# Modeling of Leakage Currents in High-κ Dielectrics

### Transcript of Modeling of Leakage Currents in High-κ Dielectrics

TECHNISCHE UNIVERSITÄT MÜNCHEN

Institut für Nanoelektronik

Modeling of Leakage Currents inHigh-κ Dielectrics

Gunther Christian Jegert

Vollständiger Abdruck der von der Fakultät für Elektrotechnik und

Informationstechnik der Technischen Universität München zur Erlangung des

akademischen Grades eines Doktors der Naturwissenschaften genehmigten

Dissertation.

Vorsitzender: Univ.-Prof. Dr. Th. Hamacher

Prüfer der Dissertation: 1. Univ.-Prof. P. Lugli, Ph.D.

2. Univ.-Prof. Dr. P. Vogl

Die Dissertation wurde am 29.09.2011 bei der Technischen Universität München

eingereicht und durch die Fakultät für Elektrotechnik und Informationstechnik am

20.12.2011 angenommen.

1. Auflage März 2012 Copyright 2012 by Verein zur Förderung des Walter Schottky Instituts der Technischen Universität München e.V., Am Coulombwall 4, 85748 Garching. Alle Rechte vorbehalten. Dieses Werk ist urheberrechtlich geschützt. Die Vervielfältigung des Buches oder von Teilen daraus ist nur in den Grenzen der geltenden gesetzlichen Bestimmungen zulässig und grundsätzlich vergütungspflichtig. Titelbild: Electronic transport mechanisms in a high-k thin film capacitor

structure Druck: Printy Digitaldruck, München (http://www.printy.de) ISBN: 978-3-941650-37-4

Contents

Abstract 1

1 Introduction 3

2 DRAM Technology 11

2.1 DRAM Basics: Operation Principle . . . . . . . . . . . . . . . . . . . 13

2.2 Introduction of High-κ Oxides . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Atomic Layer Deposition . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Electrical Characterization . . . . . . . . . . . . . . . . . . . . 25

3 Models for Charge Transport 27

3.1 Direct Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Transmission Coefficient and Tunneling Effective Mass . . . . 31

3.1.2 Thermionic Emission and Field Emission . . . . . . . . . . . . 34

3.1.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Tunneling Into and Out of Defects . . . . . . . . . . . . . . . . . . . 41

3.2.1 The Elastic Case . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.3 The Inelastic Case . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Defect-Defect Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 The Elastic Case . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.2 The Inelastic Case . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Poole-Frenkel Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 60

i

ii CONTENTS

4 Methods for Transport Simulations 63

4.1 Kinetic Monte Carlo (kMC) . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Modeling of Leakage Currents:State-of-the-Art vs. kMC . . . 64

4.1.2 History and Development of kMC . . . . . . . . . . . . . . . . 65

4.1.3 Basics of kinetic Monte Carlo . . . . . . . . . . . . . . . . . . 68

4.1.4 A Simple kMC Algorithm . . . . . . . . . . . . . . . . . . . . 73

4.2 Leakage Current Simulations via kMC . . . . . . . . . . . . . . . . . 80

4.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Time Scaling Problem . . . . . . . . . . . . . . . . . . . . . . 83

5 Leakage Current Simulations 85

5.1 High-κ Thin Film TiN/ZrO2/TiN Capacitors . . . . . . . . . . . . . 86

5.1.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.2 kMC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.3 Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.4 Contributions from Different Defect Positions . . . . . . . . . 91

5.1.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Electrode Roughness Effects . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.1 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Structural Characterization of the TiN Electrodes . . . . . . . 101

5.2.3 Generation of Random Rough Electrodes . . . . . . . . . . . . 101

5.2.4 Mesh Generation and Solution of Poisson’s Equation . . . . . 105

5.2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 106

5.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Structural Relaxation of Defects . . . . . . . . . . . . . . . . . . . . . 115

5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3.2 GW Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.3 Defect Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.4 Kinetic Monte Carlo Framework . . . . . . . . . . . . . . . . . 122

5.3.5 First-principles kMC Leakage Current Simulations . . . . . . . 124

5.3.6 Trap-assisted Tunneling Across Oxygen Vacancies . . . . . . . 125

5.3.7 TiN/ZrO2/Al2O3/ZrO2/TiN Capacitors . . . . . . . . . . . . 128

5.3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Conclusions and Outlook 131

CONTENTS iii

Bibliography 135

List of Own Publications and Contributions 154

List of Abbreviations 156

Acknowledgements 159

Abstract 1

Abstract

Leakage currents are one of the major bottlenecks impeding the downscaling efforts

of the semiconductor industry. Two core devices of integrated circuits, the tran-

sistor and, especially, the DRAM storage capacitor, suffer from the increasing loss

currents. In this perspective a fundamental understanding of the physical origin of

these leakage currents is highly desirable. However, the complexity of the involved

transport phenomena so far has prevented the development of microscopic models.

Instead, the analysis of transport through the ultra-thin layers of high-permittivity

(high-κ) dielectrics, which are employed as insulating layers, was carried out at an

empirical level using simple compact models. Unfortunately, these offer only limited

insight into the physics involved on the microscale.

In this context the present work was initialized in order to establish a framework of

microscopic physical models that allow a fundamental description of the transport

processes relevant in high-κ thin films. A simulation tool that makes use of kinetic

Monte Carlo techniques was developed for this purpose embedding the above models

in an environment that allows qualitative and quantitative analyses of the electronic

transport in such films. Existing continuum approaches, which tend to conceal

the important physics behind phenomenological fitting parameters, were replaced

by three-dimensional transport simulations at the level of single charge carriers.

Spatially localized phenomena, such as percolation of charge carriers across point-

like defects, being subject to structural relaxation processes, or electrode roughness

effects, could be investigated in this simulation scheme. Stepwise a self-consistent,

closed transport model for the TiN/ZrO2 material system, which is of outmost

importance for the semiconductor industry, was developed. Based on this model

viable strategies for the optimization of TiN/ZrO2/TiN capacitor structures were

suggested and problem areas that may arise in the future could be timely identified.

Clearly, such a simulation environment should also be of use for the study of follow-

up material systems which, already in the front-end research, show very promising

performance, however, combined with great challenges especially in gaining control

of the leakage currents.

2

Zusammenfassung

Leckströme in High-k Dielektrika sind eines der wesentlichen Hindernisse für die

weitere Miniaturisierung integrierter Schaltkreise. Ein fundamentales, physikalis-

ches Verständnis ihres Ursprungs ist daher sehr erstrebenswert.

In der vorliegenden Arbeit werden mikroskopische, physikalische Modelle für die

relevanten Leckstrommechanismen entwickelt und in einen Simulationsalgorithmus,

basierend auf kinetischem Monte Carlo, integriert. Transportsimulationen wer-

den mit einer Auflösung einzelner Ladungsträger in drei Dimensionen durchge-

führt. Dies erlaubt das Studium räumlich lokalisierter Phänomene wie Perkolation

von Ladungsträgern über punktförmige Defekte oder des Effekts von Elektroden-

rauigkeiten. Schrittweise wird ein selbstkonsistentes, abgeschlossenes Transport-

modell für das technologisch bedeutsame TiN/ZrO2-Materialsystem entwickelt und

aus diesem werden Optimierungsstrategien abgeleitet.

Chapter 1

Introduction

Introducing high-κ (κ: dielectric permittivity) materials has turned out to be one

of the most challenging tasks the semiconductor industry ever had to master. Many

problems have to be solved in order to successfully integrate these novel materials.

As this is at the very core of the present thesis, the first chapter is intended to

provide the reader with a sound overview of the topic. Major problems arising upon

the replacement of SiO2 with high-κ oxides are highlighted, and the topic of the

thesis is put in a wider context.

Referring to the last 40 years, people involved in the integrated circuit (IC) business

nostalgically talk of the "days of happy scaling". Indeed, semiconductor industry

has been able to keep pace with Moore’s lawI up to now. This strong urge to shrink

has been driven by both cost reduction due to more efficient use of substrate area

and improved device performance, e.g. switching speed, coming hand in hand as

circuit dimensions are decreased. In large part this success story is owed to the

unique properties of SiO2. SiO2 can be easily grown in very high quality, i.e. with

very low defect densities, via thermal oxidation of silicon and additionally forms

a smooth, nearly defect-free interface with the underlying silicon. In fact, this is

widely thought to be the major reason why we have a silicon industry today.

However, miniaturization lead to a point where the semiconductor industry hit some

fundamental physical boundaries, the most serious problems being associated with

the transistor gate stack, which consists of the gate electrode (polycrystalline-Si) and

the dielectric layer (SiO2) sitting on the silicon channel. Eventually, the thickness

IIn 1965 Gordon Moore, co-founder of Intel, observing the industry of integrated circuits at

that time, stated his famous law: "The number of transistors on integrated circuits doubles every

two years."

3

4 1. Introduction

of the dielectric film was scaled down to ∼ 1.4 nm, a value so thin that intrinsic

quantum mechanical tunneling currents through the dielectric exceeded a tolerable

level. Due to these currents in combination with the exponentially increasing density

of transistors on a chip power dissipation drastically increased. But, especially

nowadays, energy efficiency is extremely relevant, since low-power electronics gain

more and more importance, e.g. mobile phones, laptops etc. As a result, after more

than 40 years SiO2, at least for some applications, had to be abandonned.

New materials, enabling further device scaling, were searched for. The most urgent

problem to be addressed was the leakage current. The aim was clear: a reduction

of the leakage current without loss in device performance; and the way to go was

predetermined. Field-effect transistors are capacitance-operated devices, i.e. the

source-drain current depends on the gate capacitance C, calculated from

C = ǫ0κA

d. (1.1)

Here, ǫ0 is the permittivity of free space, κ is the dielectric permittivity, A is the

capacitor area, and d is the oxide thickness. Since tunneling currents exponentially

decrease with film thickness, one had to replace SiO2 with a physically thicker layer

of a new material with a higher dielectric permittivity, a so called "high-κ" oxide, to

retain the capacitance and thus the transistor performance. In the year 2007, Intel

introduced its 45 nm technology for their new processor generation, which uses a

completely new material combination for the transistor gate stack - a hafnium-based

high-κ gate dielectric and metal gate electrodes (e.g. TiN)[1].

The above equally applies to the DRAMII industry. Here, due to the ever shrinking

chip size, the available area for the cell capacitor, where information is stored in form

of an electrical charge, is reduced for each new product generation. However, a cer-

tain cell capacitance has to be maintained. Compared to the transistor gate stack,

discussed above, which has a more lenient leakage criterion (∼ 10−2 A/cm2 @ 1 V),

the requirements are much stricter for the DRAM capacitor to ensure that a suf-

ficiently large amount of the stored charge is preserved between two refresh cycles

(∼64 ms). Leakage currents therefore have to stay below ∼10−8-10−7 A/cm2 @ 1 V.

Again, high-κ oxides may resolve this issue, since they allow to increase the capaci-

tance.

Eventually, high-κ dielectrics have found their way to the market and, at least

in certain areas, have replaced SiO2. Today, most of the DRAM producers like

IIDRAM (Dynamic random access memory) is the primary working medium for volatile infor-

mation storage right across the range of electronic systems.

5

Samsung, Hynix, Micron, and Elpida have shifted to zirconium- and hafnium-based

dielectrics[2, 3]. This new class of materials, in principle, provides extendibility of

Moore’s law over the next technology generations. But, in contrast to the well-known

and well-behaving SiO2, these new materials are mostly unexplored. Meanwhile,

the rising commercial interest has triggered huge research efforts dealing with the

problems inherently associated with the introduction of high-κ oxides.

Above, two major fields of application for high-κ oxides in large-scale industrial

production have been identified, the transistor gate stack and the DRAM storage

capacitor. Their integration in the transistor implies a lot of problems which are ad-

dressed by current research, like the growth of unwanted interfacial layers[4], carrier

mobility degradation in the channel due to remote phonon scattering[5], and, not

to forget, reliability issues of the dielectric and Fermi level pinning effects[6], which

indicate that for the introduction of high-κ oxides in the gate stack the replacement

of the polycrystalline-Si electrode by a metal electrode will be a prerequisite. How-

ever, in transistor applications, having quite lenient leakage current requirements,

the key aspect is the electronic transport in the channel and not the transport across

the high-κ layer. This of course will change when, due to further device scaling, gate

oxide thicknesses again reach critical values.

In the following, since it is at the very core of this thesis, we comment on the appli-

cation of high-κ oxides in DRAM storage capacitors. Here, due to the rigid leakage

current criterion, the focus clearly lies on the insulating properties of the dielectric.

