′F

⟩

> 0

when ξ 6= (0, . . . , 0)T since E ′F > 0 and the set κi∂pjε | i = 0, . . . , N, j = 1, 2, 3 is

linearly independent.

With the assumption that κ0, . . . , κN is linearly independent we can also show thefollowing important property:

Theorem 3.24The map (λ0, . . . , λN) 7→ (m0, . . . , mN) is injective.

Proof Let (λ(1)0 , . . . , λ

(1)N ) and (λ

(2)0 , . . . , λ

(2)N ) be two vectors with the same image

(m0, . . . , mN), that is,

⟨

κiEF(

λ(1)0 , . . . , λ

(1)N

)⟩

= mi =⟨

κiEF(

λ(2)0 , . . . , λ

(2)N

)⟩

3.7. DIFFUSION MATRICES 31

for all i = 0, . . . , N . Building the difference, multiplying by λ(m)i , and summing over

i yieldsN∑

i=0

⟨

λ(m)i κi

(

EF(

λ(2))

− EF(

λ(1)))

⟩

= 0 .

Now we consider the difference for m = 2 and for m = 1 leading to

⟨

(

λ(2) · κ− λ(1) · κ)(

EF(

λ(2))

− EF(

λ(1)))

⟩

= 0 .

EF can be interpreted as a function mapping λ · κ ∈ R to a real number. Due toAssumption 3.9 its derivative E ′

F is positive, and thus EF is strictly monotonicallyincreasing. This means that the difference EF (λ(2) · κ) − EF (λ(1) · κ) has got thesame sign as λ(2) · κ− λ(1) · κ concluding that their product is nonnegative and thesame holds for the entire expression in the moment 〈· · · 〉, and therefore it vanishesidentically. Furthermore we get λ(1) · κ = λ(2) · κ, and the linear independence ofκ0, . . . , κN finishes the injectivity.

This allows us to arbitrarily switch between λ and m.In order to obtain simplified expressions for D and D we have to assume κi = εi

again. Then Dik can be related to Dik:

Theorem 3.25In the case κi = εi we have

Dik = kDi,k−1 and Di0 = 03×3

for all i = 0, . . . , N and k = 1, . . . , N .

Proof We calculate

DQ1(EF )ϕij = E ′F∂pjκi = iκi−1E ′

F∂pjε = iDQ1(EF )ϕi−1,j

which results in ϕij − iϕi−1,j ∈ kerDQ1(EF ). Thus there is a constant cij for eachi = 0, . . . , N and j = 1, 2, 3 such that ϕij = cijEF + iϕi−1,j . But then

−(Dik)jℓ =⟨

κiϕkℓ∂pjε⟩

= ckℓ⟨

κiEF∂pjε⟩

+ k⟨

κiϕk−1,ℓ∂pjε⟩

= −k(Di,k−1)jℓ

since pj 7→ κiEF∂pjε is odd.

With the special expressions for ϕij and ϕij in the relaxation-time operator case weobtain:

Theorem 3.26Let Q1(f) = (Ef − f)/τ1 be the relaxation-time operator.

(a) Both the 3× 3 matrices Dik and Dik are diagonal for each i and k.


(b) Let ε, κi, and τ1 be invariant under permutations of the components of themomentum vector p. (These assumption especially holds when the quanti-ties depend on |p| only which is true in several cases.) This forces us toassume simultaneously that, as soon as a vector (p1, p2, p3)

T lies in B, then(pπ(1), pπ(2), pπ(3))

T for each permutation π lies in B, too.

Then Dik and Dik are multiples of the 3× 3 unit matrix I3 and can thereforebe identified by a scalar quantity.

Proof By inserting we obtain

(Dik)jℓ = −⟨

κiϕkℓ∂pjε⟩

=⟨

τ1κiκkE ′F∂pjε ∂pℓε

⟩

,

(Dik)jℓ = −⟨

κiϕkℓ∂pjε⟩

=⟨

τ1κiE ′F∂pjε ∂pℓκk

⟩

.

(a) Again, both integrands are odd in pj if j 6= ℓ, and so we conclude (Dik)jℓ =

(Dik)jℓ = 0 which means that all the off-diagonal entries of Dik and Dik vanish.

(b) Let j, ℓ ∈ 1, 2, 3 be two distinct indices. Then the two expressions ∂pjεand ∂pℓε are the same when interchanging pj and pℓ, and since the product(τ1κiκkE ′

F )(p) is invariant under this swap, (Dik)jj and (Dik)ℓℓ have the samevalue. Hence, together with (a), the 3 × 3 matrix Dik is a multiple of theidentity matrix I3, and it holds

Dik =1

3

⟨

τ1κiκkE ′F |∇pε|2

⟩

I3 .

The same reasoning holds for Dik which yields

Dik =1

3

⟨

τ1κiE ′F (∇pε · ∇pκk)

⟩

I3 .

3.8 Drift-diffusion formulation

The drift-diffusion formulation aims for a decoupling of the current definitions. In-stead of

Ji = −N∑

k=0

Dik∇xλk + potential term

where every Ji contains a linear combination of ∇xλ0, . . . ,∇xλN , we want to rewritethem into

Ji = −∇xgi + potential term

where the g0, . . . , gN are new scalar so-called drift-diffusion variables.

3.8. DRIFT-DIFFUSION FORMULATION 33

Theorem 3.27 (“Generalized drift-diffusion” formulation)The operator equation

DQ1(EF )χj = EF∂pjεis solvable for χj , and the quantities di ∈ R3×3 given by

(di)jℓ = −⟨

κiχℓ∂pjε⟩

satisfy the equation

∇x(di)jℓ =N∑

k=0

(Dik)jℓ∇xλk

provided that the operator DQ1(EF ) does not explicitly depend on x. The currentdensity components can then be written as

Jij = −3∑

ℓ=1

∂xℓ(di)jℓ + potential term .

Proof As usual, the operator equation is solvable since [EF∂pjε, EF ]E = 0. Thecalculation

DQ1(EF )(∂xjχℓ) = ∂xj

(EF∂pℓε) =N∑

k=0

κkE ′F∂pℓε ∂xj

λk

=N∑

k=0

DQ1(EF )ϕkℓ ∂xjλk = DQ1(EF )

(

N∑

k=0

ϕkℓ∂xjλk

)

and the kernel characterization of DQ1(EF ) result in

∇xχℓ =N∑

k=0

ϕkℓ∇xλk + EF cℓ

with a vector cℓ depending on t and x. Inserting into di yields

∇x(di)jℓ = −⟨

κi∂pjε∇xχℓ

⟩

= −N∑

k=0

⟨

κiϕkℓ∂pjε⟩

∇xλk =N∑

k=0

(Dik)jℓ∇xλk

since 〈κiEF∂pjε〉 = 0. For the current densities we get

Jij = −N∑

k=0

(Dik∇xλk)j + pot. term = −N∑

k=0

3∑

ℓ=1

(Dik)jℓ∂xℓλk + pot. term

= −3∑

ℓ=1

∂xℓ(di)jℓ + pot. term . (3.8)


We remark that di is not a “new” variable in the Maxwell-Boltzmann case becauseit holds χ = ϕ0 and di = Di0 since M′

F = MF .The sum

∑

ℓ ∂xℓ(di)jℓ, however, is still not what we have really aimed for. For a

fixed index i we would like this term to be the derivative of a variable with respect toxj . This is in fact possible when the matrices di are scalar multiples of the identitymatrix which is especially true for the relaxation-time operator due to Theorem 3.26:

Theorem 3.28 (Drift-diffusion formulation)Let Q1(f) = (Ef − f)/τ1 be the relaxation-time operator, let all the assumptionsin Theorem 3.26b hold, and let τ1 be independent of x. Then DQ1(EF ) does notexplicitly depend on x and we have

(di)jℓ =⟨

τ1κiEF∂pjε ∂pℓε⟩

.

Furthermore, di is a multiple of the unit matrix such that the real-valued so-calleddrift-diffusion variables g0, . . . , gN are well-defined by

giI3 = di

and satisfy

Ji = −∇xgi + potential term .

Proof From (3.6) we get

Jij = −⟨

τ1κi(∇pε ·∇xEF )∂pjε⟩

+pot. term = −3∑

ℓ=1

∂xℓ

⟨

τ1κiEF∂pjε ∂pℓε⟩

+pot. term

which, when compared to (3.8), motivates the expression (di)jℓ = 〈τ1κiEF∂pjε ∂pℓε〉we had claimed. Copying the proof of Theorem 3.26 we get

di =1

3

⟨

τ1κiEF |∇pε|2⟩

I3 = giI3

and thus

∇xgi =1

3

N∑

k=0

⟨


⟩

∇xλk =N∑

k=0

Dik∇xλk .

For the potential term we can show:

Theorem 3.29With the assumptions of the last theorem we can write

Ji = −∇xgi − giri(g)∇xΦ

3.9. DUAL-ENTROPY FORMULATION 35

with the function

ri(g) =N∑

k=0

λkgikgi

, gikI3 = Dik

which depends on g0, . . . , gN only; we claim once again that κ0, . . . , κN is linearlyindependent (see Assumption 3.22).

Proof The quantity gik is just the diagonal entry of Dik. It is trivial that theexpression for Ji is correct, however, we have to justify that ri is a function ofg0, . . . , gN only. We show that the map (λ0, . . . , λN) 7→ (g0, . . . , gN) is injective in

the same way as we did in Theorem 3.24: Let (λ(1)0 , . . . , λ

(1)N ) and (λ

(2)0 , . . . , λ

(2)N ) be

two vectors with the same image (g0, . . . , gN), that is,⟨

τ1κiEF(

λ(1)0 , . . . , λ

(1)N

)

|∇pε|2⟩

= 3gi =⟨

τ1κiEF(

λ(2)0 , . . . , λ

(2)N

)

|∇pε|2⟩

for all i = 0, . . . , N . Building the difference, multiplying by λ(m)i , summing over i,

and building the difference for m = 2 and m = 1 yields⟨

τ1(

λ(2) · κ− λ(1) · κ)(

EF(

λ(2))

− EF(

λ(1)))

|∇pε|2⟩

= 0 .

With the same arguments – strict monotonicity of EF and linear independence ofκ0, . . . , κN – and since τ1|∇pε|2 ≥ 0 we conclude the injectivity.

Hence there is a unique preimage (λ0, . . . , λN) for a given (g0, . . . , gN) and, ofcourse, the quantities gik are all well-defined as soon as the λi are known. Writtenin a fussy way we have

ri(g) =N∑

k=0

λk(g) gik(λ(g))

gi.

3.9 Dual-entropy formulation

The motivation to develop a dual-entropy formulation is to get rid of the electricforce terms. These terms tend to become very large and then usually require specialtreatment within a numerical algorithm.

Theorem 3.30Let

≤

δik =

1 for i ≤ k ,

0 for i > k

for indices i, k = 0, . . . , N . When we define matrices A,B ∈ R(N+1)×(N+1) withentries

aik =≤

δik

(

k

i

)

(−Φ)k−i , bik =≤

δik

(

k

i

)

Φk−i

then B is the inverse of A.

Proof Since A and B are upper triangular matrices, AB is of the same shape.Thus we calculate for indices 0 ≤ i ≤ k ≤ N


(AB)ik =N∑

m=0

aimbmk =k∑

m=i

aimbmk =k∑

m=i

(

m

i

)

(−Φ)m−i

(

k

m

)

Φk−m

= Φk−ik∑

m=i

(−1)m−i

(

m

i

)(

k

m

)

=

(

k

i

)

Φk−ik∑

m=i

(−1)m−i

(

k − i

m− i

)

=

(

k

i

)

Φk−i

k−i∑

m′=0

(−1)m′

(

k − i

m′

)

=

(

k

i

)

Φk−i(1− 1)k−i = δik .

Theorem 3.31 (Dual-entropy formulation)Introducing the dual-entropy variables λ = (λ0, . . . , λN)

T, the transformed momentsm = (m0, . . . , mN)

T, and the thermodynamic fluxes J0, . . . , JN by

λ = Bλ , m = ATm, Ji =

N∑

k=0

akiJk

the model equations

∂tmi + divx Ji − iJi−1 · ∇xΦ =Wi , Ji = −N∑

k=0

(

Dik∇xλk + kλkDi,k−1∇xΦ)

which hold if κi = εi are transformed to

∂tmi + divx Ji = Wi − imi−1∂tΦ , Ji = −N∑

k=0

Dik∇xλk ,

where W = ATW and

D =

D00 · · · D0N...

. . ....

DN0 · · · DNN

= ATDA with A =

a00I3 · · · a0NI3...

. . ....

aN0I3 · · · aNNI3

.

If D is positive definite, then so is D.

Proof We calculate

∂tm = ∂t(ATm) = (∂tA

T)m+ AT∂tm = (∂tAT)BTm+ AT∂tm,

divx J0...

divx JN

=

N∑

k=0

divx(ak0Jk)...

divx(akNJk)

=

N∑

k=0

ak0 divx Jk + Jk · ∇xak0...

akN divx Jk + Jk · ∇xakN

3.9. DUAL-ENTROPY FORMULATION 37

= AT

divx J0...

divx JN

−

N∑

k=0

Jk · ∇xΦ

Φ

ak0(0− k)...

akN(N − k)

.

We write the N + 1 balance equations in vectorial form and multiply by AT fromthe left, that is

AT∂tm+ AT

divx J0...

divx JN

−AT

0J0 · ∇xΦ

...NJN−1 · ∇xΦ

= ATW .

Inserting the calculations from above yields

∂tm+

divx J0...

divx JN

= ATW + (∂tA

T)BTm

+ AT

0J0 · ∇xΦ

...NJN−1 · ∇xΦ

−N∑

k=0

Jk · ∇xΦ

Φ

ak0(0− k)...

akN(N − k)

.

It remains to show that the terms in the second line cancel each other: For theith component we calculate

N∑

k=1

akikJk−1 · ∇xΦ−N∑

k=0

Jk · ∇xΦ

Φaki(i− k)

=

(

i∑

k=1

kakiJk−1 −i−1∑

k=0

(i− k)akiJkΦ

)

· ∇xΦ

=i−1∑

k=0

(

(k + 1)≤

δk+1,i

(

i

k + 1

)

(−Φ)i−k−1 − (i− k)≤

δki

(

i

k

)

(−Φ)i−k 1

Φ

)

Jk · ∇xΦ

=i−1∑

k=0

(

(k + 1)

(

i

k + 1

)

− (i− k)

(

i

k

))

(−Φ)i−k−1Jk · ∇xΦ = 0

where the term in parentheses is zero due to the definition of the binomial coef-ficients. Since AT – and therefore ∂tA

T – and BT are lower triangular matrices,(∂tA

T)BT is of the same shape. Thus we calculate for indices 0 ≤ k ≤ i ≤ N

(

(∂tAT)BT

)

ik=

N∑

m=0

(∂tami)bkm =

i∑

m=k

(

i

m

)

∂t(−Φ)i−m

(

m

k

)

Φm−k


=i−1∑

m=k

(

i

k

)(

i− k

m− k

)

(i−m)(−Φ)i−m−1(−∂tΦ)Φm−k

=

(

i

k

)

(

i−k−1∑

m′=0

(−1)m′+1(m′ + 1)

(

i− k

m′ + 1

)

)

Φi−k−1∂tΦ

= −(i− k)

(

i

k

)

(

i−k−1∑

m′=0

(−1)m′

(

i− k − 1

m′

)

)

Φi−k−1∂tΦ .

The expression is clearly zero if k = i. Otherwise the sum in parentheses is (1 −1)i−k−1 and therefore zero if i− k − 1 ≥ 1. For all i, k = 0, . . . , N holds

(

(∂tAT)BT

)

ik= −iδi−1,k∂tΦ

and(

(∂tAT)BTm

)

i= −

N∑

k=0

iδi−1,k(∂tΦ)mk = −imi−1∂tΦ .

For the current densities we calculate

Ji = −N∑

k=0

(

k(Aλ)kDi,k−1∇xΦ+Dik∇x(Aλ)k)

= −N∑

k=0

N∑

m=0

(

kakmλmDi,k−1∇xΦ+ λmDik∇xakm + akmDik∇xλm)

= −N∑

m=0

m∑

k=0

(

k

(

m

k

)

(−Φ)m−kλmDi,k−1∇xΦ

− λm

(

m

k

)

(m− k)(−Φ)m−k−1Dik∇xΦ+ akmDik∇xλm

)

=N∑

m=0

λm

(

m−1∑

k=0

(

m

k

)

(m− k)(−Φ)m−k−1Dik −m∑

k=1

k

(

m

k

)

(−Φ)m−kDi,k−1

)

∇xΦ

−N∑

m=0

m∑

k=0

akmDik∇xλm

=N∑

m=0

λm

m−1∑

k=0

((

m

k

)

(m− k)− (k + 1)

(

m

k + 1

))

(−Φ)m−k−1Dik∇xΦ

−N∑

m=0

m∑

k=0

akmDik∇xλm

where again the binomial coefficients expression in parentheses is zero. The trans-

3.10. THE CASE N = 0 39

formed current densities then become

Jij =N∑

m=0

amiJmj = −N∑

m=0

ami

(

N∑

k=0

N∑

m′=0

am′kDmm′∇xλk

)

j

= −N∑

k=0

3∑

ℓ=1

(

N∑

m=0

N∑

m′=0

(AT)im(Dmm′)jℓ(A)m′k

)

∂xℓλk

= −N∑

k=0

3∑

ℓ=1

(Dik)jℓ∂xℓλk =⇒ Ji = −

N∑

k=0

Dik∇xλk .

Since A is an upper triangular matrix, so is A, and we see detA = 1. Thus A isregular and, by linear algebra, ATDA is positive definite when D has this property.

