Heinz

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Simulation Approaches

for NanoScale

Semiconductor Devices

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Diss. ETH No. 15435

Simulation Approaches forNanoScale Semiconductor

Devices

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of

Doctor of Technical Sciences

presented by

FREDERIK OLE HEINZ

Dipl. Phys. ETH

born 11 October 1974

citizen of Germany

accepted on the recommendation of

Prof. Dr. Wolfgang Fichtner, examiner

Prof. Dr. Giuseppe Iannaccone, co-examiner

2004

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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 coexamining 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 dessisise 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 IST199910828 (NANOTCAD) with financial support by theSwiss Federal Office for Education and Science (BBW).

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Abstract

Topic of this dissertation are the development and implementation ofa threedimensional simulation environment for semiconductor nano-electronics devices, that are dominated by quantum effects, and themodelling of the properties of various candidate structures for futureultralarge scale integrated circuits. In this context, Coulomb block-ade in the presence of strong quantum confinement, quantumballistictransport 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 dessisise.

Basis of the simulation model is an effective mass formulation ofdensity 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 Bardeens 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, aMonteCarlo approach is used.

On the classically conducting regions of the device the driftdiffu-sion model may be used for current computation. Coupling the devicesimulator dessisise with the simnad quantum mechanics simulatorresults in a simulation tool capable of modelling devices that featureboth classical dissipative currents and 3D quantum effects.

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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 CoulombBlockadeeffekte in Bauelemen-ten mit starken Quantisierungseffekten sowie uber quantenballisti-schen Transport und den Einflu 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-elementsimulatordessisise 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 TransferHamiltonianMethode 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 eineMonteCarlo Methode verwendet.

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x ZUSAMMENFASSUNG

Auf den klassisch leitenden Gebieten lassen sich Strome mittels desDriftDiffusionsmodells berechnen. Durch Koppelung des quantenme-chanischen Simulators simnad an den Bauelementsimulator dessisiseerhalt man ein Simulationswerkzeug, mit dem man sowohl klassischeLeitungsstrome als auch dreidimensionale Quanteneffekte im selbenBauelement behandeln kann.

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Contents

Acknowledgments v

Abstract vii

Zusammenfassung ix

1 Introduction 1

2 SingleElectron Tunnelling Devices 5

3 Challenges for the Simulation of NanoScale SingleElectron 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 Bardeens 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

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xii CONTENTS

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

3.7.1 Simulation of a SilicononInsulator SingleElectron Transistor . . . . . . . . . . . . . . . . 46

3.7.2 Tunnelling Rates and the Anisotropy ofm . . 483.7.3 A SplitGate IIIV HeteroStructure Single

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

4 QuantumBallistic 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 ofS . . . . . . . . . . . 804.3.6 Injected vs. Local Equilibrium Charge Density 824.3.7 Current and Charge Computation at Finite

SourceDrain Bias . . . . . . . . . . . . . . . . 834.4 Simulation Results . . . . . . . . . . . . . . . . . . . . 854.5 Comparison with NonEquilibrium Greens Functions 96

5 A coupled 3D KohnSham /DriftDiffusion Simulation Approach 99

5.1 The Test Device . . . . . . . . . . . . . . . . . . . . . 995.2 DriftDiffusion Simulation . . . . . . . . . . . . . . . . 1005.3 simnadSimulations . . . . . . . . . . . . . . . . . . . 1075.4 A Brief Excurse on Meshing . . . . . . . . . . . . . . . 109

5.4.1 TensorProduct Grids . . . . . . . . . . . . . . 1105.4.2 Finite Volume Discretisation and Delaunay

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

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CONTENTS xiii

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

5.6 Simple Coupling . . . . . . . . . . . . . . . . . . . . . 1225.7 SelfConsistent Coupling . . . . . . . . . . . . . . . . . 1235.8 Simulation Results with SelfConsistent Coupling . . . 124

6 Concluding Remarks 129

Appendices 133A Energy of the Inhomogeneous Electron Gas . . . . . . 133

B The Velocity of Bloch States . . . . . . . . . . . . . . 135

Bibliography 137

Curriculum Vitae 145

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Chapter 1

Introduction

n 1965, when typical integrated circuits (ICs) comprised justaround 50 components, Gordon Moore formulated his famouslaw, 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] (= 50.8 m) back in 1965, mainstream tech-nology has progressed to feature sizes that are limited solely by thewavelength of the light used for illumination in the photolithography.For many years the continuing success of Moores 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

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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 definingand manufacturing ever smaller structures, fundamental physical bar-riers will eventually limit further shrinkage of complementary metaloxidesemiconductor (CMOS) devices the device class on whichMoores law is based by simple scaling [2]. Even now transistorsare being scaled to a regime, in which quantum mechanical 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 fulltwodimensional quantum transport formulation.

Ultimately scaled field effect transistors (FET) will remain themost 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 stateoftheart 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 singleelectrontunnelling devices, supraconducting rapid single quantum flux logic,molecular electronics (possibly formed by selfassembly aided by DNAscaffolding), spinvalve devices and electron waveguide devices.