Shrinking the DRAM cell dimensions enforces higher capacitance densities to ensure

a sufficient cell capacitance. This is achieved either by decreasing the thickness of

the dielectric film or by utilizing high-κ materials. The ultimate goal is a metal-

insulator-metal (MIM) structure that provides a sufficient capacitance density and

concurrently fulfills the leakage current criterion. Besides this, issues arising due to

integration and reliability have to be controlled.

Since the majority of the high-κ candidate materials are not well-known, no estal-

ished recipes for large-scale production of high-quality films are known. To guarantee

sufficient insulation, the electrode material has to be chosen in accordance with the

dielectric. The electrode work function has to ensure that the potential barrier at

each band, conduction band (CB) and valence band (VB), is larger than ∼ 1 eV in

order to inhibit conduction via Schottky emission of electrons or holes into the oxide

bands. Some oxides additionally need the correct crystalline structure to reach a

high permittivity - e.g. TiO2 needs the rutile phase[7]. Typically, the crystal phase

has to be induced by the bottom electrode acting as growth substrate. In the effort

6 1. Introduction

to minimize chip area, besides downscaling feature sizes, DRAM cells are more and

more designed vertically. Nowadays, storage capacitors with aspect ratios as high as

80:1 are realized in a stack or trench geometry[8], imposing strong requirements on

the conformity of film growth. Atomic layer deposition (ALD)[9], a method of cyclic

deposition and oxidation, allows the most conformal growth of ultra-thin films, even

in these three-dimensional(3D) structures, and thus is the growth method of choice.

The aim of the current ALD research is, starting from precursor chemistry, to de-

velop well-characterized ALD processes and transfer these to the industrial scale.

This has resulted in several new processes for materials of interest to the DRAM

industry, e.g. ZrO2, HfO2, Sr(Ba)TiO3, TiN, Ru, and Pt[10, 11, 12]. Of these, ZrO2,

HfO2, and TiN have subsequently been studied worldwide and have found their way

into the newest chip generations.

Unfortunately, the high-κ oxides are materials with a high intrinsic defect concen-

tration. Due to their higher coordination number and their more ionic bonding

compared to SiO2 their chemical bonds cannot relax easily. In addition, ALD is

known to generally introduce large amounts of impurities, such as C, H, or Cl,

depending on the precursor, which further worsen the film quality.

Figure 1.1: Band gap Eg of high-κ oxides vs. permittivity κ, showing an inverse

scaling behaviour, i.e. Eg ∼ 1/κ. Currently tetragonal ZrO2 (t-ZrO2) is employed

featuring a permittivity close to 40 and a band gap of ∼ 5.5 eV.

Much of the present-day research of the high-κ oxides thus consists of pragmatic

7

strategies to reduce defect densities by processing control and annealing[13, 14, 15].

The high intrinsic and process-induced defect density comes together with the fact

that the κ-value of the candidate oxides tends to vary inversely with the band gap,

see Fig. 1.1, meaning highest-κ materials are rather weak insulators.

Against this backdrop the present work was initialized to identify immediate needs

for action. Viable strategies for leakage current reduction were searched for. A

profound understanding of electronic transport in high-κ films will be a prerequisite

for the realization of future DRAM cell capacitor generations, and it will also be

very useful for the transistor gate stack where leakage currents will become an issue

in the medium term.

Modeling of Leakage Currents: The State-of-the-art

To gain deeper insights into the important transport mechanisms, a huge amount of

experimental leakage current data has been collected for the electrode/high-κ oxide

systems of interest. However, the analysis of the data, so far, typically remained on

the level of simplified compact-models that can be evaluated very fast[16, 17, 18, 19].

Often the functional form of the leakage currents could be reproduced quite well, but

only if unrealistic or even unphysical fitting parameters were used[20, 21]. This is

due to the fact that these models, at best, only grasp certain aspects of the observed

transport and neglect essential parts, describing for instance only one single step in

a multistep transport process. A prominent example is the standard Poole-Frenkel

(PF) treatment[22], which only accounts for the field-enhanced thermal emission of

charge carriers localized at defect states, but does not describe the carrier injection

step. At worst, these models are misused in the wrong context, e.g. the often cited

Richardson-Dushman equation was originally intended to describe thermal emission

of electrons from metals into the vacuum[23, 24]. Parameters extracted from such

an erroneous analysis can be misleading.

More sophisticated transport models were proposed, e.g. the "dead layer" model,

suggesting bulk limitation of the electronic transport in SrTiO3 and (Ba,Sr)TiO3[25].

The drawback of this model is that good fitting of experimental results is only

achieved at elevated temperature (∼ 400 K) and low defect densities, since trans-

port across defect states is not considered. A similar model considers tunneling

through the interfacial layers to dominate the leakage current[26, 27]. For hafnium-

and zirconium-based dielectrics a tunneling-assisted PF conduction mechanism has

been suggested[28] that expands the standard PF treatment. Also a first effort to

embed a hopping transport model via multiple traps in a Monte Carlo algorithm

8 1. Introduction

was undertaken[29].

To summarize, existing approaches usually are tailored to describe certain material

systems and thus focus on certain (isolated) aspects of transport. They fail at

providing a closed model for the transport. Often different transport mechanisms

are present at the same time, and these turn out to be mutually interdependent,

e.g. in case of a charge carrier localized in a defect state which might either escape

via tunneling or thermal emission - competing with each other, the stronger one

mechanism is, the weaker is the other.

Towards the modeling of leakage currents via kinetic Monte Carlo

It was a major goal of this thesis to provide closed models for electronic transport

through high-κ thin films and to incorporate these into simulation tools. A global

picture encompassing all relevant transport mechanisms and their interdependence

was developed. Effects arising in systems with ultra-thin high-κ oxide films, e.g.

the influence of rough electrode/dielectric interfaces leading to local electric field

enhancement or film thinning, or structural relaxation of defects that are involved

in defect-assisted tunneling, were addressed. Modeling and simulation, in contrast

to most of the existing approaches, was carried out in three dimensions to avoid ar-

tifacts due to lower dimensionality. This allowed for the investigation of phenomena

that are localized in all three dimensions like transient degradation of the dielectric

film, i.e. the accumulation of point-like defects, which opens a percolation path for

charge carriers shortening the electrodes. A kinetic Monte Carlo (kMC) approach

was developed, which provides fascinating possibilities, since kMC simulations take

place at the level of individual charge carriers. Complex phenomena like carrier-

carrier interaction and its implications for the transport or transient phenomena

like charge trapping can be investigated in a straightforward manner.

The ultimate goal was to provide the theoretical background that allows for a deep

understanding of the electronic transport in high-κ films and thus supports the

experimental efforts in target-oriented system optimization and the identification of

future device scaling potentials.

Focus was laid on the study of the state-of-the-art TiN/ZrO2 material system for the

DRAM storage capacitor, which is employed in current chip generations. As future

material systems, based on TiO2 and SrTiO3, are still far away from being ripe

for their introduction into the production process, optimization of the TiN/ZrO2

system will be the key enabler for further scaling in the next years.

The structure of the present thesis is as follows. To get the reader, who is maybe

9

not a specialist in DRAM technology, started, chapter 2 provides the basics of

DRAM technology, beginning with the functional principle of a DRAM cell and a

general survey of nowadays DRAM business. Implications of the downsizing efforts

of the semiconductor industry are discussed and the topic of the present thesis is

put in a wider context, showing its significance for this technology field. At this

point, experimental details on the fabrication and characterization of the investi-

gated high-κ capacitor structures are supplied. The following chapter 3 sets the

physical basis for the simulations carried out in the framework of this thesis by in-

troducing the models for the charge carrier transport mechanisms that are relevant

on the nanoscale. Complementary, sensitivity analyses are carried out to establish a

deeper understanding of the system parameters and their significance for the device

performance. Chapter 4 introduces the kinetic Monte Carlo techniques. Providing

some historical background, the generic kinetic Monte Carlo algorithm is derived, its

main advantages but also its limitations are discussed. Subsequently, its implemen-

tation and application for leakage current simulations in high-κ dielectrics is pre-

sented. Chapter 5 may be regarded as the core of this thesis, the traditional "Results

and Discussion" part. Kinetic Monte Carlo leakage current simulations of different

kinds of capacitor structures are presented. They are contrasted with measurement

data. The whole process of developing a transport model for TiN/ZrO2/TiN and

TiN/ZrO2/Al2O3/ZrO2/TiN capacitors is sketched. Higher order effects, such as

electrode roughness, are systematically taken into account. Finally, a brief conclu-

sion is given, intended to sum up the most significant findings of the present thesis.

Possible fields and trends of future research are highlighted in a concise outlook, en-

compassing the application of the simulation framework, established in this thesis,

to future DRAM material systems as well as the adaption of the simulator to other

interesting devices, such as organic solar cells.

10 1. Introduction

Chapter 2

DRAM Technology

DRAM production is a highly competitive business and todays global market is

shared among a handful of huge companies, which are capable to raise the gigantic

amounts of capital needed to built up production facilities and to buy the processing

equipment. DRAM industry traditionally is very cyclic. In good times with high

market prices the top-tier companies can pile up huge profits. In the fourth quarter of

2010, e.g., market leader Samsung controlled roughly 40% of the entire market, which

amounts to a business volume of 2.5 billion euros. In such phases the companies

heavily invest into additional production capacities in order to expand their market

share. Eventually, the market starts to suffer from the resulting oversupply of DRAM

chips, causing a massive plunge of the prizes. In such a situation an intrinsic problem

of the DRAM market emerges. The amount to which the companies can react to

the market situation by reducing their chip production is very limited. Firstly,

decreasing the chip production does not lead to a significant reduction of the overall

production costs, and, secondly, beyond a certain point it is simply not possible,

as shutting down a production line would irreversibly lead to the destruction of

the equipment. Thus, the situation of oversupply remains until the market gains

enough momentum to generate a sufficient request for the chips, which drives the

prizes up again. In short words, DRAM production is one of the harshest business

environments one can think of.

As the functionality that has to be provided by the memory chips is strictly standard-

ized in nearly every detail, the companies do not have the possibility to differentiate

much from each other by providing a superior chip functionality. Thus, to gain an

advantage about the competitors, one has to manufacture the same product at lower

cost.

Grinding down the suppliers of the raw materials, i.e. especially the silicon wafers,

11

12 2. DRAM Technology

2000

2002

2004

2006

2008

2010

2012

2014

2016

2018

2020

2022

2024

0

20

40

60

80

100

120

140

160

180

200

DRAM nodes ITRS Forecast 2010

ha

lf-pi

tch

(nm

)

year of production

Figure 2.1: Evolution of DRAM node half-pitches shown for chip generations since

2000. Predicted values until 2024 are based on the ITRS.

and the manufacturers of the production tools is no viable strategy to pursue, as

the small set of these companies is shared by all the competitors. They would just

switch over to another company. So, the cost advantage has to be designed in.

A rough measure for this, neglecting differences arising to process complexity, is,

how efficiently the silicon wafer area is used. Since a wafer costs the same for all

companies, the one wins, who produces the most chips from one wafer. This urge

keeps the miniaturization process alive and its speed at a high level, as can be seen

in Fig. 2.1. Here, the evolution of the minimum feature size, the so called half-pitch,

over the past eleven years and a forecast based on estimations of the International

Technology Roadmap for Semiconductors (ITRS) are shown.

In the following, we will first, in 2.1, outline the basic operation principle of a single

DRAM cell, and then, in 2.2, we will comment on how this downscaling of DRAM

chips is realized via technology and design improvements.

2.1. DRAM Basics: Operation Principle 13

2.1 DRAM Basics: Operation Principle

Whenever information has to be stored in a way that allows quick access, Dynamic

Random Access Memory (DRAM) is the medium of choice. While the miniaturiza-

tion has driven the DRAM production process to an extreme level of sophistication,

the working principle of the DRAM cell, being very simple, has not changed during

the last decades.

Figure 2.2: Schematic picture of a DRAM cell consisting of a capacitor and a

transistor. Via the wordline a voltage can be applied to switch on the transistor and

allow access to the storage capacitor. Readout and writing of the capacitor is done

via the bitline.

As depicted in Fig. 2.2, a single DRAM cell consists of a capacitor and a transistor,

connected to a grid of perpendicular, conducting lines. The information is stored in

form of an electrical charge on the capacitor, the so called storage node or cell node.

Either the capacitor is charged ("0") or not ("1"). Thus, every cell can store one

bit of digital information. Access to the capacitor is controlled by the transistor.

Its gate is connected to the wordline (WL). By changing the bias applied to the

WL, access to the storage node via the bitline (BL) can be controlled by opening or

blocking the transistor channel.

A readout process is carried out by applying a voltage to the WL, switching on

the transistor channel. The stored information is sensed as a voltage change in the

BL, caused by the charge flowing from the storage node onto it. To write to the

memory, the BL is forced to the desired high or low voltage state, and the capacitor

is charged or discharged.

