Computation of Semiconductor Properties Using Moments of the …schenk/theses/brugger.pdf ·...

Computation of Semiconductor Properties UsingMoments of the Inverse Scattering Operator of theBoltzmann Equation

Diss. ETH No. xxxxx

Computation ofSemiconductor Properties

Using Moments of theInverse Scattering Operatorof the Boltzmann Equation

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGYZURICH

for the degree of

Doctor of Technical Sciences

presented by

SIMON CHRISTIAN BRUGGER

Dipl.-Phys. Eidgenossiche Technische Hochschule Zurichborn 17 July 1975

citizen of Auenstein AG, Switzerland

accepted on the recommendation of

Prof. Dr. Wolfgang Fichtner, examinerProf. Dr. Massimo Macucci, co-examiner

2005

Acknowledgement

First of all, I would like to express my gratitude to Prof. WolfgangFichtner, for giving me the opportunity to perform this thesis at theIntegrated Systems Laboratory (IIS) as well as for his support andencouragement. I am grateful to Prof. Massimo Macucci for readingand co-examining this thesis. Special thanks go to Prof. AndreasSchenk for his continuous scientific support troughout this project,and also to PD Dr. Fabian Bufler for teaching me almost everythingI know about Monte Carlo simulators.

I am also indebted to various people, which made this work pos-sible. I wish to thank Dr. Frederik Heinz for innumerable and fruit-ful discussions about programming, debugging, physics, mathematics,classical musics, horsemanship and lots of other importants things.Thanks to Dr. Bernhard Schmithusen who always had an open earfor my problems and helped my to retrieve important pieces of math-ematical literature. Thanks to Dr. Frank Geelhaar for interesting dis-cussions and especially for his well-furnished private library, as wellas to Dr. Stefan Rollin for his generous explanations about linearsolvers. A thank also to Timm Hohr and Lutz Schneider for the goodmood while we were writing our thesis in J76.2.

Finally, I would like to thank Dr. Dolf Aemmer, Dr. ChristophMuller, the system administrators Christoph Wicki and Anja Bohmas well as the secretary Christine Haller, for the excellent workingconditions in all respect.

This work has been partially supported by Fujitsu Ltd. and Syn-opsys Switzerland LLC.

v

Abstract

Physical objects called moments of the inverse scattering operator(MISO) of the Boltzmann equation (BE) allow computing all quanti-ties which appear in semiconductor transport theory (e.g. mobilities,hall factors, Langevin noise sources, · · · ) in an exact way, i.e. withoutusing the well-known relaxation time approximation (RTA).

In the first part, the existence and uniqueness of the MISOs areproven, and a numerical algorithm is given to actually compute all ofthem. Some of the most important applications of the MISOs includeamongst others the computation of Hall factors, and the computa-tion of exact Langevin noise sources. They are further developed tocompute the solution to the space-homogeneous Boltzmann equation(SHBE) for small field intensities to any order in the electric andmagnetic field. Another application is the derivation of a new iter-ative scheme for the one-particle Monte Carlo (MC) method, whichallows to take into account generation recombination (G-R) processes,which is not possible with the classical method. The last applicationpresented is the determination of the transition from the low- to thehigh-field regime.

The second part deals with fluctuation theory: Using the con-cept of MISO, exact formal expressions are derived for the impedancefield method (IFM) as well as for the acceleration fluctuation scheme(AFS). This allows to compare both methods on the same basis and,therefore, to better understand there meanings, implications, analo-gies and differences. It may be concluded that the IFM contains aformal flaw that seems not to affect the present applicability of themethod, but should be considered during the development of futureapplications.

vii

viii ABSTRACT

The third part introduces the numerical methods and physicalmodels underlying the calculation of a series of physical properties ofsilicon and silicon devices further illustrated in part four. The finiteelement method (FEM) was chosen to compute the solutions to thetransport model (TM) equations. Using the electric potential andquasi-Fermi potentials as variables and avoiding Bernoulli polynomi-als, leads to a suitable formulation for solving the TM equations. Incontrast to what has been claimed in the literature, the presentedmethod is appropriate to readily solve such TM equations. In orderto use the MISOs in the far-from-equilibrium regime, a MC methodhas been used to solve the BE. Coupling of a MC method with theMISO allows not only to compute stationary states, but may evengive access to Y-parameters and noise.

The last part deals with the application of the developed theoryto bulk silicon and silicon devices. First, MISOs of interest are com-puted for the physical models contained in our in-house TM and MCsimulator, SimnIC, for frequency dependent and independent scat-tering operators. Then, the MISOs are used to extract transportcoefficients and noise sources in bulk silicon for low- and high-electricfields. Simulations of a simple N+NN+ structure previously studiedin the literature underlines the applicability of MISOs for semicon-ductor small-signal analysis and noise modelling.

Zusammenfassung

Momente des inversen Streuoperators der Boltzmanngleichung (soge-nannte MISOs = Moments of the Inverse Scattering Operator) sindFunktionen, die die exakte Berechnung aller in der Halbleitertransport-theorie vorkommenden Grossen erlauben und daher die ubliche Re-laxationszeit-Naherung nicht erfordern. Im ersten Teil der Arbeit wirddie Existenz und Eindeutigkeit der MISOs bewiesen, und es wirdein Algorithmus hergeleitet, um sie im konkreten Fall berechnen zukonnen. Zu den wichtigsten Anwendungen der MISOs gehoren die Be-rechnung von makroskopischen Relaxationszeiten fur Transportmo-delle, von Hall-Faktoren und von exakten Langevin-Rauschquellen.Das Konzept der MISOs wird dann weiterentwickelt, um die Losungder raumhomogenen Boltzmanngleichung in allen Ordnungen im elek-trischen und magnetischen Feld zu berechnen. Eine weitere Anwen-dung ist die Herleitung eines neuen iterativen Verfahrens fur die Ein-teilchen-Monte-Carlo-Methode, das die Mitnahme von Generations-Rekombinationsprozessen zulasst. Als letzte Anwendung des erstenTeils wird gezeigt, wie man den Ubergang vom Bereich schwacherFelder zum Bereich starker Felder bestimmen kann.

Inhalt der zweiten Teils ist die Fluktuationstheorie: Exakte, for-male Ausdrucke werden sowohl fur die generalisierte Impedanzfeld-Methode (GIFM) als auch fur die Methode der Beschleunigungs-Fluk-tuationen mit Hilfe der MISOs hergeleitet. Dadurch konnen beideMethoden auf derselben Basis miteinander verglichen werden, wasein besseres Verstandnis ihrer Bedeutungen, Folgen, Anhlichkeitenund Unterschiede ermoglicht. Aus diesen Untersuchungen kann derSchluss gezogen werden, dass die GIFM einen Formfehler enthalt, der

ix

x ZUSAMMENFASSUNG

die praktische Anwendung der Methode jedoch vermutlich nicht inFrage stellt.

Der dritte Teil fuhrt die numerischen Methoden und physikali-schen Modelle ein, die die Grundlage fur die Berechnung der physika-lischen Grossen von Silizium sowie Silizium-Bauelementen im viertenTeil der Arbeit bilden. Fur die numerische Losung der Transportglei-chungen wurde die Methode der finiten Elemente (FEM) gewahlt.Es wird gezeigt, dass die Benutzung des elektrostatischen Potentialsund des quasi-Fermi-Potentials als Losungsvariablen zu einer geeigne-ten Formulierung fuhrt, die ohne Bernoulli-Funktionen auskommt. ImGegensatz zu Behauptungen in der Literatur uber die Unbrauchbar-keit der FEM fur die numerische Losung von Transportgleichungen,wird ihre tatsachliche Eignung in der vorliegenden Arbeit ausfuhrlichdemonstriert. Um die MISOs fur Zustande weit entfernt vom thermo-dynamischen Gleichgewicht verwenden zu konnen, wurde ein Monte-Carlo-Simulator zur Losung der Boltzmanngleichung entwickelt. DieKombination von Monte-Carlo-Simulator und MISOs erlaubt nichtnur die Berechnung von stationaren Zustanden, sondern auch die vonY-Parametern und moglicherweise sogar von Rauschparametern.

Im vierten und letzten Teil der Arbeit werden Anwendungen derdargestellten Theorie auf ”bulk”-Silizium sowie Silizium-Bauelementevorgestellt. Zuerst werden die relevanten MISOs fur das physikalischeModell berechnet, das dem entwickelten MC Simulator zugrunde liegt.Dann werden die MISOs verwendet, um Transport-Parameter undRauschquellen von ”bulk”-Silizium in schwachen und starken elektri-schen Feldern zu extrahieren. Schliesslich wird die Anwendbarkeit derMISOs fur die Bauelemente-Simulation illustriert, indem eine einfa-che, aus der Literatur bereits bekannte N+NN+-Struktur untersuchtwird.

èèèÜëÄÞó¹Boltzmann Equation: BEn qPâüáóÈMoments of the Inverse Scattering Operator: MISOh|piÏ(DShkJÆS8ÖkþYyfniÏHp ; AÕ'ÛüëÕ¡¯¿üLangevinÎ¤ºji³ÆkYjǑaéBÑ<Relaxation TimeApproximation(DZkYShLïýhj,ègoSnMISOsnX(h'<ÆWkY_np$¢ë´êºà:YMISOsnÜ(n-gÍjnhWf,kÛüëÕ¡¯¿ünWf³ÆjLangevinÎ¤ºnLRUkSzUU[Shg7nUDûLûÁL-kJQzkØjÜëÄÞó¹Spatially homogeneous Boltzmann Equationnãûn!p~gYShLïýhj _ÖnÜ(hWfoXPâóÆ«ëíÕMCÕn_n°WDÍ©¹üànÆúLBSn°WD¹üàk fS~gäxjKÕgo ïýgB _zP×í»¹generation-recombination processnYShLïýhj _kNûLßKØûLßxnwûnzkÜ(úeSh:WfD,ègoÕÖfluctuation theoryqFMISOnõ(Df¤óÔüÀó¹Õ£üëÉÕImpedance field method:IFMh Õ¹üàAcceleration fluctuation schemen_n³ÆjÆOSkSdnKÕXúkheDfÔYShLúe fns:YhS^<¹øU¹oOãYShLïýhj!n

xi

xiiÔKIFMkdDfobn eLBhPÖØQgBFSn eoSnKÕnþ(ni(ÄòíngojDFkǑLenÜ(vkJDfnUÅLB,ègop$jKÕhiâÇëkdDfðySniâÇëo,Ûègðy·ê³ójsk·ê³óÇ¤¹n#niy'nk(D 8âÇëTransport Model: TM¹ãY_koP ÕFEM(D_ûMhÕ§ëßÝÆó·ãëphWf(DÙëÌü¤n)(QShShgTM¹ãO_kiW_LHLYvgn;5hopj,vg:U_¹ÕoSnFjTM¹kþY!¿jãÕhWfigBMISOs8K'MO_ßg(D_koBEãO_kMCÕ(DMISOhhkMCÕ(DShg8¶KnLïýkjpKgjOY-Ñéáü¿Î¤ºnxSLKShkjn,Ûègo,ÖgUW_Önë¯·ê³óh·ê³óÇ¤¹xnÜ(kdDfðy~Zkn°ëü×gzW_TMûMC·ßåìü¿SimnICk+~iâÇë(DfhâpXjskhâp^Xn qPkdDfMISOsY DfMISOs(DfNûLØûLgnë¯·ê³ókJQ8Ñéáü¿hÎ¤º½úY HLvkXjN+NN+Ë n·ßåìü·çóMISOsnJÆSkJQá÷ãÎ¤ºâÇêó°xni(ïý'Æºk:WfD

Contents

Acknowledgement v

Abstract vii

Zusammenfassung ixèèè xi

1 Introduction 11.1 Inverse Scattering Operator (ISO) . . . . . . . . . . . 21.2 Moments of the ISO and the Monte Carlo method . . 31.3 Application of MISOs to noise computation . . . . . . 31.4 Numerics and results . . . . . . . . . . . . . . . . . . . 3

I Inverse Scattering Operators 5

2 Theory 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Transport models and MC simulations . . . . . . . . . 8

2.2.1 Derivation of transport models from the BE . . 82.2.2 Transport coefficients . . . . . . . . . . . . . . 11

2.3 Existence of moments of the ISO . . . . . . . . . . . . 132.3.1 Derivation of an equation for the moments of

the ISO . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Computation of the solution . . . . . . . . . . . 182.3.3 Connection between H and S−1 . . . . . . . . 21

xiii

xiv CONTENTS

2.4 Discretisation of a special class of SO . . . . . . . . . . 222.4.1 Adapted reformulation of the equations . . . . 232.4.2 Discretisation scheme . . . . . . . . . . . . . . 262.4.3 Solution algorithm for the discrete problem . . 30

2.5 Generalisation . . . . . . . . . . . . . . . . . . . . . . . 332.6 Error computation . . . . . . . . . . . . . . . . . . . . 342.7 Frequency-dependent scattering operator . . . . . . . . 37

2.7.1 Existence of T−1 and formal solution . . . . . . 382.7.2 Numerical approximation of moments of T−1 . 392.7.3 Link between T −1

g (ω) and S−1

g . . . . . . . . . 412.7.4 Remarks . . . . . . . . . . . . . . . . . . . . . . 442.7.5 Summary . . . . . . . . . . . . . . . . . . . . . 45

2.8 Boundary terms and further applications . . . . . . . . 452.8.1 Boundary terms . . . . . . . . . . . . . . . . . 452.8.2 Domain of applicability . . . . . . . . . . . . . 45

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Applications 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Low-field solution to the Boltzmann equation . . . . . 473.3 Transport coefficients . . . . . . . . . . . . . . . . . . . 493.4 Hall factor . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Bulk silicon . . . . . . . . . . . . . . . . . . . . 513.5 Relaxation times . . . . . . . . . . . . . . . . . . . . . 52

3.5.1 Introduction . . . . . . . . . . . . . . . . . . . 523.5.2 The problem of small electric fields . . . . . . . 533.5.3 The heated Maxwellian ansatz . . . . . . . . . 533.5.4 Exact solution . . . . . . . . . . . . . . . . . . 54

3.6 Langevin noise sources . . . . . . . . . . . . . . . . . . 553.6.1 Special class of SO . . . . . . . . . . . . . . . . 58

3.7 Monte Carlo and Langevin noise sources . . . . . . . . 593.7.1 Derivation of the BE with noise terms . . . . . 593.7.2 Conditions of application of the linear noise theo-

ry . . . . . . . . . . . . . . . . . . . . . . . . . 633.7.3 Existence of a stationary state . . . . . . . . . 643.7.4 Definition of transport coefficients and noise sour-

ces in a device . . . . . . . . . . . . . . . . . . 643.7.5 An interesting property of S−1

g . . . . . . . . . 66

CONTENTS xv

3.7.6 Correlation functions of Langevin noise sources 67

3.8 New iterative scheme for the one-particle MC method 72

3.8.1 Standard method . . . . . . . . . . . . . . . . . 72

3.8.2 Method based on moments of the ISO . . . . . 73

3.8.3 Advantages and drawbacks . . . . . . . . . . . 74

3.9 Determination of the low-field to high-field transition . 74

II Analytical Description of Noise Using ISO 77

4 Formal derivation of transport models 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Formal equation for a given moment . . . . . . . . . . 79

4.3 Small-signal analysis . . . . . . . . . . . . . . . . . . . 81

4.4 Common approximations . . . . . . . . . . . . . . . . 81

4.4.1 Transport coefficients . . . . . . . . . . . . . . 81

4.4.2 Parametrisation of the transport coefficients . . 82

5 Generalised Impedance Field Method 85

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Description of the generalised IFM . . . . . . . . . . . 87

5.3 Sources of noise . . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Approximations . . . . . . . . . . . . . . . . . . 94

5.3.2 Space-homogeneous case . . . . . . . . . . . . . 94

5.3.3 Bixon-Zwanzig relation . . . . . . . . . . . . . 95

5.4 Computation . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Application to common transportmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5.1 The drift-diffusion model . . . . . . . . . . . . 97

5.5.2 The energy-balance model . . . . . . . . . . . . 97

5.6 Verification . . . . . . . . . . . . . . . . . . . . . . . . 97

5.7 Two interesting basic approaches for the EB model . . 98

5.7.1 The no-derivatives ansatz . . . . . . . . . . . . 98

5.7.2 Ansatz using the gradient of the quasi-Fermi po-tential . . . . . . . . . . . . . . . . . . . . . . . 99

xvi CONTENTS

6 Systematic comparison to existing formulations 1016.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1016.2 Formal flaw of the GIFM . . . . . . . . . . . . . . . . 1016.3 The acceleration-fluctuation scheme . . . . . . . . . . 1026.4 Comparison of the GIFM with the AFS . . . . . . . . 104

III Numerics and Simulation Tools 107

7 Numerical methods for transport models 1097.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1097.2 The FEM and the DD model . . . . . . . . . . . . . . 110

7.2.1 The stationary DD equations . . . . . . . . . . 1107.2.2 The weak formulation . . . . . . . . . . . . . . 1117.2.3 Discretisation . . . . . . . . . . . . . . . . . . . 1127.2.4 Iterative algorithm . . . . . . . . . . . . . . . . 1157.2.5 Properties of the proposed discretisation . . . . 116

7.3 The FEM and the energy-balance model . . . . . . . . 1187.3.1 The stationary energy-balance equations . . . . 1187.3.2 The weak formulation . . . . . . . . . . . . . . 1197.3.3 Discretisation . . . . . . . . . . . . . . . . . . . 1207.3.4 Iterative algorithm and properties of the dis-

cretisation . . . . . . . . . . . . . . . . . . . . . 1237.4 Terminal currents . . . . . . . . . . . . . . . . . . . . . 1237.5 Small-signal analysis . . . . . . . . . . . . . . . . . . . 126

7.5.1 The Y-parameters . . . . . . . . . . . . . . . . 1267.5.2 Discretisation . . . . . . . . . . . . . . . . . . . 1267.5.3 Computation . . . . . . . . . . . . . . . . . . . 127

7.6 High-frequency noise . . . . . . . . . . . . . . . . . . . 1297.6.1 The DD Langevin equations . . . . . . . . . . . 1307.6.2 Discretisation . . . . . . . . . . . . . . . . . . . 1307.6.3 Computation . . . . . . . . . . . . . . . . . . . 131

8 Development of a suitable MC simulator for silicon 1358.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1358.2 Physical model . . . . . . . . . . . . . . . . . . . . . . 1358.3 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . 1368.4 Computation of values of interest . . . . . . . . . . . . 137

CONTENTS xvii

8.4.1 Mean values . . . . . . . . . . . . . . . . . . . . 1378.4.2 Terminal currents . . . . . . . . . . . . . . . . 1388.4.3 Correlation functions . . . . . . . . . . . . . . . 138

8.5 Iteration schemes . . . . . . . . . . . . . . . . . . . . . 1398.5.1 Iteration scheme for one-particle MC . . . . . . 1398.5.2 Iteration scheme for many-particle MC . . . . . 140

8.6 Correction of the self-forces . . . . . . . . . . . . . . . 1408.7 Calibration and validation . . . . . . . . . . . . . . . . 141

8.7.1 Low-field mobility . . . . . . . . . . . . . . . . 1418.7.2 High-field mobility . . . . . . . . . . . . . . . . 1418.7.3 Time-of-flight . . . . . . . . . . . . . . . . . . . 1438.7.4 Effective mobility . . . . . . . . . . . . . . . . . 1448.7.5 Hall factor . . . . . . . . . . . . . . . . . . . . 145

8.8 Small-signal analysis . . . . . . . . . . . . . . . . . . . 1488.8.1 One-particle MC . . . . . . . . . . . . . . . . . 1488.8.2 Many-particle MC . . . . . . . . . . . . . . . . 152

8.9 Noise and Green’s functions . . . . . . . . . . . . . . . 153

IV Applications of the MISO to Silicon 155

9 Results 1579.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Electrons in bulk silicon . . . . . . . . . . . . . . . . . 158

9.2.1 Two standard models . . . . . . . . . . . . . . 1589.2.2 Problem caused by the Dirac delta function . . 1609.2.3 Properties of the normalised SO . . . . . . . . 1619.2.4 Impact ionisation . . . . . . . . . . . . . . . . . 1619.2.5 Impurity scattering . . . . . . . . . . . . . . . . 1629.2.6 Discretisation of the energy space . . . . . . . . 1629.2.7 Model 1: Results . . . . . . . . . . . . . . . . . 1639.2.8 Model 1: Error computation . . . . . . . . . . 1639.2.9 Model 1: Langevin noise sources for moments

of interest . . . . . . . . . . . . . . . . . . . . . 1719.2.10 Model 1: Moments of interest of the frequency-

dependent ISO . . . . . . . . . . . . . . . . . . 1759.2.11 Model 2 and comparison with model 1 . . . . . 180

9.3 Monte Carlo-generated noise sources . . . . . . . . . . 183

xviii CONTENTS

9.3.1 Preliminary remark . . . . . . . . . . . . . . . 183

9.3.2 Electric field dependence . . . . . . . . . . . . . 184

9.3.3 Doping dependence . . . . . . . . . . . . . . . . 184

9.3.4 Direction dependence . . . . . . . . . . . . . . 186

9.3.5 Comparisons . . . . . . . . . . . . . . . . . . . 188

9.4 Energy relaxation times . . . . . . . . . . . . . . . . . 192

9.5 Simple N+NN+ and P+PP+ structures . . . . . . . . 198

9.5.1 MC-generated bulk transport coefficients . . . 199

9.5.2 Driving forces for the EB model . . . . . . . . 200

9.5.3 Stationary states . . . . . . . . . . . . . . . . . 200

9.5.4 Y-parameters . . . . . . . . . . . . . . . . . . . 206

9.5.5 Noise analysis . . . . . . . . . . . . . . . . . . . 209

9.5.6 An unresolved problem . . . . . . . . . . . . . 219

9.6 A case of negative mobility . . . . . . . . . . . . . . . 221

10 Conclusion 225

11 Outlook 227

Appendices 229

A Inverse Scattering Operators 231

A.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 231

A.2 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 233

A.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 233

A.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 234

A.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 234

A.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 235

A.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 236

A.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 237

A.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 238

A.2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 239

A.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 240

A.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 242

A.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 244

A.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 245

CONTENTS xix

B Space-homogeneous Boltzmann equation 247B.1 Important property of the ISO S−1

eq . . . . . . . . . . . 250B.2 Important property of Sn . . . . . . . . . . . . . . . . 251B.3 Generalisation . . . . . . . . . . . . . . . . . . . . . . . 252

B.3.1 Fermi statistics . . . . . . . . . . . . . . . . . . 252B.3.2 Boltzmann statistics . . . . . . . . . . . . . . . 253

C The EB model and RF noise 255C.1 The EB Langevin equations . . . . . . . . . . . . . . . 255

C.1.1 Discretisation . . . . . . . . . . . . . . . . . . . 257C.1.2 Computation . . . . . . . . . . . . . . . . . . . 258

C.2 MC-generated noise sources for the EB model . . . . . 261C.2.1 The bulk case . . . . . . . . . . . . . . . . . . . 261

D Correlation functions and delta functions 265

E Impurity scattering 267E.1 First numerical problem . . . . . . . . . . . . . . . . . 268E.2 Second numerical problem . . . . . . . . . . . . . . . . 268E.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 269

F The SimnIC simulator 271F.1 A realistic device . . . . . . . . . . . . . . . . . . . . . 272F.2 Many-particle MC: Benchmarks . . . . . . . . . . . . . 275

Bibliography 279

List of figures 286

List of tables 293

List of symbols 295

List of acronyms 297

Curriculum Vitæ 299

Chapter 1

Introduction

It is a fact that, when considering a single device like a MOSFET ora bipolar transistor, the concept of steady state makes no sense atall. Even if the voltage on the contacts is kept constant, the terminalcurrents will fluctuate in time. Only a mean value of the current ofa large number of devices will give a sense to the concept of steadystate. Besides mean values, other measurements are of interest tocharacterise a device or a group of devices. An example are noisemeasurements which provide indirect information about the statisticsof the current fluctuations. Two categories of methods exist in thedomain of semiconductor device simulation to reproduce noise mea-surements. The first category involves methods which try to directlycompute the fluctuations of the current of a given device. The well-known many-particle Monte Carlo method is one of these methods.The second category comprises indirect methods which try to repro-duce the comportment of a single device using transport models towhich a stochastic term is added, the so-called Langevin noise source.In the last 40 years a lot of effort was put in the development of thelatter methods, because they are much simpler than those of the firstcategory.

Four and a half years ago, the goal of this thesis was to implementthe best available model of the second category into the DESSIS devicesimulator and to compare the results with measurements. However, atan early stage of the implementation, important questions rose which

1

2 CHAPTER 1. INTRODUCTION

could not be answered using standard methods. Those questions were:

1. How reliable are bulk transport coefficients in small devices?

2. How accurate is the Einstein relation far from equilibrium?

3. What is the difference between the noise sources in a transportmodel and the Langevin noise sources?

4. How good is the relaxation time approximation (RTA)?

Tantamount to answering those questions was to find a practical meanto invert the scattering operator for any given scattering model andband structure.

After two years the problem was finally solved and well under-stood, at least mathematically and numerically, and numerous otherpotential applications of the inverse scattering operator (ISO) ap-peared. The existence, computation and application of the ISO arethe main topics of this thesis.

1.1 Inverse Scattering Operator (ISO)

Many papers about fluctuation phenomena, especially in semiconduc-tors (see all references in [21]), appeared at the end of the 60’s andthe beginning of the 70’s. Pioneering theoretical papers are those byK.M. van Vliet (see [58]) and by S.V. Gantsevich et al. (see [21]). Inan important paper by S.V. Gansevich et al. (see [20]) the concept ofthe ISO appears for the first time (to the authors knowledge) in theformal derivation of the relation between the correlation functions ofthe Langevin noise sources, the correlation functions of the currentfluctuations, and the diffusion tensor. Although the problems of thenon-unicity of the ISO and the need to construct a subspace wherethe ISO is unique, are already discussed in the paper by Gantsevichet al. (see [20] p. 280, Eqs. (4.1) to (4.4)), no proof was given of theexistence of the ISO and no algorithm was proposed to actually com-pute it. The same holds for the paper by van Vliet (see [58] p. 1989,Eq. (3.29)). After a discussion of the exact theory, both authors con-sider only simple examples which can be solved analytically and whichdo not require a complicated procedure to compute the ISO.

1.2. MISO AND THE MONTE CARLO METHOD 3

1.2 Moments of the ISO and the MonteCarlo method

In the first chapter of this thesis a proof based on the Krein-Rutmantheorem (see e.g. [18]) is given of the existence of the ISO up to aconstant and, even more important, an explicit solution is providedfor the moments of the ISO (MISO). From those mathematical con-siderations a general discretisation scheme is derived to practicallycompute the MISO. A proof of the convergence of the algorithm isgiven as well as a method to directly compute the error resulting fromthe discretisation.

In the second chapter the wide range of applications of the MISOsis discussed. The knowledge of the solution to the Boltzmann equationand of the MISOs is shown to be necessary and sufficient to computeeverything needed in the semiclassical transport and noise theories.

1.3 Application of MISOs to noise com-putation

The second category of methods to compute noise already discussedat the beginning of this introduction contains in fact two methods.The first one is called impedance field method (IFM) and was devel-oped by Shockley (see [52]) and recently generalised by Jungemannet al. (see [33],[34]). The second one is called acceleration fluctuationscheme (AFS) and was developed by Shiktorov et al. (see [51]). Inthe second part of this thesis, both methods are evaluated in a formalway. From those considerations, a simplification of the method byJungemann is proposed, and the problems of the IFM and the AFSare discussed.

1.4 Numerics and results

When the work for this thesis started, the author began to write asmall full-band Monte Carlo program to compute noise sources forelectrons and holes in bulk silicon. In order to test most of the ap-plications of the MISOs, the code was further developed. Today, the

4 CHAPTER 1. INTRODUCTION

code, called SimnIC (Simulator for Noise Computation), counts morethan 200′000 lines written in C++ and is a complete device simulatorfor silicon devices. SimnIC contains different types of drift-diffusionand energy-balance models and a full-band Monte Carlo (MC) sim-ulator. The MC simulator can perform self-consistent many-particlesimulations as well as one-particle simulations using two different it-eration schemes, one of them being able to take into account G − Rprocesses.

In the third part of this thesis, the numerical methods used tosolve the different transport models as well as the numerics used inSimnIC are described in detail. One particularity is the use of thefinite-element method (FEM) to solve the transport model equations.The FEM is often described in the literature as beeing inappropriateto solve transport model equations (see e.g. [50], [26]). However,within our formulation, the FEM seems to work as well as the boxmethod.

The fourth and last part of this thesis deals with special applica-tions of the MISOs. In order to prove the validity of the methodspresented, results of Jungemann (see [32]) for a simple N + NN+structure were systematically compared to the results from SimnIC.This lead to some interesting differences which can be explained bythe MISOs.

Part I

Inverse ScatteringOperators

5

Chapter 2

Theory

2.1 Introduction

A crucial step in the derivation of transport models is the introductionof a microscopic or macroscopic relaxation time (RT) (see e.g. [53,4]). An obvious shortcoming of the relaxation time approximation(RTA) is, among others, that a mobility tensor cannot be defined inan unambiguous way, if the dimensionality is higher than one (see e.g.[56]). Even worse, because nothing is known about the dynamics ofthis RT, it is impossible to predict its fluctuations around a stationarystate, and therefore, the computation of adequate noise sources for agiven moment of the Boltzmann equation (BE) is rendered impossible,too.

The aim of this chapter is to develop a formalism that avoids theRTA. It will allow, among others, the exact computation of trans-port coefficients and Langevin noise sources for any moment of theBE using the MC method. The key feature in this formalism is thederivation and computation of moments of the inverse scattering op-erator (MISO). Although the formalism was primarily developed tobe applied to semiconductors, it can also be used to analyse any opensystem described by a BE (linear or not).

The chapter is organised in eight sections. First, the approxima-tions done in the derivation of a transport model are recalled, and the

7

8 CHAPTER 2. THEORY

application of the MISO to the local term by term comparison of thetransport model with the outcome of a MC simulation is explained.In Section 2.3 the sufficient mathematical conditions under which theMISOs exist are exposed, and a formal solutions is presented. In Sec-tion 2.4 the problem will be restrained to scattering operators whichare only band-valley-and energy-dependent (as they appear in mostMC simulators for semiconductors), and a discretisation scheme aswell as an algorithm to approximate any MISO will be derived. InSection 2.5 the special case presented in 2.4 will be generalised to anytype of SO. In Section 2.6 the error computation will be addressed. InSection 2.7 the formalism of Section 2.4 will be extended to frequency-dependent scattering operators. It will be demonstrated that in thelimit of vanishing frequency the original formalism is recovered. Sec-tion 2.8 will address some subtleties hidden in the formalism. Finally,Section 2.9 will summarise the presented theory.

2.2 Transport models and MC simulations

2.2.1 Derivation of transport models from the BE

The first step in the derivation of any transport model from the BE

∂tf + ~v∇rf − q~~E∇kf = Sf (2.1)

is to introduce some kind of differentiable RT τ which ideally shouldcontain the whole information about the scattering operator S:

∂tf + ~v∇rf − q~~E∇kf =

f − nneq

feq

τ(f,~r, t,~k, ~E), (2.2)

τ(f,~r, t,~k, ~E)∂tf + τ(f,~r, t,~k, ~E)~v∇rf

−τ(f,~r, t,~k, ~E) q~~E∇kf = f − n

neqfeq

. (2.3)

To be able to compare (2.2) and (2.3) with the exact BE (2.1) onehas to invert the scattering operator

S−1∂tf + S−1~v∇rf − S−1 q~~E∇kf = f − n

neqfeq. (2.4)

2.2. TRANSPORT MODELS AND MC SIMULATIONS 9

Note that S−1S 6= 1! This crucial statement will be explained below.In Eqs. (2.1)–(2.4), feq denotes the Boltzmann distribution function

normalised to one (∫Bz

f2eq(~k)d3k = 1) and n :=

∫Bz

f(~k)d3k, neq :=

∫Bz

feq(~k)d3k, and Bz denotes the Brillouin zone. The symbol neq

is used here for convenience, although the actual equilibrium densitydiffers from neq by a constant due to the normalisation condition.

The second step is to build a moment of interest of (2.3) using apiece-wise continuous function g:

∫

Bz

g(~k)τ(f(~k,~r))∂tf(~k,~r)d3k +

∫

Bz

g(~k)τ(f(~k,~r))~v(~k)∇rf(~k,~r)d3k

−∫

Bz

g(~k)τ(f(~k,~r)) q~~E∇kf(~k,~r)d3k

=

∫

Bz

g(~k)

[f(~k,~r) − n

neqfeq(~k,~r)

]d3k. (2.5)

Building a moment in a similar fashion in (2.4) leads to

∫

Bz

∫

Bz

g(~k)S−1(~k,~k1, ~r)∂tf(~k1, ~r)d3k1d

3k

+

∫

Bz

∫

Bz

g(~k)S−1(~k,~k1, ~r)~v(~k1)∇rf(~k1, ~r)d3k1d

3k

−∫

Bz

∫

Bz

g(~k)S−1(~k,~k1, ~r)q~~E∇k1f(~k1, ~r)d

3k1d3k :=

∫

Bz

S−1g (~k1, ~r)∂tf(~k1, ~r)d

3k1 +

∫

Bz

S−1g (~k1, ~r)~v(~k1)∇rf(~k1, ~r)d

3k1

−∫

Bz

S−1g (~k1, ~r)

q~~E∇k1f(~k1, ~r)d

3k1 =

∫

Bz

g(~k)

[f(~k,~r) − n

neqfeq(~k,~r)

]d3k, (2.6)

10 CHAPTER 2. THEORY

where S−1g (~k1, ~r) :=

∫Bz

g(~k)S−1(~k,~k1, ~r)d3k is a moment of the ISO.

The third step is to perform a partial integration of the k-gradientterm in (2.5), and to neglect the boundary term

∫

Bz

g(~k)τ(f(~k,~r)) q~~E∇kf(~k,~r)d3k

≃ − q~~E

∫

Bz

∇k(g(~k)τ(f(~k,~r)))f(~k,~r)d3k. (2.7)

Because the function S−1g (~k1, ~r) may be discontinuous in some

points (for some silicon MC models on the boundary surfaces be-tween two valleys), the k-gradient term of (2.6) cannot be exactlytransformed as in (2.7). Instead, the domain Bz has to be decom-

posed in subdomains, where the function S−1g (~k1, ~r) is continuous and

all the boundary terms, which in general will not disappear, must bekept

∑

i

∫

Bzi

S−1g (~k,~r) q

~~E∇kf(~k,~r)d3k =

− q~~E∑

i

∫

Bzi

∇k(S−1g (~k,~r))f(~k,~r)d3k+ q

~~E∑

i

∮

∂Bzi

S−1g (~k,~r)f(~k,~r)~nda,

(2.8)

where⋃i

Bzi = Bz.

The boundary term∑i

∮∂Bzi

S−1g (~k,~r)f(~k,~r)~nda does not vanish.

To the author’s knowledge it has never been investigated numeri-cally whether this term is negligible for all moments and for all fieldstrengths of practical interest, e.g. for bulk silicon. This term will bekept in the numerical analysis.

The final step is to find closure relations in order to close thesystem of equations.

2.2. TRANSPORT MODELS AND MC SIMULATIONS 11

2.2.2 Transport coefficients

Comparing one by one the moments in well-known and widely usedmacroscopic transport models - the drift-diffusion (DD) model andthe energy-balance (EB) model (see e.g. [48], Chapter 1.1.3)- withthe corresponding terms which arise from the BE and which containthe exact ISO, allows to uniquely defined the transport coefficients(TCs).

The equation for the current in the DD model reads:

τp(∂t(〈~v〉) − µn~E −D∇rn = 〈~v〉=: −~J

q, (2.9)

where τp is the momentum relaxation time, µ the mobility tensor, Dthe diffusivity tensor, n the density and

〈~v〉 :=

∫

Bz

~vfd3k

(〈~v〉 /n is the mean velocity).Comparing term by term the lhs of (2.9) with the lhs of (2.6) leads

to the following conditions for the DD model to be exact:

τp∂t(〈vi〉) !=

∫

Bz

S−1vi

(~k1, ~r)∂tf(~k1, ~r)d3k1, i = 1, .., 3, (2.10)

(−µn~E

)i

!= q

~~E

∫

Bz

∇k(S−1vi

(~k,~r))f(~k,~r)d3k

− q~~E∑

j

∮

∂Bzj

S−1vi

(~k,~r)f(~k,~r)~nda, i = 1, .., 3, (2.11)

(−D∇rn)i!=

∫

Bz

S−1vi

(~k1, ~r)~v(~k1)∇rf(~k1, ~r)d3k1,

i = 1, .., 3, 〈vi〉eq = 0. (2.12)


Note that (2.11) gives a natural definition of the mobility tensor

(µ)ij : = − qn~

∫

Bz

∂kj(S−1vi

(~k,~r))f(~k,~r)d3k

+ qn~

∑

l

∮

∂Bzl

S−1vi

(~k,~r)f(~k,~r)(~nda)j , i = 1, .., 3. (2.13)

The corresponding equations for the EB model are:

τp∂t(〈~v〉) − µn~E − µn∇r(kBq

〈T 〉n

) −D′∇rn = 〈~v〉 , (2.14)

32kB(〈T 〉 − n

〈T 〉 eqneq︸︷︷︸

:=Teq

) =

− τε

q 〈~v〉 ~E + ∇r

[−κn∇r(

〈T 〉n

) +5kB2

〈T 〉〈~v〉n

], (2.15)

where τε is the energy relaxation time, κn the heat conduction coeffi-cient, 〈T 〉eq /neq the temperature in thermodynamic equilibrium,

D′ :=kBq

〈T 〉nµ, (2.16)

and

〈T 〉 :=m∗

3kB· Tr(

∫

Bz

(~v · ~vT )fd3k). (2.17)

Comparing (2.14) and (2.15) with (2.6) gives the following condi-tions for the EB model to be exact:

−µ(∇r(

kB 〈T 〉q

)

)

i

!=

∫

Bz

S−1vi

(~k1, ~r)~v(~k1)∇rf(~k1, ~r)d3k1, i = 1..3,

(2.18)

2.3. EXISTENCE OF MOMENTS OF THE ISO 13

τεq 〈~v〉 ~E !=−1

2m∗

3∑

i=1

q~~E

∫

Bz

∇k(S−1vivi

(~k,~r))f(~k,~r)d3k

+1

2m∗

3∑

i=1

q~~E∑

j

∮

∂Bzj

S−1vivi

(~k,~r)f(~k,~r)~nda, (2.19)

− τε∇r

[−κn∇r(

〈T 〉n

) +5kB2

〈T 〉〈~v〉n

]!=

1

2m∗

3∑

i=1

∫

Bz

S−1vivi

(~k1, ~r)~v(~k1)∇rf(~k1, ~r)d3k1. (2.20)

Although the terms in the rhs of (2.10), (2.11) and (2.18)–(2.20)

look rather cumbersome, this is not the case. As soon as S−1g (~k,~r) is

known for all required gs, those terms can be easily computed for agiven device with a MC simulation and then locally compared withthe terms of the transport model.

The usefulness of the MISOs being motivated, a general way toactually compute them is needed.

2.3 Existence of moments of the ISO

2.3.1 Derivation of an equation for the momentsof the ISO

The starting point is the BE

∂tf(~r, t,~k, b) + ~r∇rf(~r, t,~k, b) + ~k∇kf(~r, t,~k, b)

=∑

b0

∫

Vb0

f(~r, t,~k0, b0)w(~r, t)(~k0, b0|~k, b)d3k0

−∑

b0

∫

Vb0

f(~r, t,~k, b)w(~r, t)(~k, b|~k0, b0)d3k0, (2.21)


where ~r is the position in space, ~k the position in k-space, b a band-valley index, Vb the k-space of band-valley b, and w(~r, t)(~k, b|~k0, b0) is

the scattering rate from point (~k, b) to (~k0, b0) (at time t and spaceposition ~r, respectively). Note that the Pauli blocking factors (1− f)are included in the scattering rates w. Since 0 < 1 − f ≤ 1, they willnever increase the magnitude of w. This will be of some importancebelow. In the following the assumption is made that the Vbs arecompact pairwise disjoint subsets of R3, and w(~r, t)(~k, b|~k0, b0) : Vb →Vb0 , is a continuous function1 of ~k and ~k0. Therefore, the space K, of

interest here, can be defined as K :=⋃Ni=0 Vbi

.The scattering operator (SO) S is defined as

S(~r, t)(~k, b|~k0, b0) := w(~r, t)(~k0, b0|~k, b)−δ3(~k−~k0)δb,b0Wtot(~r, t)(~k, b),

(2.22)

with Wtot(~r, t)(~k, b) :=∑b′

∫Vb′

w(~r, t)(~k, b|~k′, b′) d3k′ > 0 .

By definition, S is a bound continuous operator on the Banachspace C0(K) with ||·||∞ (see e.g. [59] p. 70).

In the remainder the argument (~r, t) will be omitted, and the Diracnotation will be sometimes used for better readability (e.g. |f〉 := f).Sometimes the ”·” notation, defined as

(A · B)(~k, b|~k1, b1) :=∑

b0

∫

Vb0

A(~k, b|~k0, b0)B(~k0, b0|~k1, b1)d3k0,

(2.23)

will also be used to avoid confusion.Defining

S|f〉 :=∑

b0

∫

Vb0

S(~k, b|~k0, b0)f(~k0, b0)d3k0 =

∑

b0

∫

Vb0

(f(~k0, b0)w(~k0, b0|~k, b) − f(~k, b)w(~k, b|~k0, b0)

)d3k0, (2.24)

1The Dirac δ-function usually used in the scattering rate cannot be used herebecause it is no function. However any regularisation of the δ-function can beused (see Chapter 9).


the rhs of (2.21), i.e. the scattering term of the BE, can be rewrittenas S|f〉.

Now a definition of the inverse operatorH to the SO (i.e. the ISO)is needed.

First of all it has to be noted that one cannot define the ISO naivelyas HS|f〉 = |f〉 for all f ∈ C0(K), because S has an eigenvector witheigenvalue 0 (indeed only one, as will be shown later), namely theBoltzmann function feq.

2

Therefore, the SO has to be inverted on the space Ker⊥ perpen-dicular to its kernel Ker := λ|feq〉 | λ ∈ R.

In the following feq is chosen such that 〈feq|feq〉 = 1, where the

scalar product is naturally defined by 〈f |g〉 :=∑b

∫Vb

f(~k, b)g(~k, b)d3k.

An explicit definition of Ker⊥ is given by

Ker⊥ := Pfeq|g〉 | g ∈ C0(K), (2.25)

wherePfeq

:= 1− |feq〉〈feq|. (2.26)

By definition Pfeqand S fullfil a trivial but important property:

S|g〉 = S · Pfeq|g〉. (2.27)

The ISOH can be unequivocally defined by the properties he mustfulfil:

1. H · S|g〉 != |g〉, ∀g ∈ Ker⊥

2. H |f1〉 = 0,

with f1(~k, b) := 1rPb′

|Vb′ |= constant, where |Vb′ | is the volume of Vb′ .

Note that, using (2.27), condition 1) can be rewritten as

H · S|g〉 = H · S · Pfeq|g〉 !

=Pfeq|g〉, ∀g ∈ C0(K). (2.28)

2Also in the case of degenerate systems and/or systems with two-particle scat-tering (e.g. e-e collisions) the eigenvector with eigenvalue 0 exists and is unique(as will be shown later), but it will depend on f . This will not impact the validityof the presented approach.


The appropriateness of condition 2) will now be explained in detail.First, note that ST |f1〉 = 0 (〈f1|S = 0) by definition of the Wtot.

Without condition 2) an infinite number of ISOs could be defined,because, if H satisfies condition 1), then H + |v〉〈f1| fulfils the samecondition for any |v〉). Let H∗ be an ISO fulfilling condition 1), and|h∗〉 := H∗|f1〉. H∗ can always be rewritten as H∗ = H⊥ + |h∗〉〈f1|,where H⊥ := H∗ − |h∗〉〈f1|. Note here that because H⊥ fulfils condi-tions 1) and 2), it is unambiguously defined and, therefore, indepen-dent of H∗.

By multiplying (2.21) with 〈f1|, one obtains: ∂tn + ∇r

⟨~r⟩

= 0,

which is nothing but the current continuity equation.

(Note that for semiconductors the boundary term∮Vb0

f~k~nda al-

ways disappears due to the inversion symmetry of the Vb0).Thus, by multiplying (2.21) with H∗, and using (2.28), one obtains

H∗∂t|f〉 +H∗~r∇r |f〉 +H∗~k∇k|f〉 = |f〉 − 〈f |feq〉 |feq〉

= H⊥∂t|f〉 +H⊥~r∇r|f〉 +H⊥~k∇k|f〉. (2.29)

The last equation shows that H⊥ already contains the full informationneeded, and that it is reasonable to define H := H⊥, i.e. H |f1〉 = 0.

An equation for the ISO can now be formulated:

H · S|g〉 != |g〉 − |feq〉〈feq|g〉 , ∀g (2.30)

and finally the operator equations for H can be written as

H · S != δ3(~k − ~k0)δb,b0 − feq(~k, b)feq(~k0, b0) = 1− |feq〉〈feq|, (2.31)

H |f1〉 !=0. (2.32)

Next, (2.31) and (2.32) have to be solved. From (2.22) and (2.31)one obtains

H · S =∑

b2

∫

Vb2

H(~k, b|~k2, b2)w(~k0, b0|~k2, b2)d3k2 −Wtot(~k0, b0)H(~k, b|~k0, b0)

!= δ3(~k − ~k0)δb,b0 − feq(~k, b)feq(~k0, b0). (2.33)


By re-arranging the terms, this equation can be reformulated as

H(~k, b|~k0, b0) =∑

b2

∫

Vb2

H(~k, b|~k2, b2)w(~k0, b0|~k2, b2)

Wtot(~k0, b0)d3k2

− δ3(~k − ~k0)δb,b0

Wtot(~k0, b0)+ feq(~k, b)

feq(~k0, b0)

Wtot(~k0, b0). (2.34)

Now, remember that not the ISO, but only the MISOs

Hg(~k0, b0) :=∑

b

∫

Vb

g(~k, b)H(~k, b|~k0, b0)d3k = 〈g|H (2.35)

are of interest. The equation for the MISOs can be derived from (2.34)as

|Hg〉 := Hg(~k0, b0) =

∑

b2

∫

Vb2

Hg(~k2, b2)w(~k0, b0|~k2, b2)

Wtot(~k0, b0)d3k2−

(g(~k0, b0) − 〈g〉eq feq(~k0, b0)

Wtot(~k0, b0)

),

(2.36)

where 〈g〉eq := 〈feq|g〉.