The expressions do not simplify significantly under the assumption of the relaxation-time operator Q1(f) = (Ef − f)/τ1. When all the matrices Dik are a multiple of theunit matrix due to Theorem 3.26b, then Dik is a multiple of the unit matrix, too,and it holds

Dik =i∑

i′=0

k∑

k′=0

ai′iak′kDi′k′ =1

3

⟨

τ1

(

i∑

i′=0

ai′iκi′

)(

k∑

k′=0

ak′kκk′

)

E ′F |∇pε|2

⟩

I3 .

3.10 The case N = 0

The cause why we had to exclude the case N = 0 from the general derivation processlies in the fact that the Lagrange multiplier λ1 does not exist.

Theorem 3.32 (Equilibrium function and derivatives)There is only one fixed weight function κ0 = 1, and the entropies HM are HF aredefined as before. The equilibrium functions for Maxwell-Boltzmann and Fermi-Dirac are given by

Mf(t, x, p) = exp(

λ0(t, x)− ε(p))

and Ff(t, x, p) =1

η + exp(−λ0(t, x) + ε(p)),

respectively. The dependence on t and x is still given by λ0 : Ωt → R, but thedependence on p does not stem from the κ0 (which is constant anyway) but from εwhich explicitly occurs in the equilibrium function.

For the derivatives we analogously get

∂tEf = E ′f∂tλ0(t, x) , ∇xEf = E ′

f∇xλ0(t, x) , ∇pEf = −E ′f∇pε(p)

where E ′f again is another function acting on λ0 and ε. As in the N ≥ 1 case we

have M′f = Mf and F ′

f = Ff(1− ηFf).


Theorem 3.33 (Balance equation)Disregarding the choice of the equilibrium function Ef as long as it is even in p, weobtain the same balance equation

∂tm0 + divx J0 = 0

with m0 = 〈EF 〉 and J0 = 〈G∇pε〉 as for N ≥ 1.

For the current density J0 in the relaxation-time operator Q1 case we have to restartat the equation

J0 = −⟨

τ1(∇pε · ∇xEF +∇xΦ · ∇pEF )∇pε⟩

which becomes

J0 = −⟨

τ1(∇pε · E ′F∇xλ0 −∇xΦ · E ′

F∇pε)∇pε⟩

= −⟨

τ1E ′F (∇pε · ∇xλ0)∇pε

⟩

jej +

⟨

τ1E ′F (∇xΦ · ∇pε)∇pε

⟩

jej

= −(⟨

τ1E ′F∂pjε∇pε

⟩

· ∇xλ0)

ej +(⟨

τ1E ′F∂pjε∇pε

⟩

· ∇xΦ)

ej

= −D00∇xλ0 +D00∇xΦ = D00∇x(Φ− λ0)

with the 3× 3 diffusion matrix D00 with entries

(D00)jℓ =⟨

τ1E ′F∂pjε ∂pℓε

⟩

.

When we again impose invariance under permutations of the components of p weanalogously obtain

D00 =1

3

⟨

τ1E ′F |∇pε|2

⟩

I3 .

Theorem 3.34In the general Q1 case we get the same operator equation

DQ1(EF )G = ∇pε · ∇xEF +∇xΦ · ∇pEF

for G, the equation DQ1(EF )ϕ0j = E ′F∂pjε is solvable for ϕ0j , and it holds

G = ϕ0 · ∇x(λ0 − Φ) .

Proof We calculate

DQ1(EF )G =3∑

j=1

DQ1(EF )ϕ0j ∂xj(λ0 − Φ) = E ′

F (∇pε · ∇xλ0 −∇xΦ · ∇pε)

and are finished due to the derivatives of EF . For the diffusion matrix we get

(D00)jℓ = −⟨

ϕ0ℓ∂pjε⟩

.

3.10. THE CASE N = 0 41

Finally we look at the two alternative formulations:

Theorem 3.35 (Drift-diffusion formulation)The operator equations DQ1(EF )χj = EF∂pjε are solvable, and we have

J0j = −3∑

ℓ=1

∂xℓ(d0)jℓ + potential term

for (d0)jℓ = −〈χℓ∂pjε〉 when DQ1(EF ) does not explicitly depend on x. For therelaxation-time operator Q1(f) = (Ef − f)/τ1 we have d0 = g0I3 with

g0 =1

3

⟨

τ1EF |∇pε|2⟩

, J0 = −∇xg0 + g0r0(g0)∇xΦ , and r0(g0) =(D00)11g0

.

Proof We only have to check that (D00)11 – just one arbitrary diagonal entry ofD00 – is a well-defined function of g0. But this is easier for N = 0 since

∂λ0g0 =

1

3

⟨

τ1E ′F |∇pε|2

⟩

> 0

is sufficient for the injectivity of real-valued map λ0 7→ g0.

For the dual-entropy formulation there is nothing more to do than to define λ0 =λ0 − Φ; the general case cannot be applied since A = B = (1) which would nottransform anything. With all the other quantities left unchanged we get

∂tm0 + divx J0 = 0 , J0 = −D00∇xλ0 .


Chapter 4

Explicit models

In this chapter we want to present special models from the previous derivationchapter which are accessible for numerical simulations in the next chapter. Thuswe always choose the relaxation-time operator Q1(f) = (Ef − f)/τ1 to get explicitformulas for ϕi and ϕi, we set κi = εi, and we either look at the Maxwell-Boltzmanncase Ef = Mf or at the Fermi-Dirac case Ef = Ff .

4.1 Simplification of the integrals

Furthermore, we assume that ε – and with it κi, EF , and E ′F – and τ1 depend on

the modulus |p| of the momentum vector p only. This animates us to transfer theintegrals over B into spherical coordinates. Then we get

⟨


⟩

=

∫

B

τ(|p|) ε(|p|)i+kE ′F (|p|)

∣

∣

∣

∣

dε(|p|)d|p|

p

|p|

∣

∣

∣

∣

2

d3p

=

∫

S2

∫ |p|max (ω)

0

|p|2τ1(|p|) ε(|p|)i+k

(

dε(|p|)d|p|

)2

E ′F (|p|) d|p| d2ω

with the unit sphere S2 ⊂ R3, the solid angle ω of p, and the maximal modulus|p|max of a p ∈ B in the direction of ω. By a slight abuse of notation we use thesame name for τ1(|p|), ε(|p|), and E ′

F (|p|) as for τ1(p), ε(p), and E ′F (p), respectively.

Whenever |p|max does not depend on ω, which holds when B is a ball with radius|p|max or when B = R3 and hence |p|max = ∞, we can replace the integral over S2

by a multiplication by 4π. Usually the dependence of τ1 on p is given by

τ1 = τ1ε−βε

with a constant βε ∈ R and τ1(t, x) > 0. We use the index ε in βε to indicate thatit is the exponent for the dependence of τ1 on ε. Together we obtain

⟨


⟩

= 4πτ1

∫ |p|max

0

|p|2ε(|p|)i+k−βε

(

dε(|p|)d|p|

)2

E ′F (|p|) d|p| .

43

44 CHAPTER 4. EXPLICIT MODELS

For more simplification we assume Kane’s nonparabolic band approximationgiven by

|p| =√

2ε(1 + δε) .

We choose B = R3, and for a change of variables from |p| to ε we calculate

d|p|dε

=1 + 2δε

√

2ε(1 + δε)

and get

⟨


⟩

= 8√2πτ1

∫ ∞

0

εi+k+3/2−βε(1 + δε)3/2

1 + 2δεE ′F

(

|p|=√

2ε(1 + δε))

dε .

The same calculation can be done for the moments

mi =⟨

κiEF⟩

= 4π

∫ |p|max

0

|p|2ε(|p|)iEF (|p|) d|p|

= 4√2π

∫ ∞

0

εi+1/2√1 + δε (1 + 2δε)EF

(

|p|=√

2ε(1 + δε))

dε .

In the Maxwell-Boltzmann case we have E ′F = MF , and in the Fermi-Dirac case

E ′F = FF (1−ηFF ). In the limit δ → 0 we recover the parabolic band approximationε = 1

2|p|2, and even in this case the integrals are solvable analytically only with

Maxwell-Boltzmann statistics and N ≤ 1. For the Fermi-Dirac statistics and N ≤ 1we can make use of Fermi integrals which we are going to introduce in the following:

Definition 4.1 (Fermi integral)For every a > −1,

Ia(z) =1

Γ (a+ 1)

∫ ∞

0

sa

1 + es−zds

defines a function Ia : R→ ]0,∞[. The convergence of the integral can be seen from

0 ≤ sa

1 + es−z≤ sa

es−z= ezsae−s

since sae−s is the integrand in the definition of Γ (a+ 1), and the positivity is clear.

We summarize and prove some properties we need later on:

Theorem 4.2

(a) It holds

I0(z) = ln(ez + 1) .

4.1. SIMPLIFICATION OF THE INTEGRALS 45

(b) For z → −∞ we have the limit

Ia(z)

ez→ 1 .

(c) For z → +∞ we have the limit

Ia(z)

za+1/Γ (a+ 2)→ 1 .

(d) For every a > −1, Ia : R→ ]0,∞[ is strictly monotonically increasing, contin-uous, and bijective.

(e) For every a > 0, Ia is continuously differentiable with derivative

I ′a(z) = Ia−1(z) .

(f) The important inequality

(a + 2)Ia−1(z)Ia+1(z) > (a+ 1)Ia(z)2

holds for all a > 0 and z ∈ R.

Proof

(a) We calculate

I0(z) = −∫ ∞

0

−e−s

e−s + e−zds = − ln |e−s + e−z|

∣

∣

∣

∞

0= ln

∣

∣

∣

∣

1 + e−z

e−z

∣

∣

∣

∣

= ln(ez + 1) .

(b) With

Ia(z)

ez=

1

ez

∫ ∞

0

sa

1 + es−zds

∫ ∞

0

sa

esds

=

∫ ∞

0

1

1 + ez−s

sa

esds

∫ ∞

0

sa

esds

we get1

1 + ez≤ Ia(z)

ez≤ 1

and thus the behavior for z → −∞.

(c) Let a > −1 and z > 0. We calculate

1

za+1

∫ ∞

0

sa

1 + es−zds =

1

z

∫ ∞

0

ua

1 + euz−zz du


=

∫ 1

0

ua

1 + e(u−1)zdu+

∫ ∞

1

ua

1 + e(u−1)zdu

where we have substituted s = uz. We notice that the map

z 7→ ua

1 + e(u−1)z

is monotonically increasing for u ≤ 1 and decreasing for u ≥ 1 and that

limz→+∞

ua

1 + e(u−1)z=

ua for u < 1 ,

1/2 for u = 1 ,

0 for u > 1 .

So we can exchange the limit and the integral sign in both integrals and obtain

limz→+∞

∫ 1

0

ua

1 + e(u−1)zdu =

∫ 1

0

limz→+∞

ua

1 + e(u−1)zdu =

∫ 1

0

ua du =1

a+ 1,

limz→+∞

∫ ∞

1

ua

1 + e(u−1)zdu =

∫ ∞

1

limz→+∞

ua

1 + e(u−1)zdu = 0 .

Finally we get

Ia(z)

za+1/Γ (a+ 2)=a+ 1

za+1

∫ ∞

0

sa

1 + es−zds

z→+∞−−−−→ 1 .

(d) If z1 < z2 we get

es−z1 > es−z2 =⇒ 1

1 + es−z1<

1

1 + es−z2=⇒ Ia(z1) < Ia(z2)

and thus strict monotonicity. Continuity can be shown as for the Gammafunction, and with both properties and the limits (b) and (c) we get bijectivity.

(e) Since all occurring integrals are absolutely convergent, we can calculate

∂z

∫ ∞

0

sa

1 + es−zds =

∫ ∞

0

sa∂z1

1 + es−zds = −

∫ ∞

0

sa∂s1

1 + es−zds

=

∫ ∞

0

∂ssa 1

1 + es−zds = a

∫ ∞

0

sa−1

1 + es−zds

for each a > 0 so that

I ′a(z) =

1

Γ (a + 1)∂z

∫ ∞

0

sa

1 + es−zds =

a

Γ (a+ 1)

∫ ∞

0

sa−1

1 + es−zds = Ia−1(z) .

4.2. DRIFT-DIFFUSION MODELS 47

(f) For a fixed z ∈ R we define fz : ]−1,∞[ → ]0,∞[ by

fz(a) = Γ (a+ 2)Ia(z) = (a+ 1)

∫ ∞

0

sa

1 + es−zds =

∫ ∞

0

1

1 + es−z∂ss

a+1 ds

= −∫ ∞

0

∂s1

1 + es−zsa+1 ds =

∫ ∞

0

sa+1es−z

(1 + es−z)2ds .

With the Cauchy-Schwarz inequality we get for all a > 0

fz(a)2 =

(∫ ∞

0

sa/2√es−z

1 + es−z

sa/2+1√es−z

1 + es−zds

)2

<

(∫ ∞

0

saes−z

(1 + es−z)2ds

)(∫ ∞

0

sa+2es−z

(1 + es−z)2ds

)

= fz(a− 1)fz(a+ 1)

where the strict inequality holds due to the linear independence of the twofunctions. Division by Γ (a+ 1)Γ (a+ 2) yields

(a+ 2)Ia−1(z)Ia+1(z) > (a + 1)Ia(z)2 .

4.2 Drift-diffusion models

Drift-diffusion models are characterized by only one moment m0 and one currentdensity J0 which are interpreted as the electron density n and the electron currentdensity J . The model is given by the three equations

∂tn+ divx J = 0 , J = D∇x(Φ− λ) , D =1

3

⟨

τ1E ′F |∇pε|2

⟩

I3 .

Since there is no quantity representing the temperature the model cannot respectheating effects which means that everything is assumed to be at ambient temperaturefor all times. This is admissible for large devices with small voltages and currents.

In the Maxwell-Boltzmann case with parabolic bands we get

D =1

38√2 πτ1

∫ ∞

0

ε3/2−βεeλ−ε dε I3 =8√2 π

3Γ(5

2− βε

)

τ1eλI3 ,

n = 4√2π

∫ ∞

0

ε1/2eλ−ε dε = (2π)3/2eλ .

Thus we can eliminate λ and obtain

D =4

3√πΓ(5

2− βε

)

τ1nI3 , ∇xλ = ∇x lnn

(2π)3/2=

1

n∇xn

so that

J =4

3√πΓ(5

2− βε

)

τ1(n∇xΦ−∇xn) .

This is one of the well-known drift-diffusion models with the drift term n∇xΦ andthe diffusion term ∇xn.


In the Fermi-Dirac case with parabolic bands we get

D =8√2π

3τ1

∫ ∞

0

ε3/2−βε1

η + e−λ+ε

(

1− η

η + e−λ+ε

)

dε I3

= −8√2 π

3τ1

∫ ∞

0

ε3/2−βε∂ε1

η + e−λ+εdε I3

=8√2π

3τ1

∫ ∞

0

∂εε3/2−βε

1

η + e−λ+εdε I3

=8√2π

3

(3

2− βε

)

τ1

∫ ∞

0

ε1/2−βε1

η + e−λ+εdε I3 ,

n = 4√2π

∫ ∞

0

ε1/21

η + e−λ+εdε (4.1)

where the integration by parts formula is valid if βε <32. Rewriting the integrals as

∫ ∞

0

εa1

η + e−λ+εdε =

1

η

∫ ∞

0

εa

1 + eε−(λ+ln η)dε = Γ (a+ 1)

Ia(λ+ ln η)

η(4.2)

we get

D =8√2π

3Γ(5

2− βε

)

τ1I1/2−βε

(λ+ ln η)

ηI3 ,

n = (2π)3/2I1/2(λ+ ln η)

η,

∇xn = (2π)3/2∇xI1/2(λ+ ln η)

η= (2π)3/2

I−1/2(λ+ ln η)

η∇xλ .

Together we arrive at

J =4

3√πΓ(5

2− βε

)

τ1

(I1/2−βε(z)

I1/2(z)n∇xΦ− I1/2−βε

(z)

I−1/2(z)∇xn

)

, z = λ+ ln η .

The structure is exactly the same as in the Maxwell-Boltzmann case, however,there is one major difference: The current density still depends implicitly on λ dueto the argument of the Fermi integrals. The map λ 7→ n cannot inverted analytically,but, which is essential for the modeling and the numerics, it is strictly monotonicallyincreasing since I1/2 has this property, and thus the inverse n 7→ λ is well-defined.So both the fractions of two Fermi integrals in front of n∇xΦ and ∇xn can beunderstood as corrections that depend on n. In the limit η → 0 both fractions tendto 1 so that we recover the Maxwell-Boltzmann case as expected.

The other limit η → ∞ implies z → +∞ so that we have

n ∼ (2π)3/2z3/2

ηΓ (5/2)=

8√2π

3

(λ+ ln η)3/2

η,

4.2. DRIFT-DIFFUSION MODELS 49

λ+ ln η ∼(

3η

8√2π

n

)2/3

=1

4

(

9η2

2π2

)1/3

n2/3 ,

I1/2−βε(z)

I1/2(z)∼ z3/2−βεΓ (5/2)

Γ (5/2− βε)z3/2=

3√π

4Γ (5/2− βε)(λ+ ln η)−βε

∼ 3√π

41−βεΓ (5/2− βε)

(

2π2

9η2

)βε/3

n−2βε/3 ,

I1/2−βε(z)

I−1/2(z)∼ . . . ∼

√π

2 · 41−βεΓ (5/2− βε)

(

9η2

2π2

)(1−βε)/3

n2(1−βε)/3 ,

and inserting everything in J we obtain

J ∼ 4βε τ1n−2βε/3

((

2π2

9η2

)βε/3

n∇xΦ− 1

10

(

9η2

2π2

)(1−βε)/3

∇xn5/3

)

.

This relation is rather unsatisfactory since the coefficient of n∇xΦ vanishes and theone of ∇xn

5/3 explodes when η becomes arbitrarily large. We solve this problem byrescaling n = η−1n∞ and arrive at

J ∼ 4βε

ητ1n

−2βε/3∞

((

2π2

9

)βε/3

n∞∇xΦ− 1

10

(

9

2π2

)(1−βε)/3

∇xn5/3∞

)

.