Therefore, a simulation framework capable of handling both ulti-mate CMOS devices like nanoscale doublegate field effect transistors(FET) in the quantumballistic limit1 or nanoflash RAM devices andquantum confined singleelectron tunnelling devices in the Coulomb

blockade regime has been developed. Only semiconductorbased 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.

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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 supraconducting 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 nanodevices. In the first part (chapters 2 and 3) the main focus is on thesimulation of singleelectron devices in the Coulomb blockade regimeand on the complications that arise when device dimensions are re-duced below the critical lengthscale 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 quantumballistic trans-port. This is a transport mode in which electron transport is fullycoherent and scattering effects other than potential scattering (in-cluding electronelectron scattering by a mean field approach) are ab-sent. This transport regime is usually observed in modulation dopedheterostructures based on IIIV semiconductors at very low tem-peratures, where coherent transport may take place over distancesexceeding a micron. Here, it is also applied to ultrasmall silicononinsulator (SOI) metaloxidesemiconductor (MOS) field effect tran-sistors (FETs). This study elucidates quantumrelated 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 nonidealities.

In the third part (chapter 5) coupling between the simnad quan-tum mechanics simulator [4, 5] and a semiclassical device simulator(dessisise[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 nanoscaleflash RAM device is considered. In such a device both dissipativetransport (along the channel) and multidimensional confinement (inthe quantum dot floating gate) are present. The simulator couplingallows treatment of this situation in a fully selfconsistent way.

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Chapter 2

SingleElectron

Tunnelling Devices

ingleelectron 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 singleelectron 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. 1

To illustrate this phenomenon, let us take a look at the schematicdrawing of a singleelectron 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 nonzero 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 quarkantiquark pairs.

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6 CHAPTER 2. SINGLEELECTRON DEVICES

source drainisland

gate capacitor

tunnel junctions

Vg

Vds

Figure 2.1: Schematic drawing of a singleelectron transistor.

anism in a singleelectron 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 nonzero charging energy2

EC= e2

2Cisland. (2.1)

This charging energy is recovered during processT2. 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 nonconducting.This conductance suppression is called Coulomb blockade. Lifting theblockade at Vg = 0 requires a forward bias|Vds| > e/2Cisland. ForkBT e

2/2Cisland, Coulomb blockade effects vanish; hence, singleelectron devices must either be very small (small Cisland) or operateat cryogenic temperatures (small kBT).

2computed for the unbiased device with initially empty island

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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 processes

T1 andT2 at initial island electron number Nmay be written as

ET1(N) = e

C

e2

+ N e CgVgate-drn+ C2Vsrc-drn

, (2.2)

ET2(N) = e

C

e2 N e + CgVgate-drn+ (C1+ Cg)Vsrc-drn)

, (2.3)

withC=C1+ C2+ Cg. At zero temperature, a process will happenspontaneously only, if its associated energy change is nonpositive.Hence, at T = 0 an electron may tunnel through the island from

source 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 numbersN exist, forwhich both ET1(N) 0 and ET2(N+ 1) 0. Then the islandalternates between the occupation numbersNandN+1, and current

may flow. In linear response (i.e.with infinitesimal sourcedrain 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.

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8 CHAPTER 2. SINGLEELECTRON DEVICES

Q= +e Q= 0 Q= e

Vgate-drn

Vsrc-drn

e

C

eC

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

3e

CgVgate-drn

I

Vsrc-drn2(R1+ R2)

Figure 2.3: Conductance characteristics of a large SET.

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Chapter 3

Challenges for the

Simulation ofNanoScaleSingleElectron Devices

he 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 cutoff is cutoff(Ekin) = h/

2mEkin. In

nondegenerately doped semiconductors almost the entire kinetic en-ergy of the carriers is thermal: Ekin =

12

kBT per degree of free-

dom. Hence, the appropriate wavelength is the thermal de Brogliewavelength th =h/

mkBT of the electrons (holes

1).In degenerately doped semiconductors, the Fermi energy F 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 nonparabolicity, the approximation of a single, energyindependent effective mass forholes must be used with extreme care.

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10CHAPTER 3. NANOSCALE SINGLEELECTRON 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 cutoff wavelength is the Fermi

wavelength F,e = h/2me(F c) (F,h = h/2mh(v F) ),which under the stated circumstances will be much shorter than th.In silicon wires at room temperature the condition that the confine-ment width should be smaller thancutoffrequires structures that areonly a few nanometres in size; in GaAs with its much smaller effec-tive mass, however, the transition to the quantummechanical (QM)regime takes place at much larger structures.

In the QM regime, charge densities and capacitances are modifiedby 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 cutoffwavelength 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 singleelectrontransistor 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 conductancein singleelectron devices with quantum confinement are

1. the selfconsistent quantummechanical 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 finitetemperature generalisation of density functional theory(in an effective mass formulation),

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3.1. COMPUTATION OF THE CHARGE DENSITY 11

2. Bardeens transfer Hamiltonian method,

3. quantum statistical mechanics.

