heinz

Simulation Approachesfor Nano–ScaleSemiconductor Devices

Diss. ETH No. 15435

Simulation Approaches forNano–Scale Semiconductor

Devices

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGYZURICH

for the degree of

Doctor of Technical Sciences

presented by

FREDERIK OLE HEINZ

Dipl. Phys. ETHborn 11 October 1974citizen of Germany

accepted on the recommendation of

Prof. Dr. Wolfgang Fichtner, examinerProf. Dr. Giuseppe Iannaccone, co-examiner

2004

Acknowledgments

First of all I wish to thank Prof. Wolfgang Fichtner for the oppor-tunity to work and learn at the Institut fur Integrierte Systeme andProf. Giuseppe Iannaccone for reading and co–examining this thesis.Special thank also is due to PD Dr. Andreas Schenk for supervisingthe scientific aspects of the work.

Thanks to the members of the physical modelling group: Frank Geel-haar, Fabian Bufler, Michael Pfeiffer, Timm Hohr, Simon Brugger,Stefan Odermatt. I am indebted to Andreas Scholze for his help withthe original simnad code and many fruitful discussions, to BernhardSchmithusen for sharing his insights into numerics and for the imple-mentation work on the dessis–ise side of the simulator coupling, andto Jens Krause for his help in questions of meshing. I wish to ex-press my gratitude for the benefits I gained from conversations withMichael Stopa (NTT Atsugi Research & Development Center, Japan),Markus J. Grote (Universitat Basel), Massimo Macucci and GiuseppeIannaccone (University of Pisa). Furthermore I wish to thank LukasWorschech and Andreas Schliemann (Universitat Wurzburg) for theirhelp with experimental data.

Finally I thank the technical staff of IIS, especially our system ad-ministrators Christoph Wicki and Anja Bohm, for providing excellentworking conditions. Thanks to all the IIS people for having madework at the institute enjoyable!

This work has been carried out in the context of the EU researchproject IST–1999–10828 (NANOTCAD) with financial support by theSwiss Federal Office for Education and Science (BBW).

v

Abstract

Topic of this dissertation are the development and implementation ofa three–dimensional simulation environment for semiconductor nano-electronics devices, that are dominated by quantum effects, and themodelling of the properties of various candidate structures for futureultra–large scale integrated circuits. In this context, Coulomb block-ade in the presence of strong quantum confinement, quantum–ballistictransport and the effect of atomistic doping in aggressively scaledsemiconductor devices have been studied. The simulation frameworkpresented in this work extends the simnad quantum mechanics sim-ulator developed at the Integrated Systems Laboratory in a previousproject and couples it to the standard device simulator dessis–ise.Basis of the simulation model is an effective mass formulation of

density functional theory in local density approximation. In its gen-eralisation to finite temperatures it may be used for the computationof the quantum mechanically correct charge distribution inside thedevice. Additionally, in conjunction with Bardeen’s transfer Hamilto-nian method, it may be used to compute tunnelling currents betweenclassically insulated regions (channels, quantum dot) of the device.Doing so requires knowledge of the statistical mechanics of the quan-tum dot. To make the necessary phase space averages tractable, aMonte–Carlo approach is used.On the classically conducting regions of the device the drift–diffu-

sion model may be used for current computation. Coupling the devicesimulator dessis–ise with the simnad quantum mechanics simulatorresults in a simulation tool capable of modelling devices that featureboth classical dissipative currents and 3D quantum effects.

vii

Zusammenfassung

Thema der Dissertation sind Entwicklung und Implementierung einerdreidimensionalen Simulationsumgebung fur nanoelektronische Halb-leiterbauelemente, deren Eigenschaften von Quanteneffekten domi-niert werden, sowie die Simulation verschiedener Halbleiternanostruk-turen, die als mogliche Komponenten zukunftiger hochstintegrierterSchaltkreise vorgeschlagen worden sind. In diesem Zusammenhangwurden Untersuchungen uber Coulomb–Blockadeeffekte in Bauelemen-ten mit starken Quantisierungseffekten sowie uber quantenballisti-schen Transport und den Einfluß diskreter Dotierung auf die Leitfa-higkeit extrem miniaturisierte Halbleiterbauelemente angestellt. Dievorgestellte Simulationsumgebung basiert auf dem Quantenmechanik-simulator simnad, der im Rahmen eines Vorgangerprojektes am In-stitut fur Integrierte Systeme entwickelt wurde. simnad wurde starkerweitert und zur Erhohung der Flexibilitat an den klassischen Bau-elementsimulator dessis–ise angekoppelt.Als Grundlage des entwickelten Simulationsmodells dient eine Ef-

fektivmassenformulierung der Dichtefunktionaltheorie in der lokalenDichtenaherung. In ihrer Verallgemeinerung auf endliche Tempera-turen kann diese einerseits dazu verwendet werden, die Ladungsver-teilung innerhalb der Halbleiterstruktur quantenmechanisch korrektzu bestimmen; andererseits konnen damit in Verbindung mit Barde-ens Transfer–Hamiltonian–Methode verschiedenen im klassischen Sin-ne gegeneinander isolierten Bereichen (Quantenpunkte) des Bauele-ments berechnet werden. Dies erfordert allerdings Kenntnis der sta-tistischen Mechanik des Quantenpunktes. Um die dabei auftretendenMittelungen uber den Phasenraum bewaltigen zu konnen, wurde eineMonte–Carlo Methode verwendet.

ix

x ZUSAMMENFASSUNG

Auf den klassisch leitenden Gebieten lassen sich Strome mittels desDrift–Diffusionsmodells berechnen. Durch Koppelung des quantenme-chanischen Simulators simnad an den Bauelementsimulator dessis–ise

erhalt man ein Simulationswerkzeug, mit dem man sowohl klassischeLeitungsstrome als auch dreidimensionale Quanteneffekte im selbenBauelement behandeln kann.

Contents

Acknowledgments v

Abstract vii

Zusammenfassung ix

1 Introduction 1

2 Single–Electron Tunnelling Devices 5

3 Challenges for the Simulation of Nano–Scale Single–Electron Devices 93.1 Computation of the Charge Density . . . . . . . . . . 11

3.1.1 The Charge Density by Density FunctionalTheory . . . . . . . . . . . . . . . . . . . . . . 13

3.1.2 The Effective Mass Approximation . . . . . . . 153.1.3 Domain Decomposition and Adiabatic Decoup-

ling of the Schrodinger Equation . . . . . . . . 173.1.4 Dimensional Reduction with Geometric Confi-

nement — an Approximate Treatment . . . . . 193.2 Computation of Tunnelling Rates . . . . . . . . . . . . 23

3.2.1 Bardeen’s Transfer Hamiltonian Method . . . . 233.2.2 Transfer Hamiltonians for Arbitrary Potentials 263.2.3 Generalisation to Higher Dimensions . . . . . . 273.2.4 Transfer Hamiltonians and Charge Densities . . 31

3.3 Conductance Extraction . . . . . . . . . . . . . . . . . 343.4 Thermal Averages . . . . . . . . . . . . . . . . . . . . 37

xi

xii CONTENTS

3.5 Statistical Mechanics of the Quantum Dot . . . . . . . 393.6 Monte–Carlo Evaluation of Thermal Expectation Values 413.7 Simulation Results . . . . . . . . . . . . . . . . . . . . 46

3.7.1 Simulation of a Silicon–on–Insulator Single–Electron Transistor . . . . . . . . . . . . . . . . 46

3.7.2 Tunnelling Rates and the Anisotropy of m∗ . . 483.7.3 A Split–Gate III–V Hetero–Structure Single–

Electron Transistor . . . . . . . . . . . . . . . . 50

4 Quantum–Ballistic Transport 574.1 Theory of Quantum Ballistic Transport . . . . . . . . 58

4.1.1 Quasi 1D Transport in Quantum Wires . . . . 654.1.2 Transport in a Quantum Well . . . . . . . . . . 69

4.2 Thermionic Current Over a Barrier . . . . . . . . . . . 704.2.1 Model Description . . . . . . . . . . . . . . . . 704.2.2 Comparison with Experimental Results . . . . 71

4.3 Longitudinal Tunnelling . . . . . . . . . . . . . . . . . 734.3.1 The 3D Schrodinger Equation with Open Boun-

dary Conditions . . . . . . . . . . . . . . . . . 744.3.2 The Transfer Matrix Method . . . . . . . . . . 754.3.3 The Scattering Matrix . . . . . . . . . . . . . . 774.3.4 The Forward Construction Scheme for S . . . . 794.3.5 Recursive Construction of S . . . . . . . . . . . 804.3.6 Injected vs. Local Equilibrium Charge Density 824.3.7 Current and Charge Computation at Finite

Source–Drain Bias . . . . . . . . . . . . . . . . 834.4 Simulation Results . . . . . . . . . . . . . . . . . . . . 854.5 Comparison with Non–Equilibrium Green’s Functions 96

5 A coupled 3D Kohn–Sham /Drift–Diffusion Simulation Approach 995.1 The Test Device . . . . . . . . . . . . . . . . . . . . . 995.2 Drift–Diffusion Simulation . . . . . . . . . . . . . . . . 1005.3 simnad Simulations . . . . . . . . . . . . . . . . . . . 1075.4 A Brief Excurse on Meshing . . . . . . . . . . . . . . . 109

5.4.1 Tensor–Product Grids . . . . . . . . . . . . . . 1105.4.2 Finite Volume Discretisation and Delaunay

Meshes . . . . . . . . . . . . . . . . . . . . . . 112

CONTENTS xiii

5.5 The Coupling Strategy . . . . . . . . . . . . . . . . . . 1165.5.1 Strong Decoupling . . . . . . . . . . . . . . . . 1175.5.2 Distributed Kohn–Sham Equations . . . . . . . 1195.5.3 Mesh Merging . . . . . . . . . . . . . . . . . . 1205.5.4 Data Interpolation Between Meshes . . . . . . 121

5.6 Simple Coupling . . . . . . . . . . . . . . . . . . . . . 1225.7 Self–Consistent Coupling . . . . . . . . . . . . . . . . . 1235.8 Simulation Results with Self–Consistent Coupling . . . 124

6 Concluding Remarks 129

Appendices 133A Energy of the Inhomogeneous Electron Gas . . . . . . 133B The Velocity of Bloch States . . . . . . . . . . . . . . 135

Bibliography 137

Curriculum Vitae 145

Chapter 1

Introduction

In 1965, when typical integrated circuits (ICs) comprised justaround 50 components, Gordon Moore formulated his famous“law”, predicting, that the number of devices in commercially

available ICs would be doubling every year. In the April instalmentof the column “The experts look ahead” in “Electronics” he boldlyproclaimed [1]

Integrated circuits will lead to such wonders as home com-puters — or at least terminals connected to a central com-puter — automatic controls for automobiles, and personalportable communications equipment. The electronic wrist-watch needs only a display to be feasible today.

Since then, the world has seen almost four decades of exponentialgrowth in integration density. Mobile communication and personalcomputers are all about. From a transistor spacing of “two thou-sandths of an inch” [1] (= 50.8µm) back in 1965, main–stream tech-nology has progressed to feature sizes that are limited solely by thewavelength of the light used for illumination in the photo–lithography.For many years the continuing success of Moore’s law has been ac-companied by predictions of its imminent failure; so far, all thesepredictions have turned out to be wrong. But maintaining exponen-tial growth in IC performance does not come for free. Investments

1

2 CHAPTER 1. INTRODUCTION

into semiconductor production facilities are also growing exponentiallyas increasingly more refined technologies are needed to produce thescaled down structures of the latest technology generation.In addition to technological difficulties associated with defining

and manufacturing ever smaller structures, fundamental physical bar-riers will eventually limit further shrinkage of complementary metal–oxide–semiconductor (CMOS) devices — the device class on whichMoore’s law is based — by simple scaling [2]. Even now transistorsare being scaled to a regime, in which quantummechanical phenomenaare beginning to affect their performance. Consequently, the Interna-tional Technology Road map for Semiconductors (ITRS) [3] is callingfor the development of simulation capabilities that can handle “a fulltwo–dimensional quantum transport formulation”.Ultimately scaled field effect transistors (FET) will remain the

most important devices in the near future. There are decades of ex-perience in the field, and any newly emerging device class will haveto go a long way before it can hope to compete with current state–of–the–art devices. But, since FET performance eventually will bedegraded by quantum effects, it is nevertheless important to explorealternative devices that operate not in spite of but because of quan-tum effects. A whole range of possible device concepts have been pro-posed. The “Technology Roadmap for Nanoelectronics” [2] by the Eu-ropean Union, for example, lists devices as diverse as single–electrontunnelling devices, supra–conducting rapid single quantum flux logic,molecular electronics (possibly formed by self–assembly aided by DNAscaffolding), spin–valve devices and electron waveguide devices.Therefore, a simulation framework capable of handling both ulti-

mate CMOS devices like nano–scale double–gate field effect transistors(FET) in the quantum–ballistic limit1 or nano–flash RAM devices andquantum confined single–electron tunnelling devices in the Coulombblockade regime has been developed. Only semiconductor–based ap-proaches are discussed in this work, because they share a lot of com-mon physics and hence may be handled within a common simulationframework. Molecular electronics is basically described by a similarformalism, but different implementations with different types of ap-

1In this limit the silicon body of the device is essentially treated as an electronwaveguide.

3

proximations are required for electronic structure calculations in bulksolids on the one hand and molecules on the other hand. Rapid singlequantum flux logic, again, is based on an entirely different physi-cal foundation (the physics of supra–conducting quantum interferencedevices). Here, performance gain is expected from extremely rapidclocking rather than from enhanced integration densities.This thesis is organised in three main parts, each of which is de-

voted to a special aspect of the simulation of semiconductor nano–devices. In the first part (chapters 2 and 3) the main focus is on thesimulation of single–electron devices in the Coulomb blockade regimeand on the complications that arise when device dimensions are re-duced below the critical length–scale given by the wavelength of theelectrons inside the device. Coulomb blockade devices are interest-ing candidate structures for future ultra large scale integrated (ULSI)circuits since they provide a more efficient means for the control oftunnelling current than modulation of a single potential barrier.The second part (chapter 4) discusses quantum–ballistic trans-

port. This is a transport mode in which electron transport is fullycoherent and scattering effects other than potential scattering (in-cluding electron–electron scattering by a mean field approach) are ab-sent. This transport regime is usually observed in modulation dopedhetero–structures based on III–V semiconductors at very low tem-peratures, where coherent transport may take place over distancesexceeding a micron. Here, it is also applied to ultra–small silicon–on–insulator (SOI) metal–oxide–semiconductor (MOS) field effect tran-sistors (FETs). This study elucidates quantum–related performancelimitations encountered by ultimately scaled MOSFET devices, sincethe ballistic current is the upper limit to the current through a deviceafter removal of technology and fabrication induced non–idealities.In the third part (chapter 5) coupling between the simnad quan-

tum mechanics simulator [4, 5] and a semiclassical device simulator(dessis–ise [6]) is discussed. This coupling allows the user to combinethe strengths of the individual simulators and enables simulations in-accessible to each individual simulator. As an example a nano–scaleflash RAM device is considered. In such a device both dissipativetransport (along the channel) and multi–dimensional confinement (inthe quantum dot floating gate) are present. The simulator couplingallows treatment of this situation in a fully self–consistent way.

Chapter 2

Single–ElectronTunnelling Devices

Single–electron tunnelling devices are based on the granularnature of electrical charge — any freely observable1 particlecarries an integer multiple of the elementary charge e. Acommon feature of single–electron tunnelling devices is the

absence of a classical conductance path through the device. Chargetransport is entirely due to quantum mechanical tunnelling. In thepresence of two or more tunnelling barriers with interjacent islands,Coulomb blockade [7, 8] may take place. 1To illustrate this phenomenon, let us take a look at the schematicdrawing of a single–electron transistor (fig 2.1). This device consistsof source and drain leads and a capacitively gated island. Classically,this island is insulated from both source and drain by thin barriers(e.g. thin oxide layers). Quantum mechanically, there is a finite prob-ability for an incident electron to traverse such a barrier (tunnelling).Hence, a single barrier possesses non–zero conductance. For conduc-tance through the entire device, however, additional conditions areimposed by the island geometry. The dominant conductance mech-

1This excludes quarks, which carry fractional charge, but cannot be isolated;attempts to do so will inevitably result in the formation of additional quark–antiquark pairs.

5

6 CHAPTER 2. SINGLE–ELECTRON DEVICES

source drainisland

gate capacitor

tunnel junctions

Vg

Vds

Figure 2.1: Schematic drawing of a single–electron transistor.

anism in a single–electron transistor (SET) is sequential tunnelling,i.e. the process that transports an electron from source to drain maybe decomposed into two stages:

T1: The electron tunnels from the source lead onto the island.

T2: The electron tunnels off the island into the drain lead.

Because of the finite magnitude of the elementary charge e, processT1 requires a non–zero charging energy2

EC =e2

2Cisland. (2.1)

This charging energy is recovered during process T2. Nonetheless, thecharging energy must be available to the tunnelling electron in orderto form the intermediate state. In an unbiased SET the only sourcefor this activation energy is the thermal energy of the electron. Hence,at low temperature (kBT EC) an unbiased SET is non–conducting.This conductance suppression is called Coulomb blockade. Lifting theblockade at Vg = 0 requires a forward bias |Vds| > e/2Cisland. ForkBT e2/2Cisland, Coulomb blockade effects vanish; hence, single–electron devices must either be very small (small Cisland) or operateat cryogenic temperatures (small kBT ).

2computed for the unbiased device with initially empty island

7

As long as device extensions are “large” (according to the criterion ofchapter 3), an SET may be analysed in terms of an equivalent circuit:

C1, R1 C2, R2

CgT1 T2

source drain

gate

island

The tunnelling junctions are modelled as leaky capacitors3, which maybe described in terms of a capacitance and a tunnelling resistance. Interms of these quantities, the change in the total energy of the system(including voltage sources)4 associated with the tunnelling processesT1 and T2 at initial island electron number N may be written as

∆ET1(N) =e

CΣ

[e2+Ne− CgVgate-drn + C2Vsrc-drn

], (2.2)

∆ET2(N) =e

CΣ

[e2−Ne+ CgVgate-drn + (C1 + Cg)Vsrc-drn)

], (2.3)

with CΣ = C1+C2+Cg. At zero temperature, a process will happenspontaneously only, if its associated energy change is non–positive.Hence, at T = 0 an electron may tunnel through the island fromsource to drain only, if ∆ET1 ≤ 0 and ∆ET2 ≤ 0. This may berepresented graphically in the stability diagram (fig. 2.2). For biasconditions in the interior of the shaded diamonds (stability regions),transport is energetically forbidden. Each diamond corresponds toan integer number Nstab of excess electrons on the island; Nstab isthe unique value of N for which both ∆ET1(N) and ∆ET2(N) arepositive. Outside the stability regions, electron numbers N exist, forwhich both ∆ET1(N) ≤ 0 and ∆ET2(N + 1) ≤ 0. Then the islandalternates between the occupation numbers N and N+1, and currentmay flow. In linear response (i.e. with infinitesimal source–drain bias)an SET at T = 0 is conducting only for discrete values of the gatevoltage. At finite temperature, the conductance peaks are broadened,and the SET conductance takes the shape of figure 2.3.

3with the additional constraint, that electrons may only leak one by one4Often in literature, this quantity is referred to as “Free Energy”. This is not

entirely accurate, since the quantity is only canonical with respect to the island,but grand canonical with respect to the source/drain regions.

8 CHAPTER 2. SINGLE–ELECTRON DEVICES

Q = +e Q = 0 Q = −e

Vgate-drn

Vsrc-drn

e

CΣ

− e

CΣ

e

Cg− e

Cg

slope =Cg

C1 + Cg

slope = −CgC2

Figure 2.2: Stability plot of a “large” SET. Inside the shaded diamondsconductance is blocked.

e

Cg

3eCg

Vgate-drn

I

Vsrc-drn2(R1 +R2)

Figure 2.3: Conductance characteristics of a “large” SET.

Chapter 3

Challenges for theSimulation ofNano–ScaleSingle–Electron Devices

The equivalent circuit treatment of the preceding sectionceases to be applicable when the critical dimensions of thedevice are reduced below the wavelength associated withthe available kinetic energy of the charge carriers. If the

typical kinetic energy available to a charge carrier is Ekin, the cor-responding wavelength cut–off is λcut–off(Ekin) = h/

√2m∗Ekin . In

non–degenerately doped semiconductors almost the entire kinetic en-ergy of the carriers is thermal: Ekin = 1

2kBT per degree of free-dom. Hence, the appropriate wavelength is the thermal de Broglie–wavelength λth = h/

√m∗kBT of the electrons (holes1).

In degenerately doped semiconductors, the Fermi energy εF lies wellwithin the conduction (valence) band. This means, that in very highly

1Because of the interplay of heavy and light hole bands and the stronger non–parabolicity, the approximation of a single, energy–independent effective mass forholes must be used with extreme care.

9

10 CHAPTER 3. NANO–SCALE SINGLE–ELECTRON DEVICES

doped devices at low temperatures the kinetic energy necessitated bythe Pauli exclusion principle may be much higher than the thermalenergy 1

2kBT . Then, the relevant cut–off wavelength is the Fermi–wavelength λF,e– = h/

√2m∗e(εF − εc) (λF,h = h/

√2m∗h(εv − εF) ),

which under the stated circumstances will be much shorter than λth.In silicon wires at room temperature the condition that the confine-ment width should be smaller than λcut–off requires structures that areonly a few nanometres in size; in GaAs with its much smaller effec-tive mass, however, the transition to the quantum–mechanical (QM)regime takes place at much larger structures.In the QM regime, charge densities and capacitances are modified

by quantum depletion effects — the maximum of the charge den-sity distribution no longer coincides with the geometrical surface ofthe (semi–)conductor but is pushed back into the material. Conse-quently, it is no longer possible to compute capacitances solely fromthe device geometry. Omnidirectional confinement below the cut–offwavelength produces a discrete energy level structure (the island thenis referred to as a quantum dot), that increases the charging energy;and the tunnelling resistances are modulated by the shape of the wavefunction in the channel and island regions. Hence, in a single–electrontransistor with quantum confinement none of the quantities featuringin the equivalent circuit formulation are known. Thus, a more generalsimulation approach is needed.The necessary ingredients for the computation of the conductance

in single–electron devices with quantum confinement are

1. the self–consistent quantum–mechanical charge density and thecorresponding electrostatic potential (energy band diagram),

2. tunnelling rates between the wave functions in the channel re-gions and the island (quantum dot),

3. occupation probabilities of the individual energy levels on thedot.

The tools that will be applied in order to obtain these quantities are

1. a finite–temperature generalisation of density functional theory(in an effective mass formulation),

3.1. COMPUTATION OF THE CHARGE DENSITY 11

2. Bardeen’s transfer Hamiltonian method,

3. quantum statistical mechanics.