Defining

A(~k0, b0|~k2, b2) :=w(~k0, b0|~k2, b2)

Wtot(~k0, b0),

AT (~k0, b0|~k2, b2) :=w(~k2, b2|~k0, b0)

Wtot(~k2, b2),

|lg〉 := lg(~k0, b0) :=

(g(~k0, b0) − 〈g〉eq feq(~k0, b0)

Wtot(~k0, b0)

), (2.37)

(2.36) can be expressed as

|Hg〉 = A|Hg〉 − |lg〉 (2.38)


and (2.32) as〈Hg|f1〉 = 0. (2.39)

Note thatS(k|k0) = (AT (k|k0) − 1)Wtot(k0). (2.40)

2.3.2 Computation of the solution

One could think that by iteratively inserting the lhs of (2.38) in therhs the problem may be solved, i.e.

|Hg〉 = An|Hg〉 −n−1∑

k=0

Ak|lg〉. (2.41)

However, the way to solve (2.38) is somewhat more involved.Before going to the solution, an additional condition on A must be

discussed. The term A(k, b|k0, b0) is nothing but the probability for aparticle to end in (k0, b0) after one scattering event, having started in(k, b). Am(k, b|k0, b0) is then the probability to go to (k0, b0) after mscattering events. In the following it is assumed that there exists anM ∈ N such that AM (k, b|k0, b0) > 0, ∀(k, b), (k0, b0). It means thatit is possible, starting from any (k, b), to reach any (k0, b0) after Mscattering events. This condition is clearly related to the condition ofergodicity. Under this assumption, AM is a strong positive compactoperator.

Proposition 1 The 1st Krein-Rutman theorem assures the existenceand uniqueness of a stationary solution in thermodynamic equilibrium(∃!f ∈ C0(K)|Sf = 0).

Proof. First, the property r(AM ) = 1 must be proven.

By construction ||A|| := supx∈C0(K)\0||Ax||∞||x||∞

0 1, and A|f1〉 =

|f1〉. Therefore, r(A) 1 1, and because by definition r(A) 0 ||A||,the property M

√r(AM ) = r(A) = ||A|| = 1 is assured. The 1st

Krein-Rutman theorem assures that there is only one strict positivefunction u ∈ C0(K) such that AMu = r(AM )u = u. Of course

this function is nothing but f1. The operator (AT )M

has equiva-

lent properties: By construction (AT )M

is a strong positive compact


operator. Using again the Krein-Rutman theorem, a unique v can

be found with 0 < v ∈ C0(K) such that (AT )Mv = r((AT )

M)v.

Because u > 0 and v > 0, it is straightforward that 0 < 〈u|v〉 =

〈AMu|v〉 = 〈u|(AT )Mv〉 = r((AT )

M)〈u|v〉. Thus, r((AT )

M) = 1 and

(AT )Mv = v. Then, because (AT )

M+1v = AT v ⇐⇒ AT v = c ∗ v

with c real and cM = 1, AT v = v. Using (2.40) gives Sfeq = 0 withfeq := v

Wtot. feq is the only solution.

To solve (2.38), a sub-Banach-space Q ⊂ C0(K) must be con-structed, where the solution is unique.

Using two important properties of A:

1. A|f1〉 = |f1〉 by definition of Wtot,

2. AT |Wtotfeq〉 = |Wtotfeq〉,

because of (2.40) and S|feq〉 = 0, a projector PL can be constructed:

L :=|f1〉〈feqWtot|〈f1|feqWtot〉

, (2.42)

PL := 1− L, (2.43)

which has the following properties:

P 2L = PL, (2.44)

PL · A = A · PL = PL · A · PL. (2.45)

Thus, Q can naturally be defined as Q := PLx|x ∈ C0(K).BecauseWtotfeq and f1 are continuous functions, Q is a Banach space.

Proposition 2 A is a linear compact operator on Q.

Proof. Let be x ∈ Q. By definition ofQ and property (2.44), PLx = x.Therefore, using property (2.45), Ax = APLx = PLAx ∈ Q.

Note that |lg〉 ∈ Q, because

〈feqWtot|lg〉 = 0. (2.46)

By multiplying (2.38) with PL the problem can be reformulatedon Q:


PL|Hg〉 = PL · A · PL|Hg〉 − |lg〉 . (2.47)

Defining |H⊥g 〉 := PL|Hg〉 one obtains:

(1−A)|H⊥g 〉 = −|lg〉. (2.48)

Proposition 3 (1−AM ) is invertible on Q, and, therefore, (1−A)is also invertible on Q.

Proof. First, the property ||AM ||∞ < 1 on Q must be proven. Letx be in Q. x is a continuous function and 〈feqWtot|x〉 = 0 be-cause 〈feqWtot|PL = 0. By definition, feqWtot is a strict positivefunction, and ,therefore, x must have a positive part and a nega-tive part. Defining x+(k) := x(k) if x(k) 1 0, x+(k) = 0 else, andx−(k) := x(k) if x(k) < 0, x−(k) = 0 else, x(k) can be rewrittenas x(k) = x+(k) + x−(k), ∀k ∈ K. Without restriction on gen-erality, ||x||∞ = ||x+||∞. Then, because AM is strictly positive,|AMx| = | |AMx+| − |AMx−| | < |AMx+| 0 ||x+||∞. Therefore,||AM ||∞ < 1. It means that (1 − AM ) has an inverse on Q that can

be written as a Neumann series: (1−AM )−1 =∑∞i=0 A

Mi. Rewriting

(1−AM ) as (1−A)∑M−1

j=0 Aj and multiplying with (1−AM )−1 gives1 =∑∞

i=0 AMi

(1− A)∑M−1

j=0 Aj = (1− A)∑∞

i=0AMi∑M−1

j=0 Aj . So,

clearly (1−A)−1 =∑∞

i=0 AMi∑M−1

j=0 Aj =∑∞

i=0Ai.

Multiplying (2.48) with (1−A)−1 gives the solution:

|H⊥g 〉 = −

∞∑

i=0

Ai|lg〉. (2.49)

Thus, the problem (2.38) has a unique solution on Q, but infinitelymany of the form Hλ

g := H⊥g + λ|f1〉, λ ∈ R on C0(K). In (2.39)

the condition 〈Hg|f1〉 = 0 was introduced to ensure a unique solution.

The only Hλg fulfilling this condition is H

λgg with

λg := −〈f1|H⊥g 〉. (2.50)

Eq. (2.49) represents an iterative method to compute H⊥g , i.e. to

find an exact solution of (2.31), (2.32) for any g ∈ C0(K).


2.3.3 Connection between H and S−1

This section gives the connection between H and S−1, as well asbetween Hg and S−1

g . Letting the operator H act on both sides of(2.21) and using (2.31) gives

H∂t|f〉 +H~r ∇r|f〉 +H~k∇k|f〉 = |f〉 − 〈f |feq〉 |feq〉. (2.51)

One wants to dispose of the term 〈f |feq〉 and replace it by a termcontaining the density. By computing the 0-th moment of (2.51) oneobtains:

〈H1|∂t|f〉 + 〈H1|~r ∇r|f〉 + 〈H1|~k∇k|f〉 = n− 〈f |feq〉neq ⇔

〈f |feq〉 =n

neq− 1

neq〈H1|∂t|f〉 −

1

neq〈H1|~r∇r |f〉 −

1

neq〈H1|~k∇k|f〉,

(2.52)

whereH1(~k0, b0) :=∑b

∫Vb

H(~k, b|~k0, b0)d3k and neq :=

∑b

∫Vb

feq(~k, b)d3k.

Inserting (2.52) into (2.51) results in

(H − |feq〉〈H1|

neq

)∂t|f〉 +

(H − |feq〉〈H1|

neq

)~r ∇r|f〉

+

(H − |feq〉〈H1|

neq

)~k∇k|f〉 = |f〉 − |feq〉

n

neq. (2.53)

With the definition

S−1(~k, b|~k0, b0) := H(~k, b|~k0, b0) − feq(~k, b)H1(~k0, b0)/neq, (2.54)

(2.4) is recovered. Finally one obtains for the g-moment:

S−1g (~k0, b0) := Hg(~k0, b0) −

〈g〉eqH1(~k0, b0)

neq. (2.55)


Note that S−1g (~k0, b0) fulfils by definition the equations:

A · S−1g = S−1

g (~k, b) +

g(

~k, b) − 〈g〉eq

neq

Wtot(~k, b)

⇔S−1g = A · S−1

g (~k, b) − hg(~k, b)

(2.56)

S−1g |f1〉 = 0, (2.57)

where hg(~k, b) := (g(~k, b)− 〈g〉eq

neq)/Wtot(~k, b) (compare with (2.38) and

(2.39)).

2.4 Discretisation of a special class of SO

In the previous section it has been shown that under some importantconditions the BE has only one solution in thermodynamic equilib-rium, that under the same conditions any moment of the ISO exists,and that it can be formally written as (2.49).

To try to find a numerical approximation of H⊥g (see (2.49)) one

has two possibilities. The first one can be considered only if one knowsfeq a priory. By knowing feq the space Q is known, and, becausethe operator (1 − A) has a unique inverse on Q, one could discretiseA on Q using standard methods (see e.g. [1] chapter 3.5). If feqis unknown3 one first has to build a family of discrete collectivelycompact operators An, with limn→∞An → A and limn→∞ fn = feq.(Here fn must be the only vector fulfilling Anfn = fn. A sufficientcondition on An for fn to exist and to be unique is ∃Mn ∈ N |AMn

n > 0i.e. a strict positive matrix.)

In the following the second possibility will be considered in the casewhere the SO only depends on band index, valley index, and energy:w(~k, b|~k0, b0) =: w(ε(~k), b|ε(~k0), b0), where b is the band-valley index

and ε(~k) the energy.

3For some models used in TCAD, the Fermi-Dirac distribution is not the equi-librium distribution. One example is given in Chapter 9.

2.4. DISCRETISATION OF A SPECIAL CLASS OF SO 23

2.4.1 Adapted reformulation of the equations

To take full advantage of the symmetries of w(ε(~k), b|ε(~k0), b0), (2.36)

can be integrated on the iso-energy surface ε(~k0, b0) of the valley b0:

Hg(ε0, b0) :=

∫

Vb0

Hg(~k0, b0)δ(ε(~k0, b0) − ε0)d3k0 =

∫

Vb0

∑

b2

∫

Vb2

Hg(~k2, b2)w(~k0, b0|~k2, b2)

Wtot(~k0, b0)d3k2δ(ε(~k0, b0) − ε0)d

3k0

−∫

Vb0

(g(~k0, b0) − 〈g〉eq feq(~k0, b0)

Wtot(~k0, b0)

)δ(ε(~k0, b0) − ε0)d

3k0

= Z(ε0, b0)∑

b2

εmax(b2)∫

εmin(b2)

Hg(ε2, b2)w(ε0, b0|ε2, b2)Wtot(ε0, b0)

dε2

−g(ε0, b0) − 〈g〉eq Z(ε0, b0)feq(ε0, b0)

Wtot(ε0, b0),

(2.58)

where Z(ε0, b0) :=∫Vb0

δ(ε(~k0, b0) − ε0)d3k0, is the density of states in

band-valley b0, εmin(b0) (εmax(b0)) is the energy minimum (maximum)

in band-valley b0, and g(ε0, b0) :=∫Vb0

g(~k0, b0)δ(ε(~k0, b0) − ε0)d3k0.

Dividing (2.58) by Z(ε0, b0) and defining:

g := (Pǫg)(ε0, b0) := g/Z(ε0, b0),

B(ε0, b0|ε2, b2) :=w(ε0, b0|ε2, b2)Z(ε2, b2)

Wtot(ε0, b0)

= A(k0, b0|k2, b2)Z(ε2(k2), b2),

BT (ε0, b0|ε2, b2) := B(ε2, b2|ε0, b0),

lg(ε0, b0) :=

(g(ε0, b0) − 〈g〉eq feq(ε0, b0)

Wtot(ε0, b0)

), (2.59)


and Z(ε0, b0)Hg(ε0, b0) := Hg(ε0, b0), one obtains

Hg = B Hg − lg (2.60)

and condition (2.39) (see (2.50)) can be rewritten as

λg = − 1√∑b′

|Vb′ |

∑

b

εmax(b)∫

εmin(b)

Z(ε, b)H⊥g (ε, b)dε. (2.61)

Note that B C is now defined as

∑

b

εmax(b)∫

εmin(b)

B(ε1, b1|ε, b)C(ε, b|ε0, b0)dε (2.62)

for operators and B g as

∑

b

εmax(b)∫

εmin(b)

B(ε1, b1|ε, b)g(ε, b)dε (2.63)

for functions. It follows B |f1〉 = |f1〉, and BT |WtotfeqZ〉 =|WtotfeqZ〉.

By multiplying (2.49) with Pǫ the formal solution for H⊥g is found:

|H⊥g 〉 = −

∞∑

i=0

Bi |lg〉, (2.64)

where Bi := B ... B︸︷︷︸i times

.


By knowing Hg and Hg, Hg can be directly computed using (2.36):

Hg(~k0, b0) =

∑

b2

εmax(b2)∫

εmin(b2)

Hg(ε2, b2)w(ε0, b0|ε2, b2)Wtot(ε0, b0)

dε2 −g(~k0, b0) − 〈g〉eq feq(ε0, b0)

Wtot(ε0, b0)

= − g(~k0, b0)

Wtot(ε0, b0)+〈g〉eqfeq(ε0, b0)Wtot(ε0, b0)

+∑

b

εmax(b)∫

εmin(b)

w(ε0, b0|ε, b)Wtot(ε0, b0)

Hg(ε, b)dε

= λgf1 −g(~k0, b0)

Wtot(ε0, b0)+

〈g〉eq feq(ε0, b0)Wtot(ε0, b0)

+∑

b

εmax(b)∫

εmin(b)

w(ε0, b0|ε, b)Wtot(ε0, b0)

H⊥g (ε, b)dε

= λgf1 −g(~k0, b0)

Wtot(ε0, b0)+

〈g〉eq feq(ε0, b0)Wtot(ε0, b0)

+ B H⊥g

= λgf1 −g(~k0, b0) − g(ε0, b0)

Wtot(ε0, b0)+ H⊥

g = −g(~k0, b0) − g(ε0, b0)

Wtot(ε0, b0)+ Hg.

(2.65)

Note that in (2.65) only one term is fully ~k-dependent, namely

− g(~k, b)

Wtot(ε(~k), b). (2.66)

All other terms, like the transition probability, depend only on valleyindex, band index, and energy.

By inserting (2.65) in (2.55), and using the property H1 = H1,

one finds an analogous expression for S−1g (~k0, b0):

S−1g (~k0, b0) = −g(

~k0, b0) − g(ε0, b0)

Wtot(ε0, b0)+ S−1

g . (2.67)

Remark: If one restricts (2.67) to the case of a parabolic band struc-


ture and to g = vx, one finds

S−1vx

(~k0, b0) = − vx(~k0, b0)

Wtot(ε0, b0), (2.68)

because vx(ε0, b0) = 0. Eq. (2.68) is the well-known case where thescalar relaxation time can be exactly defined as (see e.g. [6])

τ(ε0, b0) =1

Wtot(ε0, b0).

2.4.2 Discretisation scheme

To find a numerical approximation to (2.64), B and lg have to bediscretised.

The space

C0(Vb) := Pǫg | g ∈ C0(Vb)

of the functions which only depend on the energy for a given band-valley b is the space that has to be discretised to find a numericalsolution to the special case of SO presented in this chapter. Thediscrete version of C0(Vb) will be called D(K). D(K) can be definedby:

D(K) :=⊕

b

C(b)nb, (2.69)

where

C(b)nb

:= f ∈ C0(Vb)|f |[ǫ(b)i ,ǫ(b)i+1]

is linear, ∀i = 0, .., n and ǫ(b)i ∈ I(b)

nb

(2.70)

is a subspace of continuous piecewise linear functions of C0(Vb) and

I(b)nb

:= ǫ(b)0 , ǫ(b)1 , .., ǫ

(b)nb−1, ǫ

(b)nb

(2.71)

is the ensemble of the energy points used for the discretisation, where

ǫ(b)min =: ǫ

(b)0 < ǫ

(b)1 < ... < ǫ

(b)nb−1 < ǫ(b)nb

:= ǫ(b)max,


ǫ(b)min (resp. ǫ

(b)max) is the energy minimum (resp. maximum) in Vb, and

nb(< ∞) is the number of discretisation points. From now on D(K)will be the space of interest.A projector PD from C0(K) to D(K) can be defined as

∀f ∈ C0(K), (PDf |C(b)nb

)(ǫ) :=

f(ǫ(b)i ) +

(ǫ− ǫ(b)i )

ǫ(b)i+1 − ǫ

(b)i

(f(ǫ(b)i+1) − f(ǫ

(b)i )), ǫ ∈ [ǫ

(b)i , ǫ

(b)i+1]. (2.72)

By construction PDPD = PD. Using PD, B can be discretised as

B := PDBPD (2.73)

and (2.60) as

PDHg = (PDB PD)PDHg − PD lg. (2.74)

Using the notation g := PD g, ∀g ∈ C0(K), (2.74) can be rewritten as

Hg = B Hg − lg. (2.75)

B := PDB PD is by definition a compact linear operator on thefinite-dimensional Banach space D(K). An important property of Bis that Bf1 = f1, because PDf1 = f1, and B f1 = f1.

Assuming that b = 1, .., N , where N is the number of valleys times

the number of bands, the dimension of C(b)nb is nb by construction,

and, therefore, dim(D(K)) =∑Nb=1 nb := d. Thus, B can be repre-

sented by a d × d matrix. In the following B will be used to denotea matrix representation of the operator B, f to denote the vectorassociated with the function f , and f to denote the piecewise linearfunction associated with the vector f (as illustrated in Fig. 2.1). Therepresentation is chosen such that the (b ∗ i)-th component of f isf(b, ǫbi).

The interesting property of B is by construction Bf1 = f1, wheref1 is a vector with identical components. It follows the importantproperty

d∑

j=1

(B)ij = 1, ∀i = 1, .., d. (2.76)


4(b)ε)(f 3

(b)ε)(f 2

(b)ε)(f 1

(b)ε

ff

=f~

)(f 5(b)ε)(

)

(b)ε1(b)ε0

(b)ε

f~

2

(f 0(b)ε

ε5(b)ε4

(b)ε3(b)ε

Figure 2.1: Difference between f , f and f .

Using the definition supp(g) := x ∈ K | g(x) 6= 0, ∀g ∈ C0(K), thefollowing proposition can be proven:

Proposition 4 If supp(g) ⊆ supp(f), then supp(B g) ⊆ supp(B f), ∀f ≥ 0, g ≥ 0 ∈ C0(K).

To proceed, one has to assume that I(b)nb can be chosen in such a

way that ∀g ≥ 0 ∈ D(K):

supp(B g) ⊆ supp(Bg). (2.77)

For each case it will be shown that this property can be fulfilled.Under this assumption B is a matrix with the following properties:

1. B > 0, i.e., B is a nonnegative matrix,

2. ∃P ∈ N | BP> 0.

The first property of B comes from the definition of B. The secondproperty comes from AM > 0 and (2.77).


Proposition 5 ∃P ∈ N | BP> 0.

Proof. First, AM · g = A · ... ·A︸︷︷︸M times

·g = B ... B︸︷︷︸M times

g = BM g, ∀g ∈

C0(K) ⇒ supp (AM · g) ≡ supp (BM g), ∀g ∈ C0(K). Then, be-cause AM > 0, supp (AM · g) ≡ K, ∀ 0 ≤ g ∈ C0(K) \ 0. Using(2.77) one can write K ≡ supp (BM g) ⊆ supp (BM g). Therefore,

BM> 0.

Note here that, because of Corollary 8.5.9 in [28] p. 520, P ≤min d2 − 2 ∗ d− 2,M. Matrices with properties 1) and 2) are calledprimitive.

So far a discrete form of B has been built, which takes the from

of a primitive matrix B. BM

is a compact strict positive operatorand, therefore, fulfils the hypothesis of the Krein-Rutmann theorem(in the matrix case the theorem is usually called Perron-Frobenius).

A brief repetition of the properties of a primitive matrix from thePerron-Frobenius Theorem (see e.g. [28]):

1 Theorem (Perron-Frobenius)Let B be a primitive matrix. Then

a) ρ(B) > 0.

b) ρ(B) is an eigenvalue of B.

c) There is a positive vector x such that Bx = ρ(B)x, ||x|| = 1.

d) There is a positive vector y such that BTy = ρ(B)y, ||y|| = 1.

e) ρ(B) is an algebraically simple eigenvalue of B (i.e. not degen-erated).

f) If there is a nonnegative vector x such that Bx = λx, thenλ = ρ(B).

g) limm→∞

(ρ(B)−1B)m = L, where L := xyT

yT x= |x〉〈y|

〈y|x〉 .


Remember that from the construction of B, Bf1 = f1 (becausePDf1 = f1, and Af1 = f1). Thus, f1

||f1||corresponds to the vector x

in the theorem and ρ(B) = 1.

The function space and the discrete space have some importantdifferences and similarities. First, B is a continuous operator and fa function, whereas B is a matrix and f is a vector, which (b*j)-th

component is by definition the value of the function f in ǫ(b)j . Secondly,

(Bg)(ǫ(b)j , b) = (Bf)(b∗j), ∀g ∈ D(K). Finally, two scalar products

must be distinguished:

1.⟨g|h⟩

:=∑b

∫Vb

(gh)d3k (scalar product of functions),

2.⟨g|h⟩

:= gT h (scalar product of ordinary vectors).

2.4.3 Solution algorithm for the discrete problem

Eq. (2.75) can be solved in three steps. First, the functions x and y oftheorem 1 must be computed using the so-called ”power method” (seee.g. [28] p. 523). Then a subspace of Q ⊂ D(K) must be computed,on which an approximate solution to (2.75) is unique (same idea asin Section 2.3). Finally the algorithm for the discrete case can berewritten.

From the knowledge of B, one expects the following from B:

Bx = x ⇔ x =f1∥∥∥f1

∥∥∥,

BTy = y ⇒ y ∼=

˜WtotfeqZ

|| ˜WtotfeqZ||.

Theorem 1 g) has two important consequences. First, for a given ε > 0

∃M |∀m > M, 0 6

∣∣∣∣∥∥∥∥

xm

ρ(B)m

∥∥∥∥− ‖Lx0‖∣∣∣∣ 6

∥∥∥∥xm

ρ(B)m− Lx0

∥∥∥∥ < ε.


Second,

‖Lx0‖ − ε 6

∥∥∥∥xm

ρ(B)m

∥∥∥∥ < ‖Lx0‖ + ε⇔

ln(ρ(B))+ln(‖Lx0‖ − ε)

m6

ln(‖xm‖)m

6 ln(ρ(B))+ln(‖Lx0‖ + ε)

m,

(2.78)

where

xm := Bm

f1∥∥∥f1

∥∥∥

,

ym := (BT)m(

˜WtotfeqZ

|| ˜WtotfeqZ||).

Eq. (2.78) is an iterative method to compute ln(ρ(B)) by computing

limm→∞

ln(‖xm‖)m . Because ρ(B) = 1 by construction, this method allows

to test the correctness of the implementation.

If, for a given m, ln(‖Lx0‖+ε)m is small enough, ρ(B) can be com-

puted and, by dividing xm by ρ(B)m, a vector proportional to xwithin the same precision is found:

xm

ρ(B)m= Lx0 +O(ε(m)) =

x(yTx0)

〈y|x〉 +O(ε(m)). (2.79)

Note that ‖Lx0‖ ∼= 1 is expected from the wise choice of x0.To write the algorithm in a simple way, the following definitions

are needed:

x :=xm

||xm|| ,

y :=ym

||ym|| ,

L :=xyT

〈y|x〉 ,

PL := 1 − L. (2.80)


Note that Bx = x︸︷︷︸within machine precision

and BTy = y.︸︷︷︸

within machine precision

Eq. (2.75) can be rewritten in the matrix-vector form:

Hg = BHg − lg . (2.81)

Multiplying (2.81) by y yields the condition

〈y|lg〉 != 0 (⇔ PLlg = lg). (2.82)

Condition (2.82) will usually not be fulfilled exactly, because the twoprojectors PL and PD do not commutate. Therefore, (2.81) has to beredefined as

Hg = BHg − lg , (2.83)

where lg := PL lg 6= lg (compare with the continuous case (2.46),where this was fulfilled).

As long as 〈y|lg〉 is much smaller than 1, lg remains a good approx-

imation of lg. In analogy to Section 2.3, a subset Q can be defined:

Q := PLg | g ∈ D(K). (2.84)

On Q, (2.83) has the unique solution:

H⊥g := −

∞∑

i=0

Bilg. (2.85)

Consequently, the algorithm to compute H⊥g looks as follows:

Algorithm:

1. Compute x, y, and PL

2. Compute lg

3. Compute s0 = x0 = PLlg

4. Compute xn := PLBxn−1

5. Compute sn := sn−1 + xn

6. Repeat step 4. and 5. until || |x0| − |xn| ||∞ < tol.

2.5. GENERALISATION 33

7. End ( sn ∼= H⊥g ).

The best numerical approximation to the solution of the problem(2.60), (2.39) can be defined as

|Hg〉 := |H⊥g 〉 − 〈H⊥

g |f1〉|f1〉, ∀g ∈ D(K) (2.86)

and the best numerical approximation to the solution of the problem(2.56), (2.57) can be defined as (compare with the exact expression(2.67))

|S−1g 〉

num:= |Hg〉 −

〈g〉eqneq

|H1〉 −|g〉 − |g〉Wtot

=: −|g〉 − |g〉Wtot

+ |S−1g 〉, ∀g ∈ D(K). (2.87)

In Section 2.6 the error arising from the discretisation scheme pre-sented here, will be discussed.

2.5 Generalisation

The special ISOs studied in the last two sections do not representa particular class of operators, because, any scattering operator canbe reduced to a form compatible with the hypothesis given at thebeginning of Section 2.4.

Imagine that one separates (discretises) the compact space K in afinite number of continuous, and compact subspaces Ki such that

Ki

⋂

Kj= ∅, i 6= j, (2.88)

⋃

i

Ki = K, (2.89)

as illustrated in Fig. 2.2. By redefining Vbi:= Ki, and approximating

the transition rate w(k|k′) by

wnew(k, i|k′, j) :=∫Ki

∫Kjw(k0|k1)δ(ε(k0) − ε(k))δ(ε(k1) − ε(k′))d3k0d

3k1∫Ki

∫Kjδ(ε(k0) − ε(k))δ(ε(k1) − ε(k′))d3k0d3k1

,

∀k ∈ Ki, k′ ∈ Kj, (2.90)


a new operator Anew can be defined:

Anew(k, i|k′, j) :=wnew(k, i|k′, j)Wnewtot (k, i)

, (2.91)

which has the following properties:

1. (Anew)M > 0.

2. Anew |1〉 = |1〉.

3. ATnew |Wtotfeq〉 = |Wtotfeq〉.

Property 1 is a consequence AM > 0 and because, by definition, forall positive functions g

supp (A · g) ⊆ supp (Anew · g). (2.92)

The second property is trivial and defines Wnewtot (k, i), and the third

is fulfilled by definition in the case where feq is a function of energyonly. With those properties one can again use the theory developedin Section 2.4, if A is replaced by Anew .

2.6 Error computation

The aim of this section is to answer the following question: How largeis the error contained in the numerical solution (2.86) and how doesit influence expectation values?

The error in the numerical solution can be formally defined asfollows:

|eg〉 := Wtot(A · |S−1g 〉

num− |S−1

g 〉num

− |hg〉) (2.93)

(see (2.56) for the definition of hg).If |S−1

g 〉num

is replaced by |S−1g 〉 in (2.93), then eg = 0. Therefore,

Eq. (2.93) is the deviation from the exact equation (2.56) multipliedby Wtot.Note that by definition

〈eg〉eq := 〈feq|eg〉 = 0. (2.94)

2.6. ERROR COMPUTATION 35

Figure 2.2: Possible discretisation of the first Brillouin zone.

If g is replaced by g := g + eg, one can show that |S−1g 〉num exactly

solves the equation

|S−1g 〉

num= A · |S−1

g 〉num

− |hg〉. (2.95)

This is equivalent to

|S−1g 〉

num= |S−1

g 〉. (2.96)

To better understand the physical meaning of (2.96), (2.6) can berewritten as

〈S−1g |∂tf〉 + 〈S−1

g |~v∇rf〉 + 〈S−1g | q

~~E∇kf〉 = 〈g〉 − n

neq〈g〉eq. (2.97)

(Remember that 〈g〉 := 〈g|f〉, where f is the solution of the BE and〈g〉eq := 〈g|feq〉.)


The following equation is fulfilled by |S−1g 〉

num:

num〈S−1g |∂tf〉 +

num〈S−1g |~v∇rf〉 +

num〈S−1g | q

~~E∇kf〉 =

〈g〉 − n

neq〈g〉eq + 〈eg〉.

(2.98)

Thus, |S−1g 〉

numfulfils (2.97) up to 〈eg〉. Therefore, computing trans-

port coefficients with |S−1g 〉

numas described in Section 2.2 and com-

puting the expectation value of g with these parameters, the relativeerror on 〈g〉 will be ∣∣∣∣

〈eg〉〈g〉

∣∣∣∣ . (2.99)

If noise sources are computed with |S−1g 〉

numas described in Section

3.6, the error in the correlation function of g will be

Errgg :=

∣∣∣∣∣2(〈egg〉 − 〈eg〉〈g〉) + (〈e2g〉 − 〈eg〉2)

(〈g2〉 − 〈g〉2)

∣∣∣∣∣ . (2.100)

Because g is known, |S−1g 〉

numcan be computed as well as hg. Thus,

eg is also computable. Therefore, during a MC simulation, one candirectly compute the relative errors (2.99), and (2.100).

If those errors are always small (independently of f), then |S−1g 〉

numwill give transport coefficients (resp. noise sources) which will alwaysdescribe 〈g〉 (resp. 〈Kgg〉) accurately. In this sense, |S−1

g 〉num

is a goodapproximation of |S−1

g 〉.Conditions can be given, under which eg is small in comparison

with g. eg can be separated in two parts e(1)g , e

(2)g (such that eg =

e(1)g + e

(2)g ):

e(1)g := Wtot(A · |S−1g 〉

num− |Hg〉 +

〈g〉eqneq

|H1〉 − |hg〉), (2.101)

e(2)g := Wtot(hg +g − g

Wtot− hg) = Wtot(hg − hg), (2.102)

where hg := PLhg , hg = PDhg (compare with (2.83)).

2.7. FREQUENCY-DEPENDENT SO 37

By construction of the discretisation, e(1)g is a function which is

zero in each point ǫ(b)i of the discretisation. Thus, e

(1)g is expected to

be always small.

For e(2)g to be small two conditions are needed. First hg − PDhg,

the error on hg produced by the discretisation must be small (see

Fig. 2.1). Secondly, hg must be almost perpendicular to y (The better

y is parallel to WtotfeqZ, the better hg will fulfil this condition).In this section an explicit computation of the error eg has been de-

rived, and its propagation was shown to be fully understood. There-fore, the accuracy of the discretisation can always be controlled for agiven function g.

2.7 Frequency-dependent scattering oper-ator

The starting point of this section is the Boltzmann-Langevin equation(3.43), where δs is the Langevin noise source. The electric field in ~Fand the scattering rate S are considered not to depend on time. (If~F and S only slightly vary in time around a stationary state, one canlinearise the BLE in ~F and S, and use the same formalism.)

To find a frequency-dependent scattering operator one defines

δf(ω) :=

∞∫

−∞

(f(t) − f)e−iωtdt (2.103)

as the Fourier transform of the deviation from the stationary state f ,and rearranges the terms of the BLE as follows:

~v ∇rδf(ω) +~F∇kδf(ω) = Sδf(ω) + iωδf(ω) + δs(ω) =:

T (ω)δf(ω) + δs(ω), (2.104)

where δs(ω) is the Fourier transform of δs(t), and

T (ω)(~k, b|~k0, b0) := S(~k, b|~k0, b0) + iωδ3(~k − ~k0)δb,b0 . (2.105)

By multiplying (2.104) with T−1(ω) and building a moment with afunction g, the Fourier transform of the fluctuations 〈g〉 − 〈g〉 =: δ 〈g〉


can be expressed as a function of the noise source:

〈g|T−1~v ∇rδf(ω) + 〈g|T−1~F∇kδf(ω) = δ 〈g〉 (ω) + 〈g|T−1δs(ω).

This is a generalisation of Eq. (2.86) in Ref. [32] to any BLE (norestrictions on the scattering operator S) and to arbitrary g (not onlythe moment of the velocity).

2.7.1 Existence of T−1 and formal solution

Before going to the discretisation of T−1, a proof is needed that T isalways invertible.

The Banach space C0(K,C) with the norm

||.||∞ : C0(K,C) → Rg 7→ sup

x∈K|g(x)|, (2.106)

has been chosen to work with the complex operator T (ω).The operator A defined in (2.37) can be generalised as

A(ω)(~k, b| ~k0, b0) :=w(~k, b| ~k0, b0)

Wtot(~k, b) + iω. (2.107)

Obviously, A(0) is nothing but the A defined in (2.37) and

T (ω) = (A(ω)† − 1)(Wtot − iω), (2.108)

A(ω) =Wtot

Wtot + iωA(0) := U(ω)A(0). (2.109)

Proposition 6 |A(0) · g|2(k) ≤ (A(0) · (gg∗))(k), ∀k ∈ K, ∀g ∈C0(K,C).

Proof.

|A(0) · g|2(k) = |∫

K

A(0)(k|k0)g(k0)d3k0|2

= |∫

K

√A(0)(k|k0)

√A(0)(k|k0)g(k0)d

3k0|2


Using the Schwarz inequality:

|∫

K

√A(0)(k|k0)

√A(0)(k|k0)g(k0)d

3k0|2

≤ (

∫

K

A(0)(k|k0)d3k0)(

∫

K

A(0)(k|k0)g(k0)g∗(k0)d

3k0).

Since∫K A(0)(k|k0)d

3k0 = 1 by definition of A(0), the proof is com-plete.

Proposition 7 ∀ω 6= 0, (1−A(ω))−1 exists (⇔ ρ(A(ω)) < 1).

Proof. The proposition is true iff ||A(ω)|| < 1, ∀ω 6= 0. |A(ω) · g|2 =|U(ω)A(0) · g|2 = |U(ω)|2|A(0) · g|2 ≤ |U(ω)|2(A(0) · (gg∗)). FromSection 2.3, it is clear that ||A(0)|| = 1, and by definition |U(ω)|2 < 1,∀ω 6= 0. Thus, |A(ω) · g|2 ≤ |U(ω)|2(A(0) · (gg∗)) < ||g||2∞. Therefore,||A(ω)|| < 1, ∀ω 6= 0, and the Neumann series

∑∞i=0 A(ω)i is the

inverse of (1−A(ω)).

This leads to the conclusion that T−1(ω) exists and is given by−1

Wtot−iω

∑∞i=0(A(ω)†)i.

In the following, a way to compute arbitrary moments of T−1(ω)for the special class of SO of Section 2.4 will be explained, and theresults of Section 2.4 will be recovered in the limit ω → 0.

2.7.2 Numerical approximation of moments of T−1

The goal of this section is to find an approximation to T−1g (ω) for a

given g.By definition

〈g| · T−1 · T = 〈T−1g | · T = 〈g| ⇔ T † · |T−1

g 〉 = |g〉. (2.110)

Remember that in the Dirac notation

〈y(ω)| := y(ω)† = (y(ω)⊤)∗. (2.111)

Using (2.108), (2.110) can be rewritten as

(A(ω) − 1) · |T−1g 〉 =

|g〉Wtot + iω

. (2.112)


Multiplying (2.112) with Pǫ (see. (2.59)) gives

(B(ω) − 1) |T−1g 〉 =

|g〉Wtot + iω

, (2.113)

where B(ω)(k|k0) := A(ω)(k|k0)Z(k0) (compare with (2.59)).Eq. (2.113) can be discretised in the same way as (2.60) (see

(2.74)):

PD(B(ω) − 1) PD|T−1g 〉 = PD

|g〉Wtot + iω

, (2.114)

whereD now represents the discrete spaceD(K,C) which is the canon-ical extension of D(K) to complex functions (see (2.70), and (2.69)).

Using again the definitions g := PDg, B(ω) := PDB(ω)PD, andrearranging the terms, (2.114) can be rewritten as (compare with(2.75))

|T−1g 〉 = B(ω) |T−1

g 〉− |g〉Wtot + iω

=: B(ω) |T−1g 〉−|zg(ω)〉. (2.115)

By definition B(0) is the operator B from (2.59), B(0) the operatorB from (2.75), and B(0) the matrix B from (2.76).

Proposition 8 |(B(0)g)i|2 ≤ (∑j

(B(0))ij(g)j(g∗)j), ∀i = 1, .., d, ∀g ∈Rd.

Proof. As in proposition 6, using the property (2.76).

Proposition 9 ∀ω 6= 0, (1− B(ω))−1 exists (⇔ ρ(B(ω)) < 1).

Proof. As in proposition 7 replacing U(ω) by the representation U(ω)of U(ω) := PDU(ω)PD (U(ω) is by construction a diagonal matrix).

Using proposition 9 the solution to (2.115) can be directly writtenas

T −1

g (ω) = −∞∑

i=0

B(ω)izg(ω). (2.116)

Thus, an iterative method is again found to approximate T −1

g (ω)as precisely as needed (compare with (2.85)).


2.7.3 Link between T −1

g(ω) and S−1

g

Because (1 − B(0)) is a singular operator (see Section 2.4, theorem 1),the function T −1

g will diverge when ω → 0. It is, therefore, important

to investigate the behaviour of T −1

g (ω) for ω → 0 (Remember thatTg(0) = Sg (2.105)).

The main idea of what follows is to unequivocally separate T −1

g (ω)in the sum of a divergent part and a convergent part.

First note, that, because B(ω) depends continuously on ω (B(0) =B), and because of theorem 1, Section 2.4, point e), one can find somesmall ε such that B(ω) has only one eigenvector with the maximumeigenvalue for ω ∈ [0, ε]. Let |x(ω)〉 be this eigenvector (normed to 1),α(ω) be the associated eigenvalue, and |y(ω)〉 the eigenvector (normedto 1) of B†(ω) with eigenvalue α∗(ω).

Since B(ω) depends in a continuous way on ω, so do x(ω), y(ω),and α(ω). Therefore, x(0) = x, y(0) = y, and α(0) = 1 (x, y asdefined in (2.80)).

The operator L from (2.80) can be generalised by

L(ω) :=|x(ω)〉〈y(ω)|〈y(ω)|x(ω)〉 . (2.117)

By definition L(0) is exactly the L from (2.80), and, therefore, alsothe L from Theorem 1, Section 2.4. Note that because 〈y(0)|x(0)〉 >0, one can always find a neighbourhood of 0 such that for all ω in thisneighbourhood |〈y(ω)|x(ω)〉| 6= 0.

In analogy to (2.80) a projector PL(ω) can be defined:

PL(ω) := 1− L(ω). (2.118)

By definition PL(0) is exactly the PL of (2.80).The complex matrix PL(ω) has again the nice properties (compare

with (2.44) and (2.45))

PL(ω)PL(ω) = PL(ω), (2.119)

PL(ω)B(ω) = B(ω)PL(ω) = PL(ω)B(ω)PL(ω). (2.120)

Property (2.119) comes directly from the definition of PL(ω), andproperty (2.120) can be derived using

L(ω)B(ω) = B(ω)L(ω) = α(ω)L(ω). (2.121)


In analogy to (2.84) a Banach-space Q(ω) can be defined

Q(ω) := PL(ω)g | g ∈ D(K,C). (2.122)

Eq. (2.115) can now be solved once on Q(ω) and once on the space

Q⊥(ω) : λ|x(ω)〉 | λ ∈ C (2.123)

perpendicular to Q(ω).Multiplying (2.115) with PL(ω) and using (2.120) gives

PL(ω)|T −1

g 〉 = PL(ω)B(ω)PL(ω)|T −1

g 〉 − PL(ω)|zg(ω)〉. (2.124)

Multiplying (2.115) with L(ω) and using (2.121) gives

L(ω)|T −1

g 〉 = α(ω)L(ω)|T −1

g 〉 − L(ω)|zg(ω)〉. (2.125)

Eq. (2.124) is an equation well defined on Q(ω), and (2.125) is welldefined on Q⊥(ω).

The solutions to Eq. (2.124) resp. Eq. (2.125) can be written asfunction of T −1

g 〉:

|T −1

g 〉a := PL(ω)|T −1

g 〉, (2.126)

|T −1

g 〉b := L(ω)|T −1

g 〉 = |x(ω)〉〈y(ω)|zg(ω)〉〈y(ω)|x(ω)〉 . (2.127)

By construction |T −1

g 〉 = |T −1

g 〉a + |T −1

g 〉b.The solution to (2.125) is (see Appendix A.2.3)

|T −1

g 〉b := − 1

1 − α(ω)L(ω)|zg(ω)〉. (2.128)

As one can directly see, this solution diverges for ω → 0 (Rememberthat α(0) = 1).The solution to (2.124) is naturally (see (2.116))

|T −1

g 〉a = −PL(ω)

∞∑

i=0

B(ω)iPL(ω)|zg(ω)〉. (2.129)

In Appendix A.2.1 it is shown that (2.129) is well defined, even ifω = 0.


In Appendix A.2.2 it is shown that the development of (2.128) ina Laurent-series leads to

− 1

1 − α(ω)L(ω)|zg(ω)〉 =

L(0)|zg(0)〉ωα(0)′

+1

α(0)′∂

∂ω[L(ω)|zg(ω)〉]ω=0 +O(ω), (2.130)

where α(0)′ := ∂α∂ω

∣∣ω=0

.

It follows that |T −1

g 〉b diverges likeL(0)|zg(0)〉

ωα(0)′for ω → 0, and that

the convergent part of − 11−α(ω)L(ω)|zg(ω)〉 is nothing but

1

α(0)′

∂

∂ω[L(ω)|zg(ω)〉]ω=0 . (2.131)

To summarise, the divergent part of |T −1

g 〉 behaves like

L(0)|zg(0)〉ωα(0)′

= |x(0)〉〈y(0)|zg(0)〉〈y(0)|x(0)〉ωα(0)′

(2.132)

for ω → 0 and the convergent part

|T −1∗g 〉 := lim

ω→0

[|T −1

g 〉 − L(0)|zg(0)〉ωα(0)

′

]

is given by

limω→0

|T −1

g (ω)〉a+ limω→0

1

α(ω)′

∂

∂ω[L(ω)|zg(ω)〉]ω=0

︸︷︷︸:=|u〉

= |T −1

g (0)〉a+|u〉.

(2.133)

To understand the conditions the discretisation must fulfil, thefollowing relation, derived in Appendix A.2.4, is useful:

B(0)|T −1∗g 〉 = |T −1∗

g 〉+

1− D

−1 |x(0)〉〈y(0)|⟨y(0)|D−1|x(0)

⟩

D

−1|g〉,

(2.134)

2.8. BOUNDARY TERMS AND FURTHER APPLICATIONS 45

2. For small ω, the quantities x(ω), y(ω), and α(ω) can be com-puted by the power method (see e.g. [7] p. 743).

3. The error computation can be done exactly as described in Sec-tion 2.6 defining eg(ω) := (Wtot− iω)(A(ω) · T−1

g − T−1g −zg(ω)).

2.7.5 Summary

In this section, it has been shown how to compute arbitrary momentsof the operator T−1, how the solutions behave for ω → 0, and how toretrieve an approximation to |S−1

g 〉, like that already found in Section2.4, in the limit ω = 0.

Two sufficient conditions have been found for the discretisation togive an exact solution to |S−1

g 〉 and |T−1g 〉:

1) g!= g,

2) |x(0)〉〈y(0)|D−1

|g〉Dy(0)|D

−1|x(0)

E !=

〈g〉eq

neq.

2.8 Boundary terms and further applica-

tions

2.8.1 Boundary terms

Because the transition rate is discontinuous at the interface betweentwo valleys, the odd moments of the ISO are discontinued and theeven moments are not differentiable there. Hence, for odd momentsthese interfaces are part of the boundary terms in (2.8), and eachtime a MC particle goes through one of these boundaries, it generatesa term which is not taken into account in transport models.

By knowing S−1g , one can compute the contribution of this term,

and assess its importance for each moment.

2.8.2 Domain of applicability

The formalism developed in Sections 2.4–2.7 cannot only be used tostudy interesting systems like strained semiconductors, where the SOis fully dependent on the band-valley index, but also to study electron-hole systems. To do so, the distribution function fh of the holes in


the valence bands has to be formally replaced by fe := 1− fh, i.e. thedistribution function for the electrons in the valence bands.

2.9 Summary

A formalism based on exact moments S−1g of the inverse scattering op-

erator of the Boltzmann equation has been described. This formalismis free of any relaxation time approximation. A sufficient condition forthe existence of the S−1

g s has been presented, and an explicit discreti-sation scheme for a special class of SOs has been given. The formalismwas then extended to frequency-dependent scattering operators. Thecomputation of the corresponding moments T−1

g has been fully de-scribed. Both types of moments are shown to be related to each otherin the limit of vanishing frequency.

The knowledge of the S−1g enables the exact computation of trans-

port coefficients. In Chapter 3 the usability of the moments of the ISOwill be extended to the computation of correlation functions and noisesources.

Chapter 3

Applications

3.1 Introduction

The knowledge of the moments of the ISO and of the solution f ofthe BE is necessary and sufficient to compute all transport coeffi-cients next to and far from TD equilibrium. In this chapter the mostimportant applications are given.

3.2 Low-field solution to the Boltzmann

equation

The space-homogeneous, stationary Boltzmann equation

− q~~E∇k|f〉 − q(~v ∧ ~B)∇k|f〉 = S|f〉 (3.1)

can be solved for small electric and magnetic fields in any order.In this section only the first order will be derived. The full deriva-

tion for all orders for Fermi and Boltzmann statistics is given in Ap-pendix B.

To proceed, the usual ansatz

f(k) = feq(ε(k))+ q∂feq∂ε

(ε(k)) ~E~Λa(k)+ q∂feq∂ε

(ε(k)) ~B∧ ~Λb(k) (3.2)

47

48 CHAPTER 3. APPLICATIONS

is used. Inserting (3.2) in (3.1) and considering only the first orderterms in the magnetic and electric fields, leads to

~Λb = 0, (3.3)

because (~v ∧ ~B)~v = 0, and

〈(1 − feq)(~v)i| = −〈(~Λa)i)|S, (3.4)

(see Appendix B, Eq. (B.8)). It is important that (3.4) can only bederived using the detailed balance principle (see Appendix B).