The operation of sincleelectron tunnelling devices in the Coulombblockade regime requires that the conductivity of the device mustbe small compared to the (spindegenerate) conductance quantum

G0 = 2e2

h 77.48

S. Otherwise, the notion of electronson the quan-tum dotbecomes illdefined, 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 singleelectron devices, all these quantities will be extracted neglecting cur-rent flow. The conductance, then, is extracted in a postprocessing

step following a procedure proposed by Beenakker [9].

3.1 Computation of the Charge Density

In a semiconductor, the charge density (x) depends on the localchemical potential (x) or, equivalently, on the positions of the con-duction and valence band edgesc andv relative to the electrochem-ical potential F =(x) e(x). In thermodynamic equilibrium,i.e.in the absence of net currents

0 !=j = F(x), (3.1)

the electrochemical potential is constant in space.3

Consequently, the chemical potential (x) must be modulated bythe electrostatic potential, which implies, that the charge density (x)can be written as a functional of the electrostatic potential (x), whichin turn depends on the charge density by Poissons equation. Themutual interdependence of charge density and electrostatic potential

is expressed by the nonlinear Poisson equation

0r(x)(x)= [](x). (3.2)This equation needs to be solved selfconsistently, i.e.we need to findthe potential and the charge density such that = [] is the

2Devices in this regime are discussed in chapter 4.3F may be nonconstant inside a perfect insulator ( = 0).

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potential produced by via Poissons equation, and = [] is thecharge density resulting from insertion of into the charge densitymodel. The functional form of [] 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 NewtonRaphson scheme.

The iteration procedure starts with some initial guess 0 for thepotential and the corresponding charge density 0 = [0]. If thesedo not already solve eq. (3.2), there will be a nonzero residual

r(x) := 0r(x)0(x) + 0(x). (3.3)In order to improve on the initial guess, the nonlinear Poisson equa-

tion is linearised around 0 by substituting

= 0+ (3.4)

and

= 0+

0

. (3.5)

into (3.2). This yields:

0r0 0

=:r

= 0r +

0

(3.6)

= 0r +

0

1r. (3.7)

From eq. (3.4) we obtain a new potential. The corresponding chargedensity is []. These values are used as 0 and 0 in the next iter-ation step, and the whole procedure (eq. (3.3)(3.7)) is iterated untilconvergence (i.e.until the norm ofr is small enough). In praxi, thingsare somewhat more complicated. Typically, the form of [] 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].

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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 manyparticle 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 potentialVand the electrondensityn.

EV[n] = min! n(x). (3.8)

The functionalEV[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

d3x n(x)V(x) is the potential energy of the electrons in

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

1

2

e2 d3x d3x n(x)G(x, x)n(x), G(x, x) be-ing the Greens function of the electrostatic potential, is the socalledexchange/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 selfinteraction reducing variant [15] of the local density ap-proximation (LDA) is used.

Parametrising the charge densitynatT= 0 in terms of singleparticleorbitals4 i

n=N1i=0

i/22 , (3.10)4Here, spin degeneracy is assumed. In the presence of magnetic fields, or if

spinorbit 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.

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wherex denotes the floor of x, reduces the variational problem(3.8) to a system of Schrodingerlike equations, theKohnSham equa-tions [16, 17]

2

2m2 + Vs(x)

i(x) =ii(x). (3.11)

Here, the effective potentialVs =V + VH+ Vxc takes the place of theexternal potential V in the singleparticle Hamiltonian. Together,the Hartree potential VH(x) :=e

d3x G(x, x)n(x) and the ex-

change/correlation potential Vxc := W[n]/n take care of the effectof electronelectron interaction on the singleparticle wave functions.The KohnSham orbitals i are ordered according to their Kohn

Sham energies i, starting with 0 for the lowest eigenfunction ofeq. (3.11).

At nonzero temperature [18] the above approach needs to be ex-tended. For systems strongly coupled to a reservoir5, FermiDiracstatistics may be invoked

n= 2i=0

f 1

kBT(i F)

|i|2. (3.12)

Here, F denotes the electrochemical potential of the reservoir, and fis the FermiDirac 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 KohnSham orbitals are(quasi)localised and (omitting lifetime broadening) the spectrum

becomes 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; nave application ofFermiDirac statistics using the external electrochemical potentialmay be misleading.

5The effective singleparticle Hamiltonian of such systems exhibits continuousspectrum.