In reality, due to the non-ideality of the dielectric film, acting as insulating layer

14 2. DRAM Technology

between the capacitor electrodes, the charge stored on the capacitor, i.e. the infor-

mation, is lost over time. Therefore, the information has to be refreshed in regular

intervals. The JEDEC (foundation for developing semiconductor standards) has

defined a refresh cycle length of 64 ms or less. This is the reason why this type of

memory is called "Dynamic" Random Access Memory.

It is one of the central topics of this thesis to investigate the loss mechanisms involved

in the transport of charge across the dielectric layer. These so called leakage currents

are studied in detail, invoking Monte Carlo techniques to simulate the transport

phenomena. A better understanding of the underlying physics is intended to foster

the leakage current minimization and thus to extend miniaturization potentials.

2.2 Introduction of High-κ Oxides

Figure 2.3: SEM cut through

a 90 nm deep trench DRAM cell

(source: Qimonda AG).

While the principle of a DRAM cell is very

simple, as discussed in 2.1, its realization is

very complex due to the urge of minimizing

the wafer area per cell.

Over the last years, this has lead to a growing

degree of vertical integration. Exemplarily,

Fig. 2.3 shows a scanning electron microscopy

(SEM) picture of a cut through a modern

90 nm DRAM cell with deep trench capacitor.

In order to save wafer surface area the capac-

itor is realized as a trench, which is etched

deep down into the Si wafer. The capacitor

electrode, on which the charge is stored, is

marked blue. It is separated by an ultra-thin

dielectric layer (typical thicknesses nowadays

are below 10 nm) from the plate, the grounded

counter-electrode of the capacitor, which is

shared among the individual cells. Here, the

dielectric is only visible as a thin, white line.

Also the WL, running perpendicular to the

cutting plane, and the transistor channel di-

rectly below it can be seen (marked red). In the upper part of the picture, BL and

BL contact to the cell are marked green. The ingenuity of engineers has allowed

2.2. Introduction of High-κ Oxides 15

Figure 2.4: SEM image of 90 nm deep trench capacitors and zoom on the top part

of a DRAM cell (source: Qimonda AG).

aspect ratios as high as 1:80 for the trench capacitor, as can be seen in Fig. 2.4.

Due to the shrinking DRAM cell dimensions the size of every single cell component

has to be reduced. A rigid criterion for the cell capacitor is set by the desired

minimum cell capacitance. In DRAMs the digital information is read by detecting

the change in the BL voltage ∆VBL = αVddCcell/(CBL +Ccell) that is caused by the

charge flow from the storage node onto the BL. Here, α∼ 0.5, Vdd is the operational

voltage, and Ccell and CBL are the capacitances of the cell capacitor and the BL,

respectively. To ensure a reliable information readout, a voltage change ∆VBL ≥0.1 V is necessary to differentiate between "0" and "1". This can be translated to a

minimum capacitance that has to be maintained. So, every loss of capacitor area has

to be counterbalanced by an increase of capacitance per area (capacitance density).

The situation is worsened by the fact that soon a very drastic change in the capacitor

design will inevitably take place. Currently, the cell capacitor is realized as a hollow

cylinder, as shown in Fig. 2.5, allowing to use both the inner and the outer surface

as capacitor area. Within the upcoming node generations the shrinking lateral cell

dimensions will enforce the switch to a block capacitor design. This means a massive

loss of area which will make drastic increases of the capacitance density necessary.

Let us quantify this. Capacitancy density usually is measured in terms of equivalent

oxide thickness (EOT). The EOT gives the thickness of a SiO2 dielectric film required

16 2. DRAM Technology

Figure 2.5: Hollow cylinder and block design for the cell capacitor.

2010

2012

2014

2016

2018

2020

2022

2024

0.0

0.2

0.4

0.6 ITRS forecast

EO

T (n

m)

year of production

Figure 2.6: Prediction of required EOTs for the storage capacitor in the next years

according to the ITRS.

to reach a certain capacitance density. With κ(SiO2)= 3.9 the EOT of a capacitor

using a dielectric with permittivity κ and thickness d, can be calculated from

EOT =3.9κ

·d. (2.1)

The currently best capacitor concepts, suited for large-scale production, feature

EOTs around 0.7 nm[30] and employ a trilaminate ZrO2/Al2O3/ZrO2 dielectric. Fig.

2.6 shows the EOT requirements for the next years according to ITRS estimates,

one year being roughly equivalent to one chip generation. As can be seen, the

downscaling of the capacitor has already fallen two years behind the schedule. Only a

few years after successful integration of high-κ dielectrics the semiconductor industry

2.2. Introduction of High-κ Oxides 17

seems to have reached a point where further scaling proves to be very hard. Via

clever cell architectures, up to now, the requirements could be mitigated.

But why is that? To understand what makes it so hard to increase the capacitance

density and to fulfill the leakage current criterion at the same time, let us have

another glance on the formula for the planar plate capacitor, given in equation

(1.1). In principle, one has two options.

First, one can shrink the thickness of the dielectric layer. But, as the operational

voltage is downscaled only very weakly between two node generations, this approach

leads to higher electric fields being present in the dielectric. These growing fields,

in turn, cause an increase of leakage currents, as the most transport mechanisms

strongly scale with the electric field. Additionally, higher electric fields may cause

early breakdown of the dielectric.

The alternative approach is to increase the permittivity κ, inevitably requiring the

introduction of a new dielectric material, which typically comes along with many

problems connected with, e.g., the development of suitable industry-scale growth

processes and their integration into the overall production flow.

In the past, the latter has proven to be a Herculean challenge, one only meets if

it cannot be avoided. Therefore, SiO2 was scaled down until the limit of quantum

mechanical tunneling was reached. As has been mentioned earlier in chapter 1, there

is an unfortunate correlation between the insulating properties of a material and its

dielectric properties. The higher the permittivity of a material is, the lower is its

band gap, making it a worse insulator. To get a rough feeling of the requirements

on the materials, one can make a simple estimate.

Let us assume that we have introduced a new dielectric material with permittivity

κ and a conduction band offset (CBO) Φ with respect to the Fermi level of the

employed electrodes. The CBO defines the height of the tunneling barrier, as hole

transport usually is many orders of magnitude lower due to the much larger VB

offset. We further assume that the film is perfect, i.e. free of any kind of defects. As

long as we are above the limit of quantum mechanical tunneling, ensured by a suffi-

cient film thickness, in this scenario leakage currents will be dominated by Schottky

emission, i.e. thermionic emission of charge carriers from the electrode into the CB

of the dielectric. This transport mechanism will be discussed in detail in 3.1.2. For

now, we simply estimate the amount of leakage current via the Richardson-Dushman

equation for the current density jSE due to Schottky emission:

18 2. DRAM Technology

Figure 2.7: Current density due to Schottky emission as a function of the CBO,

calculated from eqn. (2.2). The dashed green line marks the current criterion for

the storage capacitor.

jSE =emk2

B

2π2h̄3T2 · exp

− e

kBT

Φ−√

√

√

√

eF

4πǫ0ǫopt

. (2.2)

Here, m is taken as the free-electron mass, F is the applied electric field, and ǫopt is

the optical permittivity of the dielectric material. The requirement on the leakage

current for the storage capacitor is j < 10−7A/cm2 for an applied electric field of

F = 1 MV/cm and a temperature of 125◦C. ǫopt is exemplarily set to 5.6[31], the

value of t-ZrO2, currently the state-of-the-art dielectric in the storage capacitor.

With these values, a minimum CBO of ∼ 1.3 V is necessary, as can be seen in Fig.

2.7.

This value of Φ has to be understood as a minimum requirement. As will be seen,

meeting this criterion alone is by far not sufficient. But, just by regarding the evolu-

tion of the CBO with the introduction of high-κ oxides, one gets a first impression of

the problems, which may arise. In Fig. 2.8 the most promising electrode/dielectric

material combinations are listed together with an estimate of the CBO, which is to

be expected. The displayed values are calculated from the difference of the electrode

work function Φm and the electron affinity Ξa of the dielectric, i.e. it is implicitly

assumed that the interfaces are ideal. This case is called the Schottky-limit[32] and

the listed values thus only constitute an upper limit for the CBO. Realized CBOs

2.2. Introduction of High-κ Oxides 19

Figure 2.8: CBO values for electrode/dielectric material combinations in volts.

Ideal interfaces, i.e. without Fermi-level pinning or charge transfer are assumed.

Thus, the CBO is calculated as Φ = Φm −Ξa, Φm being the electrode work function

and Ξa the electron affinity of the dielectric.

can be significantly reduced due to Fermi level pinning caused by interface states in

the band gap or charge transport across the metal/insulator interface leading to the

formation of an interface dipole.

As can be seen, for the currently employed TiN/ZrO2 material, with a CBO of 2.2 V

in the Schottky-limit, no noteworthy leakage currents due to Schottky emission are

to be expected. In contrast, for the next generation of high-κ materials, i.e. TiO2

and SrTiO3, problems may arise. While Pt promises to be the best suited electrode

for these materials in terms of CBO, its integration into the DRAM production

process simply would be too costly and would also rise a lot of issues connected with

the processing of Pt. And, experimentally, the actual CBO, e.g. at the Pt/TiO2

interface, was found to be only 1.05 V[33]. TiO2 in its rutile high-κ phase has a band

gap of ∼ 3 eV[34]. The fact that the Fermi level of the metal electrode is shifted

towards the CB of TiO2 is caused by the n-type character of TiO2 due to self-doping

with oxygen vacancies[35]. Thus, p-doping is investigated as a measure to increase

the CBO for TiO2 and SrTiO3. For TiO2 it has been demonstrated that doping

with Al increases the CBO by 0.5 V to 1.54 V[36]. This is close to the optimum

value of this material, since a larger CBO would correspond to a too small VB offset

giving rise to hole injection.

Besides shrinking CBOs with increasing κ, other severe problems have to be ad-

dressed, e.g. to ensure growth in the correct crystalline phase. While rutile TiO2

has a permittivity larger than 100, in its anatase phase the permittivity is only

20 2. DRAM Technology

around 40[37]. So far, the only successful approach to reach the rutile crystalline

phase for TiO2 thin films was to induce the correct crystalline structure via the

substrate, i.e. the bottom electrode of the capacitor structure, for which Ru was

used. On top of the Ru a thin, rutile RuO2 layer was formed prior to the deposition

of the dielectric.

And, when extremely high values for the permittivity are discussed, one usually

refers to the bulk value of the materials. Unfortunately, however, especially for the

ultra-high-κ materials discussed above, the permittivity has been found to be thick-

ness dependent. At the metal/dielectric interfaces thin layers with a siginificantly

reduced dielectric constant are formed, so called "dead layers". Obviously, this is

detrimental for the scaling of the capacitor, as it reduces the effective dielectric con-

stant of the whole film. Consequently, many groups have studied this phenomenon

and tried to track down its origin. Both intrinsic and extrinsic effects have been

discussed, e.g. a distortion of the dielectric response of the insulator due to imper-

fections, secondary phases or interdiffusion at the interface, or the finite penetration

length of the electric field into the metal electrode[38]. A detailed overview of the

current knowledge is given in ref. [33].

The multitude of the above-discussed aspects, which have to be considered con-

cerning leakage currents, gives a first impression of the complexity of the task to

downscale the capacitor, while ensuring sufficiently low leakage currents. However,

the above is still just a part of the problems that one encounters on the path of

downscaling. So far, we emanated from the assumption of perfect, defect-free di-

electric thin films. Thus, the above-mentioned problem areas cover only the intrinsic

problems associated with the introduction of high−κ materials, i.e. low CBO, dead

layers, stabilization of the desired crystalline structure.

Now we drop the assumption of perfect, defect-free dielectrics, since it is far from

reality. Indeed, all high-κ candidate materials are known to suffer from high defect

densities[39]. This is due to their high coordination number, hindering the relaxation

and rebonding of broken bonds at a defect site. Often, defects create localized

electronic states in the band gap, so called defect states. These give rise to another

class of transport mechanisms, subsumed under defect-assisted transport (DAT).

Charge carriers, here, are thought to tunnel/hop across defect states, which, acting

as stop-overs, lead to drastically increased transmission probabilities of particles

through the barrier set up by the dielectric. A detailed introduction to the zoo of

transport mechanisms relevant in high-κ thin films will be given in chapter 3. For

now, it is only important to be aware of these additional transport pathways.