This allows us to rescale the current J = η−1J∞ in the same way and get the model

∂tn∞ + divx J∞ = 0 , J∞ = n−2βε/3∞

(

C1n∞∇xΦ− C2∇xn5/3∞

)

with two constants C1 and C2 that do not depend on η.We summarize all our drift-diffusion models in a final theorem:

Theorem 4.3 (Model equations)Let Q1(f) = (Ef − f)/τ1 be the relaxation-time operator with characteristic timeτ1 = τ1ε

−βε, ε = 12|p|2 the parabolic band approximation, and B = R3. Then the

balance equation is given by

∂tn+ divx J = 0 ,

and for every η > 0 we have

J =4

3√πΓ(5

2− βε

)

τ1

(I1/2−βε(z)

I1/2(z)n∇xΦ− I1/2−βε

(z)

I−1/2(z)∇xn

)

, z = λ+ ln η .

For η → 0 we obtain the Maxwell-Boltzmann model

J =4

3√πΓ(5

2− βε

)

τ1(n∇xΦ−∇xn) ,


for η → ∞ we get with scaled quantities n and J

J = 4βε τ1n−2βε/3

((

2π2

9

)βε/3

n∇xΦ− 1

10

(

9

2π2

)(1−βε)/3

∇xn5/3

)

.

In the same order we list the three common choices βε = 0

J = τ1

(

n∇xΦ− I1/2(z)

I−1/2(z)∇xn

)

,

J = τ1(n∇xΦ−∇xn) ,

J = τ1

(

n∇xΦ− 1

10

(

9

2π2

)1/3

∇xn5/3

)

,

βε =12

J =4

3√πτ1

(

ln(ez + 1)

I1/2(z)n∇xΦ− ln(ez + 1)

I−1/2(z)∇xn

)

,

J =4

3√πτ1(n∇xΦ−∇xn) ,

J = 2τ1n−1/3

((

2π2

9

)1/6

n∇xΦ− 1

10

(

9

2π2

)1/6

∇xn5/3

)

,

and βε = 1

J =2

3τ1

(I−1/2(z)

I1/2(z)n∇xΦ−∇xn

)

,

J =2

3τ1(n∇xΦ−∇xn) ,

J = 4τ1n−2/3

((

2π2

9

)1/3

n∇xΦ− 1

10∇xn

5/3

)

.

These choices for βε are used as the Gamma function can be evaluated analyticallyfor integers and half-integers. If βε < 0 then τ1 would grow with ε which is notphysically reasonable; the other way around βε ≥ 3

2would destroy our derivation,

see above below (4.1).

The Maxwell-Boltzmann case for the drift-diffusion formulation is trivial sinceg = (D)11 and thus r = (D)11/g = 1. In the Fermi-Dirac case, using (4.2), we get

g =1

3

⟨

τ1EF |∇pε|2⟩

=8√2π

3τ1

∫ ∞

0

ε3/2−βε1

η + e−λ+εdε

=8√2π

3Γ(5

2− βε

)

τ1I3/2−βε

(λ+ ln η)

η,

r =(D)11g

=I1/2−βε

(λ+ ln η)

I3/2−βε(λ+ ln η)

.

4.3. ENERGY-TRANSPORT MODELS (MAXWELL-BOLTZMANN) 51

Theorem 4.4 (Drift-diffusion formulation)In the setting of the previous theorem with τ1 independent of x the current densityis given by

J = −∇xg + gI1/2−βε

(z)

I3/2−βε(z)

∇xΦ , g =8√2π

3Γ(5

2− βε

)

τ1I3/2−βε

(z)

η.

For η → 0 we get the Maxwell-Boltzmann case

J = −∇xg + g∇xΦ , g =8√2π

3Γ(5

2− βε

)

τ1eλ .

In the both cases, respectively, we get for βε = 0

g = (2π)3/2τ1I3/2(z)

η, g = (2π)3/2τ1e

λ ,

for βε =12

g =8√2π

3τ1I1(z)

η, g =

8√2π

3τ1e

λ ,

and for βε = 1

g =2

3(2π)3/2τ1

I1/2(z)

η, g =

2

3(2π)3/2τ1e

λ .

As already mentioned in the N = 0 section of the last chapter, the dual-entropyformulation is trivial here. We simply define λ = λ−Φ and get the current definition

J = −D∇xλ .

4.3 Energy-transport models

with Maxwell-Boltzmann statistics

A model usually is called an energy-transport model when it consists of two mo-ments and two current densities which are interpreted as a particle density and anenergy density. These two moments are related to each other by a function thatdepends on the temperature which is another variable in the model. This is themajor improvement compared to the drift-diffusion model where the temperaturewas assumed to be constant.

The model in the Maxwell-Boltzmann case is given by

∂tm0 + divx J0 = 0 , J0 = −λ1D00∇xΦ−D00∇xλ0 −D01∇xλ1 ,

∂tm1 + divx J1 − J0 · ∇xΦ = W1 , J1 = −λ1D10∇xΦ−D10∇xλ0 −D11∇xλ1 ,

Dik =1

3

⟨

τ1κiκkMF |∇pε|2⟩

I3 .


It is clear that we have to impose λ1 < 0 for the convergence of∫∞

0εaMF dε. We

can change the variable s = −λ1ε, and in the parabolic band approximation we get

∫ ∞

0

εaMF dε =eλ0

(−λ1)a+1

∫ ∞

0

sae−s ds = Γ (a+ 1)eλ0

(−λ1)a+1,

Dik =1

38√2πτ1

∫ ∞

0

εi+k+3/2−βεMF dε I3

=8√2 π

3Γ(

i+ k +5

2− βε

)

τ1eλ0

(−λ1)i+k+5/2−βεI3 ,

mi = 4√2 π

∫ ∞

0

εi+1/2MF dε = 4√2πΓ

(

i+3

2

) eλ0

(−λ1)i+3/2.

Introducing the temperature

T =1

−λ1we can write

D00 =8√2 π

3Γ(5

2− βε

)

τ1T5/2−βεeλ0I3 ,

D01 = D10 =8√2 π

3Γ(7

2− βε

)


D11 =8√2 π

3Γ(9

2− βε

)


m0 = (2π)3/2T 3/2eλ0 ,

m1 =3

2(2π)3/2T 5/2eλ0 =

3

2m0T .

It is easy to see that any two of the variables m0, m1, λ0, and λ1 can describethe model in an equivalent way and that λ1 and T can be arbitrarily exchanged.Altogether we have the following models:

Theorem 4.5 (Model equations in ∇xλ0 and ∇xλ1)Let Q1(f) = (Mf − f)/τ1 be the relaxation-time operator with characteristic timeτ1 = τ1ε


balance equations in the Maxwell-Boltzmann case are given by

∂tm0 + divx J0 = 0 , ∂tm1 + divx J1 − J0 · ∇xΦ = W1 .

The current densities can be written as a linear combination of ∇xΦ, ∇xλ0, and∇xλ1

J0 = −τ1(

λ1d0∇xΦ+ d0∇xλ0 + d1∇xλ1)

,

J1 = −τ1(


4.3. ENERGY-TRANSPORT MODELS (MAXWELL-BOLTZMANN) 53

with diffusion coefficients

(

d0 d1d1 d2

)

=4

3√πΓ(5

2− βε

)

m0T1−βε

1(5

2− βε

)

T

(5

2− βε

)

T(5

2− βε

)(7

2− βε

)

T 2

.

Again we look at the three common choices βε = 0

(

d0 d1d1 d2

)

= m0T

15

2T

5

2T

35

4T 2

,

βε =12

(

d0 d1d1 d2

)

=4

3√πm0T

1/2

(

1 2T2T 6T 2

)

,

and βε = 1

(

d0 d1d1 d2

)

=2

3m0

13

2T

3

2T

15

4T 2

.

The other representation uses ∇xm0 and ∇xm1 in the current densities. Toderive it we rewrite

−λ1 =3m0

2m1

, eλ0 =m0

(2π)3/2T 3/2=

(

3

4π

)3/2m

5/20

m3/21

such that

∇xλ0 = ∇x lnm

5/20

m3/21

=5

2m0∇xm0 −

3

2m1∇xm1 ,

∇xλ1 = − 3

2m1∇xm0 +

3m0

2m21

∇xm1 .

Theorem 4.6 (Model equations in ∇xm0 and ∇xm1)In the setting of the previous theorem we can rewrite the current densities as

J0 =4

3√π

(3

2

)βε

Γ(5

2− βε

)

τ1mβε

0 m1−βε

1

·(

m0

m1

∇xΦ− 2βε3m0

∇xm0 −2(1− βε)

3m1

∇xm1

)

,


J1 =4

3√π

(2

3

)1−βε

Γ(7

2− βε

)

τ1mβε

0 m1−βε

1

m1

m0

·(

m0

m1

∇xΦ +2(1− βε)

3m0

∇xm0 −2(2− βε)

3m1

∇xm1

)

.

For βε = 0 this is

J0 = τ1m1

(

m0

m1∇xΦ− 2

3

1

m1∇xm1

)

,

J1 =5

3τ1m2

1

m0

(

m0

m1∇xΦ+

2

3

1

m0∇xm0 −

4

3

1

m1∇xm1

)

,

for βε =12

J0 = 2

√

2

3πτ1m

1/20 m

1/21

(

m0

m1∇xΦ− 1

3

1

m0∇xm0 −

1

3

1

m1∇xm1

)

,

J1 =8

3

√

2

3πτ1m

3/21

m1/20

(

m0

m1∇xΦ+

1

3

1

m0∇xm0 −

1

m1∇xm1

)

.

and for βε = 1

J0 = τ1m0

(

m0

m1∇xΦ− 2

3

1

m0∇xm0

)

, J1 = τ1m1

(

m0

m1∇xΦ− 2

3

1

m1∇xm1

)

.

For the drift-diffusion formulation we have

gi =1

3

⟨

τ1κiMF |∇pε|2⟩

=8√2 π

3Γ(

i+5

2− βε

)

τ1Ti+5/2−βεeλ0

and thus

ri =λ0(Di0)11

gi+λ1(Di1)11

gi= −(Di0)11

giT= − 1

T.

Theorem 4.7 (Drift-diffusion formulation)In the setting of the previous theorem with τ1 independent of x we can rewrite thecurrent densities as

J0 = −∇xg0 +g0T∇xΦ , J1 = −∇xg1 +

g1T∇xΦ

with g1 = (52− βε)g0T . For βε = 0 we have

g0 = (2π)3/2τ1T5/2eλ0 = τ1m0T , g1 =

5

2(2π)3/2τ1T

7/2eλ0 =5

2g0T ,

4.4. ENERGY-TRANSPORT MODELS (FERMI-DIRAC) 55

for βε =12

g0 =8√2π

3τ1T

2eλ0 =4

3√πτ1m0T

1/2 , g1 =16√2π

3τ1T

3eλ0 = 2g0T ,

and for βε = 1

g0 =2

3(2π)3/2τ1T

3/2eλ0 =2

3τ1m0 , g1 = (2π)3/2τ1T

5/2eλ0 =3

2g0T .

For the dual-entropy formulation we have the two matrices

A =

(

1 −Φ0 1

)

, B =

(

1 Φ0 1

)

so that(

m0

m1

)

=

(

m0

−Φm0 +m1

)

,

(

W0

W1

)

=

(

0W1

)

, J0 = J0 , J1 = −ΦJ0 + J1 .

For the balance equations we obtain

∂tm0 + divx J0 = 0 , ∂tm1 + divx J1 = W1 − m0∂tΦ .

4.4 Energy-transport models

with Fermi-Dirac statistics

The model equations in the Fermi-Dirac case are the same as in the Maxwell-Boltzmann case, except that the diffusion matrices are given by

Dik =1

3

⟨

τ1κiκkF ′F |∇pε|2

⟩

I3 =1

3

⟨

τ1κiκkFF (1− ηFF )|∇pε|2⟩

I3 .

Again we need λ1 < 0 for the convergence of∫∞

0εaFF dε (and

∫∞

0εaF2

F dε). Withthe same change of variables s = −λ1ε we calculate

∫ ∞

0

εaFF dε =1

(−λ1)a+1

∫ ∞

0

sa

η + es−λ0ds =

Γ (a+ 1)

(−λ1)a+1

Ia(z)

η

where z = λ0 + ln η. Then

Dik =8√2π

3τ1

∫ ∞

0

εi+k+3/2−βε1

η + e−λ0−λ1ε

(

1− η

η + e−λ0−λ1ε

)

dε I3

=8√2π

3τ1

∫ ∞

0

εi+k+3/2−βε1

λ1∂εFF dε I3


=8√2π

3τ1

1

−λ1

∫ ∞

0

∂εεi+k+3/2−βε FF dε I3

=8√2π

3

(

i+ k +3

2− βε

)

τ11

−λ1

∫ ∞

0

εi+k+1/2−βεFF dε I3

=8√2π

3Γ(

i+ k +5

2− βε

)

τ11

(−λ1)i+k+5/2−βε

Ii+k+1/2−βε(z)

ηI3 ,

mi = 4√2 π

∫ ∞

0

εi+1/2FF dε = 4√2 πΓ

(

i+3

2

) 1

(−λ1)i+3/2

Ii+1/2(z)

η.

With the same definition for the temperature T = −1/λ1 we get

D00 =8√2 π

3Γ(5

2− βε

)

τ1T5/2−βε

I1/2−βε(z)

ηI3 ,

D01 = D10 =8√2 π

3Γ(7

2− βε

)

τ1T7/2−βε

I3/2−βε(z)

ηI3 ,

D11 =8√2 π

3Γ(9

2− βε

)

τ1T9/2−βε

I5/2−βε(z)

ηI3 ,

m0 = (2π)3/2T 3/2I1/2(z)

η,

m1 =3

2(2π)3/2T 5/2I3/2(z)

η=

3

2m0T

I3/2(z)

I1/2(z).

Here, in contrast to the Maxwell-Boltzmann case, it is not so easy to see whichvariables can be used, however, it is clear that everything can be written in termsof λ0 and λ1:

Theorem 4.8 (Model equations in ∇xλ0 and ∇xλ1)Let Q1(f) = (Ff − f)/τ1 be the relaxation-time operator with characteristic timeτ1 = τ1ε


balance equations in the Fermi-Dirac case are given by

∂tm0 + divx J0 = 0 , ∂tm1 + divx J1 − J0 · ∇xΦ = W1 .

The current densities can be written as a linear combination of ∇xΦ, ∇xλ0, and∇xλ1

J0 = −τ1(


, (4.3a)

J1 = −τ1(


(4.3b)

with diffusion coefficients

(

d0 d1d1 d2

)

=4

3√πΓ(5

2− βε

)

m0T1−βε


·

I1/2−βε(z)

I1/2(z)

(5

2− βε

)I3/2−βε(z)

I1/2(z)T

(5

2− βε

)I3/2−βε(z)

I1/2(z)T(5

2− βε

)(7

2− βε

)I5/2−βε(z)

I1/2(z)T 2

. (4.4)

The three common choices are βε = 0

(

d0 d1d1 d2

)

= m0T

15

2

I3/2(z)

I1/2(z)T

5

2

I3/2(z)

I1/2(z)T

35

4

I5/2(z)

I1/2(z)T 2

,

βε =12

(

d0 d1d1 d2

)

=4

3√πm0T

1/2

ln(ez + 1)

I1/2(z)2I1(z)

I1/2(z)T

2I1(z)

I1/2(z)T 6

I2(z)

I1/2(z)T 2

,

and βε = 1

(

d0 d1d1 d2

)

=2

3m0

I−1/2(z)

I1/2(z)

3

2T

3

2T

15

4

I3/2(z)

I1/2(z)T 2

.

It is easy to see that we obtain the Maxwell-Boltzmann diffusion matrices in thelimit η → 0.

To investigate the degenerate model for very large η we rewrite the moments as

m0 ∼(2π)3/2T 3/2z3/2

ηΓ (5/2)=

8√2 π

3η(zT )3/2 , m1 ∼

3(2π)3/2T 5/2z5/2

2ηΓ (7/2)=

8√2 π

5η(zT )5/2

where we have used the asymptotic behavior of Ia(z) ∼ za+1/Γ (a + 2) for largevalues of z. But this means that

m3/21

m5/20

∼ 9√3

40√10

η

π

so that m1 is a function of m0 only. This means that the model can be completelycharacterized by one moment m0 only.

For further examination we use (4.4) and get

d0 ∼4

3√πΓ(5

2− βε

)

m0T1−βε

z3/2−βεΓ (5/2)

Γ (5/2− βε)z3/2= m0(zT )

−βεT ,


d1 ∼4

3√πΓ(5

2− βε

)

m0T1−βε

(5

2− βε

) z5/2−βεΓ (5/2)

Γ (7/2− βε)z3/2T = m0(zT )

1−βεT ,

d2 ∼4

3√πΓ(5

2− βε

)

m0T1−βε

(5

2− βε

)(7

2− βε

) z7/2−βεΓ (5/2)

Γ (9/2− βε)z3/2T 2

= m0(zT )2−βεT ,

and inserting into (4.3a) and (4.3b) yields

J0 ∼ −τ1(

− 1

Tm0(zT )

−βεT∇xΦ+m0(zT )−βεT∇xλ0 +m0(zT )

1−βεT∇xλ1

)

= −τ1m0(zT )−βε

(

−∇xΦ+ T∇xz − zT 2∇x1

T

)

= τ1m0(zT )−βε∇x(Φ− zT ) ,

J1 ∼ zTJ0 ∼ τ1m0(zT )1−βε∇x(Φ− zT ) .

We conclude that all four quantities m0, m1, J0, and J1 contain z and T only as theproduct zT = −(λ0+ln η)/λ1 and that the current equations are linearly dependent.So the limit η → ∞ does not produce a good model.