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3.1.2 The Effective Mass Approximation

The electrostatic potential inside a semiconductor consists of two con-tributions that vary on very different lengthscales. The first com-

ponenent, Vcr corresponds to the periodic lattice of the bulk crystal.It is characterised by very large spatial frequencies k > /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 nonperiodicpotential 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 allpositiondependent observables. Instead of the plane waves of the

freeelectron Hamiltonian, a periodic potential gives rise toBloch wavefunctions

x|k =uk(x) eikx. (3.14)

TheBloch factor uk(x) is a function with the periodicity of the crys-tal lattice. Adding a nonperiodic perturbation Vmod to the potentialresults in the mixing of different Bloch states. In the present worka singleband treatment is used6, i.e. it is assumed that the single

particle wave function |may be represented in terms of Bloch func-tions of a singleband0

| k

|0k 0k| . (3.15)

Substituting (3.15) into the Schrodinger equation and premultiplyingwithk| yields

0(k)0k|0+ k

k|Vmod|0k0k| =0k|0 ,(3.16)

6This singleband treatment is strictly justified only for potentials the Fouriercoefficients of which are essentially nonzero 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.

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where 0 is the band dispersion function of band 0. For a slowlyvarying potentialVmod the wave function |contains significant con-tributions only from kvectors in the vicinity of a central wavevectork0. With the approximation

x|(k0+ k) x|k0 eikx (3.17)the Bloch components 0k| of the wave function become the Fouriercoefficientsk|F0 of an envelope functionx| x|0k0x|F0,and the Bloch function matrix elements of Vmod are approximatedwithk|Vmod|k0 . Inserting this into eq. (3.16) yields after re-verse Fourier transform the following equation for the enevelope func-tion [19, 20]

(i) + Vmod(x)

F0(x) =F0(x). (3.18)

Expanding 0(k) in an extremum kv as a quadratic function

0(k) 0(kv) +2

2(k kv)

1

m0,kv

(k kv) (3.19)

introduces the reciprocal effective mass tensor corresponding to valleyv of bandas

1

m0,kv

ij

= 1

2

20(k)

kikj

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 singleparticle Schrodinger equation [21]

2

2

1

m0,kv + V(x)F0,v(x) =F0,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 nonoverlapping kspace regions.

In semiconductor device simulation there is (so far) no concernwith charge density oscillations inside a unit cell of the crystal. There-fore, the Bloch factors may be omitted in the computation of the

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charge density, and|F0,v|2 may be used instead of the probabilitydensity of the actual wave function. Electronelectron interactioneffects can be included according to the discussion of the previoussection by addition of the Hartree and exchange/correlation poten-tials to the nonperiodic potentialVmod in the envelope equation [22].Then the final expression for the electronic charge density inside asemiconductor device is

e(x) = 2ev

i

P0,v,i|F0,v,i(x)|2 (3.22)

with occupation probabilitiesP0,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 heterointerface. In the standardapproach [21] the effective mass is treated as a positiondependentparameter, and continuity of the envelope function and its normalderivative is postulated at the interface. The energy offset of thebandedges on either side of the interface (in the case of the conduc-tion band this is the difference in the electron affinity of the twomaterials) is added as a stepfunction 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,simnadstill 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 threedimensional SchrodingerPoisson prob-

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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 localbulk 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 1dimensional electron gas (1DEG) with confinement in the yz plane,a 2DEG with confinement along thezaxis and a 3DEG with slow po-tential variation in all directions (ThomasFermi gas) are [24, 25, 26]

n1DEG(r) = 12

mx

(kBT) 1

2 i

F 12 F i(x)

kBT |i,x(y, z)|2

n2DEG(r) = 1

mxmy

2 kBT

i

F0

F i(x)

kBT

|i,x,y(z)|2

n3DEG(r) = 1

32

mxmym

z

3 (kBT) 3

2 F 12

F C

kBT

. (3.23)

F denotes the FermiDirac integral of order

F () =

1

( + 1)

0

t dt

exp(t ) + 1 , > 1

1

1 + exp() , = 1(3.24)

F() = F1(), 0. (3.25)Occasionally, formal notations of the form F() with

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Figure 3.1: Domain decomposition of a silicononinsulator singleelectron 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 Schrodingers equation into an array of 1D equations alongthe main quantisation direction and a 2D equation in the remaining

in section 3.1.2.

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positionalongcut1

potential potentialpositionalongcut2

cut1

cut2

Si

SiO2

Figure 3.2: In geometrically defined quantum dots, the adiabatic de-composition of the Schrodinger equation fails, because the 1D wave

functions 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

(x,y,z) = (x, y) + (z), (3.27)

the (KohnSham) effective potential can be written as

V(x,y,z) =

=:s(x,y,z) (x,y,z) + xc(x,y,z) +(x,y,z) (3.28)

=s(x,y,z) + (z)

=:Vx,y(z)

+(x, y),

where Vx,y is a slowly varying function in x and y (i.e. the disconti-

nuities ofV in the xyplane have been excluded from V).