The operation of sincle–electron tunnelling devices in the Coulombblockade regime requires that the conductivity of the device mustbe small compared to the (spin–degenerate) conductance quantumG0 = 2e2

h ≈ 77.48 µS. Otherwise, the notion of electrons on the quan-tum dot becomes ill–defined, since the electron wave functions are nomore localised on the quantum dot but spread out all the way to thecontacts.2 Therefore, in the present simulation approach to single–electron devices, all these quantities will be extracted neglecting cur-rent flow. The conductance, then, is extracted in a post–processingstep following a procedure proposed by Beenakker [9].

3.1 Computation of the Charge Density

In a semiconductor, the charge density ρ(x) depends on the localchemical potential µ(x) or, equivalently, on the positions of the con-duction and valence band edges εc and εv relative to the electrochem-ical potential εF = µ(x)− eφ(x). In thermodynamic equilibrium, i.e.in the absence of net currents

0 != j = σ∇εF(x), (3.1)

the electrochemical potential is constant in space.3

Consequently, the chemical potential µ(x) must be modulated bythe electrostatic potential, which implies, that the charge density ρ(x)can be written as a functional of the electrostatic potential φ(x), whichin turn depends on the charge density by Poisson’s equation. Themutual interdependence of charge density and electrostatic potentialis expressed by the non–linear Poisson equation

−∇ · (ε0εr(x)∇φ(x)) = ρ[φ](x). (3.2)

This equation needs to be solved self–consistently, i.e. we need to findthe potential φ and the charge density ρ such that φ = φ[ρ] is the

2Devices in this regime are discussed in chapter 4.3εF may be non–constant inside a perfect insulator (σ = 0).


potential produced by ρ via Poisson’s equation, and ρ = ρ[φ] is thecharge density resulting from insertion of φ into the charge densitymodel. The functional form of ρ[φ] under various model assumptionswill be discussed in later sections. But regardless of the form of thecharge density functional, in the end we need to solve eq. (3.2). Thisis done iteratively in a Newton–Raphson scheme.The iteration procedure starts with some initial guess φ0 for the

potential and the corresponding charge density ρ0 = ρ[φ0]. If thesedo not already solve eq. (3.2), there will be a non–zero residual

r(x) :=∇ · (ε0εr(x)∇φ0(x))+ ρ0(x). (3.3)

In order to improve on the initial guess, the non–linear Poisson equa-tion is linearised around φ0 by substituting

φ = φ0 + δφ (3.4)

and

ρ = ρ0 +δρ

δφ

∣∣∣∣φ0

δφ. (3.5)

into (3.2). This yields:

−∇ · (ε0εr∇φ0)− ρ0︸︷︷︸=:−r

=∇ · (ε0εr∇δφ)+ δρ

δφ

∣∣∣∣φ0

δφ (3.6)

⇔ δφ = −(∇ · (ε0εr∇)+ δρ

δφ

∣∣∣∣φ0

)−1r. (3.7)

From eq. (3.4) we obtain a new potential φ. The corresponding chargedensity is ρ[φ]. These values are used as φ0 and ρ0 in the next iter-ation step, and the whole procedure (eq. (3.3)–(3.7)) is iterated untilconvergence (i.e. until the norm of r is small enough). In praxi , thingsare somewhat more complicated. Typically, the form of ρ[φ] is such,that straight iteration will result in instabilities, such as oscillationsin the charge density during the iterative cycle. Therefore, the solu-tion of (3.2), requires application of suitable damping schemes for thepotential update in order to obtain convergence [10, 11, 12, 13].


3.1.1 The Charge Density by Density FunctionalTheory

According to the work of Hohenberg and Kohn the ground state prop-erties of a system of a many–particle system of interacting electronsis uniquely represented by its ground state charge density [14]. Theground state charge density may be found by minimising a functionalEV [n] that depends solely on the external potential V and the electrondensity n.

EV [n] = min! ⇒ n(x). (3.8)

The functional EV [n] consists of three parts

EV [n] = T [n] + UV [n] + F [n]. (3.9)

The first term, T [n], is the the total kinetic energy of the electrons.UV [n] = −e

∫d3xn(x)V (x) is the potential energy of the electrons in

the external potential, and the remaining term F [n] is the interactionenergy of the electron system. Customarily, the classical electrostaticinteraction energy (Hartree energy) is split off F [n]. The remainingterm W [n] := F [n] − 1

2e2∫d3x∫d3x′ n(x)G(x,x′)n(x′), G(x, x′) be-

ing the Green’s function of the electrostatic potential, is the so–calledexchange/correlation energy. In contrast to the other terms in EV [n]there is no closed form expression for W [n]. Approximations to theexchange/correlation functional exist on various levels. In the presentwork a self–interaction reducing variant [15] of the local density ap-proximation (LDA) is used.Parametrising the charge density n at T = 0 in terms of single–particleorbitals4 ψi

n =N−1∑i=0

∣∣ψi/2∣∣2 , (3.10)

4Here, spin degeneracy is assumed. In the presence of magnetic fields, or ifspin–orbit interaction effects are included, this degeneracy is lifted. Then separatespin up and spin down densities have to be computed, and LDA is replaced bylocal spin density (LSD) approximation.


where x denotes the floor of x, reduces the variational problem(3.8) to a system of Schrodinger–like equations, the Kohn–Sham equa-tions [16, 17] (

− 2

2m∇2 + Vs(x)

)ψi(x) = εiψi(x). (3.11)

Here, the effective potential Vs = V + VH + Vxc takes the place of theexternal potential V in the single–particle Hamiltonian. Together,the Hartree potential VH(x) := −e ∫ d3x′G(x,x′)n(x′) and the ex-change/correlation potential Vxc := δW [n]/δn take care of the effectof electron–electron interaction on the single–particle wave functions.The Kohn–Sham orbitals ψi are ordered according to their Kohn–Sham energies εi, starting with ψ0 for the lowest eigenfunction ofeq. (3.11).At non–zero temperature [18] the above approach needs to be ex-

tended. For systems strongly coupled to a reservoir5, Fermi–Diracstatistics may be invoked

n = 2∞∑i=0

f( 1kBT

(εi − εF))|ψi|2. (3.12)

Here, εF denotes the electrochemical potential of the reservoir, and fis the Fermi–Dirac distribution function

f(x) =1

ex + 1. (3.13)

In the presence of omnidirectional confinement only weak couplingto the reservoir is possible. Then all the Kohn–Sham orbitals are(quasi–)localised and (omitting life–time broadening) the spectrumbecomes discrete. In this situation (as exemplified in figure 3.16 onpage 53) the grand canonical ground state must explicitly be con-structed from its constituent canonical states; naıve application ofFermi–Dirac statistics using the external electrochemical potentialmay be misleading.

5The effective single–particle Hamiltonian of such systems exhibits continuousspectrum.


3.1.2 The Effective Mass Approximation

The electrostatic potential inside a semiconductor consists of two con-tributions that vary on very different length–scales. The first com-ponenent, Vcr corresponds to the periodic lattice of the bulk crystal.It is characterised by very large spatial frequencies k > π/a (a beingthe lattice constant) and amplitudes. The second component Vmod isbrought about by external fields, gates and changes of the mean chargedensity over distances much larger than a. Usually the non–periodicpotential is much smaller in amplitude than the periodic term.In a bulk semiconductor only the periodic Vcr term is present.

The symmetry of the crystal potential gives rise to periodicity in allposition–dependent observables. Instead of the plane waves of thefree–electron Hamiltonian, a periodic potential gives rise to Bloch wavefunctions

〈x|νk〉 = uνk(x) eik·x. (3.14)

The Bloch factor uk(x) is a function with the periodicity of the crys-tal lattice. Adding a non–periodic perturbation Vmod to the potentialresults in the mixing of different Bloch states. In the present worka single–band treatment is used6, i.e. it is assumed that the single–particle wave function |ψ〉 may be represented in terms of Bloch func-tions of a single–band ν0

|ψ〉 ≈∑k′|ν0k′〉〈ν0k′|ψ〉 . (3.15)

Substituting (3.15) into the Schrodinger equation and pre–multiplyingwith 〈νk| yields

εν0(k)〈ν0k|ψ〉δνν0 +∑k′〈νk|Vmod|ν0k′〉〈ν0k′|ψ〉 = ε〈ν0k|ψ〉δνν0 ,

(3.16)

6This single–band treatment is strictly justified only for potentials the Fouriercoefficients of which are essentially non–zero only near the centre of the first Bril-louin zone. But the method is found to be successful under much more generalconditions, e.g. at potential steps.


where εν0 is the band dispersion function of band ν0. For a slowlyvarying potential Vmod the wave function |ψ〉 contains significant con-tributions only from k–vectors in the vicinity of a central wavevectork0. With the approximation

〈x|ν(k0 +∆k)〉 ≈ 〈x|νk0〉 ei∆k·x (3.17)

the Bloch components 〈ν0k|ψ〉 of the wave function become the Fouriercoefficients 〈k|Fν0〉 of an envelope function 〈x|ψ〉 ≈ 〈x|ν0k0〉〈x|Fν0〉,and the Bloch function matrix elements of Vmod are approximatedwith 〈k|Vmod|k′〉δνν0 . Inserting this into eq. (3.16) yields after re-verse Fourier transform the following equation for the enevelope func-tion [19, 20] (

ε(−i∇) + Vmod(x))Fν0(x) = εFν0(x). (3.18)

Expanding εν0(k) in an extremum kv as a quadratic function

εν0(k) ≈ εν0(kv) +2

2(k− kv) ·

[1

m∗ν0,kv

](k− kv) (3.19)

introduces the reciprocal effective mass tensor corresponding to valleyv of band ν as [

1m∗ν0,kv

]ij

=12

∂2εν0(k)∂ki∂kj

∣∣∣∣k=kv.

(3.20)

With this parabolic bandstructure, apart from replacing the electronmass m with the effective mass m∗ the envelope equation takes theform of a normal single–particle Schrodinger equation [21](

−2

2∇ ·[

1m∗ν0,kv

]∇+ V (x)

)Fν0,v(x) = εFν0,v(x). (3.21)

Wave functions obtained from effective Schrodinger equations of dif-ferent valleys are by construction orthogonal, since they are composedof Bloch functions residing in non–overlapping k–space regions.In semiconductor device simulation there is (so far) no concern

with charge density oscillations inside a unit cell of the crystal. There-fore, the Bloch factors may be omitted in the computation of the


charge density, and |Fν0,v|2 may be used instead of the probabilitydensity of the actual wave function. Electron–electron interactioneffects can be included according to the discussion of the previoussection by addition of the Hartree and exchange/correlation poten-tials to the non–periodic potential Vmod in the envelope equation [22].Then the final expression for the electronic charge density inside asemiconductor device is

ρe−(x) = −2e∑v

∑i

Pν0,v,i|Fν0,v,i(x)|2 (3.22)

with occupation probabilities Pν0,v,i as discussed in section 3.1.1.Semiconductor devices typically contain material interfaces. In

general, the bulk materials on either side of an interface have differentbandstructures and different Bloch states. The central question thenis, how to connect envelopes at a hetero–interface. In the standardapproach [21] the effective mass is treated as a position–dependentparameter, and continuity of the envelope function and its normalderivative is postulated at the interface. The energy offset of theband–edges on either side of the interface (in the case of the conduc-tion band this is the difference in the electron affinity χ of the twomaterials) is added as a step–function to the potential in the enve-lope equation. Recently, more general interface conditions based oninterface matrices have been suggested (e.g. [23]), and it has beenshown that these may give rise to additional physical effects such aslocalised interface states inside the band gap. But first principlescalculation of the interface matrix elements is extremely tedious, andphenomenological parameters are usually unavailable. Hence, simnad

still applies the standard boundary conditions; but it is important tokeep in mind, that this treatment does not automatically account forthe full interface physics.

3.1.3 Domain Decomposition and Adiabatic De-coupling of the Schrodinger Equation

A semiconductor device often contains regions, inside which potentialvariations along different directions occur on different length scales.Then, solving the full three–dimensional Schrodinger–Poisson prob-


lem7 incurs massive computational load without corresponding gainin insight — along directions of slow potential variation (classical de-grees of freedom) the charge density will depend on the potential inan essentially local fashion.Then the dimensionality of the problem may be reduced by a local

bulk approximation for the classical degrees of freedom. The Schro-dinger equation is only solved for the main confinement directions,whereas along classical directions plane waves are assumed. Theresulting expressions for the electron density n per valley for a 1–dimensional electron gas (1DEG) with confinement in the y–z plane,a 2DEG with confinement along the z–axis and a 3DEG with slow po-tential variation in all directions (Thomas–Fermi gas) are [24, 25, 26]

n1DEG(r) =1√2π·√m∗x

· (kBT ) 12 ·∑i

F− 12

(εF − εi(x)kBT

)|ψi,x(y, z)|2

n2DEG(r) =1π·√m∗xm∗y2

· kBT ·∑i

F0

(εF − εi(x)kBT

)|ψi,x,y(z)|2

n3DEG(r) =1π

32·√m∗xm∗ym∗z

3· (kBT ) 3

2 · F 12

(εF − εCkBT

). (3.23)

Fα denotes the Fermi–Dirac integral of order α

Fα (η) =

1Γ(α+ 1)

∞∫0

tα dtexp(t− η) + 1 , α > −1

11 + exp(−η) , α = −1

(3.24)

F′α(η) =Fα−1(η), α ≥ 0. (3.25)

Occasionally, formal notations of the form Fα(η) with α < −1 areencountered. Originally, this is undefined. However, it seems naturalto use relation (3.25) to extend the definition of F:

Fα(η) :=ddη

Fα+1(η) ∀α < −1. (3.26)

Usually in a semiconductor device regions of different confinement7Here and in the remainder of this work (excluding chapter 4), “Schrodinger” is

used as shorthand for the effective mass Kohn–Sham envelope equations obtained


Figure 3.1: Domain decomposition of a silicon–on–insulator single–electron transistor (micrograph from [27]; shading and labels added).

type may be identified. Then the simulation domain may be decom-posed into parts of different dimensionality as shown in figure 3.1.Computation of the tunnelling rates requires that the quantum wireand the quantum dot regions overlap. In this overlap region it is nec-essary to ensure, that only “correctly localised”wave functions (in thesense of the discussion in section 3.2.2) may contribute to the chargedensity of each constituent region. In overlap regions between 3DEG,2DEG and 1DEG regions the stronger confinement always takes prece-dence.

3.1.4 Dimensional Reduction with Geometric Con-finement — an Approximate Treatment

In geometrically defined quantum dots the usual adiabatic decompo-sition of Schrodinger’s equation into an array of 1D equations alongthe main quantisation direction and a 2D equation in the remaining

in section 3.1.2.


position along cut 1

potential potential

position along cut 2

cut1

cut2

Si

SiO2

Figure 3.2: In geometrically defined quantum dots, the adiabatic de-composition of the Schrodinger equation fails, because the 1D wavefunctions suddenly change at the material interface.

dimensions runs into problems. At the geometrical boundaries of thequantum dot (e.g. the Si/SiO2 interface) the shape of the 1D wavefunction suddenly changes, because 1D cuts that do not intersect thesilicon will not see a confining potential (cf. fig. 3.2). Thus the as-sumption of the adiabatic decomposition that the variation of the 1Dwave functions with cut position should be small is violated.However, the situation can be remedied by redistributing the dis-

continuous contribution to the confinement potential between the 1Dequations and the 2D equation: assuming that in a geometrically con-fined 2D quantum dot (with normal direction along the z axis) theband edge offset is of the form

χ(x, y, z) = χ(x, y) + χ(z), (3.27)

the (Kohn–Sham) effective potential can be written as

V (x, y, z) =

=:φs(x,y,z)︷︸︸︷φ(x, y, z) + φxc(x, y, z)+χ(x, y, z) (3.28)

= φs(x, y, z) + χ(z)︸︷︷︸=:Vx,y(z)

+χ(x, y),

where Vx,y is a slowly varying function in x and y (i.e. the disconti-nuities of V in the xy–plane have been excluded from V ).


Using this effective potential a 2D array of 1D wave functions γx,y(z)may be computed:(

−2

2∂z

1m∗z

∂z + Vx,y(z))γx,y(z) = εx,yγx,y(z) (3.29)

From this a full 3D wave function is constructed using the ansatz

Φ(x, y, z) = γx,y(z)ψ(x, y). (3.30)

Substituting this into the l.h.s. of Schodinger’s equation yields(−

2

2∇ ·[1m∗

]∇+ V (x, y, z)

)Φ(x, y, z)

=− 2

2

(ψ(x, y)

((∇T + ez∂z) ·

([1m∗

](∇T + ez∂z) γx,y(z)

))

− 2 (∇Tψ(x, y)) ·([

1m∗

]∇T γx,y(z)

)

− γx,y(z)(∇T ·([

1m∗

]∇Tψ(x, y)

)))+ V (x, y, z) ψ(x, y) γx,y(z)

=− 2

2ψ(x, y)

(∇T ·([

1m∗

]∇T γx,y(z)

))+(εx,y − Vx,y(z)

)ψ(x, y) γx,y(z)

− 2(∇Tψ(x, y)) · ([ 1

m∗

]∇T γx,y(z)

)

− 2

2γx,y(z)

(∇T ·([

1m∗

]∇Tψ(x, y)

))+(Vx,y(z) + χ(x, y)

)ψ(x, y) γx,y(z), (3.31)

where∇T =∇− ez∂z = ex∂x+ ey∂y is the transverse nabla operator,and the reciprocal effective mass tensor

[1m∗]is assumed to have its

principal axes along the coordinate directions.Since Vx,y(z) is a slowly varying function of x and y, so are the

γx,y(z), and consequently the ∇Tγx,y(z) terms may be neglected.


Thus, we obtain an effective Schrodinger equation for ψ

(−

2

2∇T ·[1m∗

]∇T + χ(x, y) + εx,y

)ψ(x, y) = ε(x, y). (3.32)

Unfortunately, the condition imposed on χ (eq. 3.27) is very restric-tive; even in simple geometries such as a cuboid silicon quantum dotsurrounded by oxide, it cannot be satisfied exactly. Often, however, itis possible to replace the true band edge offset with an offset functioncompatible with (eq. 3.27) that differs from the actual offset only inirrelevant regions (i.e. regions in which Φ is known to be small).

For example, for a prismatic quantum dot, with an z–extensionmuch smaller than its extensions in the xy–plane, the electron affinityχ(x, y, z) may approximately be decomposed into

χ(z) = χ(xcentre, ycentre, z) (3.33)χ(x, y) = χ(x, y, zcentre)− χ(xcentre, ycentre, zcentre). (3.34)

This decomposition affords to correct qualitative localisation of theelectron wave function. Inside the quantum dot there is little devia-tion of the approximate wave function from the actual wave function.In fact, unless we come too close to the edges of the bottom or toppolygon of the prism, even the decay of the wave function into thesurrounding oxide along direction normal to the faces of the prismvectors is modelled quite accurately. However, if we set out from theedge of the top (bottom) polygon and move simultaneously outwardsin the xy–plane and upwards (downwards) along the z–axis, then thewave–number for the exponential decay of the amplitude into the ox-ide is overestimated by about a factor of

√2, because the material

step is encountered both in χ(z) and χ(x, y), thereby apparently dou-bling the energy barrier. This justifies the application of the aboveapproach for the computation of charge densities and both in–planeand vertical tunnelling rates, but precludes its usage in situations inwhich tunnelling processes along “diagonal” directions are important.

3.2. COMPUTATION OF TUNNELLING RATES 23

3.2 Computation of Tunnelling Rates

3.2.1 Bardeen’s Transfer Hamiltonian Method

Bardeen’s transfer Hamiltonian method [28, 29] is a variant of first or-der time–dependent perturbation theory for the computation of tran-sition rates between states of a quantum system. It differs from theusual approach in that the initial and final states are not eigenstatesof the same Hamiltonian; instead the Hamiltonian of system with abarrier H is mapped onto a pair8 of Hamilton operators Hl and Hr(cf. fig. 3.3) such that

Hl = H on Ωl, Hr = H on Ωr, Ωl ∪ Ωr = Ω. (3.35)

The eigenstates of each of these transfer Hamiltonians are localised onthe same side of the barrier. Bardeen’s transfer Hamiltonian methodprovides matrix elements between wave functions ψl and ψr on eitherside of the barrier. These may be used to compute tunnelling ratesby Fermi’s Golden Rule

Γi→f =2π|Mi,f |2δ(Ef −Ei) (3.36)

for a transition in a discrete spectrum of final states or

Γi→f =2π|Mi,f |2 Z(Ei) (3.37)

for a transition into a continuum of states.Below the computation of the transition matrix elementsMi,f betweenan initial state ψ(l)i on the left side of the barrier and a final state ψ(r)fon the right side of the barrier will briefly be sketched. To do this, weconsider the time–dependent Schrodinger equation

i∂

∂tψ(t) = Hψ(t) (3.38)

8For systems with a double barrier the method is extended by introducing athird transfer Hamiltonian that generates eigenfunctions which are localised inbetween the two barriers (cf. e.g. [30]).


H

Hl

Hr

ψ(l)

ψ(r)

Ωl Ωr

Ω

Figure 3.3: Transfer Hamiltonians Hl/r for a system H with a squarebarrier.

with the approximate ansatz

ψ(t) ≈ a(t) e−iE

(l)i t |i〉(l) +

∑j

bj(t) e−iE

(r)j t |j〉(r) . (3.39)

Note that since we are working with eigenfunctions of Hl/r rather than

the full system Hamiltonian H, the e−iE

l/rj t |j〉(l/r) terms differ from

the product of the time evolution operator U = e−iHt with |j〉(l/r).

But, since the wave functions have their probability density concen-trated in the region in which its transfer Hamiltonian is identical withthe full Hamiltonian, this discrepancy is regarded as “small”. At t = 0we assume ψ = |i〉(l), which in terms of the expansion coefficientsreads a = 1, bk = 0 ∀k. The expansion is restricted to first order ina. This implies a = 0. Near t = 0 this yields

ibf = e−i(E

(r)f −E(l)

i ) t (r)〈f |H−E(l)i |i〉(l)︸︷︷︸=:Mi,f

. (3.40)


The remaining task is the computation of the matrix element

Mi,f = (r) 〈f |H− E(l)i |i〉(l)︸︷︷︸=:Mi,f

=∫Ω

ddx ψ(r)f∗(x)(H− E(l)i

)ψ(l)i (x).