2.3. Experimental Methods 21

The above in mind, the steps in optimizing a material system, i.e. a metal/high-κ

dielectric combination, are clear. First, it has to be ensured that intrinsic currents

are sufficiently low by providing the minimum CBO of Φ = 1.3 V. Then, extrinsic

transport has to be dealt with. The research effort, here, is focused on the reduc-

tion of defect densities, e.g. via suited annealing strategies[40] or their passivation

via doping with other elements. Doping with La, Sc, or Al, for example, has been

demonstrated to passivate oxygen vacancies in HfO2 by repelling the defect states

out of the band gap[41] and also doping with F has been shown via ab initio calcu-

lations to have an advantageous effect[42].

But, in order to guide this effort one first has to identify the defects, which open up

the transport channels, i.e. a transport model has to be set up for a material system.

In the present work, we intend to present a scheme how such a model can be system-

atically developed. We lay emphasis on the constant exchange between simulations

and experiment. After an elaborate discussion of the basic transport mechanisms in

chapter 3, we will, in chapter 4, outline a novel, kMC-based simulation approach to

the complex transport phenomena in high-κ films. Having established the routines,

a closed transport model for the currently employed state-of-the-art material system

TiN/ZrO2 will be presented. Such a model is very useful in pointing out strategies

for the minimization of leakage currents.

2.3 Experimental Methods

At this point, although the focus of this work is not on the experimental side, we will

give a concise overview of the process flow for fabrication of the TiN/ZrO2/TiN ca-

pacitors, which are investigated here with respect to their electronic properties. Also

characterization methods will be described. For further details, the reader should

refer to the dissertation of Wenke Weinreich on the "Fabrication and characteri-

zation of ultra-thin ZrO2-based dielectrics in metal-insulator-metal capacitors"[43].

There, a very detailed overview over experimental details is given, also for exactly

those capacitor structures which are investigated here from a theoretical point of

view. All samples investigated in this thesis were fabricated by the Qimonda AG in

collaboration with the Fraunhofer Center Nanoelectronic Technology in Dresden.

22 2. DRAM Technology

Figure 2.9: (a) TEM picture of 2.5 µm deep trenches, etched into a Si substrate

and subsequently coated via ALD with ZrO2 (taken from ref. [43]). (b) TEM picture

of a ZrO2/Al2O3 multi-layer structure, formed via ALD.

2.3.1 Atomic Layer Deposition

Atomic layer deposition is a method for the chemical growth of thin films from the

vapour phase. It is based on alternating, saturating surface reactions[44]. ALD guar-

antees film deposition with a high degree of conformity and homogeneity. Accurate

film thickness control and high reproducibility is ensured via a cyclic growth proce-

dure. The self-limitation of the involved chemical surface reactions allows conformal

coating of three-dimensional structures, as they are found in the case of the DRAM

storage capacitor, which is either realized as a trench, etched down in the Si-wafer

as shown in Fig. 2.9 (a), or in a stack geometry. Composition of a multi-component

material can be controlled with a resolution down to monolayers of atoms (Fig. 2.9

(b)).

As said, ALD is a cyclic process, whereas one growth cycle can be subdivided into

four steps, as sketched in Fig. 2.10. (1) Pulse of educt or precursor A until the

surface is saturated with precursor molecules. (2) Purging of the growth reactor

with an inert gas to remove excessive educts and volatile byproducts. (3) Pulse of

educt or precursor B until the surface is saturated with precursor molecules. (4)

Purging of the growth reactor with an inert gas to remove excessive educts and

volatile byproducts. Steps (1)-(4) are repeated until the desired film thickness is

deposited.

ALD is a strictly surface-controlled process. Thus, other parameters besides reac-

2.3. Experimental Methods 23

Figure 2.10: ALD flow scheme. (top) substrate surface before ALD growth, (1)

pulse of educt A, (2-1) purging of growth reactor with inert gas, (3) pulse of educt

B, (4-1) purging of growth reactor with inert gas, (2-2) repetition of step (2-1), (4-2)

repetition of step (4-1).

24 2. DRAM Technology

tants, substrate, and temperature have no or only negligible influence on the film

growth. Of course, a real ALD growth process suffers from non-idealities, such as

incomplete chemical reactions, decomposition of the precursor materials, and espe-

cially incorporation of impurities in the growing film. The latter are in the case of

high-κ dielectrics comparably high. Electrically active defect states, leading to large

leakage currents, can be the consequence. Therefore, a huge effort is undertaken

in order to optimize the ALD growth recipes (e.g. via the choice/development of

suitable precursors[44]) and minimize impurities.

Fabrication of the Capacitor Structures

For the fabrication of the TiN/ZrO2/TiN (TZT) capacitors, which are investigated

in this thesis, (001)-silicon wafers were coated with a bottom electrode consisting of

10 nm of chemical vapour-deposited TiN, with TiCl4 and NH3 as precursors, grown

at 550◦C[45]. Subsequently, the ZrO2 film was grown by ALD at 275◦C and 0.1 Torr

vapour pressure with a Jusung Eureka 3000, a single-wafer (300 mm) process-

ing equipment of Jusung Engineering (Korea). Here, Tetrakis(ethylmethylamino)-

zirconium (TEMAZr) was employed as metal precursor and ozone (O3) as oxi-

dizing agent. A high growth rate of ∼ 0.1 nm/ALD cycle was reported for this

process[46, 47, 48]. For ALD growth of Al2O3 films Trimethylaluminum (TMA)

was used as metal precursor. The top TiN electrode was deposited at 450◦C by

chemical vapour deposition.

TiN/ZrO2/Al2O3/ZrO2/TiN (TZAZT) capacitors, studied in 5.3, were additionally

subjected to a post-deposition anneal at 650◦C in N2 prior to TiN top electrode

deposition in order to crystallize the ZrO2 layers. As grown, the thinner layers of

ZrO2 (∼ 3.5-4.5 nm) employed in the TZAZT capacitors are amorphous, which is not

surprising, since crystallization of ZrO2 films is known to be thickness dependent[3].

2.3. Experimental Methods 25

2.3.2 Electrical Characterization

For the electrical characterization of the TZT capacitors, temperature-dependent

current measurements were carried out on a heatable wafer holder with a Keithley

4200 parameter analyzer, delay times of 2 s, and a voltage ramp rate of 0.05 V/s.

Additionally, capacitance measurements were carried out using an automated setup

PA300 from SüssMicroTec and an Agilent E4980A, whereas the standard measuring

frequency was 1 kHz, and the voltage amplitude was set to 50 mV. From capacitance

measurements the effective permittivity of a dielectric film can be easily extracted

via

κ=Cd

ǫ0A, (2.3)

whereas C is the measured capacitance, d the thickness of the dielectric film, and A

the capacitor area.

26 2. DRAM Technology

Chapter 3

Models for Charge Transport

In the preceding chapter the multitude of possible mechanisms by which charge

carriers can overcome the barrier for transport set up by the dielectric layer was

already mentioned and divided into two categories, intrinsic mechanims, i.e. those

which are always present, even if a dielectric film of perfect quality is assumed, and

extrinsic mechanisms, which are related to the existence of defect states. As will

be discussed in detail in chapter 4, if one plans to carry out kMC transport simu-

lations, rate models for every relevant transport process are crucial input. kMC, as

will be described, enables the study of the statistical interplay between the different

transport mechanisms. Thus, in this chapter, a detailed discussion of all important

transport mechanisms and their sensitivity towards system parameters and external

parameters is given. This is intended to form the basis for the following investiga-

tions, carried out on the currently most important material system TiN/ZrO2. In

the following we will limit the discussion to electron transport, since hole transport

in the relevant material systems is negligible due to the comparably large VB offset.

The set of transport mechanisms, which are discussed in the remainder of this chap-

ter and are implemented in the kMC simulation tool, is sketched in Fig. 3.1. These

encompass both, intrinsic mechanisms, such as (i) direct tunneling (which merges

into Fowler-Nordheim tunneling for high voltages) and (ii) Schottky emission, as well

as extrinsic mechanisms, such as (iii) elastic/inelastic tunneling of electrons into and

(iv) out of defects, (v) tunneling between defects, and (vi) field-enhanced thermal

emission of electrons from defects, so called PF emission. In an extra section we

will address the role of structural relaxation of defects upon a change of their charge

state, inducing a shift of the defect level energy (not shown).

27

28 3. Models for Charge Transport

Figure 3.1: Schematic band diagram of a MIM structure with applied voltage U .

Transport mechanisms, implemented in the kMC simulator: (i) direct/Fowler Nord-

heim tunneling, (ii) Schottky emission, (iii) elastic and inelastic tunneling (p= num-

ber of phonons with energy h̄ω) into and (iv) out of defects, (v) defect-defect tun-

neling, (vi) PF emission.

3.1. Direct Tunneling 29

3.1 Direct Tunneling

Referring to direct tunneling, one usually means coherent tunneling through an

energetically forbidden barrier region, which, in our situation, is set up by the band

gap of the dielectric material. Thus, the barrier height is determined by the CBO

of the high-κ oxide, and its width is given by the layer thickness. In the following

we will derive the standard expression for direct tunneling, the so called Tsu-Esaki

tunneling formula[49], since it is a very introductive exercise in order to get familiar

with the relevant physics. For an in-depth study the reader is referred to ref. [50].

Figure 3.2: Band diagram for a MIM

structure setting up a tunneling bar-

rier for electrons. Charge flow is indi-

cated. After tunneling thermalization

takes place.

We consider elastic, energy conserving tun-

neling through a barrier that varies only

in transport direction and is constant and

of infinite extent in the two lateral di-

mensions. Conservation of the lateral mo-

mentum of the tunneling electron is as-

sumed. Effects of inhomogeneities, impu-

rities or interface roughness only induce

minor quantitative deviations but do not

lead to a qualitatively different behaviour.

Thus, we neglect them.

The general problem we are dealing with, is

illustrated in Fig. 3.2. Here, a MIM struc-

ture with an applied bias U is shown, set-

ting up a generic tunneling barrier for elec-

trons. The two metal contacts, or, more

general, charge carrier reservoirs, are as-

sumed to be in equilibrium. A Fermi-Dirac distribution of the electrons in the

electrodes is assumed. In case of current flow, this is not exactly true, but again, we

neglect minor deviations arising from this fact. The CB bottom of the left contact is

chosen as zero-reference of the potential energy. Assuming a single, parabolic, and

isotropic CB minimum, we write the energy of the tunneling electron to the left (l)

and to the right (r) of the barrier as

El = Ex,l +Ey,z,l =h̄2k2

x,l

2m∗ +h̄2k2

y,z,l

2m∗ (3.1)

and

30 3. Models for Charge Transport

Er = Ex,r +Ey,z,r =h̄2k2

x,r

2m∗ +h̄2k2

y,z,r

2m∗ +Ec,r. (3.2)

Here, kx and ky,z are the momenta in the transport direction and the two lateral

directions. Ec,r denotes the CB minimum of the right contact. Since we assume

that the lateral momenta are conserved, i.e. ky,z,l = ky,z,r, we arrive at

Ex,l =h̄2k2

x,l

2m∗ =h̄2k2

x,r

2m∗ +Ec,r. (3.3)

After tunneling, the electron is assumed to thermalize. Via inelastic collisions excess

energy is tranferred to the reservoir and the electron loses memory of its previous

state. The overall electron current is calculated, in this picture, as net difference

between the electron current from the left to the right jl and the current from the

right to the left jr.

We consider the current flow in x direction for a given energy E with a component

Ex,l in transport direction. Particles in an infinitesimal volume element of momen-

tum space dkl around kl cause a current density ji,l impinging from the left onto

the barrier, calculated as

ji,l = −eN(kl)fl(kl)vx(kl)dkl. (3.4)

fl denotes the Fermi-Dirac distribution in the left contact and N = 1/(4π3) the

density of states in k-space. The velocity perpendicular to the barrier from the left

is

vx(kl) =1h̄

∂E(kl)∂kx,l

=h̄kx,l

m∗ . (3.5)

To calculate the current density transmitted to the right, equation (3.4) is multiplied

with the transmission probability T (kx,l), which, for the above simplifications, is only

a function of the momentum in transport direction. Thus, we have

jl = − eh̄

4π3m∗T (kx,l)fl(kl)kx,ldkx,ldky,z,l. (3.6)

In an analogous way one arrives for the current transmitted to the left for given

energies E and Ex at:

jr = − eh̄

4π3m∗T (kx,r)fr(kr)kx,rdkx,rdky,z,r. (3.7)

3.1. Direct Tunneling 31

From equation (3.3) it follows that kx,ldkx,l = kx,rdkx,r =m∗dEx/h̄2. If we addition-

ally use the symmetry of the transmission coefficient with respect to the tunneling

direction, we can calculate the total net current flow from

jDT =e

4π3h̄

∫ ∞

0dEx

∫ ∞

0dk′k′

∫ 2π

0dφT (Ex)

[

fl(Ex,k′)−fr(Ex,k

′)]

. (3.8)

As mentioned above, we assume that the electrons in the contacts are distributed

according to the Fermi-Dirac function given by

fl/r(Ex,Ey,z) =1

e(Ex+Ey,z−Ef,l/r)/(kBT ), (3.9)

whereas Ef,l =Ef,r +eU . Again using the assumption of parabolic bands, we arrive

at

jDT =em∗

2π2h̄3

∫ ∞

0dExT (Ex)

∫ ∞

0dE′E′ [fl(Ex,E

′)−fr(Ex,E′)]

(3.10)

=em∗kBT

2π2h̄3

∫ ∞

0dExT (Ex)ln

(

1+ e(Ef,l−Ex)/(kBT )

1+ e(Ef,l−eU−Ex)/(kBT )

)

, (3.11)

the Tsu-Esaki formula for tunneling currents.