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

2

2 z1

mz z+Vx,y(z) x,y(z) = x,yx,y(z) (3.29)

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

(x,y,z) =x,y(z)(x, y). (3.30)

Substituting this into the l.h.s. of Schodingers equation yields

2

2

1

m

+ V(x,y,z)(x,y,z)=

2

2

(x, y)

(T+ezz)

1

m

(T+ezz) x,y(z)

2 (T(x, y))

1

m

T x,y(z)

x,y(z)T

1

m

T(x, y)

+ V(x,y,z)(x, y) x,y(z)

= 22

(x, y)T

1m

T x,y(z)

+

x,yVx,y(z)

(x, y) x,y(z)

2T(x, y) 1m

T x,y(z)

2

2 x,y(z)

T

1

m

T(x, y)

+ Vx,y(z) + (x, y)(x, y) x,y(z), (3.31)

whereT= ezz =exx+ eyy 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 ofx and y, so are the

x,y(z), and consequently the Tx,y(z) terms may be neglected.

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Thus, we obtain an effective Schrodinger equation for

2

2T 1m T+ (x, y) + x,y(x, y) =(x, y). (3.32)

Unfortunately, the condition imposed on (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 is known to be small).

For example, for a prismatic quantum dot, with an zextensionmuch smaller than its extensions in the xyplane, the electron affinity(x,y,z) may approximately be decomposed into

(z) =(xcentre, ycentre, z) (3.33)

(x, y) =(x,y,zcentre) (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 thexyplane and upwards (downwards) along the zaxis, then thewavenumber 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 (z) and (x, y), thereby apparently dou-bling the energy barrier. This justifies the application of the aboveapproach for the computation of charge densities and both inplaneand vertical tunnelling rates, but precludes its usage in situations inwhich tunnelling processes along diagonal directions are important.

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3.2. COMPUTATION OF TUNNELLING RATES 23

3.2 Computation of Tunnelling Rates

3.2.1 Bardeens Transfer Hamiltonian Method

Bardeens transfer Hamiltonian method [28, 29] is a variant of first or-der timedependent 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 l, Hr = H on r, l

r = . (3.35)

The eigenstates of each of thesetransfer Hamiltoniansare localised onthe same side of the barrier. Bardeens transfer Hamiltonian methodprovides matrix elements between wave functions l andr on eitherside of the barrier. These may be used to compute tunnelling ratesby Fermis Golden Rule

if =2

|Mi,f|2(Ef Ei) (3.36)

for a transition in a discrete spectrum of final states or

if =2

|Mi,f|2 Z(Ei) (3.37)

for a transition into a continuum of states.

Below the computation of the transition matrix elements Mi,fbetween

an initial state(l)i on the left side of the barrier and a final state

(r)f

on the right side of the barrier will briefly be sketched. To do this, weconsider the timedependent Schrodinger equation

i

t(t) = H(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]).

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H

Hl

Hr

(l)

(r)

l r

Figure 3.3: Transfer HamiltoniansHl/r for a systemH with a squarebarrier.

with the approximate ansatz

(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 iEl/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. Att = 0

we assume =|i(l), which in terms of the expansion coefficientsreads a = 1, bk = 0k. 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)

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The remaining task is the computation of the matrix element

Mi,f = (r) f| H E(l)i |i(l)

=:Mi,f=

ddx

(r)f

(x)

H E(l)i

(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 subdomain of r, theintegration domain may be restricted to r:

Mi,f =

r

ddx (r)f

(x)

H E(l)i

(l)i (x). (3.42)

By a similar argument (and from the fact that H is selfadjoint) it isknown that

(l)i (x)

H E(r)f

(r)f

(x) = 0 x r. (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 = r ddx (r)f H E

(l)i (l)i

(l)i H E

(r)

f (r)f

=

r

ddx

(r)f

H

(l)i (l)i H (r)f

+ (E

(r)f E(l)i

(energy conservation)

= 0

) (r)f

(l)i

=

r

ddx

(r)f

H

(l)i (l)i H (r)f

. (3.44)

For a Hamiltonian of the form

H = 2

2

1

m

+ V(x) (3.45)

this may be recast as

Mi,f = 2

2

r

ddx

(r)f

1

m

(l)i (l)i

1

m

(r)f

,

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

Mi,f = 2

2 r dd1x 1

m(r)f

(l)i

n (l)

i

(r)f

n . (3.46)Here, n is the outer unit surface normal vector on ; n = n denotes the directional derivative along n, and m =

n 1

m

n1

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 onedimensional systems this condition iseasily met: localisation of a particle to the left/right of a potentialbarrier with a local maximumV(xmax) =Vmaxis 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 aquantum dotdelimited by local maxima

VxI/IImax= VI/IImax, xImax< xIImax is effected by a potentialVdot=

V(x), xImax< x < xIImax

VImax, x xImaxVIImax, x xIImax

, (3.49)

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

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V Vdot

dot

xImax xImaxx

IImax x

IImax

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 some

pitfalls.If we want to proceed in analogy to the discussion of the previoussection, we need to find an appropriate generalisation of the notion oflocal maximum of the barrier potential. In 1D, the valueVmax of thelocal maximum is the minimum escape energya 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 esc regardless of the dimensionality of . In order todo so, we need the following definition:

Definition 1

A classical path of energy from a to b is a continuous map :[0, 1] , (0) =a, (1) =b such thatV((s))< s.