(3.41)

Since Hl |i〉(l) = E(l)i |i〉(l), Hl differs from H only by a local multi-

plication operator the support of which is a sub–domain of Ωr, theintegration domain may be restricted to Ωr:

Mi,f =∫Ωr

ddx ψ(r)f∗(x)(H−E(l)i

)ψ(l)i (x). (3.42)

By a similar argument (and from the fact that H is self–adjoint) it isknown that

ψ(l)i (x)(H− E(r)f

)ψ(r)f

∗(x) = 0 ∀x ∈ Ωr. (3.43)

Thus, the l.h.s. of eq. (3.43) may be subtracted from the integrand ineq. (3.42) without changing the result, yielding

Mi,f =∫Ωr

ddx[ψ(r)f

∗ (H−E(l)i

)ψ(l)i − ψ(l)i

(H− E(r)f

)ψ(r)f

∗]=∫Ωr

ddx[ψ(r)f

∗Hψ(l)i − ψ(l)i Hψ(r)f

∗+ (E(r)f − E(l)i︸︷︷︸(energy conservation)

= 0

) ψ(r)f∗ψ(l)i

]

=∫Ωr

ddx[ψ(r)f

∗Hψ(l)i − ψ(l)i Hψ(r)f

∗]. (3.44)

For a Hamiltonian of the form

H = −2

2∇ ·[1m∗

]∇+ V (x) (3.45)

this may be recast as

Mi,f = −2

2

∫Ωr

ddx[ψ(r)f

∗∇ ·[1m∗

]∇ψ(l)i − ψ(l)i ∇ ·

[1m∗

]∇ψ(r)f

∗],


and by a variant of Green’s theorem the matrix element can be con-verted to a surface integral

Mi,f = −2

2

∫∂Ωr

dd−1x1m∗⊥

[ψ(r)f

∗ ∂ψ(l)i∂n

− ψ(l)i∂ψ

(r)f

∗

∂n

]. (3.46)

Here, n is the outer unit surface normal vector on ∂Ω; ∂∂n = n ·∇

denotes the directional derivative along n, and m∗⊥ =(n · [ 1m∗ ] n)−1

is the effective mass component normal to the surface.

3.2.2 Transfer Hamiltonians for Arbitrary Poten-tials

Besides (3.35), a necessary condition on the transfer Hamiltoniansis that their eigenfunctions must be localised on the correct side ofthe potential barrier. In one–dimensional systems this condition iseasily met: localisation of a particle to the left/right of a potentialbarrier with a local maximum V (xmax) = Vmax is afforded by transferHamiltonians with potentials

Vl(x) =V (x), x < xmaxVmax, x ≥ xmax

(3.47)

and

Vr(x) =V (x), x > xmaxVmax, x ≤ xmax

, (3.48)

respectively. Localisation in a quantum dot delimited by local maximaV(xI/IImax

)= V

I/IImax , xImax < xIImax is effected by a potential

Vdot =

V (x), xImax < x < xIImaxV Imax, x ≤ xImaxV IImax, x ≥ xIImax

, (3.49)

provided the potential well in between is deep enough to accommodatea bound state (cf. fig. 3.4).


V Vdot

Ωdot

xImax xImaxxIImax xIImax

Figure 3.4: Constructing a quantum dot transfer Hamiltonian in 1D.

3.2.3 Generalisation to Higher Dimensions

In systems of higher dimensionality [31] the construction of transferHamiltonians is somewhat less straightforward, and there are somepitfalls.If we want to proceed in analogy to the discussion of the previous

section, we need to find an appropriate generalisation of the notion oflocal maximum of the barrier potential. In 1D, the value Vmax of thelocal maximum is the minimum escape energy a particle must havein order to be able to reach a point on the other side of the barrieralong a classical trajectory, i.e. without tunnelling (the kinetic energyalong the path is always positive). If we know a point xin inside thepotential well (before the barrier)9 and a point = xout outside thepotential well (behind the barrier) then we can easily determine theescape energy εesc regardless of the dimensionality of Ω. In order todo so, we need the following definition:

Definition 1

A classical path of energy ε from a to b is a continuous map γ :[0, 1]→ Ω, γ(0) = a, γ(1) = b such that V (γ(s)) < ε ∀s.With this definition the escape energy εesc is the lowest energy

(or rather, since the set of possible energies is open, the infimum ofenergies) ε for which there exists a classical path from xin to xout.This can be used to obtain a first guess V simpledot for the quantum

dot transfer Hamiltonian Hsimpledot . For the region Ωsimpledot inside whichthe potential remains unmodified we choose the set of points thatcan be reached from xin by classical paths with energies smaller than

9The parenthesised wording refers to a simple barrier, the non–parenthesisedwording to a quantum dot.


V

=⇒

V simpledot

− : ∂Ωsimpledot

Figure 3.5: Original potential and modified quantum dot potential;V simpledot is constructed according to eq. (3.50) using classical paths.

εesc. The modified potential for the transfer Hamiltonian then maybe defined as

V simpledot (x) =V (x) x ∈ Ωsimpledot

max(V (x), εesc

)otherwise

. (3.50)

The effect of this procedure is depicted in figure 3.5.Under favourable conditions10, this construction is already suffi-

cient to eliminate states that are localised on the wrong side of thebarrier. But in general this is not the case. The reason is, that con-fined electrons are not localised point particles but reside in extendedwave functions, and their energy contains strictly positive kinetic en-ergy contributions even for directions along which their momentumexpectation value is zero. Thus, not all classical paths are accessibleby quantum mechanical electrons. This is the origin of pure geomet-ric confinement, which cannot be detected by the above approach.Consider for example an oxide coated silicon nano–wire with a con-striction (cf. fig. 3.6). If we assume the potential VSi inside the siliconto be constant, a “classical” electron of energy ε > VSi is allowed totravel freely throughout the silicon, and no potential well is found.Quantum–mechanically, however, the electron wave function is later-ally squeezed inside the constriction. This leads to an increase in the

10e.g. if there is very little lateral confinement, or if lateral confinement is con-stant (quasi–1D).


Figure 3.6: A constriction in a silicon nano–wire may give rise tolocalised states even with constant V inside the silicon.

quantum–mechanical transverse kinetic energy of the electron, andconsequently the energy available for longitudinal motion is reduced.If the constrictions are sufficiently narrow (and the distance betweenthem is not to short) this may give rise to quasi–bound states betweenthe constrictions.This effect must be included in the construction scheme for the

transfer Hamiltonian; it results in a change of the class of allowablepaths11 in the computation of the escape energy. We are still onlyinterested in paths, by which the particle may leave the potential wellwithout tunnelling; therefore the extension of the electron wave func-tion along the classical direction of the electron path is disregarded.For an electron at classical position x and propagation direction v wecompute the (d − 1)–dimensional ground–state wave function |0〉(p)in the plane p :=

r∣∣ (x− r) · v = 0

with the restriction of the

Hamiltonian to p

H|p = −2

2

(∇ ·[1m∗

]∇− ∂

∂v1m∗v

∂

∂v

)︸︷︷︸

=:Tv

+V (x). (3.51)

The transverse kinetic energy εtrans (x, v) for a particle at x with clas-sical direction v then is defined as the expectation value of the trans-verse kinetic energy operator Tv

εtrans (x, v) := (p)〈0|Tv |0〉(p) (3.52)

=ε(p)0 − (p)〈0|V |0〉(p) . (3.53)

11Although we know that the electron is in fact delocalised, for technical reasonsit is still convenient to assign to it a “classical” position and propagation direction.


In order to enter a point x from direction v, a particle must possesthis extra energy εtrans (x, v) in addition to the normal potential en-ergy V (x). Paths that traverse the constriction “diagonally” in or-der to lower the transverse kinetic energy are unphysical and arisefrom the artificial distinction between transport direction and trans-verse directions that was introduced to prohibit tunnelling throughthe barrier. Hence, the energy necessary to reach a point x is thequantum–corrected potential

V (x) := V (x) + maxv

εtrans(x, v). (3.54)

The allowable paths for the construction of Ωdot are laterally confinednon–tunnelling paths as defined below.

Definition 2

A laterally confined non–tunnelling path (LCNTP) of energy ε from ato b is a differentiable map γ : [0, 1] → Ω, γ(0) = a, γ(1) = b suchthat V (γ(s)) + maxv εtrans

(γ(s), ddsγ(s)

)< ε ∀s.

By comparison with definition 1 it can be seen that these paths arejust the classical paths in the modified potential landscape V (x).The quantum corrected escape energy then is defined as

εesc = inf ε| ∃ a LCNTP of energy ε from xin to xout , (3.55)

and the interior of the quantum well is given by

Ωdot = x ∈ Ω| ∃ a LCNTP of energy ε < εesc from xin to x .(3.56)

Whereas these definitions are identical to the definitions in the “sim-ple” treatment up to replacing the term classical trajectory with LC-NTP, the formula for the potential of the quantum–corrected transferHamiltonian Hdot actually differs from the expression found in (3.50).The reason for this difference is that in the “simple” case the samepotential field V was used for the construction of the paths and in theHamiltonian H. In the quantum corrected case, however, a differentfield V is used for determining the accessibility of points, and it is thisfield that must be kept above εesc outside of Ωdot in order to preventthe emergence of spurious localised states; the Hamiltonian, however,


still features V. Therefore, the condition on V must be translated toa condition on V, and thus we obtain

Vdot =V (x), x ∈ Ωdot or V (x) ≥ εescεesc −maxv εtrans(x, v), otherwise

. (3.57)

The success of this construction scheme is illustrated by figure 3.7.The undesirable states, which arise outside the potential well of thequantum dot when the original Hamiltonian H is used, are eliminatedaltogether. But shape end energies of the correctly localised statesare only negligibly affected by the transition from H to Hdot.

3.2.4 Transfer Hamiltonians and Charge Densities

Besides the computation of tunnelling rates, transfer Hamiltoniansmay be used to extend the range of applicability of the domain de-composition approach of section 3.1.3. In systems with quantum dotswhich are separated from neighbouring semiconductor regions by verythin dielectric layers, or in the presence of potential barriers, whosemaximum position shifts as gate voltages are changed, the naıve ap-plication of the decomposition approach may get into trouble: if thedomain for the quantum dot is chosen too small, the proximity of theboundary conditions may compromise the charge density inside thedot; if it is chosen too large, the Schrodinger solver might find spuri-ous solutions which are localised in the artificial potential well betweenthe tunnelling barrier and the hard wall of the Dirichlet condition.Without need to increase the volume of the Schrodinger box, the

situation may be remedied by using more physical boundary condi-tions such as the evanescent boundary condition (smooth transitionto a decaying WKB wave function)

∂

∂nlogψ = −

√− 1ψ∇⊥ ·([

1m∗

]∇⊥ψ)+2m∗⊥(V −E)

2. (3.58)

But with its non–linear dependence on the (initially unknown) eigen-energy E this condition cannot be enforced in the context of a ma-trix eigenvalue solver. For 1D problems, it is possible to resort toalternative methods of eigenpair construction such as the shootingmethod [32]; these methods, however, are unsuited for 3D problems.


Using the quantum dot transfer Hamiltonian Hdot instead of the re-striction of the global Hamiltonian to the 3D Schrodinger box of thequantum dot eliminates these spurious states by construction. Thus,only the true quantum dot charge density is computed. The homoge-neous Dirichlet condition imposed on the quantum dot wave functionscauses the quantum dot charge density to be zero at the boundariesof the 3D Schrodinger domain. In order to reduce the impact of theboundary conditions on the quantum dot wave functions, this domainboundary may lie a considerable distance beyond the maximum of thepotential barrier, that delimits the quantum dot. There, neigbouringcharge models (typically quantum wires) may already supply consid-erable charge. Therefore, in the overlap region of the 3D Schrodingerbox with the domains of neighbouring charge models the previouslystated hierarchy of charge density models (“the strongest confinementtakes precedence”, cf. section 3.1.3), must be relaxed: the quantumdot charge density must not simply overwrite the channel density, any-more. Instead, the charge densities provided by the quantum dot andthe quantum wire charge models must be combined in a physicallysensible way.A computationally expensive method but generally valid method

to do this consists in computing the charge density injected from thefar end of the channel by means of the open boundary 3D Schrodingerequation as discussed in section 4.3.6.For quantum wires with slowly varying cross–section, a computa-

tionally less involved method based on a local equilibrium approxi-mation for the charge density may be used. The charge density iscomputed by evaluating the expression for n1DEG in eq. (3.23) oneach cross–sectional plane of the quantum wire. In the overlap region,correct decay properties of the wire charge density are imposed man-ually by making the quantum wire non–increasing along the directionof the interior normal vector of the boundary face of the quantum dotintersected by the quantum wire. Alternatively, the quantum wirecharge density could be computed directly using 2D slices of the po-tential of the quantum wire transfer Hamiltonian for the computationof the 2D wave functions.With each of these approaches the total charge density in the over-

lap region is the sum of quantum dot charge and quantum wire charge.


a) b)

280 300 320

x [nm]

0.04

0.05

0.06

0.07

ener

gy [e

V]

280 300 320

x [nm]

0.04

0.05

0.06

0.07

Figure 3.7: One–dimensional cuts through the eigenstates ofa) the original Hamiltonian Hb) the improved transfer Hamiltonian Hdot[note the suppression of the spurious states by Hdot].


3.3 Conductance Extraction

If the lifetime of the (quasi–)localised quantum–dot states exceeds theinelastic scattering time inside the quantum dot, the dominant mecha-nism for charge transport from source to drain is sequential tunnellingthrough the quantum dot. For a given occupation configuration (i.e.a fixed vector of occupation numbers ni of the single–particle orbitalsof the quantum dot) this transport process is schematically depictedin fig. 3.8. In order for electron transport through a single–particleorbital |α〉 in the quantum dot (with total electron number N) totake place, the orbital must initially be unoccupied nα = 0. Then,the source must supply an electron of energy ε′α, which is the energyneeded to add an electron to the originally unoccupied orbital |α〉.Finally, there needs to be an unoccupied final state in the drain. Inthe constant interaction model — the interaction energy is assumedto depend only on the total electron number N not the individualoccupation numbers nα of the single–electron orbitals — this may be

EsrcF

EdrnF

Γsrc3 Γdrn3

ε1

ε2

ε3

ε4ε′3

ε′4

Vsrc-drn

∆U

∆U no free state

no occupied state

Figure 3.8: Sequential tunnelling mechanism for the transport of anelectron through a fixed occupation configuration of the quantum dotstates.

3.3. CONDUCTANCE EXTRACTION 35

approximated by

ε′α,ni ≈ ε(N)α +∆UN→N+1, (3.59)

where ε(N)α is the eigenenergy of the Kohn–Sham orbital |α〉 at totalelectron number of N assuming Fermi–Dirac filling of the individualorbitals according to [18], and ∆UN→N+1 is the change in Kohn–Shamtotal energy of the quantum dot as the electron number is increasedfrom N to N + 1 subject to the same assumption.The current onto the quantum dot may be written as a weighed

sum over all possible single–particle occupation configurations ni as

Iin = −e∑ni

P (ni)∑α

Γsrcα f(−β(ε′α,ni −EsrcF )

)δni,0 (3.60)

The corresponding expression for the current out of the quantum dotis

Iout = −e∑ni

P (ni)∑α

Γdrnα

(1− f(β(ε′α,ni − EdrnF )

))δni,1

(3.61)

The density of states of the leads is already incorporated into thetunnelling rates Γ.If the tunnelling rates Γ for the individual barriers are low — this

is a necessary condition for device operatation in Coulomb blockaderegime — inelastic scattering events will thermalise the electron dis-tribution inside the quantum dot, before the second tunnelling processtakes place. Therefore, in the limit of strong scattering, it is possibleto employ equilibrium statistical mechanics on the quantum dot. Thecondition that the total current onto the quantum dot and the totalcurrent off the quantum dot should cancel may be used to define achemical potential for the quantum dot.In the general case, computation of the quantum dot chemical po-

tential that results in current conservation is computationally expen-sive. Evaluation of the phase space average for Iin and Iout is expensiveeven for a single value of µdot. Solving a non–linear equation in µdoteven more so — not even analytical derivatives for P (nα = 1|N) areknown. Also, with increasing source–drain voltage the assumption of


full thermalisation of the electron gas on the quantum dot becomes in-creasingly questionable. Therefore, the treatment of the conductanceof single–electron transistors is restricted to the linear response regime(infinitesimal difference between EsrcF and EdrnF ). After lengthy cal-culation that has been reported elsewhere [9, 13] the linear responseconductance of an SET is found to be

G =dIdV

∣∣∣∣V=0

=e2

kBT

∑nα

∑α

Γsrcα Γdrnα

Γsrcα + Γdrnα

Peq(ni)δnα,0f

(β(ε′α,ni −EF)

).

(3.62)

In the context of the constant interaction model, this expression sim-plifies to

G =e2

kBT

∞∑N=0

Peq(N)∑α

Γsrcα Γdrnα

Γsrcα + Γdrnα

×

× Peq(nα = 0|N)f(β(ε′α,ni −EF)

), (3.63)

or, equivalently,

G =e2

kBT

∞∑N=0

Peq(N)∑α

Γsrcα Γdrnα

Γsrcα + Γdrnα

×

× Peq(nα = 1|N)(1− f(β(ε′α,ni − EF)

)). (3.64)

Expression (3.63) describes the current from the source onto the quan-tum dot, (3.64) the drain current, both of which are identical by con-struction. In (3.63) the tunnelling rates Γα are evaluated using thesingle–particle orbitals |α〉(N+1), since these are the orbitals that con-stitute the target state. Accordingly, in (3.64), the |α〉(N) have to beused, since these occur in the initial state.For small electron numbers, the conditional probability Peq(nα =

1|N) for the occupation of single–particle orbital α, provided that thetotal electron number is N , differs in shape from the Fermi–Diracdistribution f

(β(εα − εF )

)In the limit of large N it approaches a

3.4. THERMAL AVERAGES 37

shifted Fermi–Dirac distribution. An efficient numerical method forthe computation of P (nα = 1|N) is described in section 3.6. This wasused in the computation of the statistical terms in the conductanceformulae; for the computation of the charge densities associated withthe various possible N on the dot, shifted Fermi–Dirac occupationfactors were assumed.

3.4 Thermal Averages

The thermal average of the expectation value of an observable A ina quantum–mechanical system described by a Hamiltonian H is givenby

〈A〉 = tr (AP) , (3.65)

where P is the equilibrium statistical operator.For a grand canonical ensemble P takes the form

Pg.c. =1Ξe−β(H−µN), (3.66)

while for a canonical ensemble it reads

Pcan =1Ze−βH. (3.67)

Here, Ξ = tr (Pg.c.) and Z = tr (Pcan) denote the grand canonical andcanonical partition functions, respectively; β = 1/kBT , and N is thetotal particle number operator.If H and N commute, there exists a basis of simultaneous eigen-

states of N and H. Consequently the grand canonical statistical oper-ator simplifies to

Pg.c. =1Ξe−βHeβµN

=1Ξ

⊕N

eβµNe−βHN , (3.68)

where HN denotes the restriction of the system Hamiltonian to thesub–space of N–particle states. That means, that the grand canonical


state may be regarded as a statistical mixture of canonical states. If Aconserves the total particle number, the grand canonical expectationvalue of A can be expressed by the expectation values of A in thevarious canonical states as

〈A〉g.c. =∑N

P (N) 〈A〉N , (3.69)

where 〈A〉N denotes the expectation value of A in the canonical statewith total particle number N , and P (N) is the probability for theoccurrence of this total particle number in the grand canonical state.In a (simultaneous) eigenbasis |α〉 of the Hamiltonian H (and the

number operator N) with eigenvalues εα (Nα)

H |α〉 = εα |α〉 (3.70)(N |α〉 = Nα |α〉

)(3.71)

and expectation values

〈α|A |α〉 = 〈α|Aα |α〉 , (3.72)

the equilibrium statistical operator becomes diagonal.12

Thus, taking the trace of the operator product A exp (−βH) reducesto a simple sum

tr(Aexp (−βH)) =∑

α

〈α|A |α〉〈α| exp (−βH) |α〉

=∑α

Aα exp (−βεα) . (3.73)

Then, the expectation value of A for a canonical ensemble takes theform

〈A〉can =∑

αAα exp (−βεα)∑α′ exp (−βεα′)

, (3.74)

12Taking the matrix exponential (or more generally any analytical function ofan operator) preserves diagonality, since all powers of a diagonal operator arediagonal.

3.5. STATISTICAL MECHANICS OF THE QUANTUM DOT 39

whereas for a grand canonical ensemble we obtain

〈A〉g.c. =∑

αAα exp(−β (εα − µNα)

)∑α′ exp(−β (εα′ − µNα′)

) , (3.75)

which may be re–arranged in accordance with eq. (3.69) as shownbelow.

3.5 Statistical Mechanics of the QuantumDot

Now we apply the results of the preceding section to the quantumdot of our SET. Assume that the energy of the system (consisting ofthe quantum dot and its environment) can be decomposed into a partthat depends on the occupation of the individual levels and a partthat depends only on the total electron number in the quantum dot13

Esystem(ni) = ε(ni) +E(∑i

ni). (3.76)

Density functional theory strictly yields only the ground state en-ergy and density, and this ground state is characterised by the occu-pation number vector

ni =1, i < N0, i ≥ N

. (3.77)

Nevertheless, arbitrary occupation number vectors ni|ni ∈ 0, 1 ∀ihere are treated as eigenstates of the the many–particle HamiltonianH.Within this approximation, both the statistical operators and the

orbital electron number operators become diagonal in the ni base.Then, the equilibrium probability terms in Beenakker’s conductanceformula (3.62) may be rewritten as shown below.

13By the constant interaction model, all interaction terms — either within thequantum dot or between the quantum dot and the environment — will later beregarded as part of the latter term.


Probability for a fixed vector of occupation numbers ni:

Peq(ni) = 1Ξe−β(ε(ni)+E(

∑i ni)−µ

∑i ni), (3.78)

where the grand canonical partition function Ξ takes the form

Ξ =∑N

∑niN

e−β(ε(ni)+E(∑

i ni)−µ∑

i ni)

=∑N

eβµN∑niN

e−β(ε(ni)+E(∑

i ni))

︸︷︷︸canonical partition function Z(N)

=∑N

Z(N) eβεµN

=∑N

e−β(F (N)−µN). (3.79)

Here, F (N) = −kBT logZ(N) denotes the Free Energy for thecanonical N–particle state. For notational convenience we haveintroduced the symbol niN :=

ni∣∣∑i ni = Nfor the

sub–space of N–particle states.

Please note, that — despite superficial similarity — the quantityω(T, µ,N) := F (N) − µN in the exponent of (3.79) is not thegrand canonical potential

Ω(T, µ) = F(T, V,N(T, µ)

)− εFN(T, µ). (3.80)

In the grand canonical potential Ω, N = N(T, µ) is implicitlydefined by specifying T and µ. ω, however, explicitly dependsboth on µ and on N .

3.6. MONTE–CARLO / EXPECTATION VALUES 41

Probability that the system has total particle number∑i

ni = N :

Peq(N) =∑ni

Peq(ni) δ∑i ni,N

=1Ξ

∑ni

e−β(ε(ni+E(N)−µN)δ∑i ni,N

=1Ξ

∑niN

e−β(ε(ni+E(N)−µN)

=e−β(F (N)−µN)∑

N ′e−β(F (N

′)−µN ′). (3.81)

Probability that the single–particle orbital |k〉 is occupied, pro-vided that the total electron number is N :

Peq(nk = 1 |N) = 1Peq(N)

∑ni

Peq(ni)δnk,1δ∑

i ni,N

=

∑ni

Peq(ni)δnk,1δ∑

i ni,N∑ni

Peq(ni)δ∑i ni,N

= eβ(F (N)−E(N))∑niN

e−βε(ni)δnk,1. (3.82)

3.6 Monte–Carlo Evaluation of Thermal

Expectation Values

As shown in the preceding section, the probabilities P (ni = 1|N) canbe evaluated as phase space averages 〈ni〉N over the N–particle sub–space of the phase–space of the quantum dot. Likewise, the P (N) ∝exp(−β(F (N)−Nµ)) may be expressed in terms of phase–space av-

erages 〈·〉, by re–writing the Free Energy F (N) = −kBT logZ(N)


as

F (N) = 〈E〉N − T · S(N) (3.83)

in terms of the energy expectation value 〈E〉N and the entropy S(N) ofthe canonical N–particle state. The entropy may in turn be expressedas an expectation value as shown below.