We return to the model with arbitrary finite η. For the representation of themodel in m0 and m1 and, as we are going to see later on, for the numerics, we canshow:

Theorem 4.9The map (λ0, λ1) 7→ (m0, m1) is injective. Thus, for suitably given m0 and m1, wecan find uniquely defined λ0 and λ1.

Proof We calculate

m50

m31

=(2

3

)3

(2π)3I1/2(z)

5

η2I3/2(z)3=⇒ fη(z) =

I1/2(z)5

η2I3/2(z)3=( 3

4π

)3m50

m31

.

Due to Fermi integral inequality for a = 12

we have

f ′η(z) =

5I1/2(z)4I−1/2(z)I3/2(z)

3 − 3I1/2(z)5I3/2(z)

2I1/2(z)

η2I3/2(z)6

=(

5I−1/2(z)I3/2(z)− 3I1/2(z)2) I1/2(z)

4

η2I3/2(z)4> 0

and therefore fη is strictly monotonically increasing for each η > 0. In the limitz → −∞ we get

fη(z) =e2z

η2

(I1/2(z)

ez

)5(I3/2(z)

ez

)−3z→−∞−−−−→ 0

and for z → +∞

fη(z) =Γ (7/2)3

Γ (5/2)5η2

(

Γ(5

2

)I1/2(z)

z3/2

)5(

Γ(7

2

)I3/2(z)

z5/2

)−3z→+∞−−−−→ 250

9πη2.

So fη : R→ ]0, 250/9πη2[ is bijective.


Thus, for given m0 and m1, we can find a unique z and with it λ0 = z− ln η. λ1 (orT ) can then be easily calculated from m0 = (2πT )3/2I1/2(z)/η.

We apply the gradient on m0 and m1 which yields

∇xm0 =(2π)3/2

(−λ1)3/2I−1/2(z)

η∇xλ0 +

3

2

(2π)3/2

(−λ1)5/2I1/2(z)

η∇xλ1 ,

∇xm1 =3

2

(2π)3/2

(−λ1)5/2I1/2(z)

η∇xλ0 +

15

4

(2π)3/2

(−λ1)7/2I3/2(z)

η∇xλ1 .

∇xm0 and ∇xm1 are given as a linear combination of ∇xλ0 and ∇xλ1. Invertingthis relationship we arrive at

∇xλ0 = 5(−λ1)3/2(2π)3/2

ηI3/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

− 2(−λ1)5/2(2π)3/2

ηI1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1 ,

∇xλ1 = −2(−λ1)5/2(2π)3/2

ηI1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

+4

3

(−λ1)7/2(2π)3/2

ηI−1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

which is well-defined since the denominator is positive due to the Fermi integralinequality for a = 1

2. Together with

−λ1 =3

2

m0

m1

I3/2(z)

I1/2(z)

we get:

Theorem 4.10 (Model equations in ∇xm0 and ∇xm1)The current densities from Theorem 4.8 can be rewritten as

J0 =4

3√πΓ(5

2− βε

)

τ1T−βε

(

m0

I1/2−βε(z)

I1/2(z)∇xΦ

− T5I1/2−βε

(z)I3/2(z)− (5− 2βε)I3/2−βε(z)I1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

− 2

3

(5− 2βε)I3/2−βε(z)I−1/2(z)− 3I1/2−βε

(z)I1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

)

,

J1 =4

3√πΓ(7

2− βε

)

τ1T−βε

(

2

3m1

I3/2−βε(z)

I3/2(z)∇xΦ

− T 2 5I3/2−βε(z)I3/2(z)− (7− 2βε)I5/2−βε

(z)I1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0


− 2

3T(7− 2βε)I5/2−βε

(z)I−1/2(z)− 3I3/2−βε(z)I1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

)

where

T =2

3

m1

m0

I1/2(z)

I3/2(z).

Again we choose βε = 0

J0 = τ1

(

m0∇xΦ− 2

3∇xm1

)

,

J1 =5

3τ1

(

m1∇xΦ+3

2T 2 7I1/2(z)I5/2(z)− 5I3/2(z)

2

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

− T7I−1/2(z)I5/2(z)− 3I1/2(z)I3/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

)

,

βε =12

J0 =4

3√πτ1T

1/2

(

m0

T

I0(z)

I1/2(z)∇xΦ− 5I0(z)I3/2(z)− 4I1/2(z)I1(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

− 2

3T−14I1(z)I−1/2(z)− 3I0(z)I1/2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

)

,

J1 =16

9√πτ1T

1/2

(

m1

T

I1(z)

I3/2(z)∇xΦ− 3

2T5I1(z)I3/2(z)− 6I1/2(z)I2(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm0

− 6I−1/2(z)I2(z)− 3I1/2(z)I1(z)

5I−1/2(z)I3/2(z)− 3I1/2(z)2∇xm1

)

,

and βε = 1

J0 =2

3τ1

(

m0

T

I−1/2(z)

I1/2(z)∇xΦ−∇xm0

)

, J1 =2

3τ1

(

m1

T

I1/2(z)

I3/2(z)∇xΦ−∇xm1

)

.

Note that J0 and J1 are very symmetric for βε = 1.

Proof Inserting the terms for d0, d1, d2, ∇xλ0, and ∇xλ1 into the expressions forJ0 and J1 results in the lengthy calculation

J0τ1

=4

3√πΓ(5

2− βε

)

m0T1−βε

(

1

T

I1/2−βε(z)

I1/2(z)∇xΦ

− I1/2−βε(z)

I1/2(z)

(

5ηI3/2(z)

(2π)3/2T 3/2C∇xm0 −

2ηI1/2(z)

(2π)3/2T 5/2C∇xm1

)

−(5

2− βε

)I3/2−βε(z)

I1/2(z)T

(

− 2ηI1/2(z)

(2π)3/2T 5/2C∇xm0 +

4ηI−1/2(z)

3(2π)3/2T 7/2C∇xm1

))

,

=4

3√πΓ(5

2− βε

)

m0

(

T−βεI1/2−βε

(z)

I1/2(z)∇xΦ


− ηT−1/2−βε

(2π)3/2I1/2(z)

5I1/2−βε(z)I3/2(z)− (5− 2βε)I3/2−βε

(z)I1/2(z)

C∇xm0

− ηT−3/2−βε

(2π)3/2I1/2(z)

23(5− 2βε)I3/2−βε

(z)I−1/2(z)− 2I1/2−βε(z)I1/2(z)

C∇xm1

)

,

J1τ1

=4

3√πΓ(7

2− βε

)

m0T1−βε

(

1

T

I3/2−βε(z)

I1/2(z)T∇xΦ

− I3/2−βε(z)

I1/2(z)T

(

5ηI3/2(z)

(2π)3/2T 3/2C∇xm0 −

2ηI1/2(z)

(2π)3/2T 5/2C∇xm1

)

−(7

2− βε

)I5/2−βε(z)

I1/2(z)T 2

(

− 2ηI1/2(z)

(2π)3/2T 5/2C∇xm0 +

4ηI−1/2(z)

3(2π)3/2T 7/2C∇xm1

))

=4

3√πΓ(7

2− βε

)

m0

(

T 1−βεI3/2−βε

(z)

I1/2(z)∇xΦ

− ηT 1/2−βε

(2π)3/2I1/2(z)

5I3/2−βε(z)I3/2(z)− (7− 2βε)I5/2−βε

(z)I1/2(z)

C∇xm0

− ηT−1/2−βε

(2π)3/2I1/2(z)

23(7− 2βε)I5/2−βε

(z)I−1/2(z)− 2I3/2−βε(z)I1/2(z)

C∇xm1

)

where C = 5I−1/2(z)I3/2(z)− 3I1/2(z)2.

For the drift-diffusion formulation we have

gi =1

3

⟨

τ1κiFF |∇pε|2⟩

=8√2π

3Γ(

i+5

2− βε

)

τ1Ti+5/2−βε

Ii+3/2−βε(z)

η

and thus

ri =λ0(Di0)11

gi+λ1(Di1)11

gi= −(Di0)11

giT= −Ii+1/2−βε

(z)

Ii+3/2−βε(z)

1

T.

Theorem 4.11 (Drift-diffusion formulation)In the setting of the previous theorem with τ1 independent of x we can rewrite thecurrent densities as

J0 = −∇xg0 +I1/2−βε

(z)

I3/2−βε(z)

g0T∇xΦ , J1 = −∇xg1 +

I3/2−βε(z)

I5/2−βε(z)

g1T∇xΦ .

For βε = 0 we have

g0 = (2π)3/2τ1T5/2I3/2(z)

η=

I3/2(z)

I1/2(z)τ1m0T ,

g1 =5

2(2π)3/2τ1T

7/2I5/2(z)

η=

5

2

I5/2(z)

I3/2(z)g0T ,


for βε =12

g0 =8√2π

3τ1T

2I1(z)

η=

4

3√π

I1(z)

I1/2(z)τ1m0T

1/2 ,

g1 =16√2 π

3τ1T

3I2(z)

η= 2

I2(z)

I1(z)g0T ,

and for βε = 1

g0 =2

3(2π)3/2τ1T

3/2I1/2(z)

η=

2

3τ1m0 , g1 = (2π)3/2τ1T

5/2I3/2(z)

η=

3

2

I3/2(z)

I1/2(z)g0T .

The formulas for the dual-entropy formulation determine how to calculate thetransformed variables mi, Ji, etc. out of the original ones mi, Ji, etc. They dependon N only and especially not on EF , and thus the results are the same as in theMaxwell-Boltzmann case.

4.5 Extended energy-transport models

with Maxwell-Boltzmann statistics

We call a model with more than two densities and currents an extended energy-transport model. The equations in the Maxwell-Boltzmann case are

∂tm0 + divx J0 = 0 ,

∂tm1 + divx J1 − J0 · ∇xΦ = W1 ,

∂tm2 + divx J2 − 2J1 · ∇xΦ = W2 ,

Ji = −λ1Di0∇xΦ− 2λ2Di1∇xΦ−Di0∇xλ0 −Di1∇xλ1 −Di2∇xλ2 ,

mi =⟨

κiMF

⟩

= 4√2 π

∫ ∞

0

εi+1/2 exp(λ0 + λ1ε+ λ2ε2) dε ,

Dik =1

3

⟨

τ1κiκkMF |∇pε|2⟩

I3 =8√2 π

3τ1

∫ ∞

0

εi+k+3/2−βε exp(λ0 + λ1ε+ λ2ε2) dε I3 .

Note that, strictly speaking, the model cannot be derived directly for parabolicbands since the entropy maximization problem may have no solution. We can useKane’s nonparabolic ansatz, however, and formally perform δ → 0 at the end whichleads to the same result as if we had started with parabolic bands from the veryfirst. This is why the energy integrals for mi and Dik are the exact counterpartcompared to the energy-transport model.

Whereas the integrals in the Maxwell-Boltzmann energy-transport models couldbe computed analytically, they cannot anymore when generalizing to Fermi-Dirac

4.5. EXTENDED ET MODELS (MAXWELL-BOLTZMANN) 63

as just discussed or when adding one more moment. Similar to the Fermi integralswe can simplify the expressions which yields

∫ ∞

0

εa exp(λ0 + λ1ε+ λ2ε2) dε = eλ0

∫ ∞

0

(

s√−λ2

)a

exp

(

λ1s√−λ2

− s2)

ds√−λ2

=eλ0

(−λ2)(a+1)/2

∫ ∞

0

sa exp

(

sλ1√−λ2

− s2)

ds

where the substitution ε = s/√−λ2 was used which is well-defined since λ2 < 0 is

necessary for the integral to converge. As an analogon to Ia(z) we can define

Ja(z) =

∫ ∞

0

saesz−s2 ds

which converges for all z ∈ R and a > −1 and which is differentiable with derivativeJ ′

a(z) = Ja+1(z) – note that the index increases in contrast to I ′a(z) = Ia−1(z).

Rewriting mi and Dik yields

mi =4√2 πeλ0

(−λ2)i/2+3/4Ji+1/2

(

λ1√−λ2

)

,

Dik =8√2πτ1e

λ0

3(−λ2)(i+k−βε)/2+5/4Ji+k+3/2−βε

(

λ1√−λ2

)

I3 =2

3τ1mi+k+1−βε

I3 .

Theorem 4.12 (Currents with moments)The current densities can we written as

Ji = τ1

(

2i+ 3

3mi−βε

∇xΦ− 2

3∇xmi+1−βε

)

.

Proof For simplification we set z = λ1/√−λ2. Using integration by parts we get

mi =4√2πeλ0

(−λ2)i/2+3/4

∫ ∞

0

∂ssi+3/2

i+ 3/2esz−s2 ds

= − 4√2πeλ0

(−λ2)i/2+3/4

∫ ∞

0

si+3/2

i+ 3/2(z − 2s)esz−s2 ds

= − λ1i+ 3/2

mi+1 +2(−λ2)i+ 3/2

mi+2 = − 2

2i+ 3(λ1mi+1 + 2λ2mi+2) ,

and with the representation from the previous page

∇xmi = 4√2 π

∫ ∞

0

εi+1/2∇x exp(λ0 + λ1ε+ λ2ε2) dε

= mi∇xλ0 +mi+1∇xλ1 +mi+2∇xλ2 .


Inserting into the current density results in

Ji = −λ1Di0∇xΦ− 2λ2Di1∇xΦ−Di0∇xλ0 −Di1∇xλ1 −Di2∇xλ2

= −2

3τ1

(

(λ1mi+1−βε+ 2λ2mi+2−βε

)∇xΦ

+mi+1−βε∇xλ0 +mi+2−βε

∇xλ1 +mi+3−βε∇xλ2

)

= τ1

(

2i+ 3

3mi−βε

∇xΦ− 2

3∇xmi+1−βε

)

.

In these formulas J0, J1, and J2 are expressed in terms of m−βε, m1−βε

, m2−βε,

and m3−βε. In the case that λ0, λ1, and λ2 are given we can clearly calculate all the

moments. The other way round, when m0, m1, and m2 are given, the Lagrange mul-tipliers are uniquely defined due to the injectivity property in Theorem 3.24. Whenthree consecutive moments mi, mi+1, and mi+2 are given (i ∈ −1,−1

2, 0, 1

2, 1)

we can again show as in Theorem 3.24 that the Lagrange multipliers are uniquelydefined: Let

∫ ∞

0

εi+k+1/2 exp(

λ(1)0 + λ

(1)1 ε+ λ

(1)2 ε2

)

dε

=mi+k

4√2 π

=

∫ ∞

0

εi+k+1/2 exp(

λ(2)0 + λ

(2)1 ε+ λ

(2)2 ε2

)

dε

for k = 0, 1, 2. Again summing and building differences yield

∫ ∞

0

εi+1/2

(

2∑

k=0

λ(2)k εk −

2∑

k=0

λ(1)k εk

)

·(

exp(

λ(2)0 + λ

(2)1 ε+ λ

(2)2 ε2

)

− exp(

λ(1)0 + λ

(1)1 ε+ λ

(1)2 ε2

)

)

dε = 0

and again with the monotonicity of the exponential function, εi+1/2 ≥ 0, and thelinear independence of 1, ε, ε2 everything is proven.

We remark that Grasser et al. presented a higher-order model in [13, Equations(124)–(129)], called the six-moments transport equations, which fits in our settingwhen identifying

m0 = n , m1 =3

2nT , m2 =

15

4nT 2βn ;

the quantity βn is called curtosis.

Chapter 5

Discretization of the equations

To prepare the model equations for the numerical algorithms, we have to discretizethem, and we are going to use hybridized mixed finite elements. In general we use[7] as a guide for this discretization.

5.1 Model equations

We recall the Fermi-Dirac energy-transport model in drift-diffusion formulation

∂tm0 + divx J0 = 0 , J0 = −∇xg0 +g0T0

∇xΦ , T0 =I3/2−βε

(z)

I1/2−βε(z)

T ,

∂tm1 + divx J1 − J0 · ∇xΦ = W1 , J1 = −∇xg1 +g1T1

∇xΦ , T1 =I5/2−βε

(z)

I3/2−βε(z)

T

together with the Poisson equation

λ2D∆xΦ = m0 − ndop .

Additionally, m0, m1, g0, and g1 are given by

m0 = (2π)3/2T 3/2I1/2(z)

η, g0 =

4

3√πΓ(5

2− βε

)

τ1I3/2−βε

(z)

I1/2(z)m0T

1−βε ,

m1 =3

2(2π)3/2T 5/2I3/2(z)

η, g1 =

8

9√πΓ(7

2− βε

)

τ1I5/2−βε

(z)

I3/2(z)m1T

1−βε .

These are eleven equations for the eleven quantities m0, m1, g0, g1, J0, J1, λ0 (orz = λ0 + ln η), λ1 (or T = −1/λ1), T1, T2, and Φ.

We also recall that we use the relaxation-time operator Q1(f) = (Ff − f)/τ1with the relaxation time τ1 = τ1ε

−βε with exponent 0 ≤ βε <32

and that τ1 maynot depend on x in the drift-diffusion formulation. We are going to consider thestationary model only so that, in contrast to the transient model, we set ∂tm0 =

65

66 CHAPTER 5. DISCRETIZATION OF THE EQUATIONS

∂tm1 = 0. Thus there is no need to make τ1 dependent on time so that it is justa positive constant. The constant βε is another parameter for Q1, and λD is theso-called scaled Debye length that depends on the semiconductor device.

The quantity still to be specified is W1. We look at the quotient

g1g0

=5− 2βε

3

I1/2(z)I5/2−βε(z)

I3/2(z)I3/2−βε(z)

m1

m0=

5− 2βε2

I5/2−βε(z)

I3/2−βε(z)

T . (5.1)

W1 must be an expression of g0 and g1 that vanishes at the thermal equilibrium whichis T = 1, and moreover that relaxes the system towards the thermal equilibrium.So we consider

W1 =1

τ2

(

5− 2βε2

I5/2−βε(z)

I3/2−βε(z)

g0 − g1

)

with a relaxation time τ2; we use the index “2” here since the expression stems fromthe operator Q2 – in the modeling chapter we had defined W1 = 〈εQ2(F )〉. We aregoing to abbreviate the expression to W1 = c0g0 − c1g1 in the following since theexact structure of the coefficients c0 and c1 is not needed during the discretizationprocess.