With this definition the escape energy esc is the lowest energy(or rather, since the set of possible energies is open, the infimum of

energies) for which there exists a classical path from xin to xout.This can be used to obtain a first guess Vsimpledot for the quantum

dot transfer Hamiltonian Hsimpledot . For the region simpledot inside which

the 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 nonparenthesisedwording to a quantum dot.

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V

=

Vsimpledot

:simpledot

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

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

Vsimpledot (x) =

V(x) x simpledotmax

V(x), esc

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 nanowire with a con-

striction (cf.fig. 3.6). If we assume the potentialVSi inside the siliconto be constant, a classical electron of energy > VSi is allowed totravel freely throughout the silicon, and no potential well is found.Quantummechanically, 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 (quasi1D).

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Figure 3.6: A constriction in a silicon nanowire may give rise tolocalised states even with constantV inside the silicon.

quantummechanical transverse kinetic energy of the electron, andconsequently the energy available for longitudinal motion is reduced.If the constrictions are sufficiently narrow (and the distance between

them is not to short) this may give rise to quasibound states betweenthe constrictions.

This effect must be included in the construction scheme for thetransfer 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 directionvwe

compute the (d 1)dimensional groundstate wave function|0(p)

in the plane p :=

r (x r)v = 0 with the restriction of the

Hamiltonian top

H|p= 2

2

1

m

v

1

mv

v

=:Tv

+V(x). (3.51)

The transverse kinetic energytrans (x, v) for a particle atx with clas-

sical directionv then is defined as the expectation value of the trans-verse kinetic energy operator Tv

trans (x, v) :=(p)0| Tv |0(p) (3.52)

=(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.

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In order to enter a point x from direction v, a particle must possesthis extra energy trans (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 thequantumcorrected potential

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

trans(x, v). (3.54)

The allowable paths for the construction of dot arelaterally confinednontunnelling pathsas defined below.

Definition 2

A laterally confined nontunnelling path (LCNTP) of energy fromato b is a differentiable map : [0, 1] , (0) = a, (1) = b suchthatV((s)) + maxvtrans

(s), d

ds(s)

< 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

esc= inf{| a LCNTP of energy from xin to xout} , (3.55)

and the interior of the quantum well is given by

dot= {x | a LCNTP of energy

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

Vdot = V(x), x dot orV(x)

esc

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

nlog = 1 1m +2m

(V

E)

2 . (3.58)

But with its nonlinear 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.

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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 methodto 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 crosssection, 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 crosssectional 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 nonincreasing 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 thesumof quantum dot charge and quantum wire charge.

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a) b)

280 300 320

x [nm]

0.04

0.05

0.06

0.07

energy[eV]

280 300 320

x [nm]

0.04

0.05

0.06

0.07

Figure 3.7: Onedimensional cuts through the eigenstates ofa) the original HamiltonianHb) the improved transfer HamiltonianHdot

[note the suppression of the spurious states byHdot].

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3.3 Conductance Extraction

If the lifetime of the (quasi)localised quantumdot states exceeds the

inelastic 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 singleparticle orbitalsof the quantum dot) this transport process is schematically depictedin fig. 3.8. In order for electron transport through a singleparticleorbital| in the quantum dot (with total electron number N) totake place, the orbital must initially be unoccupied n = 0. Then,the source must supply an electron of energy , which is the energy

needed to add an electron to the originally unoccupied orbital|.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 numbersn of the singleelectron orbitals this may be

EsrcF

EdrnF

src3 drn3

1

2

3

43

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.

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3.3. CONDUCTANCE EXTRACTION 35

approximated by

,{ni} (N) + UNN+1, (3.59)

where (N) is the eigenenergy of the KohnSham orbital| at totalelectron number ofNassuming FermiDirac filling of the individualorbitals according to [18], and UNN+1is the change in KohnShamtotal 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 weighedsum over all possible singleparticle occupation configurations {ni} as

Iin=

e {ni}P({ni})

src f(

,{ni}

EsrcF ) ni,0 (3.60)

The corresponding expression for the current out of the quantum dotis

Iout= e{ni}

P({ni})

drn

1 f(,{ni} EdrnF ) ni,1

(3.61)

The density of states of the leads is already incorporated into thetunnelling rates .

If the tunnelling rates for the individual barriers are low thisis 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 Iinand Ioutis expensiveeven for a single value ofdot. Solving a nonlinear equation in doteven more so not even analytical derivatives for P(n = 1|N) areknown. Also, with increasing sourcedrain voltage the assumption of

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full thermalisation of the electron gas on the quantum dot becomes in-creasingly questionable. Therefore, the treatment of the conductanceof singleelectron transistors is restricted to the linear response regime(infinitesimal difference between Esrc

F

and Edrn

F

). After lengthy cal-culation that has been reported elsewhere [9, 13] the linear responseconductance of an SET is found to be

G= dI

dV

V=0

= e2

kBT

{n}

src drn

src + drn

Peq{ni}n,0f(,{ni} EF).