S(N) =1T

(〈E〉N − F (N))=1T

( ∑niN

P(ni|N)E(ni))+ kB logZ(N) (3.84)

Since∑niN

P(ni|N) = 1, the second term may be included in

the sum to yield

S(N) = −kB∑niN

P(ni|N) (−βE(ni)− logZ(N))

= −kB∑niN

P(ni|N) log( 1

Z(N)e−βE(ni))

= −kB∑niN

P(ni|N) logP(ni|N). (3.85)

In this expression we recognise the structure of an expectation value,and thus

S(N) = −kB⟨log(P (ni|N)

)⟩N. (3.86)

In order to make the task of phase–space averaging tractable, constantinteraction is assumed inside each N–particle segment of phase–space,i.e. EH , Exc and Exc−pot are assumed to depend only on N =

∑i ni,

not on the specific occupation number configuration ni. Thus, thedifference in energy between two occupation number configurationsni at equalN reduces to the difference in Kohn–Sham orbital energyEorbital

(ni) =∑i niεi,N .Still, the computational effort for direct evaluation of 〈·〉N will be

prohibitive for large electron numbers, especially at elevated temper-ature. Then, the number of single–particle orbitals #orb that need


to be taken into account will be considerably larger than the num-ber of electrons, thus making the number of possible configurations,card(niN) = (#orbN

), very large. This has given rise to approxi-

mations, e.g. restriction to a T = 0 ground–state (“Slater rule”) orreplacement of the Free Energy F with the internal energy E and useof a shifted Fermi function (with εF chosen such that the total electronnumber N is reproduced) instead of the Gibbs distribution.This called for the development of a Monte–Carlo (MC) sampling

scheme that allows for full evaluation of the statistical mechanics ofthe quantum dot at moderate computational effort even in the case ofvery large phase spaces [33]. Since a properly designed Monte–Carloscheme automatically detects the important segments of the phasespace [34, 35], it eliminates the need for ad hoc approximations, andtherefore allows us to study the validity of customary approximations.The thermal average of A in the canonical state with total electron

number N may be written as

〈A〉N =∑niN

P(ni|N)A(ni)

=1

card(MN )

∑ni∈MN

A(ni), (3.87)

provided that the set MN is chosen in such a way, that the num-ber of occurrences of a micro–state ni in MN is proportional toP(ni|N). It has been shown by Metropolis [34] that this condition

is automatically met, if MN is constructed as a sequence of statess1, s2, . . . ∈ niN according to the following rules:

Transition: sn → sn+1:

1. Choose a random state s ⇒ energy: ε(s).

2. Accept s as sn+1 with probability

P =

1, if ε(s) < ε(sn),

exp(− ε(s)−ε(sn)

kBT

), otherwise.

3. If s is not accepted, go back to step 1.


The finite sub–sequences MnN := s1, . . . , sn of MN may be used to

obtain estimates 〈A〉nN for 〈A〉N .When estimating the error

∣∣〈A〉nN − 〈A〉N ∣∣, it is important to notethat the samples si are correlated. Therefore, the error is augmentedby a factor of

√2τ + 1 relative to the uncorrelated case, τ being the

correlation “time” of the sequence A(si). In our implementation, onlytransitions that transfer a single electron to a different orbital are con-sidered. This results in typical correlation times of about 4 sequencesteps; the worst correlation time observed was about 50. A less restric-tive transition matrix might help reduce correlation times and thusspeed up convergence. But in contrast to the deterministic evaluationof phase space averages the Monte–Carlo scheme never dominated thetotal simulation time; thus, little need was felt for optimisation at thisstage.The algorithm has been tested between N = 1 and N 100 (i.e.

in phase–spaces with more than 1025 micro–states!). Fig. 3.9 shows acomparison of the occupation numbers for a system of 84 electrons ina QD obtained with the MC method and with a Fermi distribution,respectively. The self–consistent chemical potential in the latter tookits minimum value w.r.t. a variation of the gate voltage. Thus, theoffset of the Fermi curve from zero is related to the self–capacitance ofthe QD. The agreement in fig. 3.9 is excellent and proves the accuracyof the MC scheme.


-0.001 -0.0005 0 0.00050

0.5

1

1.5

2

-0.006 -0.004 -0.002 0 0.002

E - EF [eV]

0

0.5

1

1.5

2

Occ

upat

ion

prob

abili

ty (

x 2)

Figure 3.9: Occupation number (including spin degeneracy) for a sys-tem of 84 electrons calculated by MC integration (circles) and fromthe shifted Fermi distribution corresponding to N= 84 (lines).


3.7 Simulation Results

3.7.1 Simulation of a Silicon–on–InsulatorSingle–Electron Transistor

The simulations in this section are modelled on an experimental de-vice manufactured at the Universitat Tubingen [27]. The simulationgeometry is depicted in figure 3.10. The active region consists ofa silicon nano–wire with two constrictions defined by electron beamlithography and subsequent seize reduction by oxidation. Becauseof the increased lateral confinement energy, the constrictions in thequantum wire serve as tunnelling barriers, and a quantum dot 20 nmin diameter is formed between them. The quantum dot is capacitivelycontrolled by a pair of side gates. The whole structure is populatedwith electrons by applying a positive bias of 5.2V to the back gate.All silicon in this structure is doped with 3× 1018 cm−3 of arsenic.

Figure 3.10: Simulation geometry of the single–electron transistorof [27]. Both the silicon surface and an iso–surface of the charge den-sity are shown to illustrate quantum depletion. The whole structure issurrounded by oxide and sits on top of a silicon substrate serving as aback gate (not shown).

3.7. SIMULATION RESULTS 47

Figure 3.11 shows the Coulomb charging staircase and the linear re-sponse conductance curve of the silicon–on–insulator (SOI) single–electron transistor (SET). In this plot, the tunnelling rates were notincluded in the computation of the conductance, because contrary toexperiment each conductance peak in the simulated curve was aboutone order of magnitude higher than the previous one. This suggeststhat the tunnelling barriers in the simulated device are lower than inthe experimental device. The input data for the simulation geometryconsisted solely of a two–dimensional micrograph (fig. 3.1 on page 19)of the device and capacitance estimates. Therefore, it is not surpris-ing that tunnelling rates with their exponential dependance on bothbarrier height and width should not be reproduced. It seems likelythat the quantum dot is not solely defined by the geometry — in the

0

1

2

3

4

5

cond

ucta

nce

[a.u

.]

10 K4.2 K

−2.5 V −2.4 V −2.3 V −2.2 V0

1

2

3

4

5

#ele

ctro

ns

a)

b)

Figure 3.11: a) Coulomb charging staircase and b) conductance curveof the SOI SET.


simulated device the quantum dot potential well cannot hold 12 elec-trons in quasi–bound states as suggested by the experiment. Probablyimpurities or stray charges have increased the effective barrier heightresulting in a more uniform peak height.

3.7.2 Tunnelling Rates and the Anisotropy of m∗

The effective mass anisotropy has a pronounced effect on the shapeof the wave functions: depending on the orientation of the reciprocaleffective mass tensor their spread along the transport direction variesso strongly that the tunnelling rates of corresponding states in differ-ent valleys diverge by up to 8 orders of magnitude. Figure 3.12 showsthe eigenenergies and tunnelling rates associated with the differentsingle–particle wave functions of the quantum dot. Where applica-ble, particle–in–a–box quantum numbers nxnynz are used to labelthe wave functions; nx is the number of lobes along the transport di-rection, ny the number of lobes in the horizontal and nz in the veticaltransverse direction. For wave functions that cannot be classified inthis way, an iso–surface of the probability density is shown.There is strong suppression of tunnelling for ny = 2 states rela-

tive e.g. to nz = 2 states. This results from the symmetry of thestructure in y–direction: at ny = 2 the maximum of the symmetric|nynz〉 = |00〉 wave function of the dominant subband in the chan-nel coincides with a node of the quantum dot wave function. Inz–direction the situation is different. The quantum wire enters thequantum dot in the cylindrical bottom section, but is centred alongthe y–axis (cf. fig. 3.10). Therefore, there is no suppression of tun-nelling for wave functions that have even nz . The straight lines joiningseries of states (e.g. 111–211–311–411–511 for the m∗max = mx orien-tation) correspond to an exponential increase of Γ with single–particleenergy.


0.03 0.04 0.05 0.06 0.07 0.08Energy [eV]

10−8

10−6

10−4

10−2

100

102

104

106

108

Tun

nelli

ng r

ate

Γ [

s−1 ]

m*max = m*x

m*max = m*y

m*max = m*z

111 211

121

221

121

121

ny=2

311

411

511113112

212

Figure 3.12: Source–dot tunnelling rates of the single–particle wavefunctions (particle–in–a–box quantum numbers nxnynz shown whereappropriate).


3.7.3 A Split–Gate III–V Hetero–Structure Single–Electron Transistor

Whereas in the silicon device, quantum dot and wires of the SET weredefined by structuring the silicon, in this device a two–dimensionalelectron gas (2DEG) forms at the hetero–interface between the GaAssubstrate and the AlGaAs layer separating it from the δ–doping layer.This 2DEG is locally depleted by means of negatively biased surfacegates. Gate layout and simulation geometry are shown in figure 3.13.If negative bias is applied to all six gates, it operates as a single–electron transistor (cf. fig. 3.14). Alternatively, only a single gate–pairmay be used to deplete the 2DEG. Then, a quantum point contact(QPC) is formed. The simulation results for this mode of operationare shown in section 4.2.2.Charging staircase and conductance for the GaAs/AlGaAs hetero–

structure operated in SET mode are shown in figure 3.15. The increasein peak conductance with increasing electron number is in agreementwith the experiment. So are the capacitances extracted from mea-sured and simulated data. The absolute values for the conductance,however, are far off. This is due to incomplete knowledge of the bound-ary conditions of the device: it is not known how the Fermi energy

Figure 3.13: FEM image (courtesy A. Forchel) and simulation ge-ometry of the GaAs/AlGaAs split gate structure manufactured at theUniversitat Wurzburg.


Figure 3.14: Charge density inside the GaAs/AlGaAs hetero–structurewhen operated as an SET.

varies parallel to the device surface at bias conditions far from equilib-rium. It was tried to calibrate the surface energy for use in the SETsimulations by comparison of simulated and measured conductancesof the QPCs. But it turned out that QPC conductance is consider-ably less susceptible to changes in the surface energy than the SETconductance. Therefore, QPC conductance data insufficient for thecalibration of an SET simulation.For the SET simulation it was assumed that the first conductance

peak corresponds to the N = 0 → 1 transition in the quantum dot.Consequently, the surface pinning energy was adjusted to a value suchthat14 the first electron is found inside the quantum dot in the vicinityof Vgate = 0.75V. In SET operation mode minor alterations of thesurface energy were seen to cause strong changes in the occupation ofthe quantum dot — modifying the surface energy by as little as 2mVwould change the electron number on the quantum dot by 1. Thisshows that large–scale integration of such devices would require very

14with the assumption of Fermi–Dirac occupation factors


-1.8 -1.6 -1.4 -1.2 -1 -0.8gate voltage [V]

33.5

44.5

55.5

66.5

77.5

88.5

99.510

10.511

11.512

12.513

13.514

14.515

#ele

ctro

ns

-1.8 -1.6 -1.4 -1.2 -1 -0.8gate voltage [V]

10-44

10-42

10-40

10-38

10-36

10-34

10-32

10-30

10-28

cond

ucta

nce

[G0]

Figure 3.15: Charging curve (solid line) and conductance (dashed) ofthe GaAs/AlGaAs SET.

rigid control of such process details as surface pinning energies.Prior to starting the actual conductance simulations for the SET,

we ran a series of simulations on the change of the electron numberon the quantum dot (without inclusion of the discrete charging mech-anism) with both gate voltage and surface pinning energy. These sim-ulations were performed at a device temperature of 0.3K as reportedby the experimental group in Wurzburg [36]. At this low tempera-ture, however, convergence turned out to be difficult: for some voltagevalues convergence occurred readily, but minor alterations would leadto situations in which convergence stalled. This prompted us to in-crease the simulation temperature from 300mK to values of 1K and4.2K. At 1K the charge distribution was almost indistinguishablefrom the situation at 300mK; a temperature of 4.2K, however, re-sulted in significant alterations of the charge distribution and, hence,the electrostatic situation.Since simulations at 300mK and 1K produce almost identical elec-


−0.9 −0.8 −0.7 −0.6Control gate voltage (V)

0

2

4

6

elec

tron

num

ber

with Fermi functionfrom full statistical mechanics

Figure 3.16: Comparison of the electron number on the quantum dotobtained from the full partition function with the electron number thatis obtained by assuming Fermi occupancy of the Kohn–Sham orbitals.

trostatics, no significant alteration of the single–particle eigenstatesat equal total electron number in the quantum dot is expected. Thus,and in order to exploit the much faster convergence at 1K, the self–consistent single–particle orbitals obtained from a simulation at 1Kwere used for extracting Coulomb charging and conductance oscilla-tions both at 1K and 0.3K (cf. fig. 3.16 and fig. 3.17).Interestingly, when the simulated conductance data then was com-

pared to the experimental results, it turned out, that the shape ofthe first conductance peak with T = 1K perfectly matches the firstconductance peak of the experimental curve, whereas the same con-ductance peak computed with the reported experimental temperatureof 300mK is much narrower than the experimental peak (cf. fig. 3.18).From this it was inferred, that the actual temperature inside the de-vice during SET conductance measurement exceeds the 300mK of thecoolant inside the cryostat; thus, by simulations it was possible to ex-tract information on an internal physical quantity of the device thatis inaccessible by measurement.In fig. 3.17 the tunnelling rates Γ are excluded from the conduc-

tance calculation such that more conductance peaks can be displayedon a linear scale. If the effect of the variation of the tunnelling rates


−0.9 −0.85 −0.8 −0.75 −0.7 −0.65 −0.6Control gate voltage [V]

0.00e+00

2.00e−11

4.00e−11

6.00e−11

8.00e−11C

ondu

ctan

ce w

ithou

t Γ (

a.u.

) T = 0.3 KT = 1.0 K

Figure 3.17: Simulated SET conductance at different temperatures (ex-clusive of tunnelling rates).

with increasing electron number inside the dot is taken into account,there is a strong increase in peak height as the quantum dot gets fuller— from the first to the 18th oscillation the peak height increases by20 orders of magnitude.The height ratio of subsequent conductance peaks is about 50 for

low filling numbers. But for higher electron numbers this ratio de-creases: near N ≈ 15 we obtain peak height ratios similar to those inthe experimental curve (cf. fig. 3.19 — the conductance axis is still la-belled with [a.u.] because of the incomplete knowledge of the pinningenergy and its strong influence on the conductance.15

These observations suggest that the onset of conduction in theexperimental curve is in fact not identical with the N = 0 → 1 tran-sition of the quantum dot: while being energetically allowed, the lowN conductance peaks are kinetically suppressed due to the minisculetunnelling matrix elements that arise from the small effective size ofthe quantum dot at low filling.

15The apparently better agreement with experimental data of the absolute con-ductance values of the simulations contained in [13] resulted from a bug in theimplementation of the tunnelling rates.


−0.74 −0.72 −0.70 −0.68control gate voltage (V)

0.00

0.05

0.10

0.15

cond

ucta

nce

(2e2 /h

)

measurementsimulation (1 K)simulation (0.3 K)

Figure 3.18: Shape of the first conductance peak: comparison betweenexperiment and simulation results for T = 300mK and 1K (simula-tion data shifted and scaled). Outer gate bias: −1.1V.

-0.45 -0.4 -0.35Control gate voltage [V]

0

1×10-33

2×10-33

3×10-33

Con

duct

ance

[a.u

.]

-0.45 -0.4 -0.35Control gate voltage [V]

10

12

14

16

18

Ele

ctro

n nu

mbe

r

Figure 3.19: light line: charging curve; heavy line and circles: SETconductance (including tunnelling rates).

Chapter 4

Quantum–BallisticTransport

Quantum–ballistic transport is phase conserving transportwithout inelastic scattering processes: electrons injectedinto the device through a contact are propagated coher-ently until they re–emerge through either the same or a

different contact. This type of transport requires that the mean freepath between inelastic scattering events be significantly greater thanthe device extensions. Please note that despite the similarity in namethis transport mode is different from quasi–ballistic transport, a trans-port regime, in which mean free path and device extensions are similarin magnitude — in this mode scattering is very important and is ex-plicitly accounted for by solving the Boltzmann transport equation orthe Wigner–Boltzmann equation.There is ample experimental evidence for quantum–ballistic trans-port (e.g. [37, 38, 39, 40]). In III–V high electron mobility tran-sistor (HEMT) structures quantum–ballistic transport has been ob-served over distances of several microns. Prerequisite for such largecoherence lengths are epitactic material stacks of very high quality inconjunction with non–local modulation doping [41].In modulation doping a very thin layer of doped semiconductor is

incorporated into the material stack in the vicinity of a quantum well.

57

58 CHAPTER 4. QUANTUM–BALLISTIC TRANSPORT

Because of the small extensions of the doping layer (high localisationenergy required!) it is energetically more favourable for the electronsto reside in the quantum well than in the doped layer. This results inthe formation of a two–dimensional electron gas (2DEG) at a high–quality (epitactic!) interface of undoped material layers. Thus theoccurrence of inelastic scattering events is much reduced in comparisonwith locally doped structures; this leads to a strong increase in theelectron coherence length.

4.1 Theory of Quantum BallisticTransport

The treatment of conductance as the quantum–ballistic transmissionof electrons has become known by the name of Landauer–Buttikerformalism [42, 43, 44, 45]. The expectation value for the particlecurrent1 through a (d−1)–dimensional hyper–surface Q in a quantumsystem with d spatial dimensions is given by

〈I〉 =∫Q

dd−1x n · tr(j (x)P), (4.1)

where j (x) is the second quantised current density operator at posi-tion x, and P is the statistical operator of the system. In contrast tothe treatment in section 3.5, here we are using non–equilibrium statis-tical operators, since the grand canonical state of a quantum systemdescribed by the equilibrium statistical operator exp

(−β(H − µN))

carries zero current.For an arbitrary basis of single–particle states |i〉 with correspond-

ing creation and annihilation operators c†i and ci , the second–quantisedcurrent operator takes the form

j (x) =∑i,j

c†i cj 〈i|j 1(x)|j〉 . (4.2)

Here, j 1(x) denotes the current operator for a single–particle state. Bythe correspondence principle, this operator must be the self–adjoint

1The electrical current is obtained by multiplying with the charge of the par-ticles.

4.1. THEORY OF QUANTUM BALLISTIC TRANSPORT 59

operator analogue of the classical current density expression

j(x) = ρ(x)v(x) =ρ(x)p(x)

m. (4.3)

The particle density ρ(x) corresponds to the projection operator ρx =|x〉〈x| onto the position eigenstate |x〉 (in position representation thisis a Dirac–δ at x), and the momentum operator takes its usual formp =

i∇. Thus, the one–particle current density operator reads

j 1(x) =12m(p† |x〉〈x|+ |x〉〈x|p) , (4.4)

or alternatively

j 1(x) =

m|x〉〈x| ∇. (4.5)

From the form (4.5) it can be seen that it is the phase gradient∇ arg(ψ) = 1/ψ(∇ψ) of the wave function that drives the cur-rent. Hence, a purely real wave function carries zero current, whereasthe current density of a plane wave aeik·x equals j(x) = |a|2k/mregardless of the choice of x.In a semiconductor, wave functions are more likely to be expanded

in terms of Bloch functions than in a plane wave basis. Therefore, itis useful to look at the Bloch function

〈x |ν k〉 = uν,k(x) exp(ik · x). (4.6)

The mean velocity of an electron in this Bloch state is (cf.Appendix B)

vν(k) =

m〈ν k|1

i∇|ν k〉 = 1

∇kεν(k), (4.7)

where εν(k) denotes the dispersion relation inside the νth band. Thisis equal to the group velocity of a wave packet centred around k.For a parabolic band structure

ε(k) = εk0 +2

2(k− k0) ·

[1m∗

](k− k0), (4.8)


the velocity becomes

v(k) =[1m∗

]k =

crystal momentumeffective mass

. (4.9)

Within this approximation, the correct current is obtained by apply-ing the single–particle current operator (4.4) to the envelope wavefunction from the effective mass equation rather than the true wavefunction, if simultaneously all occurrences of 1m are replaced with thereciprocal effective mass tensor

[1m∗].

For the construction of an explicit form of the current expression(4.1) it is useful to introduce an eigenbasis of the (effective) single–particle Hamiltonian h1. For easier handling of the injection condi-tions at the terminals2 of our system it is useful to choose the basisinside each of the degenerate eigenspaces of h1 in such a way that thebasis vectors satisfy scattering boundary conditions (cf. section 4.3.3).This allows us to classify each single–particle wave function by its en-ergy, by the terminal from which it is injected, and (except for one–dimensional systems) by its transverse structure at a selected termi-nal, its spin, and (in indirect semiconductors) its valley index. For theremainder of this section the vectors of this special eigenbasis of h1will be denoted by |i α〉, where α indicates the injecting terminal (e.g.α ∈ src, drn for a two–terminal device) and i is a collective label forthe remaining quantum numbers.Now, each terminal α is allowed to supply particles to the system ata different chemical potential µα. All reservoirs are assumed to havea common temperature T (β = 1/kBT ). Then, in the absence ofscattering the statistical operator takes the form

P =1Ξexp(−β(H−∑i,α

c†i,αci,α µα))

︸︷︷︸=:P

, (4.10)

with the normalisation factor Ξ := tr(P).2A terminal α is a section of the system boundary with boundary conditions

that allow current flow normal to the boundary. Often these boundary conditionsare chosen as though an ideal waveguide with full translational invariance alongits transport direction connected the system to a particle reservoir of chemicalpotential µα.


The expectation value for the current density at position x can bewritten as

〈j (x)〉 = tr (j (x)P)= tr( ∑

i,α; j,β

c†i,αcj,β 〈i α|j 1(x)|j β〉P)

=∑

i,α; j,β

〈i α|j 1(x)|j β〉 tr(c†i,αcj,βP

)(4.11)

By introducing the one–particle density matrix M with matrix ele-ments

Mi,α; j,β := tr(c†i,αcj,βP

), (4.12)

this may be rewritten as the trace3 of the product of j 1(x) with thetranspose MT of the one–particle density matrix.For non–interacting electrons4, it is possible to compute the one–particle density matrix for the density operator (4.10) in closed form.The Hamiltonian of the interaction free system has the form

H =∑i,α

c†i,αci,αεi. (4.13)

It gives rise to a density operator

P =1

tr(P)exp(−β∑i,α

c†i,αci,α(εi − µα))

︸︷︷︸=:P

. (4.14)

Since this density operator contains only pairs of creation and anni-hilation operators with identical quantum numbers, the off–diagonalterms of the one–particle density matrix vanish:

Mi,α; j,γ = ni,α δi,jδα,γ . (4.15)

3This trace is evaluated over the space of single–particle wave functions,whereas in (4.10) and (4.11) the trace is taken over the whole Fock–space.