All information on the contacts is contained in the effective mass and the logarithmic

term, while information on the tunneling barrier is contained in the transmission

coefficient. As said, some uncritical assumptions were made concerning the electron

distribution within the contacts and the band structure - cf. page 29. Dropping these

does not lead to major modifications of the tunneling current. The crucial point in

carrying out studies of tunneling currents is how one deals with the transmission

coefficient.

3.1.1 Transmission Coefficient and Tunneling Effective Mass

The transmission coefficient T (E) is defined as the ratio of the quantum mechanical

(qm.) current density due to a wave impinging on a potential barrier V (x) and

the transmitted current density. It is a measure for the probability of a particle,

described by a wave package, to penetrate a potential barrier, whereas E is the

energy perpendicular to the barrier.

Tsu and Esaki developed the transfer-matrix method[49], which allows the straight-

forward calculation of T (E). As sketched in Fig. 3.3, calculation of the transmission

through an arbitrarily shaped barrier is carried out by approximating the barrier

as a series of piecewise constant rectangular barriers, for which the transmitted

32 3. Models for Charge Transport

Figure 3.3: Tunneling of a particle with energy E through an arbitrarily shaped

barrier V (x). For numerical computation of the transmission coefficient the barrier

is approximated by a series of rectangular barriers with constant potential.

qm. current density can be easily determined. The integration over this series of

infinitesimal barriers is also reflected in the well-known Wentzel-Kramers-Brillouin

(WKB) approximation (e.g. ref. [51]):

T (E) ≈ e−2∫ x1

x0dx√

2m∗

h̄2(V (x)−E)

1+ 14e

−2∫ x1

x0dx√

2m∗

h̄2(V (x)−E)

2 ≈ e−2∫ x1

x0dx√

2m∗

h̄2(V (x)−E)

. (3.12)

Here, the integration is performed between the classical turning points x0 and x1, at

which E = V (x). The huge advantage of the WKB transmission coefficient is that

its numerical evaluation involves only small computational effort.

However, a point that is often not addressed is the importance of the tunneling

mass m∗, which has frequently been misused as an adjustable parameter to fit ex-

perimentally observed tunneling currents (e.g. ref. [52]). The correct treatment of

the tunneling mass, in contrast, leads to an energy-dependent mass that, just as the

band masses, can be calculated for every material, as will be outlined below.

A particle, say an electron, that tunnels through the band gap of an insulator, is

described by a wavefunction with complex wave vector k, whereas the imaginary

part of k describes the dampening of the evanescent wave within the tunneling

barrier. It enters the above formula for the transmission coefficient, equation (3.12),

and is given by k =√

2m∗

h̄2 (V (x)−E). Thus, T (E) strongly depends on the relation

k(E), which can be extracted from the dispersion relation in the band gap, i.e. the

imaginary band structure (IBS) of the insulator. The IBS is the complex extension

3.1. Direct Tunneling 33

Figure 3.4: Energy dependence of the effective tunneling mass of cubic ZrO2 in the

upper part of the band gap, as it is derived from the dispersion models in equations

(3.13) and (3.14) and tight binding (TB) calculations. The figure is taken from ref.

[53].

of the lowest real conduction band of the insulator and determines the decay of the

tunneling wave function[53].

The simplest possible assumption, i.e. a parabolic (pb) dispersion in the band gap

leads to

kpb(E′) =

√

2mpbE′

h̄, (3.13)

with a constant tunneling mass mpb throughout the whole band gap, as can be seen

in Fig. 3.4. Here, E′ is the negative difference between electron energy and the CB

bottom.

Assuming a Franz-type dispersion[54], one arrives at[55]

kF =√

2mF

h̄

√E′

√

√

√

√1− E′

Eg. (3.14)

Equation (3.14) reduces to a parabolic dispersion close to the CB edge (E′ → 0)

and the VB edge (E′ → Eg, Eg being the size of the band gap) with the same

34 3. Models for Charge Transport

effective mass (see Fig. 3.4). Since VB and CB masses are different in most cases,

Dressendorfer proposed a modified dispersion

kD =√

2mc

h̄

√E′

√

√

√

√

√

1− E′

Eg

1−(

1− mcmv

)

E′

Eg

. (3.15)

This dispersion reduces to a parabolic one with the appropriate masses mc close to

the CB and mv close to the VB. The accuracy can be further increased by performing

theoretical calculations of the IBS, as it has been done for SiO2[56] and recently also

for ZrO2[53], unfortunately in the cubic crystalline phase and not in the tetragonal

phase, which features the highest permittivity.

Since the dispersion of the wavevector in the band gap, equivalent to an effective

tunneling mass, enters the exponents in equation (3.12), the tunneling mass has

a strong impact on the transmission coefficient. This issue will again be referred

to in the following sections, where we introduce rate models, which are necessary

to perform kMC simulations and frequently involve the calculation of transmission

coefficients.

3.1.2 Thermionic Emission and Field Emission

As CBOs of high-κ materials are comparably low (see Fig. 2.8), emission of charge

carriers over the tunneling barrier into the CB of the high-κ film, called thermionic

emission, plays an important role. This process has been studied in detail[57]. In

the metal electrode a free-electron gas is assumed, whereas the electrons follow

the Fermi-Dirac statistics. The major task in order to determine the current due

to thermionic emission is to calculate the transmission probability for the tunneling

barrier depicted in Fig. 3.5. Here, a rectangular barrier V0 (grey) with a CBO of EB

with respect to the electrode Fermi level at EF is shown. Application of an electric

field F tilts the barrier by introducing the additional potential VF (x) = −eFx. Then

the barrier has a slope −eF (dash-dotted line). The study can be extended to take

image charge effects into account. An electron which is emitted from the metal

electrode induces a positive image charge within the metal electrode that attracts

the emitted electron. The attractive force follows a screened Coulomb law, whereas a

traveling distance x of the electron from the metal surface corresponds to a distance

2x between the electron and its image charge. With ǫopt we denote the optical

permittivity of the barrier material. Thus, the image force can be written as

3.1. Direct Tunneling 35

Figure 3.5: Effective barrier shape (black) at a metal-insulator interface. The

rectangular barrier V0 is tilted, if an electric field F is applied (VF ), and further

reduced due to the image potential Vim.

Fim(x) =−e2

4πǫ0ǫopt(2x)2(3.16)

and the image potential Vim (dashed line) is then

Vim(x) = −∫ x

∞Fim(x′)dx′ =

−e2

16πǫ0ǫoptx. (3.17)

Addition of the different contributions to the total potential energy leads to an

effective tunneling barrier with a reduced CBO (drawn in black). We can write the

total potential as

V (x) = V0(x)+VF (x)+Vim(x) = EB − eFx− e2

16πǫ0ǫoptx. (3.18)

From dV (x)/dx= 0 for x= xm we conclude that V (x) has its maximum at

xm =√

e/(16πǫ0ǫoptF ) with a value of

V (xm) = EB −√

√

√

√

e3F

16πǫ0ǫopt−√

√

√

√

e3F

16πǫ0ǫopt= EB −

√

√

√

√

e3F

4πǫ0ǫopt. (3.19)

Thus, the barrier is reduced by ∆ESE =√

e3F4πǫ0ǫopt

.

To accurately calculate the current through the barrier, equation (3.11) has to be

solved for the given barrier shape. While this, in general, is only possible numerically,

for some limiting cases well-known analytical approximations exist[58, 59].

36 3. Models for Charge Transport

Figure 3.6: Comparison of analytical formulas for SE, equation (3.20), and FN

tunneling, equation (3.21), with an accurate numerical solution. A CBO of 1 eV and

a barrier thickness of 10 nm was assumed. SE describes the limit of low electric fields

and high temperatures and FN the limit of high electric fields and low temperatures.

For high temperatures and low electric fields emission of electrons over the tunneling

barrier delivers the dominant current contribution, while for low temperatures and

high electric fields tunneling of electrons close to the Fermi levels dominates. The

former case is called Schottky emission (SE) - if barrier lowering due to the image

potential is taken into account. This limit is described by the Richardson-Dushman

equation[57]

jSE =em∗k2

BT2

2π2h̄3 exp

− 1kBT

EB −√

√

√

√

e3F

4πǫ0ǫopt

. (3.20)

The latter case is known as Fowler-Nordheim (FN) tunneling. An approximate

formula is found, if one calculates the tunneling of electrons close to the Fermi level

at zero Kelvin through a triangular barrier, which is realized at high electric fields.

Then, the current is described by[60]

jF N =e3F 2

8πhEBexp

−4√

2m∗E3B

3h̄eF

. (3.21)

In Fig. 3.6 a comparison of the approximations given in equations (3.20) and (3.21)

3.1. Direct Tunneling 37

with a numerical solution of equation (3.11), as it is implemented in the kMC simu-

lator, is shown. Calculations were carried out for different temperatures and a fixed

CBO of EB = 1 eV. As can be seen, in this case equation (3.21) is a reasonable

approximation for electrical fields larger than 3 MV/cm and temperatures of 200 K

and lower. For higher temperatures, the approximation fails, as it does not account

for temperature scaling of the current at all. Equation (3.20), on the other hand,

is a good approximation for temperatures above 350 K and electrical fields below

∼ 1.5 MV/cm. In the intermediate transition regime only the numerical solution

delivers accurate results.

There is often the misunderstanding that direct tunneling, FN tunneling, and SE

describe qualitatively different transport mechanisms. Thus, to conclude, we point

out that they are all contained in the general treatment of electron emission through

a tunneling barrier leading to equation (3.11). Sometimes it is convenient to study

them separately. This can be done, e.g. by splitting the integral over Ex, the electron

energy perpendicular to the barrier, in equation (3.11), as discussed in [59]. SE, for

example, denotes emission over the tunneling barrier, and thus one only integrates

over energies Ex close to the maximum energy of the barrier. However, if not stated

otherwise, in the remainder of this thesis we will refer to all these three regimes as

direct tunneling.

3.1.3 Sensitivity Analysis

If one aims at developing transport models based on simulations, it is important

to know how sensitive the simulation results are towards changes of the adjustable

parameters. In general, changes in simulation parameters should be carried out

with outmost care, and the plausibility of the parameter values should always be

scrutinized, especially, if the simulations react very sensitively towards the change

of a certain parameter. In the case of kMC the simulation algorithm is a purely

stochastical procedure in the sense that it works correctly from a mathematical point

of view. Physics enter this stochastic framework via rate models, which are discussed

in this chapter. The validity of the models itself and of the input parameters which

enter the simulations with the rate models has to be ensured by the scientist carrying

out the simulations, cf. chapter 4. A sensitivity analysis of the model parameters

has proven to be very useful in this context and thus is provided for all models

implemented in the kMC simulator, beginning with the above-discussed model for

direct tunneling.

We regard formula (3.11), stated again below for convenience:

38 3. Models for Charge Transport

jDT =em∗kBT

2π2h̄3

∫ ∞

0dExT (Ex)ln

1+ e(Elf −Ex)/(kBT )

1+ e(El

f −eU−Ex)/(kBT )

.

Apart from the effective electron mass in the contacts, which we keep fixed during

all simulations, there is no further explicit dependence on input parameters. All

information on the investigated structure is implicitly included in the transmission

coefficient T (Ex). T (Ex) contains information on the barrier shape, i.e. the bar-

rier thickness, its height (CBO), and the barrier material, determining, e.g., the

tunneling effective mass or ǫopt.

In Fig. 3.7 the effect of a change of the CBO between electrode and dielectric

material on the current flow is illustrated. For a barrier of 10 nm thickness EB was

varied between 0.8 eV and 1.6 eV. The tunneling effective mass was treated according

to equation (3.15), whereas for mc and mv values of 1.16·m0 and 2.5·m0 were used,

corresponding to t-ZrO2[61], and the transmission coefficient was calculated in the

WKB approximation, given in equation (3.12). ǫopt in equation (3.18) was set to

the literature value 5.6 for t-ZrO2[31].