(3.62)

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

G= e2

kBT

N=0

Peq(N)

src drn

src + drn

Peq(n= 0|N)f

(,{ni} EF)

, (3.63)

or, equivalently,

G= e2

kBT

N=0

Peq(N)

src drn

src + drn

Peq(n = 1|N)

1 f(,{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 are evaluated using the

singleparticle orbitals |(N+1), since these are the orbitals that con-stitute the target state. Accordingly, in (3.64), the|(N) have to beused, since these occur in the initial state.

For small electron numbers, the conditional probability Peq(n =1|N) for the occupation of singleparticle orbital , provided that thetotal electron number is N, differs in shape from the FermiDiracdistribution f

( F)

In the limit of large N it approaches a

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3.4. THERMAL AVERAGES 37

shifted FermiDirac distribution. An efficient numerical method forthe computation ofP(n = 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 FermiDirac occupationfactors were assumed.

3.4 Thermal Averages

The thermal average of the expectation value of an observable A ina quantummechanical 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.= 1

e(HN), (3.66)

while for a canonical ensemble it reads

Pcan= 1Z

eH. (3.67)

Here, = tr (Pg.c.) andZ= tr (Pcan) denote the grand canonical andcanonical partition functions, respectively; = 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.= 1

eHeN

= 1

N

eNeHN, (3.68)

where HNdenotes the restriction of the system Hamiltonian to thesubspace ofNparticle states. That means, that the grand canonical

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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

Ag.c.=N

P(N) AN, (3.69)

whereANdenotes 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| of the Hamiltonian H (and thenumber operator N) with eigenvalues (N)

H | = | (3.70)N | =N |

(3.71)

and expectation values

| A | = | A | , (3.72)

the equilibrium statistical operator becomes diagonal.12

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

tr

Aexp(H)=

| A | | exp(H) |

=

Aexp () . (3.73)

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

form

Acan=

Aexp ()exp ()

, (3.74)

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

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3.5. STATISTICAL MECHANICS OF THE QUANTUM DOT 39

whereas for a grand canonical ensemble we obtain

A

g.c.=

Aexp

( N)

exp( N) , (3.75)which may be rearranged 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}) =({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 < N 0, i N . (3.77)

Nevertheless, arbitrary occupation number vectors {ni|ni {0, 1} i}here are treated as eigenstates of the the manyparticle HamiltonianH.

Within this approximation, both the statistical operators and theorbital electron number operators become diagonal in the{ni} base.Then, the equilibrium probability terms in Beenakkers 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.

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Probability for a fixed vector of occupation numbers{ni}:

Peq({ni}) = 1

e

(({ni})+E( ini) ini), (3.78)

where the grand canonical partition function takes the form

=N

{ni}N

e(({ni})+E(

ini)

ini)

= N eN {ni}N e(({ni})+E( ini))

canonical partition function Z(N)

=N

Z(N) eN

=N

e(F(N)N). (3.79)

Here, F(N) =kBTlog Z(N) denotes the Free Energy for thecanonicalNparticle state. For notational convenience we haveintroduced the symbol{ni}N :=

{ni}i ni = N for thesubspace ofNparticle states.

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

(T, ) =F

T , V , N (T, ) FN(T, ). (3.80)

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

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3.6. MONTECARLO / EXPECTATION VALUES 41

Probability that the system has total particle numberi

ni =N:

Peq(N) = {ni}Peq({ni}) ini,N=

1

{ni}

e(({ni}+E(N)N)ini,N

= 1

{ni}N

e(({ni}+E(N)N)

= e(F(N)N)

N e(F(N)N)

. (3.81)

Probability that the singleparticle orbital|k is occupied, pro-vided that the total electron number is N:

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

{ni}

Peq({ni})nk,1 ini,N

=

{ni}P

eq({ni})nk,1 ini,N{ni}

Peq({ni})ini,N

= e(F(N)E(N)){ni}N

e({ni})nk,1. (3.82)

3.6 MonteCarlo Evaluation of Thermal

Expectation Values

As shown in the preceding section, the probabilities P(ni = 1|N) canbe evaluated as phase space averagesniNover the Nparticle subspace of the phasespace of the quantum dot. Likewise, theP(N)exp

(F(N) N ) may be expressed in terms of phasespace av-erages, by rewriting the Free Energy F(N) =kBTlog Z(N)

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as

F(N) = EN T S(N) (3.83)

in terms of the energy expectation value ENand the entropy S(N) ofthe canonicalNparticle state. The entropy may in turn be expressedas an expectation value as shown below.