4According to [45], this treatment may still be used to handle interaction onthe level of a mean–field description.


This simplifies the expression for the expectation value in (4.11) to

〈j (x)〉 =∑i,α

ni,α 〈i α|j 1(x)|i α〉 . (4.16)

The average occupation numbers ni,α can be computed from the den-sity operator (4.14) as follows:

tr(P) =∑i,α

1∑ni,α=0

∏j,γ

e−βnj,γ(εj−µγ)

=∏j,γ

1∑nj,γ=0

e−βnj,γ (εj−µγ), (4.17)

ni,α =tr(c†i,αci,αP

)tr(P)

=1

tr(P)

∏(j,γ) =(i,α)

1∑nj,γ=0

e−βnj,γ (εj−µγ )e−β(εi−µα)

=e−β(εi−µα)

1 + e−β(εi−µα)

= f(β(εi − µα)

). (4.18)

Thus, the average occupation of each single–particle orbital is just theFermi function evaluated with the chemical potential of the injectingterminal.For a two–terminal system, α ∈ src, drn, (in the absence of magneticfields) the current densities 〈i src|j 1(x)|i src〉 and 〈i drn|j 1(x)|i drn〉are equal in magnitude but opposite in direction. This allows us todefine ji(x) = 〈i src|j 1(x)|i src〉 and rewrite (4.16) as

〈j (x)〉 =∑

i

ji(x)(f(β(εi − µsrc)

)− f(β(εi − µdrn))). (4.19)

Next, we turn to the computation of the current expectation values〈i src|j 1(x)|i src〉. Usually (cf. e.g. [46]), it is assumed that the ter-minals, that couple our system to the reservoirs, can be described as


αβ

γ

eα

eβ

eγ

system

Figure 4.1: The simulation domain (“system”) is coupled to the reser-voirs by ideal waveguides.

perfect waveguides (cf. fig. 4.1). This means that inside the associatedwaveguide a wave function injected from terminal α may be written asa sum of products of a plane wave propagating along the waveguidedirection eα with a transverse wave function. From the scatteringboundary conditions it is known, that each wave function inside itsinjecting waveguide contains only a single wavevector pointing intothe device (k · eα > 0); but due to scattering inside the device, theremay be multiple contributions (at most one per transverse mode) atwavevectors pointing out of the device. Consequently, each wave func-tion may be identified by specifying its injection terminal α and bothits transverse mode t and its incident wavenumber kt

‖,α > 0 insidethe injecting waveguide. This wavenumber can immediately be deter-mined from the longitudinal kinetic energy in the injecting waveguide

εkin‖,α,t := ε− εtα, (4.20)

where εtα denotes the subband energy of the transverse mode t of


terminal waveguide α. εtα is a function of the total energy ε andthe quantum numbers α and t, that have been specified separately.Therefore, the double reference to injection terminal and subband in|α t kt

‖,α〉 may be eliminated by choosing the total energy as a newindependent quantum number and writing |α t ε〉.The wavenumber kt

‖,α (and consequently also the total energy ε) is acontinuous quantum number, giving rise to a Dirac–δ normalisation ofthe wave functions. Except for an infinitesimal contribution the wholeprobability density of each wave function resides in the infinitely longwaveguides outside the simulation domain5.For convenience, let us introduce an arbitrary normalisation length

L, such that the asymptotic amplitude of the incident plane wave inits supplying waveguide is 1/

√L. In addition, we attach coordinates

to the injecting waveguide α, such that x is the coordinate alongthe transport direction eα and y and z are coordinates in the cross–sectional plane of the waveguide. Then, inside waveguide α, each wavefunction |t ε α〉 that is injected into the system from terminal α takesthe form

〈x y z | tα ε〉 = 1√L

( injected wave︷︸︸︷eik

t‖,α(ε)xψt(y, z)

+∑t′r(α,t)→(α,t′)(ε) e

−ikt′‖,α(ε)xψt′(y, z)︸︷︷︸

reflected waves

),

(4.21)

5This is important for the discussion of the assumption that wave functions in-jected from different waveguides are orthogonal. The wave functions are boundedinside the system. Therfore, the overlap of the wave functions inside the finitesystem volume is finite. Also finite is the overlap in each waveguide: the scatter-ing boundary conditions ensure that a wave function injected through waveguideα has no contribution at k–vectors pointing into the device at terminal α′ = α.The integral of exp(i∆k), ∆k ∈ R \ 0 over an arbitrary half–space is finite. Afinite overlap of δ–normalised functions, however, signifies orthogonal states (asdiscussed e.g. in [47, 48]). The only terms that might yield non–vanishing con-tributions are overlap integrals of outwards propagating waves in the waveguides,e.g. between reflected parts of a wave injected at terminal α and transmitted partsof a wave injected at terminal α′ = α. In the two–terminal case with symmetricpotential, it can be strictly shown, that these contributions exactly cancel (fromtime–reversal symmetry) [49].


whereas a wave function injected from terminal β = α may be writtenas

〈x y z | tβε〉 = 1√L

∑t′t(β,t)→(α,t′)(ε) e

−ikt′‖,α(ε)xψt′(y, z)︸︷︷︸

transmitted waves

, (4.22)

with the subband dispersion relations kt‖,α(ε). The sum in the two–

terminal current density expression (4.16) can be decomposed into asum over transverse modes and an integral over the k–value kt

‖,α ofthe injected wave

∑α,i

=

spin!↓2 · L2π

∑α

∑t

∫ ∞0

dkt‖,α. (4.23)

4.1.1 Quasi 1D Transport in Quantum Wires

In a quantum wire, the transverse wave functions ψt(x, y) are localised,and therefore may be normalised to 1. The multi–index t of the trans-verse modes, then, simplifies to a scalar index i ≡ t, that countsthe subbands at the injecting terminal. In this situation, the currentIα(i, α, ε) carried by the wave function |i α ε〉 through a cross–sectionalplane x = x0 of waveguide α is

Iα(i, α, ε) =∫dy dz eα · 〈i α ε| j 1(x0, y, z) |i α ε〉

=1L

m

(ki‖,α(ε)−

∑i′|r(α,i)→(α,i′)(ε)|2ki

′‖,α(ε)), (4.24)

whereas for a wave function injected from a terminal β = α, thecurrent takes the form

Iα(i, β, ε) = − 1L

m

∑i′|t(β,i)→(α,i′)(ε)|2ki

′‖,α(ε). (4.25)

The simple form of these expressions arises because eα · j 1(x0, y, z)contains only derivatives with respect to x. Hence, for wave functionsof the form 〈x y z|f i〉 = f(x)ψi(y, z) we may write∫

dy dz eα · 〈f i| j 1(x0, y, z) |g i′〉 =

m(f∗(x0) ddxg(x0)) 〈i|i′〉 .

(4.26)


This eliminates all mixed terms between transverse modes i = i′. Inthe remaining terms, f and g are equal. In most terms of eq. (4.24)and all terms of eq. (4.25) f is a simple exponential a exp(ikx), yieldinga current contribution |a|2k/m. The only term that requires closerattention is the transverse mode i in eq. (4.24). In this mode both aforward and a backward contribution are present

f(x) =1√L

(eik‖,α(ε)x + r(α,i)→(α,i′)(ε) e−ik‖,α(ε)x

). (4.27)

The associated current is (indices dropped for brevity)

Ii =1L

m(e−ikx + r∗eikx)(ikeikx − r ike−ikx))

=1L

k

m

(1− |r|2 + (re−2ikx − r∗e2ikx)︸︷︷︸

=0

)=1L

k

m(1− |r|2), (4.28)

and (4.24) is recovered. The total current at terminal α, then, is

Iα=1L

m

∫dε(f( ε− µαkBT

)×

×∑i

Z+α,i(ε)(ki‖,α(ε)−

∑i′|r(α,i)→(α,i′)(ε)|2ki

′‖,α(ε))

−∑β =α

f(ε− µβkBT

)∑i

Z+β,i(ε)∑i′|t(β,i)→(α,i′)(ε)|2ki

′‖,α(ε)),

(4.29)

where Z+α,i is the 1D density of (forward propagating) states of sub-band i in the waveguide connected to terminal α.

Alternatively, the current may be expressed in terms of the k values atwhich the electrons are injected into the system instead of the values,


at which they are extracted

Iα=1L

m

∫dε

(f( ε− µαkBT

)×

×∑i

Z+α,i(ε)(1−∑i′|r(α,i)→(α,i′)(ε)|2

ki′‖,α(ε)

ki‖,α(ε)

)ki‖,α(ε)

−∑β =α

f(ε− µβkBT

)∑i

Z+β,i(ε)(∑

i′|t(β,i)→(α,i′)(ε)|2

ki′‖,α(ε)

ki‖,β(ε)

)ki‖,β(ε)

).

(4.30)

In this form, each term is a multiple of the current 1L

mki‖,·(ε) carried

by the incident plane wave component in the wave function | · i ε〉.6This allows the interpretation of the coefficients of the partial currentsas transmission and reflection probabilities

T(β,i)→(α,i′)(ε) := |t(β,i)→(α,i′)(ε)|2ki′‖,α(ε)

ki‖,β(ε), (4.31)

R(α,i)→(α,i′)(ε) := |r(α,i)→(α,i′)(ε)|2ki′‖,α(ε)

ki‖,α(ε). (4.32)

In terms of these probabilities, the total terminal current for terminalα may be recast as

Iα=1L

m

∫dε(f(ε− µαkBT

)×

×∑i

Z+α,i(ε)(1−∑i′R(α,i)→(α,i′)(ε)

)ki‖,α(ε)

−∑β =α

f(ε− µβkBT

)∑i

Z+β,i(ε)(∑

i′T(β,i)→(α,i′)(ε)

)ki‖,β(ε)).

(4.33)

A true 1D system has no transverse subband structure, which elimi-nates the need for i/i′ summation. Also, there are only two terminals

6Sign convention: current into the device is positive.


with identical densities of states

Z+(ε) =

spin!↓2 · L2π

∫k>0

δ(ε− ε(k))dk

=L

π

∫k>0

δ(k − k∗)|∇ε(k)|k=k∗

dk, k∗ such that ε = ε(k∗)

=L

π· 1|∇ε(k)|k=k∗

(4.34)

In that case, the expression for the current simplifies to

I1D→ =2h

∫dε(f(ε− µ→

kBT

)(1−R→(ε)

)− f(ε− µ←kBT

)T←(ε)).

(4.35)

For µ→ = µ←, the current must vanish. This implies that we mayintroduce a transmission probability T (ε), that is valid for both trans-port directions.

1−R→(ε) = T←(ε) =: T (ε), (4.36)

from which the current takes the form

I1D→ =2h

∫dε T (ε)

(f( ε− µ→

kBT

)− f(ε− µ←

kBT

)). (4.37)

On the level of the effective mass approximation — the Bloch coef-ficients of the wave function are approximated by the Fourier coeffi-cients of the envelope function (cf. e.g. [19, 20]) —, this result, whichwas derived in the absence of a periodic potential, remains valid insidea semiconductor even for non–parabolic band–dispersion. The reasonfor this lies in the fact, that the mean electron velocity associated witha Bloch function of wavevector k is the group velocity 1

∇ε(k); and,

except for a factor of L/π, this is just the reciprocal of the (one–sided) density of states. In any case, the integration domain for theenergy in eq. (4.37) must be the support of the density of states.Often, experiments are performed for very small voltage differences

across the device. Then, the relevant quantity is the linear response


conductance G = −e dIdV

∣∣V=0

.7 By setting µ← = µ→+eV ,8 we obtain

G1D = − 2e2

hkBT

∫dε T (ε)f ′

(ε− µ→kBT

). (4.38)

4.1.2 Transport in a Quantum Well

The current through the two–dimensional electron gas (2DEG) insidea quantum well can be computed in much the same way as in thequantum wire. The only difference is the existence of a degree of free-dom orthogonal to both the confinement direction of the well and thetransport direction. Therefore, the transverse mode label t ≡ (i, k⊥)acquires an additional continuous quantum number, the wavenum-ber k⊥. Hence, when applied to quantum wells, in the expressions ofthe preceding section all summations over transverse modes must beaugmented by a k⊥ integration —∑

t

→ W

2π

∑i

∫dk⊥. (4.39)

Here, W is the normalisation length for plane wave states in the ad-ditional direction. In contrast to the L for the longitudinal states,this W will not cancel out. It represents the lateral extension of theconductor, and therefore the current should be expected to be pro-portional to W .By applying (4.39) to equations (4.37) and (4.38), particle current

and linear response conductance inside a quantum well (with only asingle state in the quantisation direction) are found as

I2D→ =2h

√πW

√2m∗⊥kBTh

∞∫εsub

dε T (ε)[F− 1

2

(µ− εkBT

)]µ→µ=µ←

, (4.40)

G2D =2e2

hkBT

√πW

√2m∗⊥kBTh

∞∫εsub

dε T (ε)F′−12

(µ→ − εkBT

). (4.41)

7All the currents I in this section are probability density currents. Hence,electrical current (when carried by electrons) acquires a pre–factor of −e.

8Note, that f ′ < 0. Therfore, G > 0 for µ→ < µ← (the electrical currentdirection is anti–parallel to the particle current direction).


Here, the integration variable ε is not the total energy of the particle;it is only the “forward”energy, i.e., the energy that remains, when thekinetic energy associated with motion in the “free” direction ε⊥(k⊥)is subtracted from the total energy.The expressions for the quantum well differ from their 1D equiva-

lent by replacing Fermi functions with Fermi–Dirac integrals, and byan additional factor of

√πW

√2m∗⊥kBTh

=√πW

λT,

with the (transverse) thermal wavelength λT = h/√2m∗⊥kBT . This

factor may be interpreted as the effective number of k⊥–modes thatcontribute to the current. The minus sign disappears on the transitionfrom (4.38) to (4.41) because of the different sign conventions in thearguments of f and F (cf. eq. (3.24)).

4.2 Thermionic Current Over a Barrier

4.2.1 Model Description

Except for the transmission probability T (ε) — or, in the more generalform, T(α,t)→(α′,t′)(ε) —, all the quantities in the current/conductanceformulae are known or easily calculated, e.g. by solving the transverseSchrodinger equation for subband energies and transverse eigenmodes.The full computation of the transmission probabilities T is more

involved and will be discussed in section 4.3.1. Before doing so, someapproximation methods for the treatment of T shall be discussed.When applied appropriately, they can tremendously reduce the com-putational effort involved in the current computation without degrad-ing the quality of the results.The simplest possible treatment for the transmission probability

T is thermionic emission over a barrier (cf. e.g. [37], [50]). Despite itssimplicity, it sometimes yields strikingly good agreement with exper-iment — especially in the presence of energy barriers that are widerather than high, such that tunnelling processes are strongly sup-pressed.

4.2. THERMIONIC CURRENT OVER A BARRIER 71

In the thermionic model, tunnelling and inter–subband scatteringare disregarded. Neglecting inter–subband terms allows us to write theconductance for a system with multiple subbands as a sum of single–mode conductances (eq. (4.38) for 1D, eq. (4.41) for 2D systems).Disregarding tunnelling processes results in a classical treatment

of the transmission probability. Particles for which a classical trajec-tory through the device exists are always transmitted; if the terminalsbetween which transmission is studied are separated by a classicallyforbidden region, transmission is zero.More formally, the energy ε of the particle is compared to the max-

imum εmax of the potential (or rather the maximum of the subbandenergy of the transverse mode in which it is injected, if the system isnot purely one–dimensional) between the injecting and the extractingterminal. If ε ≥ Vmax the particle is transmitted, otherwise it is al-ways reflected. Thus, the expression for the transmission probabilitymay be written as

T (ε) = Θ(ε− εmax). (4.42)

In this approximation, the linear response conductance of a (quasi)one–dimensional (spin–degenerate) system may be written in closedform as

G =2e2

h

∑i

f(β(εF − ε(i)max)

), (4.43)

with i denoting the transverse subband index. The pre–factor G0 =2e2

h = (12.9 kΩ)−1 is the quantum conductance of a single (spin–degenerate) 1D channel. At low temperatures (subband spacing greaterthan kBT ), the channels are activated one–by–one, giving the conduc-tance curve a staircase–like appearance.

4.2.2 Comparison with Experimental Results

The split–gate GaAs/AlGaAs structure, that has already been dis-cussed in the context of SET simulation (cf. section 3.7.3) can alsobe operated as a quantum point contact (QPC). In this mode of op-eration, only one pair of gates is operated at a time (the remaininggates were left floating). The selected gate pair is negatively biased


Figure 4.2: Simulated charge density inside a quantum point contact(QPC).

−1.5 −1.0 −0.5 0.0Gate voltage (V)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

Con

duct

ance

(G

o)

upper gates (meas.)lower gates (meas.)simulation (1 Kelvin)

Figure 4.3: Thermionic current through a QPC — experimental datafor two nominally identical QPCs and simulation results.

4.3. LONGITUDINAL TUNNELLING 73

in order to deplete the 2DEG underneath (cf. the charge density plotin fig. 4.2). Since the upper and the lower gate pair in the device are(nominally) identical, the corresponding QPCs should have the sameconductance. In fig. 4.3 experimental results from the two nominallyidentical QPCs (labelled“upper gates”and“lower gates”, respectively)are shown together with the linear response conductance obtainedfrom the thermionic emission model. Conductance is graphed versusthe bias of the gates that define the QPC. The difference between thetwo “identical” contacts is of the same order as the difference betweenexperimental and simulated conductance. Better agreement cannotbe hoped for, since that would require the simulation results to bemore accurate than the device description on which the simulation ismodelled. The work function Φs of the exposed GaAs surface maybe used as a fitting parameter to shift the simulated curve to the po-sition of each of the experimental curves. But because of the largeuncertainty in the experimental value of the QPC turn–on voltage theusefuleness of such a calibration would be limited.

4.3 Longitudinal Tunnelling

For systems with a high barrier (compared to kBT ) having a width ofthe order of the electron wavelength the thermionic emission modelstrongly underestimates the current. Here, tunnelling effects dominatethe current. Hence, a tunnelling inclusive model for the transmissionprobability is nedded.Transmission models that include tunnelling may be constructed ondifferent levels of complexity. The simplest possibility (as used in[51]) neglects inter–subband terms and treats longitudinal tunnellingon the level of the WKB approximation:

Ti(ε) =vextri (ε)vinji (ε)

exp(−2∣∣∣∣xextr∫xinj

κi(ε− εi(x)

)dx∣∣∣∣), (4.44)

where κi(εkin) := 1Θ(−εkin) ·

√−2m∗i,‖εkin is the imaginary wavenumber corresponding to the negative kinetic energy εkin, and εi(x) isthe ith subband energy from the transverse Schrodinger equation atposition x.


Since the WKB method discards the exponential increasing part ofthe wave function, it is blind to resonance phenomena. This may beremedied by using the full 1D Schrodinger equation for the computa-tion of T . The most general treatment of the transmission probability(at least within the context of quantum–ballistic transport) consistsin solving the full 2D/3D (for quantum wells/wires) Schrodinger equa-tions with open boundary conditions. This restores all inter–subbandcoupling terms, thus allowing for full evaluation of the current for-mula (4.33). One final remark before stepping over to the discussionof the open 3D Schrodinger equation: in the numerical evaluation ofthe conductivity with the inclusion of tunnelling contributions it isimportant to have high enough upper integration limits. Since someof the methods for the computation of T (ε) are numerically intense, itis tempting to use too low a value for the upper limit of the energy in-tegration, e.g. the higher of the two Fermi energies involved plus a fewkBT . While, this may be a good choice (especially for very thin, veryhigh barriers), it can lead to a massive underestimation of the currentin situations in which the transmission probability grows faster withincreasing energy than the thermal occupation factor decreases (ther-mally assisted tunnelling). To avoid this, it is necessary to raise theupper integration limit to the barrier maximum of the lowest subband(plus a few kBT ).

4.3.1 The 3D Schrodinger Equation with OpenBoundary Conditions

The Hamiltonian of an open quantum system possesses a continuousspectrum. Therefore, the Schrodinger equation needs not be treatedas an eigenvalue problem. Instead, it is possible to select an energy ε(which is known to reside in the spectrum of H) and solve the Schro-dinger equation as a boundary value problem for a partial differentialequation. For example, on a cuboidal simulation domain

(x, y, z) ∈ [0, xmax]× [0, ymax]× [0, zmax],one might formulate the problem(

−2

2∇ ·[1m∗

]∇+ V (r)− ε

)ψ(r) = 0, (4.45)


with boundary conditions

ψ(0, y, z) = f(y, z),∂

∂xψ(0, y, z) = g(y, z),

ψ∣∣y=0

= ψ∣∣y=ymax

= ψ∣∣z=0

= ψ∣∣z=zmax

= 0. (4.46)

Starting at x = 0,9 ψ(r) may be computed by using the “propagator”

∂2

∂x2ψ(r) =

2m∗x2

V (r)− m∗xm∗y

∂2

∂y2ψ(r)− m∗x

m∗z

∂2

∂z2ψ(r). (4.47)

Equation (4.47) could immediately be mapped onto a finite differencesexpression on a tensor–product grid. However, doing so places verystrict restrictions on the grid. The spacing of the x–points must beso small, that ∆x · k is small for all wavenumbers k that figure in theproblem. Along the y– and z– directions the situation is typically lessproblematic, because in a typical quantum wire the extensions alongthe transport direction (x) are much larger than in the transversedirections (y, z). Therefore, it is a good idea to restrict the real spacerepresentation of the wave function ψ to the transverse directions andtreat the transport direction in a plane–wave expansion10, as shownin the next section.

4.3.2 The Transfer Matrix Method

On each interval x ∈ In := [ 12 (xn−1 + xn), 12 (xn + xn+1)], the poten-tial V (x, y, z) is approximated by V (xn, y, z). Then, the transverseHamiltonian

H⊥(x) ≡ H(n)⊥ = −2

2

(∂

∂y

1m∗y

∂

∂y+

∂

∂z

1m∗z

∂

∂z

)+ V (xn, y, z)

(4.48)

9and assuming a homogeneous reciprocal effective mass tensor with its principalaxes along the coordinate axes

10Alternatively, Airy functions could be used; this would result in the potentialbeing treated as piecewise linear instead of piecewise constant.


also is constant on the interval, and the x–derivative of its eigenfunc-tions |i〉(n) vanishes. Therefore, for x ∈ In the adiabatic decompo-sition of the Schrodinger equation is exact, and for each i and ε weobtain solutions of the Schrodinger equation of the form

ψ(x ∈ In, y, z) = e±ik(n)i x〈y, z|i〉(n), (4.49)

where the wavenumbers are given by

k(n)i =

√2m∗x(ε− εi(xn)

)

, ε− εi(xn) ≥ 0

i

√2m∗x(εi(xn)− ε

)

, otherwise.