The solution to (3.4) is

(~Λa)i(k) = −S−1(1−feq)vi

(k), (3.5)

as computed in Appendix B. Therefore, the solution of the low-fieldBE is

f(k) = feq

(1 +

q

kBT(1 − feq) ~ES

−1(1−feq)~v

)(3.6)

in the case of the Fermi-Dirac statistics, and

f(k) = feq

(1 +

q

kBT~ES−1

~v

)(3.7)

in the case of the Boltzmann statistics.In the low-field case the solution of the BE is, therefore, only

determined by the equilibrium distribution feq and by S−1(1−feq)~v resp.

S−1~v .

If the principle of detailed balance is not fulfilled1, there is nosimple general solution in the case of Fermi-Dirac statistics, whereasin the case of Boltzmann statistics the solution is given by

f(k) = feq +q

kBT~ES−T

feq~v, (3.8)

where S−Tg is the g-moment of the transposed ISO, defined as

|S−Tg 〉 := S−T |g〉. (3.9)

The S−Tg s can also be computed using an iterative method, but this

is beyond the scope of this thesis.

1In Chapter 9 a standard model for impact ionization is used, which does notfullfil the principle of detailed balance.

3.3. TRANSPORT COEFFICIENTS 49

3.3 Transport coefficients

As already mentioned in Chapter 2.2, tensorial transport coefficientscan be exactly computed using inverse moments of the ISO. For ex-ample, the mobility is given by

µij :=q

n~

∫

K

S−1vi∂kj

fd3k (3.10)

and the diffusivity tensor by

Dij := − 1

n

∫

K

S−1vivjfd

3k. (3.11)

Note that in the case of Boltzmann statistics, setting f = feq in (3.10)and (3.11), yields the well-known Einstein relation for all componentsof the tensors

kBT

qµij = Dij . (3.12)

3.4 Hall factor

If a constant voltage is applied between A and B (see Fig. 3.1) anda constant magnetic field Bz′ is present in the z′-direction, then twoHall factors can be defined:

RH :=V21

d1Jx′Bz′, (3.13)

R∗H :=

V43

d2Jx′Bz′, (3.14)

where V21 is the voltage between the points 1 and 2, V43 the voltagebetween the points 3 and 4, and Jx′ is the current density in the x′-direction. Under the assumption that the electric fields Ey′ in they′- and Ez′ in the z′-direction are constants, the definitions can berewritten as

RH :=Ey′

Jx′Bz′, (3.15)

R∗H :=

Ez′

Jx′Bz′. (3.16)


To derive expressions for RH and R∗H , the current density equation

in the space-homogeneous case can be written as

~J

nq= µ~E + α~B, (3.17)

where µ is defined in (3.10) and α is defined as

αij :=1

n

∫

K

S−1vi

(~v ∧∇kf)j d3k. (3.18)

For better readability, only the case of Boltzmann statistics will beconsidered.

In the low-field case, f can be replaced by (3.7) leading to

αij =1

n

q

kBT

∫

K

S−1vi

(~v ∧∇k(feq ~ES

−1~v ))jd3k, (3.19)

where the term independent of the electric field disappeared because~v∧~v = 0. Taking advantage of the linearity of (3.19) in ~E, (3.17) canbe rewritten as

~J

nq= µ~E +Bxγx ~E +Byγy ~E +Bzγz ~E (3.20)

with the matrices γi defined as

(γl)ij :=1

n

q

kBT

∫

K

S−1vi

(~v ∧∇k(feqS

−1vj

))ld3k. (3.21)

Therefore,

~E = (µ+Bxγx + Byγy +Bzγz)−1

~J

nq. (3.22)

If R is the matrix that transforms the (x, y, z)-coordinate system intothe (x′, y′, z′) one, then the Hall factors can be written as

RH =(R (µ+Bxγx +Byγy +Bzγz)

−1R−1)yxqnBz′

, (3.23)

R∗H =

(R (µ+Bxγx +Byγy +Bzγz)−1R−1)zx

qnBz′, (3.24)

because the current ~J ′ flows only in the x′-direction (see Fig. 3.1).Eq. (3.23) is more general than the formula given in [30] and reducesto the formula given by [38] in special cases.

3.4. HALL FACTOR 51

d

1

21

2

4

3

BA

d

xy

z

x’

y’’z

Figure 3.1: Piece of bulk material.

3.4.1 Bulk silicon

In the case of unstrained bulk silicon, due to the symmetries of thecrystal, (3.20) takes the form:

~J

neqq= µeq ~E − γeq ~B ∧ ~E, (3.25)

where

µeq :=q

neq~

∫

K

S−1vx∂kx

feqd3k (3.26)

and

γeq :=1

neq

q

kBT

∫

K

S−1vx

(~v ∧∇k(feqS

−1vy

))zd3k. (3.27)

The Hall factors are then

RH =1

qneq

γeqµ2eq + γ2

eqB2z′, (3.28)

R∗H = 0, (3.29)

where RH and R∗H are independent of the transformation matrix R,

i.e. of the crystal orientation.


3.5 Relaxation times

3.5.1 Introduction

Using the Monte Carlo method, the relaxation times in bulk materialare usually computed using the formula (see e.g. [32]):

τg = −∫g(k)(f(k) − feq(k)

nneq

)d3k∫ ∫

g(k)S(k, k′)f(k′)d3k′d3k, (3.30)

where f is the solution of the stationary BE, feq is the equilibriumdistribution, n (resp. neq) is the integral of f (resp. feq) over thek-space, and g is the moment for which the relaxation time is needed.This formula, however, becomes problematic in the case of small elec-tric fields especially for an even function g.

Based on the moment of the ISO, another formula for the relax-ation times can be derived. By multiplying the space-homogeneousBE

− q~~E · ∇kf = S|f〉 (3.31)

with g and S−1g and integrating one obtains

− q

~~E ·∫

(∇kf)g(k)d3k =

∫ ∫g(k)S(k, k′)f(k′)d3k′d3k, (3.32)

− q

~~E ·∫

(∇kf)S−1g (k)d3k =

∫g(k)(f(k) − feq(k)

n

neq)d3k. (3.33)

Inserting the rhs of (3.32) and (3.33) in (3.30) gives:

τg = −~n ·∫

(∇kf)S−1g (k)d3k

~n ·∫

(∇kf)g(k)d3k, (3.34)

where ~n is the vector pointing in the direction of the electric field.Note that (3.34) can be easily extended to the space-inhomogeneous

3.5. RELAXATION TIMES 53

BE. Using (3.34) allows to compute exact relaxation times particularlyin the limit of a vanishing electric field. For high electric fields theMonte Carlo method can be used to evaluate either (3.30) or (3.34).

3.5.2 The problem of small electric fields

For small electric fields the distribution function f can be written as

f(k) = feq(k)

(1 +

q

kBTeq~E · S−1

~v (k)

)+O(E2), (3.35)

where S−1~v is the vector whose i-th component is the (~v)i-moment of

the ISO. The important point for the following is that using standardmethods, the O(E2) terms cannot be computed.

Inserting (3.35) into (3.30) gives:

τg =

−∫g(k)

(feq(k) ~E · S−1

~v (k) +O(E2))d3k

∫ ∫g(k)S(k, k′)

(feq(k′) ~E · S−1

~v (k′) +O(E2))d3k′d3k

.

(3.36)

For any system with an even band structure (ǫ(k) = ǫ(−k)) the func-

tion feq(k) ~E · S−1~v (k) is odd, because S−1

~v has by construction thesame symmetry properties as ~v. Thus, if the function g is an evenfunction, (3.36) reduces to

τg = −∫g(k)O(E2)d3k∫ ∫

g(k)S(k, k′)O(E2)d3k′d3k. (3.37)

Because O(E2) cannot be computed in the general case using standardmethods, (3.37) cannot be evaluated. Therefore, formula (3.30) isunusable at all for even functions g in the limit of a vanishing electricfield.

3.5.3 The heated Maxwellian ansatz

Jungemann et al.[32] tackled the problem by using a heated Maxwellianas ansatz for f :

f(k) := e− ǫ(k)

kBT , (3.38)


and computed τg in the limit T → Teq. Plugging this into (3.30) leadsto

τg = −∫ǫ(k)feq(k)

(g(k) − geq

neq

)d3k

∫ ∫g(k)S(k, k′)ǫ(k′)feq(k′)d3k′d3k

, (3.39)

where geq := 〈g|feq〉.The scattering operator S(k, k′) can be formally separated in one

elastic part S(k, k′)elastic and one inelastic part S(k, k′)inelastic. Theelastic part has the property

∫S(k, k′)elastich(ǫ(k

′))d3k′ = 0, (3.40)

for all functions h of the energy. Thus, the denominator of (3.39) doesnot depend on the elastic scattering. This is unfortunate, because inmost models the impurity scattering is an elastic process.

Therefore, the heated Maxwellian ansatz comes off badly in thecase of silicon, in particular because contributions from the impurityscattering automatically drop out, which yields relaxation times inde-pendent of the doping concentration.

3.5.4 Exact solution

Inserting (3.35) in (3.34) gives

τg =

−~n ·∫

(∇k(feq

(1 + q

kBTeq

~E · S−1~v

)))S−1

g d3k + O(E2)

~n ·∫

(∇k(feq

(1 + q

kBTeq

~E · S−1~v

)))gd3k +O(E2)

.

(3.41)

Considering again an even function g and neglecting the O(E2) terms,(3.41) reduces to

τg = −~n ·∫∇k(feq(k)~n · S−1

~v (k))S−1g (k)d3k

~n ·∫

(∇k(feq(k)~n · S−1~v (k))g(k)d3k

, (3.42)

because S−1g (k) has by construction the same symmetry properties as

g(k).

3.6. LANGEVIN NOISE SOURCES 55

As (3.42) is independent of E, there is no problem anymore toevaluate τg in the limit E → 0. One can easily show that, if ǫ(k) isan even function, than (3.42) does also not depend on ~n, i.e. on thedirection of the electric field.

Thus, the knowledge of S−1g (k) allows one to compute relaxation

times for all moments of the Boltzmann equation, even in the limit ofa vanishing electric field.

Some examples will be given in Chapter 9.

3.6 Langevin noise sources

The so-called ”Boltzmann-Langevin” equation (BLE) is the BE withan additional stochastic term, called Langevin term:

∂tf(~r, t,~k, b)+~r ∇rf(~r, t,~k, b)+~k∇kf(~r, t,~k, b) = S ·f+δs(~r, t,~k, b).

(3.43)

The BE describes the average state of an infinite number of systemswith identical initial conditions, whereas the BLE describes the evolu-tion of one of these systems. The Langevin source term is responsiblefor the deviation from the average state.

By multiplying (3.43) with S−1g one obtains

S−1g ·∂tf +S−1

g · ~r ∇rf +S−1g · ~k∇kf = 〈g〉− 〈g〉eq

n

neq+S−1

g · δs .(3.44)

An expression which is usually needed is the Fourier transform ofthe correlation functions of S−1

g · δs around a stationary state for thehomogeneous BLE with constant density n, i.e.

Cgg′ (ω) :=

∞∫

−∞

limT→∞

1

2T

T∫

−T

(S−1g · δs)(t)(S−1

g′ ·δs)(t+s)dte−iωsds,

(3.45)


for constant n. The expression for Cgg′ (ω) can be rewritten as

Cgg′ (ω) =∑

b,b0

∫

Vb

∫

Vb0

S−1g (~k, b)S−1

g′ (~k0, b0)C(ω)(~k, b|~k0, b0)d3kd3k0,

(3.46)

where C(ω)(~k, b|~k0, b0) is the correlation function of the Langevinstochastic terms:

C(ω)(~k, b|~k0, b0) :=

∞∫

−∞

limT→∞

1

2T

T∫

−T

δs(t)(~k, b)δs(t+ s)(~k0, b0)dte−iωsds .

(3.47)

In [21] p. 21, Eq. (1.55a) 2 an analytic expression for (3.47) isderived in the case of an homogeneous nondegenerate system with aconstant density:

C(ω)(~k, b|~k0, b0) = δ3(~k − ~k0)δb,b0∑

b1

∫

Vb1

w(~k, b|~k1, b1)f(~k, b)d3k1

+ δ3(~k − ~k0)δb,b0∑

b1

∫

Vb1

w(~k1, b1|~k, b)f(~k1, b1)d3k1

− w(~k0, b0|~k, b)f(~k0, b0) − w(~k, b|~k0, b0)f(~k, b) , (3.48)

where f(~k, b) is the stationary homogeneous solution to (2.21).

By plugging (3.48) into (3.46) and rearranging terms, one obtains

Cgg′ (ω) =∑

b

∫

Vb

Kgg′(~k, b)f(~k, b)d3k = 〈Kgg′ 〉 , (3.49)

2In the case of particle-particle scattering the corresponding additional contri-bution to the correlation function has to be added (see [21] p. 21, Eq. (1.55b)).

3.6. LANGEVIN NOISE SOURCES 57

where

Kgg′ (~k, b) :=

∑

b0

∫

Vb0

w(~k, b|~k0, b0)

S−1g (~k, b)S−1

g′ (~k, b)

−S−1g (~k, b)S−1

g′ (~k0, b0)

−S−1g (~k0, b0)S

−1g′ (~k, b)

+S−1g (~k0, b0)S

−1g′ (~k0, b0)

d3k0.

(3.50)

Note that (3.50) is invariant under the transformation

S−1g (~k, b) → S−1

g (~k, b) + αgf1.

Therefore, Cgg′ (ω) is independent of condition (2.32) as it should be.

Thus, the function Cgg′ (ω) is nothing but the expectancy of Kgg′ .Since Kgg′ can be computed, 〈Kgg′〉 can also be computed with a MCsimulation in a very efficient way. The function Cgg′ (ω) is called theLangevin noise source of the functions g, g′. It describes white noise,because it does not depend on ω.

At TD equilibrium (~k = 0) in the space-homogeneous case (n =const.), the following important relation exists between Cgg′ and thecorrelation function of the fluctuations of g, g′ at ω = 0:

Cgg′ (0) =

∞∫

−∞

limT→∞

1

2T

T∫

−T

δg(t)δg′(t+ s)dte−iωsds|ω=0. (3.51)

Eq. (3.51) means that at TD equilibrium the correlation function ofthe fluctuations of g, g′ at zero frequency are directly related to (3.49).


3.6.1 Special class of SO

Because of the symmetries of the transition probability of the SO ofSection 2.4, Kgg′ can be rewritten as (see (3.50))

Kgg′ =gg′

Wtot+

[Lg

〈g′〉eqneq

+ Lg′〈g〉eqneq

]−WtotLgLg′

+C(gg′

Wtot− [Lgg

′ + Lg′ g] +WtotLgLg′

)− 1

Wtot

[g〈g′〉eqneq

+ g′〈g〉eqneq

],

(3.52)

where all notations are the same as in Section 3.6 and

Lg :=

(Hg −

〈g〉eqneq

H1 +g

Wtot

)=

(S−1g +

g

Wtot

),

C(ε, b|ε0, b0) :=w(ε, b|ε0, b0)Z(ε0, b0)

Wtot(ε0, b0). (3.53)

The derivation is given in Appendix A.1.In TD equilibrium a simple expression for 〈Kgg′〉eq (see (3.49))

can be found. Since 〈feq|C = 〈feqZ|, one obtains

〈Kgg′〉eq = 〈feq|Kgg′〉 =

〈feqZ| (

2gg′

Wtot− 1

Wtot

[g〈g′〉eqneq

+ g′〈g〉eqneq

])

+ 〈feqZ| [Lg(

〈g′〉eqneq

− g′) + Lg′(〈g〉eqneq

− g)

]=

2〈feqZ| ( ˇgg′

Wtot− gg′

Wtot

)

+ 〈feqZ| [S−1g (

〈g′〉eqneq

− g′) + S−1g′ (

〈g〉eqneq

− g)

].

(3.54)

This equation is very important because it allows to find out, bysymmetry arguments only, and without computing anything, whichcorrelation functions disappear in TD equilibrium, and which do not.

3.7. MC AND LANGEVIN NOISE SOURCES 59

3.7 Monte Carlo and Langevin noise sour-ces

In this section the concept of the Langevin noise sources in a de-vice will be defined, and a method will be given to compute thesenoise sources and all relevant transport coefficients from a transientMonte Carlo (MC) device simulation.

3.7.1 Derivation of the BE with noise terms

In the case of a device, the distribution function is given by

fMC(~r, t)(~k, b) :=

Np∑

i=1

δ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi, (3.55)

where Np is the number of particles in the system (device R and

environment), (~ki(t),bi) the position in k-space of the i-th particle attime t, and ~ri(t) the position in real space of the i-th particle at timet.

If the particle m undergoes a scattering event at time t1, (3.55)must be corrected:

fMC(~r, t)(~k, b) :=

Np∑

i=1,i6=m

δ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi

+ θ(t− t1)δ3(~k+m(t) − ~k)δb,b+mδ

3(~rm(t) − ~r)

+ (1 − θ(t− t1))δ3(~k−m(t) − ~k)δb,b−mδ3(~rm(t) − ~r), (3.56)

where

~k−m(t) := ~km(t), t 6 t1, ~k−m(t) := const = limt→t1

−

~km(t), t > t1, (3.57)

~k+m(t) := ~km(t), t > t1 ~k+

m(t) := const = limt→t1+

~km(t), t 6 t1. (3.58)

This definition is illustrated in Fig. 3.2. (The definition is valid onlyin a small time interval around t1, where only one scattering takesplace.) For t 6= t1 (3.55) and (3.56) are identical.


~k

t1 t

~k−m(t)

~k+m(t)

Figure 3.2: Illustration of the functions ~k−m(t) and ~k+m(t).


It is always possible to decompose the history of a MC simula-tion in time intervals Ii,j := [ti,j , ti,j+1] such that either nothinghappens or only one scattering happens during Ii,j , and I = [0, T ] =⋃Ni(T )j=1 Ii,j , ∀i, where T is the simulated time and Ni(T ) is the number

of intervals since the beginning of the simulation.

Supposing that fMC(~r, t)(~k, b) describes a possible state of thedevice at time t, a parallel can be made between fMC , f and δs in(3.44). Computing the total time derivative of (3.55) and (3.56) oneobtains

d

dtfMC(~r, t)(~k, b) =

d

dt

Np∑

i=1

δ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi

= −Np∑

i=1

~ki(t)∇kδ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi

−Np∑

i=1

~ri(t)∇rδ3(~ri(t) − ~r)δ3(~ki(t) − ~k)δb,bi

= −Np∑

i=1

~F(~ri, ~ki)∇kδ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi

−Np∑

i=1

~v(~ri, ~ki)∇rδ3(~ri(t) − ~r)δ3(~ki(t) − ~k)δb,bi

= −~F(~r,~k) Np∑

i=1

∇kδ3(~ki(t) − ~k)δ3(~ri(t) − ~r)δb,bi

− ~v(~r,~k)

Np∑

i=1

∇rδ3(~ri(t) − ~r)δ3(~ki(t) − ~k)δb,bi

= −~F(~r,~k)∇kfMC(~r, t)(~k, b) − ~v(~r,~k)∇rfMC(~r, t)(~k, b)

(3.59)


for (3.55) and

d

dtfMC = −~F∇kfMC − ~v∇rfMC

+δ(t−t1)[δ3(~k+

m(t1) − ~k)δb,b+m − δ3(~k−m(t1) − ~k)δb,b−m

]δ3(~rm(t1)−~r)

(3.60)

for (3.56). By rearranging (3.60) one finds

d

dtfMC + ~v∇rfMC +~F∇kfMC =

δ(t− t1)[δ3(~k+


]δ3(~rm(t1)−~r).

(3.61)

The lhs of (3.61) looks like the lhs of (3.43) but the right-handsides are different.

The Langevin noise sources δsMC can be naturally defined suchthat (3.61) takes the form of (3.43):

d

dtfMC + ~v∇rfMC +~F∇kfMC =: S(fMC)fMC + δsMC

= δ(t−t1)[δ3(~k+


]δ3(~rm(t1)−~r).

(3.62)

Here, an important restriction on the dynamics of S(fMC(t)) and~F(fMC(t)) has to be made. S(fMC(t)) and ~F(fMC(t)) are not allowedto react instantaneously to a change in the system, i.e. if a scatteringevent takes place, the values of S(fMC(t)) and ~F(fMC(t)) must be thesame just before and just after the scattering.

Eq. (3.62) gives the unique definition of δsMC

δsMC(t) := −S(fMC(t))fMC(t), (3.63)

if there is no scattering at time t, and

δsMC(t) := −S(fMC(t))fMC(t)

+δ(t−t1)[δ3(~k+


]δ3(~rm(t1)−~r),

(3.64)


if particle m scatters at time t1.

3.7.2 Conditions of application of the linear noisetheory

By making an ensemble average of (3.62), δsMC should disappear

(it does in the case, where S and ~F are time-independent) and thefollowing equation remains:

d

dt〈fMC〉 + ~v∇r 〈fMC〉 +

⟨~F∇kfMC

⟩= 〈S(fMC)fMC〉 . (3.65)

Here 〈〉 denotes the ensemble average. Remember, that ~F andS(fMC) depend on fMC under the previously explained restriction.

If the variations of ~F and S(fMC) are small over the ensemble,(3.65) can be approximated by

d

dt〈fMC〉 + ~v∇r 〈fMC〉 +

⟨~F⟩∇k 〈fMC〉 ≃ 〈S(fMC)〉〈fMC〉 , (3.66)

where⟨~F⟩ ≃ ~F(〈fMC〉), 〈S(fMC)〉 ≃ S(〈fMC〉), which is nothing but

the BE.So, only under the assumptions

⟨~F∇kfMC

⟩≃⟨~F⟩∇k 〈fMC〉 (3.67)

〈S(fMC)fMC〉 ≃ 〈S(fMC)〉〈fMC〉 (3.68)⟨~F⟩ ≃ ~F(〈fMC〉) (3.69)

〈S(fMC)〉 ≃ S(〈fMC〉) (3.70)

does the ensemble average 〈fMC〉 solve the BE. If the fluctuations

of ~F and S(fMC) are too strong (in a real device and/or in a self-consistent ensemble MC simulation), then the solution of (3.66) doesnot describe what can actually be measured (simulated), and therewill be no BE and, therefore, no transport model able to accuratelyreproduce the measurements (MC simulations).

The four conditions (3.67)–(3.70) can be easily verified by a MCsimulation. In the remainder those conditions are assumed to be ful-filled.


3.7.3 Existence of a stationary state

The stationary state f is defined as the state fulfilling

~v ∇r f − q~~E(f)∇kf = S(f)f . (3.71)

An ensemble device MC simulation converges towards this state,if and only if the changes in ~E and S are time-uncorrelatedcoefficientwith the change in fMC . If ~E and S are updated as described inSection 3.7, and if the time interval between two updates is larger thanthe autocorrelation time of fMC , then the simulation will convergetowards the stationary state. If not, then it will converge towardssomething else.

In a real device the changes in ~E and S are correlated to thechange in f , the stronger the lower the density. Therefore, in order tomatch the behaviour of a real device with a MC device simulation, asmuch MC particles as there are in the real device have to be simulatedand ~E and S must be updated locally, the more often the smaller thenumber of particles.

3.7.4 Definition of transport coefficients and noisesources in a device

By multiplying (3.62) with a moment of the ISO one finds

S−1g (fMC) · d

dtfMC +~vS−1

g (fMC) ·∇rfMC +S−1g (fMC) ·~F∇kfMC =

〈g〉MC − nMC

〈g〉eqneq

+ S−1g (fMC) · δsMC = δ(t− t1)×

[S−1g (fMC)(~k+

m(t1), b+m) − S−1g (fMC)(~k−m(t1), b−m)

]δ3(~rm(t1) − ~r)

(3.72)

and, therefore,

S−1g (fMC)δsMC = −〈g〉MC + nMC

〈g〉eqneq

+ δ(t− t1)×[S−1g (fMC)(~k+

m(t1), b+m) − S−1g (fMC)(~k−m(t1), b−m)

]δ3(~rm(t1) − ~r),

(3.73)


where 〈g〉MC is the expectancy of g and nMC the density. This givesan explicit definition of the noise source for the g-moment of the BE.

All the terms of (3.72) as well as S−1g (fMC) can be computed with

a transient MC device simulation. First, (3.72) has to be rewrittento make it ”MC-friendly”, i.e. all derivatives of fMC except the timederivative must disappear:

S−1g · dfMC

dt+ S−1

g · ~v∇rfMC − 〈g〉MC + 〈g〉eqnMC

neq

−∑

b

∫

Vb

∇k(S−1g (~k,~r)~F)fMC(~k,~r)d3k

+∑

b

∮

∂Vb

~FS−1g (~k,~r)fMC(~k,~r)~nbda = S−1

g · δsMC , (3.74)

where the ∇k-term has been integrated by parts:

S−1g ·~F∇kfMC =︸︷︷︸

Integration by parts

−∑

b

∫

Vb

∇k(S−1g (~k,~r)~F)fMC(~k,~r)d3k+

∑

b

∮

∂Vb

~FS−1g (~k,~r)fMC(~k,~r)~nda.

(3.75)

Now imagine a MC simulation being run on the space domain Rwith a discretisation D(R) := Rii=1,..,n, and that S−1

g |Ri, ~v |Ri

,

~F |Ri:= ~Fi(~k) (e.g. ~Fi(~k) := −q

(1~~Ei − ~Bi ∧ ~v(~k)

)) are constants in

real space and S−1g |Ri

only depends on mean values Mi,j on Ri, e.g.the mean density ni or the mean energy 〈ǫi〉.

In usual MC simulations, S−1g |Ri

and ~F |Ri:= ~Fi(~k) are updated

at given discrete time steps tupdatei,j , j = 1...M .

The mean value of h on Ri can be defined as

(h)i(t)(~k, b) :=1

|Ri|

∫

Ri

h(~r, t)(~k, b)d3r. (3.76)


By further integrating (3.72) and (3.73) on Ri, one finds

(S−1g · dfMC

dt)i − (〈∇k(S

−1g~Fi)〉MC)i

− (〈g〉MC)i +〈g〉eqneq

(nMC)i +1

|Ri|

∮

∂Ri

〈S−1g ~v〉MC~nida

+ (∑

b

∮

∂Vb

~FiS−1g (~k,~r)fMC(~k,~r)~nbda)i = (S−1

g · δsMC)i (3.77)

and

(S−1g · δsMC)i = −(〈g〉MC)i +

〈g〉eqneq

(nMC)i

+ δ(t− t1)1

|Ri|[S−1g (~k+

m(t1), b+m) − S−1g (~k−m(t1), b−m)

]. (3.78)

If one is interested in the stationary case, one can, by averaging (3.77)in time, compute transport coefficients as described in Section 2.3.Using (3.78), the noise sources can be computed as described in thenext section.

3.7.5 An interesting property of S−1g

S−1g is naturally related to (3.78) and has, therefore, an interesting

property: In the stationary state, the time average of (3.78) mustvanish. The time average of

(〈g〉MC)i −〈g〉eqneq

(nMC)i (3.79)

can, thus, be replaced by the time average of

δ(t− t1)1

|Ri|[S−1g (~k+

m(t1), b+m) − S−1g (~k−m(t1), b−m)

]. (3.80)

During a MC simulation one can compute

S−1g (~k+

m(t1), b+m) − S−1g (~k−m(t1), b−m)


for all scattering events. If the mean value over all events is computed,

it will then be equal to the time average of (〈g〉MC)i − 〈g〉eq

neq(nMC)i.

This gives a method to compute time averages by computing meanvalues over the scattering events (like the ”just before scattering”method, see e.g. [43]).

Summary

In this section it has been shown that (S−1g · δsMC)i(t) can be com-

puted during a transient MC device simulation, that the four con-ditions (3.67)–(3.70) have to be fulfilled such that a transport modelexists for the considered device, and that transport coefficients for anymoment of the BE can be locally computed, provided these conditionsare fulfilled.

3.7.6 Correlation functions of Langevin noise sour-ces

In this section the computation of the auto- and cross-correlationfunctions of S−1

g (fMC) will be discussed and their importance willbe explained.

The correlation functional is defined as

Cabgg′ [ϕ] := limδ→0

∞∫

−∞

limT→∞

1

T

T∫

0

[(S−1g · δsMC)a,δ(t)

]

×[(S−1g′ · δsMC)b,δ(t+ s)

]dtϕ(s)ds (3.81)

∀ϕ ∈ S, where S denotes the Schwarz-space, and

(S−1g · δsMC)i,δ :=

1

|Ri,δ|

∫

Ri,δ

(S−1g · δsMC)(~r)d3r, (3.82)

Ri,δ ∩ ∂Ri ≡ ∅, Ri,δ1 ⊂ Ri,δ, ∀δ1 > δ, and limδ→0

Ri,δ ≡ Ri. The

geometrical definition ofRa, Rb, Ra,δ andRb,δ is illustrated in Fig. 3.3.


Ra

Rb

Rb,δ

Ra,δ

Figure 3.3: Geometrical definition of Ra,δ and Rb,δ.

For a = b one obtains

Caagg′ [ϕ] =

∞∫

−∞

limT→∞

1

T

T∫

0

[(S−1g · δsMC)a(t)

]

×[(S−1g′ · δsMC)a(t+ s)

]dtϕ(s)ds. (3.83)

From a theoretical and practical point of view it is important toknow Cabgg′ [ϕ], because e.g. the impedance field method works if and

only if Cabgg′ [ϕ] = 0, a 6= b (space-uncorrelated), Caagg′ [ϕ] = ϕ(0)∗const.(time-uncorrelated).

Definition (3.81) is important for a 6= b, Ra ∩ Rb 6= ∅ (i.e. Ra,Rbneighbors), because for the definition

Cabgg′ [ϕ] :=

∞∫

−∞

limT→∞

1

T

T∫

0

[(S−1g · δsMC)a(t)

]

×[(S−1g′ · δsMC)b(t+ s)

]dtϕ(s)ds (3.84)

the functional would have no chance to become zero, due to commonevents on the common boundary Ra ∩Rb.


Eq. (3.81) is the only reasonable way to define correlation func-tionals of

(S−1g · δsMC)i(t). (3.85)

In the case a = b and in the interval I = [0, T ], (S−1g · δsMC)i(t)

can be written as

Fg(t) +

Ni(T )∑

j=1

A(g)jθ(t− tj)

︸︷︷︸:=Gg(t)

+

Ni(T )∑

j=1

B(g)jδ(t− tj), (3.86)

where Fg(t) is a continuous function3 , A(g)j , B(g)j are constants,and tj is the time of an event. Fg(t), A(g)j , and B(g)j can becomputed from (3.78) (the θ functions originate from the terms 〈g〉MC

and 〈n〉MC).

Thus, the correlation functional can be computed provided that

∞∫

−∞

limT→∞

1

T

T∫

0

Fg(t) +

Ni(T )∑

j=1

A(g)jθ(t− tj) +

Ni(T )∑

j=1

B(g)jδ(t− tj)

×

Fg′(t+ s) +

Ni(T )∑

k=1

A(g′)kθ(t+ s− tk)

+

Ni(T )∑

k=1

B(g′)kδ(t+ s− tk)

dtϕ(s)ds

(3.87)

can be computed.

3For example, if g is the velocity and if the particles move ballistically, thenFg(t) is the mean velocity density, which is time- and space-correlated.


Eq. (3.87) can be decomposed into 6 parts:

∞∫

−∞

limT→∞

1

T

T∫

0

Fg(t) +

Ni(T )∑

j=1

A(g)jθ(t− tj)

×

Fg′ (t+ s) +

Ni(T )∑

k=1


dtϕ(s)ds, (3.88)

∞∫

−∞

limT→∞

1

T

T∫

0

[Fg(t)] ×

Ni(T )∑

k=1


dtϕ(s)ds, (3.89)

∞∫

−∞

limT→∞

1

T

T∫

0

Ni(T )∑

j=1

B(g)jδ(t− tj)

× [Fg′(t+ s)]dtϕ(s)ds, (3.90)

∞∫

−∞

limT→∞

1

T

T∫

0

Ni(T )∑

j=1

A(g)jθ(t− tj)

×

Ni(T )∑

k=1


dtϕ(s)ds, (3.91)

∞∫

−∞

limT→∞

1

T

T∫

0

Ni(T )∑

j=1

B(g)jδ(t− tj)

×

Ni(T )∑

k=1


dtϕ(s)ds, (3.92)


∞∫

−∞

limT→∞

1

T

T∫

0

Ni(T )∑

j=1

B(g)jδ(t− tj)

×

Ni(T )∑

k=1


dtϕ(s)ds, (3.93)

which contain the following functionals:

(3.88) →

limT→∞

1T

T∫0

Fg(t)Fg′ (t+ s)dt+

1T

∑j,k

A(g)jA(g′)kT∫0

θ(t+ s− tk)θ(t − tj)dt

+ 1T

∑k

A(g′)kT∫0

Fg(t)θ(t + s− tk)dt

+ 1T

∑k

A(g)jT∫0

Fg′ (t+ s)θ(t − tj)dt

,

(3.94)

(3.89) + (3.90) →

limT→∞

1

T

∑

k

B(g′)kFg(tk − s) +1

T

∑

j

B(g)jFg′(tj + s)

, (3.95)

(3.91) + (3.92) →

limT→∞

1T

∑j,k

A(g)jB(g′)kθ(tk − tj − s)θ(s− tk + T )

+ 1T

∑j,k

B(g)jA(g′)kθ(tj − tk + s)θ(s− tk + T )

,

(3.96)

(3.93) → limT→∞

1

T

∑

j,k

B(g)jB(g′)kδ(tj − tk + s), (3.97)


where the t-integration has been performed in Eqs. (3.95)–(3.97).For a 6= b the same expression is obtained except that the sums in

(3.94)–(3.97) do not have diagonal terms (there are no tj such thattj = tk).

Because the first term of (3.94) is space-correlated, the suppositionthat the Langevin noise sources are space uncorrelated is wrong for allmoments (even if one uses the exact Langevin noise sources), and theimpedance field method must be modified to take this into account.

In bulk, (3.81) must give the same result as (3.49).

Summary

The knowledge of S−1g allows one to locally compute the Langevin

noise sources using a MC simulation. The Langevin noise sources areby nature always space-correlated.

3.8 New iterative scheme for the one-par-ticle MC method

One major drawback of the MC method is the apparent inability toinclude generation-recombination (G-R) processes. In this section, anew iterative scheme is described for the one-particle MC method thatallows the inclusion of any G-R process, like e.g. Shockley-Read-Hallrecombination or impact ionisation.

3.8.1 Standard method

The usual way to iterate self-consistently the Poisson equation andthe Boltzmann equation, when using the one-particle MC method, isdescribed in [13]. It can be summarised as follows:

1. Frozen-field simulation.

2. Extraction of the new quasi-Fermi potentials ψn from the newMC density nnewMC and the old potential ϕold using the formula

nnewMC = nieq

kBTL(ϕold−ψn)

, (3.98)

3.8. NEW ITERATIVE SCHEME 73

where ni is the local intrinsic density and TL the lattice temper-ature.

3. Computation of the new electrostatic potential ϕnew by solvingthe non-linear Poisson equation using the quasi-Fermi potentialextracted in the previous step.

4. Computation of the new electric field.

5. ϕnew becomes ϕold and continue with 1.

This method, however, does not allow to include G-R processes.

3.8.2 Method based on moments of the ISO

The main idea of this method is to compute the new electric fieldby solving the coupled system of the Drift-Diffusion (DD) equationsand the linear Poisson equation, using transport coefficients extractedfrom the one-particle MC simulation.

As already shown in Section 2.2, the mobility tensor µMC as well asthe diffusion tensor DMC can be exactly extracted from a MC simula-tion using the ~v-moment of the ISO. With these transport coefficients,the stationary DD equation can be written as

qnµMC~E + q∇T

r (DMCn) = ~Jn. (3.99)

To solve this equation coupled with the linear Poisson equation, onealways uses the continuity equation

−∇r~Jn = q(G−R). (3.100)

Inserting (3.99) into (3.100) one finds the well known equation

−∇r

(nµMC

~E + q∇Tr (DMCn)

)= q(G−R). (3.101)

Eq. (3.101) is interesting, because it contains information from theMC simulation and G-R processes at the same time. Thus, by solving(3.101) and its equivalent for holes coupled with the linear Poissonequation, a new electric field can be computed, which contains thewhole MC physics coupled with G-R processes.

Based on these considerations, the following iterative scheme isproposed:


1. Frozen-field simulation.

2. Extraction of the tensorial transport coefficients µMC andDMC ,and also of the impact ionisation rate GII .

3. Computation of the new electrostatic potential by solving thelinear Poisson equation coupled with (3.101) and its equivalentfor holes.

4. Computation of the new electric field, and continue with 1.

In practical examples this method seems to be at least as stableas the standard method.

3.8.3 Advantages and drawbacks

The new method has two main advantages:

1. It allows to use the MC method coupled with G-R processes.

2. Because the transport coefficients are not proportional to thedensity, the results of the iterative scheme are fundamentally in-dependent on the statistical weight of the MC particles. There-fore, the total charge content of the device can change as itshould, which is not the case using the standard method.

The only drawback is that the transport coefficients must be com-puted, which can slow down the simulation. This is, however, nota serious problem, because the one-particle MC method can be verywell parallelised.

Note that quantum corrections to the DD model can be easilyadded in the new method.

3.9 Determination of the low-field to high-field transition

When computing tables of MC generated transport coefficients and/orcorrelation functions, one is confronted with the question:”What is themaximum electric field intensity Fmax up to which the thermodynamic

3.9. LOW-FIELD TO HIGH-FIELD TRANSITION 75

considerations are still valid?”. This question is important, becausefor electric fields smaller than Fmax the transport parmeters and/orthe noise sources can be quickly computed without using MC simula-tions. It seems that in bulk semiconductors the above question may bereplaced by:”At which field strength Fmax will the carrier temperatureexceed the lattice temperature by one degree?”.

This second question can be easily answered using (2.15) and(2.19). Inserting (2.19) in (2.15) and considering the space-homo-geneous case, one finds

T − Teq = − qm∗

3kBn~~E

∫

Bz

S−1v2 (k)∇kf(k)d3k. (3.102)

Using Boltzmann statistics, (3.102) can be computed up to the orderO(E2) by inserting (3.7):

T −Teq = −E2 q2m∗

3k2Bn~Teq

~n

∫

Bz

S−1v2 (k)∇k(feq(k)~nS

−1~v (k))d3k, (3.103)

where E is the electric field intensity and ~n its direction. In (3.103)the first term of (3.7) (feq) disappears because of the symmetry ofthe band structure (ǫ(k) = ǫ(−k)). Note that (3.103) is an exactexpression for T − Teq as function of the electric field intensity forsmall E. It can be compared with the approximation given in theliterature (see e.g. [27]). From (3.103), Fmax can be computed bysetting T − Teq = 1K:

Fmax :=

√1K

− q2m∗

3k2Bn~Teq

~n

∫

Bz

S−1v2 (k)∇k(feq(k)~nS

−1~v (k))d3k

− 12

[V/m].

(3.104)

As will be shown in Chapter 9, Fmax is strongly dependent on thedoping concentration as already known from simpler models (see e.g.[27] p. 111, Eq. (9.31)).

Part II

Analytical Descriptionof Noise Using ISO

77

Chapter 4

Formal derivation oftransport models

4.1 Introduction

In this chapter a general equation for any moment of the BE using theformalism developed in Chapter 2 will be derived. From this equationexact expressions for the small-signal analysis and for noise will becomputed. At the end, some common approximations used in thederivation of the DD and HD models will be discussed.

4.2 Formal equation for a given moment

Starting from the BE:

∂f(t, r, k)

∂t+ ~v(k)∇rf(t, r, k) +~F(t, r, k)∇kf(t, r, k) = (Sf)(t, r, k)

(4.1)

79

80 CHAPTER 4. DERIVATION OF TRANSPORT MODELS

and contracting it with S−1g (t, r, k) gives

∫

Bz

S−1g (t, r, k)

∂f(t, r, k)

∂td3k +

∫

Bz

S−1g (t, r, k)~v(k)∇rf(t, r, k)d3k

+

∫

Bz

S−1g (t, r, k)~F(t, r, k)∇kf(t, r, k)d3k =

∫

Bz

g(k)

(f(t, r, k) − feq(t, r, k)

n

neq

). (4.2)

Note that for g = 1 one obtains 0 = 0 because S−11 = 0 by definition

(see (2.56) and (2.57)). To obtain the continuity equation instead,S−1

1 can be redefined as the constant function 1. The distributionfunction f can be developed in an arbitrary basis αi(k):

f(t, r, k) =: n(r, t)∑

i

ci(t, r)αi(k). (4.3)

We define

gi :=

∫

Bz

g(k)αi(k)d3k, (4.4)

Q0g,i(r, t) :=

∫

Bz

S−1g (t, r, k)αi(k)d

3k, (4.5)

~Q1g,i(r, t) :=

∫

Bz

S−1g (t, r, k)~v(k)αi(k)d

3k, (4.6)

~Q2g,i(r, t) :=

∫

Bz

S−1g (t, r, k)∇kαi(k)d

3k. (4.7)

For example, the basis αi could be composed of Legendre polynomials.With (4.3)–(4.7), Eq. (4.2) takes the form

n∑

i

Q0g,ici + n

∑

i

Q0g,ici +

∑

i

~Q1g,ici∇rn+ n

∑

i

~Q1g,i∇rci+

n∑

i

~Q2g,ici

~F = n∑

i

gici − ngeqneq

. (4.8)

The∑

iQmg,ici are, by definition, the well-known transport coefficients.

Administrator

Highlight

4.3. SMALL-SIGNAL ANALYSIS 81

Note that (4.8) is an expansion of the g-moment of the BE. There-fore, (4.8) has little to do with the usual method of moments.

The well-defined and exact expression (4.8) will be the basis forthe further investigations in this chapter.

4.3 Small-signal analysis

In this section it is assumed that the scattering operator only dependson the local density n and on the local mean energy T . Linearising

(4.8) around the stationary state ci, Q0g,i,

~Q1g,i and ~Q2

g,i gives

˙δn∑

i

Q0g,ici + n

∑

i

Q0g,iδci +

∑

i

[∂ ~Q1

g,i

∂nδn+

∂ ~Q1g,i

∂TδT

]∇r (cin)

+∑

i

~Q1g,i∇r (δcin) +

∑

i

~Q1g,i∇r (ciδn)

+ n∑

i

~Q2g,iciδ

~F + n∑

i

[∂ ~Q2

g,i

∂nδn+

∂ ~Q2g,i

∂TδT

]ci~F + n

∑

i

~Q2g,iδci

~F+ δn

∑

i

~Q2g,ici

~F = δn

(∑

i

gici −geqneq

)+ n

∑

i

giδci, (4.9)

where δci(r, t) := ci(r, t)− ci and δn(r, t) := n(r, t)−n. Note that thederivatives of Q0

g,i naturally do not contribute to (4.9). Eq. (4.9) willbe the starting point to derive a method to compute Y-parametersusing the one-particle MC method. For the noise analysis, the rhs of(4.9) must be replaced by the rhs of (3.72).

4.4 Common approximations

In this section, the approximations that must be made to find theusual equations for the DD model and for the HD model are described.

4.4.1 Transport coefficients

For the seek of simplicity only the case g = ~v(k) will be considered.Usual transport models, like the DD and HD models, use scalars for

82 CHAPTER 4. DERIVATION OF TRANSPORT MODELS

the mobility and diffusivity. This means for (4.8), that the followingapproximations must be made:

(∑

i

~Q1vk,i

ci

)

j

≈D, j = k = 1, 2, 30, j 6= k

(4.10)

(∑

i

~Q2vk,ici

)

j

≈µ, j = k = 1, 2, 30, j 6= k

(4.11)

∑

i

Q0vj ,ici ≈ −µm

∗

qn

(~J)j, j = 1, 2, 3, (4.12)

where µ is the scalar mobility, D the scalar diffusion coefficient, T themean particle energy, m∗ the effective mass, and ~J the current density.Note that in the case of the small-signal analysis the derivatives of(4.12) do not occur, as pointed out in the Section 4.3. This will be ofsome importance later.

For the DD model the additional assumptions

∂ ~Q1g,i

∂TδT ≈ 0, (4.13)

∂ ~Q2g,i

∂TδT ≈ 0 (4.14)

have to be made.

4.4.2 Parametrisation of the transport coefficients

Transport coefficients (mobility, diffusivity,...) are usually parame-terised by the norm F of the gradient of the quasi-Fermi potential inthe case of the DD model and by the mean local energy T in the caseof the HD model.

For the DD model this parametrisation has the consequence thatthe fluctuations of the ci are proportional to the fluctuactions of F ,i.e.

δci ≈∂ci∂F

δF, δci ≈∂ci∂F

δF , (4.15)

Administrator

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4.4. COMMON APPROXIMATIONS 83

and for the HD model:

δci ≈∂ci∂T

δT, δci ≈∂ci∂T

δT . (4.16)

Chapter 5

Generalised ImpedanceField Method

5.1 Introduction

When dealing with the linear fluctuations of the terminal currentsIl(t) around a stationary state in a device, one is often interested inthe correlation functions of the fluctuations of these currents. Fig. 5.1shows the typical time dependence of the current, when it fluctuatesaround a stationary state. The correlation function of the fluctuationsof the terminal currents is defined as

SδIkδIl(w) := lim

u→∞

1

u

u∫

0

[Ik(t) − Ik][Il(t+ w) − I l]dt. (5.1)

Fig. 5.2 shows the auto-correlation function SδIδI(w) for the exampleshown in Fig. 5.1. The function SδIkδIl

(w) can be easily computedusing the data from the many-particle MC simulation. However, itscomputation is a problem when using transport models. Thus, theimpedance field method (IFM) was developed, which allows the com-putation of the Fourier transform of SδIkδIl

based on transport models.In this chapter, a generalised form of the IFM will first be formally

derived from the theory of the previous chapter, and the difference

85

86 CHAPTER 5. GENERALISED IMPEDANCE FIELD

0 1e-10 2e-10 3e-10time [s]

2.25e-03

2.30e-03

2.35e-03

2.40e-03

2.45e-03

2.50e-03

2.55e-03

0 1e-13 2e-13 3e-13 4e-13 5e-13 6e-13 7e-13 8e-13 9e-13 1e-12time [s]

2.25e-03

2.30e-03

2.35e-03

2.40e-03

I(t

)[A/µm

]

I(t

)[A/µm

]

Figure 5.1: Typical fluctuation of a terminal current around a sta-tionary state.