We have to specify boundary conditions for several quantities. The Dirichlet datafor the potential is Φ(0) and Φ(1) at the left and right end of the device, respectively.The behavior will depend on the difference Uext = Φ(0)−Φ(1), the applied externalvoltage, only, so that we can claim Φ(0) = Uext ≥ 0 and Φ(1) = 0 without loss ofgenerality. The temperature is assumed to be at ambient temperature at the bound-ary, that is, T (0) = T (1) = 1. The electron density m0 is considered to coincide withthe doping profile at the boundary, that is, m0(0) = ndop(0) and m0(1) = ndop(1).Then, due to the expression for m0, it holds ndop = (2π)3/2I1/2(z)/η at the boundarywhich leads to the Dirichlet data for z. This finally determines the boundary valuesfor the remaining quantities m1, g0, and g1.

In the following, the discretization and numerics are carried out in one spacedimension only.

5.2 Thermal equilibrium and Poisson equation

The thermal equilibrium is defined by the static state with vanishing applied voltageUext = 0. As a consequence all currents are 0 and the temperature is constant 1. Inthe Maxwell-Boltzmann case η → 0 two of the model equations become

0 = −∂xg0 + g0∂xΦ and g0 =4

3√πΓ(5

2− βε

)

τ1m0 . (5.2)

The first equation can be analytically solved yielding g0 = ceΦ. Due to the secondequation, m0 is just a constant multiple of eΦ, too, which suggests that m0 dependsexponentially on Φ in general. Since this makes up a strong nonlinearity within

5.2. THERMAL EQUILIBRIUM AND POISSON EQUATION 67

the Poisson equation, it would be too naïve to directly solve a discretized form ofλ2D∂xxΦ = m0 − ndop for Φ with a given m0 inside the complete iteration process.

The better approach replaces m0 by e(Φ−Ψ)/T . This can be seen as a definitionof an unknown and lately unimportant quantity Ψ by Ψ = Φ − T lnm0. It couldin principle be calculated from given values for Φ, T , and m0 before calculating anupdate for Φ, but we will see that this is not necessary. We are going to use theNewton scheme on

f(Φ) = −λ2D∂xxΦ+m0(Φ)− ndop

where m0(Φ) means that m0 is locally given as a function of Φ. For the derivativewe get

Df(Φ)Φinc = limδ→0

f(Φ+ δΦinc)− f(Φ)

δ

= limδ→0

1

δ

(

−δλ2D∂xxΦinc +m0(Φ+ δΦinc)−m0(Φ))

= −λ2D∂xxΦinc + Φinc limδ→0

m0(Φ+ δΦinc)−m0(Φ)

δΦinc

= −λ2D∂xxΦinc +∂m0

∂ΦΦinc ,

and thus Df(Φ)Φinc = −f(Φ) becomes

(

−λ2D∂xx +∂m0

∂Φ

)

Φinc = λ2D∂xxΦ−m0 + ndop (5.3)

which has to be solved for the Newton increment Φinc . The substitution from aboveleads to

∂m0

∂Φ≈ 1

Te(Φ−Ψ)/T =

m0

T

where the dependence of T and Ψ on Φ is neglected. We will see later on that thepure exponential behavior works quite well in the Maxwell-Boltzmann case, however,does not serve as a suitable approximation in the Fermi-Dirac case. The term m0/Tfor ∂m0/∂Φ will have to be replaced by a more elaborate expression.

The spatial interval [0, 1] is decomposed into n sub-intervals I1, . . . , In whereIi = ]xi−1, xi[ with nodes 0 = x0 < x1 < · · · < xn−1 < xn = 1. As usual theinterval lengths are denoted by hi = xi−xi−1, and we denote by Λi the hat functionassociated to the node xi; so Λi is continuous and piecewise linear with Λi(xj) =δij. Multiplying (5.3) by Λi (i = 1, . . . , n − 1), integrating over [0, 1], and usingintegration by parts results in

λ2D

∫ 1

0

∂xΦinc(x) ∂xΛi(x) dx+

∫ 1

0

m0(x)

T (x)Φinc(x)Λi(x) dx

= −λ2D∫ 1

0

∂xΦ(x) ∂xΛi(x) dx+

∫ 1

0

(ndop −m0)(x)Λi(x) dx .


Since Φinc, Φ, and ndop−m0 are linear combinations of Λ0, . . . , Λn, all integrals exceptthe second are of the form

∫

Λj(x)Λi(x) dx or∫

∂xΛj(x) ∂xΛi(x) dx and can be imme-

diately calculated. In the second integral we approximate m0/T by∑

k m0,k/Tk Λk

so that∫

Λk(x)Λj(x)Λi(x) dx occurs. Calculating those we obtain

−λ2D

hiΦinc,i−1 +

(

λ2Dhi

+λ2Dhi+1

)

Φinc,i −λ2Dhi+1

Φinc,i+1

+ai−1+ai

12hiΦinc,i−1 +

(

ai−1+3ai12

hi +3ai+ai+1

12hi+1

)

Φinc,i +ai+ai+1

12hi+1Φinc,i+1

=λ2DhiΦi−1 −

(

λ2Dhi

+λ2Dhi+1

)

Φi +λ2Dhi+1

Φi+1

+hi6(ndop,i−1 − m0,i−1) +

hi + hi+1

3(ndop,i − m0,i) +

hi+1

6(ndop,i+1 − m0,i+1)

where ai = m0,i/Ti. For each variable v the notation vi = v(xi) means evaluation atthe node point xi.

In matrix representation we get

1

12

d1 (a1 + a2)h2

(a1 + a2)h2. . .

. . .. . .

. . . (an−2 + an−1)hn−1

(an−2 + an−1)hn−1 dn−1

+ λ2D

1h1

+ 1h2

− 1h2

− 1h2

. . .. . .

. . .. . . − 1

hn−1

− 1hn−1

1hn−1

+ 1hn

Φinc,1...

Φinc,n−1

=1

6

2(h1 + h2) h2

h2. . .

. . .. . .

. . . hn−1

hn−1 2(hn−1 + hn)

ndop,1 − m0,1...

ndop,n−1 − m0,n−1

− λ2D

− 1h1

1h1

+ 1h2

− 1h2

− 1h2

. . .. . .

. . .. . . − 1

hn−1

− 1hn−1

1hn−1

+ 1hn

Φ0

Φ1...

Φn−1

(5.4)

with di = (ai−1+3ai)hi+(3ai+ ai+1)hi+1. Note that Φinc,0 = Φinc,n = ndop,0−m0,0 =ndop,n − m0,n = Φn = 0, but Φ0 = Uext ≥ 0.

5.3. TRANSFORMATION AND ANSATZ SPACES 69

5.3 Transformation and ansatz spaces

We are going to describe in detail the discretization of the balance/definition equa-tion pair

∂xJ1 + c1g1 = J0∂xΦ+ c0g0 and J1 = −∂xg1 +g1T1∂xΦ .

The quantities g1 are J1 the unknowns, and g0 and J0 are given by the previousiteration step that solved the similar but easier pair

∂xJ0 = 0 and J0 = −∂xg0 +g0T0∂xΦ .

The other quantities c0, c1, and T1 are approximated by piecewise constant functionsdenoted by c0, c1, and T1. The same holds for ∂xΦ which is piecewise constant sinceΦ is continuous and piecewise linear as defined in the last section – we denote thisby ∂xΦ.

We transform the variable g1 to a so-called local Slotboom variable y1 by

y1 = e−Φ/T1g1

which yields

∂xJ1 + c1g1 = J0∂xΦ + c0g0 and e−Φ/T1J1 = −∂xy1 (5.5)

on each Ii. By the transformation from g1 to y1, which is called exponential fittingmethod, we have managed to get rid of the electric field as a factor in front of theunknown g1. This is comparable to the idea of the dual-entropy formulation in thederivation chapter.

Finally, the function spaces for the unknowns g1 and J1 have to be replacedby finite dimensional spaces which is the key idea in all finite element methods.Furthermore, instead of inserting the definition for J1 into the balance equationto obtain a second order differential equation for g1 as the only variable, differentspaces are introduced for the approximation of g1 and J1 which is then referred toas a mixed scheme. This might look artificial mathematically, however, the currentdensity is the far more important variable compared to the density, and without themixed scheme J1 containing the derivative of g1 would have even lower regularitythan g1. A possible choice of spaces could be again piecewise constant functionsfor g1 and Raviart-Thomas elements for J1 (as for Φ) which then would be globallycontinuous – a mandatory condition on the current.

We are going to use the Marini-Pietra elements for the current instead like in [7]whereas the invention and more detailed investigation of the elements can be foundin Marini’s and Pietra’s paper [22]. Compared to the Raviart-Thomas elements, thelinear part is exchanged by a quadratic one in the following manner:


. ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

..

.

.

..

.

..

.

..

.

.

xi−1

.

..

.

.

..

.

..

.

..

.

.

xi.

..

.

.

..

.

..

.

..

.

.

xi+1Ii Ii+1

.

.

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

.

..

..

.

.

..

..

..

...

..

..

..

..

..

..

...

..

..

..

..

.

.

..

..

..

..

..

..

..

..

..

...

..

..

..

..

.

..

...

..

...

..

...

..

...

..

...

..

..

.

...

..

...

..

...

..

..

...

..

...

...

...

..................

.

....................

...................

................

..............

.............

............

.......................

............

.............

..

..

..........

..............

..

.

..

..

..

.

..

..

..

.

.

..

.

..

..

..

.

..

.

..

..

.

pi

.

.............................

............................

...........................

..........................

.........................

...............

.........

................................................

.........................

..........................

...........................

............................

............................

.............................

..............................

...............................

.................................

...................................

..

....................................

..

......................................

pi+1

.

..............................................................................................................................................................................................................................................................

.

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Φ

x

Figure 5.1: two adjacent intervals Ii and Ii+1 with a sample potential Φ and the associatedquadratic “Marini-Pietra polynomials” pi and pi+1

Definition 5.1The polynomial pMP(x) = 3x2 − 2x is the uniquely defined polynomial of degree 2that satisfies

pMP(0) = 0 , pMP(1) = 1 , and

∫ 1

0

pMP(x) dx = 0 .

Furthermore it holds

∫ 1

0

p′MP(x) dx = 1 and

∫ 1

0

pMP(x)2 dx =

2

15.

We assign a polynomial function pi to each interval Ii depending on the behavior ofthe potential Φ on this interval:

Definition 5.2For each i = 1, . . . , n we define

pi(x) =

−pMP

(

xi − x

hi

)

if Φ(xi−1) < Φ(xi) ,

pMP

(

x− xi−1

hi

)

if Φ(xi−1) ≥ Φ(xi) .

Since Φ is linear on Ii it attains its maximum at either of the boundary points of Ii,and the polynomial pi is zero at this node. Due to the property of pMP it holds

∫

Ii

pi(x) dx = 0 ,

∫

Ii

p′i(x) dx = 1 , and

∫

Ii

pi(x)2 dx =

2

15hi

in both cases of the definition of pi. Two adjacent example intervals are plotted inFigure 5.1.

5.4. THE DISCRETIZED EQUATIONS FOR G1 AND J1 71

With the help of p1, . . . , pn we define the following finite-dimensional spaces:

Definition 5.3Let the underlying grid Z = x0, . . . , xn and the potential Φ : [0, 1] → R be given.Then we define the Marini-Pietra ansatz space

QZ,Φ =

Q : [0, 1] → R ∣∣∣Q|Ii(x) = Qi + Qipi(x) , Qi, Qi ∈ R ,

an ansatz space for constant functions

CZ =

C : [0, 1] → R ∣∣∣ C|Ii(x) = Ci , Ci ∈ R ,and an ansatz space for functions on the grid

DZ,f =

d : Z → R ∣∣∣d(0) = f(0) , d(1) = f(1)

.

The space QZ,Φ will especially serve for the approximation of the current density,however, its definition does not ensure continuity of its functions. To overcome thisdeficit a continuity condition will be introduced in a weak form making use of thediscrete space DZ,f .

5.4 The discretized equations for g1 and J1

We start with the first equation of (5.5). We replace J0(x) and J1(x) by theirapproximations in QZ,Φ, that is, J0,i+ J0,ipi(x) and J1,i+ J1,ipi(x) on Ii, respectively.Furthermore, g0 and g1 are approximated by piecewise constant functions so thaton each Ii we have

J1,ip′i(x) + c1,ig1,i =

(

J0,i + J0,ipi(x))

(∂xΦ)i + c0,ig0,i .

To obtain the weak form of the equation, we multiply by a test function C ∈ CZand integrate over Ii which yields

n∑

i=1

∫

Ii

(

J1,ip′i(x) + c1,ig1,i

)

C(x) dx

=n∑

i=1

∫

Ii

(

(

J0,i + J0,ipi(x))

(∂xΦ)i + c0,ig0,i

)

C(x) dx . (5.6)

The second equation of (5.5) is multiplied by a test function Q ∈ QZ,Φ and thenintegrated over Ii. We insert the approximation for J1, use integration by parts, andchange back from y1 to g1 so that


n∑

i=1

∫

Ii

e−Φ(x)/T1,i(

J1,i + J1,ipi(x))

Q(x) dx

=n∑

i=1

∫

Ii

e−Φ(x)/T1,ig1(x)Q′(x) dx−

n∑

i=1

[

e−Φ(x)/T1,ig1(x)Q(x)]xi

xi−1

.

Here the idea of hybridization comes into play: Inside the interval Ii the quantityg1 is approximated by a constant function g1,i, in the boundary term we use anapproximation g1,i at the grid nodes. The expression e−Φ(x)/T1,i inside both integralsis approximated in two different ways, in the first integral by its integral average, inthe second by its maximum. This leads to

n∑

i=1

1

hi

∫

Ii

e−Φ(x)/T1,i dx

∫

Ii

(

J1,i + J1,ipi(x))

Q(x) dx

=n∑

i=1

e−Φmin,i/T1,i

∫

Ii

g1,iQ′(x) dx−

n∑

i=1

(

e−Φi/T1,i g1,iQ(xi)− e−Φi−1/T1,i g1,i−1Q(xi−1))

(5.7)

with Φmin ,i = inf Φ|Ii = minΦi−1, Φi.The third equation describes the continuity of the current which has already

been mentioned. For each test function d ∈ DZ,0 it should hold

n∑

i=1

[

(

J1,i + J1,ipi(x))

d(x)]xi

xi−1

= 0 . (5.8)

So the complete finite-element task is: Find g1,1, . . . , g1,n−1 (g1,0 and g1,n are deter-

mined by boundary conditions), g1,1, . . . , g1,n, J1,1, . . . , J1,n, and J1,1, . . . , J1,n suchthat the equations (5.6)–(5.8) are satisfied for all C ∈ CZ , Q ∈ QZ,Φ, and d ∈ DZ,0.

Now we start choosing special test functions in all three equations. TakingC = χIj ∈ CZ in (5.6) yields

J1,j + hj c1,j g1,j = (Φj − Φj−1)J0,j + hj c0,j g0,j . (5.9)

Taking Q = χIj ∈ QZ,Φ in (5.7) yields

(∫

Ij

e−Φ(x)/T1,j dx

)

J1,j = −e−Φj/T1,j g1,j + e−Φj−1/T1,j g1,j−1 . (5.10)

Taking Q = pjχIj ∈ QZ,Φ in (5.7) yields

2

15

(∫

Ij

e−Φ(x)/T1,j dx

)

J1,j

= e−Φmin,j/T1,j g1,j − pj(xj)e−Φj/T1,j g1,j + pj(xj−1)e

−Φj−1/T1,j g1,j−1 . (5.11)


Taking d = χxj ∈ DZ,0 in (5.8) yields

J1,j + J1,jpj(xj)− J1,j+1 − J1,j+1pj+1(xj) = 0 . (5.12)

The equations (5.9)–(5.12) are now used to derive a linear system in g1,1, . . . ,g1,n−1 only by the so-called static condensation process. The remaining integral in(5.10) and (5.11) can be calculated explicitly to

∫

Ii

e−Φ(x)/T1,i dx =

hie−Φi/T1,i − e−Φi−1/T1,i

(−Φi/T1,i)− (−Φi−1/T1,i)if Φi−1 6= Φi ,

hie−Φi/T1,i if Φi−1 = Φi .

(5.13)

In the case Φi−1 6= Φi (5.10) becomes (changing back to i)

J1,i =2e−Φi/T1,i g1,i − 2e−Φi−1/T1,i g1,i−1

e−Φi/T1,i − e−Φi−1/T1,i

Φi − Φi−1

2hiT1,i

=(e−Φi/T1,i − e−Φi−1/T1,i)(g1,i + g1,i−1) + (e−Φi/T1,i + e−Φi−1/T1,i)(g1,i − g1,i−1)

e−Φi/T1,i − e−Φi−1/T1,i

· Φi − Φi−1

2hiT1,i

=

(

g1,i + g1,i−1 +1 + e(Φi−Φi−1)/T1,i

1− e(Φi−Φi−1)/T1,i

(g1,i − g1,i−1)

)

Φi − Φi−1

2hiT1,i

=Φi − Φi−1

hi

g1,i + g1,i−1

2T1,i− Φi − Φi−1

2T1,i

(

cothΦi − Φi−1

2T1,i

)

g1,i − g1,i−1

hi. (5.14)

This last expression is far better for the numerics than the first one. Due to thelimit y coth y → 1 for y → 0 the expression is correct in the case Φi−1 = Φi, too.The last formula can be interpreted as a nonlinear Scharfetter-Gummel scheme, forexample see [4].