(4.50)

An arbitrary wave function on the whole simulation domain, then,may be written as

ψ(r) =∑n

χIn(x)∑i

(a(n)i eik

(n)i (x−xn) + b(n)i e−ik

(n)i (x−xn)

)〈y, z|i〉(n).

(4.51)

Here, χIn denotes the characteristic function of the interval I. Forconvenience, the exponential terms have been rewritten in such a way,that they vanish at the grid planes xn.For a potential without δ–singularities, the wave functions must be

continuous and possess a continuous first derivative. Enforcing theseconditions at the interval boundaries yields 11∑

j

(a(n+1)j e−

12 ik

(n+1)j ∆xn + b

(n+1)j e

12 ik

(n+1)j ∆xn

)|j〉(n+1)

!=∑i

(a(n)i e

12 ik

(n)i ∆xn + b

(n)i e−

12 ik

(n)i ∆xn

)|i〉(n) , (4.52)

and∑j

k(n+1)j

(a(n+1)j e−

12 ik

(n+1)j ∆xn − b(n+1)j e

12 ik

(n+1)j ∆xn

)|j〉(n+1)

!=∑i

k(n)i

(a(n)i e

12 ik

(n)i ∆xn − b(n)i e−

12 ik

(n)i ∆xn

)|i〉(n) . (4.53)

11Notation: ∆xn := xn+1 − xn


With the orthogonality relation (n)〈i|i′〉(n) = δi,i′ , one of the twosummations in (4.52) and (4.53) may be collapsed. Then, providedthat all the k(n+1)j are non–zero, the coefficients on slice n+ 1 can bewritten in terms of the coefficients on slice n as

a(n+1)j =

12e

12 ik

(n+1)j ∆xn ×

×∑i

[(1 +

k(n)i

k(n+1)j

)e

12 ik

(n)i ∆xna

(n)i

(n+1)〈j|i〉(n)

+(1− k

(n)i

k(n+1)j

)e−

12 ik

(n)i ∆xnb

(n)i

(n+1)〈j|i〉(n)], (4.54)

and

b(n+1)j =

12e−

12 ik

(n+1)j ∆xn ×

×∑i

[(1− k

(n)i

k(n+1)j

)e

12 ik

(n)i ∆xna

(n)i

(n+1)〈j|i〉(n)

+(1 +

k(n)i

k(n+1)j

)e−

12 ik

(n)i ∆xnb

(n)i

(n+1)〈j|i〉(n)]. (4.55)

Equations (4.54) an (4.55) may be combined into a vector equation[a(n+1)

b(n+1)

]= M

(n)

[a(n)

b(n)

]. (4.56)

The matrix M(n) is called the (partial) transfer matrix.

4.3.3 The Scattering Matrix

In the formulation of the conductance formulae in section 4.1, thewave functions were assumed to fulfil scattering boundary conditionsat the terminals. In this type of boundary condition, the wave func-tion is decomposed into terms propagating into the system and termspropagating out of the system. The coefficients for terms propagat-ing into the system are fixed, whereas no restriction is placed on theterms propagating out of the system.


In the notation of the preceding section, this means, that the coef-ficients for forward propagating states at x = 0, a(0), as well as forbackward propagating states at x = xmax, b(nmax), are specified. Thistype of boundary conditions contains exactly the same amount of in-formation as the specification of both ψ and ∂

∂xψ (Cauchy boundaryconditions) at x = 0 (this fixes the a(0) and the b(0)) or Dirichletconditions at both x = 0 and x = xmax (this fixes a(0) + b(0) anda(nmax) + b(nmax)).12 This proofs, that for fixed a(0) and b(nmax) andat an arbitrary energy ε a unique solution of Schrodinger’s equationalways exists.13 In the expression for the current (4.33), summationruns over states that inject only into a single subband i at a singleterminal. In the two–terminal case, such functions are specified by

a(0)j = δi,j , b

(nmax)j = 0 (4.57)

for injection from the left terminal, and

a(0)j = 0, b

(nmax)j = δi,j (4.58)

for injection from the right.

The transmission and reflection probabilities (4.31) and (4.32) forthese wave functions take the form

Ti→j =k(nmax)j

k(0)i

∣∣∣a(nmax)j

∣∣∣2 (4.59)

Ri→j =k(0)j

k(0)i

∣∣∣b(0)j ∣∣∣2 (4.60)

12The lateral Dirichlet conditions in (4.46) were already applied during the com-putation of the lateral wave functions.

13Note, that this statement is not in disagreement with the possibility of adiscrete spectrum of the Hamiltonian, when restricted to the state of physicalstates (e.g. elimination of formal solutions that exponentially increase towardsinfinity and are not even Dirac–δ normalisable).


for injection from the left, and

Tj←i =k(0)j

k(nmax)i

∣∣∣b(0)j ∣∣∣2 (4.61)

Rj←i =k(nmaxj )

k(nmax)i

∣∣∣a(nmax)j

∣∣∣2 (4.62)

for injection from the right.Therefore, the coefficient vectors a(nmax) and b(0) for the outward

propagating states must be computed from the the injection coeffi-cients a(0) and b(nmax). This is conventionally written in matrix form[

a(nmax)

b(0)

]= S

(0,nmax)

[a(0)

b(nmax)

]. (4.63)

The matrix S is called scattering matrix.14

4.3.4 The Forward Construction Scheme for S

Formally, the scattering matrix may be obtained by writing[a(nmax)

b(nmax)

]= M

(nmax−1) · · ·M(1)M(0)

[a(0)

b(0)

]

=: Mtot

[a(0)

b(0)

], (4.64)

with the total transfer matrix Mtot.In block matrix notation this equation reads[

a(nmax)

b(nmax)

]=[

Mtot0,0 Mtot

0,1

Mtot1,0 M

tot1,1

] [a(0)

b(0)

]. (4.65)

14Unfortunately, terminology is not uniform in this branch of physics. For ex-ample, [52] uses the above definition for the scattering matrix, whereas in [53] theinverse of the transfer matrix is called scattering matrix. Other authors use yetother definitions.


Assuming a(0) and b(nmax) as known and solving for a(nmax) and b(0)

yields

b(0) = M−11,1

(b(nmax) −M1,0 a(0)

), (4.66)

a(nmax) = M0,0 a(0) +M0,1 b(0)

= M0,0 a(0) +M0,1M−11,1 b

(nmax) −M0,1M−11,1M1,0 a(0). (4.67)

Hence, the scattering matrix is

S =

[M0,0 −M0,1M

−11,1M1,0 M0,1M

−11,1

−M−11,1 M1,0 M

−11,1

]. (4.68)

Unfortunately, this scheme is numerically unstable in long de-vices — especially, if we are not only interested in the transmissionand reflectance coefficients a(nmax) and b(0) but in the whole wavefunction, and therefore need a(n) and b(n) for all n = 0, 1, . . . , nmax.The reason for this is numerical overflow/underflow that may arisebecause in the multiplication of the partial transfer matrices exponen-tially growing and exponentially decaying terms exp(±κ∆x) are keptseparately, and loss of precision when at the end they are combinedinto the scattering matrix.

4.3.5 Recursive Construction of S

In order to improve on the numerical stability of the “forward” ordirect multiplication method, a recursive construction scheme for thescattering matrix can be used. In this scheme, the scattering matrixis assembled slice by slice. The transfer matrix for each new sliceis immediately incorporated into the scattering matrix, before thenext slice is added. That way, forward and back–propagating termsare mixed as soon as possible, which makes the numerics much moredocile.In the preceding section it was shown, how a transfer matrix can be

converted to a scattering matrix. When applied to the transfer matrixM(0) (as defined in (4.56)), equation (4.63) yields the scattering matrix[

a(1)

b(0)

]= S(0, 1)

[a(0)

b(1)

]. (4.69)


This matrix shall serve as the starting point for the recursive con-struction scheme (

S(0, n),M(n))→ S(0, n+ 1). (4.70)

For greater ease of notation the backward transfer matrix

B(n+ 1) :=(M(n))−1 (4.71)

is introduced (all matrix arguments refer to the slice indices in thecoefficient vectors on which the matrix regularly operates).Then, the starting equations for the insertion of the slice (xn, xn+1]

into the scattering matrix are[a(n)

b(0)

]= S(0, n)

[a(0)

b(n)

], (4.72)

and [a(n)

b(n)

]= B(n+ 1)

[a(n+1)

b(n+1)

]. (4.73)

From these equations, a(n) and b(n) may be eliminated to yield [52][ −S1,1B1,0 1

B0,0 − S0,1B1,0 0

] [a(n+1)

b(0)

]

=[

S1,0 S1,1B1,1

S0,0 S0,1B1,1 − B0,1

] [a(0)

b(n+1)

]. (4.74)

For brevity, the notation S = S(n), B = B(n+ 1) has been used.This may be rewritten in more explicit albeit less compact form as

S0,0(0, n+ 1) = (B0,0 − S0,1B1,0)−1 S0,0

S0,1(0, n+ 1) = (B0,0 − S0,1B1,0)−1 (S0,1B1,1 − B0,1)S1,0(0, n+ 1) = S1,0 − S1,1B1,0 S0,0(0, n+ 1)S1,1(0, n+ 1) = S1,1B1,1 − S1,1B1,0 S0,1(0, n+ 1). (4.75)

Successive application of these equations starting with S(0, 1) fromeq. (4.69) (or alternatively with S(0, 0) = 1 ⊕ 1) yields the full scat-tering matrix S(0, nmax) after nmax − 1 (or nmax) steps.


4.3.6 Injected vs. Local EquilibriumCharge Density

When computing the charge density inside ballistic devices, there aretwo possibilities for the treatment of the charge density. A local equi-librium charge density can be used on each slice; this corresponds to apurely classical treatment and, therefore, is usually appropriate whencomputing the properties of devices inside which tunnelling currentsare small. Alternatively, the injected charge density may be computed.This charge density is obtained by explicit integration of the chargedensity of the full wave functions from the open boundary Schrodingersolver

ρ(r) = gval∑α

∑v

∑i

∞∫εv,i,α

Zincv,i,α(ε)f(ε− µαkBT

)|ψv,i,α,ε(r)|2 dε. (4.76)

Here, α is the injecting terminal, v the valley index15 and i the trans-verse mode index. εv,i,α is the subband bottom energy of subband iin valley v at terminal α. The densities of state Zinc are one–sideddensities of state inclusive of spin degeneracy. The index ε in the wavefunction ψ denotes injection energy.Evaluation of the injected wave function ψv,i,α,ε at arbitrary x–

positions is equivalent to the computation of a(n) and b(n) for anyn. In principle, these coefficient vectors can be obtained by using thescattering matrix S(0, nmax) to convert scattering boundary condi-tions(a(0),b(nmax)

)to Cauchy boundary conditions

(a(0),b(0)

). The

remaining coefficients follow from equation (4.56). But as in the for-ward construction for the scattering matrix, this forward scheme forthe coefficient vectors suffers from exponential error propagation and,therefore, is unsuited for the actual computation of the wave func-tion. Again, this may be remedied by using a scattering matrix ap-proach [52]. To link a(n) and b(n) to the scattering boundary vectorsa(0) and b(nmax), the two scattering matrix equations[

a(n)

b(0)

]= S(0, n)

[a(0)

b(n)

],

[a(nmax)

b(n)

]= S(n, nmax)

[a(n)

b(nmax)

](4.77)

may be employed.15A separate effective mass equation is solved in each valley.


From these equations the quantities a(nmax) and b(0) may be elimi-nated. The resulting expressions for a(n) and b(n) are

a(n) =(1− S0,1(0, n)S1,0(n, nmax)

)−1 ××(S0,0(0, n)a(0) + S0,1(0, n)S1,1(n, nmax)b(nmax)

), (4.78)

b(n) =(1− S1,0(n, nmax)S0,1(0, n)

)−1 ××(S1,0(n, nmax)S0,0(0, n)a(0) + S1,1(n, nmax)b(nmax)

). (4.79)

While the injected charge density includes longitudinal coherent ef-fects, scattering is omitted entirely. This results in strict energy con-servation inside the device; and consequently bound states of energieslower than the lowest transverse subband energy at either of the ter-minals do not contribute to the charge density, because there is noenergy conserving way to populate them. Since even minimal scatter-ing is sufficient to populate such states, a more physical model of thedevice might be obtained if their filling were enforced explicitely.

4.3.7 Current and Charge Computation at FiniteSource–Drain Bias

If a non–negligible bias is applied between the terminals of a ballisticdevice, the Fermi energy inside the device ceases to be defined, andthe charge density may differ significantly from an equilibrium chargedensity. For devices in which tunnelling electrons have little influ-ence on the electrostatics, this non–equilibrium charge density maybe modelled on the level of a classical treatment of the transport di-rection [54] as shown in fig. 4.4. Up to the barrier peak energy εmaxboth left and right propagating states are populated according to εsrcF

for x < xmax (εdrnF for x ≥ xmax) because all injected electrons in theclassical picture are reflected back to their injection terminal. For en-ergies greater than εmax all electrons are transmitted over the barrier.Therefore, right propagating states are populated according to εsrcF ,whereas left propagating states are populated with εdrnF [50, 55]. The


xmax

εsrcF

εdrnF

εmax

: both +k(ε) and −k(ε) contribute to the charge density;: only the right travelling (+k) plane wave states are occupied.

Figure 4.4: Ballistic non–equilibrium charge density (at T = 0; tun-nelling neglected).

charge density ρ then may be written as16

ρsplit(r) =∑v

∑i

∣∣∣〈y, z|i〉(x)∣∣∣2 ××( εv,i

max∫εv,i(x)

dε Z‡v,i(ε)f(ε− ε‡FkBT

)

+12

∞∫εv,imax

dε(Zsrcv,i f( ε− εsrcF

kBT

)+ Zdrnv,i f

(ε− εdrnF

kBT

))), (4.80)

with the notation

‡ :=

src, x < xv,imaxdrn, otherwise.

Here, densities of states encompass both positive and negative k–values.For devices with sizable tunnelling currents, this treatment may betoo rough. Here, the charge density has to be computed from the

16The charge density expression given in [55] only applies for x ≤ xmax.


xmax

εsrcF

εdrnF

εmax

tunnelling contribution to ρ

Figure 4.5: The injected charge density automatically accounts fordifferent contact Fermi energies. The lighter shading of the tunnellingcontribution indicates filling factors δ < f < 1 for some non–zero δ.

full 2D/3D wave functions according to equation (4.76). Please note,that this expression automatically treats different Fermi energies atthe terminals in the correct way (cf. fig 4.5).

4.4 Simulation Results

The results in this section were obtained from hypothetical double–gate MOSFET devices similar to the one studied by Lundstrom etal. in [56].17 The geometry of the devices studied can be seen fromfigure 4.6. In all simulated devices gate oxide thickness was 6 A. Thegate length was varied between 3 nm and 30 nm. Two values (1 nm,3 nm) were used for the silicon body thickness18.In the derivation of the expressions for the current/conductivity

we make use of the fact that the wave functions19 asymptotically ap-proach a linear combination of a forward and a backward propagatingplane wave. The implementation of the conductance expressions in

17In contrast to the use in this work, Lundstrom takes“ballistic”to mean“quasi–ballistic” and solves the Boltzmann transport equation, thereby neglecting longi-tudinal quantum effects.

18Recently, an experimental group at the NTT laboratories (Japan) actu-ally succeeded in fabricating double–gate MOSFETs with these small film–thicknesses [57].

19in the multi–band case: the coefficient function to each transverse mode


10 nm

1.0 nm

0.6 nm

0.6 nm

10 nm

ND = 2× 1020 cm−3ND = 2× 1020 cm−3 NA = 1014 cm−3

source

drain

gate

silicon SiO2Bgate

Figure 4.6: Geometry of the SOI double–gate MOSFET structure.

simnad assumes, that this asymptotic behaviour exactly describes thewave functions at the terminal planes. The validity of this assumptionwas checked by simulations, results of which are depicted in fig. 4.7.The probability density (integrated over the cross–section) of a wavefunction injected from the left terminal at ε = 0 is shown togetherwith the conduction band profile. Please note the small number ofsampling points used in the computation of the wave function. Thisis only possible because of the analytical ansatz for the wave functionon each slice. With a finite difference treatment, a much greater num-ber of sampling points ( 5 per half–wave) would have been necessary.Far enough from the barrier, the summed up charge density of all thewave functions varies only slowly with distance. Therefore, the coarsediscretisation shown here is sufficient to yield all the relevant data.At Vgate = 0 the channel is closed, and the entire incident probabilitydensity flux is expected to be reflected at the barrier. Therefore (sinceinter–subband coupling terms turned out to be negligible in this de-vice), the wave function must asymptotically approach a phase–shiftedsine wave of twice the amplitude of the incident wave. Using the cir-cled points of the wave function for a least–square fit to a sine–waveresulted in the dashed curve. It can clearly be seen, that the sine–likebehaviour is reached very quickly outside the barrier. For the presentdevice, the assumptions of flat bands and plane waves at the terminalsare clearly justified.In figure 4.8 the thermionic excitation current over the barrier (cf.

section 4.2.1) is compared to the current from the full scattering ma-trix. This latter treatment results in the inclusion of both tunnelling


|Ψ|2

potential[eV]

distance [nm]

0

0 10 20 30 40

0.0

0.5

1.0

1.5

2.0

0.2

0.4

-0.2

: sine function fit to the wave function outside the barrier.: data points for which Ψ was computed.: data points used in fitting the sine.

· · · : energy at which the wave function was computed.Figure 4.7: Numerical results for a left–incident wave function Ψ inthe SOI double–gate MOSFET.

and quantum reflection effects. Due to the neglect of tunnelling con-tributions to the current, the I–V –characteristic for the thermioniccurrent begins to rise later than the current obtained from the fullscattering matrix (labelled “tunnelling current” in fig. 4.8). As soonas the gate voltage is high enough to obtain a flat band condition alongthe channel, the thermionic current becomes independent of the gatevoltage, because the potential maximum featuring in equation (4.42)is fixed by the doping and does not depend on Vgate beyond flat band.In contrast, the scattering matrix current in the turned–on channel


Vgate [V]

I drain[A/µ

m]

full S–matrixthermionic approx.

3 nm MOSFET proposed by Lundstrom

-10

0 1 2 3 4 5 6 7 8

2× 10−8

4× 10−8

6× 10−8

8× 10−8

1× 10−7

Figure 4.8: Full scattering matrix current and thermionic currentthrough the SOI MOSFET with 1 nm body thickness and 3 nm gatelength. The full formalism affords both to tunnelling contributions tothe current in the sub–threshold regime and to quantum oscillations inthe on–current (in real devices typically masked by series resistance).

typically is smaller than the thermionic current, because quantum–mechanically there is some probability for quantum reflection even atenergies above the barrier maximum. Full transmission is only ob-tained as discrete resonance energies; hence, the oscillations in theon–current of the MOSFET.20

In the simulations leading to figure 4.8, a local equilibrium chargedensity was used; the scattering matrix computation for the cur-rent extraction was applied as a post–processing step, once conver-

20In these simulations, contact resistances were not taken into account. Thisseries resistance is likely to mask the quantum oscillations in real devices.


0 10 20 30 40 50distance along channel [nm]

−1.5

−1

−0.5

0

0.5

cond

uctio

n ba

nd e

dge

[eV

]

Vsd=0.5 V

0 10 20 30 40 5010

5

1010

1015

1020

elec

tron

den

sity

[cm

−3 ]

contact doping: 1019

cm−3

, 200 samples for kforward

‘‘adiabatic’’ rho (Vsd = 0)injected rho (Vsd = 0)injected rho (Vsd = 0.5 V)

20 21 22 230.28

0.3

0.32

0.34

Figure 4.9: SOI MOSFET: conduction band edge and charge densityalong the centre of the channel at Vgate = 0.

gence of the adiabatically decomposed Kohn–Sham equations has beenreached. Furthermore, the device was operating in linear response,allowing us to use a single Fermi energy throughout the channel. Fig-ure 4.9 shows the effect on charge density and potential if one or bothof these simplifications are abandoned. The local equilibrium chargedensity (dotted lines) exhibits a very deep ditch between source andrain: charge density drops by ten orders in magnitude, correspond-ing to a barrier height of about 0.6V.21 If longitudinal tunnellingcontributions to the charge density are taken into account by insert-ing the scattering matrix computation for the wave functions into theself–consistency loop (solid line, “injected rho”), the drop in chargedensity inside the barrier is strongly reduced. Applying a forward

21Simulations were performed for room temperature.


5 10 15 20 25 30 35Position [nm]

108

1010

1012

1014

1016

1018

1020E

lect

ron

Den

sity

[cm

−3]

tSi=1nm, VG=0V, VD=1µV, depth=0.5nm

1D−SP2D−QB

31020

Figure 4.10: Lateral density profiles in the middle of the channel.Local equilibrium charge density from 1D Schrodinger–Poisson andinjected charge density from self–consistent quantum–ballistic (QB)simulations. In very short devices, wave function penetration into thebarrier may have a marked effect on the electrostatics.

bias of half a volt shifts the minimum of the charge density distri-bution towards the drain, because electron injection from the drainside is reduced — electrons are supplied by the drain at lower energieshence, they “see” a much wider and higher potential barrier. With re-spect to the injected charge density at zero bias, the right flank of theinjected charge density at Vsd = 0.5V is shifted outward by 3.7 nm.This is accompanied by a lowering of the barrier maximum (“draininduced barrier lowering”) as can be seen from the insert.In figure 4.10 local equilibrium (dashed lines) and self–consistently

injected (solid lines) charge density profiles are shown for double–gateMOSFETs of various gate lengths. Figure 4.11 shows a comparisonof transfer characteristics obtained by different models for MOSFETs


gate length: 3 nm gate length: 5 nm

1D−SP2D−QB (sc)2D−QB (nsc)

I[A/µ

m]

Vgate [V]

10−7

10−9

10−11

10−13

10−15

10−17

10−19−0.5 0.0 0.5 1.0 1.5


Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5



Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5

I[A/µ

m]


Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5



Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5

I[A/µ

m]


Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5

Figure 4.11: Transfer characteristics of SOI double gate MOSFETsby various models. Vdrain = 1 µV. Silicon body thickness: 1 nm.




I[A/µ

m]

Vgate [V]

10−6

10−7

10−8

10−9

10−10

10−11−0.5 0.0 0.5 1.0 1.5


Vgate [V]

10−6

10−7

10−8

10−9

10−1010−11

10−12

10−13

10−14

10−15

10−16−0.5 0.0 0.5 1.0 1.5



Vgate [V]

10−610−710−810−910−1010−1110−1210−1310−1410−1510−1610−1710−18−0.5 0.0 0.5 1.0 1.5

I[A/µ

m]


Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5



Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5

I[A/µ

m]


Vgate [V]

10−6

10−8

10−10

10−12

10−14

10−16

10−18

10−20−0.5 0.0 0.5 1.0 1.5

Figure 4.12: Transfer characteristics of SOI double gate MOSFETsby various models. Vdrain = 1 µV. Silicon body thickness: 3 nm.