0 5e-14 1e-13 1.5e-13 2e-13 2.5e-13w [s]

0

1e-06

2e-06

3e-06

4e-06

5e-06

SδIδI(w

)[A

2s/µm

2]

Figure 5.2: Auto-correlation function of the terminal current.

5.2. DESCRIPTION OF THE GENERALISED IFM 87

between pure Langevin noise sources and the noise sources actuallyneeded for the transport models will be explained. Then, their imple-mentation in the DD and the HD models is outlined. Finally, a generalexact scheme to test the IFM with the many-particle MC method isdeveloped.

5.2 Description of the generalised IFM

As already explained at the end of Section 4.3, the exact stochasticdifferential equation for the g-moment of the BE in the linear regimeis

˙δn∑

i

Q0g,ici + n

∑

i

Q0g,iδci +

∑

i

[∂ ~Q1

g,i

∂nδn+

∂ ~Q1g,i

∂TδT

]∇r (cin)

+∑

i

~Q1g,i∇r (δcin) +

∑

i

~Q1g,i∇r (ciδn)

+ n∑

i

~Q2g,iciδ

~F + n∑

i

[∂ ~Q2

g,i

∂nδn+

∂ ~Q2g,i

∂TδT

]ci~F + n

∑

i

~Q2g,iδci

~F+δn

∑

i

~Q2g,ici

~F =

∞∑

m=0

δ(t−tm)δ3(r−rnm(tm))

[S−1g (k+

nm) − S−1

g (k−nm)]

− n

(∑

i

gici −geqneq

), (5.2)

where tm is the time at which the m-th scattering events occurs, nmis the index of the particle that actually scatters at time tm, rnm

(tm)its position in real space, k−nm

its position in k-space just before scat-tering, and k+

nmits position in k-space just after scattering. By def-

inition, the Langevin noise source δsg for the g-moment is (comparewith Eq. (3.73))

δsg(t, r) :=

∞∑

m=0

δ(t− tm)δ3(r − rnm(tm))

[S−1g (k+

nm) − S−1

g (k−nm)]

− δn

(∑

i

gici −geqneq

)− n

∑

i

giδci − n

(∑

i

gici −geqneq

). (5.3)


Therefore, the rhs of (5.2) can be replaced by

δn

(∑

i

gici −geqneq

)+ n

∑

i

giδci + δsg. (5.4)

Suppose a system of (d + 1)(N + 1) equations built by the mo-ments of the functions g0 = h0, ~g1 = S−1

h0~v, g1 = h1, ~g2 = S−1

h1~v,

... , g2N = hN , ~g2N+1 = S−1hN~v, where h0 = 1 and h1 = ε or v2,

and d is the dimension in real space (d = 1, 2 or 3). The functionS−1

1 is again redefined to be the constant function with value 1 (seeChapter 4). The g0-moment is the contraction of the stochastic BE(3.61) with S−1

1 = 1 which is by definition the continuity equation.The other moments are obtained by contracting the stochastic BE(3.61) with S−1

gias shown in (3.72), which leads to (5.2). For exam-

ple, if N = 0, one obtains the current continuity equation from theh0-moment, and the velocity equation from the ~g1 = h0~v-moment.These two equations are the starting point for the derivation of theDD model. N = 1 produces the starting equations for the HD modeland N = 2 the equations for the six-moments model (see [24]). Thesystem is parameterised by the functions φi

1, and for each function hia ”driving force”2 Fi(φ0, ..., φN , φN+1) is introduced, where φN+1 isthe electrostatic potential φ, F0 is usually the norm of the gradient ofthe quasi-Fermi potential, and F1 is the mean energy of the particlesor the velocity squared.

To find a closed system of equations, the approximations

δci ≈N∑

j=0

∂ci∂Fj

δFj , δci ≈N∑

j=0

∂ci∂Fj

δFj (5.5)

are needed.

1In the case of the DD model, φ0 can be the quasi-Fermi potential. In the HDmodel, φ0 = 0 and φ1 is the mean temperature. The six-moments model (see [24])uses the same parameters as the HD model, but with the kurtosis as additionalparameter for φ2.

2In the DD model, the gradient of the quasi-Fermi potential is commonly usedas driving force to parametrise the transport coefficients, whereas in higher-ordermoments of the BE the mean particle energy plays this role. As the energy is nota driving force in the conventional sense, quotation marks were used.

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Adding these sums to both sides of (5.2) and rearranging terms,leads to the still exact equation

˙δn∑

i

Q0g,ici + n

∑

i

Q0g,i

N∑

j=0

∂ci∂Fj

δFj

+∑

i

[∂ ~Q1

g,i

∂nδn+

∂ ~Q1g,i

∂TδT

]∇r (cin) +

∑

i

~Q1g,i∇r

n

N∑

j=0

∂ci∂Fj

δFj

+∑

i

~Q1g,i∇r (ciδn)+n

∑

i

~Q2g,iciδ

~F+n∑

i

[∂ ~Q2

g,i

∂nδn+

∂ ~Q2g,i

∂TδT

]ci~F

+ n∑

i

~Q2g,i

N∑

j=0

∂ci∂Fj

δFj

~F + δn

∑

i

~Q2g,ici

~F =

δn

(∑

i

gici −geqneq

)+ n

∑

i

giδci + δsg

+n∑

i

Q0g,i

N∑

j=0

∂ci∂Fj

δFj − δci

+

∑

i

~Q1g,i∇r

(

N∑

j=0

∂ci∂Fj

δFj − δci)n

+ n∑

i

~Q2g,i

N∑

j=0

∂ci∂Fj

δFj − δci

~F. (5.6)

From (5.6) a natural definition for the noise sources δζg in transportmodels can be found:

δζg := δsg + n∑

i

Q0g,i

N∑

j=0

∂ci∂Fj

δFj − δci

+∑

i

~Q1g,i∇r

(

N∑

j=0

∂ci∂Fj

δFj − δci)n

+ n∑

i

~Q2g,i

N∑

j=0

∂ci∂Fj

δFj − δci

~F. (5.7)


The relation between the Langevin and the noise sources (NS) of thetransport models is fully described by (5.7). It is interesting to notethat in the case of a full basis of gi, i.e. in the case of an infinitenumber of moments of the BE, δζg = δsg. Hence the transport modelNS are equivalent to the Langevin NS only in the limiting case of aninfinite number of moments of the BE. In Chapter 9 both kinds of NSwill be compared with each other in the case of bulk silicon.

In the order m there are 1 + d moments: the hm-moment and thed components of the S−1

hm~v-moment. By inserting the equations for

the S−1hm~v-moment into the ∇r-term of the lhs of the equation for the

hm-moment, taking the Fourier transform in time, and leaving onlyexpressions with δζgm

on the rhs, one finds

N+1∑

j=0

(Λ)mj (ω, r)δφj(ω, r) = δζhm(ω, r)+∇rδζS−1

hm~v(ω, r), ∀m = 0...N,

(5.8)where δφj(t, r) := φj(t, r) − φj(r). To include the Fourier transformin time of the Poisson equation into the system of equations, an ex-pression for m = N + 1 can be written as

N+1∑

j=0

(Λ)N+1j (ω, r)δφj(ω, r) = 0, (5.9)

where the rhs is zero because there are no noise sources for the electro-static potential. The linear operator Λ(ω, r) is unequivocally definedby (5.6) and by the Poisson equation.

The interesting functions are the fluctuations of the terminal cur-rents δIl(ω), where the index l designates the l-th contact δΩl of thedevice Ω. To compute the δIl(ω), differentiable functions hTFl (r),called test functions, with the property

hTFl (r)∣∣δΩk

= δl,k (5.10)

can be introduced. With these hTFl (r) the fluctuations δIl(ω) can be

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written as

δIl(ω) =:

∫

δΩl

(δ ~J(ω, r) − iωǫδ ~E(ω, r)

)~nda

=

∫

δΩ


)hTFl (r)~nda

=Gauss

∫

δΩ

∇r

[(δ ~J(ω, r) − iωǫδ ~E(ω, r)

)hTFl (r)

]d3r

=

∫

Ω


)∇rh

TFl (r)d3r

+

∫

Ω

∇r


)hTFl (r)d3r,

(5.11)

where ~n is the normal to the l-th contact, ~E is the electric field, and~J the current density. The last term on the rhs of (5.11) can berewritten as

∫

Ω

∇r


)hTFl (r)d3r =

q

∫

Ω

δζh0(ω, r)hTFl (r)d3r. (5.12)

δζh0 can be computed from (3.61). The result is δζh0 = 0 because nogeneration-recombination processes were considered in the derivationof (3.61). Nevertheless, the term (5.12) will be kept, in order that theformulation stays valid even in the presence of G-R processes. Usingthe current density equation, the dependence of ~E on φ, and (5.12),

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one can rewrite (5.11) as

δIl(ω) =∫

Ω

N+1∑

j=0

Hlj(ω, r)δφj(ω, r)d3r − q

∫

Ω

δζS−1h0~v(ω, r)∇rh

TFl (r)d3r

+ q

∫

Ω

δζh0(ω, r)hTFl (r)d3r, (5.13)

where Hlj(ω, r) is fully defined by (5.11). To express δIl(ω) as afunction of the noise sources only, the inverse operator Λ−1 to Λ (see(5.8) and (5.9)) must be computed. This can be done numerically (seeChapter 7). Thus, the δIl(ω) can be written as a function of the NSin the form

δIl(ω) =:

∫

Ω

∫

Ω

∑

jm

Hlj(ω, r)(Λ−1(ω, r, r′)

)jmδζhm

(ω, r′)d3r′d3r

+

∫

Ω

∫

Ω

∑

jm

Hlj(ω, r)(Λ−1(ω, r, r′)

)jm

∇r′δζS−1hm

~v(ω, r′)d3r′d3r

− q

∫

Ω

δζS−1h0~v(ω, r)∇rh

TFl (r)d3r + q

∫

Ω

δζh0(ω, r)hTFl (r)d3r.

(5.14)

In Chapter 7, Eq. (5.14) will be directly used. However, to find ageneralisation of the usual expression used in the IFM (5.14), mustbe slightly modified. Under the assumption that Λ−1(ω, r, r′) is com-puted with Dirichlet boundary conditions on the contacts and thatthere are no fluctuations perpendicular to the Neumann boundaries,the term of (5.14) containing ∇r and Λ−1 can be partially integratedwithout remaining boundary terms. Doing so and using the definitions

Glm(ω, r) :=

∫ ∑

j

Hlj(ω, r′)(Λ−1(ω, r′, r)

)jmd3r′ + qhTFl (r)δm,0

(5.15)

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5.3. SOURCES OF NOISE 93

gives the final expression for δIl(ω)

δIl(ω) =:

∫

Ω

∑

m

Glm(ω, r)δζhmd3r −

∫

Ω

∑

m

∇rGlm(ω, r)δζS−1hm

~vd3r.

(5.16)

To compute the Fourier transform SδIkδIl(ω) of the correlation func-

tions of the fluctuations of the terminal currents, the Wiener-Khin-tchine theorem is used together with (5.16) to give

SδIkδIl(ω) =∫ ∫ ∑

mp

Glm(ω, r)Gkp(ω, r′)Sδζhm δζhp

(ω, r, r′)d3rd3r′

−∫ ∫ ∑

mp

Glm(ω, r)∇r′Gkp(ω, r′)Sδζhmδζ

S−1hp

~v(ω, r, r′)d3rd3r′

−∫ ∫ ∑

mp

∇rGlm(ω, r)Gkp(ω, r′)Sδζ

S−1hm

~vδζhp

(ω, r, r′)d3rd3r′

+

∫ ∫ ∑

mp

∇rGlm(ω, r)∇r′Gkp(ω, r′)Sδζ

S−1hm

~vδζ

S−1hp

~v(ω, r, r′)d3rd3r′,

(5.17)

where Sδζgmδζgp(ω, r, r′) is the Fourier transform of the correlation

function of δζgmwith δζgp

. Eq. (5.17) fully describes the generalisedIFM (GIFM). It contains the Green’s functions Glm(ω, r) which canbe directly computed for any given transport model, and the noisesources Sδζgmδζgp

(ω, r, r′).

5.3 Sources of noise

In this part the general assumptions about the Sδζgmδζgp(ω, r, r′) will

be given. Also, two special cases, where the noise sources can berelated to transport coefficients, will be discussed and the generalmethods to actually compute these noise sources for bulk materialwill be described.

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5.3.1 Approximations

To simplify the problem, one often assumes that the Sδζgmδζgp(ω, r, r′)

are spatially uncorrelated, i.e.

Sδζgmδζgp(ω, r, r′) = Sδζgmδζgp

(ω, r, r)δ3(r − r′), (5.18)

and that they do not depend on the frequency (white noise). Thefrequency-independence is certainly justified for frequencies smallerthan 10 GHz (see Chapter 2). The assumption that the noise sourcesare spatially uncorrelated should be taken with more care (see Chap-ter 6).

5.3.2 Space-homogeneous case

In the case of the space-homogeneous BE, the noise sources for themoments g = vi are, in TD equilibrium, directly related to the diffu-sion tensor, in the limit of vanishing frequencies:

Sδζviδζvj

(ω = 0) = Sδsviδsvj

= n(Dij +Dji). (5.19)

Using the theory developed in Chapter 2, the proof is simple. In the

space-homogeneous case at TD equilibrium (~F = 0) it follows from(5.7), that the last two terms disappear. The second term, contain-ing the time derivatives, will be zero at zero frequency. Therefore,Sδζvi

δζvj(ω = 0) = Sδsvi

δsvj. Putting f = feq in (3.49), using the

property

〈feq|C = 〈feqZ|

used in (3.54), the detailed balance principle, and the definition of thediffusion tensor (see (3.11)) gives

Sδsviδsvj

=

−∫

K

S−1vi

(k)vj(k)feq(k)d3k−

∫

K

S−1vj

(k)vi(k)feq(k)d3k = n(Dij+Dji)

2. (5.20)

5.4. COMPUTATION 95

5.3.3 Bixon-Zwanzig relation

In a paper by Bixon and Zwanzig (see [3]) an expression for Sδsv2viδsv2vj

for the case of an isotropic fluid is found:

m∗2

9Sδsv2vi

δsv2vj= kBT

2κδij , (5.21)

where κ is the thermal conductivity which can be expressed as functionof the mobility using the Wiedemann-Franz law (see e.g. [38]). InChapter 9, a comparison between the exact expression and (5.21) willshow that the Bixon-Zwanzig approximation works rather poorly atleast in silicon.

5.4 Computation

In bulk3, (5.6) reduces to

n∑

i

Q0g,i

N∑

j=1

∂ci∂Fj

δFj + n∑

i

[∂ ~Q2

g,i

∂TδT

]ci~F

+ n∑

i

~Q2g,i

N∑

j=1

∂ci∂Fj

δFj

~F = n

∑

i

giδci + δsg

+ n∑

i

Q0g,i

N∑

j=1

∂ci∂Fj

δFj − δci

+ n∑

i

~Q2g,i

N∑

j=1

∂ci∂Fj

δFj − δci

~F, (5.22)

where δsg is as in (5.3), but without the δn-term. To find (5.22), all

the terms in (5.6) containing ∇r, δn, or δ~F must be disregarded, aswell as all terms containing derivatives by F0.

3In the MC community the term ”bulk” is used for a zero-dimensional system,i.e. a system with constant density and constant electric field.


From (5.22) a natural definition for the bulk noise sources δζbulkg

of transport models can be found (see (5.7)):

δζbulkg := δsg + n∑

i

Q0g,i

N∑

j=1

∂ci∂Fj

δFj − δci

+ n∑

i

~Q2g,i

N∑

j=1

∂ci∂Fj

δFj − δci

~F.

(5.23)

The usual assumption for the simulation of devices is

δζg ≈ δζbulkg . (5.24)

It will be fulfilled, as long as the third term on the rhs of (5.7) doesnot dominate (small gradients). This has, to the authors knowledge,never been verified for realistic devices.

To go further, one assumes that the parametrisation of the trans-port coefficients has been done such that Fj = hj , ∀j 6= 0. In thiscase the Fourier transform of (5.22) can be rewritten as

∑

l

(Γ(ω))jl δgl = δζbulkgj, (5.25)

where Γ(ω) is a ((d+ 1)(N + 1)) × ((1 + d)(N + 1)) matrix, which isunequivocally defined by (5.22).

Using the Monte Carlo method, one usually computes Sδgiδgj(ω).

In fact, Γ(ω) can also be computed using MC or a given transportmodel. Thus, using the relation

Sδζbulkgi

δζbulkgj

(ω) =∑

lm

(Γ(ω))il (Γ(ω))∗jm Sδgmδgl

(ω), (5.26)

the bulk noise sources of transport models can be computed.

Note that if G-R processes are neglected, the dimension of thematrix Γ(ω) is reduced by one.

5.5. APPLICATION TO TRANSPORT MODELS 97

5.5 Application to common transportmodels

In the following, G-R processes will be neglected.

5.5.1 The drift-diffusion model

The approximations to find the DD model from (5.6) are given inSection 4.4. The resulting equations are given in Section 7.6. Sincethe DD model turns out for N = 0, i.e. when not more than h0 = 1is considered, the (d× d)-matrix Γ(ω) is the matrix −1, i.e.

Sδζbulkvi

δζbulkvj

(ω) = Sδviδvj(ω). (5.27)

5.5.2 The energy-balance model

The approximations in the derivation of the EB model from (5.6) arepartly given in Section 4.4. For a full derivation see e.g. [4]. Theresulting equations are given in Appendix C.1. The model turns out,when h0 = 1 and h1 = v2 are considered (N=1). The (2d+1)×(2d+1)matrix Γ can be rewritten as shown in (C.42) and (C.43).

5.6 Verification

To verify that the bulk noise sources (5.23) are good approximations ofthe true noise sources (5.7) in a device, one can directly compute (5.7)during a many-particle transient MC simulation. To do so, one needsthe derivatives wrt all Fi of the transport coefficients. They can becomputed using the same method as proposed in 8.8.1. The functionsδζgi

(t) usually contain delta functions of the time, which hamper thenumerical computation of the correlation functions Sδζgmδζgp

(w, r, r′).A simple trick to solve this problem is to fold the δζgi

(t) with a Gaus-sian (see Appendix D).


5.7 Two interesting basic approaches forthe EB model

Using the theory of the previous chapter for the EB model leads toequations containing (2d + 1) × (2d + 1) noise sources. Are all thesenoise sources really needed as suggested in [32]?

By carefully inspecting Eqs. (5.6) and (5.7) one can see, that foreach new moment hi the transport equations and the noise sourcesare modified, i.e. each new moment modifies all equations and noisesources, and adds noise sources to the system. One possible interpre-tation of this fact is that each new derivative of the transport coef-ficients adds fluctuations in the equation of a given order that mustbe compensated by new noise sources in this equation. In particular,the equation for the velocity fluctuation is extended by derivativesof the mobility/diffusivity by higher-order ”driving forces”, whichdo not represent additional contributions to the current fluctuations,but must be exactly compensated by the corresponding noise-sourceterms!

Following this hypothesis, two new systems of equations containingless noise sources can be written for the EB model.

5.7.1 The no-derivatives ansatz

Guided by the hypothesis of the previous section, the most simpleapproach is to remove all derivatives with respect to the driving forcesfrom (5.6) and (5.7). This decouples all moments from each other:

˙δn∑

i

Q0g,ici +

∑

i

[∂ ~Q1

g,i

∂nδn

]∇r (cin) +

∑

i

~Q1g,i∇r (ciδn)

+ n∑

i

~Q2g,iciδ

~F + n∑

i

[∂ ~Q2

g,i

∂nδn

]ci~F

+ δn∑

i

~Q2g,ici

~F = δn

(∑

i

gici −geqneq

)+ n

∑

i

giδci + δζg, (5.28)

5.7. TWO INTERESTING APPROACHES 99

where

δζg = δsg − n∑

i

Q0g,i

N∑

j=1

δci

−∑

i

~Q1g,i∇r (nδci) − n

∑

i

~Q2g,i

N∑

j=1

δci

~F. (5.29)

As one is interested in the fluctuations of the terminals currents,only the d-moments describing the current density are needed: g =vx, vy or vz.

Note that removing the∑Nj=0-terms on both sides of (5.6) is not

an approximation, but it changes the validity of assumption (5.24).

After some numerical experiments, it turned out that this ansatzdoes not work, presumably because the assumption (5.24) is poorlyfulfilled. For this, two reasons can be given. Firstly, δζg (5.29) con-tains non-compensated fluctuations of the density which are absent inδζbulkg . Secondly, the ∇r-term of (5.28) might not be negligible. Thisleads to the second ansatz which should better fulfil (5.29) withoutthe need of more noise sources.

5.7.2 Ansatz using the gradient of the quasi-Fermipotential

To try to save the no-derivatives ansatz without additional noisesources, the approximation (4.15) may be used, where F denotes the


norm of the gradient of the quasi-Fermi potential:

˙δn∑

i

Q0g,ici + n

∑

i

Q0g,i

∂ci∂F

δF

+∑

i

[∂ ~Q1

g,i

∂nδn+

∂ ~Q1g,i

∂T

∂T

∂FδF

]∇r (cin) +

∑

i

~Q1g,i∇r

(n∂ci∂F

δF

)

+∑

i

~Q1g,i∇r (ciδn) + n

∑

i

~Q2g,iciδ

~F+ n

∑

i

[∂ ~Q2

g,i

∂nδn+

∂ ~Q2g,i

∂T

∂T

∂FδF

]ci~F + n

∑

i

~Q2g,i

(∂ci∂F

δF

)~F

+ δn∑

i

~Q2g,ici

~F = δn

(∑

i

gici −geqneq

)+ n

∑

i

giδci + δζg, (5.30)

where

δζg := δsg + n∑

i

Q0g,i

(∂ci∂F

δF − δci

)

+∑

i

~Q1g,i∇r

((∂ci∂F

δF − δci)n

)+ n

∑

i

~Q2g,i

(∂ci∂F

δF − δci

)~F,

(5.31)

and g = vx, vy or vz.One expects from this ansatz that it partially solves the problems

of the density fluctuations and of the ∇r-term. In fact, as describedin Chapter 9, this ansatz gives very good results when compared tomany-particle MC simulations.

Chapter 6

Systematic comparisonto existing formulations

6.1 Introduction

In this short chapter the major flaw of the GIFM, which the atten-tive reader must already have suspected ever since Chapter 3, willbe explained. Then the alternative method to compute noise called”Acceleration Fluctuation Scheme” (see e.g. [51]) will be shortly ex-plained. Finally, both method will be compared with each other.

6.2 Formal flaw of the GIFM

The master equations for the GIFM are given in (5.2). Consideringthe velocity-moment (i.e. g = vj , j = x, y or z), the rhs of (5.2) canbe rewritten as

∆S−1vj

(r, t) − n(r)vj(r), (6.1)

where

∆S−1vj

(r, t) :=

∞∑

m=0

δ(t− tm)δ3(r− rnm(tm))

[S−1vj

(k+nm

) − S−1vj

(k−nm)].

(6.2)

101

102 CHAPTER 6. SYSTEMATIC COMPARISON

The function ∆S−1vj

(r, t) has by construction the property (see Sec-tion 3.7.5)

∆S−1vj (r) = n(r)vj(r), (6.3)

i.e. the mean value of ∆S−1vj

is the same as the mean value of the ve-locity density. The important point is that, altough the mean valuesof ∆S−1

vjand n(r, t)vj(r, t) are the same, their fluctuations are differ-

ent. This is the main problem of the GIFM, i.e. the method startsfrom an equation for the fluctuations of ∆S−1

vjaround its stationary

state nvj and not for the fluctuations of the velocity density (cur-rent density) as needed. It is impossible to derive an expression forn(r, t)vj(r, t) from the BE (3.61). It is only possible to find an equa-tion for d(nvj)(r, t)/dt, by integrating (3.61) with the velocity. Thiswill be the subject of the next section. To save the method, one has toadd n(r, t)vj(r, t)−n(r, t)vj(r, t) = 0 on the rhs of (5.2), absorbing thefirst term in the transport equation and the second in the noise source(see (5.3) and (5.4)). This leads to Langevin noise sources which arespace- and time-correlated, even though the function ∆S−1

vjitself is

space- and time-uncorrelated by construction. This is the major flawof the GIFM.

6.3 The acceleration-fluctuation scheme

To correct the flaw of the GIFM, the acceleration-fluctuation scheme(AFS) was proposed (see [51]). The main idea is to write a masterequation which contains the fluctuations of the velocity density ex-plicitly. To do so, (3.61) (resp. (4.1)) is integrated directly with thefunction g and not with the function S−1

g as in (3.72) (resp. (4.2)).The resulting equation can be written in the same basis as in (4.3).Using the definitions (4.4) and

Q3g,i(r, t) :=

∫

Bz

STg (t, r, k)αi(k)d3k, (6.4)

~Q4g,i(r, t) :=

∫

Bz

g(k)~v(k)αi(k)d3k, (6.5)

~Q5g,i(r, t) :=

∫

Bz

g(k)∇kαi(k)d3k, (6.6)

6.3. THE ACCELERATION-FLUCTUATION SCHEME 103

an analogon to (4.8) can be written:

n∑

i

Q3g,ici −

∑

i

~Q4g,ici∇rn− n

∑

i

~Q4g,i∇rci−

n∑

i

~Q5g,ici

~F = n∑

i

gici + n∑

i

gici. (6.7)

In the case of noise the analogon to (5.2) is

−∑

i

[∂ ~Q4

g,i

∂nδn+

∂ ~Q4g,i

∂TδT

]∇r (cin) −

∑

i

~Q4g,i∇r (δcin)

−∑

i

~Q4g,i∇r (ciδn) − n

∑

i

~Q5g,iciδ

~F− n

∑

i

[∂ ~Q5

g,i

∂nδn−

∂ ~Q5g,i

∂TδT

]ci~F− n

∑

i

~Q5g,iδci

~F− δn

∑

i

~Q5g,ici

~F = ˙δn∑

i

gici + n∑

i

giδci

−∞∑

m=0


[g(k+

nm) − g(k−nm

)]+ n

∑

i

Q3g,ici,

(6.8)

and the corresponding equation to (5.6) reads

−∑

i

[∂ ~Q4

g,i

∂nδn+

∂ ~Q4g,i

∂TδT

]∇r (cin)−

∑

i

~Q4g,i∇r

N∑

j=0

∂ci∂Fj

δFjn

−∑

i

~Q4g,i∇r (ciδn) − n

∑

i

~Q5g,iciδ

~F− n

∑

i

[∂ ~Q5

g,i

∂nδn−

∂ ~Q5g,i

∂TδT

]ci~F− n

∑

i

~Q5g,i

N∑

j=0

∂ci∂Fj

δFj~F− δn

∑

i

~Q5g,ici

~F = ˙δn∑

i

gici + n∑

i

giδci + δςg, (6.9)


where

δςg :=

−∞∑

m=0


[g(k+

nm) − g(k−nm

)]+ n

∑

i

Q3g,ici

−∑

i

~Q4g,i∇r

(

N∑

j=0

∂ci∂Fj

δFj − δci)n

−n

∑

i

~Q5g,i(

N∑

j=0

∂ci∂Fj

δFj − δci)~F.(6.10)

The function δςg is the equivalent to δζg (see (5.7)), i.e. it is the noisesource of the transport models for the the transport equation (6.8).As described in [51], the first part of (6.10),

−∞∑

m=0

δ(t−tm)δ3(r−rnm(tm))

[g(k+

nm) − g(k−nm

)]+n∑

i

Q3g,ici =: δug,

(6.11)is a function which is uncorrelated in time and space. Unfortunately,δug is equal to δςg only in the limit of an infinit number of drivingforces. In the case of a finite number of driving forces, the noisesources of the transport models are space- and time-correlated as inthe case of the GIFM. This is the main problem of the accelerationfluctuation scheme. In the literature the usual assumption is

δug ≈ δςg, (6.12)

(see [51]) independently of the number of driving forces. At leastin the case of silicon, taking only moments up to the energy currentdensity (N = 1) is not sufficient to fulfil the approximation (6.12) ina satisfying way, as discussed in the Chapter 9.

6.4 Comparison of the GIFM with the AFS

A common problem of the GIFM and of the AFS is that the noisesources (5.7) and (6.10) containe a ∇r-term that can only be min-imised by taking more moments of the BE into account. The influ-ence of this term in different types of devices would be an interestingresearch subject.

6.4. COMPARISON OF THE GIFM WITH THE AFS 105

The advantage of the AFS over the GIFM is that in the asymptoticcase of an infinit number of moments of the BE, the noise sourcesbecome space- and time-uncorrelated.

The advantage of the GIFM over the AFS is perhaps that in thecase of a small number of moments of the BE, the approximation(5.24) is much better fulfilled as the approximation (6.12), becauseδζbulkg already contains all the terms in δci and δFj , which is not thecase for δug.

As a summary, one can say that the AFS looks formally betterthan the GIFM, but that in the practical cases of the DD and EBmodels (small number of moments), the GIFM may still be better.This however must be verified in practical cases, which is beyond thescope of this thesis.

Part III

Numerics andSimulation Tools

107

Chapter 7

Numerical methods fortransport models

7.1 Introduction

In this chapter the different numerical techniques which were usedto produce all the results in this thesis will be addressed. The wholenumerics in this chapter is based on the finite element method (FEM).The FEM was used, because its formulation allows one to use differentboundary conditions in a very easy way, the formation of the rigiditymatrix is straightforward and still very readable after implementation.It also allows to use potentials as solution variables, and last but notleast, the approximations intrensic to the method are very clear. It iswell known that the box method would be more appropriate due toits pretended better convergence properties, but for all the examplestreated in this thesis the FEM did the job as well. First, the methodused to solve the DD and the EB models will be described. Then,the implementation of the computation of the Y-parameters will beexplained. Finally, the way to compute numerically relevant Green’sfunctions and noise figures for RF noise will be fully described.

109

110 CHAPTER 7. NUMERICAL METHODS

7.2 The FEM and the DD model

This section describes the discretisation of the DD equations for elec-trons and holes using the FEM, the algorithm to use to solve theequations, and the appropriatness of the discretisation for all quanti-ties that are needed in this work.

7.2.1 The stationary DD equations

The system of Poisson and the DD equations are given by (7.1)–(7.3):

−∇ǫ∇ϕ = q(p− n+N − P ), (7.1)

~Jnq

= µn

(εL∇n− n∇(ϕ− εg

2)), (7.2)

~Jpq

= −µp(εL∇p+ p∇(ϕ+

εg2

)), (7.3)

where ϕ is the electrostatic potential, ǫ the dielectric constant, µ themobility, n (p) the electron (hole) density, N the acceptor concentra-tion, P the donor concentration, εL the lattice temperature in eV,εg the band gap narrowing, and q the absolute value of the electroncharge.

The potential ϕ and the quasi-Fermi potentials ψn and ψp havebeen chosen as solution variables. These variables were selected be-cause they should behave in a linear way on the discretisation element.Using the relations

n(x) = ni exp

(−ϕ(x) − εg(x)/2 − ψn(x)

εL

), (7.4)

p(x) = ni exp

(ϕ(x) + εg(x)/2 − ψp(x)

εL

), (7.5)

where ni is the semiconductor intrinsic density, one can rewrite (7.1)–(7.3) as

−∇ǫ∇ϕ = q(ni exp

(ϕ+ εg/2 − ψp

εL

)

− ni exp

(−ϕ− εg/2 − ψn

εL

)+N − P ), (7.6)

7.2. THE FEM AND THE DD MODEL 111

~Jnq

= −µnni exp

(−ϕ− εg/2 − ψn

εL

)∇ψn = −µnn∇ψn, (7.7)

~Jpq

= −µpni exp

(ϕ+ εg/2 − ψp

εL

)∇ψp = −µpp∇ψp. (7.8)

7.2.2 The weak formulation

The main idea of the FEM is to integrate (7.6)–(7.8) on the device Ωwith test functions chosen in a proper function space and to requirethat for each of the test functions the equations (7.6)–(7.8) are ful-filled. Let T be a shape-regular, affine triangulation of the device Ω.In this work, the function space VN of the continuous piecewise linearfunctions on the triangulation T has been chosen. In order to satisfythe Dirichlet boundary conditions, one has to restrict the functionspace VN to the space UN of the continuous piecewise linear functionswhich are zero on the Dirichlet boundary.

Using the continuity equations

∇~Jnq

= R, (7.9)

∇~Jpq

= −R, (7.10)

one obtains for (7.6)–(7.8) the following results:

∫

Ω

−u(x)(∇(ǫ(x)∇ϕ))d2x =

q

∫

Ω

u(x) p(x) − n(x) +N(x) − P (x) d2x, (7.11)

∫

Ω

−v(x)∇(µn(x)n(x)∇ψn(x))d2x =

∫

Ω

v(x)R(x)d2x, (7.12)

∫

Ω

−w(x)∇(µp(x)p(x)∇ψp(x))d2x =

∫

Ω

−w(x)R(x)d2x, (7.13)

where R(x) is the function describing generation-recombination pro-cesses, and u, v and w are functions of UN . By partially integrating


the lhs of (7.11)–(7.13) and by using the property that the test func-tions are zero on the Dirichlet boundary, one finds:∫

Ω

∇u(x)(ǫ(x)∇ϕ)d2x = q

∫

Ω

u(x) p(x) − n(x) +N(x) − P (x) d2x,

(7.14)∫

Ω

∇v(x)(µn(x)n(x)∇ψn(x))d2x =

∫

Ω

v(x)R(x)d2x, (7.15)

∫

Ω

∇w(x)(µp(x)p(x)∇ψp(x))d2x =

∫

Ω

−w(x)R(x)d2x. (7.16)

Thus (7.14)–(7.16) must be solved for all u, v and w in UN .

7.2.3 Discretisation

The system can be solved under the assumption that ϕ(x), ψn(x) andψp(x) are functions in VN . The mobility can be locally parameterisedby the norm of the gradient of the quasi-Fermi potential, which is aconstant inside each triangle.

Local parametrisation

Eq. (7.14) integrated on a given triangle T ∈ T takes the form∫

T

∇u(x)(ǫ(x)∇ϕ)d2x =

q

∫

T

u(x) p(x) − n(x) +N(x) − P (x) d2x. (7.17)

A triangle T is illustrated in Fig. 7.1. To simplify the problem, theintegral on T can be expressed as an integral on the unitary triangleT shown in Fig. 7.1. To do so, the following variable transformationmust be performed:

~x :=

(x1

x2

)= ~P0 + ξ1( ~P1 − ~P0) + ξ2( ~P2 − ~P0) =: ~P0 +BTT

~ξ, (7.18)

BT :=

(( ~P1 − ~P0)

T

( ~P2 − ~P0)T

), (7.19)

~ξ :=

(ξ1ξ2

). (7.20)


ξ1

ξ2

0 1

1

x2

x1

~P0

~P1

~P2

Figure 7.1: The unity triangle T (left) and a given triangle T (right).

In this new coordinate system:

d2x = det (BT )d2ξ, (7.21)

∇~xf = B−1T ∇~ξf. (7.22)

Any linear function f(x) on the triangle T can now be written as

f(x) =2∑

i=0

f iξi(x), (7.23)

where ξ0(x) := 1 − ξ1(x) − ξ2(x) and fi is the value of f(x) in the

point ~Pi.

Under the assumption that ǫ(x) is a constant on each triangle,


Eq. (7.17) can be rewritten as

det (BT )ǫ2∑

i=0

2∑

j=0

uiϕj∫

bT(B−1

T ∇~ξξi)T (B−1

T ∇~ξξj)d2ξ =

det (BT )q2∑

i=0

ui∫

bTξi

ni exp

2∑

j=0

(ϕj + εjg/2 − ψjp)ξj

εL

d2ξ

−det (BT )q2∑

i=0

ui∫

bTξi

ni exp

−

2∑

j=0

− (ϕj − εjg/2 − ψjn)ξj

εL

d2ξ

+ det (BT )q

2∑

i=0

ui∫

bTξi (N(ξ) − P (ξ)) d2ξ. (7.24)

Discrete form of the equations

Using the definitions

~bk := B−1T ∇~ξξk = const., (7.25)

Aij := ~biT ~bj, (7.26)

n0(ϕ, ψn, εg) :=

∫

bTn(ξ)d2ξ, p0(ϕ, ψp, εg) :=

∫

bTp(ξ)d2ξ, (7.27)

n1(i, ϕ, ψn, εg) :=

∫

bTξini exp

2∑

j=0

− (ϕj − εjg/2 − ψjn)ξj

εL

d2ξ,

(7.28)

p1(i, ϕ, ψp, εg) :=

∫

bTξini exp

2∑

j=0

(ϕj + εjg/2 − ψjp)ξj

εL

d2ξ, (7.29)

ρi :=

∫

bTξi (N(ξ) − P (ξ)) d2ξ, (7.30)



∑

T∈T

det (BT )2∑

i=0

2∑

j=0

ǫ

2uiϕjAij + quin1(i) − quip1(i)

=

∑

T∈T

det (BT )q

2∑

i=0

uiρi. (7.31)

An important property of n0(ϕ, ψn, εg), p0(ϕ, ψp, εg), n1(i, ϕ, ψn, εg)and p1(i, ϕ, ψp, εg) and of their derivatives is that they can be com-puted analytically.

Assuming that the mobility is constant within a triangle, Eqs. (7.15)and (7.16) can be written in the same way as Eq. (7.14):

∑

T∈T

det (BT )

2∑

i=0

2∑

j=0

µnviψjnAijn0 − viR1(i)

= 0, (7.32)

∑

T∈T

det (BT )

2∑

i=0

2∑

j=0

µpwiψjpAijp0 + wiR1(i)

= 0, (7.33)

R1(i, ϕ, ψn, ψp, εg) :=

∫

bTξiR(~ξ)d2ξ. (7.34)

Note that in general R1(i, ϕ, ψn, ψp, εg) cannot be computed ana-lytically.

7.2.4 Iterative algorithm

To solve the nonlinear system consisting of Eqs. (7.31)–(7.33), a simpleNewton-based algorithm is used.

The system can be formally rewritten as

~sTX(~φ)~φ− ~sT~l(~φ) = 0, ∀~s⇔ X(~φ)~φ −~l(~φ) = 0, (7.35)


with

~s :=

u1

v1

w1

.

.

.uN

vN

wN

, ~φ :=

ϕ1

ψ1n

ψ1p

.

.

.ϕN

ψNnψNp

. (7.36)

Let ~φs be the solution of Eq. (7.35), and ~φi be an approximation

of the solution such that ~φi =: ~φs + δ~φi. Then, Eq. (7.35) can beapproximated by

X(~φi − δ~φi)(~φi − δ~φi) −~l(~φi − δ~φi) = 0,

⇒ X(~φi)~φi −~l(~φi)︸︷︷︸:=~ri(~φi)

≃ (δX(~φ)~φi

δ~φ+X(~φi) −

δ~l(~φ)

δ~φ)

︸︷︷︸:=M(~φi)

δ~φi,

⇔ δ~φi = M(~φi)−1~ri(~φi). (7.37)

Eq. (7.37) gives us the Newton algorithm. In rare cases, the directNewton does not converge, and an alternative method must be used.Fig. 7.2 shows the flowchart of the algorithm used in these cases. Inorder to control the solution locally, all errors are computed using themaximum norm.

It is important to note that M(~φi) can be computed analytically

up to the δ~l(~φ)

δ~φ-term which has to be computed numerically, if a

generation-recombination model is used. However, this term does notseem to be critical for the convergence.

7.2.5 Properties of the proposed discretisation

It is easily shown that the discretisation reduces to the proper equa-tions in the limiting case of pure drift (n = const., p = const.) and ofpure diffusion (∇ϕ = 0). In the 1D case the equations do not reduce


No

Start with charge neutrality

Iterate with the previous mobilityand the previous G−R rate.

Converged?

Iterate

Converged?

Switch the mobility update on

Switch the G−R update on

Iterate

Converged?

Reduce the step size

Step size is too small?

goal is reached?

The End

No

No

Increase the step size

Yes

Yes

Yes

Yes

Yes

No

No

Figure 7.2: Flowchart of the iterative solver for the DD model.


to the Scharfetter-Gummel case, but they do not reduce to the well-known nonstable case, either. The curl condition ∇× ~Jn = − ~E × ~Jnis automatically fulfilled. In the case R = 0 the model does not give∇ ~Jn = 0. However, for all test functions in UN , Eqs. (7.12) and (7.13)are fulfilled, which means that as long as the solution to Eq. (7.35) isintegrated with a test function, the divergence condition is fulfilled.Since in the following the solution to Eq. (7.35) will never be used onits own, but always integrated with a test function, the approach iswell justified.

7.3 The FEM and the energy-balance model

This section describes the discretisation of the EB model using theBløtekjær approach as an example (see [53, 4]), the variables used,and the algorithm to solve the equations.

7.3.1 The stationary energy-balance equations

The Poisson equation is given by Eq. (7.1). The equations for thecurrent density and the energy current density are

~Jnq

= µn

(∇(εnn) − n∇(ϕ− εg

2)), (7.38)

~Jpq

= −µp(∇(εpp) + p∇(ϕ+

εg2

)), (7.39)

∇~Snq

= −~Jnq∇(ϕ − εg

2)) − 3n(εn − εL)

2τεn

− 3

2εnR, (7.40)

∇~Spq

= −~Jpq∇(ϕ+

εg2

) − 3p(εp − εL)

2τεp

− 3

2εpR, (7.41)

~Snq

= −5

2µnnεn∇εn − 5

2εn

~Jnq, (7.42)

~Spq

= −5

2µppεp∇εp +

5

2εp~Jpq, (7.43)

7.3. THE FEM AND THE ENERGY-BALANCE MODEL 119

where εn resp. εp is the electron (hole) temperature in eV, and τεn

resp. τεpthe electron (hole) energy relaxation time.

To solve this system, the solution variables ϕ, ψn, ψp, εn and εphave been used. The relationship between the electrostatic potential,the quasi-Fermi potentials and the densities is as in Eqs. (7.4),(7.5).With these solution variables, Eqs. (7.38) and (7.39) can be rewritten:

~Jnq

= nµn

((εnεL

− 1)∇(ϕ− εg(x)

2) − εn

εL∇ψn + ∇εn

), (7.44)

~Jpq

= pµp

((εpεL

− 1)∇(ϕ+εg(x)

2) − εp

εL∇ψp −∇εp

). (7.45)

The following shortcuts will be used for convenience:

~Fn(x) := (εn(x)

εL− 1)∇(ϕ(x) − εg(x)

2) − εn(x)

εL∇ψn(x) + ∇εn(x),

(7.46)

~Fp(x) := (εp(x)

εL−1)∇(ϕ(x)+

εg(x)

2)− εp(x)

εL∇ψp(x)−∇εp(x). (7.47)

7.3.2 The weak formulation

The idea is the same as in Section 7.2.2. Integrating the divergenceof Eq. (7.38) resp. Eq. (7.39) with a test function and performing apartial integration gives

−∫

Ω

∇v(x)[µn(x)n(x)~Fn(x)

]d2x =

∫

Ω

v(x)R(x)d2x, (7.48)

−∫

Ω

∇w(x)[µp(x)p(x)~Fp(x)

]d2x =

∫

Ω

−w(x)R(x)d2x. (7.49)

Integrating Eqs. (7.40) and (7.41) with test functions and by partiallyintegrating the lhs, one finds∫

Ω

∇g(x)[5

2µn(x)n(x)εn(x)

(∇εn(x) + ~Fn(x)

)]d2x

= −∫

Ω

g(x)

[µn(x)n(x)~Fn(x)∇(ϕ(x) − εg(x)

2)

]d2x

−∫

Ω

g(x)

[3n(x)(εn(x) − εL)

2τεn(x)

+3

2εn(x)R(x)

]d2x, (7.50)


∫

Ω

∇h(x)[5

2µp(x)p(x)εp(x)

(∇εp(x) − ~Fp(x)

)]d2x

= −∫

Ω

h(x)

[µp(x)p(x)~Fp(x)∇(ϕ(x) +

εg(x)

2)

]d2x

−∫

Ω

h(x)

[3p(x)(εp(x) − εL)

2τεp(x)

+3

2εp(x)R(x)

]d2x. (7.51)

The task is now to solve Eq. (7.14) and Eqs. (7.48)–(7.51) for allu, v, w, g and h in UN .


Again (see Section 7.2.3), to solve the system, the assumption is madethat ϕ(x), ψn(x), ψp(x), εn(x) and εp(x) are functions in VN .

Local parametrisation of the mobility

The mobility can be locally parameterised using one of the followingdriving forces:

1. | ~Jn,p

q(n,p)µn,p| = |~Fn,p|

2. Gradient of the quasi-Fermi potential: |∇ψn,p|3. the local energy εn,p.

The first possibility is the most natural, because in the case of bulksilicon it gives the norm of the electric field, whereas in the case ofthe DD model it gives the norm of the gradient quasi-Fermi potential.Therefore, using |~Fn,p| as driving force is the most natural parametri-sation.