At 125◦C and for lower CBOs also at 25◦C SE dominates in the whole field region

studied. For higher barriers (EB → 1.6 eV) FN tunneling becomes dominant for F >

2.5 MV/cm, visible as an increase of the slope of the current-voltage(IV)-curves. The

higher the CBO, the more important becomes FN tunneling. As to the sensitivity

of the tunneling current, a reduction of ∆E = 0.1 eV in the CBO corresponds to

an increase of the current by one and a half orders of magnitude. If FN tunneling

dominates, a slightly weaker current increase by one order of magnitude is observed.

Both SE and FN tunneling do not depend on the barrier thickness, provided that

for different thicknesses identical electric fields, and not identical voltages, are com-

pared. This can be understood regarding Fig. 3.8. SE (Fig. 3.8 (a)) only depends

on the effective barrier height and the electron distribution within the contact, since

the electrons are emitted over the barrier. In the case of FN tunneling (Fig. 3.8

(b)), electrons penetrate through a triangular barrier and are injected into the CB of

the dielectric. The tunneling distance is determined by the classical turning points

and thus does not depend on the total thickness of the dielectric. Any thickness

dependence observed for direct tunneling has its origin in the electrons at lower

energies Ex, which tunnel through a trapezoidal barrier, as shown in Fig. 3.8 (c).

In this case an exponential dependence on the barrier thickness is to be expected.

However, this contribution to the total direct tunneling current only becomes dom-

inant for extremely thin films (e.g. ∼ 3 nm for t-ZrO2 on TiN electrodes) and very

3.1. Direct Tunneling 39

Figure 3.7: Direct tunneling current was calculated at temperatures of (a) 25◦C

and (b) 125◦C for a 10 nm thick tunneling barrier consisting of t-ZrO2. EB was

varied between 0.8 eV and 1.6 eV.

40 3. Models for Charge Transport

Figure 3.8: Different mechanisms subsumed under the general term direct tun-

neling. (a) SE. Electrons with large thermal energy are emitted over the tunneling

barrier. (b) FN tunneling. At high electric fields electrons with a lower energy tun-

nel through a triangular barrier (marked orange). (c) At low electric fields, electrons

with energies around the electrode Fermi level face a trapezoidal tunneling barrier

(marked orange).

3.2. Tunneling Into and Out of Defects 41

low temperatures, i.e. for conditions which have not been part of this work.

3.2 Tunneling Into and Out of Defects

Besides intrinsic conduction mechanisms, as discussed in 3.1, especially in high-κ

oxides with large defect densities extrinsic conduction mechanisms are important.

A first step in DAT is, naturally, always the injection of electrons from the electrode

into a defect state. First studies of such processes appeared in the early seventies[62,

63].

Figure 3.9: (a) Injection and (b) emission of an electron with energy Ee + ph̄ω

into/out of a defect state with depth ED. Both elastic tunneling (red) and inelastic,

multiphonon-assisted tunneling (black) under emission or absorption of p phonons

are shown.

As shown in Fig. 3.9, tunneling can be both an elastic process or an inelastic

one, in the latter case under emission or absorption of p lattice phonons of energy

h̄ω (single-mode approximation), whereas ω is a typical phonon frequency. Both

kinds of electron injection into and emission out of defects will be discussed in the

following.

3.2.1 The Elastic Case

Lundström et al. determined the probability for a metal electron to tunnel into

a defect state[62] using Bardeen’s method[64], according to which the transition

matrix element can be calculated as

TED =∫

ψ∗D(H−Ee)ψEdr, (3.22)

where ψE and ψD are the wave functions of the electron in the metal electrode

and of the trapped electron, respectively. H is the complete Hamiltonian of the

42 3. Models for Charge Transport

system and Ee the total electron energy. The transition probability per unit time p

for a constant energy E|| = h̄2k2||/(2mE) in the lateral dimensions then follows from

Fermi’s golden rule, i.e.

p=2πh̄

|TED|2(

dNE

dE

)

E=Ee

. (3.23)

Here, NE(E) is the electron density of states and mE the electron mass in the

electrode. Integration over k|| delivers the total transition probability per unit time

P . Assuming a three-dimensional delta-like defect potential, Lundström arrived

at[62]

P =4h̄mE

3m2i

Kk30

K2 +k20

(

1− 25x0k

20

K

)

exp(−2Kd). (3.24)

Here, mi is the electron mass in the insulator, K2 = 2miED/h̄2, with the defect

depth ED (see Fig. 3.9), k20 = 2mEEe/h̄

2, and the distance x0 of the defect from

the electrode. One can also derive an alternative expression for P , where P is

reformulated as a product of a time constant and a transmission coefficient[65],

which can be easily computed, e.g. in the WKB approximation. Then, we obtain

P =43h̄mEk

30

m2iK

T (Ee) =(

me

mi

)5/2

8E3/2e

3h̄√ED

T (Ee), (3.25)

an expression which, for convenience, was used in later works, e.g. to study the

enhanced conductivity of nitrided SiO2 films[66, 67]. To arrive at the final rate for

electron injection into the defect, P has to be weighted with the probability that

the electronic states in the electrode at energy Ee are occupied, i.e. f(Ee), given in

equation (3.9). Thus, we arrive at

RED =(

me

mi

)5/2

8E3/2e

3h̄√ED

f(Ee)T (Ee) = C ·f(Ee)T (Ee), (3.26)

with C =(

memi

)5/2(

8E3/2

e3h̄

√ED

)

. In an analogues manner, release of trapped electrons

to the electrode is calculated via

RDE =(

me

mi

)5/2

8E3/2e

3h̄√ED

(1−f(Ee))T (Ee) = C · (1−f(Ee))T (Ee), (3.27)

whereas (1−f(Ee)) is the probability that the electronic states in the electrode at

energy Ee are empty.

3.2. Tunneling Into and Out of Defects 43

3.2.2 Sensitivity Analysis

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0103

104

105

106

107

108

109

1010

R

ED(s

-1)

ED(eV)

(a)

Figure 3.10: (a) Rate RED for injection of electrons into defects as a function of

defect depth ED. The defect was placed in a distance of 2 nm from the electrode.

EB was set to 1 eV, and temperature was set to 25◦C. (b) Sketch of electron injection

into a defect for varying defect depth.

Since C is a slowly varying function of Ee and ED, the product f(Ee)T (Ee) is in

the focus of the sensitivity analysis. The interplay of the two factors governs the

behaviour of the injection rate RED. We calculated RED for a barrier with a CBO

44 3. Models for Charge Transport

of EB = 1 eV and a defect in a distance of 2 nm from the electrode. The defect

depth ED was varied between 0.3 eV and 1.7 eV, as can be seen in Fig. 3.10 (b). A

maximum rate is observed, for a defect aligned with the electrode Fermi level. The

rate strongly decreases for shallower defects and weakly decreases for deeper defects.

For shallow defects the tunneling barrier is small, thus T (Ee) is large. But this is

overcompensated by the low occupation probability f(Ee) for energies that are far

above the Fermi level. For defects below the Fermi level and defects aligned with

the Fermi level the number of electrons impinging on the tunnel barrier is nearly

equal. Thus, a moderate reduction of the injection rate is observed, which is due

to the higher tunneling barrier for the deeper defects. The higher the temperature,

the more the maximum rate will shift towards shallower defects.

It is important to keep in mind that, if an electric field is applied, the bands of the

insulator are tilted. Consequently, for growing fields, shallow defect will move closer

to the Fermi level, while deeper defects will be pushed further down, as shown in

Fig. 3.11. In Fig. 3.11 (a), calculated injection rates under an applied electric field

of up to 4 MV/cm are shown for a shallow defect with ED = 0.5 eV and a deeper

one with ED = 1.0 eV. For F ∼ 2.5 MV/cm we observe a maximum injection for

the shallow defect. At a distance of x= 2 nm from the electrode the potential drop

due to F amounts to ∆V = eFx= 0.5 eV. Thus, the defect is perfectly aligned with

the Fermi level. For higher fields, as can be seen in Fig. 3.11 (b), the tunneling

barrier increases and the injection rate is slightly reduced. The deeper defect with

ED = 1 eV, accordingly, has its maximum injection rate at F ∼ 0 and RED slightly

decreases for higher fields.

3.2. Tunneling Into and Out of Defects 45

0 1 2 3 4104

105

106

107

108

109

1010

1011

1012

0.5 eV 1.0 eV

RE

D(s

-1)

F (MV/cm)

(a)

ED

Figure 3.11: (a) Rate RED as a function of the electric field F for two defect depths

ED = 0.5 eV and ED = 1.0 eV. The defect was placed in a distance of 2 nm from the

electrode. EB was set to 1 eV, and the temperature was set to 25◦C. (b) Change

of the barrier shape due to the applied field. Orange colour marks the barrier for

ED = 0.5 eV and green colour for ED = 1.0 eV.

46 3. Models for Charge Transport

3.2.3 The Inelastic Case

We describe inelastic tunneling from an electrode contact into a defect and the

reverse process of emission of trapped electrons into the electrode as a multi-phonon-

assisted tunneling process. A first analysis of tunnel-assisted release of trapped

electrons into the electrode was carried out by Kiveris et al.[68], which was later

on augmented by Dalidchik[69], who refined the multi-phonon transition theory and

coupled it to WKB tunneling expressions. Herrmann further expanded the model by

calculating Shockley-Read-Hall lifetimes, assuming that electron capture/emission

is a multi-phonon-assisted transition between a free-electron state in the electrode

and a localized state at the defect site[70]. We followed his treatment, adapting it

to our needs.

As depicted in Fig. 3.9, we consider a defect state at energy Ee(x), which, in presence

of an electrical field F , depends on the x coordinate of the defect. Coupling to the

phonons of the insulator is described in the single-mode approximation, i.e. we

replace the coupling to the various phonon modes of the insulator by a coupling to

only one effective phonon mode, characterized by the effective phonon energy h̄ω

and two coupling constants, the lattice relaxation energy Elattice and the Huang-

Rhys factor S. Consequently, we obtain a series of virtual states of the coupled

electron-phonon system at energies Ep = Ee + ph̄ω. Each of these states can be

initial or final state of a multi-phonon-assisted tunnel process.

We express the rates for multi-phonon-assisted tunneling into (RMPED ) and out of

defects (RMPDE ) via characteristic times, needed to occupy or to empty a defect,

i.e.[70]

RMPED =

1τMP

ED

, (3.28)

RMPDE =

1τMP

DE

. (3.29)

Unlike in former treatments where the interest was focused on the steady-state con-

ditions of the system, i.e. the equilibrium occupation probability of the defect[70],

we intend to explicitly study the dynamical behaviour of the system. Besides, a

steady-state analysis in the framework of our kMC simulations would also have to

encompass other transport mechanisms that are present at the same time and, com-

peting with multi-phonon-assisted tunneling, influence the steady-state occupation

probability of the defect.

The time constants given in equations (3.28) and (3.29) are calculated from[70]

3.2. Tunneling Into and Out of Defects 47

1τMP

ED

=∫ ∞

−∞NE(E)f(E)TED(E,x)cED(E,x)dE (3.30)

1τMP

DE

=∫ ∞

−∞NE(E) [1−f(E)]TDE(E,x)cDE(E,x)dE. (3.31)

Here, NE(E) is the density of states in the electrode,

NE(E) =1

2π2

(2mE

h̄2

)3/2√

E−Ec ·Θ(E−Ec) (free-electron gas), (3.32)

where Θ is the Heaviside function andEc is the CB edge of the electrode. TED(E,x) =

TDE(E,x) ≡ T (E,x) is the transmission coefficient through the tunnel barrier be-

tween the electrode and the defect at position x for electron energy E. cED and

cDE denote the characteristic electron-phonon interaction rates,

cED(E,x) = c0

∞∑

p=−∞,p 6=0

Lp(z)δ (E−Ep(x))

exp(

−Ee(x)−EkBT

)

, if Ee(x)−E > 0,

1, otherwise,(3.33)

and

cDE(E,x) = c0

∞∑

p=−∞,p 6=0

Lp(z)δ (E−Ep(x))

exp(

−E−Ee(x)kBT

)

, if E−Ee(x)> 0,

1, otherwise.(3.34)

For a 3D, delta-like defect potential c0 has been estimated to[71, 72]

c0 =(4π)2r3

D

h̄Eg(h̄E0)3. (3.35)

Here, rD = h̄/√

2miED is the localization radius of the electron in the defect with

depth ED. h̄E0 denotes the electro-optical energy in the insulator,

h̄E0 =

(

e2h̄2F 2

2mi

)1/3

, (3.36)

and Eg is the band gap of the insulator material.