0 5 10 15 20Gate Length [nm]

10−28

10−26

10−24

10−22

10−20

10−18

10−16

10−14

10−12

10−10

I D [

A]

tSi=1nm, VDS=1µV

VGS=0VVGS=−0.25VVGS=−0.50V

QB

QDD

Figure 4.13: Drain current versus gate length in the sub–thresholdrange for a silicon body thickness of 1 nm. Anisotropic effective masswas emulated in the QDD simulations by compressing the geometryalong the x axis by a factor of

√mlong/mtrans ≈ 2.2.

with a silicon body thickness of 1 nm and various channel lengths.Each of the graphs depicts the current computed from a drift–diffusionsimulation on top of a 1D Schrodinger–Poisson simulation (1D–SP)22.Heavy dashed lines are used for the results of a quantum–ballisticcurrent computation on top of a 1D Schrodinger–Poisson charge den-sity; heavy solid lines denote the results of quantum–ballistic (QB)current computations using the self–consistently (sc) injected charge(no Fermi energy defined inside the device). Figure 4.12 shows thecorresponding graphs for a silicon body thickness of 3 nm.In both cases it can be seen that the current in the devices with a

gate length of 20 nm is entirely thermionic: the drift–diffusion and thequantum–ballistic simulations exhibit the same sub–threshold swing.

22dessis–ise with the quantum mechanical mobility model of [32].


Bgate = 5nm Bgate = 7nm

1

Vsd = 50 mVVsd = 200 mVVsd = 800 mV

10−0

10−0

10−2

10−4

10−6

10−8−0.5 0.5


10−0

10−2

10−4

10−6

10−8

10−10−0.5 0.0 0.5 1.0Bgate = 10 nm Bgate = 14 nm


10−0

10−2

10−4

10−6

10−8

10−10

10−12−0.5 0.0 0.5 1.0


10−0

10−2

10−4

10−6

10−8

10−10

10−12

10−14

10−16−0.5 0.0 0.5 1.0

Figure 4.14: Idrain [A/µm] vs. Vgate [V] of double–gate MOSFETs of1 nm body thickness and gate length B at various source–drain voltages.

At 3 nm and 5 nm gate length the sub–threshold current is dominatedby the tunnelling current — the sub–threshold slope of the quantum–ballistic current is much smaller than that of the drift–diffusion cur-rent. Between 7 and 14 nanometres the I–V curves show mixed char-acteristics. As the channel is closed by increasingly negative gatevoltages, the current at first drops with a slope close to that of thethermionic characteristic. Eventually, however, the curve bends upand approaches the tunnelling slope for more negative gate voltages.In the three nanometre case the sub–threshold swing is further re-duced by a pronounced electrostatic short–channel effect.Comparison with quantum drift–diffusion (QDD) simulations us-

ing the density gradient model of [32] has shown that at least for


devices with a silicon body thickness of 1 nm the density gradientmethod is capable of predicting the source–drain tunnelling of thesenano–scale MOSFETs correctly provided that the source-drain volt-age is not too large (cf. fig. 4.13). Since at the time of writing theQDD implementation inside dessis–ise is restricted to isotropic effec-tive mass, the anisotropy was manually incorporated into these simu-lations by reducing the longitudinal extensions of the transistors by afactor of

√mlong/mtrans ≈ 2.2. This scaling of course introduces some

error in the electrostatics of the device. But this turned out to be asmall effect compared to the huge impact of a modified tunnelling masson the tunnelling current. For devices with a silicon body thicknessof 3 nm, the agreement between quantum–ballistic results and QDDresults is much worse. The reason for this is, that in the transistorswith 1 nm body thickness only the valley with the large effective masscomponent along the quantisation direction contributes to the current.In the 3 nm case, it can clearly be seen from quantum–ballistic sim-ulations, that the valley with small effective mass components bothalong the quantisation and along the transport direction contributessignificantly to the current. In such a situation a full implementa-tion of QDD with anisotropic effective mass is necessary in order toaccount correctly for tunnelling currents.To conclude the discussion, let us take a look at the current at largersource–drain voltages. Figure 4.14 shows transfer characteristics ob-tained for various values for the source drain bias in transistors withgate lengths ranging from 5 to 14 nanometres. For all channel lengths,it can be seen that both at positive and at negative gate voltagesthere is a considerable increase in drain current as the drain voltageincreases from 50mV to 200mV. But with a further increase of thedrain voltage the current near the positive end of the I–V curve ofthe long transistors saturates, whereas in the short transistors the cur-rent continues to increase23. Regardless of length, there is no currentsaturation at the negative end of the I–V curve.The reason for this is that increasing the source–drain voltage re-

sults only in a small decrease in barrier height — in the 14 nm tran-sistor the drain induced barrier lowering at a forward bias of 800mV

23For all voltages shown, there is still a small potential barrier left inside thechannel.


ConductionBand Vsd = 0.8VConductionBand Vsd = 0VElectronDensity Vsd = 0.8VElectronDensity Vsd = 0V

0 20 40

−1

−0.5

0

1E13

1E14

1E15

1E16

1E17

1E18

1E19

1E20

1E21

Figure 4.15: Electron density (in cm−3) and conduction band edge (ineV) along the transport direction (x in nm) in a double–gate MOSFETwith Bgate = 14 nm and silicon body thickness of 1 nm at a gate voltageof 0.1V.

was found to be less than a millivolt; but the width of the barrier is re-duced significantly. Therefore, the tunnelling contribution to the cur-rent increases much stronger with increasing source–drain bias thanthe thermionic current over the barrier (cf. fig. 4.15). In very shorttransistors source–drain tunnelling dominates the sub–threshold cur-rent throughout the sub–threshold regime. In longer devices, however,tunnelling dominates only at very negative gate voltages. Otherwisethe current is mostly thermionic emission current over the barrier.This discussion applies only to voltages Vsrc–drn 100mV. For

smaller voltages backward electron injection from the drain reducesthe current at low forward bias, leading to an approximately lineardependency of the current on the source drain voltage (cf. fig. 4.16).

4.5 Comparison with Non–Equilibrium

Green’s Functions

Recently, there has been considerable interest in the simulation ofnano–scale MOSFETs with the method of non–equilibrium Green’sfunctions (NEGF) (e.g. [58, 59, 60, 61]). Therefore, it seems appropri-

4.5. COMPARISON WITH NEGF FORMALISM 97

0 0.2 0.4 0.6Vsrc–drn [V]

I[A

/m

m]

0.0

0.1

0.2

0.3

0.4

Figure 4.16: Output characteristics of a 14 nm MOSFET transistorwith 1.5 nm body thickness (same geometry as in [58]) at Vgate = 0.

ate to relate the present work to the NEGF approach. In the absenceof coherence breaking scattering processes (i.e. in the quantum bal-listic limit) the NEGF approach used in nanoMOS [61] is equivalentto the scattering matrix method24 of the present work when appliedto devices in which the different subbands do not mix (i.e. the over-lap matrices (n)〈i|j〉(n+1) are diagonal for all n). In simnad subbandmixing is taken fully into account, whereas in nanoMOS the differentsubbands are assumed to be independent — an assumption well suitedto thin double–gate MOSFETs but not applicable to more general de-vices. In addition, simnad uses a piecewise analytical representationof the wave function as opposed to the finite difference representationused with the NEGF formalism. This allows simnad to work witha considerably smaller number of sampling positions along the trans-port direction — an important prerequisite for keeping 3D simulationstractable.

24with the injected charge density of eq. (4.76)


Coherence breaking scattering has not yet been included in simnad.But there is no fundamental obstacle to doing so on the level of But-tiker probes (cf. e.g. [61]). The imaginary diagonal self–energy termsarising in the treatment of inelastic scattering in the NEGF formalismmay formally be treated as imaginary contributions to the electro-static potential. The resulting non–Hermitian Hamiltonian gives riseto wave functions that violate current conservation. In the Buttikerprobe method virtual particle reservoirs are coupled to each point ofthe system in order to compensate the divergence of the current den-sity.25 An electrochemical potential is assigned to each of the virtualparticle reservoirs. By a self–consistent computation, the electrochem-ical potentials of all the reservoirs have to be adjusted in such a wayas to make the total current density divergence–free.

25This is done by injection of “wave functions” that carry a current densitythat both in positive and in negative transport direction travels away from theinjection point. In NEGF formulation such functions (in 1D) are given by ψ(x) =G(x, x′, E), x′ being the injection point. In the Schrodinger formulation thiscorresponds to piecewise defined functions

ψ(x) =

ψ+(x), x > x′

ψ−(x), x < x′

using boundary conditions ddx

logψ±(x)|x=x′ = ±ikx(E, x′).

Chapter 5

A coupled3D Kohn–Sham /Drift–DiffusionSimulation Approach

5.1 The Test Device

A promising structure for large–scale integration of semiconductor de-vices is the single–electron memory (SEM). It is often assumed thatthis class of devices could be within the reach of technologies usedin current mass production of advanced Si–based MOSFETs. Ourtest device for the integration of the SET simulator simnad with thedrift–diffusion (and energy–balance) simulator dessis–ise [62] is anSOI MOSFET with a 7×7×2 nm3 poly–Si floating gate (quantum dotflash memory). Structure and doping information are based on exper-imental papers by Guo et al. [63, 64, 65, 66, 67] and by J. J. Welser etal. [68]. A similar structure is presented in [69, 70]. Figure 5.1 showsmicrographs of the real device. For their device Guo et al. report anincrease in threshold voltage of ∆Vth = 55mV per electron. Sincethey do not supply information on the doping conditions inside the

99

100 CHAPTER 5. COUPLING WITH A DD SIMULATOR

Gate

Channel

Dot

OxideDrain

Source

Figure 5.1: SEM (left) and TEM (right) of the QD flash memory(from [68]).

channel region, the channel doping density was treated as a variableparameter.The geometry for the drift–diffusion simulation is presented in

fig. 5.2. The entire MOSFET is several microns long, whereas thefloating gate (white region) has nanometre dimensions. Resolving thewave function inside the floating gate requires mesh refinement on anAngstrom scale.This imposes an extreme challenge on the mesh generation. Fur-

thermore, it is well–known from TCAD of conventional CMOS thatin order to obtain grid–independent I–V characteristics, the inversionlayer must be refined, at least, on a nanometre scale, too. These re-quirements on the grid were one of the reasons to develop an interfacebetween nano–scale and conventional devices.

5.2 Drift–Diffusion Simulation

Before attempting a coupled simulation of the structure, the devicewas studied with each simulator, separately. In the 3D drift–diffusion

5.2. DRIFT–DIFFUSION SIMULATION 101

Figure 5.2: Geometry of the simulated device. Total length: 5 µm,length of gate: 3 µm, channel width: 10 nm, channel height: 26 nm.

simulations the main focus was put on the variation of the thresholdvoltage of the device with the amount of charge stored on the floatinggate. I–V characteristics were calculated at various (homogeneous)doping levels inside the channel region. Fig. 5.3 shows the classicalcurrent density (high current density in bright regions) in the siliconchannel. Since a drift–diffusion simulation does not take into accountquantisation in the inversion layer, the current density has its maximain the edges of the channel.Only a minor (about 1–2mV) increase in threshold voltage was

found to result from the addition of each extra electron to the float-ing gate. The effect does not visibly depend on the doping level (cf.fig. 5.4). This result seems to be in sharp contrast to the measured in-crease in threshold voltage of about 55mV per single–electron broughtonto the floating gate.It is important to note that the independence of ∆Vth per–electron

on the doping concentration is only obtained with a sufficient refine-ment of the edge regions of the channel. Too coarse a grid produces


Figure 5.3: Current density in the homogeneously doped device.

the artefact of a strong increase of ∆Vth per–electron with increasingdoping.Due to the small device dimensions — the channel cross section

is 10 nm × 26 nm — the simplifying assumption of a homogeneousdoping concentration inside the channel material loses its justification(cf. e.g. [71]). At a doping concentration of 1018 cm−3 there are onaverage less then 3 dopant atoms in the channel section underneaththe floating gate. Therefore, the simulation model was modified suchas to allow for the inclusion of atomistic doping effects [72]. Thehomogeneous doping density was replaced by a random distributionof dopant atoms. In order to reduce the computational expense athigh boron concentrations, the discrete doping was restricted to thecentral section of the channel. The I–V curves obtained from thisatomistic doping model were found to be very sensitive to the positionof the dopant atom closest to the floating gate (cf. fig. 5.6): displacingthis atom resulted in effects as large as 150mV. The self–consistentcurrent density brought about by the discrete doping distribution isdepicted in fig. 5.5: doping centres are surrounded by zones of highresistivity. Therefore, at high gate voltages the source–drain current


through the discretely doped device is smaller than in the continuousdoping case. The onset of conduction, however, takes place at lowergate voltages with discrete doping: since regions far away from thedopant atoms are effectively undoped, a conducting path avoiding thelocations of the dopant atoms can form without the need for inversionof p–Si. In addition, it was seen that in the atomistic doping case alarger ∆Vth per–electron might occur: fig. 5.7 shows results from adiscrete doping model with an average doping density of 1018 cm−3;the threshold voltage is increased by ca. 40mV per electron on thefloating gate. While this is still smaller than the reported 55mVincrease, it is considerably larger than the result obtained with thecorresponding homogeneous doping density.


0 1 2 3 4 5 6 7Control gate voltage (V)

10−11

10−10

10−9

10−8

10−7

Dra

in c

urre

nt (

A)

0 electrons1 electron2 electrons

2e17

1.2e18

5e181e19

0.09 0.10 0.11Control gate voltage [V]

9e−11

1e−10

1.1e−10

Dra

in c

urre

nt [A

]

0 electrons1 electron2 electrons

Figure 5.4: Top: Idrain [A] vs. Vgate [V] for different electron numberson the floating gate at various (homogeneous) doping levels. Bottom:zoom for the lowest (2× 1017 cm−3) doping case.


j[A/cm2]106

3000 10

Figure 5.5: Current density with atomistic doping (156 boron atoms= 2 × 1018 cm−3): j is suppressed by 5 orders of magnitude near thedopant atoms.


p doping restricted to central 100nm of channel[i.e. discrete: 27 boron atoms ; continuous: 10

18 cm

−3]

discrete doping; atom directly under floating gatediscrete doping; atom near edge of channeldiscrete doping; atom near bottom af channelcontinuous doping

Vgate [V]

I drain[µ

A]

0.5 1.0 1.5 2.0 2.5 3.00

0.05

0.10

0.15

Figure 5.6: Discrete channel doping: sensitivity of the I–V character-istics to changes in the position of a single dopant atom.

#e− = 0

#e− = 1

#e− = 2

#e− = 3

#e− = 4

#e− = 5

Vgate [V]

I drain[µ

A]

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.02

0.04

0.06

0.08

0.10#[fixed atoms]=27; extra atom near edge

Figure 5.7: Single–electron shift in the discretely doped device.

5.3. SIMNAD SIMULATIONS 107

5.3 simnad Simulations

The separate simnad simulations were performed on the central partof the structure that contains the quantum dot (QD) floating gate andsome of the surrounding oxide.Points addressed in this section are the charging of the QD, the en-

ergy levels of the Kohn–Sham orbitals, the self–consistent probabilitydensities of the orbitals, and their deformation as the total electronnumbers on the quantum dot is increased. All results in this sectionwere obtained for a simulation temperature of T = 300K.Fig. 5.8 shows the number of electrons on the QD as function of the

control gate voltage. A constant channel doping of 2× 1018cm−3 wasused. Allowing for non–integer electron numbers on the dot resultsin a continuous curve (thin line), derivation of which reveals threefeatures. The first feature appears when the QD occupation numberbecomes non–zero, the second near an electron number of 4, and the

0 5 10 15 20 25 30Gate voltage [V]

0

4

8

12

16

20

24

28

Ele

ctro

n n

um

ber

#e−

d(#e−)/dV [x20]

Figure 5.8: Number of electrons on the QD (heavy line) as function ofcontrol gate voltage and the corresponding capacitance (dashed line)obtained from the derivative of the continuous charging curve (thinline).


third when the QD is occupied by 12 electrons. The linear branchesbetween the structures correspond to the monotonous increase of theCoulomb energy (mainly Hartree contribution) with increasing chargeon the dot. The structures related to the fillings with 4 and 12 elec-trons (the second “shell” consists of two almost degenerate spatialorbitals each with a spin/valley degeneracy of 4) originate from theconfinement effect: once the first energy level is fully populated, fur-ther charging is inhibited until the Fermi energy approaches the nextlevel. This suppression of the increase in Coulomb energy is smearedout on the scale of the thermal energy kBT . Fig. 5.9 gives additionalinformation on the energies of the Kohn-Sham orbitals as functionof the control–gate voltage. Observe that the degeneracy of the sec-ond “shell” is slightly lifted due to the asymmetrical shape of the gatecontact in the xy–plane.

Fig. 5.10 shows the probability density plots of the first 12 states in theempty QD. In this figure εF = 0. The above discussed degeneracies are

4 6 8 10 12 14 16 18 20Gate voltage [V]

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Lev

el e

ner

gy

[eV

]

0

2

4

6

8

10

12

14

16

18

Ele

ctro

n n

um

ber

Figure 5.9: Number of electrons on the QD (thick black line) andenergies of the Kohn-Sham orbitals (thin black lines) relative to theFermi level (dotted). The dashed lines indicate εF ± kBT .

5.4. A BRIEF EXCURSE ON MESHING 109

Figure 5.10: Probability density distributions of the first 12 states inthe empty QD (εF = 0, no electrons on the dot).

easily seen. For comparison, the situation with charging is presented infig. 5.11. In this figure, εF ≈ 3.4 eV, and there are about 24 electronson the dot. One observes that the Coulomb repulsion pushes theelectrons towards the walls of the QD and makes the charge envelopemore similar to the geometrical shape of the poly QD. Furthermore,due to the self–consistent interaction terms the charge distribution ofthe lowest state in the x– (y–)valley acquires a “p–like” appearance incontrast to the “s–like” shape seen in the empty dot.

5.4 A Brief Excurse on Meshing

The two simulators in our coupling scheme operate on different typesof discretisation grids: simnad uses a finite difference scheme on arectilinear tensor–product grid; dessis–ise works on a box methodconforming Delaunay mesh.


Figure 5.11: Probability density distributions of the first 12 states atεF ≈ 3.4 eV, about 24 electrons on the dot.

5.4.1 Tensor–Product Grids

A tensor–product grid (in 3 dimensions) is the generalisation of a(uniform) Cartesian grid. It is constructed by selecting a set of 3linearly independent directions ex, ey, ez (axis directions). For eachdirection we choose a finite set of sampling positions:

X = xi | i = 1, . . . , nxY = yj | j = 1, . . . , nyZ = zk | k = 1, . . . , nz. (5.1)

The Cartesian product X × Y × Z = (x, y, z) ∣∣x ∈ X, y ∈ Y, z ∈ Zof these one–dimensional partitions defines the set of discretisationvertices in 3–dimensional space:

P = x ex + y ey + z ez | x ∈ X, y ∈ Y, z ∈ Z. (5.2)

The name tensor–product grid relates to the space FP of discretisedfunctions on P . Each discretised function shall be uniquely defined


by its values on the vertices. Then the one dimensional partitions X,Y and Z generate function–spaces

FX =(f1, . . . , fnx)

∣∣ fi ∈ F ∼= F

nx ,

FY =(f1, . . . , fny )

∣∣ fj ∈ F ∼= F

nx ,

FZ =(f1, . . . , fnz )

∣∣ fk ∈ F ∼= F

nx , (5.3)

respectively (F is either R or C). The vector–space of discretisedfunctions over P , then, is the tensor–product space

FP = FX ⊗ FY ⊗ FZ=(f111, . . . fnx11, . . . , fnxnynz

)∣∣ fijk ∈ F

∼= F

nx·ny·nz . (5.4)

In general, the spanning vectors of a tensor–product grid may enclosearbitrary angles as long as they are linearly independent; if they areorthonormal, the corresponding grid is called rectilinear.A rectilinear grid with axis directions parallel/normal to the main

confinement directions of the device allows us to use the dimensionaldecomposition approach discussed in section 3.1.3. In addition, forsuitable crystal orientations the principal axes of the reciprocal ef-fective mass tensor

[1m∗]are parallel to the axis directions, which

simplifies the discretised expression for ∇ · [ 1m∗ ]∇.Unfortunately this convenience does not come for free: the price

to be paid is smaller flexibility in the grid construction, and typicallyfor the same quality of the solution a tensor–product grid must haveconsiderably more vertices than an unrestricted discretisation meshbecause it does not allow for local refinement. Additional lines in-troduced for refinement in important regions (e.g. the quantum dot)cannot be terminated in a tensor–product grid. Consequently refine-ment may introduce a large number of additional vertices in regionsof the simulation domain, where they are not needed. For example infig. 5.12 the additional lines needed to resolve the cylindrical quantumdot structure result in a large number of extra points in the substrate,that do not add to the quality of the solution.


Figure 5.12: In a tensor–product grid, local refinement (e.g. in thecylindrical quantum dot region) introduces a large number of extrapoints in regions where they do not improve the quality of the result.

5.4.2 Finite Volume Discretisation and DelaunayMeshes

The finite volume discretisation scheme employed by dessis–ise (boxmethod) is a generalisation of finite differences from a regular gridto more general point meshes. In this generalisation the notion ofaxis–directions is lost; but a great increase in flexibility is gained.In a tensor–product grid all (interior) vertices have equivalent neigh-


i

j

dij

Aij

Vi

: Delaunay mesh : Voronoı diagram: Voronoı cell of vertex i •: nearest neighbours of i

Figure 5.13: Finite volume discretisation.

bourhoods: each vertex has 2 nearest neighbours per axis direction;only the distances to the neighbouring vertices may vary. In the boxmethod the number of nearest neighbours for each vertex may be dif-ferent. Figure 5.13 illustrates the set–up for the box discretisation.Each vertex i is associated with a Voronoı box — the region of pointsthat are closer to vertex i than to any other vertex (in periodic gridsthis corresponds to the Wigner–Seitz cell). The discretised versionof a differential equation, then, is obtained by integration over thevolume of the Voronoı cell of each vertex. We will illustrate the pro-cedure with the example of Poisson’s equation ∇2f(x) + u(x) = 0.


α β

i

j

ij

Aij

Vi

Figure 5.14: A Delaunay mesh may contain obtuse angles as long asα+ β ≤ 180.