Inserting (5.9) into (5.11) yields

2

15

(∫

Ii

e−Φ(x)/T1,i dx

)

(

(Φi − Φi−1)J0,i + hi(c0,ig0,i − c1,ig1,i))

= e−Φmin,i/T1,i g1,i − pi(xi)e−Φi/T1,i g1,i + pi(xi−1)e

−Φi−1/T1,i g1,i−1

which we are going to solve for g1,i. In order to get rid of the polynomial terms, wedefine on each interval Ii the index ımin ,i by

ımin ,i =

i− 1 if Φi−1 < Φi ,

i if Φi−1 ≥ Φi .


If ımin ,i = i − 1 we have pi(xi) = 0 and pi(xi−1) = −1, if ımin ,i = i we havepi(xi) = 1 and pi(xi−1) = 0. Thus in both cases the two summands containing pi are−e−Φmin,i/T1,i g1,ımin,i

. We collect the terms with g1,i on the right-hand side so that

2

15

(∫

Ii

e(Φmin,i−Φ(x))/T1,i dx

)

(

(Φi − Φi−1)J0,i + hic0,ig0,i

)

+ g1,ımin,i

=

(

2hic1,i15

∫

Ii

e(Φmin,i−Φ(x))/T1,i dx+ 1

)

g1,i

and eventually

g1,i =

γ1,ic1,i

(

Φi−Φi−1

hiJ0,i + c0,ig0,i

)

+ g1,ımin,i

γ1,i + 1, γ1,i =

2hic1,i15

∫

Ii

e(Φmin,i−Φ(x))/T1,i dx .

(5.15)In the final step we insert the last relationship into (5.9) and get

J1,i = (Φi − Φi−1)J0,i + hic0,ig0,i − hic1,i

γ1,ic1,i

(

Φi−Φi−1

hiJ0,i + c0,ig0,i

)

+ g1,ımin,i

γ1,i + 1

=(Φi − Φi−1)J0,i + hic0,ig0,i

γ1,i + 1− hic1,iγ1,i + 1

g1,ımin,i. (5.16)

We insert this expression for J1,i and (5.14) for J1,i into the continuity (5.12) for J1and obtain

Φi − Φi−1

hi

g1,i + g1,i−1

2T1,i− Φi − Φi−1

2T1,i

(

cothΦi − Φi−1

2T1,i

)

g1,i − g1,i−1

hi

+

(

(Φi − Φi−1)J0,i + hic0,ig0,iγ1,i + 1

− hic1,iγ1,i + 1

g1,ımin,i

)

pi(xi)

=Φi+1 − Φi

hi+1

g1,i+1 + g1,i2T1,i+1

− Φi+1 − Φi

2T1,i+1

(

cothΦi+1 − Φi

2T1,i+1

)

g1,i+1 − g1,ihi+1

+

(

(Φi+1 − Φi)J0,i+1 + hi+1c0,i+1g0,i+1

γ1,i+1 + 1− hi+1c1,i+1

γ1,i+1 + 1g1,ımin,i+1

)

pi+1(xi)

which is a linear system of n − 1 equations (i runs from 1 to n − 1) for the n − 1unknowns g1,1, . . . , g1,n−1.

We can slightly simplify these equations by removing the minimal indices: Theexpression pi(xi) is nonzero if and only if ımin ,i = i, and in this case it is 1. Theother expression pi+1(xi) is nonzero if and only if ımin ,i+1 = i, and in this case it is−1. Rearranging the terms we get

(

w1,i+1 + v1,i+1 + w1,i − v1,i − s1,i+1pi+1(xi) + s1,ipi(xi))

g1,i

− (w1,i + v1,i)g1,i−1 − (w1,i+1 − v1,i+1)g1,i+1 = −r1,i+1pi+1(xi) + r1,ipi(xi) (5.17)


with the abbreviations

v1,i =Φi − Φi−1

2hiT1,i, w1,i =

Φi − Φi−1

2hiT1,icoth

Φi − Φi−1

2T1,i,

r1,i =(Φi − Φi−1)J0,i + hic0,ig0,i

γ1,i + 1, s1,i =

hic1,iγ1,i + 1

which is a tridiagonal linear system with the positive contribution s1 of the zerothorder term appearing in the diagonal only. Due to w1,i > |v1,i| all the diagonalentries are positive and all nondiagonal entries are negative.

5.5 The discretized equations for g0 and J0

The model equations for g0 and J0 are obtained from those for g1 and J1 by settingc0, c1 (and thus γ1), and J0 to zero and thereafter relabeling g1, J1, and T1 to g0,J0, and T0, respectively. Applying the same substitutions to (5.17) yields

Φi − Φi−1

hi

g0,i + g0,i−1

2T0,i− Φi − Φi−1

2T0,i

(

cothΦi − Φi−1

2T0,i

)

g0,i − g0,i−1

hi

=Φi+1 − Φi

hi+1

g0,i+1 + g0,i2T0,i+1

− Φi+1 − Φi

2T0,i+1

(

cothΦi+1 − Φi

2T0,i+1

)

g0,i+1 − g0,ihi+1

which again is a linear system of n−1 equations for n−1 unknowns g0,1, . . . , g0,n−1.Once they are calculated (5.15) can be rewritten for g0,1, . . . , g0,n resulting in

g0,i = g0,ımin,i

which can be seen as an upwind scheme.Note that (5.16) simplifies to J0,i = 0; this is not surprising since ∂xJ0 = 0 holds

on each interval Ii and J0 is supposed to be continuous. Thus J0 is constant on [0, 1]and the quadratic part vanishes. J0,i can be calculated by the exact copy of (5.14)after exchanging g1 by g0 and T1 by T0, and up to numerical errors the value of J0,idoes not depend on i.

In summary, when z (that is λ0), T (that is λ1), and Φ are given, we can calculatethe auxiliary quantities T0 and T1, and with these at first g0 and J0 and thereafterg1 and J1. From g0 and g1 updates for z and T are calculated allowing for anupdate of m0. With this data we enter into the Poisson equation producing a newapproximation for Φ.


Chapter 6

Numerical results

6.1 Numerical setting and scaling

We are going to simulate a one-dimensional ballistic silicon diode which is a semi-conductor device consisting of three layers. The regions near the two contacts arehighly n-doped whereas the intermediate region is also n-doped but with a lowerconcentration. This n+nn+ device serves as a simple model for the channel of an n-channel junction field effect transistor. The contacts are called source (S) and drain(D), and the characteristics of the channel are determined by the third contact ofthe transistor called the gate (G). For more details see for example [26]. The devicedata are given in Table 6.1, and the doping profile is shown in Figure 6.1. In oursetting the applied voltage Uext is the potential difference between the drain andsource contact, and the gate is omitted as we only look at the channel itself. Thecurrent density J0 in our models corresponds to the current IS in the circuit.

In order to return to unscaled physical quantities, we have to retract the scaling,but we cannot just use the same reference quantities: If we kept τR as the time scalingconstant, we would conclude τC = α2τR → 0 since α → 0. But because λC/τC mustbe equal to λR/τR (both are an expression for the mean electron velocity) we get thatλC/λR = τC/τR → 0 and thus

√λCλR cannot serve as the space scaling constant

anymore. As an alternative we use the device length L for the space scaling –which justifies that the numerics is done on the interval [0, 1] – and a suitable

Length of the device: L = 0.6 µm

Length of the n+ region: = 0.4 µm

Doping concentration in the n+ region: ndop,max = 5 · 1023 m−3

Doping concentration in the n region: ndop,min = 2 · 1021 m−3

Relative permittivity of silicon: εr = 11.68

Electron mobility constant: µ = 0.15 m2/Vs

Table 6.1: Data of the semiconductor crystal

77

78 CHAPTER 6. NUMERICAL RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

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pro

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Figure 6.1: Doping profile ndop and a basic example circuit for a JFET

Temperature T : Tamb = 300 K

Energy: kBTamb = 4.14195 · 10−21 J = 25.8520 meV

Voltage Uext , potential Φ: Uth = kBTamb/e = 25.8520 mV

Space x: L = 0.6 µm

Time: L2/µUth = 92.8360 ps

Velocity: µUth/L = 6463.01 m/s

Electron density m0: ndop,max = 5 · 1023 m−3

Energy density m1: ndop,maxkBTamb = 2070.98 J/m3

Table 6.2: Scaling constants

term containing the electron mobility µ is used for the time scale. We collect thescaling constants for each quantity in Table 6.2. The physical constants are listedin Table 6.3, and the value of the so-called squared scaled Debye length λ2D in thePoisson equation is

λ2D =ε0εrkBTamb

e2L2ndop,max

= 9.27050 · 10−5 .

For the two relaxation times we choose

τ1 = τ2 = 0.4 ps .

Since the values of g0 and g1 are not important we plot them as the scaledquantities, or in other words, in arbitrary units. But we have to be careful sinceJ0 and J1 are current densities belonging to g0 and g1, and thus we have to rescalethem back so that we can interpret their values. Due to (5.2) the scaling constantdepends on βε in the following way: For βε = 0 we have

1

τ1

L2

µUth

eµUthndop,max

L= 1.20163 · 1011 A

m2,

6.2. FERMI INTEGRALS 79

Elementary charge: e = 1.60218 · 10−19 C

Vacuum permittivity: ε0 = 8.85419 · 10−12 As/Vm

Boltzmann constant: kB = 1.38065 · 10−23 J/K

Table 6.3: Physical constants

1

τ1

L2

µUth

eµU2thndop,max

L= 3.10646 · 109 W

m2,

for βε =12

3√π

4τ1

L2

µUth

eµUthndop,max

L= 1.59738 · 1011 A

m2,

3√π

4τ1

L2

µUth

eµU2thndop,max

L= 4.12955 · 109 W

m2,

and for βε = 1

3

2τ1

L2

µUth

eµUthndop,max

L= 1.80245 · 1011 A

m2,

3

2τ1

L2

µUth

eµU2thndop,max

L= 4.65970 · 109 W

m2

for J0 and J1, respectively.The Matlab code allows for an arbitrary grid that does not have to be equidis-

tant. The simulation is done with n = 300 subintervals I1, . . . , I300 and thus 301equidistant nodes x0, . . . , x300, though. A tolerance of 10−6 is used throughout theentire simulation.

6.2 Fermi integrals

When η becomes very small or even zero in the Maxwell case, the variable z =λ0 + ln η numerically becomes intractable. Thus we are going to use λ0 within thecode and calculate Ia(z) as ηeλ0 when z is very negative. This is not yet satisfactoryif η = 0 since then the quotient of two Fermi integrals would evaluate to “0/0”. Toovercome this problem we define

Ia(λ0) =

eλ0 if η = 0 ,

Ia(λ0 + ln η)

ηif η > 0 .

In principle the model can be studied for each 0 ≤ βε <32, however, it is sufficient

to limit oneself to the three cases βε = 0, βε = 12, and βε = 1 as in the modeling


a −12

12

1 32

2 52

z− −6.6 −5.9 −5.6 −5.2 −4.9 −4.5

z+ 20.4 35.2 57.4 78.5 99.3 120.0

Table 6.4: Left and right limits for z in Ia(z) at tolerance 10−3

chapter. So we have to calculate the Fermi integrals Ia for a ∈ −12, 0, 1

2, 1, 3

2, 2, 5

2

only. In Section 4.1 we have derived the asymptotic behavior for Ia, and we aregoing to use them instead of a numerical approximation of the integral as soon asthe relative error of the asymptotics is small enough. Numerical evaluations withMaple show that the relative error of the approximations Ia(z) ≈ ez or Ia(z) ≈za+1/Γ (a + 2) is at most 10−3 if z ≤ z− or z ≥ z+, respectively, given by theTable 6.4. The case a = 0 is omitted since I0 can be calculated analytically.

In summary for the numerical algorithm we get

Ia(λ0)

= eλ0 if η = 0 ,

=ln(ηeλ0 + 1)

ηif η > 0 , a = 0 ,

≈ eλ0 if η > 0 , a 6= 0 , z ≤ z−(a) ,

≈ za+1

ηΓ (a+ 2)if η > 0 , a 6= 0 , z ≥ z+(a) ,

≈ some approximation ofIa(z)

ηif η > 0 , a 6= 0 , z−(a) < z < z+(a) .

The remaining part is to define a suitable approximation for the last case. Themethod that entails the fewest problems is to create a list of precalculated Fermiintegrals and use them as a look-up table. In the program a step size of ∆z = 0.1is used with a linear interpolation between two neighboring values to cover each z.Due to the values for z− and z+ the table contains 271 entries for a = −1

2up to

1,246 entries for a = 32.

The relative error of 10−3 is acceptable inside the iteration but is too large whenwe finally want to converge to a result. To guarantee a tolerance of 10−6 we usethe step size ∆z = 2−14 with the same linear interpolation, and as soon as a Fermiintegral value is needed the Fermi integrals at the neighboring nodes are calculatedand stored in the table or are looked up from this table. When the precision of 10−6

is needed, the data in Table 6.4 are irrelevant.In order to numerically approximate the improper integral one could use the

Gauss-Laguerre quadrature rule, but it is difficult to estimate the number of nodesneeded to guarantee the prescribed error. Thus we use a decomposition

Γ (a + 1)Ia(z) = I0 +

∞∑

n=1

In

6.3. THERMAL EQUILIBRIUM AND ITERATION 81

with I0 =

∫ s0

0

sa

1 + es−zds and In =

∫ s0+n∆s

s0+(n−1)∆s

sa

1 + es−zds

with an initial step s0 and a step size ∆s that are adapted to the integrand. Theintegrals are evaluated using Matlab’s quadl function, and the infinite series istruncated as soon as In+1 ≤ (I0 + I1 + . . . + In)tol/10. To specify the parameterswe differentiate sa/(1 + es−z) with respect to s which yields

sa−1ea−z

(1 + es−z)2(

aez−a − (s− a)es−a)

.

Since u 7→ ueu is a strictly monotonically increasing function [0,+∞[ → [0,+∞[,this derivative has got a unique positive zero for a > 0 and none for a < 0.

For a > 0 the integrand is bounded at s = 0 and thus we can choose s0 = 0so that I0 = 0. The zero of the derivative can be written as ∆s = a +W (aez−a)where W denotes the Lambert W-function uniquely satisfying W (u)eW (u) = u foru > 0. Thus the integrand reaches its maximum in s = ∆s and the step size ∆sis well-adapted. For a < 0 the integrand is unbounded near s = 0 and we chooses0 = 0.05. Since the integrand is strictly monotonically decreasing we do not needa sophisticated choice for ∆s and simply set ∆s = 1.

6.3 Thermal equilibrium and iteration

In the thermal equilibrium it holds

∂xg0 = g0I1/2−βε

(λ0)

I3/2−βε(λ0)

∂xΦ and g0 =8√2π

3Γ(5

2− βε

)

τ1I3/2−βε(λ0)

which results in ∂xλ0 = ∂xΦ. Thus λ0−Φ is constant on [0, 1], and using the formulafor m0 we get

(λ0 − Φ)|∂ = λ0|∂ = I−11/2

(

m0|∂(2π)3/2

)

= I−11/2

(

ndop |∂(2π)3/2

)

at the boundary.In order to calculate the thermal equilibrium we use Φi = ln ndop,i as an initial

guess for the potential. Then λ0 and m0 are given by

λ0,i = Φi + (λ0 − Φ)|∂ and m0,i = (2π)3/2I1/2(λ0,i) ,

respectively. With m0,i and Φi we enter into the linear system (5.4) and execute oneNewton step in order to update the potential. Then λ0 and m0 are recalculated,the Poisson equation is solved again, and everything is repeated until the desiredaccuracy is achieved.


Although – due to the definition of the thermal equilibrium – T , J0, and J1 aretrivial, Φ and m0 show some characteristic behavior. We use this data to enter intoan outer loop iterating over the applied voltage Uext with a step size ∆Uext . At everychosen voltage we use an inner loop, a so-called Gummel iteration, which avoids thefull Newton scheme. Although the latter one usually is very robust and quicklyconverging, its implementation is very costly and thereafter quite cumbersome toadept to a changed setting. In contrast, the Gummel iteration itself is easy toimplement, but converges much slowlier, and it is prone to failing to converge at all.This means that the voltage step ∆Uext has to be chosen more carefully than in theNewton scheme which, however, does not seem to be critical since ∆Uext = 0.1 Vworks well in the Maxwell-Boltzmann case. We are going to see later on that we arefacing difficulties when successively increasing η.

In our situation, the idea of the Gummel iteration is to put the whole nonlinearityinto the Poisson equation and to update the rest of the data by solving the otherequations exactly. We are now going to describe one Gummel iteration in moredetail. We assume that the following data is known: the electron density at thenodes (m0,i), λ0 piecewise constant (λ0,i), the temperature at the nodes (Ti) andpiecewise constant (Ti), and the potential at the nodes (Φi).

At first we use m0,i, Ti, Φi, and the linear system (5.4) to execute one Newtonstep in order to update the potential. See the next section for the approximation of∂m0/∂Φ. Then we calculate the auxiliary quantities

T0,i =I3/2−βε

(λ0,i)

I1/2−βε(λ0,i)

Ti and T1,i =I5/2−βε

(λ0,i)

I3/2−βε(λ0,i)

Ti ,

the coefficients for W1

c0,i =5− 2βε2τ2

I5/2−βε(λ0,i)

I3/2−βε(λ0,i)

and c1,i =1

τ2,

and γ1,i with (5.15). For the numerically stable implementation see the next section.

Using T0,i and Φi we calculate g0,i, g0,i, J0,i, and J0,i (which is zero) according to

Section 5.5, and thereafter we calculate g1,i, g1,i, J1,i, and J1,i using T1,i, Φi, c0,i,c1,i, γ1,i, g0,i, and J0,i according to Section 5.4. See the next section also for thenumerical treatment of the term containing the hyperbolic cotangent.

In the next step λ0 and the temperature have to be updated. Using the modelequations in Section 5.1 we have

g0 =8√2π

3Γ(5

2− βε

)

τ1I3/2−βε(λ0)T

5/2−βε ,

g1 =8√2π

3Γ(7

2− βε

)

τ1I5/2−βε(λ0)T

7/2−βε .