Apparently, only energies E = Ep = Ee(x) + ph̄ω, where p is the integer phonon

number, are allowed for the multi-phonon-assisted processes. This is an immediate

consequence of the single-mode approximation. In this picture the multi-phonon

transition probability Lp(z), introduced in equation (3.33), can be calculated as

48 3. Models for Charge Transport

Lp(z) =

(

fBE +1fBE

)p/2

exp [−S(2fBE +1)]Ip(z), (3.37)

where Ip(z) denotes the modified Bessel function of order p with the argument

z = 2S√

fBE(fBE +1), and fBE is the Bose-Einstein distribution, i.e.

fBE =1

exp(

h̄ωkBT

)

−1. (3.38)

Inserting equations (3.33) and (3.34) into equations (3.30) and (3.31) delivers

RMPED =

∞∑

p<0

c0NE(Ep)f(Ep)T (Ep,x)Lp(z) · eph̄ωkBT

+∞∑

p>0

c0NE(Ep)f(Ep)T (Ep,x)Lp(z)(3.39)

RMPDE =

∞∑

p<0

c0NE(Ep) [1−f(Ep)]T (Ep,x)Lp(z)

+∞∑

p>0

c0NE(Ep) [1−f(Ep)]T (Ep,x)Lp(z) · e−ph̄ωkBT .

(3.40)

3.2.4 Sensitivity Analysis

In general, the above-presented model for inelastic, multiphonon-assisted tunneling

introduces a multitude of parameters. However, most of these can be chosen to

match experimentally extracted values, as stated below. Compared to elastic injec-

tion of electrons into defects, the inelastic injection is usually suppressed by some

orders of magnitude. This is at least the case for the above-chosen approximation,

i.e. especially the chosen defect potential shape which determines the magnitude of

c0. To study the sensitivity towards the model parameters, we calculated RMPED for

a defect in a distance of x= 2 nm from the electrode, a CBO of 1 eV, and a defect

depth of ED = 1 eV.

In Fig. 3.12, the sensitivity of RMPED on the most important variables, i.e. (a)

temperature, (b) Huang-Rhys factor, and (c) phonon energy is illustrated, whereas

the contributions from individual phonon numbers, calculated from

c0NE(Ep)f(Ep)T (Ep,x)Lp(z) ·

exp(

−Ee(x)−EkBT

)

, if p > 0,

1, otherwise,(3.41)

3.2. Tunneling Into and Out of Defects 49

-10 -5 0 5 1010-6

10-4

10-2

100

102

104

106

T 25°C 125°C 225°C

R

MP

ED(s

-1)

p

(a)

-10 -5 0 5 1010-6

10-4

10-2

100

102

104

106

S 0.5 1.0 10.0

RM

PE

D(s

-1)

p

(b)

-20 -15 -10 -5 0 5 10 15 2010-6

10-4

10-2

100

102

104

106

h 0.02 eV 0.04 eV 0.06 eV

RM

PE

D(s

-1)

E/0.04 (eV)

(c)

Figure 3.12: Rate RMPED for inelastic, multi-phonon-assisted injection into a defect

in a distance of x = 2 nm from the electrode. A CBO of 1 eV, a defect depth

ED = 1 eV , and an electrical field F = 1 MV/cm were used. The dependence of the

rate on (a) temperature, (b) Huang-Rhys factor, and (c) phonon energy is depicted.

are resolved. For electron injection negative phonon numbers correspond to phonon

absorption and positive numbers to phonon emission (and vice versa for electron

emission from the defect).

Increasing the temperature leads to an increased injection rate (see Fig. 3.12 (a)), as

intuitively expected, since the number of available phonons increases. An increase in

temperature of 100◦C leads to an increase of RMPED by 1-2 orders of magnitude. Addi-

tionally, the maximum contribution to the total rate slightly shifts to higher phonon

numbers. While for T = 25◦C it lies at p= −1, it shifts to p= 1 for T = 225◦C. This

shift is due to the broadening of the Fermi-distribution, i.e. the increased probabil-

ity to find occupied electron states with higher energy in the electrode, which can

50 3. Models for Charge Transport

tunnel under emission of phonons into the defect.

Coupling to oxide phonons, here represented by an effective phonon energy h̄ω, leads

to a thermal broadening of the defect level, whereas the Huang-Rhys factor S is a

measure for the coupling strength of the electron-phonon interaction. If one would

set S to zero, thus eliminating the electron-phonon interaction, one would observe

a delta-like peak in Fig. 3.12 (b). The higher S, the more this peak is broadened,

i.e. the probability for transitions involving higher numbers of phonons increases.

In principle, S can be calculated for a certain material, if its phonon modes and the

electron-phonon coupling potential are known[73, 74]. Experimentally, via studies

of temperature-dependent photoluminescence of zirconia films, a Huang-Rhys factor

of up to 40 was found[75], indicating very strong electron-phonon coupling in these

films. In most common binary semiconductor materials like GaAs, GaP, GaSb,

InAs, InP, or InSb S ranges from 1 to 3[76]. In our kMC simulations we used values

between 1 and 10 for S.

The dependence ofRMPED on the assumed effective phonon mode is shown in Fig. 3.12 (c),

where RMPED is depicted as a function of the energy change ∆E during an inelastic

transition. This energy ∆E has to be either absorbed by or emitted from the elec-

tron via phonon absorption or emission. For a given ∆E the transition probability,

i.e. RMPED , decreases with decreasing phonon energy h̄ω, since this means that a

higher number of phonons is necessary to supply or to dissipate this energy, thus

making it a less likely process. In agreement with theoretically and experimentally

determined values for ZrO2 h̄ω was kept in the interval 30 − 50 meV[75, 31] for all

kMC simulations.

3.3. Defect-Defect Tunneling 51

3.3 Defect-Defect Tunneling

In the preceding section electron injection into defects was investigated, forming the

first step in transport across defect states. In a subsequent step trapped electrons

can be 1) reemitted to the electrode, they came from, or tunnel to the counter-

electrode, described by RMPDE in equation (3.40), 2) reach the CB of the insulator via

thermal emission, as discussed in 3.4, or 3) tunnel to another defect in their vicinity.

For the latter kind of transition, unlike trapping and detrapping processes involving

delocalized states in the electrode, which are dealt with in 3.2, both initial and final

state, have to be described by strongly localized wave-functions.

Figure 3.13: Schematic of (a) elastic and (b) inelastic tunneling between two

defects i and j in the band gap of an insulator. In (a) no electric field is applied,

thus assuming two identical defects, the defects have the same energy level. The

exponentially decaying wave functions of the trap states are sketched. In (b) an

electrical field is applied, which tilts the bands and shifts the defect level energies

with respect to each other. The electron transition from defect i to j can only occur

under phonon emission.

As sketched in Fig. 3.13, defect-defect tunneling can be (a) an elastic process,

e.g. between two defects with the same energy without an applied electric field or

for transport perpendicular to an applied electric field, or (b) an inelastic, multi-

phonon-assisted process, e.g. between defects energetically shifted with respect to

each other due to an applied electric field.

52 3. Models for Charge Transport

3.3.1 The Elastic Case

Elastic defect-defect transitions are modeled as a hopping process between isolated

electron states. For elastic hopping from defect i to defect j Mott derived the

transition rate[77]

Rij = ν · exp(−2rij

rD

)

. (3.42)

Here, ν is a typical phonon frequency, i.e. of the order of 1013 s−1, rij is the distance

between the two defects, and rD is defined by the wave function of the localized

states, decreasing as exp(−r/rD). For a delta-like defect potential the normalized

wave funtion of the trapped electron is given by[71]

ψ =√

rD

2πe−r/rD

r(r > 0), (3.43)

as sketched in Fig. 3.13, with the localization radius

rD =h̄√

2miED, (3.44)

which has already been introduced in 3.2. Here, ED is the depth of the defect, i.e.

the deeper a defect level, the stronger the electron localization.

3.3.2 The Inelastic Case

Inelastic defect-defect tunneling, depicted in Fig.3.13 (b), analogous to the case

of inelastic electron injection into a defect, described in 3.2, is a phonon-assisted

process, where the electron exchanges energy with the crystal lattice via electron-

phonon interaction. Phenomenologically, the inelasticity can be incorporated into

the rate expression, given in equation (3.42) by multiplication with an exponential

weighting factor. Then the rate for inelastic tunneling from defect i to defect j is[78]

Rij = ν · exp(−2rij

rD

)

·

exp(

− ∆EkBT

)

, if ∆E > 0

1, otherwise,

where ∆E is the energy difference between final and initial defect state. The above

expression for Rij corresponds to the Miller-Abrahms hopping model, which was de-

veloped to describe impurity conduction in doped semiconductors[79]. The inelastic

defect-defect transitions can also be regarded as a two-step process: 1) elastic tun-

neling from defect i into a virtual state at the position of defect j and 2) subsequent

3.3. Defect-Defect Tunneling 53

energetic relaxation to Ej , the energy of defect j, moderated by the electron-phonon

interaction. Then, one arrives at

Rij = ν ·Tij ·

exp(

− ∆EkBT

)

, if ∆E > 0

1, otherwise,(3.45)

whereas Tij is the transmission coefficient through the tunneling barrier separating

the two defects.

Alternatively, we can calculate the rate Rij from expressions analogue to equations

(3.30) and (3.31). Adopted to the current situation, one can write

Rij =∫ ∞

−∞

∫ ∞

−∞Ni(E)fiNj(E′)[1−fj ]Tij(E,rij)cij(E,E′)dEdE′. (3.46)

Here, Ni(E) = δ(E−Ei) and Nj(E) = δ(E−Ej) are the densities of states for the

two defects at energies Ei and Ej , and fi and fj are there occupations, i.e. either

zero or one, if multiple occupation is forbidden. A transition from defect i to defect

j, in this picture, is only possible, if defect i is occupied and defect j is empty. Thus,

we arrive at

Rij = Tij(Ei, rij)cij(Ei,Ej). (3.47)

Ridley carried out quantum-mechanical calculations of the rate cij for multi-phonon-

assisted, non-radiative defect-defect transitions[76]. Adopting the single-mode ap-

proximation, he arrived at a widely applicable, simple expression for cij , given by

cemij =R0(fBE +1)pexp(−2fBES) (3.48)

for a transition under emission of p phonons and

cabsij =R0f

pBEexp(−2fBES) (3.49)

for a transition under absorption of p phonons. The prefactor R0 can be approxi-

mately written as

R0 =π

h̄2ω

Sp−1e−S

(p−1)!

[

0.26|V |2 +0.18|∆|2S

(p−1)

]

, (3.50)

with

|V |2 =∑

q

|Vq|2 (3.51)

54 3. Models for Charge Transport

and

|V∆|2 = |∑

q

Vq∆q|2. (3.52)

Here, Vq and ∆q are defined as

Vq =< ψj |Uq|ψi > (3.53)

and

∆q =< ψj |Uq|ψj >−< ψi|Uq|ψi > . (3.54)

Uq denotes the electron-phonon coupling potential, which may depend on the phonon

wave vector q. ψi and ψj are the wave functions of the initial defect i and the final

defect j.

So far, no assumptions have been made on the localized electron states or the

electron-phonon interaction. If detailed models for Uq, ψi and ψj are supplied, the

matrix elements |V |2 and |V∆|2 can be evaluated and, thus, also RDD in equation

(3.48), a task which has been carried out in the past for various models[80, 81, 82].

For our purposes the calculation of the above matrix elements was performed numer-

ically by the simulator assuming for the defect wave function a constant probability

density in a cubic volume centered around the defect position and a vanishing prob-

ability density outside the cube, i.e. for a defect at r = 0,

ψ =

[

4π3 r

3D

]−1/2, if− R

2 ≤ x,y,z ≤ R2

0, otherwise.(3.55)

The edge length of the cube was chosen to be

R = (4π/3 · rD)1/3, (3.56)

and for the electron-phonon interaction an optical deformation potential

Uq = UD · exp(iq · r) (3.57)

was adopted.

3.3.3 Sensitivity Analysis

In equations (3.45) and (3.47) two models for inelastic, multi-phonon-assisted tun-

neling between two defect states were introduced, whereas the first can be evaluated

3.3. Defect-Defect Tunneling 55

straightforwardly, while the second involves much more mathematics and also more

assumptions, which are reflected in additional model parameters. As the phonon

vibration frequency ν was kept fixed to 1013 s−1 in all our simulations, it is not

discussed here. Anyway, the dependence of Rij on it is trivial.

Figure 3.14: Sketch of two defect levels i and j in the band gap of an insulator. (a)

If U1 > 0, then Ei >Ej . Thus, tunneling from i to j can only occur under emission

of p1 = ∆E1/h̄ω phonons. (b) If U2 > U1, then ∆E2 > ∆E1. Thus, the number

of emitted phonons increases, i.e. p2 > p1. The rate Rij for non-radiative, multi-

phonon-assisted transitions from i to j was calculated for rij = 2 nm, ED = 1 eV,

h̄ω = 40 meV and T = 125◦C.