Discrete form of the equations

Using the definitions

n2(i, j, ϕ, ψn, εg) :=

∫

bTξiξjn(ξ)d2ξ, (7.52)

p2(i, j, ϕ, ψp, εg) :=

∫

bTξiξjp(ξ)d

2ξ, (7.53)

7.3. THE FEM AND THE ENERGY-BALANCE MODEL 121

Ainϕ :=

2∑

j=0

Aij(ϕj − εjg

2), Aipϕ :=

2∑

j=0

Aij(ϕj +

εjg2

), (7.54)

Aiψn,p:=

2∑

j=0

Aijψjn,p, A

iεn,p

:=2∑

j=0

Aijεjn,p, (7.55)

Anϕϕ :=

2∑

j=0

Ajnϕ(ϕj − εjg2

), Apϕϕ :=

2∑

j=0

Ajpϕ(ϕj +εjg2

), (7.56)

Aψnϕ :=

2∑

j=0

Ajψn(ϕj − εjg

2), Aψpϕ :=

2∑

j=0

Ajψp(ϕj +

εjg2

), (7.57)

Aεnϕ :=2∑

j=0

Ajεn(ϕj − εjg

2) Aεpϕ :=

2∑

j=0

Ajεp(ϕj +

εjg2

), (7.58)

Hεn:=

2∑

j=0

εjnεLn1(j, ϕ, ψn, εg), Hεp

:=

2∑

j=0

εjpεLp1(j, ϕ, ψp, εg), (7.59)

Hiεn

:=

2∑

j=0

εjnεLn2(i, j, ϕ, ψn, εg), H

iεp

:=

2∑

j=0

εjpεLp2(i, j, ϕ, ψp, εg),

(7.60)

Hεnεn:=

2∑

i=0

εinεLHiεn, Hεpεp

:=

2∑

i=0

εipεLHiεp, (7.61)

one can rewrite Eqs. (7.48)–(7.51) as

∑

T∈T

det (BT )

2∑

i=0

viµn[Hεn

(Ainϕ −Aiψn) − n0(A

inϕ −Aiεn

)]

+∑

T∈T

det (BT )

2∑

i=0

viR1(i) = 0, (7.62)


∑

T∈T

det (BT )

2∑

i=0

wiµp

[Hεp

(Aipϕ −Aiψp) − p0(A

ipϕ +Aiεp

)]

−∑

T∈T

det (BT )

2∑

i=0

wiR1(i) = 0, (7.63)

∑

T∈T

det (BT )2∑

i=0

gi5

2εLµn

[Hεn

(Ainϕ − 2Aiεn) −Hεnεn

(Ainϕ −Aiψn)]

+∑

T∈T

det (BT )

2∑

i=0

giµn[n1(i) (Anϕϕ −Aεnϕ) −Hi

εn(Anϕϕ −Aψnϕ)

]

−∑

T∈T

det (BT )2∑

i=0

gi3

2

εLτεn

(Hiεn

− n1(i))

+2∑

j=0

εjnR2(i, j)

= 0,

(7.64)

∑

T∈T

det (BT )

2∑

i=0

hi5

2εLµp

[Hεpεp

(Aipϕ −Aiψp) −Hεp

(Aipϕ − 2Aiεp)]

+∑

T∈T

det (BT )2∑

i=0

hiµp

[p1(i)

(Apϕϕ +Aεpϕ

)−Hi

εp

(Apϕϕ −Aψpϕ

)]

−∑

T∈T

det (BT )

2∑

i=0

hi3

2

εLτεp

(Hiεp

− p1(i))

+

2∑

j=0

εjpR2(i, j)

= 0,

(7.65)

R2(i, j, ϕ, ψn, ψp, εg) :=

∫

bTξiξjR(~ξ)d2ξ. (7.66)

Again, an important property of the quantities in Eqs. (7.52)–(7.61) and of their derivatives is that they can be computed analyt-ically. Note however, that R2(i, j, ϕ, ψn, ψp, εg) cannot be computedanalytically in general.

7.4. TERMINAL CURRENTS 123

7.3.4 Iterative algorithm and properties of the dis-cretisation

To solve the nonlinear consisting of the Eqs. (7.31) and (7.62)–(7.65),a simple Newton-based algorithm is used. In some rare cases, wherethe algorithm does not converge, a modified version is used. It isdepicted in the flowchart of Fig. 7.3. In order to control locally thesolution, all errors are computed using the maximum norm.

Again, the solution fulfils the divergence condition when it is in-tegrated with a test function of UN .

7.4 Terminal currents

The total current Il on the l-th contact δΩl is the integral over thecontact of the total current density. It can be expressed as (comparewith (5.13))

Il :=

∫

δΩl

(~Jn + ~Jp + ǫ

δ ~E

δt

)~nda, (7.67)

where ~n is the unit vector perpendicular to the contact. To evaluate(7.67), the Ramo-Shockley test functions (RSTF) can be used (seee.g. [32]). The l-th RSTF hRSl has the following properties:

∇(ǫ∇hRSl (x)) = 0, ∀x ∈ Ω (7.68)

hRSl (x) = 1, ∀x ∈ δΩl, hRSm (x) = 0, ∀x ∈ δΩm, m 6= l. (7.69)

One can show that

Il =

∫

Ω

(~Jn(x, t) + ~Jp(x, t)

)(∇hRSl (x))d2x−

Ncont∑

m=1

Cl,mδVm(t)

δt,

(7.70)where

Cl,m :=

∫

Ω

ǫ(x)(∇hRSl (x))(∇hRSm (x))d2x (7.71)

and Vm(t) is the potential on the m-th contact at time t.The RSTFs are computed with the FEM under the assumption

that hRSl ∈ VN .


Switch the temperature update on

Start with charge neutrality

Converged?

Iterate

Switch the mobility update onReduce the step size

Step size is too small?

Iterate with the previous mobility,

and G−R rate.the previous temperature

Converged?

Iterate

Converged?

Iterate

Converged?

goal is reached?

The End

Increase the step size

Switch the G−R update on

No

No

Yes

Yes

No

No

YesNo

YesNo

Yes

Yes

Figure 7.3: Flowchart of the iterative solver for the EB model.

7.4. TERMINAL CURRENTS 125

All terminal currents in this thesis where computed using Eq. (7.70).The terminal currents must fulfil the current continuity:

Ncont∑

l=0

Il = 0. (7.72)

To prove that the discretisation scheme actually has this property,hRSl can be unequivocally separated in the sum of a function ul ofUN and a function rl of VN . rl is the function which takes the valueone on the l-th contact points and zero on all other points. Therefore,ul = hRSl − rl is a function of UN . Because the numerical solutions~Jn and ~Jp solve exactly the equation

∫

Ω

∇v( ~Jn + ~Jp)d2x = 0, ∀v ∈ UN , (7.73)


Il =

∫

Ω

(~Jn(x, t) + ~Jp(x, t)

)(∇rl(x))d2x−

Ncont∑

m=1

Cl,mδVm(t)

δt. (7.74)

Using the function 1 −∑Ncont

l=0 rl(x) := p(x) ∈ UN one can write

0 =

∫

Ω

∇1( ~Jn(x) + ~Jp(x))d2x

=

∫

Ω

∇(p(x) +

Ncont∑

l=0

rl(x))( ~Jn(x) + ~Jp(x))d2x

=

Ncont∑

l=0

∫

Ω

∇rl(x)( ~Jn(x) + ~Jp(x))d2x. (7.75)

In the same way one proves

Ncont∑

l=0

Cl,m = 0, ∀m = 1...Ncont. (7.76)

Therefore, Eq. (7.72) is true.



7.5.1 The Y-parameters

Linearising Eq. (7.70) around a stationary state using the notations

~Jn,p(x, t) =: ~Jn,p(x) + δ ~Jn,p(x, t), (7.77)

Vm(t) =: Vm + δVm(t), (7.78)

where ~Jn,p(x) is the mean current density and Vm is the mean voltageon the m-th contact, and applying the Fourier transformation, definesthe Y -parameters:

δIl(ω) =

∫

Ω

(δ ~Jn(x, ω) + δ ~Jp(x, ω)

)∇hRSl (x)d2x

+ iω

Ncont∑

m=1

Cl,mδVm(ω) =:

Ncont∑

m=1

Yl,m(ω)δVm(ω). (7.79)


In the following only the case of the DD model will be considered.The generalisation to the EB model is straightforward.

The divergence of the fluctuations of the currents are:

∇δ ~Jn = −q∇[δµnn∇ψn + µnδn∇ψn + µnn∇δψn

]

+ iωq∇[αnδn∇ψn + δαnn∇ψn + αnn∇δψn

]= −iωδn, (7.80)

∇δ ~Jp = −q∇[δµpp∇ψp + µpδp∇ψp + µpp∇δψp

]

+ iωq∇[αpδp∇ψp + δαpp∇ψp + αpp∇δψp

]= iωδp, (7.81)

where αn := µ2nme/q and αp := µ2

pmh/q. δµ represents the totalderivative of µ whereas δα represent the total derivative of α withoutthe derivatives by the densities. The origin and signification of αn,pand δα are explained in Chapter 4.

The terms containing R were not considered, because they disap-pear in the expression δ ~Jn + δ ~Jp.


The linearisation of the Poisson equation (see (7.1)) reads

−∇ǫ∇δϕ = q(δp− δn). (7.82)

The weak formulation for the FEM is found by integrating eachof the equations (7.80)–(7.82) with a test function of UN and by per-forming a partial integration of the lhs. Then, the system of equationsgiven by (7.80)–(7.82) takes the form

~sT

B(ω) − iωC︸︷︷︸

:=Λ(ω)

δ~φ(ω) = 0, ∀~s, (7.83)

where N is the number of points in the discretisation including thepoints on the contacts. The matrix B(ω) originates from the lhs of(7.80) and (7.81), and from eq. (7.82). The matrix C comes fromthe rhs of (7.80) and (7.81) and is, therefore, real. One will see belowwhy the separation of the matrix Λ in these two matrices is reasonable.The vectors ~s and δ~φ are of the form

~s :=

u1

v1

w1

.

.

.uN

vN

wN

, δ~φ :=

δϕ1

δψ1n

δψ1p

.

.

.δϕN

δψNnδψNp

, (7.84)

where f i is the value of the function f ∈ VN on the point i. u,v andw are test functions of UN .

7.5.3 Computation

In the following the l-th contact is assumed to have Ml points andthe

∑Ncont

l=1 Ml =: Π-first points are assumed to be on the contacts

i.e. the 3Π-first components of the vectors ~s and δ~φ come from the


contact points. Those assumptions are illustrated in (7.85):

~sT =

~sT1 · · ·~sTM1︸︷︷︸

first contact

· · ·~sTΠ−MNcont· · ·~sTΠ︸︷︷︸

last contact

~sTΠ+1 · · ·~sTN︸︷︷︸all other points

,

~sTj :=(ujvjwj

). (7.85)

This leads to a natural box form of the matrices Λ, B and C whichall have the structure

Γ =

(Γ11 Γ12

Γ21 Γ22

), (7.86)

where the Γ11 component is a 3Π× 3Π, the Γ12 component is a 3Π×3(N−Π), the Γ21 component is a 3(N−Π)×3Π and the Γ22 componentis a 3(N − Π) × 3(N − Π) matrix.

Boundary conditions

For the AC analysis, the device under test is stimulated with a smallperiodic signal of frequency ω and amplitude δVm(ω) on its m-thcontact. On the other contacts no change of the electric and quasi-Fermi potentials are allowed. Taking these conditions into accountleads to the system

( 1 0Λ21 Λ22

)δ~φm = δVm(ω)~em, (7.87)

where ~em is such that the components arising from the m-th contactare all equal to 1, but all others are 0. Solving (7.87) for δ~φm gives theresponse of the device to the stimulation on the m-th contact in eachpoint of the discretisation. However, one is not directly interestedin δ~φm, but one wants to use it to compute the term in the lhs ofEq. (7.79) which contains the δ ~Jn,p. This term is

~hTl B(ω)δ~φm(ω) = δVm(ω)~hTl B(ω)

( 1 0Λ21 Λ22

)−1

~em, (7.88)

~hTl :=(0 −(hRSl )1 −(hRSl )1 · · · 0 −(hRSl )N −(hRSl )N

).

(7.89)

7.6. HIGH-FREQUENCY NOISE 129

Putting (7.88) in Eq. (7.79) and setting δVm = δm,k gives the solutionfor Yl,k:

δYl,k(ω) = ~hTl B(ω)

( 1 0Λ21 Λ22

)−1

~ek + iωCl,k. (7.90)

Numerical enhancement

Although the direct computation of (7.90) gives not too bad results,it is numerically advantageous to transform it. Using the definitions

(1)ij = (−1)(i mod 3+1)δij , (7.91)

~el := 1~el, (7.92)


~hTl B(ω)

( 1 0Λ21 Λ22

)−1

~ek

= ~hTl

((Λ11 Λ12

Λ21 Λ22

)+ iωC

)( 1 0Λ21 Λ22

)−1

~ek

= ~hTl

(( 1 0Λ21 Λ22

)+

(Λ11 − 1 Λ120 0 )+ iωC

)( 1 0Λ21 Λ22

)−1

~ek

= ~hTl

((Λ11 − 1 Λ120 0 )+ iωC

)( 1 0Λ21 Λ22

)−1

~ek, (7.93)

which gives smoother numerical results.

7.6 High-frequency noise

This section explains how to compute the correlation functions (CF) ofthe fluctuations of the terminal currents using the IFM (see Chapter 5)and the FEM. The equations will again be derived for the DD model.The EB model is outlined in the Appendix C. The transformationfrom the CF of the current fluctuations to the CF of the potentialfluctuations will still be exact within the FEM framework when usingthe Y -parameters derived in the previous chapter. This section doesnot consider G−R noise.


7.6.1 The DD Langevin equations

The Poisson equation (7.82) stays unchanged, because there are nonoise sources for the electrostatic potential. The equations for thecurrent continuity are modified by adding ∇δ~ςvn

to (7.80) and adding∇δ~ςvp

to (7.81):

∇δ ~Jn = −q∇[δµnn∇ψn + µnδn∇ψn + µnn∇δψn

]

+ iωq∇[αnδn∇ψn + δαnn∇ψn + αnn∇δψn

]+ ∇δ~ςvn

= −iωδn, (7.94)

∇δ ~Jp = −q∇[δµpp∇ψp + µpδp∇ψp + µpp∇δψp

]

+ iωq∇[αpδp∇ψp + δαpp∇ψp + αpp∇δψp

]+ ∇δ~ςvp

= iωδp, (7.95)

where δ~ςvnand δ~ςvp

are the noise sources described in Chapter 5.


The discretisation is again achieved by integrating (7.80), (7.94) and(7.95) with a test function of UN and then carrying out a partialintegration of the lhs. This leads to the system

~sT

B(ω) − iωC︸︷︷︸

:=Λ(ω)

δ~φ(ω) = ~sT δ~ς, ∀~s, (7.96)

where the matrices have been defined in Section 7.5.2. The compo-nents of δ~ς are by construction

(δ~ς)i =

0 if i mod 3 = 1,∑

T∈T |pj∈T

det(BT )~bTσ(j)

∫

bTδ~ςvn

(ξ)d2ξ if i mod 3 = 2,

∑

T∈T |pj∈T

det(BT )~bTσ(j)

∫

bTδ~ςvp

(ξ)d2ξ if i mod 3 = 0,

(7.97)


where ~pj is the j-th point of the grid T with i = 3 ∗ j + (i mod 3),

σ(j) is the index (0, 1 or 2) of the j-th point in the triangle T , and ~bkwas already defined in (7.25).

7.6.3 Computation

The spectral intensity of the fluctuation of the terminal currents SδIkδIl

is defined as the Fourier transform of the correlation function:

SδIkδIl(s) := lim

T→∞

1

2T

∫ T

−T

δIk(t)δI∗l (t+ s)dt. (7.98)

The same assumptions as in Section 7.5.3 are made for the orderingof the components of the vectors.

Solving (7.96) with the conditions that the charge densities on thecontacts stay constant and that the potential variation δVm on them-th contact is uniform on the contact, gives

δ~φ =

( 1 0Λ21 Λ22

)−1(δ~ς0 +

Ncont∑

l=1

δVl(ω)~el

), (7.99)

where δ~ς0 has the same components as δ~ς up to the first 3M compo-nents which are 0 in δ~ς0.

Rewriting (7.67) using (7.94), (7.95) and (7.99) gives

δIl(ω) = ~hTl B(ω)

( 1 0Λ21 Λ22

)−1(δ~ς0 +

Ncont∑

m=1

δVm(ω)~em

)− ~hTl δ~ς

+ iω

Ncont∑

m=1

Cl,mδVm(ω). (7.100)

Using (7.90) one can exactly rewrite (7.100) as

δIl(ω) = ~hTl B(ω)

( 1 0Λ21 Λ22

)−1

δ~ς0 − ~hTl δ~ς

+

Ncont∑

m=1

Yl,m(ω)δVm(ω). (7.101)


A useful vectors ~lm can be defined as

~lTm(ω) := ~hTl B(ω)

( 1 0Λ21 Λ22

)−1(0 00 1)− ~hTl . (7.102)

With this vector one can rewrite (7.101) as

δIl(ω) = ~lTl (ω)δ~ς +

Ncont∑

m=1

Yl,m(ω)δVm(ω). (7.103)

The numerical computation of ~lm will be explained at the end of thissection.

Boundary conditions

The expression (7.103) still contains the unknowns δVm, which mustbe fixed (as boundary conditions) to define a unique problem. Thereare two standard ways to do this. The first possibility (the mostnatural one) is to require that δVm = 0, ∀m. The second is to usethe arbitrary conditions δIl = 0, ∀l. Using δVm = 0, ∀m, will reduce(7.103) to

δIl(ω) = ~lTl (ω)δ~ς =: δIIl (ω), (7.104)

whereas using δIl = 0, ∀l, leads to

−Ncont∑

m=1

Yl,m(ω)δVm(ω) = ~lTl (ω)δ~ς =: −Ncont∑

m=1

Yl,m(ω)δV IIm (ω).

(7.105)Thus the important relation

δ ~II = −Y δ ~V II (7.106)

holds, where Y is the matrix of the Y -parameters, δ ~II the vector of

the δIIl (ω)s and δ ~V II the vector of the δV IIm (ω)s.

In the following the natural conditions δVm = 0, ∀m, will be used.


Spectral intensity

Using the Wiener-Khintchin theorem, the Fourier transform of (7.98)can be written as

SδIkδIl(ω) = ~lTk (ω)Sδςδς(ω)~l∗l (ω), (7.107)

whereSδςδς(ω) :=

∫ ∞

−∞

limT→∞

1

2T

∫ T

−T

δ~ς(t)δ~ςT∗(t+ s)dte−iωsds. (7.108)

The usual assumptions for the spectral intensities of the noisesources δ~ςvn

and δ~ςvpare that they are neither cross-correlated, nor

space correlated, and that they are white (frequency independent)with the properties

∫ ∞

−∞

limT→∞

1

2T

∫ T

−T

(δ~ςvn

(x, t)δ~ςT∗vn

(x′, t+ s))ijdte−iωsds

= 4q2n(x)(Dn)ij(x)δ3(x − x′), (7.109)

∫ ∞

−∞

limT→∞

1

2T

∫ T

−T

(δ~ςvp

(x, t)δ~ςT∗vp

(x′, t+ s))ijdte−iωsds

= 4q2p(x)(Dp)ij(x)δ3(x − x′), (7.110)

where Dn,p(x) is the position-dependent diffusion tensor.With these properties, the components of Sδςδς(ω) can be easily

computed:

(Sδςδς)ij =

0 if a 6= b, or a = 1,

4q2∑

T∈T |pl,pm∈T

det(BT )n0

2∑

c,d=1

(~bσ(l))c(~bσ(m))d(Dn)cd if a = 2,

4q2∑

T∈T |pl,pm∈T

det(BT )p0

2∑

c,d=1

(~bσ(l))c(~bσ(m))d(Dp)cd if a = 0,

a := i mod 3, b := j mod 3, l := (i− a)/3, and m := (j − b)/3.(7.111)


With (7.107) and (7.111), SδIkδIl(ω) can be computed. Note that

in this FEM formalism there is no need for an explicit definition andcomputation of Green’s functions or to work with the response of thesystem to a δ-like excitation as is usually done in the literature (seee.g. [5]).

Numerical enhancement

The direct computation of (7.102) gives quite poor results. It is,therefore, necessary to transform it. Since

( 1 0Λ21 Λ22

)−1

=

( 1 0−Λ−1

22 Λ21 Λ−122

), (7.112)


~lTm = ~hTm

(B(ω)

(0 00 Λ−122

)−(1 00 1))

= ~hTm

([(Λ11 Λ12

Λ21 Λ22

)+ iω

(C11 C12

C21 C22

)](0 00 Λ−122

)−(1 00 1))

= ~hTm

(−1 (Λ12 + iωC12) Λ−1

220 iωC22Λ−122

). (7.113)

Computing ~lm using the expression on the rhs of (7.113) gives verystable and smooth results.

Chapter 8

Development of asuitable MC simulatorfor silicon

8.1 Introduction

This chapter will first give a brief overview of the physics used in theSimnIC MC simulator. Then, the technique to simulate a magneticfield will be described. The numerical methods to obtain differentquantities of interest will be detailed. Then, the iteration scheme ofthe one-particle and of the ensemble MC simulator will be described.The calibration and tests (validation) of the simulator will be pre-sented next. Finally, the computation of Y-parameters and of theGreen’s functions using many-particle and one-particle MC simula-tions will be given.

8.2 Physical model

The model used is a seven band model with three bands for holes andfour bands for electrons. The bands are computed with the pseudo-potential method (see e.g. [16],[17]). Phonon (see [9]), impurity (see

135

136 CHAPTER 8. MONTE CARLO SIMULATOR

[11]) and impact ionisation (see [15]) scattering are taken into account.The impurity scattering model takes screening by the carriers intoaccount as described in [11]. The influence of an external magneticfield is modelled based on the simple expression for the Lorentz force.The scattering due to boundaries is modelled using a specular-elasticmodel which can be found in [47]. The injected carriers on the contactsare assumed to have a velocity weighted Maxwellian distribution

P (~k) =Θ(~v(~k)~nc)~v(~k)~nce

−βε(~k)

∫Bz

Θ(~v(~k′)~nc)~v(~k′)~nce−βε(~k′)d3k′

(8.1)

as proposed in [22] (~nc is the normal to the contact boundary pointinginside the device).

A detailed description of the model for electrons in bulk silicon isgiven in Section 9.2.1.

8.3 Magnetic fields

The two major problems when dealing with magnetic fields are relatedto loops. The first problem is a physical one and originates from thefact that electrons in a semiconductor exposed to a magnetic field canmove on closed loops. The second problem is a numerical one and iscaused by the discretisation of the Brillouin zone. Certain combina-tions of velocity, magnetic field and electric field lead to a frenetic inand out from pairs of discretisation boxes (see Fig. 8.1). Without anymeasure to treat these loops, the computational overhead becomesparalysing. An algorithm was, therefore, developed which detectsboth kinds of loops. In the case of physical loops, the algorithm com-putes the time to the next scattering and the position of the electronon the loop at this time. For the numerical loops, the mean averageof the electron is evaluated and the electron is the propagated onlyalong this direction. The benefit of this algorithm is that simulationswith external magnetic fields converge as fast as simulations without.

8.4. COMPUTATION OF VALUES OF INTEREST 137

Box 1

Box 2

mean direction

particle path

Figure 8.1: Typical path of a particle due to the discretisation.

8.4 Computation of values of interest

8.4.1 Mean values

In this thesis, all mean values were computed using path integralstatistics (PIS). Within the PIS the mean value of a function g ona time interval [t1, t2] in a given triangle T of the discretisation iscomputed with the formula

gj :=1

t2 − t1

t2∫

t1

∫

T

ξj(x)

∫

Bz

g(x, k)δ2(x− x(t))δ3(k − k(t))d3kd2x,

(8.2)where ξj(x) is a test function as needed in the FEM. The meaningof Eq. (8.2) is illustrated in Fig. 8.2. With this method there is noneed to stop the simulation at given time intervals to gather informa-tions about functions of interest. The whole available information isautomatically taken into account.


T

~x(t)

0

1

2

Figure 8.2: Path integral statistics.

8.4.2 Terminal currents

The terminal currents are computed using the Ramo-Shockley testfunctions (see (7.70)), which enables to compute the total current(carrier currents and displacement current) very efficiently.

8.4.3 Correlation functions

Noise sources

The noise sources for the transport models are computed using themethod described in [32] p. 147-148. The method consist in directlycomputing the integral of a correlation function, which is the coeffi-cient of the Fourier transform at zero frequency, i.e. by definition thenoise source.

Terminal currents

The auto-correlation and cross-correlation functions of the terminalscurrents are computed using a history of the terminal currents with aresolution of 0.1fs and a linear interpolation between the points.

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8.5. ITERATION SCHEMES 139

8.5 Iteration schemes

8.5.1 Iteration scheme for one-particle MC

The one-particle MC method is well described in [13]. It can be sum-marised as follows:

1. Compute the density in each triangle with a frozen field one-particle MC simulation.

2. Extract the quasi-Fermi potentials from the newly computeddensity using the ”old” potential (potential from step 1).

3. Solve the non-linear Poisson equation using the new quasi-Fermipotentials.

4. Compute the new electric field and run a new frozen-field sim-ulation (back to 1).

Point 2 is not trivial when using linear functions for the potentials ona triangle. The MC density njMC on a triangle is computed with (8.2)

setting g = 1. To extract the quasi-Fermi potential from njMC , theequation (see Eq. (7.28))

njMC = n1(j, ϕ, ψ, ǫg) (8.3)

must be solved for ψ, for all triangles. However, because (8.3) has ingeneral no exact solution, it must be replaced by an equation for thebest possible approximation:

∑

Tj∋pi

(njMC − n1(j, ϕ, ψ, ǫg))2 = min, ∀pi, (8.4)

where Tj is the j-th triangle and pi the i-th point of the discretisation.Under the condition that the QF potential is constant on the contactsthere follow the equations:

∑

Tj∋pi

(njMC − n1(j, ϕ, ψ, ǫg))δn1(j, ϕ, ψ, ǫg)

δψi= 0, ∀pi,

pi not on a contact. (8.5)

This non-linear system can be solved using a Newton like iterationscheme.

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8.5.2 Iteration scheme for many-particle MC

The iteration scheme used in the many-particle MC simulator is thefollowing:

1. Choose a time step ∆t compatible with the criteria given inRambo et al. [44].

2. Propagate all particles during ∆t and compute the mean densityusing (8.2).

3. Inject particles on the contacts using the probability given in(8.1) until the density on the contacts reaches the equilibriumdensity, and propagate the new particles during ∆t.

4. Update the scattering rates using the new mean density andenergy.

5. Solve the linear Poisson equation with the new mean density.

6. Ramp the voltage on the contacts if, needed.

7. Return to point 2.

8.6 Correction of the self-forces

When solving the linear Poisson equation with the charge density pro-vided by the MC simulator, the electric field produced by all particlescan be computed. However, during the propagation of a particle, thismean electric field should not directly be used, because each particlewould produce a force on itself. This non-physical force, called self-force (SF), must be removed. The self-forces on each particle duringthe propagation can be removed by the superposition principle. Byusing pre-computed Green’s functions as described in [32] p. 112-113,the self-force can be efficiently removed without noticeable slowdownof the simulator.

8.7. CALIBRATION AND VALIDATION 141

8.7 Calibration and validation

The calibration of the bulk mobilities and of the time-of-flight mea-surements was already done in e.g. [9], [12] and [10]. The dopingdependence of the low-field mobility was again fitted to check forconsistency with previously published results (see [23], p. 30). Theimpact ionisation rate was taken over from [15] but, was not furthercalibrated. The next sections deal with the validation of the MCsimulator and the calibration of the effective mobility.

8.7.1 Low-field mobility

Temperature dependence

As all simulations were performed for 300 K, the accuracy of the tem-perature dependence was not verified at other temperatures.

Doping- and carrier concentration dependence

The Monte Carlo low-field mobilities were fitted to experimental databy multiplying the scattering rate from [11] with a function s(N)of the doping concentration. To quickly determine s(N), the low-field mobility was computed directly using (3.10). Then, it was againcomputed with a MC simulation to check for consistency. To achievea good accuracy of the mean velocity at small electric fields, one hasto simulate a very long time. In order to avoid this problem, themobility was evaluated from the diffusivity using the Einstein relation.The diffusivity itself was computed via the auto-correlation functionof the fluctuations of the velocity. This method is exact for smallfield intensities and gives good results in a much shorter time thanby a direct computation of the mobility. Fig. 8.3 shows the low-field mobility for electrons as function of the doping concentrationmeasured by different groups in comparison with the MC results.

8.7.2 High-field mobility

The high-field mobility was determined by dividing the mean veloc-ity in the field direction by the field strength. A comparison withexperimental data is shown in Fig. 8.4.

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1e+13 1e+14 1e+15 1e+16 1e+17 1e+18 1e+19 1e+20 1e+21

doping concentration [cm-3

]

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

Thurber (C-V)Mousty et al. (NTT)Thurber (NTT)Natsuki et al. (EMA)SimnIC

elec

tron

mobility

[cm

2/V

s]

Figure 8.3: Low-field mobility as function of the doping concentrationfor electrons.

100 1000 10000 1e+05 1e+06electric field [V/cm]

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

Canali et al.SimnIC

elec

tron

mobility

[cm

2/V

s]

Figure 8.4: High-field mobility as function of the electric field forelectrons.


~E

~E

dy

dx

t = 0

t = 0...dt

Figure 8.5: Time-of-flight measurements.

8.7.3 Time-of-flight

The time-of-flight measurement method is important because it yieldsdirect information about the auto-correlation function of the fluctua-tions of the velocity. As these functions are crucial for noise computa-tions, any MC simulator used to extract noise data should be able toreproduce them. The idea behind the method is illustrated in Fig. 8.5.At time t = 0, a well-localised bunch of electrons is generated (e.g.with a short LASER pulse) on one side of a piece of silicon exposed

to an electric field ~E. Then the bunch drifts in field direction andchanges its shape. This alteration of the shape can be measured onthe other side of the sample at time dt. The new shape has a directrelation to the auto-correlation function of the fluctuations of the ve-locity. Fig. 8.6 shows the MC results compared with measurementsavailable from the literature (see. [9] and [46]).


100 1000 10000 1e+05electric field [V/cm]

10

15

20

25

30

35

40

SimnICBrunetti et al.Rolland et al.

elec

tron

diff

usi

vity

[cm

2/s]

Figure 8.6: Time-of-flight measurements and simulation results forelectrons.

8.7.4 Effective mobility

The effective mobility (µeff ) as function of the effective electric field(Eeff ) is a figure of merit that can be measured in MOSFETs. Theeffective mobility is defined as

µeff :=

∫ L0n(x)µpar (x)dx∫ L

0n(x)dx

, (8.6)

and the effective electric field as

Eeff :=

∫ L0 n(x)Eper (x)dx∫ L

0n(x)dx

. (8.7)

The meaning of µpar (x) and Eper (x) is given in Fig. 8.7. µeff gives anindication for the degradation of the mobility at the Silicon/Silicon-Oxide interface. To reproduce this degradation, a model proposed bySangiorgi et al. (see [47]) was used. The idea behind this model isto allow for two types of scattering when a particle reaches the oxide


Gate contact

Body

Oxide

Substrate contact

~Epar~Eper (x)

µpar (x)

0

Lx

Figure 8.7: Vertical cut in a MOSFET channel.

boundary. The first type is the so-called specular scattering, wherethe momentum parallel to the interface as well as the energy mustbe conserved. The second type is the so-called diffusive scattering,where only energy conservation is required. By varying the ratio α ofdiffusive to specular scattering, one can fit the experimental results.As µeff depends strongly on the crystal orientation, α must be de-termined for each crystal orientation. Fig. 8.8 shows the fits for twodifferent values of α (0.14 and 0.19) compared with experimental datafrom [54],[55] for a 〈110〉-orientation of the crystal. The results shownin Fig. 8.8 were computed using 105 particles in a self-consistent way.This is in contrast to previously published non-self-consistent resultsby Jungemann et al. (see [31]).

8.7.5 Hall factor

The Hall factor rH has been computed for low magnetic fields andlow electric fields using the theory of Chapter 3. The results shown inFigs. 8.9 and 8.10 were then compared with many-particle simulationsof a van der Pauw structure. The latter is illustrated in Fig. 8.11,whereas the simulation results are shown in Fig. 8.12.

A very good agreement is achieved for doping concentrations higherthan 1014 cm−3. The discrepancy at 1014 cm−3 is not yet understoodand should be further investigated.


1e+05 1e+06E

eff [V/cm]

200

300

400

500

600

700

800

900

1000

Takagi et al. (1994)SimnIC 14% diffusive scatteringSimnIC 19% diffusive scattering

µeff

[cm

2/V

s]

Figure 8.8: µeff as function of Eeff for a 〈110〉-orientation of thecrystal.

100 200 300 400 500temperature [K]

0

0.5

1

1.5

N=0.0N=1e14N=1e15N=1e16N=1e17N=1e18N=1e19N=1e20N=1e21

r H

Figure 8.9: rH for electrons as function of the lattice temperature fordifferent doping concentrations.


100 200 300 400 500temperature [K]

0

1

2

3N=0.0N=1e14N=1e15N=1e16N=1e17N=1e18N=1e19N=1e20

r H

Figure 8.10: rH for holes as function of the lattice temperature fordifferent doping concentrations.

Figure 8.11: Van der Pauw structure for the extraction of rH .


1e+10 1e+11 1e+12 1e+13 1e+141e+151e+16 1e+17 1e+18 1e+19 1e+20 1e+21


]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

theorymany-particle MC

r H

Figure 8.12: Hall factor computed by an ensemble MC simulationcompared to results from the theory of Section 3.4.


8.8.1 One-particle MC

Introduction

To compute Y-parameters by one-particle MC simulations, Eq. (4.9)developed in Chapter 4.3 and the approximation (4.15) have beenused. Since only the current fluctuations are of interest, the momentg = ~v(k) has to be considered. For better readability the followingabbreviations are introduced:

(L~v)j :=∑

i

Q0vj ,i

ci =

∫

Bz

S−1vj

(n, T )(k)f(k)

nd3k, (8.8)

∂L~v∂F

∣∣∣∣ n=n

T=T

j

:=∑

i

Q0vj ,i

∂ci∂F

=

∫

Bz

S−1vj

(n, T )(k)∂ f(k)

n

∂Fd3k,

(8.9)


(D~v)jl :=∑

i

(~Q1vj ,i

)lci =

∫

Bz

vlS−1vj

(n, T )(k)f(k)

nd3k, (8.10)

∂D~v∂F

∣∣∣∣ n=n

T=T

jl

:=∑

i

(~Q1vj ,i

)l

∂ci∂F

=

∫

Bz

vlS−1vj

(n, T )(k)∂ f(k)

n

∂Fd3k, (8.11)

∂D~v∂n

∣∣∣∣ F=F

T=T

jl

:=∑

i

(∂ ~Q1

vj,i

∂n

)

l

ci =

∫

Bz

vl∂S−1

vj(n, T )(k)

∂n

∣∣∣∣∣n=n

f(k)

nd3k, (8.12)

(µ~v)jl :=q

~

∑

i

(~Q2vj ,i

)lci =

q

~

∫

Bz

S−1vj

(n, T )(k)∂kl

(f(k)

n

)d,3 k

(8.13)

∂µ~v∂F

∣∣∣∣ n=n

T=T

jl

:=q

~

∑

i

(~Q2vj ,i

)l

∂ci∂F

=

q

~

∫

Bz

S−1vj

(n, T )(k)∂∂kl

(f(k)n )

∂Fd3k, (8.14)

∂µ~v∂n

∣∣∣∣ F=F

T=T

jl

:=q

~

∑

i

(∂ ~Q2

vj,i

∂n

)

l

ci =

q

~

∫

Bz

∂S−1vj

(n, T )(k)

∂n

∣∣∣∣∣n=n

∂kl

(f(k)

n

)d3k. (8.15)


The function D~v is the diffusion tensor, µ~v is the mobility tensor,whereas L~v has, to the authors knowledge, no special name. The ideaconsists in computing (8.8)–(8.15) with the one-particle MC methodand then to solve the equation

˙δnL~v+n∂L~v∂F

˙δF +∇⊤r

(n∂D~v∂n

δn

)+∇⊤

r

(n∂D~v∂F

δF

)+∇⊤

r (D~vδn)

+ nµ~vδ ~E + n

(∂µ~v∂n

δn

)~E + n

(∂µ~v∂F

δF

)~E + δnµ~v ~E = ∓δ

~J

q(8.16)

coupled with the linearised continuity equation and the Poisson equa-tion. In Eq. (8.16) ”−” stands for electrons and ”+” for holes.

Computation

As S−1~v is known, ~L~v, D~v and µ~v can be computed directly. To obtain

∂D~v

∂n and ∂µ~v

∂n , the functions

∂S−1vj

(n, T )(k)

∂n

∣∣∣∣∣n=n

are needed. In the model used here, the functions S−1vj

can be writtenas (see Eq. (2.67))

S−1vj

= − vj − vjWph +Wii +Wimp

+ S−1vj, (8.17)

where Wph + Wii + Wimp = Wtot and Wph is the part of the totalscattering Wtot caused by the phonons, Wii is the part which de-scribes impact ionisation, and Wimp scattering at ionized impurities.In (8.17), the only term depending on n and T is Wimp (see [11]).Therefore

∂S−1vj

∂n(k) =

vj(k) − vj(k)

Wtot(k)2∂Wimp(k)

∂n. (8.18)

Because∂Wimp(k)

∂n is an analytical function, ∂D~v

∂n and ∂µ~v

∂n can bequite easily computed (for a discussion of numerical problems see Ap-pendix E).


The last terms needed to compute the linear system containing

(8.16) are ∂~L~v

∂F , ∂D~v

∂F , and ∂µ~v

∂F . These terms are computed based onthe following algorithm:

1. Compute the stationary state and extract n, T .

2. Start with the first contact: i = 1.

3. Add ∆V Volts to the i-th contact and compute F+ := F , ~L+ :=~L~v, D

+ := D~v and µ+ := µ~v without updating the scatteringrate from the previous state.

4. Restart from the stationary state.

5. Add −∆V Volts to the i-th contact and compute F− := F ,~L− := ~L~v, D

− := D~v and µ− := µ~v again without updating thescattering rate from the previous state.

6. Approximate

(a) ∂~L~v

∂F by (~L+ − ~L−)/(F+ − F−),

(b) ∂D~v

∂F by (D+ −D−)/(F+ − F−),

(c) ∂µ~v

∂F by (µ+ − µ−)/(F+ − F−).

7. Go to the next contact (i → i+ 1) and repeat steps 3.-6. untilthe last contact is reached.

8. Compute the mean values on the contacts of all the computedfunctions.

The Y-parameters are then computed using the FEM as described inChapter 7, but with tensorial transport coefficients, instead of scalars.

This method is very complicated and time consuming and shouldnever be used to compute Y-parameters alone. However, in Sec-tion 8.9, it will become clear that also the noise analysis can be donewith the functions computed in this section.


8.8.2 Many-particle MC

The computation of Y-parameters using the many-particle MC methodis performed by a slightly modified version of the method describedin Reiser et al. [45] (see also [37]).

The Reiser method consists in applying a sudden voltage changeof amplitude ∆Vj on the j-th contact. Inspecting the terminal cur-rent Ii(t) on the i-th contact, one can directly compute Yij using theformulas

ℜ[Yij ] =Ii(∞) − Ii(0)

∆Vj+

ω

∆Vj

∞∫

0

(Ii(t) − Ii(∞)) sin (ωt)dt, (8.19)

ℑ[Yij ] =ω

∆Vj

∞∫

0

(Ii(t) − Ii(∞)) cos (ωt)dt. (8.20)

Because applying a positive or a negative offset ∆Vj will often givedifferent results, a simple algorithm which provides a mean value ofthe two configurations has been developed:

1. n = 1. Simulate until the stationary state is reached.

2. Apply a voltage change ∆Vj on the j-th contact and wait untilthe system has reached a stationary state.

3. Use the above described method to compute Y 1ij := Yij for all i.

4. Apply a voltage change -∆Vj on the j-th contact and wait untilthe system has reached a stationary state.

5. Compute Y 2ij := Yij for all i.

6. Apply a voltage change -∆Vj on the j-th contact and wait untilthe system has reached a stationary state.


8. Apply a voltage change ∆Vj on the j-th contact and wait un-til the system has reached the same stationary state as at thebeginning.

8.9. NOISE AND GREEN’S FUNCTIONS 153

0 1e-11 2e-11 3e-11 4e-11 5e-11time [s]

2e-04

3e-04

4e-04

5e-04

6e-04

7e-04

8e-04I(t

)[A/µm

]

Figure 8.13: Typical progression of I(t).


10. Compute Yij(n) :=Y 1

ij+Y2

ij+Y3

ij+Y4

ij

4 .

11. Approximate Yij by Y meanij := 1n

∑nl=1 Yij(l).

12. n → n + 1. Go back to 2., until the error on Y meanij is smallerthan the desired precision.

A typical output for I(t), when using the above-described algorithmis shown in Fig. 8.13.

8.9 Noise and Green’s functions

One-particle MC

The correlation functions of the fluctuations of the terminal currentsare computed using the impedance field method (IFM) described inChapter 5. The transport coefficients and their derivatives were de-fined and computed in Section 8.8.1. The noise sources are taken from


bulk MC simulations as described in Chapter 5. This method doesnot only allow to compute correlation functions, but also to localisethe noisy parts of a device. Since the transport coefficients used areexact, the only two approximations left are due to the parametrisationof the transport coefficients (see Section 4.4.2) and the application ofbulk noise sources.

Many-particle MC

The many-particle MC method directly provides the terminal cur-rents with there fluctuations. Therefore, the method described inSection 8.4.3 is used. The advantage of the latter is that it can alsobe applied in the non-linear case. A drawback is that the origins ofthe noise cannot be localised.

Part IV

Applications of theMISO to Silicon

155

Chapter 9

Results

9.1 Introduction

All results presented in this thesis were obtained using the simulatorSimnIC (Simulator for noI se Computations), which has been devel-oped by the author. SimnIC is able to compute stationary solutionsto the DD and EB transport models, as well as to perform small sig-nal and noise analysis using the generalised IFM (see Chapter 5) anddifferent kinds of noise sources. The numerical methods used weredescribed in Chapter 7 and in Appendix C. An example for the simu-lation of a realistic device using the EB model is given in Appendix F.SimnIC is at the same time a full-band MC simulator for silicon de-vices, using both the many-particle, and the one-particle approaches,as described in Chapter 8. The intrinsic speed of SimnIC along withits capability to run on many processors in shared-memory environ-ments, makes it an ideal simulation tool for realistic devices usingmany millions of MC particles. Benchmarks are given at the end ofAppendix F.

In Section 9.2 the theory outlined in Sections 2.4 and 3.6.1 will beapplied to two full-band models for electrons in silicon. In Section 9.3the Monte Carlo-generated noise sources are discussed and comparedwith both the Langevin noise sources and with the usual approxi-mations. In Section 9.4 relaxation times are computed based on the

157

158 CHAPTER 9. RESULTS

theory of Section 3.5. In Section 9.5 a simple N+NN+ structure andits P+PP+ equivalent are simulated using all methods described anddeveloped in the previous chapters (an analysis of a more involved de-vice is given in [49]). Results of the DD and EB models are comparedto those of MC simulations. The Nyquist theorem is recovered andthe different contributions of the noise sources are discussed in detail.

In Section 9.6 an interestingNIN structure with a locally negativemobility is presented.

9.2 Electrons in bulk silicon

The general formalism developed in Chapter 2 is used to computemoments of the ISO for electrons in a full-band model for silicon.

9.2.1 Two standard models

The first four conduction bands are taken into account, computedwith the empirical pseudo-potential method (see [16], [17]).

The energy boundaries of these bands are

band 1: ε(1)min = 0.00 eV, ε

(1)max = 3.83 eV,

band 2: ε(2)min = 0.13 eV, ε

(2)max = 5.81 eV,

band 3: ε(3)min = 2.31 eV, ε

(3)max = 11.18 eV,

band 4: ε(4)min = 3.23 eV, ε

(4)max = 11.18 eV.

For the scattering operators (SO) the following two models have beenused:

Model 1:

1. T = 300 K

2. Phonon scattering (as described in [29]): One acoustic (6.26meV) intravalley, intraband and inter-band. Two acoustic (12.07meV, 18.53meV) g-type intervalley, in-traband and interband.

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9.2. ELECTRONS IN BULK SILICON 159 One optical (62.05meV) g-type intervalley, intraband andinterband. Two acoustic (18.96meV, 47.40meV) f-type intervalley, in-traband and interband. One optical (59.03meV) f-type intervalley, intraband andinterband.

3. Impurity scattering (elastic intravalley, intraband scattering asdescribed in [11]).

4. Impact ionisation (intervalley, intraband and interband scatter-ing as described in [15]).

The state after an impact ionisation event is homogeneously cho-sen on the space of all allowed energies, i.e. if the electron has anenergy larger than εgap = 1.12 eV before scattering, then the finalstate is chosen in the energy interval [0, ε− εgap], with a probabilityproportional to the density of states at the energy of the final state.Thus, the transition probability per time unit for impact ionisationcan be written as

wII(k, b|k0, b0) :=I(ε)Θ(ε(k, b) − ε(k0, b0) − εgap)

N∑i=0

ε(k,b)−εgap∫

ε(i)min

Z(ε′, i)dε′

, (9.1)

where I(ε) is the function given in [15], Eq. (1), and N is the numberof bands times the number of valleys.

Model 2:

The same as model 1, but without impact ionisation.

In these two models the transition probability depends only onvalley index, band index, and energy. Therefore, the formalism intro-duced in Sections 3.6–2.7 can be used.

When model 1 is chosen within a Monte Carlo (MC) simulator, allrelevant silicon bulk measurements are reproduced in a very satisfyingway (see e.g. [10]).


9.2.2 Problem caused by the Dirac delta function

A function belonging to a family of continuous functions δγ(ε) depend-ing on a continuous parameter γ is called a regularisation of the delta

function iff limγ→0

∞∫−∞

δγ(ε)f(ε)dε = f(0) for all continuous functions of

the energy f(ε), and∞∫

−∞

δγ(ε)dε = 1 for all γ. In the following, the

continuous function of the energy δ(a)γ (ε) will be used. This function

is such that supp (δ(a)γ ) = [−γ, γ], i.e. δ

(a)γ (ε) is different from 0 only

in the energy interval [−γ, γ].In the mathematical formulation of the electron-phonon scattering

rate, use is commonly made of the Dirac delta function in order toinsure the conservation of energy. However, three reasons can be givenwhy one should not take these delta functions too seriously.

The mathematical reason

The delta function in the semiconductor Boltzmann equation orig-

inates from the regularisation δ(b)γ (ε) = (γ sin( εγ )2)/(πε2) as conse-

quence of Fermi’s Golden Rule. As long as δ(b)γ (ε) is used, the scatter-

ing operator is compact. But the limit γ → 0 leads to a non-compactoperator. Therefore, calculations involving the scattering operatorwill depend on whether the limit is accomplished prior or after thesecalculations. Because a delta function only turns out in the limit ofinfinite time, one should take the limit γ → 0 only at the end of thecalculation of any physical quantity and, therefore, compute every-thing with the compact scattering operator. Otherwise one is facedwith a lot of spurious problems.

The physical reason

Model 2 with delta functions has infinitely many positive solutionsin thermodynamic (TD) equilibrium (see Appendix A.6), which doesnot make any physical sense. Model 2, however, has only one solution

at TD equilibrium, if one uses δ(b)γ (ε) (γ > 0) instead of the delta

function. This point will be clarified below.

9.2. ELECTRONS IN BULK SILICON 161

The numerical reason

A MC simulator (or any algorithm) working with double-precisionarithmetic is unable to distinguish between a delta function and e.g.

δ(a)10−306 , or δ

(b)10−306 . Thus, the numerics sets a limit on γ and, therefore,

the computer will only be able to approximate a compact scatteringoperator.