6.4. SOME IMPLEMENTATION DETAILS 83

Eliminating the temperature yields

f(λ0) =I5/2−βε

(λ0)5/2−βε

I3/2−βε(λ0)7/2−βε

− 8√2π

3

Γ (5/2− βε)7/2−βε

Γ (7/2− βε)5/2−βετ1g5/2−βε

1

g7/2−βε

0

= 0

which is a nonlinear equation for λ0. The derivative of f is

f ′(λ0) =

(52− βε)I5/2−βε

(λ0)3/2−βε I3/2−βε

(λ0)I3/2−βε(λ0)

7/2−βε

− I5/2−βε(λ0)5/2−βε(7

2− βε)I3/2−βε

(λ0)5/2−βε I1/2−βε(λ0)

I3/2−βε(λ0)7−2βε

=I5/2−βε

(λ0)3/2−βε

I3/2−βε(λ0)9/2−βε

·(

(5

2− βε

)

I3/2−βε(λ0)

2 −(7

2− βε

)

I1/2−βε(λ0)I5/2−βε

(λ0)

)

which is negative for all λ0 and βε due to the Fermi integral inequality for a = 32−βε.

So the zero of f is unique and the Newton scheme is well-defined, and with λ0 weget T from (5.1). In the program we will both calculate the nodal values λ0,i and Tias well as the piecewise constant values λ0,i and Ti.

Finally the nodal values m0,i of the electron density are calculated using m0,i =(2π)3/2I1/2(λ0,i)Ti

3/2. The energy density m1 is not needed during the iteration andis calculated afterwards for the diagrams only.

6.4 Some implementation details

The continuous form of the Newton step in the Poisson equation was derived in(5.3). The approximation ∂m0/∂Φ ≈ m0/T works well for the Maxwell-Boltzmanncase in [7], but the Gummel iteration fails to converge for larger η. To get an ideafor the dependence of m0 on Φ we choose βε = 1 and Uext = 1 V and plot the pairs(Φi, m0,i) for η = 0 and η = 50 in the Figures 6.2 and 6.3, respectively. We definethe index set

I =

i ∈ 1, 2, . . . , n− 1∣

∣ 0.1 < m0,i < 0.9

.

For each i ∈ I we define

bi =m0,i+1 − m0,i−1

Φi+1 − Φi−1

≈ ∂m0

∂Φ

∣

∣

∣

∣

xi

.

The Figures 6.4 and 6.5 show these approximated derivatives bi and the quotientsm0,i/Ti for all i ∈ I. We can assume that both data coincide sufficiently well forη = 0 but only poorly for η = 50. The idea is to replace m0 = e(Φ−Ψ)/T by

m0 =1

γ + e−(Φ−Ψ)/T


−10 0 10 20 30 400

0.5

1

potential Φ

elec

tron

den

sity

m0

Figure 6.2: Plot of the pairs (Φi, m0,i) for βε = 1, Uext = 1 V, and η = 0

−10 0 10 20 30 400

0.5

1

potential Φ

elec

tron

den

sity

m0

Figure 6.3: Plot of the pairs (Φi, m0,i) for βε = 1, Uext = 1 V, and η = 50

which follows the principle of generalizing Mf to Ff . Differentiating leads to

∂m0

∂Φ≈ e−(Φ−Ψ)/T

(γ + e−(Φ−Ψ)/T )21

T=m2

0

T

(

1

m0− γ

)

=m0

T(1− γm0)

having again neglected the dependence of T and Ψ on Φ. Clearly, we recover m0/Tfor γ = 0.

The final task is to choose γ. To do so we minimize the function

f(γ) =∑

i∈I

(

m0,i

Ti(1− γm0,i)− bi

)2

which describes the least square distance of the derivative entering into the Newton

6.4. SOME IMPLEMENTATION DETAILS 85

0.12 0.14 0.16 0.180

0.5

1

position x

der

ivat

ive

bm0/T

0.8 0.82 0.84 0.860

0.5

1

position x

der

ivat

ive

bm0/T

Figure 6.4: Plot of bi and m0,i/Ti for βε = 1, Uext = 1 V, and η = 0

0.12 0.14 0.16 0.180

0.5

1

position x

der

ivat

ive

bm0/T

0.8 0.82 0.84 0.860

0.5

1

position x

der

ivat

ive

bm0/T

Figure 6.5: Plot of bi and m0,i/Ti for βε = 1, Uext = 1 V, and η = 50

step and the derivative numerically obtained by the difference quotient. We calculate

f ′(γ) = −2∑

i∈I

(

m0,i

Ti(1− γm0,i)− bi

)

m20,i

Ti= 2

∑

i∈I

(

m40,i

T 2i

γ −m3

0,i

T 2i

+ bim2

0,i

Ti

)

with the unique zero

γ =

∑

i∈I(m0,i/Ti − bi)m20,i/Ti

∑

i∈I m40,i/T

2i

.

This value for γ is calculated from m0, T , and Φ before each Newton step, and then

ai =m0,i

Ti(1− γm0,i)

is used in (5.4). The better coincidence of ai and bi is shown in Figure 6.6 with theoptimal γ = 0.54.


0.12 0.14 0.16 0.180

0.5

1

position x

der

ivat

ive

ba = (1− γm0)m0/T

0.8 0.82 0.84 0.860

0.5

1

position x

der

ivat

ive

ba = (1− γm0)m0/T

Figure 6.6: Plot of bi and ai for βε = 1, Uext = 1 V, and η = 50 (γ = 0.54)

The next details refer to the equations for (g0, J0) and (g1, J1): The definitionof γ1,i contains the integral

∫

Iie−Φ(x)/T1,i dx which can be analytically calculated by

(5.13). The formula is numerically unstable when v = −Φi/T1,i and u = −Φi−1/T1,iare almost equal. If this is the case we rewrite the fraction as

ev − eu

v − u= e

v+u2

ev−u2 − e−

v−u2

v − u= e

v+u2

sinh v−u2

v−u2

where the quotient can be continued analytically at v−u2

= 0 with the power series

sinh v−u2

v−u2

= 1 +

(

1

6+

(

1

120+

1

5040ε

)

ε

)

ε+O(ε4) with ε =(v − u

2

)2

.

This truncated expansion is used when |v − u| < 10−5.We are facing a similar situation in the term

Φi − Φi−1

2Ticoth

Φi − Φi−1

2Ti

since coth u has got a pole at u = 0. But again, the singularity of u cothu isremovable with the power series

u cothu = 1 +

(

1

3+

(

− 1

45+

2

945ε

)

ε

)

ε+O(ε4) with ε = u2 .

This truncated expansion is again used when |u| < 10−5.The function λ0 7→ f(λ0) from the end of the last section has got a negative

derivative, and moreover a numerical plot suggests that it is convex, see Figure 6.7for βε = 1. Note that the convexity cannot be shown easily with the second derivative

6.5. RESULTS FOR MAXWELL-BOLTZMANN 87

−10 −8 −6 −4 −2 0

×104

0

1

2

Lagrange multiplier λ0

I 3/2(λ

0)3

/2/I

1/2(λ

0)5

/2

Figure 6.7: Plot of the function λ0 7→ I3/2(λ0)3/2/I1/2(λ0)

5/2 for η = 50

since I ′1/2−βε

would be needed which cannot be expressed by I−1/2−βεbecause the

Fermi integral only converges when its index is greater than −1. Both properties off lead to the fact that the sequence of the Newton iteration monotonically increasestowards the zero of f as soon as one element of the sequence is smaller than thezero. When starting with a λ0 larger than the zero, however, it can happen that theiteration jumps across the zero way too far with the consequence that Fermi integralshave to be computed for useless values of λ0. To overcome this inconvenience, weuse the value of λ0 at the corresponding node of the previous Gummel iteration,but when f is negative at this point we keep on stepping to the left until we havecrossed the zero. After few iterations of bisection, because evaluating f is muchcheaper than its derivative, we start the Newton method until convergence. Forstability purposes we apply a bit of a relaxation for λ0, that means if λ0,new is thecalculated and λ0,old is the old value, we use

λ0 = 0.9 λ0,new + 0.1 λ0,old .

6.5 Results for Maxwell-Boltzmann

In the Maxwell-Boltzmann case, that is η = 0, we present in the following Figures 6.8to 6.37 the discrete values at x0, . . . , x300 of all the quantities m0, m1, g0, g1, J0,J1, λ0, T , Φ, and the electron velocity v0. The calculations were done for βε = 0,βε =

12, and βε = 1 up to the applied voltage Uext = 1.5 V, and the data is plotted

for 0.5 V, 1 V, and 1.5 V. m1 is calculated after reaching the three voltages only,g0, g1, and λ0 are presented in their scaled values. The velocity is calculated byv0 = J0/(m0e), and J0 is actually constant – J0,i = 0 is put into the code directly,but J0,i is calculated by the copy of (5.14). The currents J0 and J1 and the velocityv0 are inverted since positive values are more convenient in the plots; anyway, theelectrons travel from the right to the left.


0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3

Uext = 0.5 VUext = 1 VUext = 1.5 V

Figure 6.8: Electron density m0 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3


Figure 6.9: Electron density m0 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3




0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3


Figure 6.11: Energy density m1 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3


Figure 6.12: Energy density m1 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3




0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0


Figure 6.14: Electron density g0 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0


Figure 6.15: Electron density g0 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0




0 0.1 0.2 0.3 0.4 0.5 0.610−4

10−3

10−2

10−1

position x in µm

ener

gyden

sity

g 1


Figure 6.17: Energy density g1 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.610−4

10−3

10−2

position x in µm

ener

gyden

sity

g 1


Figure 6.18: Energy density g1 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

ener

gyden

sity

g 1




0 0.1 0.2 0.3 0.4 0.5 0.6108

109

position x in µm

char

gecu

rren

tden

sity

J0

inA

/m2


Figure 6.20: Negative charge current density −J0 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6108

109

position x in µm

char

gecu

rren

tden

sity

J0

inA

/m2


Figure 6.21: Negative charge current density −J0 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6108

109

position x in µm

char

gecu

rren

tden

sity

J0

inA

/m2


Figure 6.22: Negative charge current density −J0 for βε = 1 and η = 0


0 0.1 0.2 0.3 0.4 0.5 0.6107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2


Figure 6.23: Negative energy current density −J1 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6106

107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2


Figure 6.24: Negative energy current density −J1 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6106

107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2




0 0.1 0.2 0.3 0.4 0.5 0.6−10

−8

−6

−4

−2

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V

Uext = 1 VUext = 1.5 V

Figure 6.26: Lagrange multiplier λ0 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6−10

−8

−6

−4

−2

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V


Figure 6.27: Lagrange multiplier λ0 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6−10

−8

−6

−4

−2

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V




0 0.1 0.2 0.3 0.4 0.5 0.60

500

1000

1500

2000

position x in µm

tem

per

ature

Tin

K


Figure 6.29: Temperature T for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.60

500

1000

1500

2000

position x in µm

tem

per

ature

Tin

K


Figure 6.30: Temperature T for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6

×103

0

1

2

3

position x in µm

tem

per

ature

Tin

K




0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V


Figure 6.32: Potential Φ for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V


Figure 6.33: Potential Φ for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V




0 0.1 0.2 0.3 0.4 0.5 0.6

×105

0

5

10

position x in µm

elec

tron

velo

city

v 0in

m/s Uext = 0.5 V


Figure 6.35: Negative electron velocity −v0 for βε = 0 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6

×105

0

1

2

3

4

position x in µm

elec

tron

velo

city

v 0in

m/s Uext = 0.5 V


Figure 6.36: Negative electron velocity −v0 for βε =12 and η = 0

0 0.1 0.2 0.3 0.4 0.5 0.6

×104

0

5

10

15

position x in µm

elec

tron

velo

city

v 0in

m/s




0 0.5 1 1.5

×108

0

5

10

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2

Figure 6.38: Current-voltage characteristic for βε = 0 and η = 0

0 0.5 1 1.5

×108

0

2

4

6

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2

Figure 6.39: Current-voltage characteristic for βε =12 and η = 0

0 0.5 1 1.5

×108

0

1

2

3

4

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2


6.6. RESULTS FOR FERMI-DIRAC 99

We compare our results for βε = 1 and Uext = 1.5 V with the Chen model in[7] for nonparabolicity parameter α = 0 which means parabolic bands. We get amaximal temperature of 2338 K at 162 nm which coincides with 2330 K of thereference, the maximal velocity is 140 km/s at 144 nm compared to 144 km/s.

The maybe technically most interesting behavior of a semiconductor device is itscurrent-voltage characteristic. In steps of ∆Uext = 0.01 V we increased the appliedvoltage and calculated the current J0 which is plotted in the Figures 6.38 to 6.40.Again the slightly sublinear curvature coincides well with the plot in [7].

6.6 Results for Fermi-Dirac

In the Fermi-Dirac case we present in the following Figures 6.41 to 6.67 the samequantities in the same manner – we omit J0 since it is constant anyway. The quan-tities are plotted for (βε, η) with values (1

2, 50), (1, 50), and (1, 100). In the case

βε = 12

the iteration fails to converge for η = 100, with βε = 0 is even fails forη = 50; see the next section for a discussion.

Additionally, we plot the two Fermi integral quotient modifiers in the definitionof T0 and T1, that is

T0 =I3/2−βε

(z)

I1/2−βε(z)

T , T1 =I5/2−βε

(z)

I3/2−βε(z)

T ,

respectively, in the Figures 6.68 to 6.73. As done in the Maxwell-Boltzmann casethe current-voltage characteristic in shown in the Figures 6.74 to 6.76.

6.7 Convergence analysis

Although we have replaced ∂m0/∂Φ by a better approximation, the complete algo-rithm does not work for high values of η which may be of interest. One can observethat the Gummel iteration produces an increasing sequence of Poisson right-handsides for η ' 110 at βε = 1 even for Uext = 0.1 V. As mentioned before this iseven worse for the lower values of βε: For βε = 1

2the limit is around η ≈ 85, and

for βε = 0 it is around η ≈ 30. It seems that the Gummel iteration then loses itscontractive property.

This may be somehow surprising since the plots pretend that there is almost nodifference between η = 0 and η = 100 at βε = 1. When looking at the Figures 6.68 to6.73 showing T0/T and T1/T , however, there is some difference since they have valuesaround 2 or 3 which enter into the drift-diffusion equations. Since the Scharfetter-Gummel discretizations for (g0, J0) and (g1, J1) are very stable and well-understood,the problem might still be found in the Newton step.

Returning to the figures, we look at two different types of errors: the right-handside in the Poisson equation and the error for different spatial mesh sizes.


0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3


Figure 6.41: Electron density m0 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3



0 0.1 0.2 0.3 0.4 0.5 0.61021

1022

1023

1024

position x in µm

elec

tron

den

sity

m0

inm

−3



6.7. CONVERGENCE ANALYSIS 101

0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3


Figure 6.44: Energy density m1 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3



0 0.1 0.2 0.3 0.4 0.5 0.6101

102

103

104

position x in µm

ener

gyden

sity

m1

inJ/m

3




0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0


Figure 6.47: Electron density g0 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0



0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

elec

tron

den

sity

g 0




0 0.1 0.2 0.3 0.4 0.5 0.610−4

10−3

10−2

10−1

position x in µm

ener

gyden

sity

g 1


Figure 6.50: Energy density g1 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.610−5

10−4

10−3

10−2

position x in µm

ener

gyden

sity

g 1



0 0.1 0.2 0.3 0.4 0.5 0.610−4

10−3

10−2

position x in µm

ener

gyden

sity

g 1




0 0.1 0.2 0.3 0.4 0.5 0.6106

107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2


Figure 6.53: Negative energy current density −J1 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6106

107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2



0 0.1 0.2 0.3 0.4 0.5 0.6106

107

108

109

position x in µm

ener

gycu

rren

tden

sity

J1

inW

/m2




0 0.1 0.2 0.3 0.4 0.5 0.6−10

−5

0

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V


Figure 6.56: Lagrange multiplier λ0 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6−10

−5

0

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V



0 0.1 0.2 0.3 0.4 0.5 0.6−10

−5

0

position x in µm

Lag

range

multip

lier

λ0 Uext = 0.5 V




0 0.1 0.2 0.3 0.4 0.5 0.60

500

1000

1500

2000

position x in µm

tem

per

ature

Tin

K


Figure 6.59: Temperature T for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6

×103

0

1

2

3

position x in µm

tem

per

ature

Tin

K



0 0.1 0.2 0.3 0.4 0.5 0.6

×103

0

1

2

3

position x in µm

tem

per

ature

Tin

K




0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V


Figure 6.62: Potential Φ for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V



0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

position x in µm

pot

ential

Φin

V




0 0.1 0.2 0.3 0.4 0.5 0.6

×105

0

1

2

3

4

position x in µm

elec

tron

velo

city

v 0in

m/s Uext = 0.5 V


Figure 6.65: Negative electron velocity −v0 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.6

×104

0

5

10

15

position x in µm

elec

tron

velo

city

v 0in

m/s



0 0.1 0.2 0.3 0.4 0.5 0.6

×104

0

5

10

15

position x in µm

elec

tron

velo

city

v 0in

m/s




0 0.1 0.2 0.3 0.4 0.5 0.61

1.2

1.4

1.6

1.8

position x in µm

I 3/2−βε(z)/I 1

/2−βε(z) Uext = 0.5 V


Figure 6.68: Fermi integral quotient modifier in T0 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.61

1.5

2

2.5

position x in µm

I 3/2−βε(z)/I 1

/2−βε(z) Uext = 0.5 V


Figure 6.69: Fermi integral quotient modifier in T0 for βε = 1 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.61