S(N) = 1

T

EN F(N)=

1

T

{ni}N

P{ni}|NE{ni} + kBlog Z(N) (3.84)

Since{ni}N P{ni}|N = 1, the second term may be included inthe sum to yield

S(N) = kB{ni}N

P{ni}|N E{ni} log Z(N)

= kB{ni}N

P{ni}|N log 1

Z(N)eE

{ni}

=

kB{ni}NP{ni}|N log P{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 phasespace averaging tractable, constantinteraction is assumed inside eachNparticle segment of phasespace,i.e.EH,Exc andExcpot 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 configurations{ni} at equalNreduces to the difference in KohnSham orbital energyEorbital

{ni}=i nii,N.Still, the computational effort for direct evaluation ofNwill be

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

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to be taken into account will be considerably larger than the num-ber of electrons, thus making the number of possible configurations,card

{ni}N

=

#orbN

, very large. This has given rise to approxi-mations, e.g. restriction to a T = 0 groundstate (Slater rule) orreplacement of the Free Energy Fwith the internal energy Eand useof a shifted Fermi function (with Fchosen such that the total electronnumber Nis reproduced) instead of the Gibbs distribution.

This called for the development of a MonteCarlo (MC) samplingscheme 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 MonteCarloscheme automatically detects the important segments of the phasespace [34, 35], it eliminates the need for ad hocapproximations, andtherefore allows us to study the validity of customary approximations.

The thermal average of A in the canonical state with total electronnumber Nmay be written as

AN={ni}N

P{ni}|NA{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 microstate{ni} in MN is proportional toP{ni}|N. It has been shown by Metropolis [34] that this condition

is automatically met, ifMN is constructed as a sequence of statess1, s2, . . . {ni}Naccording to the following rules:

Transition: sn sn+1:1. Choose a random states

energy: (s).

2. Accepts assn+1 with probability

P =

1, if(s)< (sn),

exp (s)(sn)

kBT

, otherwise.

3. Ifs is not accepted, go back to step 1.

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The finite subsequences MnN :={s1, . . . , sn} ofMNmay be used toobtain estimatesAnN forAN.

When estimating the error

AnN AN

, it is important to note

that the samplessi are correlated. Therefore, the error is augmentedby a factor of 2+ 1 relative to the uncorrelated case, being thecorrelation time of the sequenceA(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 MonteCarlo 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 phasespaces with more than 1025 microstates!). 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 selfconsistent 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 selfcapacitance ofthe QD. The agreement in fig. 3.9 is excellent and proves the accuracyof the MC scheme.

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-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

Occupationprobability(x2)

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).

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3.7 Simulation Results

3.7.1 Simulation of a SilicononInsulator

SingleElectron TransistorThe 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 nanowire 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 capacitively

controlled by a pair of side gates. The whole structure is populatedwith electrons by applying a positive bias of 5.2 V to the back gate.All silicon in this structure is doped with 3 1018 cm3 of arsenic.

Figure 3.10: Simulation geometry of the singleelectron transistorof [27]. Both the silicon surface and an isosurface 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).

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3.7. SIMULATION RESULTS 47

Figure 3.11 shows the Coulomb charging staircase and the linear re-sponse conductance curve of the silicononinsulator (SOI) singleelectron 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 twodimensional 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

conductance[a.u.]

10 K

4.2 K

2.5 V 2.4 V 2.3 V 2.2 V0

1

2

3

4

5

#electrons

a)

b)

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

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simulated device the quantum dot potential well cannot hold 12 elec-trons in quasibound 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 ofm

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 different

singleparticle wave functions of the quantum dot. Where applica-ble, particleinabox 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 andnz in the veticaltransverse direction. For wave functions that cannot be classified inthis way, an isosurface 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 ydirection: 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. Inzdirection the situation is different. The quantum wire enters thequantum dot in the cylindrical bottom section, but is centred alongthe yaxis (cf. fig. 3.10). Therefore, there is no suppression of tun-nelling for wave functions that have evennz. The straight lines joiningseries of states (e.g.111211311411511 for the mmax=mx orien-tation) correspond to an exponential increase of with singleparticleenergy.

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0.03 0.04 0.05 0.06 0.07 0.08

Energy [eV]

108

106

104

102

100

102

104

106

108

Tunnelling

rate

[s

]

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: Sourcedot tunnelling rates of the singleparticle wavefunctions (particleinabox quantum numbers nxnynz shown whereappropriate).

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3.7.3 A SplitGate IIIV HeteroStructure SingleElectron Transistor

Whereas in the silicon device, quantum dot and wires of the SET weredefined by structuring the silicon, in this device a twodimensionalelectron gas (2DEG) forms at the heterointerface between the GaAssubstrate and the AlGaAs layer separating it from thedoping 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 singleelectron transistor (cf.fig. 3.14). Alternatively, only a single gatepairmay 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 heterostructure 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.

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Figure 3.14: Charge density inside the GaAs/AlGaAs heterostructurewhen 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 conductancepeak corresponds to the N = 0 1 transition in the quantum dot.Consequently, the surface pinning energy was adjusted to a value such

that

14

the first electron is found inside the quantum dot in the vicinityof Vgate = 0.75 V. 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 2 mVwould change the electron number on the quantum dot by 1. Thisshows that largescale integration of such devices would require very

14with the assumption of FermiDirac occupation factors

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-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

#electrons

-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

conductance[G

0]

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 0.3 K as reportedby the experimental group in Wurzburg [36]. At this low tempera-ture, however, converge

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