9.2.3 Properties of the normalised SO

In Section 2.3 the operatorA(k, b|k0, b0) has been defined. A(k, b|k0, b0)is the probability to scatter from (k, b) to (k0, b0) in one scatteringevent. The theory there was developed under the condition that∃M ∈ N | AM is a strong positive operator, i.e. that it is possi-ble, starting from any (k, b) ∈ K, to reach any (k0, b0) ∈ K after Mscattering events.

Appendix A.4 explains why this property is fulfilled for model 2

using δ(a)γ (ε) (thus, also for model 1 using δ

(a)γ (ε)), and gives an upper

limit for M(γ).

9.2.4 Impact ionisation

Use of (9.1) gives a SO whose eigenvector with eigenvalue 0 is not aMaxwell-Boltzmann distribution anymore.

The eigenvector WtotfeqZ with the largest eigenvalue of the ma-

trix M (M is a representation of M := PDBTPD) has been used as an

approximation to the function WtotfeqZ, and feq has been computed

using the approximation WtotfeqZ/(WtotZ) ≃ feq.

The functions WtotfeqZ/(WtotZ) for model 1 and 2 have beencomputed using the discretisation described in Section 9.2.6.

Fig. 9.1 shows that for low energies the functions are almost identi-cal, whereas for energies higher than 4 eV the equilibrium distributionfor model 1 goes faster towards zero than the Maxwell-Boltzmann dis-tribution.

Hence model 1 has a TD distribution which differs from the Maxwelldistribution, and which does not fulfil the principle of detailed balance.This is an artefact of the model.


0 1 2 3 4 5 6 7 8 9 10 11ε [eV]

10-300

10-275

10-250

10-225

10-200

10-175

10-150

10-125

10-100

10-75

10-50

10-25

100

f eq

model 1model 2Maxwell distribution

Figure 9.1: The equilibrium distribution feq for the two models.

9.2.5 Impurity scattering

In Appendix A.5 a proof is given that, if the scattering ratew(ε, b|ε0, b0)is replaced by w′(ε, b|ε0, b0) := w(ε, b|ε0, b0) + t(ε, b)δ(ε − ε0)δb,b0 (t(ε, b) a given function), then Hg does not change. Hence, if the sys-tem is modified by adding any intraband-intravalley elastic scattering,Hg will not change (This is however not true for Hg).

Because impurity scattering is described, by intraband-intravalleyelastic scattering, it will not influence Hg at all, but only the Hg asshown in Appendix A.5, (A.73).

Thus, Hg (and, therefore, S−1g ) is a function which describes the

system independently of the doping and carrier concentrations.

9.2.6 Discretisation of the energy space

To fulfil condition (2.77), the energy intervals of the four differentbands were discretised as follows: Band 1: 2000 points, band 2: 2966points, band 3: 4631 points, band 4: 4152 points. The points aredistributed uniformly between εmin and εmax. Appendix A.3 explainswhy this discretisation fulfils (2.77), i.e. why B is a primitive matrix.

This discretisation is rather ”naive” and one could certainly find


others which require fewer points. Nevertheless, the discretisation isusable and allows to compute large tables of Sg in a reasonable amountof time.

9.2.7 Model 1: Results

The matrix B was first computed for model 1. In this special case,the operator B acting on a function that only depends on energy willproduce a result which also depends on energy only.

After 182 iterations x,y, and ρ(B) were determined within ma-chine precision. The computed value for the spectral radius of thematrix was ρ(B) = 1.000000000000000± 10−16.

The functions H⊥1 and S−1

g have then been computed for g =ε, vx, v

2, vxv2. The results are presented in Figs. 9.3–9.7. Fig. 9.2

defines the names of the valleys.The oscillations in Fig. 9.3 are no artefacts. They originate from

the strong localisation of the function feq. The ”+” correspond to thepoints of the discretisation in the first band.

Generally one could observe that odd moments (vx (88 iter.), vxv2

(77 iter.)) converge faster than even moments (1 (267 iter.), ε (241iter.), v2 (251 iter.)). Because the functions feq(ε) and ε(ε) depend

only on energy and not on band-valley index, the functions H⊥1 and

S−1ε will also only depend on energy.

Even though v2 strongly depends on the band-valley index b, S−1v2

does not show a strong dependence on b. This is because Bi lv2 doesnot depend anymore on b for all i > 0 and because in the summationof (2.64) the terms with i > 0 give the dominant contribution to thesolution.

9.2.8 Model 1: Error computation

In this section the eg are computed as described in Section 2.6.As pointed out at the end of Section 2.6 and in Appendix A.2.5 (A.33),an important condition for the error to be small is

|y(0)〉 ≃ | ˜WtotfeqZ〉|| ˜WtotfeqZ||

. (9.2)


kx

ky

kz

va

ey

f

ll

g−valley valley of reference

Figure 9.2: Notation for the valleys.

0.001 0.01 0.1 1 10ε [eV]

-1e-12

0

1e-12

2e-12

H⊥ 1

[s]

Figure 9.3: H⊥1 as function of energy.


0.001 0.01 0.1 1 10ε [eV]

-2e-13

0

2e-13

4e-13

6e-13

8e-13S−

1ε

[eV·s

]

Figure 9.4: S−1ε as function of energy.

0.01 0.1 1 10ε [eV]

-5e-09

0

5e-09

1e-08 band 1, ref. valleyband 1, g-valley band 2, ref. valleyband 2, g-valleyband 3, ref. valleyband 3, g-valleyband 4, ref. valleyband 4, g-valley

S−

1v

x[m

]

Figure 9.5: S−1vx

as function of energy, band, and valley index.


0.001 0.01 0.1 1 10ε [eV]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

band 1band 2band 3band 4

S−

1v2

[m2s−

1]

Figure 9.6: S−1v2 as function of energy.

0.01 0.1 1 10ε [eV]

-6e+03

-4e+03

-2e+03

0

2e+03

4e+03

6e+03

band1, ref. valleyband 2, ref. valleyband 3, ref. valleyband 4, ref. valley

S−

1v

xv2[m

3s−

2]

Figure 9.7: S−1vxv2

as function of energy, band and valley index.


0.001 0.01 0.1 1 10ε [eV]

0.05

0.1

0.15

0.2

0.25

0.3

[s]

feqZ

|| ˜WtotfeqZ||y(0)Wtot

Figure 9.8: Comparison of |y(0)〉/Wtot with |feqZ〉/|| ˜WtotfeqZ||.

In Fig. 9.8 |y(0)〉/Wtot and |feqZ〉/|| ˜WtotfeqZ|| are depicted. One

can observe that (9.2) is well fulfilled and, therefore, expects e(2)g to

be small.

The following figures compare eg with g (by definition ε = ε).Fig. 9.9 shows eε in the first band. Remember that eε(ε) is the abso-lute error made on ε(ε) using |S−1

ε (ε)〉num

instead of the exact functionS−1ε (ε). This error is always small compared to ε(ε), and there are

only a few points where the error is much larger than the mean error.However, these points will have only little influence on the expressions(2.99) and (2.100), because eε(ε) only appears in mean values.Figs. 9.10, 9.11, and 9.12 show the absolute error on vx, v2, and ˇvxv2

in the first band. The error functions had to be magnified 100 times inorder to see them in the graphics. Next to van Hove singularities (seee.g. [2]) the error is especially large because, there, piecewise linearfunctions cannot approximates the g well.

As described in Section 2.6, one can compute the error on meanvalues with a MC simulation, if one uses |S−1

g (ε)〉num

to compute trans-port coefficients instead of S−1

g (ε). Figs. 9.13–9.15 show the relativeerror on 〈vx〉, 〈ε〉, and 〈v2〉 as function of the electric field (the fieldis parallel to 〈100〉).

Administrator

Highlight

Administrator

Highlight


0 1 2 3ε [eV]

-0.04

-0.02

0

0.02

0.04e ε

[eV

]

Figure 9.9: eε in the first band.

0 1 2 3 4ε [eV]

-4e+05

-2e+05

0

2e+05

4e+05

e vx[m/s]

,v x

[m/s]

vx100 · evx

Figure 9.10: evxin the first band.


0 1 2 3 4ε [eV]

-1e+12

-5e+11

0

5e+11

1e+12

1.5e+12

2e+12e v

2[m

2/s],v2[m

2/s]

v2

100 · ev2

Figure 9.11: ev2 in the first band.

0 1 2 3 4ε [eV]

-2e+17

0

2e+17

4e+17

e vxv2[m

3/s2

],ˇv xv2[m

3/s2

] ˇvxv2

100 · evxv2

Figure 9.12: evxv2 in the first band.


100 1000 10000 1e+05 1e+06 1e+07electric field [V/cm]

-1e-06

0

1e-06

2e-06〈ev

x〉/〈vx〉

Figure 9.13: Relative error 〈evx〉/〈vx〉 as function of the electric field.


3e-04

4e-04

〈eε〉/〈ε〉

Figure 9.14: Relative error 〈eε〉/〈ε〉 as function of the electric field.



4e-04

5e-04

6e-04〈ev2〉/〈v

2〉

Figure 9.15: Relative error 〈ev2〉/〈v2〉 as function of the electric field.

9.2.9 Model 1: Langevin noise sources for mo-ments of interest

Using the theory of Section 3.6, the noise sources Cgg for g = vx, ε,and v2 have been computed with a MC simulation for an electric fieldpointing in 〈100〉-direction. The results are shown in Figs. 9.16–9.18.

Note that for a small electric field Cvxvx/n/2 has the same value

as the auto-correlation function of the velocity (see e.g. [9] Fig. 5).

Figs. 9.19–9.21 show the relative error on the auto-correlation func-tions (see (2.100)) in dependence on the electric field. These functionswere computed using the method described in Section 2.6. Remark:After computation of all functions in (3.52) for Kvxvx

it was noticedthat the contribution of the term S−1

vxwas negligible, i.e. that one can

compute the correlation function of the Langevin noise source of thefirst moment of the velocity without knowing the ISO (i.e. by settingS−1vx

= 0 in (3.52))! This is only true for this moment.

Generally it seems that the contribution of the ISO to (3.52) issmaller for odd moments than for even ones.



25

30

35

40

45[c

m2/s]

Cvxvx/n/2

Cvyvy/n/2

Figure 9.16: Cvv/n/2 as function of the electric field.


1e-52

1e-51

1e-50

1e-49

Cεε/n

[J2s]

Figure 9.17: Cεε/n as function of the electric field.



1e+18

1e+19

1e+20Cv2v2/n

[cm

4/s3

]

Figure 9.18: Cv2v2/n as function of the electric field.


1e-08

1e-07

1e-06

1e-05

rela

tive

erro

r

Errvxvx

Errvyvy

Figure 9.19: Errvv as function of the electric field.



5e-04

1e-03

2e-03Errεε

Figure 9.20: Errεε as function of the electric field.


7e-04

8e-04

9e-04

1e-03

Errv2v2

Figure 9.21: Errv2v2 as function of the electric field.


0 1 2 3 4ε [eV]

1e+12

1e+13

1e+14

1e+15Wtot(ε,1

)[s−

1]

Figure 9.22: Wtot in the first band as function of energy.

9.2.10 Model 1: Moments of interest of the frequency-dependent ISO

This subsection will show some graphs of the functions α(ω), x(ω),y(ω), T−1

vx(ω)a and T−1

v2 (ω)a, computed as described in Section (2.7).From (2.107) one expects three domains of interest:

(1) ω ≪Wtot, (9.3)

(2) ω ≃Wtot, (9.4)

(3) ω ≫Wtot. (9.5)

Fig. 9.22 gives an idea of the magnitude ofWtot. For ω ≪ 1012s−1,T−1g (ω) will be in the domain (9.3), for ω ≃ 1012s−1 in the domain

(9.4) and for ω ≫ 1012s−1 in the domain (9.5).In the first domain one expects the functions x(ω), y(ω), and

T−1g (ω)a to be almost frequency-independent (because ω only appears

in the sum Wtot+ iω). In the second and third domains, the functionsshould depend very much on the frequency. This is exactly what onecan observe in Figs. 9.24–9.31.


107

108

109

1010

1011

1012

1013

1014

1015

frequency [Hz]

-0.2

0

0.2

0.4

0.6

0.8

1

Re(α)Im(α)

α(ω

)

Figure 9.23: α(ω) as function of frequency.

0 1 2 3 4ε [eV]

-0.006

-0.004

-0.002

0

0.002

0.004

0.006 0 Hz10

9 Hz

1010

Hz10

11 Hz

1013

Hz10

15 Hz

Re(x(ω

))

Figure 9.24: Re(x(ω)) in the first band as function of frequency.


0 1 2 3 4ε [eV]

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006 0 Hz10

9 Hz

1010

Hz10

11 Hz

1013

Hz10

15 Hz

Im

(x(ω

))

Figure 9.25: Im(x(ω)) in the first band as function of frequency.

0 1 2 3 4ε [eV]

-0.02

0

0.02

0.04

0.06 0 Hz10

10 Hz

1011

Hz10

12 Hz

1013

Hz10

14 Hz

1015

Hz

Re(y(ω

))

Figure 9.26: Re(y(ω)) in the first band as function of frequency.


0 1 2 3 4ε [eV]

-0.02

0

0.02

0.04

0.06 0 Hz10

10 Hz

1011

Hz10

12 Hz

1013

Hz10

14 Hz

1015

Hz

Im

(y(ω

))

Figure 9.27: Im(y(ω)) in the first band as function of frequency.

0 1 2 3 4ε [eV]

-2e-09

-1e-09

0

1e-09

2e-09

0 Hz10

11 Hz

1012

Hz10

13 Hz

1014

Hz10

15 Hz

Re(T

−1

vx

(ω) a

)[m

]

Figure 9.28: Re(T−1vx

(ω)a) in the first band and first valley as functionof frequency.


0 1 2 3 4ε [eV]

-1e-09

-5e-10

0

5e-10

0 Hz10

11 Hz

1012

Hz10

13 Hz

1014

Hz10

15 HzI

m(T

−1

vx

(ω) a

)[m

]

Figure 9.29: Im(T−1vx

(ω)a) in the first band and first valley as functionof frequency.

0 1 2 3 4ε [eV]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 Hz10

10 Hz

1011

Hz10

12 Hz

1014

Hz10

15 Hz

Re(T

−1

v2

(ω) a

)[m

2/s]

Figure 9.30: Re(T−1v2 (ω)a) in the first band as function of frequency.


0 1 2 3 4ε [eV]

-0.4

-0.2

0

0.2

0.4

0.6

0 Hz10

9 Hz

1010

Hz10

11 Hz

1012

Hz10

14 Hz

1015

Hz

Im

(T−

1v2

(ω) a

)[m

2/s]

Figure 9.31: Im(T−1v2 (ω)a) in the first band as function of frequency.

Fig. 9.23 shows the dependence of α(ω) on frequency (rememberthat B(ω)|x(ω)〉 = α(ω)|x(ω)〉). Up to 10 Ghz there is no visibledependence on the frequency (compare e.g. with [32], Chapter 7.5).

9.2.11 Model 2 and comparison with model 1

In the case of the TD equilibrium distribution, even though impactionisation starts at an energy of Eg ≃ 1.12 eV, its effect becomesvisible only for energies larger than 4 eV (see Fig. 9.1) as alreadydiscussed in Section 9.2.4.

Figs. 9.32 and 9.35 show that this is also true for S−1vx

(ε, 2) and

S−1vxv2

(ε, 2).Figs. 9.33 and 9.34 show that the difference between the two mod-

els shows up at all energies for the moments S−1ε and S−1

v2 .Even though the difference between the two models is very pro-

nounced in Figs. 9.33 and 9.34, it becomes obvious that for energiesbelow 2.5 eV the two curves only differ by a constant, which is physi-cally irrelevant as already discussed in Section 2.3.


0 1 2 3 4 5 6ε [eV]

0

2e-09

4e-09

6e-09

8e-09

1e-08

model 1model 2

S−

1v

x(ε,2

)[m

]

Figure 9.32: S−1vx

in the second band and first valley as function ofenergy.

0 1 2 3 4 5 6ε [eV]

0

1e-12

2e-12

3e-12

model 1model 2

S−

1ε

(ε,2

)[e

V·s

]

Figure 9.33: S−1ε in the second band as function of energy.


0 1 2 3 4 5 6ε [eV]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4model 1model 2

S−

1v2

(ε,2

)[m

2/s]

Figure 9.34: S−1v2 in the second band as function of energy.

0 1 2 3 4 5 6ε [eV]

0

2e+03

4e+03

6e+03

8e+03

1e+04 model 1model 2

S−

1v

xv2(ε,2

)[m

3/s2

]

Figure 9.35: S−1vxv2

in the second band and first valley as function ofenergy.

9.3. MONTE CARLO-GENERATED NOISE SOURCES 183

1e+14 1e+15 1e+16 1e+17 1e+18 1e+19 1e+20 1e+21


]

0

1000

2000

3000

4000

5000

6000

electrons at 300Kholes at 300K

Fcrit

[V/cm

]

Figure 9.36: Fcrit as function of doping concentration.

9.3 Monte Carlo-generated noise sources

In Chapter 5, the concept of noise sources for transport model wasintroduced. Here, some examples for the DD and EB models will beshown, characterised and compared with different approximations andexact expressions.

9.3.1 Preliminary remark

The results presented in this section were computed using the methodof Section 3.6 for low electric fields. To determine the low-to-high-fieldtransition, the theory of Section 3.9 was applied. Fig. 9.36 depicts thecritical electric field Fcrit (defined in 9.36) as function of the dop-ing concentration in the case of charge neutrality. One can see thatfor doping concentrations higher than 1018 cm−3 the MC simulationsmust be used only for field strengths higher than 1.5 kV/cm. This isan important information, because it allows to save a lot of computa-tion time.



0

20

40

60

80

Sδvxδvx

Sδvyδvy

[cm

2/s]

Figure 9.37: Auto-correlation function of the velocity fluctuations inundoped bulk silicon as function of the electric field. The field pointsin 〈100〉-direction.

9.3.2 Electric field dependence

For better comparability with already published results, only the func-tions Sδgδg′ found in the rhs of (5.26) are considered for g, g′ =v2, vx, vy, vxv

2, and vyv2. Figs. 9.37–9.40 show the results for un-

doped bulk silicon and an electric field pointing in 〈100〉-direction.

For the fluctuations of the velocity, Fig. 9.37, the auto-correlationfunction in field direction is in general always smaller than the oneperpendicular to the field. The cross-correlation functions shown inFig. 9.40 do not vanish at TD equilibrium and are strongly dependenton the electric field. Therefore, these term cannot be neglected as willbe shown later.

9.3.3 Doping dependence

The doping dependence of the correlation functions reveals two impor-tant features. Firstly, for strong electric fields the functions become



1e+17

1e+18

1e+19

Sδv2δv2

[cm

4/s]

Figure 9.38: Auto-correlation function of the v2 fluctuations in un-doped bulk silicon as function of the electric field. The field points in〈100〉-direction.


1e+31

1e+32

1e+33

1e+34

Sδvxv2δvxv2

Sδvyv2δvyv2

[cm

6/s]

Figure 9.39: Auto-correlation function of the energy current fluctu-ations in un-doped bulk silicon as function of the electric field. Thefield points in the 〈100〉-direction.



1e+16

1e+17

1e+18

Sδvxv2δvx

Sδvyv2δvy

[cm

4/s]

Figure 9.40: Cross correlation function of the energy current fluctua-tions with the velocity fluctuations in undoped bulk silicon as functionof the electric field. The field points in the 〈100〉-direction.

weakly doping-dependent as shown in Figs. 9.41–9.43. This is becauseat high energies the total scattering rate is almost independent fromthe presence of impurities. The second important feature shown inFigs. 9.41–9.43 is that for g = v2 the dependence is much weaker thanfor the other moments. This comes from the fact that the impurityscattering rate is an elastic process in the models used.

9.3.4 Direction dependence

The dependence on the direction of the electric field of the correla-tion functions is shown in Figs. 9.44–9.46. The legends of the figuresdesignate the angle between the field vector and the 〈100〉-directionin the 〈100〉–〈010〉 plane. For all moments, the direction dependencecannot be neglected as soon as the field strength exceeds 5 kV/cm.



0

10

20

30

40

50

60

70

80

N=0N=1e15N=1e16N=7e16N=2e17N=1e18N=1e19N=1e20

Sδv

xδv

x[c

m2/s]

Figure 9.41: Auto-correlation function of the velocity fluctuations inthe field direction as function of the electric field for different dopingconcentrations.


1e+17

1e+18


Sδv

xv2δv

xv2[c

m6/s]

Figure 9.42: Auto-correlation function of the v2 fluctuations as func-tion of the electric field for different doping concentrations.



1e+30

1e+31

1e+32

1e+33

1e+34


Sδv

xv2δv

xv2[c

m6/s]

Figure 9.43: Auto-correlation function of the energy current fluctu-ations in field direction as function of the electric field for differentdoping concentrations.

9.3.5 Comparisons

In commercially available device simulators, like DESSIS, Taurus-Medici from Synopsys Ltd or S-Pisces from Silvaco Ltd, the noisesources are computed using the expressions (5.19) and (5.21), whichare only valid at thermodynamic equilibrium. Furthermore, the cross-correlation functions are assumed to be zero. These approximationscome off badly as discussed in the next two subsections. In the lastsubsection noise sources of the transport models are compared withthe Langevin noise sources.

Generalised Einstein relation

The generalised Einstein relation, applied to noise sources, leads tothe approximation

Sδviδvj(ω) ≈ 2n

kBT

qµδij , (9.6)



0

20

40

60

80

Sδv

xδv

x[c

m4/s]

09182736

45

Figure 9.44: Auto-correlation function of the velocity fluctuations infield direction as function of the field strength for different field ori-entations.


1e+17

1e+18

Sδv

xv2δv

xv2[c

m6/s]

0918273645

Figure 9.45: Auto-correlation function of the v2 fluctuations as func-tion of the field strength for different field orientations.



1e+32

1e+33

1e+34

Sδv

xv2δv

xv2[c

m6/s]

0918273645

Figure 9.46: Auto-correlation function of the energy current fluctu-ations in field direction as function of the field strength for differentfield orientations.

where µ is the mobility in field direction. In the DD model, T is thelattice temperature, whereas in the EB model T is the mean particle”temperature” as defined in (2.17). Fig. 9.47 clearly demonstratesthat in the case of undoped silicon, the Einstein relation (ER) failsfor electric field intensities higher than 1 kV/cm. For typical dopingconcentrations it becomes even worse. In general, one observes thatthe ER with the lattice temperature (resp. carrier temperature) givessmaller (resp. larger) values than 1

2Sδviδvj(ω).

Bixon-Zwanzig relation

Using the thermal conductivities κ from the Bløtekjær model

κ := n5

2

k2B

qµT (9.7)

and inserting it in (5.21) leads to the approximation (see [3])

m∗2

9nSδs

v2vxδs

v2vx≈ 5

2

k3BT

3

qµ. (9.8)



0

10

20

30

40

MCDD-EinsteinEB-Einstein

[cm

2/s]

Figure 9.47: Comparison of 12Sδviδvj

(ω) with the rhs of Eq. (9.6) asfunction of the electric field in undoped silicon for the DD and EBmodel, respectively.

In Fig. 9.48 both sides of (9.8) are compared with each other. Evenif next to TD equilibrium the result looks not so bad (about a factor4/5), the situation deteriorates at higher field intensities and also athigher doping concentrations.

Fig. 9.49 shows that in the case of a vanishing field, the Bixon-Zwanzig relation holds reasonably over a wide range of doping con-centrations.

Note that the Stratton model [53], which defines κ as

κ := n3

2

k2B

qµT, (9.9)

would give worse results.

Langevin noise sources

As proven in Chapter 5, the noise sources of a transport model andthe corresponding Langevin noise sources must be the same in the



1e-43

1e-42

1e-41

1e-40

MCBixon-Zwanzig

[J2m

2/s]

Figure 9.48: m∗2

9n Sδsv2vxδs

v2vx(0) compared with the rhs of the Bixon-

Zwanzig relation as function of the electric field intensity in undopedsilicon.

limiting case of an infinite number of moments of the BE. Figs. 9.50–9.51 clearly show that more than four moments must be taken intoaccount to approach this limit.

9.4 Energy relaxation times

In order to improve the EB model, non constant energy relaxationtimes (ERTs) were used. They were computed using the theory ofSection 3.5 in the low-field regime and by bulk MC simulations inthe high-field regime based on the formula (3.34). Then, they wereparameterised by the doping concentration, carrier concentration, andthe mean energy of the particles.

Figs. 9.52 and 9.53 show the values of the ERT in bulk silicon fordifferent lattice temperatures and doping concentrations. The boldlines designate the results from (3.34), whereas the thin lines repre-sent the approximation proposed by Jungemann et al. [32]. Figs. 9.52and 9.53 clearly demonstrate that the dependence on the doping con-

9.4. ENERGY RELAXATION TIMES 193

1e+12 1e+13 1e+14 1e+15 1e+16 1e+17 1e+18 1e+19 1e+20 1e+21


]

1e-47

1e-46

1e-45

1e-44

1e-43

1e-42

MCBixon-Zwanzig

[J2m

2/s]

Figure 9.49: m∗2

9n Sδsv2vxδs

v2vx(0) compared with the the rhs of the

Bixon-Zwanzig relation for the Bløtekjær model as function of thedoping concentration at thermodynamic equilibrium.



0

5

10

15

20

25

30

35

40

45

LangevinDDEB

1 2Sδζ

vxδζ

vx

[cm

2/s]

Figure 9.50: Sδζvx δζvxfor the DD and the EB models compared with

the corresponding Langevin term as function of the electric field.


1e-16

1e-15

1e-14

1e-13

1e-12

LangevinMCEB

Sδζ

v2δζ

v2

[eV

2/s]

Figure 9.51: Sδζv2δζv2 for the EB model compared with Sδv2δv2 (MC)and with the corresponding Langevin term as function of the electricfield.


1012

1013

1014

1015

1016

1017

1018

1019

1020

1021


]

10-13

10-12

10-11

10-10

50K100K150K200K250K300K350K400K450K500K

τ v2[s−

1]

Figure 9.52: Low-field energy relaxation times for electrons in bulksilicon as function of lattice temperature and doping concentration.

centration cannot be neglected contrarily to what is suggested in theliterature (see e.g. [25]). Relaxation times for higher moments can befound in [8].

Fig. 9.54 presents the ERTs for the EB model as function of themean electron temperature in bulk silicon at 300 K. For temperaturesbetween 300 K and 600 K there is a strong dependence on the dopingconcentration. Between 600 K and 3500 K the RT is almost constantwith a value of 1.8×10−13 s independently of the doping concentration.From 3500 K on it diminishes quickly. In the case of holes (9.55) thedoping dependence is even visible up to 800 K. From 800 K to 2000 Kthe RT is almost independent of the doping concentration with a valueof about 9.5 × 10−14 s. Then it becomes smaller.

To let the reader receive an impression of the shape of the S−1g ,

one example has been chosen. Fig. 9.56 shows the x-component of

the functiondS−1

ε

dε in dependence of energy in the first band and thevalley along the 〈100〉-direction for the case of electrons. This functionappears in (3.42) in the numerator, if one performs an integration byparts.

Administrator

Highlight


1012

1013

1014

1015

1016

1017

1018

1019

1020

1021


]

10-13

10-12

10-11

10-10

50K100K150K200K250K300K350K400K450K500K

τ v2[s−

1]

Figure 9.53: Low-field energy relaxation times for holes in bulk siliconas function of lattice temperature and doping concentration.

1 10 100 1000T-T

eq [K]

0

2e-13

4e-13

6e-13

8e-13

N=0.0N=1e15N=1e16N=1e17N=1e18N=1e19N=1e20

τ ε[s−

1]

Figure 9.54: Energy relaxation time for electrons in bulk silicon at300 K as function of mean particle energy and doping concentration.


1 10 100 1000T-T

eq [K]

0

1e-13

2e-13

3e-13

4e-13

N=0.0N=1e15N=1e16N=1e17N=1e18N=1e19N=1e20

τ ε[s−

1]

Figure 9.55: Energy relaxation time for holes in bulk silicon at 300 Kas function of mean particle energy and doping concentration.

0.001 0.01 0.1 1ε [eV]

-3e-12

-2e-12

-1e-12

0

1e-12

2e-12

3e-12

4e-12

dS

−1

ε

dε

[s]

Figure 9.56:dS−1

ε

dε as function of energy in the first band and in thevalley along the 〈100〉-direction for electrons.


10nm

100nm

400nm

100nm

2e15 −3cm 5e17 −3cm5e17 −3cm

Figure 9.57: Geometry and doping concentration of the N+NN+

structure.

The strongly oscillating behaviour ofdS−1

ε

dε yields a better under-standing for the pronounced oscillations of the ERT in regions wherethe carriers behave quasi-ballistically: In the quasi-ballistic regime thedistribution function is only shifted towards higher energies. There-fore, integrating a strong oscillating function of the energy with thisshifting distribution function (a kind of moving windows) will lead torelaxation times which oscillate in the real-space.

9.5 Simple N+NN

+ and P+PP

+ structures

In this section most of the theory of the previous chapters and mostof the functions of the previous sections will be used to investigate theproperties of a simple N+NN+ structure and of its P+PP+ equiva-lent.

The N+NN+ device and its doping profile are shown in Fig. 9.57.Fig. 9.58 shows the homogeneous grid (as used by Jungemann et al.(see [32]) p.182) used for the discretisation as well as the effectiveintrinsic density on the first 120 nm.

These structures have already been extensively studied in a bookby Jungemann (see [32]) and are, therefore, a good starting point tocheck for the consistency of the approaches.

First, the stationary state and the transport coefficients will bestudied. To do so, the DD, EB, many-particle MC method, and the

Administrator

Highlight

9.5. SIMPLE N+NN+ AND P+PP+ STRUCTURES 199

EffectiveIntrinsicDensity: 1.2E+10 1.3E+10 1.4E+10 1.5E+10 1.6E+10 1.7E+10

Figure 9.58: Discretisation grid and effective intrinsic density for theN+NN+ structure.

new one-particle MC method have been employed. Then, the small-signal analysis will be conducted and the role of the ERTs will beemphasised. Finally, the noise characteristics will be studied.

9.5.1 MC-generated bulk transport coefficients

To compare the DD and EB models with the MC method on a fairbasis, tables of mobilities and ERTs generated by bulk MC simulationshave been used. Both transport coefficients were parametrised usingfive parameters:

1. Doping concentration,

2. Carrier density,

3. Driving force,

4. Minority-majority,

5. Crystallographic orientation.

A smooth transition between majority- and minority-carrier regimeswas achieved using the heuristic formula

µn :=n

n+ pµmajn +

p

n+ pµminn , (9.10)

for the electron mobilities, and

µp :=p

n+ pµmajp +

n

n+ pµminp (9.11)


for the hole mobilities, where µn (resp. µp) is the electron (resp. hole)mobility, n (resp. p) the electron (resp. hole) density, and µmaj (resp.µmin) the mobility for the majority (resp. minority) carriers. Ananalogous formula has been used for the ERTs.

9.5.2 Driving forces for the EB model

The EB model implemented in SimnIC uses three different drivingforces:

1. The norm of Jn

qµnnor

Jp

qµpp, respectively,

2. The norm of the gradient of the quasi-Fermi potential |∇ψn,p|,

3. The mean energy εn,p.

The author is not aware of other works investigating the firstdriving force, even though it represents the natural generalisation of|∇ψn,p| to the EB model. This driving force evidently reduces to thenorm of the electric field in bulk and to the norm of the gradient ofthe quasi-Fermi potential, when carrier temperature and lattice tem-perature are equal (e.g. in the DD model).

In the following, the difference between these parametrisations willbe systematically investigated.

9.5.3 Stationary states

All results presented here were obtained using the in-house devicesimulator SimnIC developed by the author. The device is oriented inthe 〈100〉-direction.

Many-particle MC simulations

A mean number of 2000 particles has been simulated. Due to thecontact boundary conditions, the total number of particles inside thedevice was fluctuating during the simulation. To avoid plasma oscilla-tions, a time step of 5 fs was chosen according to the paper by Ramboet al. (see [44]). The self-forces were corrected using the methodbriefly described in 8.6. The simulation time for each operation pointwas 0.5µs.


One-particle MC simulations

The method described in Section 3.8.2 has been used to compute thestationary states. For each operation point 10 steps of 0.8,mus were simulated to allow the device to reach the stationary state.Then the device was simulated during 16µs with updates of the scat-tering rates and the Poisson equation every 0.4µs.

Transport-model simulations

For the DD and EB simulations, two different mobility models wereused. The first is the Philips Unified Mobility model as describedin [35],[36] for the low-field mobility, coupled with the Canali (see[14]) or the Meinerzhagen-Engl (see [40]) model for different drivingforces. The second model is the MC-generated table of bulk transportcoefficients as discussed in Section 9.5.1.

I-V characteristics

Fig. 9.59 (resp. 9.61) shows the I-V curves in the range between 1 V to5 V for electrons (resp. holes) for the different models. The transportcoefficients (TCs) for the EB model were parametrised by the localmean particle temperature. The many-particle MC curves are consid-ered as reference. The excellent agreement between the many-particleand the one-particle MC results was expected and confirms the va-lidity of the new iteration scheme. A comparison with the transportmodels (TMs) shows that the use of MC-generated TCs gives betterresults in the case of electrons, whereas the analytical model is a bitbetter for holes. Surprisingly, in both cases the simple DD model givesbetter results than the more sophisticated EB model. This comes infact from the parametrisation of the TCs in the EB model.

Figs. 9.60 and 9.62 show the I-V curves for the three differentparametrisations of the TCs as discussed in 9.5.2. In the legends,the driving forces (DFs) correspond to the list given in 9.5.2. It isobvious that the DF 1 gives better results than the parametrisationby the energy, and yields results almost identical to those of the DDmodel. The same observations can be made in the case of the small-signal analysis and for noise.


1 2 3 4 5bias [V]

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

7e-06

8e-06

many-particle MCone-particle MCEB, MC tableEB, PhilipsDD, MC tableDD, Philips

curr

ent

[A/µm

]

Figure 9.59: I-V curves of the N+NN+ structure for different models.

1 2 3 4 5bias [V]

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

7e-06

8e-06

many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3DD, MC table

curr

ent

[A/µm

]

Figure 9.60: I-V curves of the N+NN+ structure for different drivingforces.


1 2 3 4 5bias [V]

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

many-particle MCone-particle MCEB, MC tableEB, PhilipsDD, MC TableDD, Philips

curr

ent

[A/µm

]

Figure 9.61: I-V curves of the P+PP+ structure for different models.

1 2 3 4 5bias [V]

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3DD, MC table

curr

ent

[A/µm

]

Figure 9.62: I-V curves of the P+PP+ structure for different drivingforces.


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

1e+16

1e+17

one-particle MCEB, driving force 3EB, driving force 1

elec

tron

den

sity

[cm

−3]

Figure 9.63: Electron density profile of the N+NN+ structure fordifferent models (VCA = 2 V).

Bulk transport coefficients vs exact transport coefficients

To better understand the advantage of the first DF compared to theparametrisation by the mean carrier temperature, a look inside thedevice may be helpful. Figs. 9.63–9.66 show the profile of the den-sity, temperature, mobility, and ERT for electrons in the N+NN+

structure at a voltage of 2 V. The MC results are that of the newone-particle MC method. The terminal current of the device is pri-marily determined by the physics in the middle part of the device,i.e. in the zone of high resistivity. Figs. 9.65 and 9.66 show that theparametrisation with the first DF results in a better agreement to MCresults in this part of the device.

The oscillating behaviour of the ERT after the N+N transition,which is not a numerical artefact, can be explained by the fact thatthe carriers in this region fly quasi-ballistically (see Section 9.4).

The Einstein relation, which comes into play in the TMs whenexpressing the diffusion constant as function of the mobility, workspoorly in transport direction (x-direction) as shown in Fig. 9.67. Inthe perpendicular direction (y-direction) the Einstein relation works


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

200

400

600

800

1000

1200

1400

1600


elec

tron

tem

per

atu

re[K

]

Figure 9.64: Electron temperature profile of the N+NN+ structurefor different models (VCA = 2 V).

-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

500

1000

1500

one-particle MC µyy

one-particle MC µxx

EB, driving force 3EB, driving force 1

elec

tron

mobility

[cm

2/V

s]

Figure 9.65: Electron mobility profile of the N+NN+ structure fordifferent models (VCA = 2 V).


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

2e-13

4e-13

6e-13

8e-13


elec

tron

ERTτ v

2[s

]

Figure 9.66: Electron energy relaxation time profile of the N+NN+

structure for different models (VCA = 2 V).

much better. However, this is irrelevant for the terminal currents.

9.5.4 Y-parameters

The small-signal analysis for the TMs was performed based on the the-ory of Section 7.5. For the many-particle MC simulations the methoddescribed in Section 8.8.2 was applied using ∆V = 0.05 V for appliedvoltages smaller than 2 V and of ∆V = 0.1 V otherwise.

Figs. 9.68 and 9.69 present the results. For voltages smaller than10 mV the same behaviour can be observed for electrons and holes:The MC simulations produce the smallest values for the Y-parameters,whereas the DD model gives the highest values. The EB simulationresults are found somewhere in the middle. Concerning the differentDFs, the same conclusions as for the stationary states can be drawn:The first two DFs yield better matches compared to the parametrisa-tion by the mean carrier temperature.

As explained in Section 9.4, the ERTs used in the EB model wereexactly computed in the low-field regime, in contrast to the work by


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

10

20

30

40

50

60

one-particle MC Dyy

one-particle MC Dxx

one-particle MC µxx

q-1

kBT

e

one-particle MC µyy

q-1

kBT

e

elec

tron

diff

usi

on

coeffi

cien

t[c

m2/s]

Figure 9.67: Comparison between the Einstein relation and exact dif-fusion tensor for electrons in the N+NN+ structure (VCA = 2 V).

0.0001 0.001 0.01 0.1 1bias [V]

1

1.5

2

2.5

3

many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3EB, driving force 1 (heated Maxwellian τε)EB, driving force 3 (heated Maxwellian τε )DD, MC table

Y11[A/V

m]

Figure 9.68: Y-parameters at zero frequency of theN+NN+ structurefor different models as function of VCA.


0.0001 0.001 0.01 0.1 1bias [V]

0.7

0.8

0.9

1

1.1

1.2

1.3

many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3EB, driving force 1 (heated Maxwellian τε) EB, driving force 3 (heated Maxwellian τε) DD, MC table

Y11[A/V

m]

Figure 9.69: Y-parameters at zero frequency of the P+PP+ structurefor different models as function of VCA.

Jungemann et al. ([32] and [19]). As illustrated in Figs. 9.52 and 9.53,a noticeable difference exists at 300 K between the exact expressionand the heated Maxwellian approximation. Figs. 9.68 and 9.69 demon-strate the real impact of these differences on a device. Curves with thelabels Heated Maxwellian τε result from simulations where a cutoff hasbeen used for the low-field ERT. For electrons the cutoff was 0.28 fsas chosen by Jungemann et al. (see [32] p. 155, Fig. 7.2). For holesan ERT of 0.11 fs was used, as obtained by the heated Maxwellianansatz. In the N+NN+ structure a maximal relative deviation of2.9% is observed between the third and the fifth curve. It seems thatthe introduced cutoff plays a more important role, when the TCsare parametrised by the third DF. In the P+PP+ structure a max-imal relative deviation of 1.8% is present at low voltages (< 10 mV)whereas for voltages in the range 10 mV to 1 V discrepancies as highas 15% are observed when the TCs are parametrised by the meancarrier temperature. Thus, the effect of a cutoff in the ERT on thesestructures strongly depends on the DF and is at least 1.8% in thelow-field domain.


0 1 2 3 4 5bias [V]

0

5e-20

1e-19

1.5e-19

2e-19

2.5e-19 many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3DD, MC tableEB, driving force 1, EinsteinEB, driving force 3, EinsteinDD, Einstein

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.70: Spectral intensity of the current at zero frequency of theN+NN+ structure for different models as function of VCA.

9.5.5 Noise analysis

The theory of Section 7.6 and Appendix C.2 has been used to com-pute noise properties of both devices using TMs. The results of many-particle MC simulations were evaluated using the method briefly dis-cussed in Section 8.9. The results are presented in Figs. 9.70–9.73.

Many-particle MC

Figs. 9.74 and 9.75 show the integrals from −t to t of the correlationfunction of the fluctuations of the current as function of t for differentvoltage offsets. By definition, the spectral intensity of the current(SIC) at zero frequency is the value found for t = ∞. However, thesefunctions are expected to become almost constant for t = t1 < ∞.From Figs. 9.74 and 9.75 one can see that t1 is about 10 ps. Fromt = 10 ps to t = 100 ps the functions are almost constants. Then,from t = 100 ps to t = 1 ns the functions fluctuate. This is dueto poor statistics (finite simulation time of 0.5µs. To enable a faircomparison with the results obtained by Jungemann et al., the values


0 1 2 3 4 5bias [V]

0

5e-20

1e-19

1.5e-19

2e-19

many-particle MCEB, driving force 1EB, driving force 1 (heated Maxwellian τε)EB, driving force 3EB, driving force 3 (heated Maxwellian τε)DD, MC table

0 0.01 0.02

3.5e-20

4e-20

4.5e-20

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.71: The influence of an ERT cutoff on the spectral intensityof the current at zero frequency of the N+NN+ structure as functionof VCA.

0 1 2 3 4 5bias [V]

0

1e-20

2e-20

3e-20

4e-20

5e-20

6e-20

7e-20

8e-20

9e-20

1e-19

1.1e-19

1.2e-19

1.3e-19 many-particle MCEB, driving force 1EB, driving force 2EB, driving force 3DD, MC tableEB, driving force 1, EinsteinEB, driving force 3, EinsteinDD, Einstein

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.72: Spectral intensity of the current at zero frequency of theP+PP+ structure for different models as function of VCA.


0 1 2 3 4 5bias [V]

0

2e-20

4e-20

6e-20

8e-20

many-particle MCEB, driving force 1EB, driving force 1 (heated Maxwellian τε)EB, driving force 3EB, driving force 3 (heated Maxwellian τε)DD, MC table

0 0.01 0.02

1.25e-20

1.5e-20

1.75e-20

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.73: The influence of an ERT cutoff on the spectral intensityof the current at zero frequency of the P+PP+ structure as functionof VCA.


1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09

t+10-15

[s]

0

5e-20

1e-19

1.5e-19

2e-19

2.5e-19

3e-19

10-4

V1V2V3V4V5V

[A2s/

m]

Figure 9.74: Integrals from −t to t of the correlation function of thefluctuations of the current of the N+NN+ structure as function of t.

of the noise were extracted at t = 100 ps.

The Einstein relation

In Figs. 9.70 and 9.72 the curved labeled Einstein were obtained usingthe Einstein relation to compute the noise sources as discussed inSection 9.3.5. As expected from the discussion in Section 9.3.5, thesenoise sources together with the DD model give a lower bound to theother models, whereas using them together with the EB model yieldsan upper bound.

Reproducibility of the Nyquist theorem

Because of the Nyquist theorem (see e.g. [42]) one expects the samebehaviour of the spectral intensity of the current (SIC) and the Y-parameters at small voltages. This is in fact the case: Figs. 9.71and 9.73 show that the MC simulations give the smallest values forthe SIC, whereas the DD model yields the highest values. The EBsimulation results are found somewhere in the middle.


1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09

t+10-15

[s]

0

5e-20

1e-19

1.5e-19

10-4

V1V2V3V4V5V

[A2s/

m]

Figure 9.75: Integral from −t to t of the correlation function of thefluctuations of the current of the P+PP+ structure as function of t.

Tables 9.1 and 9.2 give a detailed analysis of the reproducibilityof the Nyquist theorem for all considered models. As the Nyquisttheorem is an exact result for the DD model, the last line of bothtables gives an idea of the precision of the numerics and the nextto last lines an idea of the precision of the interpolation in the MC-generated table. It is worth noting that the Nyquist theorem is alwaysbetter reproduced when the exact ERTs are used (compare the threefirst lines with lines 6 and 7). Thus, using exact ERTs permits toreproduce the Nyquist theorem to better than 1% independently ofthe chosen DF.

Check of the gradient of the quasi-Fermi potential ansatz

In Section 5.7.2 an ansatz was made to compute the SICs with only asmall number of noise sources, and this for any transport model. Themain idea of the method is to use only the 0-th moment and the veloc-ity moments of the Boltzmann equation. To do so, the derivatives ofthe transport coefficients by the gradient of the quasi-Fermi potentialappearing in these moments are needed. To check the method, the


Nyquist Theorem: N+NN+ DeviceModel

4kBTℜ(Y ) SIC rel. errorEB DF 1

4.1365 · 10−20 4.0960 · 10−20 0.98%EB DF 2

4.1365 · 10−20 4.0960 · 10−20 0.98%EB DF 3

4.1357 · 10−20 4.0953 · 10−20 0.98%EB DF 1, ER

4.1365 · 10−20 4.2675 · 10−20 3.2%EB DF 3, ER

4.1357 · 10−20 4.2702 · 10−20 3.2%EB DF 1,ERT cutoff 4.2564 · 10−20 4.3386 · 10−20 1.9%

EB DF 3,ERT cutoff 4.2564 · 10−20 4.3386 · 10−20 1.9%

DD4.5116 · 10−20 4.5159 · 10−20 0.09%

DD, ER4.5116 · 10−20 4.5116 · 10−20 0.00085%

Table 9.1: Nyquist Theorem: N+NN+ Device


Nyquist Theorem: P+PP+ DeviceModel

4kBTℜ(Y ) SIC rel. errorEB DF 1

1.5568 · 10−20 1.5536 · 10−20 0.20%EB DF 2

1.5568 · 10−20 1.5536 · 10−20 0.20%EB DF 3

1.5568 · 10−20 1.5536 · 10−20 0.20%EB DF 1, ER

1.5568 · 10−20 1.5449 · 10−20 0.77%EB DF 3, ER

1.5568 · 10−20 1.5454 · 10−20 0.74%EB DF 1,ERT cutoff 1.5856 · 10−20 1.6097 · 10−20 1.5%

EB DF 3,ERT cutoff 1.5857 · 10−20 1.6097 · 10−20 1.5%

DD1.6072 · 10−20 1.6107 · 10−20 0.22%

DD, ER1.6072 · 10−20 1.6072 · 10−20 0.00085%

Table 9.2: Nyquist Theorem: P+PP+ Device


0 1 2 3 4 5bias [V]

0

5e-20

1e-19

1.5e-19

2e-19

2.5e-19

many-particle MCEB, driving force 3EB, driving force 3, alternative method

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.76: Comparison of the gradient of the quasi-Fermi potentialansatz with the previous method for the N+NN+ structure.