1.5

2

2.5

3

position x in µm

I 3/2−βε(z)/I 1

/2−βε(z) Uext = 0.5 V




0 0.1 0.2 0.3 0.4 0.5 0.61

1.1

1.2

1.3

1.4

position x in µm

I 5/2−βε(z)/I 3

/2−βε(z) Uext = 0.5 V


Figure 6.71: Fermi integral quotient modifier in T1 for βε =12 and η = 50

0 0.1 0.2 0.3 0.4 0.5 0.61

1.2

1.4

1.6

position x in µm

I 5/2−βε(z)/I 3

/2−βε(z) Uext = 0.5 V



0 0.1 0.2 0.3 0.4 0.5 0.61

1.5

2

2.5

position x in µm

I 5/2−βε(z)/I 3

/2−βε(z) Uext = 0.5 V




0 0.5 1 1.5

×108

0

2

4

6

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2

Figure 6.74: Current-voltage characteristic for βε =12 and η = 50

0 0.5 1 1.5

×108

0

1

2

3

4

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2


0 0.5 1 1.5

×108

0

1

2

3

4

voltage Uext in V

char

gecu

rren

tden

sity

J0

inA

/m2



0 5 10 15 20 25 3010−7

10−6

10−5

10−4

10−3

number of iterations

Poi

sson

equat

ion

righ

t-han

dsi

de


Figure 6.77: Convergence for βε = 0 and η = 0

0 10 20 30 4010−7

10−6

10−5

10−4

10−3


Poi

sson

equat

ion

righ

t-han

dsi

de


Figure 6.78: Convergence for βε =12 and η = 0

0 20 40 60 80 10010−7

10−6

10−5

10−4

10−3


Poi

sson

equat

ion

righ

t-han

dsi

de




0 5 10 15 2010−7

10−6

10−5

10−4

10−3


Poi

sson

equat

ion

righ

t-han

dsi

de


Figure 6.80: Convergence for βε =12 and η = 50

0 5 10 15 2010−7

10−6

10−5

10−4

10−3


Poi

sson

equat

ion

righ

t-han

dsi

de



0 10 20 30 40 5010−7

10−6

10−5

10−4

10−3


Poi

sson

equat

ion

righ

t-han

dsi

de




βε 0 0 0 12

12

12

1 1 1

Uext 0.5 V 1 V 1.5 V 0.5 V 1 V 1.5 V 0.5 V 1 V 1.5 V

q 0.392 0.682 0.833 0.426 0.749 0.893 0.495 0.830 0.954

Table 6.5: Linear convergence rates q of the Poisson right-hand side for η = 0

βε12

12

12

1 1 1 1 1 1

η 50 50 50 50 50 50 100 100 100

Uext 0.5 V 1 V 1.5 V 0.5 V 1 V 1.5 V 0.5 V 1 V 1.5 V

q 0.541 0.632 0.753 0.561 0.627 0.787 0.874 0.886 0.874

Table 6.6: Linear convergence rates q of the Poisson right-hand side for η > 0

For the right-hand side of the Poisson equation in the Gummel iteration we plotthe discrete L2 norm of the right-hand side in (5.3), that is the Euclidean norm ofthe right-hand side in (5.4), for the usual three βε and three Uext . The results forη = 0 are given in the Figures 6.77 to 6.79. The convergence is linear in all ninesettings which is expected for the Gummel iteration – in contrast, the full Newtonscheme would certainly converge quadratically. The observed convergence rates qare presented in Table 6.5.

The Figures 6.80 to 6.82 show these errors for the usual parameters in the Fermi-Dirac case. It is not obvious why the convergence is faster – at least for Uext = 1.5 V– than in the case η = 0. Even though the shapes in the first two plots may suggestthat the convergence is less stable than for η = 0 – there are much more oscillations– the third does not reveal any troubles; instead the convergence is even better thanfor η = 0. Again the observed convergence rates q are presented in Table 6.6.

For the discretization error we consider different mesh sizes. We use the approx-imation performed on 2401 nodes as a reference called v and calculate the relativeL2 errors by

‖vh − v‖L2

‖v‖L2

where vh denotes the approximation of v on a coarser grid with step h. We calculate

Quantity m0 m1 T Φ

n = 150 3.16 · 10−2 3.31 · 10−2 2.60 · 10−2 3.92 · 10−3

n = 300 1.50 · 10−2 1.57 · 10−2 1.23 · 10−2 1.84 · 10−3

n = 600 6.42 · 10−3 6.77 · 10−3 4.64 · 10−3 7.65 · 10−4

n = 1200 2.16 · 10−3 2.26 · 10−3 1.67 · 10−3 2.61 · 10−4

Order 1.28 1.28 1.33 1.30

Table 6.7: Relative errors for different numbers n of subintervals for η = 0


Quantity m0 m1 T Φ

n = 150 2.94 · 10−2 3.14 · 10−2 2.49 · 10−2 4.71 · 10−3

n = 300 1.38 · 10−2 1.48 · 10−2 1.17 · 10−2 2.24 · 10−3

n = 600 6.96 · 10−3 6.39 · 10−3 4.97 · 10−3 9.68 · 10−4

n = 1200 1.99 · 10−3 2.14 · 10−3 1.59 · 10−3 3.19 · 10−4

Order 1.29 1.28 1.31 1.29

Table 6.8: Relative errors for different numbers n of subintervals for η = 50

the errors for the quantities m0, m1, T , and Φ for the numbers n ∈ 150, 300, 600,1200 of subintervals. The results together with the orders of convergence are pre-sented in the Tables 6.7 and 6.8 for the Maxwell-Boltzmann and Fermi-Dirac case,respectively. All orders lie around 1.3 and they are almost identical for η = 0 andη > 0.


Chapter 7

Summary and outlook

We have derived a hierarchy of diffusive semiconductor models based on the Boltz-mann equation. The steps are:

• Start with the Boltzmann equation in diffusive scaling and the usual Poissonequation. Use a small parameter α on the distribution function.

• Define an entropy and a maximizer of this entropy under the constraint thatcertain moments built from given weight functions are prescribed which leadsto Lagrange multipliers. Use the maximizer as the equilibrium state.

• The papers [16] and [17] deal with the Maxwell-Boltzmann and Fermi-Diracstatistics, respectively. The derivation presented here is even more general andincludes both cases.

• Decompose the collision operator in a dominant part and a small part. Assumecertain physically reasonable conservation properties on both parts.

• Insert a Chapman-Enskog expansion consisting of the equilibrium and a cor-rection term.

• Assume certain properties of the Fréchet derivative of the dominant collisionoperator to finally define N +1 current densities Ji to the N +1 moments mi.

We have succeeded in proving the following:

• The diffusion matrices are positive definite which justifies to use the word“diffusion” from the physical point of view.

• From the mathematical point of view, the positivity leads to a system of partialdifferential equations which is of parabolic type when denoted in the Lagrangemultipliers as the variables.

117

118 CHAPTER 7. SUMMARY AND OUTLOOK

• The mapping from the N + 1 Lagrange multipliers to the N + 1 moments isinjective which allows us to write the model equations with the moments asthe variables in an equivalent way.

• Under weak additional assumptions, the model can be denoted in the so-calleddrift-diffusion formulation or in the dual-entropy formulation eliminating theelectric field.

Some disadvantages are:

• The limits, especially α → 0, are done in a formal way. Rigorous existenceanalysis would require the exact definition of function spaces which was notdone in this context.

• When only the more or less minimal assumptions are used, the model equationsremain rather abstract since they contain unknown functions satisfying someoperator equations, however, the existence and uniqueness of these solutionsis proven.

• The other way round, the relaxation-time operator as an example seems to bequite special. It would be an interesting task to derive some class of operatorswhich are more general than the relaxation-time operator but special enoughto get rid of the abstract formulation.

Some other extensions would be a bipolar model which should arise by copying themodel equations, adjusting some signs, and adding both densities in the Poissonequation – or a transient model where then the moment m1 is needed during theiteration since it appears explicitly as ∂tm1 in the second balance equation.

From the numerical point of view, the drift-diffusion formulation in the Fermi-Dirac case with the relaxation-time operator was implemented. The discretizationfollowing [7] is:

• Define an arbitrary one-dimensional spatial mesh, not necessarily equidistant,and use a standard Raviart-Thomas discretization for the Poisson equation.

• Define a hybridized mixed finite element scheme with Marini-Pietra elementsfor both of the pairs (g0, J0) and (g1, J1). Their advantage is their M-matrixproperty.

• Use the static condensation procedure to arrive at a linear system for thedensity gi at the nodes which can be seen as a Scharfetter-Gummel scheme.

The numerical algorithm has been run for:

• A one-dimensional ballistic diode served as a sample device.

119

• The applied voltage was increased step by step, and for each step a Gummeliteration was executed. Each iteration consists of one Newton step for thePoisson equation, two linear systems for g0 and g1, and another node-wisenonlinear equation to calculate the Lagrange multipliers λ0 and λ1 (or thetemperature T ).

• The numerics work well for the Maxwell-Boltzmann case up to 1.5 volts andreproduce the results which were presented in the reference [7].

• The Fermi integrals are computed numerically. Some behavior of the integrandis used to decompose the interval, and the integrals over the subintervals arecalculated using an internal integrator. Values that are already computed arestored in a look-up table.

Nevertheless, the proper Fermi-Dirac case works in principal, too, but there aresome drawbacks:

• The Gummel iteration fails to converge for η ' 100. This problem cannot beovercome by just reducing the applied voltage and/or by refining the mesh.

• It is unclear up to now whether this is a numerical problem of the Gummeliteration or anything else. At least, it can be easily shown that η = 100is still far away from the fact that the Fermi-Dirac energy-transport modeldegenerates to a one-moment-one-current model which happens for η → ∞.

• The laborious full Newton scheme carries the additional problem that, in con-trast to the Maxwell-Boltzmann case, not only the equations are much morecomplicated due to the Fermi integrals, but that the derivative of I−1/2 isneeded. Since I−3/2 does not exist, one has to either develop some specialtechnique or use an approximation by a difference quotient.

• Even for η = 100 which should be significantly far away from Maxwell-Boltzmann, the results do not differ from the ones for η = 0 in such a waythat one could justify that it is another model. Especially the current doesnot change conspicuously.

120 CHAPTER 7. SUMMARY AND OUTLOOK

Bibliography

[1] N. W. Ashcroft and N. D. Mermin: “Solid State Physics”, Brooks/Cole –Thomson Learning, 1976.

[2] I. Babuška and J. E. Osborn: “Generalized Finite Element Methods: TheirPerformance and Their Relation to Mixed Methods”, SIAM Journal on Numer-ical Analysis, volume 20, number 3, 1983, pages 510–536.

[3] N. Ben Abdallah and P. Degond: “On a hierarchy of macroscopic modelsfor semiconductors”, Journal of Mathematical Physics, volume 37, number 7,1996, pages 3306–3333.

[4] F. Brezzi, L. D. Marini, and P. Pietra: “Méthodes d’éléments finismixtes et schéma de Scharfetter-Gummel”, Comptes Rendus de l’Académie dessciences, Série Mathématique, volume 305, 1987, pages 599–604.

[5] D. Chen, E. C. Kan, U. Ravaioli, C.-W. Shu, and R. W. Dutton:“An Improved Energy Transport Model Including Nonparabolicity and Non-Maxwellian Distribution Effects”, IEEE Electron Device Letters, volume 13,number 1, 1992, pages 26–28.

[6] J. B. Conway: “A Course in Functional Analysis”, second edition, SpringerScience + Business Media LLC, 2007.

[7] P. Degond, A. Jüngel, and P. Pietra: “Numerical Discretization ofEnergy-Transport Models for Semiconductors with Nonparabolic Band Struc-ture”, SIAM Journal on Scientific Computing, volume 22, number 3, 2000, pages986–1007.

[8] P. Degond, F. Méhats, and C. Ringhofer: “Quantum Energy-Transportand Drift-Diffusion Models”, Journal of Statistical Physics, volume 118, num-bers 3/4, 2005, pages 625–667.

[9] W. Dreyer, M. Junk, and M. Kunik: “On the approximation of the Fokker-Planck equation by moment systems”, Nonlinearity, volume 14, number 4, 2001,pages 881–906.

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[10] S. Gadau, A. Jüngel, and P. Pietra: “A mixed finite-element scheme ofa semiconductor energy-transport model using dual entropy variables”, Pro-ceeding in Tenth International Conference on Hyperbolic Problems: Theory,Numerics, Applications 2004, Yokohama Publisher, 2006, pages 139–146.

[11] T. Grasser, R. Kosik, C. Jungemann, B. Meinerzhagen, H. Kosina,and S. Selberherr: “A Non-Parabolic Six Moments Model for the Simulationof Sub-100 nm Semiconductor Devices”, Journal of Computational Electronics,volume 3, numbers 3/4, 2004, pages 183–187.

[12] T. Grasser, H. Kosina, M. Gritsch, and S. Selberherr: “Using sixmoments of Boltzmann’s transport equation for device simulation”, Journal ofApplied Physics, volume 90, number 5, 2001, pages 2389–2396.

[13] T. Grasser, H. Kosina, and S. Selberherr: “Hot Carrier Effects withinMacroscopic Transport Models”, International Journal of High Speed Electron-ics and Systems, volume 13, number 3, 2003, pages 873–901.

[14] S. Ihara: “Information Theory for Continuous Systems”, World Scientific Sin-gapore, 1993.

[15] A. Jüngel: “Transport Equations for Semiconductors”, Springer-Verlag, 2009.

[16] A. Jüngel, S. Krause, and P. Pietra: “A Hierarchy of Diffusive Higher-Order Moment Equations for Semiconductors”, SIAM Journal on Applied Math-ematics, volume 68, number 1, 2007, pages 171–198.

[17] A. Jüngel, S. Krause, and P. Pietra: “Diffusive semiconductor momentsequations using Fermi-Dirac statistics”, ZAMP Journal of Applied Mathematicsand Physics, accepted for publication October 2010.

[18] M. Junk: “Domain of Definition of Levermore’s Five-Moment System”, Journalof Statistical Physics, volume 93, numbers 5/6, 1998, pages 1143–1167.

[19] C. Kittel: “Introduction to Solid State Physics”, eighth edition, John Wiley& Sons Inc, 2005.

[20] D. Levermore: “Moment Closure Hierarchies for Kinetic Theories”, Journalof Statistical Physics, volume 83, 1996, pages 1021–1065.

[21] M. Lundstrom: “Fundamentals of carrier transport”, second edition, Cam-bridge University Press, 2000.

[22] L. D. Marini and P. Pietra: “New mixed finite element schemes for currentcontinuity equations”, COMPEL, volume 9, number 4, 1990, pages 257–268.

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[23] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser: “SemiconductorEquations”, Springer-Verlag, 1990.

[24] W. van Roosbroeck: “Theory of the Flow of Electrons and Holes in Germa-nium and Other Semiconductors”, Bell System Technical Journal, volume 29,number 4, 1950, pages 560–607.

[25] K. Seeger: “Semiconductor Physics”, ninth edition, Springer-Verlag, 2004.

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124 BIBLIOGRAPHY

Acknowledgments

There are so many people I would like to and have to thank, either for supportingthe progress of my thesis or for just being part of my life during the last years – orsome suitable linear combination thereof.

Mein allgemeiner Dank geht an meine deutschen und österreichischen FreundeFranz Achleitner, Tobias Berg, Michael Bogner, Christian Breitenberger, Felix Brei-tenecker, Martin Bruckner, Markus Denz, Xenia Descovich, Alexander Dick, Phil-ipp Dörsek, Sofie Esterhazy, Verena Goldammer, Mathias Grimm, Marcel Gruner,Pascal Gussmann, Karl Hollaus, Bernhard Hametner, Birgit Hischenhuber, Son-ja Höllrigl-Binder, Andreas Hula, Christoph Keller, Carmen Kintscher, MaximilianKleinert, Andreas Körner, Gregor Lasser, Jens Mandavid, Maja Miletic, ChristopherMuth, Nico Ondracek, Christian Pausch, Martin Pausch, Matthias Reinhardt, Cle-mens Ruhl, Daniel Saure, Holger Schier, Stephan Schmitz, Günter Schneckenreither,Felix Schneider, Stefan Schuchnigg, Christopher Schütz, Albrecht Seelmann, MarcSirotzki, Konrad Steiner, Dominik Strecker, Dominik Stürzer, René Tavernier, Chri-stopher Walz, Matthias Wastian, Florian Xaver, Hayato Yamato und Anna Zechner.

Für die sowohl fachliche als auch soziale Unterstützung geht mein Dank an mei-ne Bürokollegen Markus Brunk und Peter Kristöfel und an alle weiteren Mainzerund Wiener Arbeitsgruppenmitglieder, nämlich Mario Bukal, Bertram Düring, Phil-ipp Fuchs, Stephan Gadau, Jutta Gonska, Jan Haskovec, Sabine Hittmeir, StefanHolst, Daniel Matthes, Jan-Frederik Mennemann, Pina Milisic, Karl Rupp, BirgitSchörkhuber und Ines Stelzer.

Un agradecimiento especial va a varios amigos de la UBA en Buenos Aires, porejemplo a Pablo Amster, Manuel Maurette y Leonardo Vicchi, y creo que no olvidarénunca mi estancia en Argentina.

Ein selbstverständlicher Dank geht an meine Eltern, die mir mein Studium er-möglicht haben und mir immer zur Seite standen.

Schließlich möchte ich mich bei meinem Doktorvater, Betreuer und Erstgutach-ter Ansgar Jüngel bedanken, der mich über eine lange Zeit hinweg in Mainz undWien fachlich und finanziell unterstützt hat, mir das interessante Thema Halblei-termodellierung vorgeschlagen hat und mir bei Fragen stets zur Seite stand.

L’ultimo grazie va a Paola Pietra per i miei soggiorni in Pavia, per la sua dispo-nibilità ad aiutare e per il fatto che giudica la mia tesi di dottorato.

125

Documents

DISSERTATIONdie Maxwell-Boltzmann- und Fermi-Dirac-Statistik als Spezialfälle enthalten sind. Die Lagrange-Multiplikatoren, die von der Maximierung stammen, können ebenfalls als