The discretised equation arising from vertex i is

0 !=∫Vi

(∇2f(x) + u(x))ddx

=∫∂Vi

(n ·∇)f(x) dd−1x+∫Vi

u(x) ddx

≈∑j∈Ni

Aijfj − fidij

+ ||Vi|| ui, (5.5)

where Ni denotes the set of the nearest neighbours of vertex i, dij isthe distance between vertices i and j, and Aij denotes the (oriented)length/area of the edge/face of the Voronoı box, that is normally in-tersected by the line segment ij. Here, “generalised intersection” asin fig. 5.14 is allowed (albeit undesirable). The one–to–one corre-spondence between grid edges ij and Voronoı edges/faces Aij thatwas assumed in writing eq. (5.5) is non–trivial. Discretisation meshes


α β

i

j

ij

Vi

Figure 5.15: This triangulation is non–Delaunay grid (α+β > 180).Here, the Voronoı edge Aij does not exist.

i

Vi

Figure 5.16: Using a different triangulation with the same set of ver-tices restores the Delaunay condition.


that fulfil these conditions are called Delaunay meshes. A more for-mal definition is found in [73]. There, a triangulation is defined asDelaunay, if and only if all simplices (triangles in 2D, tetrahedra in3D) contained in it are Delaunay simplices, i.e. the interior of theircircumsphere does not contain any vertices.Given a set of vertices, there always exists a Delaunay triangula-

tion. In device modelling, however, additional restrictions may ap-ply. At material interfaces, for example, not only the vertices but alsosome of the edges/faces of the triangulation are fixed. Such additionalrequirements may result in the need to insert additional vertices toensure the existence of a Delaunay triangulation compatible with thedevice geometry (cf. section 5.5.3).

5.5 The Coupling Strategy

The strategy for the simulator coupling between simnad and des-

sis–ise has undergone several revisions before it took its current shape.In its earliest version the coupling scheme read like this:

The simulation domain is sub–divided into regions for (semi–)classical (Ωclassical) and regions for quantum–mechanical treat-ment (Ωquantum).

dessis–ise computes charge density, electrical potential and qua-si–Fermi energy in the classical regions. Initially the QM regionsare assumed to be charge free (e.g. by filling them with ‘perfectinsulator’ of the same dielectric permittivity as the actually usedmaterial).

simnad computes the charge densities inside the quantum re-gions using the external potential originating from the dessis–ise

charge density in the classical regions.

– In quantum regions that are in ‘classical’ electrical contactwith classical regions, the local quasi–Fermi energy is de-fined by its value on the boundary with the classical region.

– In quantum regions, that are isolated from their environ-ment except for tunnelling, the quasi–Fermi energy may bedefined by requiring that the current into and out of eachquantum point should cancel (stationary state).

5.5. THE COUPLING STRATEGY 117

simnad computes tunnelling currents between quantum regions.dessis–ise uses the simnad charge density inside the quantumregions and the tunnelling currents to compute a new solutionfor the classical region. The simnad charge density might bemapped to a surface charge density on the boundary of thequantum region; tunnelling currents might be represented bysurface recombination velocities at the domain interfaces.

iterate self–consistently!

5.5.1 Strong Decoupling

The idea in the original approach was to distribute both the chargedensity and the electrostatic potential betwen the two simulators

ρ = ρdessis + ρsimnad, (5.6)φ = φdessis + φsimnad, (5.7)

such that

ρdessis = 0, ∀x ∈ Ωquantum, (5.8)ρsimnad = 0, ∀x ∈ Ωclassical, (5.9)

−∇ · (ε0εr(x)∇φdessis (x)) = ρdessis (x), (5.10)

−∇ · (ε0εr(x)∇φsimnad(x)) = ρsimnad(x). (5.11)

Such a decomposition decouples the self–consistency iteration of theKohn–Sham equations in a fixed external potential from the reac-tion of the external charge density to the potential of the quantum–mechanical charge density. Communication between the simulatorsis only needed when the Kohn–Sham equations are converged, andρdessis and φdessis need to be updated. Since inter–process commu-nication always incurs some overhead, it is desirable to limit it to aminimum. Also, at this stage it was hoped, that it might be possibleto achieve coupling of the simulators on the level of (automaticallygenerated) command files, i.e. without having to modify the actualdessis–ise simulator.It turned out, however, that such a coupling scheme is infeasi-

ble. The reason is the long–range nature of the Coulomb potential.


The potential of the quantum–mechanical charge density ρsimnad hasto honour the boundary condition on the surface of the global sim-ulation domain (i.e. the surface of the dessis–ise grid). As usual inelectrostatics, this boundary ∂Ω is the disjoint union of surfaces ∂ΩDwith Dirichlet boundary conditions (e.g. contacts) and surfaces ΩNwith Neumann boundary conditions. Since eventual inhomogeneousboundary conditions are already satisfied by φdessis, φsimnad must sat-isfy homogeneous Dirichlet conditions on ∂ΩD and homogeneous Neu-mann boundary conditions on ∂ΩN. If we want to restrict the regionhandled by simnad to the quantum–mechanical confinement regionΩquantum — doing so is strongly advisable, since meshing the full de-vice geometry with a tensor–product grid usually results in a hugevertex count —, the boundary of the dessis–ise domain lies outsidethe simnad simulation domain. Therefore, we have to “transport” theboundary conditions from the dessis–ise boundary to the boundary ofthe simnad domain. This may be done by means of the Dirichlet–to–Neumann map (also known as Stekhov–Poincare operator) [74]. Thisis an operator M with the property(

(n ·∇)φ)∣∣Γ=M(φ∣∣Γ

), (5.12)

where Γ is the boundary for which boundary conditions are sought,and n is the surface normal on Γ. In our case Γ = ∂Ωquantum. TheoperatorM may be found by postulating pointwise Dirichlet boundaryconditions on Γ (i.e. φ

∣∣Γis zero everywhere except for a single vertex

on Γ). With each one of these boundary conditions on Γ, Poisson’sequation then is solved on Ωclassical with ρclassical = 0. The valuesof (n · Γ)φ|Γ in the resulting potential are one column in the matrixrepresentation ofM — in essence we are constructing the operatorMby studying its action on a basis of the (discretised) function spaceover Γ. Since dessis–ise solves the non–linear Poisson’s equation, inorder to construct M , all semiconductors have to be replaced withinsulators of equal εr, to make sure that no charge density may arisein Ωclassical.While it is principally possible to construct M by the method

described above, it is tedious and very time–consuming to do so. Theprocedure requires one run of dessis–ise per vertex on the boundary ofΩquantum. In addition, the matrix of M is not sparse. Consequently,incorporating eq. (5.12) as generalised boundary conditions into the


dessis–ise: global Delaunay mesh simnad: local tensor grid

¿ How to exchange data between these ?

Figure 5.17: With the Kohn–Sham equations and Poisson’s equationbeing distributed between the simulators, there is need for frequent dataexchange between the simulators.

matrix for Poisson’s equation in simnad destroys its sparsity. Forthese reasons, the coupling strategy was modified.

5.5.2 Distributed Kohn–Sham Equations

Since decomposing the potential into contributions stemming from theclassical charge density and contributions of the quantum–mechanicalcharge density turned out to be impractical, it was decided that des-

sis–ise should handle the whole electrostatics of the device, whereassimnad should be solely responsible for the computation of quantummechanical charge densities. This requires a closer coupling of thesimulators: since simnad now relies on dessis–ise for the computationof the electrostatic potential — including the potential generated byρsimnad — dessis–ise and simnad must communicate charge densities,potentials and the derivative of the charge density with respect to theelectrostatic potential to each other for each iteration of the Kohn–Sham equations. This makes it too time–consuming to start a new


←

Figure 5.18: Embedding of the tensor grid on Ωquantum in the des-

sis–ise Delaunay mesh.

simulation for each iteration step. Therefore, an interface for directdata exchange at run–time was programmed for each of the simulators.Since simnad and dessis–ise operate on different grid types (cf.

section 5.4), but simnad now requires knowledge of the potential com-puted by dessis–ise in the quantum region, we must define a methodfor the data exchange between the two simulators.

5.5.3 Mesh Merging

One possibility for a data interface between dessis–ise and simnad

consists in mesh merging. Here, the tensor product used by simnad

in the quantum region is embedded as a submesh into the dessis–ise

mesh as shown in fig. 5.18. This mesh merging is a service providedby noffset3d [62, 73]. The procedure is as follows: all vertices insideΩquantum are eliminated from the dessis–ise mesh. Then the verticesand edges of simnad’s tensor product grid are inserted into the des-

sis–ise mesh. Any vertices which because of their proximity to theboundary of Ωquantum are incompatible with the Delaunay criterion,unless edges are eliminated from the tensor product grid, are removed.Then the geometry is re–triangulated keeping the edges of the embed-ded tensor grid fixed.The main advantage of this approach is that inside the quantum

region there exists a one–to–one correspondence between vertices in


the merged dessis–ise mesh and the simnad grid. This eliminatesthe need for interpolation during data exchange, thereby avoiding apotential convergence hazard.Disadvantages of the method are that the merged mesh often is

very large; also, the merging procedure may conflict with refinementcriteria used in the preparation of the original dessis–ise mesh. Inthe nano–flash device example, having the tensor–product grid ex-tend into the channel silicon eliminates the refinement of the originaldessis–ise mesh in the inversion region of the MOSFET. This ruinsthe accuracy of the simulation. Terminating the tensor grid at thematerial interface does not solve the problem, because the first refine-ment layer inside the silicon is so thin that it may introduce pointsthat make the Si/SiO2 interface incompatible with Delaunay criterion(cf. fig. 5.15). Removing the boundary of the tensor grid from thematerial interface by making the simnad region smaller, has a detri-mental effect on the quantum dot wave functions, because they cannotrecover from the artificial Dirichlet condition.This shows that there are situation, for which a Delaunay mesh

compatible with the device geometry, the refinement requirement foraccurate simulation of the classical regions of the device and the re-quirements of the quantum mechanical charge density computationdoes not exist. This means that the mesh merging sacrifices gen-erality of the simulation framework. Hence, alternatives should bepursued.

5.5.4 Data Interpolation Between Meshes

The obvious alternative to using a common (sub–)grid for both simula-tors is maintaining separate grids for each simulator. This introducesthe need to interpolate data during transfer — simple multilinear in-terpolation is applied for mapping of the data between the grids. Theadvantage of this approach is its full generality. Also, each simulatorcan operate on a grid that was tailored towards its needs. simnadmayuse a fine grid in order to resolve high–order single–particle orbitalswithout incurring a run–time penalty on dessis–ise. On the other handdessis–ise does not need to be able to resolve the individual orbitals —typically the total charge density inside a quantum region can be rep-resented with much fewer vertices than the single–particle probability


.bnd/.cmd↓

.cmd/.dop _des.cmd preparation ↓ ↓

coupling⇐⇒.dat

simulation

↓ ↓

visualisation

Figure 5.19: Coupled dessis–ise/simnad simulation framework.

densities, from which it is composed. Since the meshing processesare independent, there are no artificial constraints on the refinementof the dessis–ise grids. It was feared, that interpolating data severaltimes during the Newton iterations might compromise convergence.But no detrimental effects were observed. When the convergence ofthe self–consistent Kohn–Sham equations with simnad alone and inthe coupled dessis–ise/simnad framework were compared, equal it-eration numbers were obtained either way. This coupling scheme iscurrently implemented in dessis–ise (as of release 9.5) and simnad

(cf. fig 5.19).

5.6 Simple Coupling

In early coupling experiments only a one–way data transfer from sim-

nad to dessis–ise was used. A simnad computation was run to ob-tain a quantum mechanical charge density distribution, which wasexported to a file. Then this file would be read by dessis–ise andtreated as a frozen charge density. Figure 5.20 shows the simnad

5.7. SELF–CONSISTENT COUPLING 123

Figure 5.20: The quantum–mechanical charge density has been trans-ferred from simnad to dessis–ise.

charge density after transfer to dessis–ise.

5.7 Self–Consistent Coupling

In self–consistent coupling mode, dessis–ise acts as a master simula-tor, that calls on simnad, whenever need for the computation of aquantum–mechanical charge density arises. Both simulators are runas separate processes in an environment that allows both simulatorsto access common files (i.e. the processes may run either on the samemachine or on different machines sharing a common file system vianfs or samba). A special XSIMNAD block in the command file for thedessis–ise simulator instructs dessis–ise to request data from simnad

at each (or every nth) charge density update. This is done by cre-ating a semaphore file on the shared file system. simnad (runningin server mode) detects the semaphore, resets it, reads in the des-


sis–ise potential and starts the charge–density computation. Whenthe computation is complete, simnad exports charge density and itslocal derivative with respect to the local potential to a file and notifiesdessis–ise of the availability of new charge density data. dessis–ise re-sumes control and continues with the simulation. The procedure isiterated until convergence.

5.8 Simulation Results with Self–ConsistentCoupling

The threshold voltage Vth was defined as the control gate voltagewith Idrain = 10−10A. The increase of Vth brought about by addinga single–electron on the floating gate is shown in fig. 5.21. We found∆V 0→1th = 34mV, ∆V 1→2th = 42mV, ∆V 2→3th = 48mV, and ∆V 3→4th =55mV, which indicates a monotonous increase of ∆Vth with risingnumber of electrons on the dot. The same behavior was found in theexperimental paper by Guo et al. [63]: ∆V 0→1th = 45mV, ∆V 1→2th =55mV, ∆V 2→3th = 65mV.Fig. 5.9 shows that the charging of the floating gate QD starts at

about 6.5V, which is much larger than the threshold voltage Vth ofthe MOSFET. An increase of the channel doping to about 1019cm−3

would result in a Vth of about 6V. The channel doping must be re-garded as a free parameter. Moreover, the random distribution ofdopants has a strong effect in nano–scale devices. At a doping con-centration of 1018 cm−3 there are on average less then 3 dopant atomsin the channel section underneath the floating gate. Against this back-ground, the agreement between simulation and measurement is satis-factory.In order to study the read–out characteristics of the single–electron

memory, the above simulations were run in inhibited equilibrium, i.e.the Fermi energy of the quantum dot was allowed to vary in orderto keep the total quantum dot charge fixed. With the original de-vice of [63] no significant difference was found between characteristicsobtained with

a constant surface charge density,

a constant bulk charge density,

5.8. SELF–CONSISTENT COUPLING RESULTS 125

0 1 2 3 4 5 6 7Control gate voltage [V]

10−19

10−17

10−15

10−13

10−11

10−9

10−7

10−5

Dra

in c

urr

ent

[A]

1 electron3 electrons

0.8 0.9 1.0 1.1 1.2 1.3Control gate voltage [V]

10−11

10−10

10−9

10−8

10−7

Dra

in c

urr

ent

[A]

empty1 electron2 electrons3 electrons4 electrons

Figure 5.21: Top: Idrain vs. Vgate for two electron numbers at Vdrain =50mV. Bottom: close up showing ∆Vth for different discrete chargeson the QD.


a frozen quantum–mechanical charge density (single simnad

run), and

fully self–consistent dessis–ise–simnad coupling,

with the same total charge on the quantum dot.This is probably due to the fact, that the channel edges, where

the current density (in the Drift–Diffusion model without quantumcorrection) is concentrated, are not underneath the quantum dot, butshifted out sideways. Therefore, the geometry was modified slightly,by reducing the width of the channel. In the modified device, constantsurface charge and quantum–mechanical charge density do give riseto visible differences in the transfer characteristics as apparent fromfigure 5.22.These simulations are intended as proof of principle; they illus-

trate, that self–consistently coupled dessis–ise/simnad simulationsdo indeed work. For actual predictive device–simulation more re-fined physical models could be enabled on the dessis–ise side. Thefull functionality of the versatile dessis–ise simulation environmentremains accessible in coupled mode. The new possibility to treatmulti–dimensional confinement effects in the framework of a state–of–the–art semi–classical device simulator will hopefully be greetedby the simulation community as a useful addition to the availablesimulation capabilites.

5.8. SELF–CONSISTENT COUPLING RESULTS 127

1.1

Vgate [V]

I drain[µ

A]

0.8 0.85 0.9 0.95 1 1.050.06

0.07

0.08

0.09

0.10

fully coupled simulationconst. surface charge

#e− = 0

#e− = 1

#e− = 2

Figure 5.22: Transfer characteristics of the modified nano–flash de-vice; comparison of I–V curves obtained with a constant surfacecharge density (dashed) and fully coupled simulations with self–consistent polarisation of the quantum–mechanical charge density(solid).

Chapter 6

Concluding Remarks

This work describes a framework for the simulation of various typesof nano–electronics devices, each with their unique advantages andweaknesses. Device types that can be modelled comprise single–electron transistors, single–electron flash–memory devices and ulti-mately scaled MOSFETs in the ballistic limit.To achieve this, the pre–existing simnad program had to be ex-

tended in various respects. Whereas previously only electrostatic con-finement was supported and many parts of the simulator relied on asingle–valley bandstructure (III–V materials), now there is full sup-port also for pure geometric and for mixed geometric and electrostaticconfinement. Correct treatment of the six–valley bandstructure of sili-con is implemented throughout the program. The domain decomposi-tion approach used for dimensional reduction of the geometry has beenrevisited. Instead of the spatial restriction of the effective Kohn–ShamHamiltonian to each domain the corresponding transfer Hamiltonianis used for the computation of the charge density inside the quan-tum dot; in quantum wires correct decay properties of the charge–density1 as per the transfer Hamiltonian are enforced by a posteriorimodification of the quantum wire charge density. An approximativemethod has been developed to enable adiabatic decomposition of theSchrodinger equation even in the presence of the band–edge disconti-

1when computed in a local–equilibrium approximation

129

130 CHAPTER 6. CONCLUDING REMARKS

nuities, that arise at the material interfaces of geometrically definedquantum dots. By these extensions it is made possible to simulatesingle–electron devices in silicon–on–insulator technology.

Lower temperatures have been made accessible to single–electron tran-sistor simulations by the possibility to use different temperatures forthe computation of the charge density and the computation of the con-ductance. This approach is justified by the observation, that at lowtemperatures the influence of temperature changes on charge densityand wave–functions is minimal, whereas the convergence propertiesof the temperature dependent Kohn–Sham equations are severly de-graded by further lowering of the temperature. In an SET simulationsimnad stores all the quantities present in Beenakker’s conductanceformula in files to make them available for re–computation of theconductance (and equilibrium electron number) with modified tem-perature.

The slowing–down of the conductance computation in large phase–spaces at elevated temperatures has been countered by the introduc-tion of a Monte–Carlo sampling scheme for the computation of thermalaverages.

To study devices that contain both quantum dots and (semi–)classicalchannels, a coupling scheme between the simnad quantum simulatorand the semi–classical simulator dessis–ise was implemented. Betweeniterations the simulators can exchange data, which allows for full self–consistency of the coupled simulation results. This implementation isthe first operational simulation framework capable of the simultaneousdescription of dissipative transport and multi–dimensional quantumconfinement.

In order to estimate the upper limit to the performance of ultimatelyscaled MOSFET devices, a quantum–ballistic transport model hasbeen implemented. While such a model tends to overestimate the on–current, results for the blocking capability of the gate are predictive.Since the quantum–ballistic model describes transport in idealised de-vices, it may serve as a reference for mobility models — a mobilitymodel that predicts currents in excess of the ballistic limit is unlikelyto be correct.

131

This thesis was begun by quoting the famous and tremendously suc-cessful prediction by Gordon Moore. To conclude it, let us take a lookat a prognosis on the future of electronics in computing dating backto 1949.

Where a calculator on the ENIAC is equipped with 18,000vacuum tubes and weighs 30 tons, computers in the futuremay have only 1,000 vacuum tubes and perhaps weigh 1 1/2tons.

— Popular Mechanics, March 1949

Appendices

Appendix A:

Energy of the Inhomogeneous Electron Gas

In density functional theory [14], the total energy of an electron systemE[ρ] contained by an external potential Vext can be written as as afunctional of electron density ρ =

∑i ni|ψi|2 in the following way1:

E[ρ] = T [ρ] + Uext[ρ] + Ee-e[ρ] +Exc[ρ] (A.1)

=∑i

niεkini +∫d3x ρ(x)Vext(x)

+12

∫d3x ρ(x)

∫d3x′ ρ(x′)G(x,x′)︸︷︷︸

=:VH(x)

+Exc[ρ],

where the self–consistent Kohn–Sham orbitals ψi (with associated en-ergies εsci ) are eigensolutions of the Schrodinger–like Kohn–Sham equa-tion (

− 2

2m∇2 + Vs(x)

)ψi(x) = εsci ψi(x) (A.2)

1The potential terms V∗ as well as the Green’s function G of the electrostaticpotential have the dimension of energy rather than of electrostatic potential

133

134 APPENDICES

with the effective potential

Vs = Vext + VH︸︷︷︸=−eφ(x)

+Vxc. (A.3)

If each orbital ψi is occupied by ni electrons, the total Kohn–Shamorbital energy is

∑i

niεsci =

∑i

niεkini +∫d3x ρ(x)Vext(x)

+∫d3x ρ(x)VH(x) +

∫d3x ρ(x)Vxc(x). (A.4)

With this result, the total energy of the electron gas may be recast as

E[ρ] =∑i

niεsci −

12

∫d3x ρ(x)VH(x)

−∫d3x ρ(x)Vxc(x) +Exc[ρ]. (A.5)

135

Appendix B:The Velocity of Bloch States

In this appendix it is shown, that the velocity expectation value of anelectron residing in a Bloch state |ν k〉 may be written in terms of thek–gradient of the bandstructure. The periodic term uν,k of the Blochfunction fulfils the Schrodinger–like equation (cf. [75])(

2

2m

(1i∇+ k)2+ V (r)︸︷︷︸

=:Hk

)uν,k(r) = εν(k)uν,k(r). (B.1)

A small change δk in the k–vector may be treated as a perturbationto the Hamiltonian.

Hk+δk = Hk +2

mδk(1i∇+ k)+O(δk2). (B.2)

The associated energy change is in leading order in δk

δεν =2

mδk∫d3r u∗ν,k(r)

(1i∇+ k)uν,k(r), (B.3)

which may alternatively be written as

∇kεν =2

m

∫d3r u∗ν,k(r)

(1i∇+ k)uν,k(r). (B.4)

The full Bloch state 〈r|ν k〉 is the product of uν,k with exp(ik · r).With the product rule, eq. (B.4) takes the form

1∇kεν = 〈ν k| 1

m

i∇︸︷︷︸

=v

|ν k〉 , (B.5)

with the velocity operator v .2

This shows, that the velocity of an electron in a Bloch state |ν k〉is the group velocity of the dispersion relation εν of band ν evaluatedat the k–vector of the Bloch state.

2That v is the correct expression for a velocity operator may be justified from

its correspondence with the classical expression v = p/m. In a more formal way

one may write v = ddt

r = 1i[r ,H]. For H = 1

2mp2 + V (r), the commutator is

[r ,H] = 2

m∇. Hence, v = 1

m

i∇.

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Curriculum Vitae

Frederik Heinz was born in Konstanz, Germany, on October 11, 1974.He studied physics at the Swiss Federal Institute of Technology (ETH)in Zurich. His studies were supported by the Studienstiftung des deut-schen Volkes. In the academic year 1997/98 he attended PembrokeCollege, Cambridge (UK) with a scholarship by the Dr. Jurgen Ul-derup foundation. In 2000 he received his Dipl. Phys. Degree fromETH Zurich. In the same year he joined the Integrated Systems Lab-oratory as a research and teaching assistant. The current focus of hiswork is on methods and algorithms for the simulation of nano–scalesemiconductor devices. Previous projects comprised theoretical andcomputational topics as well as an extensive experimental study intothe properties of metallic nano–particles.

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