EB model was used. These derivatives were computed by the methoddescribed in Section 8.8.1 pretending that the transport coefficientsfor the EB model were those that the one-particle MC method shouldhave produced.

In the case of the P+PP+ device the method gives slightly betterresults, whereas for the N+NN+ device there is no winner. Thesefindings leads to the conclusion that the gradient of the quasi-Fermipotential ansatz is usable. However, more devices should be studied toget a clear understanding of the domain of applicability of the method.

Failure of the EB model

Compared with the very good results obtained by Jungemann et al.(see [32] p. 188, Fig. 9.11), the results presented here are rather dis-appointing. Although good agreement was obtained for the N+NN+

device, the results obtained for the P+PP+ structure are puzzling.

To understand the main cause(s) of the discrepancies, a list of dif-ferences between the here presented MC simulations and those done


0 1 2 3 4 5bias [V]

0

1e-20

2e-20

3e-20

4e-20

5e-20

6e-20

7e-20

8e-20many-particle MCEB, driving force 3EB, driving force 3, alternative method

spec

tralin

tensi

tycu

rren

t[A

2s/

m]

Figure 9.77: Comparison of the gradient of the quasi-Fermi potentialansatz with the previous method for the P+PP+ structure.

by Jungemann et al. is helpful: As already indicated in Section 9.5.3,the main differences between the two many-particle MC simulationsare the time steps of 5 fs instead of 10 fs (to avoid unphysical plasmaoscillations), the mean number of carriers (2000 instead of 1000), anda different impact ionisation model. Another difference is the useof a bandgap narrowing model, whereas Jungemann et al. used nobandgap narrowing. After a more careful inspection of the condi-tions of the MC simulations by Jungemann et al., it even turned outthat they do not allow the particles to reach the lateral boundariesof the device, avoiding, therefore, a reduction of the current due tosurface roughness scattering. This can explain the differences for theI-V curves, but will probably be only responsible for the noise dis-crepancies. The only plausible explanation left is that, even in suchsimple structures, the more detailed hydrodynamic model developedby Thoma (see [57]) and used by Jungemann et al. captures thephysics much better than the Bløtekjær model used here. Neverthe-less the considerations about the ERTs in the low-field regime are stillvalid, because both models are equivalent at TD equilibrium.


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

2e-13

4e-13

6e-13

8e-13

1e-12

1.2e-12

1.4e-12

1.6e-12

1.8e-12

2e-12

5V4V3V2V1V0V

den

sity

ofnois

e[A

2s/

m2]

Figure 9.78: Local density of noise in the N+NN+ structure for dif-ferent biases.

A look inside the devices

As already pointed out in Section 8.9, the IFM allows to detect thespatial origin of the noise in a device.

Figs. 9.78 and 9.79 show the local density of noise (LDN) for dif-ferent biases, as computed with the EB model using the third DF. At0 V most of the noise originates from the middle of the devices, wherethe conductivity is lowest. At higher biases, the maximum of the LDNshifts towards the N+N transition. This is in very good agreementwith the findings of Jungemann et al. and shows that the hot carriersonly negligibly contribute to noise as long as the electron-hole pairsproduced by impact-ionisation are disregarded.

Figs. 9.80 and 9.81 show the LDN at a bias of 3 V as computedwith the DD and EB models for the three proposed driving forces.The last curve in the graphs presents the result obtained with themethod of Section 5.7.2. Altough the general shape of the LNDs isthe same for all models, some small differences may be noticed. TheEB model with the first two driving forces produces small peaks at thefirst N+N (resp. P+P ) transition (x ≃ −0.2µm), whereas a strange


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

1e-13

2e-13

3e-13

4e-13

5e-13

6e-13

7e-13

8e-13

9e-13

1e-12

1.1e-12

0V1V2V3V4V5V

den

sity

ofnois

e[A

2s/

m2]

Figure 9.79: Local density of noise in the P+PP+ structure for dif-ferent biases.

oscillatory behaviour is observed near to the middle of the device,when the TCs are parametrised with the mean carrier energy. Noneof these features appears when the alternative method is used. Thisshould be further investigated.

9.5.6 An unresolved problem

In this section, the one-particle MC method has only been used totreat steady states (see Fig. 9.59 and 9.61), and not to compute Y -parameters or noise. This is due to a numerical problem. The theorypresented in Section 8.8.1 requires the accurate computation of thederivatives of the transport coefficients by the norm of the gradientof the quasi-Fermi potential. In highly doped regions of a device, thevariation of the norm of the gradient of the quasi-Fermi potential isvery small, which makes it difficult to accurately compute the deriva-tives. However, as the electric field is small in such regions, it shouldbe possible to replace the true derivatives by numerical expressions.This is still an open issue.


-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

5e-13

1e-12

1.5e-12

2e-12

2.5e-12

DD, MC tableEB, driving force 1EB, driving force 2EB, driving force 3EB, driving force 3, alternative method

den

sity

ofnois

e[A

2s/

m2]

Figure 9.80: Local density of noise in the N+NN+ structure for dif-ferent models at VCA = 3V .

-0.3 -0.2 -0.1 0 0.1 0.2 0.3position [µm]

0

1e-13

2e-13

3e-13

4e-13

5e-13

6e-13

7e-13

8e-13

9e-13

DD, MC tableEB, driving force 1EB, driving force 2EB, driving force 3EB, driving force 3, alternative method

den

sity

ofnois

e[A

2s/

m2]

Figure 9.81: Local density of noise in the P+PP+ structure for dif-ferent models at VCA = 3V .

9.6. A CASE OF NEGATIVE MOBILITY 221

9.6 A case of negative mobility

In the case of the scattering model used here (see Eq.(2.67)), theexpression (3.10) for the mobility can be rewritten as

µij ≃ − q

n~

∫

K

f∂kjS−1vid3k, (9.12)

1

~∂kj

S−1vi

= −m−1ij

Wtot+ vj

∂

∂ε

(viWtot

)+vivjW 2tot

∂Wtot

∂ε+∂S−1

i

∂εvj . (9.13)

For an isotropic band structure with effective mass m∗ and a constanttotal scattering rate τ−1 := Wtot, only the first term of the rhs of(9.13) contributes:

1

~∂kj

S−1vi

= −m−1ij

Wtot= − τ

m∗δi,j . (9.14)

Pluggin (9.14) in (9.12) yields the usual macroscopic definition of themobility:

µij ≃qτ

m∗δi,j , (9.15)

which is always positive.For a non-parabolic band structure (9.13) may become positive

and, therefore, the mobility negative. This fact can be illustratedusing a simple NIN structure with the following characteristics:

1. The two N -parts are 0.1µm long, whereas the intrinsic part is0.14µm long.

2. The doping concentration of the N -regions is of 5 × 1018 cm−3.

When simulating this device with the new one-particle MC methodintroduced in Section 3.8 and applying a bias of 2 V, the mobilityprofile exhibits a strange behaviour directly behind the NI-transitionas visible from Fig. 9.82. Fig. 9.83 shows the contribution of the fourterms of the rhs of (9.13). The first term is always positive as must be.The two terms containing functions averaged on iso-energy surfaces(term 2 and 4) do almost not contribute, whereas the 3rd term isalone responsible for the negative mobility. This negative mobility

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0 0.1 0.2 0.3position [µm]

-200

0

200

400

600

800

1000

1200

µxx

[cm

2/V

s]

Figure 9.82: Spatial profile of the component of the mobility alongthe transport direction for the NIN structure at a bias of 2 V.

appends in a domain where the particles comming from the left of thedevice are flying quasi-balistically and where the current is dominatedby a strong diffusion current due to a large gradient of the density.It is therefore difficult to physically understand the meaning of thisnegative mobility. This should be further investigated.

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9.6. A CASE OF NEGATIVE MOBILITY 223

0 0.1 0.2 0.3position [µm]

-1250

-1000

-750

-500

-250

0

250

500

750

1000

1250

1500

1750

1st term2nd term3rd term4th termsum

contr

ibutions

toµxx[c

m2/V

s]

Figure 9.83: Spatial profile of the four terms of the rhs of (9.13) forthe NIN structure at a bias of 2 V.

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Chapter 10

Conclusion

The numerical computation of semiconductor properties can now sig-nificantly benefit from coupling the theoretical concept of ISO to nu-merics and powerful computers as outlined in the present work. Morespecifically, the concept of MISO permits to combine transport mod-els with MC methods in a unified framework without the need of therelaxation time approximation. As shown in this thesis, the MISOscan be applied to the computation of

1. tensorial transport coefficients,

2. relaxation times,

3. Langevin noise sources,

4. Hall factors,

5. low-field solutions to the space-homogeneous Boltzmann equa-tion in any order in the electric and magnetic field, for Fermi-and Boltzmann statistics,

and gives the possibilities to

1. include G-R processes in the framework of a MC simulation,

2. check the assumptions underlying the linear response theory,

225

226 CHAPTER 10. CONCLUSION

3. generalise the just-before-scattering method.

Thus, computational methods using MISOs offer a viable alternativeto conventional RTA-based approaches.

Together with the new possibilities offered by the MISOs, newquestions appear which seem quite difficult to answer and must behandled with care. In Chapter 9, the comparison between exact trans-port coefficients (TCs) and transport models showed that even thoughthe TCs are very different, the currents, Y-parameters and noise char-acteristics are almost identical. This is rather disturbing and a betterreason than serendipity should be found. It could be a hint that other,more appropriate definitions of drift and mobility tensors exist.

Chapter 11

Outlook

The FEM seems to be well applicable to device simulation. However,much more devices should be simulated to underpin this statement.In this thesis only a few tens of realistic devices from partners in theindustry have been successfully tested.

As pointed out in Chapter 9 the computation of noise using theone-particle MC method is still hampered by numerical problems.However, no fundamental problems exist for the computation of thederivatives of the transport coefficients. Thus, the one-particle MCmethod based on the new iterative algorithm could become a powerfultool to compute not only steady states but also correlation functionsof the fluctuations of the current. Although this algorithm is slowerthan the standard one, the possibility to include G-R processes shouldbe esteemed. Furthermore, in comparison with the many-particle MCmethod, the one-particle method can be very efficiently parallelisedusing e.g. the MPI2 library. It hypothetically allows speed-ups pro-portional to the number of available processors, and the real speed-upsare close to that.

In view of the experience acquired during this thesis, one shouldgo further into the following questions:

1. Are there alternative definitions for the transport coefficientscompared to those given in this work?

2. How many moments are needed to cancel the ∇r-term in the

227

228 CHAPTER 11. OUTLOOK

noise sources of a transport model?

3. Is the acceleration-fluctuation scheme superior to the impedancefield method for realistic devices?

4. What is the impact of exactly computed relaxation times ontransport models for different materials?

5. How can the theory exposed in this thesis be generalised to theBoltzmann-Wigner equation?

The last question is the most interesting one and should have priority.

Appendices

229

Appendix A

Inverse ScatteringOperators

A.1 Appendix

Using (2.67), S−1g (k0, b0) can be rewritten as

S−1g (~k0, b0) = −g(

~k0, b0) − g(ε0, b0)

Wtot(ε0, b0)+S−1

g (~k0, b0) = −g(~k0, b0)

Wtot+Lg.

(A.1)

Kgg′(~k, b) is by definition:

Kgg′ (~k, b) =∑

b0

∫

Vb0

w(~k, b|~k0, b0)

S−1g (~k, b)S−1

g′ (~k, b)

−S−1g (~k, b)S−1

g′ (~k0, b0)

−S−1g (~k0, b0)S

−1g′ (~k, b)

+S−1g (~k0, b0)S

−1g′ (~k0, b0)

d3k0.

(A.2)

231

232 APPENDIX A. INVERSE SCATTERING OPERATORS

Eq. (A.2) can be separated into four terms:

Kgg′(~k, b) = Wtot(~k, b)S−1g (~k, b)S−1

g′ (~k, b) (A.3)

−S−1g (~k, b)

∑

b0

εmax(b0)∫

εmin(b0)

w(ǫ(~k), b|ǫ0, b0)Z(ǫ0, b0)S−1g′ (ǫ0, b0)dε0 (A.4)

−S−1g′ (~k, b)

∑

b0

εmax(b0)∫

εmin(b0)

w(ǫ(~k), b|ǫ0, b0)Z(ǫ0, b0)S−1g (ǫ0, b0)dε0 (A.5)

+∑

b0

εmax(b0)∫

εmin(b0)

w(ǫ(~k), b|ǫ0, b0)Z(ǫ0, b0)

[gg′

W 2tot

−Lgg′

Wtot− Lg′

g

Wtot+ LgLg′

]dε0, (A.6)

where the relations

∑

b

∫

Vb

f(ǫ(~k), b)g(~k, b)d3k

=∑

b0

ǫmax(b0)∫

ǫmin(b0)

f(ǫ0, b0)∑

b

∫

Vb

g(~k, b)δ(ǫ0 − ǫ(~k))δb,b0d3kdǫ0

=∑

b0

ǫmax(b0)∫

ǫmin(b0)

f(ǫ0, b0)Z(ǫ0, b0)g(ǫ0, b0)dǫ0, (A.7)

have been used.By definition (A.6) is equal to

C (gg′

Wtot− [Lgg

′ + Lg′ g] +WtotLgLg′). (A.8)

One rewrites (A.4) and (A.5) as

−Wtot(~k, b)S−1g (~k, b)(B S−1

g′ ), (A.9)

−Wtot(~k, b)S−1g′ (~k, b)(B S−1

g ). (A.10)

A.2. APPENDIX 233

Because of (2.56), S−1g solve the equation

S−1g = B S−1

g −g − 〈g〉eq

neq

Wtot, (A.11)

and, thus,

B S−1g = S−1

g +g − 〈g〉eq

neq

Wtot= Lg −

〈g〉eq

neq

Wtot. (A.12)

Using (A.12) one can rewrite (A.9) and (A.10) as

−Wtot(~k, b)S−1g (~k, b)Lg′ + S−1

g (~k, b)〈g′〉eqneq

, (A.13)

−Wtot(~k, b)S−1g′ (~k, b)Lg + S−1

g′ (~k, b)〈g〉eqneq

. (A.14)

Replacing (A.4) and (A.5) by (A.13) and (A.14) in the expressionfor Kgg′ and rearranging gives (3.52).

A.2 Appendix

A.2.1

To prove:

limω→0

|T −1

g (ω)〉a = limω→0

−PL(ω)

∞∑

k=0

(B(ω)PL(ω))kPL(ω)|zg(ω)〉

= −PL(0)∞∑

k=0

(B(0)PL(0))kPL(0)|zg(0)〉 <∞. (A.15)

First note thatlimω→0

|zg(ω)〉 = |zg(0)〉 (A.16)

is well defined.Secondly lim

ω→0PL(ω)B(ω)PL(ω) = PL(0)B(0)PL(0) is also well

defined, and by construction ρ(PL(ω)B(ω)PL(ω)) < 1, ∀ω.


Therefore,

(1− PL(ω)B(ω)PL(ω))−1

=

∞∑

k=0

(PL(ω)B(ω)PL(ω)

)k, ∀ω.

(A.17)

Eq. (A.15) is proven by setting ω = 0 in (A.17) and using prop-erty (A.16).

A.2.2

To prove (2.130):

− 1

1 − α(ω)L(ω)|zg(ω)〉 =

L(0)|zg(0)〉ωα′

+|L(ω)zg(ω)〉′

α′+O(ω) ,

(A.18)

where f ′ := ∂f(ω)∂ω

∣∣∣ω=0

.

By making a Laurent expansion of |L(ω)zg(ω)〉 and α(ω) one finds

− 1

1 − α(ω)L(ω)|zg(ω)〉 =

L(0)|zg(0)〉ωα′ +O(ω2)

+|L(ω)zg(ω)〉′α′ +O(ω)

+O(ω)

=L(0)|zg(0)〉

ωα′

(1 + O(ω2)

)+

|L(ω)zg(ω)〉′α′

(1 +O(ω)) +O(ω)

=L(0)|zg(0)〉

ωα′+

|L(ω)zg(ω)〉′α′

+O(ω). (A.19)

A.2.3

To prove: The solution to (2.125) is

|T −1

g 〉b = − 1

1 − α(ω)L(ω)|zg(ω)〉. (A.20)

A.2. APPENDIX 235

Definitions:

γ :=〈y(ω)|T −1

g 〉〈y(ω)|x(ω)〉 , (A.21)

ξ :=〈y(ω)|zg〉

〈y(ω)|x(ω)〉 . (A.22)

Using (A.21) and (A.22) one can rewrite (2.125) as

γ|x(ω)〉 = α(ω)γ|x(ω)〉 − ξ|x(ω)〉. (A.23)

From (A.23) one finds

γ = − ξ

1 − α(ω)(A.24)

and, thus, (A.20).

A.2.4

To compute: B(0)|T −1∗g 〉, where

|T −1∗g 〉 := |T −1

g (0)〉a + |u〉. (A.25)

By definition (see (2.75))

B(0)|T −1

g (0)〉a = |T −1

g (0)〉a + PL(0)|zg(0)〉. (A.26)

In Appendix A.2.6 it is shown that

B(0)|u〉 =

|u〉 + L(0)|zg(0)〉 − D−1

L(0)|zg(0)〉〈x(0)|y(0)〉⟨y(0)|D−1|x(0)

⟩ . (A.27)

Adding (A.26) to (A.27) and using the property PL(0) + L(0) = 1,leads to the result

B(0)|T −1∗g 〉 = |T −1∗

g 〉+

1− D

−1L(0)

〈x(0)|y(0)〉⟨x(0)|D−1|y(0)

⟩

|zg(0)〉.

(A.28)


A.2.5

Discussion of the conditions under which

|x(0)〉〈y(0)|D−1|g〉⟨y(0)|D−1|x(0)

⟩ !=

〈g〉eqneq

(A.29)

is fulfilled.By construction

|x(0)〉 =|f1〉||f1||

, (A.30)

|x(0)〉 =|f1〉||f1||

, (A.31)

D diagonal, (D−1

)ii =1

Wtot(ǫ(b)j , b)

, i :=

b−1∑

l=0

nl + j + 1. (A.32)

The first condition is

|y(0)〉 ≃ | ˜WtotfeqZ〉|| ˜WtotfeqZ||

. (A.33)

Under condition (A.33) one can write

D−1|y(0)〉 ≃ |feqZ〉

|| ˜WtotfeqZ||(A.34)

and, thus,

|x(0)〉〈y(0)|D−1|g〉⟨y(0)|D−1|x(0)

⟩ ≃ |f1〉

⟨feqZ|g

⟩

⟨feqZ|f1

⟩ . (A.35)

Now assume that the discretisation is homogeneous in the domainwhere |feqZ〉 is essentially non-zero. Under this assumption one canwrite ⟨

feqZ|g⟩

⟨feqZ|f1

⟩ ≃

⟨feq|g

⟩

⟨feq|f1

⟩ ≃ 〈feq|g〉〈feq|f1〉

. (A.36)

A.2. APPENDIX 237

Then, the better conditions (A.34) and (A.36) are fulfilled, thebetter

|f1〉

⟨feqZ|g

⟩

⟨feqZ|f1

⟩ ≃ |f1〉

⟨feq|g

⟩

⟨feq|f1

⟩ ≃ |f1〉〈feq|g〉〈feq|f1〉

= |f1〉〈g〉eqneq

√∑

b′

|Vb′ | =〈g〉eqneq

(A.37)

will hold.Remember that f1

√∑b′

|Vb′ | = 1 by definition.

A.2.6

To prove:

B(0)|u〉 = |u〉+L(0)|zg(0)〉−D−1

L(0)|zg(0)〉〈x(0)|y(0)〉⟨y(0)|D−1|x(0)

⟩ .

(A.38)

In the following, the notations

f ′ :=∂f(ω)

∂ω

∣∣∣∣ω=0

(A.39)

and

|z〉 :=〈y(ω)|zg(ω)〉〈y(ω)|x(ω)〉 |x(ω)〉 = L(ω)|zg(ω)〉. (A.40)

will be used.By definition |u〉 := |z〉′

α′. Thus, one needs to compute B(0) |z〉

′

α′.

Note thatB(ω)|z(ω)〉 = α(ω)|z(ω)〉 (A.41)

(see (2.121)) and that α(0) = 1.Deriving (A.41) by ω and setting ω = 0 gives

(B(ω)|z(ω)〉

)′= B

′|z(0)〉 + B(0)|z〉′ = α′|z(0)〉 + |z〉′. (A.42)


Dividing (A.42) by α′ and rearranging leads to

B(0)|z〉′α′

=|z〉′α′

+ |z(0)〉 − B′

α′|z(0)〉

=|z〉′α′

+ L(0)|zg(0)〉 − B′

α′L(0)|zg(0)〉. (A.43)

The rhs of (A.43) already contains two terms one is looking for(compare with the rhs of (A.38)). It remaines to prove that

B′

α′L(0)|zg(0)〉 = D

−1L(0)|zg(0)〉〈x(0)|y(0)〉⟨

x(0)|D−1|y(0)⟩ . (A.44)

In Appendix A.2.7 one proves

B′= iD

−1B(0). (A.45)

In Appendix A.2.8 one proves

α′ =

⟨y(0)|B′|x(0)

⟩

〈y(0)|x(0)〉 . (A.46)

Inserting (A.45) into (A.46) and remembering that B|x(0)〉 =|x(0)〉, one finds

α′ =i⟨y(0)|D−1|x(0)

⟩

〈y(0)|x(0)〉 . (A.47)

Inserting (A.47) and (A.45) in the lhs of (A.44) and using theproperty B(0)L(0) = L(0) leads to the rhs of (A.44).

A.2.7

To prove:

∂B(ω)

∂ω

∣∣∣∣∣ω=0

= iD−1

B(0). (A.48)

Remembering that

B(ω) = U(ω)B(0) = (D − iω1)−1DB(0),

A.2. APPENDIX 239

one can write

∂B(ω)

∂ω

∣∣∣∣∣ω=0

=∂((D − iω1)−1

)

∂ω

∣∣∣∣∣∣ω=0

DB(0). (A.49)

It is trivial that

(∂(D − iω1)−1

∂ω

∣∣∣∣∣ω=0

)

ii

=∂

∂ω

1

dii − iω

∣∣∣∣ω=0

=i

d2ii

⇔

∂(D − iω1)−1

∂ω

∣∣∣∣∣ω=0

= iD−1

D−1, (A.50)

where dii :=(D)ii.

Hence one finds

∂B(ω)

∂ω

∣∣∣∣∣ω=0

= iD−1

D−1

DB(0) = iD−1

B(0). (A.51)

A.2.8

To prove:

∂α(ω)

∂ω

∣∣∣∣ω=0

=

⟨y(0)|B′|x(0)

⟩

〈y(0)|x(0)〉 . (A.52)

Starting from

B(ω)|x(ω)〉 = α(ω)|x(ω)〉 , (A.53)

where α(0) = 1, and performing a Taylor expansion of both sides, oneobtains

[B(0) + ωB

′+O(ω2)

] [|x(0)〉 + ω|x〉′ +O(ω2)

]=

[1 + ωα′ +O(ω2)

] [|x(0)〉 + ω|x〉′ +O(ω2)

]. (A.54)


Collecting all terms which contain the first power of ω yields

B(0)|x〉′ + B′|x(0)〉 = |x〉′ + α′|x(0)〉 . (A.55)

Multiplying (A.55) with 〈y(0)| and using the property 〈y(0)|B(0) =〈y(0)| one obtains

〈y(0)|B′|x(0)〉 = α′ 〈y(0)|x(0)〉 ⇔ α′ =〈y(0)|B′|x(0)〉〈y(0)|x(0)〉 . (A.56)

A.3 Appendix

To prove:supp(Bf) ⊆ supp(Bf), ∀f ∈ D(K). (A.57)

An argument using the Dirac delta function will be given. Using

a regularisation (e.g. δ(a)γ (ǫ), see Section 9.2.3) will yield the same

results.First, one needs a basis of D(K):

f(b)i ∈ D(K), f

(b)i (ǫ

(b′)j ) := δbb′δij . (A.58)

An example of such a function is shown in Fig. A.1.

If one can prove (A.57) for all f(b)i , then the claim is proven.

Without restriction of generality, only the case

w(ǫ, b|ǫ0, b0) = δ(ǫ− ǫ0 + Ω)δb,b0 (A.59)

is considered. Then

B =δ(ǫ− ǫ0 + Ω)δb,b0Z(ǫ0, b0)

Z(ǫ, b)= δ(ǫ− ǫ0 + Ω)δb,b0 . (A.60)

What is the effect of B on f(b)3 ? Part (I) of Fig. A.1 shows f

(b)3 ,

and the vertical ticks represent the discretisation, i.e the points of I(b)14

(see (2.71)). Part (II) shows the function B f (b)3 the most important

feature of which is not its shape, but the fact that there is more than

A.3. APPENDIX 241

one point of I(b)14 , where B f (b)

3 is different from 0. Part (III) shows

the function PDB f (b)3 = B f (b)

3 (see (2.73)). One sees that PD can

only let supp(Bf (b)3 ) increase. Thus, supp(Bf (b)

3 ) ⊆ supp(Bf (b)3 ).

B

^

supp( )f3(b)

B

^

f3(b)

B~

supp( ) supp(

(b)

1

f3(b)

f3

)

∆ε(b)

ε0

Ω

ε

ε~

f3(b)

f3(b)

B

^

B~U

supp( )f3(b)

B

1

ε(b)min ε(b)

max

II

I

III

ε(b)min ε(b)

maxB

^

PD

ε(b)min ε(b)

max

1

Figure A.1: Effect of the operator B on a function f(b)i .

This can be done for all f(b)i . The argument also holds, if the rhs of

(A.59) is multiplied with a strict positive function a(ǫ, b)

δ(ǫ− ǫ0 + Ω)δb,b0 → a(ǫ, b)δ(ǫ− ǫ0 + Ω)δb,b0 .

The function a(ǫ, b) will change the shape of B f (b)i but not its

support.

If one allows inter-band-valley scattering events, the argument alsoholds under the condition that max

b,b′|ǫ(b) −ǫ(b′)| is smaller than the

smallest (non zero) phonon energy. (ǫ(b) is the discretisation energyinterval in the band-valley b, see Fig. A.1.)

Therefore, if w(ǫ, b|ǫ0, b0) can be written as a countable sum of


delta functions, i.e.

w(ǫ, b|ǫ0, b0) :=

∞∑

k=0

ak(ǫ, b)δ(ǫ− ǫ0 + Ωk)δb,b0 , (A.61)

then, because the claim holds for each term of the sum, it also holdsfor the sum (supp (f + g) = supp (f)

⋃supp (g)).

If one adds impact ionisation as described in (9.1), an analogousargument can be given.

A.4 Appendix

To prove:

∃M ∈ N | AM is a strong positive operator (A.62)

and one can explicitly write an upper limit for M .

Without restriction of generality, (supp (f + g) =supp (f)⋃

supp (g))only the simple single-phonon case is considered:

w(ǫ, b|ǫ0, b0) = δ(a)γ (ǫ− ǫ0 ± Ω)δb,b0 . (A.63)

Then

A(~k, b| ~k0, b0) =δ(a)γ (ǫ(~k) − ǫ( ~k0) ± Ω)δb,b0

Wtot(ǫ(~k), b). (A.64)

What is the effect of the repeated application of A on g0 := δ(ǫ−ǫ(b)min)δb,b0?

Consider a simple example, where γ := Ω/11 and Ω as shown inFig. A.2.

Fig. A.2 depicts the evolution of supp(AM ·g0) as a function of M .

One can see, e.g., that supp(A · g0) = ( ~k0, b0) ∈ K | b = b0, ǫ(k) ∈[ǫ

(b)min + Ω − γ, ǫ

(b)min + Ω + γ] (step 0 → 1 in Fig. A.2).

A.4. APPENDIX 243

g

11109876543210

0ε

)−ε(δ (b)minε0

g :=

:= ).MAsupp( 0

ε

M

Num

ber

of s

catte

ring

even

ts

Ω

totε0γ(a)δδb, 0b),|b,ε(A 0b W

γ= /11Ω

/)Ω±−ε(= )b,ε(

(b)minε (b)

maxε

Figure A.2: Effect of the operator A repeatedly applied on the ”func-

tion” g0(ǫ, b) := δ(ǫ− ǫ(b)min).

Fig. A.2 shows that for M = 11, supp(A11 · g0) = [ǫ(b)min, ǫ

(b)max], i.e.,

that after 11 scattering events the probability to reach any state in

the band-valley b, starting in the point (ǫ(b)min, b), is positive.

The same argument can be used to show that this claim holds forany starting point (ǫ(b), b) in the band-valley b. Thus, A11 is a strongpositive operator.

The important point is that

M = 11 = Ω/γ. (A.65)

From this observation one can generalise the argumentation tosystems with multiple band-valleys (as long as there is at least a pos-sibility to go from any band-valley to any other), and it is easy tounderstand that one can take

M = max ((Ωmin/γ) + 1, |ǫ(0)min − ǫ(bmax)max |/Ωmin), (A.66)

where γ is given (e.g. γ = 10−306), Ωmin is the smallest (non zero)

phonon energy in the system, and ǫ(0)min (resp. ǫ

(bmax)max ) the smallest

(resp. the highest) energy in the system.


In this thesis the following numerical values have been used:

γ = 10−306 eV, (A.67)

Ωmin = 6.26 meV, (A.68)

ǫ(bmax)max − ǫ

(0)min = 11.8 eV − 0.0 eV = 11.8 eV. (A.69)

One can, therefore, take M = Ωmin/γ + 1 = 6.26 · 10303 + 1 and besure that AM is a strong positive operator.

Note that in the case of model 1 (phonon+impact ionisation) onecan show that there is a M that does not depends on γ, but only on

ǫ(bmax)max − ǫ

(0)min and Ωmin.

A.5 Appendix

To prove: Under the transformation

w(ǫ, b|ǫ0, b0) −→ w′(ǫ, b|ǫ0, b0) := w(ǫ, b|ǫ0, b0)+t(ǫ, b)δ(ǫ, ǫ0)δb,b0(A.70)

in (2.60) (where t(ǫ, b) is a given function), the solution stays un-changed.

The transformation (A.70) modifies Wtot in the following way:

Wtot(ǫ, b) −→W ′tot(ǫ, b) := Wtot(ǫ, b) + t(ǫ, b)Z(ǫ, b). (A.71)

Multiplying (2.60) with Wtot(ǫ, b) gives

Wtot(ǫ, b)Hg(ǫ, b) =

∑

b0

εmax(b0)∫

εmin(b0)

w(ǫ, b|ǫ0, b0)Z(ǫ0, b0)Hg(ǫ0, b0)dǫ0 − g − 〈g〉eqfeq(ǫ, b).

(A.72)

Replacing w by w′ in (A.72) does not change the equation, becausethe t(ǫ, b)Z(ǫ, b) terms are the same on both sides.

Thus, (A.72) is invariant under (A.70).

A.6. APPENDIX 245

Note that the function Hg from (2.65) is modified by the transfor-mation (A.70) in the simple way

Hg(~k0, b0) = −g(~k0, b0) − g(~k0, b0)

Wtot(ε0, b0)+ Hg −→

H ′g(~k0, b0) = −g(

~k0, b0) − g(~k0, b0)

W ′tot(ε0, b0)

+ Hg. (A.73)

The argument stays valid in the frequency-dependent case (2.113).

A.6 Appendix

By construction model 2 has the Maxwell distribution as equilibriumdistribution. If one multiplies the Maxwell distribution with a positivefunction F which has the property

F (ǫ+ Ωi) = F (ǫ), ∀i = 1, .., Nphonon, (A.74)

where Nphonon is the number of phonons in the model and Ωi is theenergy of the i-th phonon, the product would be clearly another solu-tion of S · f = 0, and, therefore, a possible equilibrium distribution.

As long as the phonon energies are rational numbers (this is alwaysthe case as soon as one uses a computer), there exists an infinitenumber of such F s.

Thus, for model 2, an infinite number of solutions exists to theproblem S · f = 0.

This argument is based on an argument in [39].

Appendix B

Space-homogeneousBoltzmann equation

For small electric and magnetic fields, the homogeneous BE can besolved to any order using the moments of the ISO.

In this chapter, above statement will be first proved in the case ofFermi statistics in the presence of an electric field only. The generalisa-tion to mixed magnetic and electric fields as well as the specialisationto Boltzmann statistics will be given at the end.

To solve the BE, the following ansatz for the solution f and thescattering operator S are used:

f =∞∑

i=0

fi, (B.1)

S(k, k′) =

∞∑

i=0

Si(k, k′). (B.2)

The fi and Si are chosen such that they only contain terms propor-tional to Ei, where E is the magnitude of the electric field. Thus,f0 is the Fermi distribution feq. Under these constraints, Si can be

247

248 APPENDIX B. BOLTZMANN EQUATION

unequivocally written as

S0(k, k′) =

w(k′, k)(1 − feq(k)) − δ3(k − k′)

∫w(k, k′)(1 − feq(k

′))d3k′, (B.3)

Si(k, k′) = −w(k′, k)fi(k)+δ

3(k−k′)∫w(k, k′)fi(k

′)d3k′, i > 0.

(B.4)

Using (B.1) and (B.2), the space-homogeneous BE (SHBE) reads

−q ~E~

∇k

(∞∑

i=0

|fi〉)

=

∞∑

l=0

Sl ·∞∑

i=0

|fi〉. (B.5)

In the following, S0 will be replaced by the more natural notationSeq, because in fact S0 is by definition the scattering operator atthermodynamic equilibrium.

For the terms of order En (n > 0) the following equation can bewritten:

−q ~E~

∇k|fn−1〉 =

n∑

l=0

Sl · |fn−l〉. (B.6)

A possible way to solve (B.6) is to rewrite it as an equation for fnas function of f0 · · · fn−1. In this way, because f0 is known, one coulditeratively solve all equations by induction.

Putting all the terms of (B.6) containing fn in the lhs and allothers in the rhs leads to

Seq · |fn〉 + Sn · |feq〉 =−q ~E

~∇k|fn−1〉 −

n−1∑

l=1

Sl · |fn−l〉. (B.7)

Using the nice property Sn|feq〉 = Seq · | fnfeq

(1−feq)〉 (see B.2) one finds

Seq ·(|fn〉 + | fnfeq

(1 − feq)〉)

= Seq · |fn

(1 − feq)〉 =

−q ~E~

∇k|fn−1〉 −n−1∑

l=1

Sl · |fn−l〉. (B.8)

249

To simplify the notations, two definitions can be introduced:

P1 := 1− |feq〉〈1|neq

, (B.9)

gn :=1

feq

(−q ~E

~∇k|fn−1〉 −

n−1∑

l=1

Sl · |fn−l〉). (B.10)

Multiplying (B.8) with the ISO S−1eq leads to

P1 · |fn

(1 − feq)〉 = S−1

eq |feqgn〉 (B.11)

Using the property S−1(k, k′)feq(k′) = feq(k)S

−1T (k, k′), (see Ap-pendix B.1) one can rewrite (B.11) in its almost final form

P1 · |fn

(1 − feq)〉 = feq|S−1

gn〉, (B.12)

where S−1gn

is the gn-moment of the ISO S−1eq . Eq. (B.12) is an equation

for the part of fn

(1−feq) living in the Banach space Q1 built by P1. The

other part must be proportional to feq. Therefore, the following ansatzfor fn can be made:

fn =: fn + αnfeq(1 − feq), (B.13)

where fn lives in Q1, i.e. P1 · |fn〉 = |fn〉.Using (B.12) gives the solution for fn:

fn(k) = feq(k)(1 − feq(k))S−1gn

(k). (B.14)

The constant αn can be unequivocally computed by requiring thedensity (the norm of f) to be constant in any order, i.e. 〈fn|1〉 = 0for all n > 0. Thus, one finds

αn = −∫feq(k)(1 − feq(k))S

−1gn

(k)d3k∫feq(k)(1 − feq(k))d3k

. (B.15)

Finally, the solution for fn can be written

fn = feq(k)(1 − feq(k))(S−1gn

(k) + αn), (B.16)


and the exact expression for f is given by:

f(k) = feq(k)

(1 + (1 − feq(k))

∞∑

i=1

[S−1gi

(k) + αi]). (B.17)

B.1 Important property of the ISO S−1eq

To be able to easily derive the solution to the space-homogeneousBE to any order in the electric and magnetic field magnitude, thefollowing property of the ISO S−1

eq is needed:

S−1eq (k, k′)feq(k

′) = feq(k)S−1eq

T(k, k′). (B.18)

This property is fulfilled only, if the principle of detailed balance holds,i.e.

w(k, k′)(1 − feq(k′))feq(k) = w(k′, k)(1 − feq(k))feq(k

′), (B.19)

where w(k, k′) is the transition probability per unit time. Rememberthat the transition probability w(k, k′) per unit time to scatter fromk to k′ was redefined as

w(k, k′) = w(k, k′)(1 − feq(k′)). (B.20)

Using (2.22), (B.19) and (B.20) one can write

Seq(k, k′)feq(k

′) = feq(k)STeq(k, k

′). (B.21)

Multiplying (B.21) on the left by S−1eq one finds

δ3(k−k′)feq(k)−feq(k)feq(k

′)

neq=

∫S−1eq (k, k′′)feq(k

′′)STeq(k′′, k′)d3k′′

(B.22)

Multiplying (B.22) on the right by S−1eq

T(k′, k′′′) and integrating over

k′ leads to

feq(k)S−1eq

T(k, k′′′) − (S−1

eq |feq〉)(k′′′)feq(k)neq

=

S−1eq (k, k′′′)feq(k

′′′) − (S−1eq |feq〉)(k)feq(k′′′)

neq. (B.23)

Administrator

Highlight

B.2. IMPORTANT PROPERTY OF SN 251

By definition of S−1eq one has

Seq · S−1eq |feq〉 = 0. (B.24)

Because the only right eigenvector with eigenvalue 0 of Seq is |feq〉,(B.24) tells us

S−1eq |feq〉 ∝ |feq〉. (B.25)

Using this property in (B.23) directly leads to (B.18).

B.2 Important property of Sn

To prove:

Sn|feq〉 = Seq|feqfn

(1 − feq)〉. (B.26)

Writing (B.26) explicitly and using the principle of detailed balance,one finds

Sn|feq〉 = −∫feq(k

′)w(k′, k)fn(k)d3k′+feq(k)

∫w(k, k′)fn(k′)d3k′

= −∫w(k, k′)

(1 − feq(k′))feq(k)fn(k)

(1 − feq(k))d3k′

+

∫w(k′, k)

(1 − feq(k))feq(k′)fn(k′)

(1 − feq(k′))d3k′

= Seq|feqfn

(1 − feq)〉. (B.27)


B.3 Generalisation

B.3.1 Fermi statistics

In the case of the SHBE with an electric and a magnetic field, (B.17)can be easily generalised. Using the ansatz

f =

∞∑

i,j=0

fi,j , (B.28)

S(k, k′) =∞∑

i,j=0

Si,j(k, k′), (B.29)

where fi,j and Si,j are such that they only contain terms proportionalto EiBj , where E is the magnitude of the electric field and B themagnitude of the magnetic field, one finds the general form

S0,0(k, k′) = w(k′, k)(1−feq(k))−δ3(k−k′)

∫w(k, k′)(1 − feq(k

′))d3k′,

(B.30)

Si,j(k, k′) = −w(k′, k)fi,j(k) + δ3(k − k′)

∫w(k, k′)fi,j(k

′)d3k′,

i or j > 0. (B.31)

gn,m :=

−q ~E~feq

∇k|fn−1,m〉− q

feq(~v∧ ~B)∇k|fn,m−1〉−

1

feq

n∑

l=0

m−1∑

s=1

Sl,s · |fn−l,m−s〉

− 1

feq

n∑

l=1

Sl,0 · |fn−l,m〉 − 1

feq

n−1∑

l=0

Sl,m · |fn−l,0〉. (B.32)

The general expression for f is therefore

f(k) = feq(k)

1 + (1 − feq(k))

∞∑

i,j=1

[S−1gi,j

(k) + αi,j

] . (B.33)

B.3. GENERALISATION 253

B.3.2 Boltzmann statistics

In the case of Boltzmann statistics all Si,j for i or j > 0 disappearand the (1 − feq) must be replaced by 1. Doing this, one finds

gn,m :=−q ~E~feq

∇k|fn−1,m〉 −q

feq(~v ∧ ~B)∇k|fn,m−1〉, (B.34)

and the solution for f is:

f(k) = feq(k)

1 +

∞∑

i,j=1

[S−1gi,j

(k) + αi,j

] . (B.35)

Appendix C

The EB model and RFnoise

In this appendix the computation of RF noise starting from the EBis presented in analogy to Section 7.6.

C.1 The EB Langevin equations

The Langevin equations for the EB (Bløtekjær [4]) model are:

−∇ǫ∇δϕ = q(δp− δn+N − P ), (C.1)

∇δ ~Jnq

= ∇[δµnn~Fn + µnδn~Fn + µnnδ ~Fn

]

− iω∇[αnδn~Fn + δαnn~Fn + αnnδ ~Fn

]+ ∇δ~ςjn

q

= −iωδn+ δR+ δςRn, (C.2)

255

256 APPENDIX C. THE EB MODEL AND RF NOISE

∇δ ~Jpq

= ∇[δµpp~Fp + µpδp∇~Fp + µppδ ~Fp

]

− iω∇[αpδp ~Fp + δαpp~Fp + αppδ ~Fp

]+ ∇δ~ςjp

q

= iωδp− δR+ δςRp, (C.3)

∇δ ~Sn =

− q5

2∇ [δµnnεn∇εn + µnδnεn∇εn + µnnδεn∇εn + µnnεn∇δεn]

− iωq5

2∇ [δαnnεn∇εn + αnδnεn∇εn + αnnδεn∇εn + αnnεn∇δεn]

− 5

2∇[δεn ~Jn + εnδ ~Jn

]+ ∇δ~ςSn

= −δ ~Jn∇(ϕ− εg2

) − ~Jn∇δϕ

− 3q

2τεn

[δn(εn(1 − iωτεn) − εL) + nδεn(1 − iωτεn

)]

− 3q

2[δεnR+ εnδR] +

3q

2τεn

δςεn, (C.4)

∇δ ~Sp =

− q5

2∇ [δµppεp∇εp + µpδpεp∇εp + µppδεp∇εp + µppεp∇δεp]

− iωq5

2∇ [δαppεp∇εp + αpδpεp∇εp + αppδεp∇εp + αppεp∇δεp]

+5

2∇[δεp ~Jp + εpδ ~Jp

]+ ∇δ~ςSp

= −δ ~Jp∇(ϕ+εg2

) − ~Jp∇δϕ

− 3q

2τεp

[δp(εp(1 − iωτεp

) − εL) + pδεp(1 − iωτεp)]

− 3q

2[δεpR+ εpδR] +

3q

2τεp

δςεp, (C.5)

where δ~ςjn,pare the noise sources (NS) for the current density, δ~ςSn,p

the NS for the energy current density, δςεn,pthe NS for the energy,

and δςRn,pthe NS for generation-recombination processes.

C.1. THE EB LANGEVIN EQUATIONS 257

C.1.1 Discretisation

By integrating each of the equations (C.1)–(C.5) with a test functionof UN and by performing a partial integration of the lhs, one obtainsthe discrete form of the equations:

~sT

B(ω) − iωC︸︷︷︸

:=Λ(ω)

δ~φ(ω) = ~sT δ~ς, ∀~s. (C.6)

The matrix C arises from the rhs of (C.2) and (C.3), the matrix Boriginates from the rest setting the noise sources to 0, and the vectorδ~ς contains the noise sources. The vectors ~s and δ~φ are similar tothose defined in Section 7.5.2:

~s :=

u1

v1

w1

g1

h1

.

.

.uN

vN

wN

gN

hN

, δ~φ :=

δϕ1

δψ1n

δψ1p

δε1nδε1p...

δϕN

δψNnδψNpδεNnδεNp

, (C.7)

where f i is the value of the function f ∈ VN on the point i. u,v, w,g, and h are test functions of UN as defined in Section 7.3.2. Oneassumes here that the 5M first components of the vectors come fromthe M points that belong to a contact. By redefining ~em as the vectorwhich components are all 0 except the components coming from thepoints on the m-th contact which are 1 for the potentials and 0 forthe temperatures, one can compute the Y -parameters as described in


Section 7.5. Redefining ~hl as

~hTl :=(0 − (hRSl )1 − (hRSl )1 0 0 · · · 0 − (hRSl )N − (hRSl )N 0 0

)

(C.8)

allows to use, without modifications, the theory developed in Sec-tion 7.6.3 to compute the spectral density of the terminal currentfluctuations.

C.1.2 Computation

To compute SδIkδIl(ω) one needs the explicit form of δ~ς and then ofSδςδς(ω). Each point pl, l = 1, .., N of the discretisation determines

by construction five components of the vector δ~ς, i.e. (δ~ς)5l−4 is thecomponent coming from the Poisson equation, (δ~ς)5l−3 originates fromthe current continuity equation for electrons, and so one. The FEMscheme yields the following components to δ~ς:

(δ~ς)5l−4 := 0, (C.9)

(δ~ς)5l−3 :=∑

T∈T |pl∈T

det(BT )

∫

bT

[−~bTσ(l)δ~ςjn(ξ) − qδςRn

(ξ)ξσ(l)

]d2ξ,

(C.10)

(δ~ς)5l−2 :=∑

T∈T |pl∈T

det(BT )

∫

bT

[−~bTσ(l)δ~ςjp(ξ) + qδςRp

(ξ)ξσ(l)

]d2ξ,

(C.11)

(δ~ς)5l−1 :=∑

T∈T |pl∈T

det(BT )

∫

bT~bTσ(l)

[−δ~ςSn

(ξ) +5

2εn(ξ)δ~ςjn (ξ)

]d2ξ

+∑

T∈T |pl∈T

det(BT )

∫

bTξσ(l)

[− 3q

2τεn

δςεn(ξ) + δ~ςjn(ξ)∇(ϕ − εg

2)

]d2ξ,

(C.12)

C.1. THE EB LANGEVIN EQUATIONS 259

(δ~ς)5l :=∑

T∈T |pl∈T

det(BT )

∫

bT~bTσ(l)

[−δ~ςSp

(ξ) − 5

2εp(ξ)δ~ςjp (ξ)

]d2ξ

+∑

T∈T |pl∈T

det(BT )

∫

bTξσ(l)

[− 3q

2τεp

δςεp(ξ) + δ~ςjp(ξ)∇(ϕ +

εg2

)

]d2ξ.

(C.13)

In the following all G-R processes will be neglected. To computeSδςδς(ω) one needs the following ansatz and definitions:

S[f(x, t), g(x′, t)](ω) :=

∫ ∞

−∞

limT→∞

1

2T

∫ T

−T

f(x, t)g(x′, t+ s)dte−iωsds,

(C.14)kl∑

T

[f(T, k, l)] :=∑

T∈T |pk,pl∈T

det(BT )f(T, k, l), (C.15)

S[δςεn(x, t), δςεn

(x′, t)](ω) =: n(x)Dεnεn(ω)δ3(x− x′), (C.16)

S[δςεp(x, t), δςεp

(x′, t)](ω) =: p(x)Dεpεp(ω)δ3(x− x′), (C.17)

S[(δ~ςjn)i(x, t), (δ~ςjn )j(x′, t)](ω) =: n(x)Dij

jnjn(ω)δ3(x− x′), (C.18)

S[(δ~ςjp)i(x, t), (δ~ςjp )j(x′, t)](ω) =: p(x)Dij

jpjp(ω)δ3(x− x′), (C.19)

S[(δ~ςSn)i(x, t), (δ~ςSn

)j(x′, t)](ω) =: n(x)Dij

SnSn(ω)δ3(x − x′), (C.20)

S[(δ~ςSp)i(x, t), (δ~ςSp

)j(x′, t)](ω) =: p(x)Dij

SpSp(ω)δ3(x− x′), (C.21)

S[δςεn(x, t), (δ~ςjn )i(x

′, t)](ω) =: n(x)Diεnjn(ω)δ3(x− x′), (C.22)

S[δςεp(x, t), (δ~ςjp )i(x

′, t)](ω) =: p(x)Diεpjp(ω)δ3(x− x′), (C.23)

S[δςεn(x, t), (δ~ςSn

)i(x′, t)](ω) =: n(x)Di

εnSn(ω)δ3(x− x′), (C.24)

S[δςεp(x, t), (δ~ςSp

)i(x′, t)](ω) =: p(x)Di

εpSp(ω)δ3(x− x′), (C.25)

S[(δ~ςjn)i(x, t), (δ~ςSn)j(x

′, t)](ω) =: n(x)DijjnSn

(ω)δ3(x− x′), (C.26)

S[(δ~ςjp )i(x, t), (δ~ςSp)j(x

′, t)](ω) =: p(x)DijjpSp

(ω)δ3(x − x′). (C.27)

One assumes that the Ds are constants on a given triangle, and thatthe NS of electrons are not cross-correlated with the NS of holes.Using these new notations and remembering that

(Sδςδς(ω))ij = S[(δ~ς)i(t), (δ~ς)j(t)](ω) (C.28)


the components of Sδςδς(ω) can be written as follows

case: i = 5k − 3, j = 5l− 3

(Sδςδς)ij =

kl∑

T

[n0~bTσ(k)Djnjn

~bσ(l)], (C.29)

case: i = 5k − 3, j = 5l− 1

(Sδςδς)ij =

kl∑

T

[n0~bTσ(k)DjnSn

~bσ(l) −5

2εLHεn

~bTσ(k)Djnjn~bσ(l)

+3q

2τεn

n1(σ(l))~bTσ(k)Djnεn− n1(σ(l))~bTσ(k)Djnjn∇(φ− εg

2)], (C.30)

case: i = 5k − 1, j = 5l− 1

(Sδςδς)ij =

kl∑

T

[n0~bTσ(k)DSnSn

~bσ(l) +25

4ε2LHεnεn

~bTσ(k)Djnjn~bσ(l)

+9q2

4τ2εn

n2(σ(k), σ(l))Dεnεn+n2(σ(k), σ(l))∇T (φ− εg

2)Djnjn∇(φ− εg

2)

+ (n1(σ(l))~bTσ(k) + n1(σ(k))~bTσ(l))(3q

2τεn

DSnεn−DSnjn∇(φ− εg

2))

− 5

2εL(Hσ(l)

εn

~bTσ(k) +Hσ(k)εn

~bTσ(l))(3q

2τεn

Djnεn−Djnjn∇(φ− εg

2))

−5

2εLHεn

~bTσ(k)(DjnSn+DT

jnSn)~bσ(l)−

3q

τεn

n2(σ(k), σ(l))∇T (φ−εg2

)Djnεn],

(C.31)

case: i = 5k − 2, j = 5l− 2

(Sδςδς)ij =kl∑

T

[p0~bTσ(k)Djpjp

~bσ(l)], (C.32)

C.2. MC-GENERATED NS FOR THE EB MODEL 261

case: i = 5k − 2, j = 5l

(Sδςδς)ij =

kl∑

T

[p0~bTσ(k)DjpSp

~bσ(l) +5

2εLHεp

~bTσ(k)Djpjp~bσ(l)

+3q

2τεp

p1(σ(l))~bσ(k)Djpεp− p1(σ(l))~bTσ(k)Djpjp∇(φ+

εg2

)], (C.33)

case: i = 5k, j = 5l

(Sδςδς)ij =kl∑

T

[p0~bTσ(k)DSpSp

~bσ(l) +25

4ε2LHεpεp

~bTσ(k)Djpjp~bσ(l)

+9q2

4τ2εp

p2(σ(k), σ(l))Dεpεp+p2(σ(k), σ(l))∇T (φ+

εg2

)Djpjp∇(φ+εg2

)

+ (p1(σ(l))~bTσ(k) + p1(σ(k))~bTσ(l))(3q

2τεp

DSpεp−DSpjp∇(φ+

εg2

))

+5

2εL(Hσ(l)

εp

~bTσ(k) +Hσ(k)εp

~bTσ(l))(3q

2τεp

Djpεp−Djpjp∇(φ+

εg2

))

+5

2εLHεp

~bTσ(k)(DjpSp+DT

jpSp)~bσ(l)−

3q

τεp

p2(σ(k), σ(l))∇T (φ+εg2

)Djpεp].

(C.34)

All other elements of Sδςδς are 0.

C.2 MC-generated noise sources for the

EB model

In this section the relation between the noise sources that are com-puted with a MC bulk simulation (see Chapter 5) and the Ds of theprecedent section are presented.

C.2.1 The bulk case

In the case of bulk material (i.e. when n and p are constant in spaceand time, ∇ψn=∇ψp=∇ϕ = constant in space and time, εn and εp


are constants in space), equations (C.1)–(C.5) and the linearised formof equations (7.38), (7.39), (7.42) and (7.43) reduce to

~Fp = ~Fn = −∇ϕ, (C.35)

δ ~Jn = qn~Fn

(δµnδεn

− iωδαnδεn

)δεn + δ~ςjn , (C.36)

δ ~Jp = qp ~Fp

(δµpδεp

− iωδαpδεp

)δεp + δ~ςjp , (C.37)

δ ~Sn = −5

2

[δεn ~Jn + εnδ ~Jn

]+ δ~ςSn

, (C.38)

0 = −δ ~Jn∇ϕ− 3q

2τεn

nδεn(1 − iωτεn) +

3q

2τεn

δςεn, (C.39)

δ ~Sp =5

2

[δεp ~Jp + εpδ ~Jp

]+ δ~ςSp

, (C.40)

0 = −δ ~Jp∇ϕ− 3q

2τεp

pδεp(1 − iωτεp) +

3q

2τεp

δςεp. (C.41)

These equations can be rewritten as

1 −~Fn(δµn

δεn− iω δαn

δεn

)0

− 23τεn

~Fn (1 − iωτεn) 0

52εn

52µn

~Fn 1︸︷︷︸

:=Γn(ω)

δ ~Jn

q

nδεnδ~Sn

q

︸︷︷︸:=δ ~mn

=

δ~ςjn

q

δςεn

δ~ςSn

q

︸︷︷︸:=δ~ςn

,

(C.42)

1 −~Fp(δµp

δεp− iω

δαp

δεp

)0

− 23τεp

~Fp (1 − iωτεp) 0

− 52εp − 5

2µp~Fp 1

︸︷︷︸:=Γp(ω)

δ ~Jp

q

pδεpδ~Sp

q

︸︷︷︸:=δ ~mp

=

δ~ςjp

q

δςεp

δ~ςSp

q

︸︷︷︸:=δ~ςp

.

(C.43)

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Highlight

C.2. MC-GENERATED NS FOR THE EB MODEL 263

With a bulk MC simulation one can compute

M ijn (ω) := S[(δ ~mn(t))i, (δ ~mn(t))j ](ω) (C.44)

andM ijp (ω) := S[(δ ~mp(t))i, (δ ~mp(t))j ](ω). (C.45)

Using (C.42), (C.43) and the Wiener-Khintchin theorem, one canwrite the relation between the D and the M in the form

ΓT∗n MnΓn = n

Djnjn

q2Djnεn

qDjnSn

q2Dεnjn

q Dεnεn

DεnSn

qDSnjn

q2DSnεn

qDSnSn

q2

(C.46)

ΓT∗p MpΓp = p

Djpjp

q2Djpεp

q

DjpSp

q2Dεpjp

q Dεpεp

DεpSp

qDSpjp

q2DSpεp

q

DSpSp

q2

(C.47)

In this section it has been shown that by solving (C.44), (C.45)with a bulk MC simulation, one can compute Sδςδς(ω) and, therefore,the power spectrum of the fluctuations of the terminal currents.

Appendix D

Correlation functionsand delta functions

Let δζi(t) be a family of functions containing delta functions of thetime. When computing the Fourier transform of the correlation func-tions

Sδζiδζj(ω) := lim

T→∞

1

2T

T∫

−T

∞∫

−∞

δζi(t)δζj(t+ w)e−iωwdwdt (D.1)

one is numerically confronted with the problem that as long as thelimit T → ∞ is not reached, the function Sδζiδζj

(w) will not be con-tinuous at all.

A simple folding of the δζi(t) with a Gaussian will remedy thisproblem. Defining

δζi(t) :=

√a

π

∞∫

−∞

e−au2

δζi(t− u)du, (D.2)

the fundamental relation

Sδζiδζj(ω) = e

ω2

2a Sδζiδζj(ω) (D.3)

265

266 APPENDIX D. CORRELATION FUNCTIONS

can be proven as follows:

Sδζiδζj(ω) =

a

πlimT→∞

1

2T

T∫

−T

∞∫

−∞

∞∫

−∞

∞∫

−∞

e−au2

δζi(t− u)e−av2

δζj(t+ w − v)

e−iωwdvdudwdt =

a

πlimT→∞

1

2T

T∫

−T

∞∫

−∞

∞∫

−∞

∞∫

−∞

e−a(u2+v2)+iω(u−v)e−iωs

δζi(t− u)δζj(t− u+ s)dvdudsdt =

Sδζiδζj(ω)

a

π

∞∫

−∞

∞∫

−∞

e−a(u2+v2)+iω(u−v)dvdu = Sδζiδζj

(ω)e−ω2

2a .2

(D.4)

The variable transformation s = w+ u− v was used in the transitionfrom the second to the third line. In the limit a→ ∞, the exact result

lima→∞

Sδζiδζj(ω) = lim

a→∞e−

ω2

2a Sδζiδζj(ω) = Sδζiδζj

(ω), −∞ < ω <∞(D.5)

is recovered.

Appendix E

Impurity scattering

As already mentioned in Chapter 8.2, the expression found in [11] hasbeen used. In two cases this expression is numerically very challengingand also very interesting. Two developments in series have been foundto overcome these difficulties.

The expression given in [11] can be rewritten as

τ−1(ε, n, T ) =

F (ε)

[exp

(γη

1 + η

)Ei(−γη) − Ei(

−γη1 + η

)

− 1

γη

1 − exp

(−γη2

1 + η

)], (E.1)

where

η = 2.234 · 1022ε(1 + α)T

n, (E.2)

γ = 1.005 · 10−16 (1 + 2αε)2

(ε(1 + αε))2N2/3, (E.3)

F = 2.4608 · 10−8 (1 + 2αε)

(ε(1 + αε))3/2

N, (E.4)

267

268 APPENDIX E. IMPURITY SCATTERING

and Ei is the exponential integral function

Ei(x) :=

x∫

−∞

et

tdt. (E.5)

N is the sum of the absolute values of the donor and acceptor con-centrations in cm−3, n the sum of the absolute values of the electronand hole concentrations in cm−3, T the mean temperature of the par-ticles in eV, and ε the energy in eV. α is the nonparabolicity factorin eV−1.

E.1 First numerical problem

The first numerical problem occurs when η is smaller than 1. Definingx := γη, and y := η, one can prove by a rather long and uninterestingcomputation that

exp

(x

1 + y

)Ei(−x) − Ei(

−x1 + y

)

− 1

x

1 − exp

( −xy1 + y

)=

∑

n

Pn(x)

n!(−1)nyn, (E.6)

where Pn(x) can be computed using the following recursive expression:

Pn+1(x) = Pn(x)(x + 2n) − n(n− 1)Pn−1(x),

P0(x) = P1(x) = 0 and P2(x) = 1. (E.7)

E.2 Second numerical problem

The second problem occurs when z := γη1+η is large and positive (typi-

cally z > 100). Defining a := (1+η), one can prove, again by a ratherlong computation that

exp z Ei(−za)− Ei(−z) −1

za1 − exp (−za+ z) =

1

z

∞∑

n=0

(−1)n(

1

z

)nn! − 1

za. (E.8)

E.3. DERIVATIVES 269

E.3 Derivatives

The development of τ−1(ε, n, T ) in δn and δT is often needed in MCsimulators. To take advantage of the properties derived in the twosections above, one can prove:

τ−1(ε, n+ δn, T + δT ) = τ−1(ε, n, T )+(δT

T− δn

n

)γη

(1 + η)2τ−1(ε, n, T )

−F(

η

(1 + η)2− 1

γη

[1 − exp

(−γη2

1 + η

)]). (E.9)

Appendix F

The SimnIC simulator

The SimnIC (Simulator for noIse Computation) simulator was entirelydeveloped during this thesis. There are three different approaches todevice simulation. The first one is the usage of transport models(TMs). So far, the DD model and the EB model were implemented.They can be used in conjunction with the Philips mobility model orwith MC-generated tables for the transport coefficients. The numericsfor the computation of transport parameters (TPs) is exactly as hasbeen explained in Chapter 7. In Chapter 9 some results for very simpledevices were presented. However, to prove that the FEM can actuallybe applied to bigger problems too, some results for a more realisticdevice will be shown in this appendix.

The second approach to device simulation is the use of the many-particle MC method as described in Chapter 8. The many-particleMC algorithm in SimnIC can be run on many processors in a sharedmemory environment. To give an idea of the performance of the code,some benchmarks will be shown in the second part of this appendix.

The third approach is the use of the one-particle MC method.The method has been parallelised for shared memory architecturesand also for distributed memory using the MPI2 library.

271

272 APPENDIX F. THE SIMNIC SIMULATOR

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2V

B in [V]

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

[A/µ

m]

Basis electron currentBasis hole currentCollector electron currentCollector hole currentEmitter electron currentEmitter hole current

Figure F.1: I −V characteristics of the Toshiba bipolar transistor forVCE = 2V .

F.1 A realistic device

Fig. F.2 shows the geometry of a bipolar transistor from Toshibaas presented in [41]. This transistor was simulated using the EBmodel together with MC-generated tables for the mobilities and theenergy relaxation times. The mobilities and relaxation times wereparametrised by the mean carrier energy. The Shockley-Read-HallRecombination model was used to properly describe generation-re-combination processes. Fig. F.1 shows the terminal currents as func-tion of VB for a constant emitter-collector bias of 2 V. The results arevery smooth even for very small currents. The strange shape of theelectron current in the basis at biases higher than 0.96 V results fromthe formation of an electron-hole plasma in the basis. Fig. F.3 showsa profile of the electron energy relaxation time inside the device. Theactual influence of these non-constant energy relaxation times on thecharacteristics of the device has not been studied in detail yet.

Administrator

Highlight

F.1. A REALISTIC DEVICE 273

Figure F.2: Geometry and spatial profile of the doping concentration.


eTauE

7.0E-13

6.0E-13

5.0E-13

4.0E-13

3.0E-13

2.0E-13

Figure F.3: Spatial profile of the electron energy relaxation time.

F.2. MANY-PARTICLE MC: BENCHMARKS 275

F.2 Many-particle MC: Benchmarks

The typical simulation flow of a parallel many-particle MC simulationis shown in Fig. F.4: A master process starts n child processes, wheren is usually the number of available CPU(s) minus one. After syn-chronisation, all processes solve the linear Poisson equation and thesimulate 1/(n + 1) of the total number of particles for a given timestep. Then, the processes are synchronised again and exchange a min-imal amount of useful data, usually only the information about thecharge densities. This cycle is repeated until the specified simulationtime is reached.

To evaluate the performance of the proposed algorithm, theN+NN+

structure presented in Section 9.5 has been simulated at a bias of 2 Vfor 50 ps with a time step of 5 fs. The simulation was launched ona four 2.4 GHz Opteron 850 machine with 1 megabyte of cache perprocessor and 32 gigabytes of main memory.

Fig. F.5 shows the simulation time and the speedup as obtainedwith a mean number of 2000 particles. Fig. F.6 and F.7 show the samefor a mean number of 20′000 resp. 200′000 particles. Obviously thereis no slowdown even for a small number of particles (2000). For alarger number of particles (20′000 and 200′000) comfortable speedupsare already achieved.

These observations have been confirmed by simulations of biggerdevices (e.g. the Toshiba transistor introduced in the first part of thisappendix) on up to 32 processors using millions of particles. SimnICis, therefore, a tool of choice for the simulation of realistic devices ina reasonable amount of time using MC methods.


Synchronisation

Synchronisation

Master Process

Child process 1 Child process n. . .

Data exchange

SimulationPoisson equation

Figure F.4: Typical simulation flow of a parallel many-particle MCsimulation.

111

81.4

71.168

1 2 3 4number of CPU(s)

0

20

40

60

80

100

120

sim

ulat

ion

time

[s]

1

1.371.57 1.64


0

0.5

1

1.5

2

2.5

3

3.5

4

spee

dup

Figure F.5: Simulation of the N+NN+ structure using 2000 particles:Simulation time and speedup.

F.2. MANY-PARTICLE MC: BENCHMARKS 277

726

401

290

238


0

100

200

300

400

500

600

700

800

sim

ulat

ion

time

[s]

1

1.81

2.51

3.06


0

0.5

1

1.5

2

2.5

3

3.5

4

spee

dup

Figure F.6: Simulation of the N+NN+ structure using 20′000 parti-cles: Simulation time and speedup.

6810

3427

2328

1777


0

1000

2000

3000

4000

5000

6000

7000

sim

ulat

ion

time

[s]

1

1.99

2.92

3.83


0

0.5

1

1.5

2

2.5

3

3.5

4

spee

dup

Figure F.7: Simulation of the N+NN+ structure using 200′000 par-ticles: Simulation time and speedup.

Bibliography

[1] P. Anselone, Collectively compact operator approximation theory,Automatic computation, Prenctice-Hall, 1971.

[2] N. W. Ashcroft and N. D. Mermin, Solid state physics, SaundersCollege Publishing, Harcourt Brace College Publishers, ForthWorth, Philadelphia, San Diego, New York, Orlando, Austin,San Antonio, Toronto, Montreal, London, Sydney, Tokyo, 1976.

[3] M. Bixon and R. Zwanzig, Boltzmann-Langevin equation and hy-drodynamic fluctuations, Physical Review 187 (1969), no. 1, 267–272.

[4] K. Bløtekjær, Transport equations for electrons in two-valleysemiconductors, IEEE Transactions on Electron Devices ED–17(1970), 38–47.

[5] F. Bonani, G. Ghione, M. Pinto, and R. K. Smith, An efficientapproach to noise analysis through multidimensional physics-based models, IEEE Transactions on Electron Devices 45 (1998),no. 1, 261–269.

[6] W. Brauer and H.-W. Streitwolf, Theoretische Grundlagen derHalbleiterphysik, Reihe Wissenschaft, Vieweg, Braunschweig,1977.

[7] I. N. Bronstein and K. A. Semendjajew, Taschenbuch der Math-ematik, Harri Deutsch, Thun und Frankfurt am Main, 1985.

279

280 BIBLIOGRAPHY

[8] S. C. Brugger and A. Schenk, First-principle computation of re-laxation times in semiconductors for low and high electric fields,Simulation of Semiconductor Processes and Devices (SISPAD)(Tokyo, Japan), September 2005, pp. 151–154.

[9] R. Brunetti, C. Jacoboni, F. Nava, and L. Reggiani, Diffusioncoefficient of electrons in silicon, Journal of Applied Physics 52(1981), no. 11, 6713–6722.

[10] F. M. Bufler, Y. Asahi, H. Yoshimura, C. Zechner, A. Schenk,and W. Fichtner, Monte Carlo simulation and measurement ofnanoscale n-MOSFETs, IEEE Transaction on Electron Devices50 (2003), no. 2, 418–424.

[11] F. M. Bufler, A. Schenk, and W. Fichtner, Efficient MonteCarlo device modeling, IEEE Transaction on Electron Devices47 (2000), no. 10, 1891–1897.

[12] F. M. Bufler, A. Schenk, and W. Fichtner, Simplified model forinelastic acoustic phonon scattering of holes in Si and Ge, Journalof Applied Physics 90 (2001), no. 5, 2626–2628.

[13] F. M. Bufler, A. Schenk, and W. Fichtner, Single-particle ap-proach to self-consistent Monte Carlo device simulation, IEICETransactions on Electron Devices (2003), 308–313.

[14] C. Canali, G. Majni, R. Minder, and G. Ottaviani, Electron andhole drift velocity measurement in silicon and their empirical re-lation to electric field and temperature, IEEE Transactions onElectron Devices (1975), 1045–1047.

[15] E. Cartier, M. Fischetti, E. Eklund, and F. McFeely, Impactionization in silicon, Applied Physics Letters 62 (1993), no. 25,3339–3341.

[16] M. Cohen and J. Chelikowsky, Nonlocal pseudopotential calcu-lations for the electronic structure of eleven diamond and zinc-blende semiconductors, Physical Review B 14 (1976), 556–582.

[17] M. Cohen and J. Chelikowsky, Electronic structure and opticalproperties of semiconductors, second ed., Springer, New York,1989.

BIBLIOGRAPHY 281

[18] R. Dautray and J.-L. Lions, Spectral theory and applications,Mathematical Analysis and Numerical Methods for Science andTechnology, vol. 3, Springer-Verlag, Berlin Heidelberg, 1990.

[19] S. Decker, Numerische Simulation von Si/SiGe-Hochfrequenztransistoren unter besonderer Berucksichtigungdes elektronischen Rauschens, Ph.D. thesis, Universitat Bremen,published at Shaker Verlag, Aachen, 2002.

[20] S. V. Gantsevich, V. L. Gurevich, and R. Katilius, Fluctuationsin semiconductors in a strong electric field and scattering of lightby ”hot” electrons, Soviet Physics JETP 30 (1970), no. 2, 276–284.

[21] S. V. Gantsevich, V. L. Gurevich, and R. Katilius, Theory offluctuations in nonequilibrium electron gas, Rivista del nuovo ci-mento 2 (1979), no. 5, 1–87.

[22] T. Gonzalez and D. Pardo, Physical models of ohmic contact forMonte Carlo device simulation, Solid-State Electronics 39 (1996),no. 4, 555–562.

[23] P. Graf, Entwicklung eines Monte–Carlo-Bauelementsimulatorsfur Si/SiGe–heterobipolartransistoren, Ph.D. thesis, UniversitatBremen, published at Herbert Utz Verlag, Munchen, 1999.

[24] T. Grasser, H. Kosina, M. Gritsch, and S. Selberherr, Using sixmoments of Boltzmann’s transport equation for device simulation,Journal of Applied Physics 90 (2001), no. 5, 2389–2396.

[25] T. Grasser, T. Tang, H. Kosina, and S. Selberherr, A reviewof hydrodynamic and energy-transport models for semiconductordevice simulation, Proc. of the IEEE 91 (1999), no. 2, 251–274.

[26] G. Heiser, Design and implementation of a three-dimensional,general purpose semiconductor device simulator, Ph.D. thesis,ETH Zurich, published at Hartung Gorre Verlag, Konstanz, 1991.

[27] K. Hess, Advanced theory of semiconductor devices, Solid StatePhysical Electronics, Prentice Hall, Englewood Cliffs, New Jersey07632, 1988.

282 BIBLIOGRAPHY

[28] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge Uni-versity Press, 1985.

[29] C. Jacoboni and L. Reggiani, The Monte Carlo method for the so-lution of charge transport in semiconductors with applications tocovalent materials, Reviews of Modern Physics 55 (1983), no. 3,645–705.

[30] C. Jungemann, M. Bartels, S. Keith, and B. Meinerzhagen, Effi-cient methods for Hall factor and transport coefficient evaluationfor electrons and holes in Si and SiGe based on a full-band struc-ture, Extended Abstracts IWCE–6, Osaka, Japan, 1998, IEEECat. No. 98EX116, pp. 104–107.

[31] C. Jungemann, A. Emunds, and W. L. Engl, Simulation of linearand nonlinear electron transport in homogeneous silicon inversionlayers, Solid-State Electronics 36 (1993), 1529–1540.

[32] C. Jungemann and B. Meinerzhagen, Hierarchical device simula-tion. the Monte-Carlo perspective, Computational Microelectron-ics, Springer, Wien, New York, 2003.

[33] C. Jungemann, B. Neinhus, and B. Meinerzhagen, Hierarchical2-D DD and HD noise simulations of Si and SiGe devices–part I:Theory, IEEE Transactions on Electron Devices 49 (2002), no. 7,1250–1257.

[34] C. Jungemann, B. Neinhus, and B. Meinerzhagen, Hierarchical2-D DD and HD noise simulations of Si and SiGe devices–partII: Results, IEEE Transactions on Electron Devices 49 (2002),no. 7, 1258–1264.

[35] D. B. M. Klaassen, A unified mobility model for device simu-lation: I. model equations and concentration dependence, Solid-State Electronics 35 (1992), no. 7, 953–959.

[36] D. B. M. Klaassen, A unified mobility model for device simulation:II. temperature dependence of carrier mobility and lifetime, Solid-State Electronics 35 (1992), no. 7, 961–967.

BIBLIOGRAPHY 283

[37] S. Laux, Techniques for small-signal analysis of semiconductordevices, IEEE Transaction on Computer-Aided Design (1985),no. 4, 472–481.

[38] O. Madelung, Introduction to solid-state theory, Solid-State Sci-ences, Springer, Berlin Heidelberg New York, 1996.

[39] A. Majorana, Space homogeneous solutions of the Boltzmannequation describing electron-phonon interactions in semiconduc-tors, Transport theory and Statistical Physics 20 (1991), no. 4,261–279.

[40] B. Meinerzhagen and W. L. Engl, The influence of the ther-mal equilibrium approximation on the accuracy of classical two-dimensional numerical modeling of silicon submicrometer MOStransistors, IEEE Transactions on Electron Devices 35 (1988),no. 5, 689–697.

[41] H. Nii, C. Yoshino, S. Yoshitomi, K. Inoh, H. Furuya, H. Nakjima,H. Sugaya, H. Naruse, Y. Katsumata, and H. Iwai, An 0.3− µmSi epitaxial base BiCMOS technology with 37-Ghz fmax and 10-VBVceo for RF telecommunication, IEEE Transaction on ElectronDevices 46 (1999), no. 4, 712–721.

[42] H. Nyquist, Thermal agitation of electric charge in conductors,Physical Review 32 (1928), 110–113.

[43] P. Price, Monte Carlo calculation of electron transport in solids,Semiconductors and Semimetals 14 (1979), 249.

[44] P. W. Rambo and J. Denavit, Time stability of Monte Carlodevice simulation, IEEE Transactions on Computer-Aided Designof Integrated Circuits and Systems 12 (1993), no. 11, 1734–1741.

[45] M. Reiser, A two-dimensional numerical FET model for dc, ac,and large-signal analysis, IEEE Transaction on Electron Devices(1973), 35–45.

[46] M. Roland and J. P. Nougier, Proceedings of the 13th Interna-tional Conference on the Physics of Semiconductors (1976), 1227.

284 BIBLIOGRAPHY

[47] E. Sangiorgi and M. R. Pinto, A semi-empirical model of surfacescattering for Monte Carlo simulation of silicon n-MOSFET’s,IEEE Transactions on Electron Devices 39 (1992), 356–361.

[48] A. Schenk, Advanced physical models for silicon device simu-lation, Computational Microelectronics, Springer, Wien, NewYork, 1998.

[49] A. Schenk, B. Schmithusen, A. Wettstein, A. Erlebach, S. Brug-ger, F. Bufler, T. Feudel, and W. Fichtner, Simulation of RFnoise in MOSFETs using different transport models, IEICETransactions on Electronics E86-C (2003), no. 3, 481–489.

[50] S. Selberherr, Analysis and simulation of semiconductor devices,Springer-Verlag, Wien, New York, 1984.

[51] P. Shiktorov, E. Starikov, V. Gruzinskis, T. Gonzalez, J. Mateos,P. D., L. Reggiani, L. Varani, and J. C. Vaissiere, Langevin forcesand generalized transfer fields for noise modeling in deep submi-cron devices, IEEE Transactions on Electron Devices 47 (2000),no. 10, 1992–1998.

[52] W. Shockley, J. A. Copeland, and R. R. James, Quantum theoryof atoms, molecules and solid state, Academic Press, New York,1966.

[53] R. Stratton, Diffusion of hot and cold electrons in semiconductorbarriers, Physical Review 126 (1962), 2002–2014.

[54] S. Takagi, M. I. Toriumi, and H. Tango, On the universalityof inversion layer mobility in Si MOSFET’s: Part I–effects ofsubstrate impurity concentration, IEEE Transactions on ElectronDevices 41 (1994), no. 12, 2357–2362.

[55] S. Takagi, M. I. Toriumi, and H. Tango, On the universality of in-version layer mobility in Si MOSFET’s: Part II–effects of surfaceorientation, IEEE Transactions on Electron Devices 41 (1994),no. 12, 2363–2368.

[56] T.-W. Tang and H. Gan, Two formulations of semiconductortransport equations based on spherical harmonic expansion of the

BIBLIOGRAPHY 285

Boltzmann equation, IEEE Transactions on Electron Devices 47(2000), no. 9.

[57] R. Thoma, Entwicklung eines hydrodynamischen Gleichungssys-tems zur Beschreibung von Halbleiterbauelementen, Ph.D. thesis,Rheinisch-Westfallischen Technischen Hochschule Aachen, 1991.

[58] K. M. van Vliet, Markov approach to density fluctuations due totransport and scattering: I. mathematical formalism, Journal ofMathematical Physics 12 (1971), no. 9, 1981–1998.

[59] D. Werner, Funktionalanalysis, second ed., Springer, 2000.

List of Figures

2.1 Difference between f , f and f . . . . . . . . . . . . . . 28

2.2 Possible discretisation of the first Brillouin zone. . . . 35

3.1 Piece of bulk material. . . . . . . . . . . . . . . . . . . 51

3.2 Illustration of the functions ~k−m(t) and ~k+m(t). . . . . . 60

3.3 Geometrical definition of Ra,δ and Rb,δ. . . . . . . . . 68

5.1 Typical fluctuation of a terminal current around a sta-tionary state. . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Auto-correlation function of the terminal current. . . . 86

7.1 The unity triangle T (left) and a given triangle T (right).113

7.2 Flowchart of the iterative solver for the DD model. . . 117

7.3 Flowchart of the iterative solver for the EB model. . . 124

8.1 Typical path of a particle due to the discretisation. . . 137

8.2 Path integral statistics. . . . . . . . . . . . . . . . . . 138

8.3 Low-field mobility as function of the doping concentra-tion for electrons. . . . . . . . . . . . . . . . . . . . . . 142

8.4 High-field mobility as function of the electric field forelectrons. . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.5 Time-of-flight measurements. . . . . . . . . . . . . . . 143

8.6 Time-of-flight measurements and simulation results forelectrons. . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.7 Vertical cut in a MOSFET channel. . . . . . . . . . . 145

287

288 LIST OF FIGURES

8.8 µeff as function of Eeff for a 〈110〉-orientation of thecrystal. . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.9 rH for electrons as function of the lattice temperaturefor different doping concentrations. . . . . . . . . . . . 146

8.10 rH for holes as function of the lattice temperature fordifferent doping concentrations. . . . . . . . . . . . . . 147

8.11 Van der Pauw structure for the extraction of rH . . . . 1478.12 Hall factor computed by an ensemble MC simulation

compared to results from the theory of Section 3.4. . . 1488.13 Typical progression of I(t). . . . . . . . . . . . . . . . 153

9.1 The equilibrium distribution feq for the two models. . 1629.2 Notation for the valleys. . . . . . . . . . . . . . . . . . 1649.3 H⊥

1 as function of energy. . . . . . . . . . . . . . . . . 1649.4 S−1

ε as function of energy. . . . . . . . . . . . . . . . . 1659.5 S−1

vxas function of energy, band, and valley index. . . 165

9.6 S−1v2 as function of energy. . . . . . . . . . . . . . . . . 166

9.7 S−1vxv2

as function of energy, band and valley index. . . 166

9.8 Comparison of |y(0)〉/Wtot with |feqZ〉/|| ˜WtotfeqZ||. 1679.9 eε in the first band. . . . . . . . . . . . . . . . . . . . . 1689.10 evx

in the first band. . . . . . . . . . . . . . . . . . . . 1689.11 ev2 in the first band. . . . . . . . . . . . . . . . . . . . 1699.12 evxv2 in the first band. . . . . . . . . . . . . . . . . . . 1699.13 Relative error 〈evx

〉/〈vx〉 as function of the electric field. 1709.14 Relative error 〈eε〉/〈ε〉 as function of the electric field. 1709.15 Relative error 〈ev2〉/〈v2〉 as function of the electric field. 1719.16 Cvv/n/2 as function of the electric field. . . . . . . . . 1729.17 Cεε/n as function of the electric field. . . . . . . . . . 1729.18 Cv2v2/n as function of the electric field. . . . . . . . . 1739.19 Errvv as function of the electric field. . . . . . . . . . 1739.20 Errεε as function of the electric field. . . . . . . . . . 1749.21 Errv2v2 as function of the electric field. . . . . . . . . 1749.22 Wtot in the first band as function of energy. . . . . . . 1759.23 α(ω) as function of frequency. . . . . . . . . . . . . . . 1769.24 Re(x(ω)) in the first band as function of frequency. . . 1769.25 Im(x(ω)) in the first band as function of frequency. . . 1779.26 Re(y(ω)) in the first band as function of frequency. . . 177

LIST OF FIGURES 289

9.27 Im(y(ω)) in the first band as function of frequency. . . 178

9.28 Re(T−1vx

(ω)a) in the first band and first valley as func-tion of frequency. . . . . . . . . . . . . . . . . . . . . . 178

9.29 Im(T−1vx

(ω)a) in the first band and first valley as func-tion of frequency. . . . . . . . . . . . . . . . . . . . . . 179

9.30 Re(T−1v2 (ω)a) in the first band as function of frequency. 179

9.31 Im(T−1v2 (ω)a) in the first band as function of frequency. 180

9.32 S−1vx

in the second band and first valley as function ofenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.33 S−1ε in the second band as function of energy. . . . . . 181

9.34 S−1v2 in the second band as function of energy. . . . . . 182

9.35 S−1vxv2

in the second band and first valley as function ofenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.36 Fcrit as function of doping concentration. . . . . . . . 183

9.37 Auto-correlation function of the velocity fluctuationsin undoped bulk silicon as function of the electric field.The field points in 〈100〉-direction. . . . . . . . . . . . 184

9.38 Auto-correlation function of the v2 fluctuations in un-doped bulk silicon as function of the electric field. Thefield points in 〈100〉-direction. . . . . . . . . . . . . . . 185

9.39 Auto-correlation function of the energy current fluctua-tions in un-doped bulk silicon as function of the electricfield. The field points in the 〈100〉-direction. . . . . . . 185

9.40 Cross correlation function of the energy current fluctu-ations with the velocity fluctuations in undoped bulksilicon as function of the electric field. The field pointsin the 〈100〉-direction. . . . . . . . . . . . . . . . . . . 186

9.41 Auto-correlation function of the velocity fluctuationsin the field direction as function of the electric field fordifferent doping concentrations. . . . . . . . . . . . . 187

9.42 Auto-correlation function of the v2 fluctuations as func-tion of the electric field for different doping concentra-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.43 Auto-correlation function of the energy current fluctu-ations in field direction as function of the electric fieldfor different doping concentrations. . . . . . . . . . . . 188

290 LIST OF FIGURES

9.44 Auto-correlation function of the velocity fluctuationsin field direction as function of the field strength fordifferent field orientations. . . . . . . . . . . . . . . . 189

9.45 Auto-correlation function of the v2 fluctuations as func-tion of the field strength for different field orientations. 189

9.46 Auto-correlation function of the energy current fluctu-ations in field direction as function of the field strengthfor different field orientations. . . . . . . . . . . . . . . 190

9.47 Comparison of 12Sδviδvj

(ω) with the rhs of Eq. (9.6) asfunction of the electric field in undoped silicon for theDD and EB model, respectively. . . . . . . . . . . . . 191

9.48 m∗2

9n Sδsv2vxδsv2vx

(0) compared with the rhs of the Bixon-Zwanzig relation as function of the electric field inten-sity in undoped silicon. . . . . . . . . . . . . . . . . . . 192

9.49 m∗2

9n Sδsv2vxδs

v2vx(0) compared with the the rhs of the

Bixon-Zwanzig relation for the Bløtekjær model as func-tion of the doping concentration at thermodynamic equi-librium. . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.50 Sδζvx δζvxfor the DD and the EB models compared

with the corresponding Langevin term as function ofthe electric field. . . . . . . . . . . . . . . . . . . . . . 194

9.51 Sδζv2 δζv2 for the EB model compared with Sδv2δv2 (MC)

and with the corresponding Langevin term as functionof the electric field. . . . . . . . . . . . . . . . . . . . . 194

9.52 Low-field energy relaxation times for electrons in bulksilicon as function of lattice temperature and dopingconcentration. . . . . . . . . . . . . . . . . . . . . . . . 195

9.53 Low-field energy relaxation times for holes in bulk sili-con as function of lattice temperature and doping con-centration. . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.54 Energy relaxation time for electrons in bulk silicon at300 K as function of mean particle energy and dopingconcentration. . . . . . . . . . . . . . . . . . . . . . . . 196

9.55 Energy relaxation time for holes in bulk silicon at 300 Kas function of mean particle energy and doping concen-tration. . . . . . . . . . . . . . . . . . . . . . . . . . . 197

LIST OF FIGURES 291

9.56dS−1

ε

dε as function of energy in the first band and in thevalley along the 〈100〉-direction for electrons. . . . . . 197

9.57 Geometry and doping concentration of the N+NN+

structure. . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.58 Discretisation grid and effective intrinsic density for theN+NN+ structure. . . . . . . . . . . . . . . . . . . . . 199

9.59 I-V curves of the N+NN+ structure for different models.202

9.60 I-V curves of the N+NN+ structure for different driv-ing forces. . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.61 I-V curves of the P+PP+ structure for different models.203

9.62 I-V curves of the P+PP+ structure for different drivingforces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.63 Electron density profile of the N+NN+ structure fordifferent models (VCA = 2 V). . . . . . . . . . . . . . . 204

9.64 Electron temperature profile of the N+NN+ structurefor different models (VCA = 2 V). . . . . . . . . . . . . 205

9.65 Electron mobility profile of the N+NN+ structure fordifferent models (VCA = 2 V). . . . . . . . . . . . . . . 205

9.66 Electron energy relaxation time profile of the N+NN+

structure for different models (VCA = 2 V). . . . . . . 206

9.67 Comparison between the Einstein relation and exactdiffusion tensor for electrons in the N+NN+ structure(VCA = 2 V). . . . . . . . . . . . . . . . . . . . . . . . 207

9.68 Y-parameters at zero frequency of the N+NN+ struc-ture for different models as function of VCA. . . . . . . 207

9.69 Y-parameters at zero frequency of the P+PP+ struc-ture for different models as function of VCA. . . . . . . 208

9.70 Spectral intensity of the current at zero frequency ofthe N+NN+ structure for different models as functionof VCA. . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.71 The influence of an ERT cutoff on the spectral inten-sity of the current at zero frequency of the N+NN+

structure as function of VCA. . . . . . . . . . . . . . . 210

9.72 Spectral intensity of the current at zero frequency ofthe P+PP+ structure for different models as functionof VCA. . . . . . . . . . . . . . . . . . . . . . . . . . . 210

292 LIST OF FIGURES

9.73 The influence of an ERT cutoff on the spectral inten-sity of the current at zero frequency of the P+PP+

structure as function of VCA. . . . . . . . . . . . . . . 211

9.74 Integrals from −t to t of the correlation function of thefluctuations of the current of the N+NN+ structure asfunction of t. . . . . . . . . . . . . . . . . . . . . . . . 212

9.75 Integral from −t to t of the correlation function of thefluctuations of the current of the P+PP+ structure asfunction of t. . . . . . . . . . . . . . . . . . . . . . . . 213

9.76 Comparison of the gradient of the quasi-Fermi poten-tial ansatz with the previous method for the N+NN+

structure. . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.77 Comparison of the gradient of the quasi-Fermi poten-tial ansatz with the previous method for the P+PP+

structure. . . . . . . . . . . . . . . . . . . . . . . . . . 217

9.78 Local density of noise in the N+NN+ structure fordifferent biases. . . . . . . . . . . . . . . . . . . . . . . 218

9.79 Local density of noise in the P+PP+ structure for dif-ferent biases. . . . . . . . . . . . . . . . . . . . . . . . 219

9.80 Local density of noise in the N+NN+ structure fordifferent models at VCA = 3V . . . . . . . . . . . . . . . 220

9.81 Local density of noise in the P+PP+ structure for dif-ferent models at VCA = 3V . . . . . . . . . . . . . . . . 220

9.82 Spatial profile of the component of the mobility alongthe transport direction for the NIN structure at a biasof 2 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

9.83 Spatial profile of the four terms of the rhs of (9.13) forthe NIN structure at a bias of 2 V. . . . . . . . . . . 223

A.1 Effect of the operator B on a function f(b)i . . . . . . . 241

A.2 Effect of the operatorA repeatedly applied on the ”func-

tion” g0(ǫ, b) := δ(ǫ− ǫ(b)min). . . . . . . . . . . . . . . 243

F.1 I − V characteristics of the Toshiba bipolar transistorfor VCE = 2V . . . . . . . . . . . . . . . . . . . . . . . 272

F.2 Geometry and spatial profile of the doping concentration.273

F.3 Spatial profile of the electron energy relaxation time. . 274

LIST OF FIGURES 293

F.4 Typical simulation flow of a parallel many-particle MCsimulation. . . . . . . . . . . . . . . . . . . . . . . . . 276

F.5 Simulation of the N+NN+ structure using 2000 parti-cles: Simulation time and speedup. . . . . . . . . . . . 276

F.6 Simulation of the N+NN+ structure using 20′000 par-ticles: Simulation time and speedup. . . . . . . . . . . 277

F.7 Simulation of theN+NN+ structure using 200′000 par-ticles: Simulation time and speedup. . . . . . . . . . . 277

List of Tables

9.1 Nyquist Theorem: N+NN+ Device . . . . . . . . 2149.2 Nyquist Theorem: P+PP+ Device . . . . . . . . . 215

294

List of symbols

f distribution function, solution of theBoltzmann equation

feq right eigenvector with eigenvalue 0 of thescattering operator

f1 left eigenvector with eigenvalue 0 of thescattering operator

n particle densityp hole density~v group velocityS scattering operatorS−1 inverse scattering operatorw(k|k′) transition rate from k to k′

Wtot(k) scattering rate~E electric fieldEeff effective electric field~B magnetic fieldq absolut value of the electron chargekB Boltzmann constant~ Planck’s constant divided by 2πµ mobilityD diffusion coefficientS−1g g-moment of the inverse scattering oper-

atorτp impulse relaxation timeτε energy relaxation timeτg relaxation time for the g-moment of the

Boltzmann equation

295

296 LIST OF SYMBOLS

µij i,j component of the mobility tensorµeff effective mobilityDij i,j component of the diffusion tensor~J current densityT temperatureTeq equilibrium temperatureTL lattice temperature〈g〉 expectation value of the function g〈g〉eq value of the function g integrated with

feqκ thermo-diffusion coefficientm∗ effective massZ density of statesfMC Monte Carlo distribution functionalϕ electrostatic potentialψ quasi-Fermi potentialni intrensic densityΘ(t) step functionδ(t) Dirac δ-distributiong time derivative of the function gǫ dielectric function1 unity matrixrH Hall factorRH Hall coefficientg time average of the function gi imaginary partε energysupp(g) support of the function gdet(M) determinant of the matrix Mr(A) spectral radius of the operator APD projector on the discretised spacePL projector on the space where the SO has

a unique inverse

List of acronyms

AFS Acceleration Fluctuations Scheme

BE Boltzmann Equation

BLE Boltzmann Langevin Equation

CF Correlation Function

DD Drift Diffusion

DF Driving Force

EB Energy Balance

ER Einstein Relation

ERT Energy Relaxation Time

GIFM Generalised Impedance Field Method

HD Hydro Dynamic

IFM Impedance Field Method

ISO Inverse Scattering Operator

LBE Linear Boltzmann Equation

LDN Local Density of Noise

MC Monte Carlo

297

298 LIST OF ACRONYMS

MISO Moment of the Inverse Scattering Operator

mod Modulo

NS Noise Sources

RF Radio Frequency

RSTF Ramo-Shockley Test Function

RT Relaxation Time

RTA Relaxation Time Approximation

SF Self-Force

SHBE Space Homogeneous Boltzmann Equation

SIC Spectral Intensity of the Current

SimnIC Simulator for Noise Computation

SO Scattering Operator

TC Transport Coefficient

TD Thermodynamic Equilibrium

TM Transport Model

Curriculum Vitæ

Simon Brugger was born in Geneva, Switzerland, on July 17, 1975.From 1993 to 1995 he took part three consecutive times in the interna-tional chemistry olympiad (IChO) as member of the Swiss team. Hestudied physics at the Swiss Federal Institute of Technology (ETH)in Zurich. During an exchange year at the Humboldt University inBerlin he worked in the laboratory of Prof. T.W. Masselink in thedomain of Quantum dots. He finished his studies at ETH Zurich withthe master thesis entitled Two Homogeneous Cosmological Models ofQuintessence in general relativity which was supervised by Prof. N.Straumann. In February 2001 he joined the Integrated Systems Labo-ratory at ETH, where he received his PhD degree in 2006. His researchinterests focus on semiconductor transport theory, fluctuation theory,numerical device simulation, and numerical mathematics.

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