Instead, we compare the two models introduced in the preceding subsection with

respect to their scaling with phonon number p. To this end, we investigated the

following situation, shown in Fig. 3.14. We regard two defect states i and j in

the bulk of an insulator separated rij = 2 nm from each other. The defect depth

is chosen to be ED = 1 eV. Electric fields between 0 MV/cm and 4 MV/cm are

applied in order to shift the defect levels with respect to each other. The rate for

non-radiative, multi-phonon-assisted tunneling Rij from defect i to j was computed.

The higher the applied bias, the larger ∆E, the energy difference between the defect

levels. Thus, for higher fields higher phonon numbers are desired for the transition

to take place.

56 3. Models for Charge Transport

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0102

103

104

105

eqn. (3.45) eqn. (3.47)

R

ij(s-1)

F (MV/cm)

Figure 3.15: Rij was computed for the defect-defect transition sketched in Fig. 3.14

for varying electric fields. The two models given in equations (3.45) and (3.47) are

compared.

As Fig. 3.15 shows, rates Rij calculated from equations (3.45) and (3.47) are in

close agreement, both, in quantitative terms and in their scaling behaviour with the

electric field, or, equivalently, the phonon number. The minor differences, which are

observed, have their origin in the assumptions made for the defect potential and the

localized electron wave functions, which enter equation (3.47).

In Fig. 3.16 Rij is shown for different defect depths. ED was varied between

0.5 eV and 1.5 eV. In general, for all investigated defect depths the Miller-Abrahams-

like (MA) model (equation (3.45)) scales stronger with the electric field than the

model derived from Ridley’s considerations (equation (3.47)), as already observed

in Fig. 3.15. The rate calculated according to the MA model also scales stronger

with the defect depth. For both models, the scaling behaviour is dominated by the

scaling of the transmission coefficient, thus it is very similar. However, in equation

(3.47) also cij depends on ED. The matrix elements in equation (3.50) depend

on the shape of the wave functions which again depends on ED. In the chosen

approximation the distance, to which the localized states extend into the insulating

material, is determined by the edge length R, which is calculated from (3.56). The

dependence of cij on ED enters the model via the localization radius rD.

3.4. Poole-Frenkel Emission 57

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

102

103

104

105

106

107

108 ED = 0.5 eV eqn. (3.45)

ED = 0.5 eV eqn. (3.47)

ED = 1.0 eV eqn. (3.45)

ED = 1.0 eV eqn. (3.47)

ED = 1.5 eV eqn. (3.45)

ED = 1.5 eV eqn. (3.47)

Rij(s

-1)

F (MV/cm)

Figure 3.16: Rij was computed for the defect-defect transition sketched in Fig. 3.14

for different defect depths ED between 0.5 eV and 1.5 eV for varying electric fields.

The two models given in equations (3.45) and (3.47) are compared.

3.4 Poole-Frenkel Emission

Discussing DAT, we so far only considered tunneling processes, where the electrons

never reach the CB of the insulator. We regard an electron trapped in a defect

state. Besides tunneling to other nearby defects or to the contacts, it can also leave

the defect via thermal emission to the CB. In the late 1930s, Frenkel developed a

model for this kind of process, motivated by the observation that in insulators and

semiconductors the conductivity increases, if large electric fields are applied[22]. A

field-dependent mobility of the charge carriers could be ruled out, since the same

increase of conductivity was also found upon illumination of the materials, whereas

the applied electric field was left constant. Thus, the reason for this enhancement

was assumed to be an increase of the charge carrier density. Frenkel envisioned

a field-assisted detrapping, freeing immobile charge carriers, which subsequently

contribute to the conduction. In a medium with high defect densities, as for instance

the high-κ oxides, defect states are assumed to be the sources of the additional charge

carriers. The increased conductivity, in this picture, is explained by a field-assisted

barrier lowering, whereas for an electron trapped in a defect the barrier height equals

the defect depth.

58 3. Models for Charge Transport

A simple model, describing the field-enhanced detrapping of electrons from defect

states, can be obtained from the following considerations: We regard a defect state,

which, if occupied with an electron, is neutral. Thus, as soon as the electron leaves

the defect, the latter becomes positively charged. The resulting Coulomb attraction

between electron and charged defect is sufficient to explain a barrier lowering in

presence of an electric field, as can be understood from Fig. 3.17.

Figure 3.17: Effective electron potential (black) in the vicinity of a positively

charged defect under an applied electric field F . The constant crystal potential V0

is tilted by the electrostatic potential VF . Combined with the screened Coulomb

potential VD of the defect, this leads to a field-dependent barrier reduction ∆EP F

for emission of trapped electrons.

A constant crystal potential

VC(x) = V0, (3.58)

describing, e.g., the CB edge of the insulator, is assumed. We add an electric field

F , which gives rise to an electrostatic potential

VF (x) = −eFx. (3.59)

The attractive Coulomb interaction between the positively charged defect at posi-

tion x0 and the emitted electron at position x is described by a screened Coulomb

3.4. Poole-Frenkel Emission 59

potential

VD(x) = − e2

4πǫ0ǫopt|x−x0| . (3.60)

ǫopt is the optical permittivity of the insulator material. Its usage at this place has

been subject to discussion[83]. The overall potential V , shown as continuous line in

Fig. 3.17, is calculated by superimposing VC , VF , and VD:

V (x) = VC(x)+VF (x)+VD(x) = V0 − eFx− e2

4πǫ0ǫopt|x−x0| . (3.61)

The combination of the band tilting due to the electric field and the attractive

Coulomb potential leads to the barrier reduction ∆E, which can be easily deter-

mined. From the condition

(

dV (x)dx

)

x=xm

= 0 (3.62)

we extract the coordinate of the maximum of the barrier for electron emission xm,

xm = x0 +√

e

4πǫ0ǫoptF. (3.63)

Back in equation (3.61) we obtain

V (xm) = V0 − eFx0 −√

√

√

√

e3F

πǫ0ǫopt(3.64)

and thus an effective barrier height of

ED −∆EP F = V0 − eFx0 −√

√

√

√

e3F

πǫ0ǫopt− (V0 −ED − eFx0) = ED −

√

√

√

√

e3F

πǫ0ǫopt(3.65)

with a field-dependent barrier reduction ∆EP F =√

e3Fπǫ0ǫopt

. This expression is quite

similar to ∆ESE , the barrier reduction for thermionic emission, calculated on page

35, which resulted from the interaction of the electron with its image charge in the

electrode, see 3.1.2.

∆EP F = 2∆ESE , (3.66)

whereas the factor 2 is due to the fact that, in the case of PF emission, the positive

charge is spatially fixed at the defect site, while, in case of thermionic emission, if

the electron moves a certain distance in the x-direction, its image charge moves the

same distance in the −x-direction.

60 3. Models for Charge Transport

Emission of the trapped electron into the CB can be modeled as a thermally excited

process, i.e. one can calculate the probability that, at a certain point in time, a

sufficient amount of thermal energy is transferred to the electron so that it can

overcome the barrier. The corresponding transition rate is

RP F = ν · exp

− 1kBT

ED −√

√

√

√

e3F

πǫ0ǫopt

. (3.67)

ν denotes the lattice vibration frequency (∼ 1013 s−1), which can be physically

interpreted as number of times per second the electron interacts with the surrounding

crystal lattice[84].

If the above one-dimensional treatment, which only considers electron emission par-

allel to the applied electric field, is expanded to three dimensions, allowing electron

emission under every angle with respect to the electric field, one arrives at[85]

RP F = ν · exp(

− ED

kBT

)

(

kBT

β√F

)2 [

1+

(

β√F

kBT−1

)

exp

(

β√F

kBT

)]

+12

, (3.68)

with β =√

e3/(πǫ0ǫopt).

3.4.1 Sensitivity Analysis

The major, and in fact the only, parameter governing the rate for field-enhanced

thermal emission of trapped electrons from defects into the insulator CB is the trap

depth ED, since we adopted throughout all simulations a value of 1013 s−1 for ν.

In Fig. 3.18 the rate for PF emission has been calculated for electrical fields F up

to 4 MV/cm varying ED. ǫopt = 5.6, the value for t-ZrO2[31], was adopted. The

comparison between the 1D model, given in equation (3.67), and the 3D model,

given in equation (3.68), shows that both have the same scaling behaviour with

respect to ED. Rates calculated according to the 3D model are slightly smaller,

which is due to the fact that in this case the net rate was obtained by averaging

over the rates for electron emission in all angles with respect to the applied field,

whereas the 1D model only treats emission parallel to the field. In the latter case

the barrier reduction takes on its maximum value, resulting in a higher emission

rate.

Variation of ǫopt has a huge impact on the calculated emission rates, as shown in

Fig. 3.19. Smaller values of ǫopt lead to a stronger increase of RP F for increasing

electric fields. Unfortunately, ǫopt has been frequently misused as a free fitting

3.4. Poole-Frenkel Emission 61

400 600 800 1000 1200 1400 1600 1800 200010-4

10-2

100

102

104

106

108

1010

1012

1014

1016

0.6 eV 0.8 eV 1.0 eV 1.2 eV 1.4 eV

RP

F(s-1)

F0.5 (V/cm)0.5

ED

Figure 3.18: RP F is calculated varying ED between 0.6 eV and 1.4 eV. Continuous

lines correspond to the 3D model given in equation (3.68), dashed lines to the 1D

model given in equation (3.67).

400 600 800 1000 1200 1400 1600 1800 2000101

102

103

104

105

106

107

108

4.6 5.6 6.6

RP

F(s-1)

F0.5 (V/cm)0.5

opt

Figure 3.19: RP F is calculated for ED = 1 eV varying ǫopt. The 3D model from

equation (3.68) was used.

62 3. Models for Charge Transport

parameter in order to match experimental current-voltage data, e.g. in refs. [86, 87].

In contrast to this, we fix ǫopt to literature values for the respective insulator material.

Chapter 4

Methods for Transport

Simulations

4.1 Kinetic Monte Carlo (kMC)

As outlined in 2.2, the downscaling of semiconductor devices inevitably lead to the

introduction of high-κ materials, e.g. as gate dielectric in transistors or as insulating

film in the DRAM storage capacitor. In general, they allow higher capacitance

densities and thus enable further downscaling. Unfortunately, these high-κ thin films

typically suffer from high defect densities, giving rise to defect states that open up

pathways for undesirable leakage currents. Especially the DRAM storage capacitor

poses high requirements to the insulating properties of the dielectric. Hence, further

scaling requires a detailed understanding of the transport across these films.

It has been the major task of this thesis to develop a deeper insight into the transport

in high-κ thin film capacitors for the usage as DRAM storage capacitors. To this

end, models, describing the individual charge transport mechanisms, see chapter

3, were developed and, in a next step, integrated in a simulation algorithm, which

allows us to mimic the charge carrier flow in the capacitors, i.e. to simulate the

movement of charge carriers through the insulating high-κ layer, which separates

the two capacitor electrodes. This framework was used to develop and validate

transport models for the state-of-the-art material systems employed in the storage

capacitors of the most recent DRAM chip generations.

Since transport through high-κ films is very complex and involves numerous different

microscopic processes, the algorithm had to be very versatile, in order to be able to

unite the individual processes in one simulator. Especially DAT so far was described

by continuum approaches. However, in the face of the peculiarities of the system

63

64 4. Methods for Transport Simulations

under investigation, i.e. the high permittivity, the high defect density, and the

small dielectric film thickness, such treatments are questionable. Many phenomena

connected with high-κ thin films require a simulation of charge transport at the

level of single charge carriers that resolves their point-like nature, as well as point-

like defects and other spatially localized phenomena, such as localized electric field

enhancement due to electrode roughness, cf. 5.2, or transient degradation of the

dielectric via defect accumulation[88, 89].

Therefore, we decided to carry out our simulations in a kinetic Monte Carlo (kMC)

framework. The advantages, but also the difficulties, that come along with this

choice will be outlined in this chapter.

First, we will give a concise survey of the state-of-the-art as what concerns modeling

of leakage currents in high-κ films in order to show up the deficiencies in the estab-

lished analysis techniques and in how far kMC promises to bring an improvement.

The reader will be introduced to Monte Carlo techniques in general and especially

into the specialties of the subtype kMC. We will briefly discuss the origins of kMC,

as far as it impacts the design of the algorithm, and we will outline the basic ideas

of a kMC simulation, before we move on to the practical implementation of the

algorithm for our purposes.

4.1.1 Modeling of Leakage Currents:

State-of-the-Art vs. kMC

Up to now transport in the most established high-κ materials, ZrO2 and HfO2, has

been studied using simplified compact-models, e.g. refs. [90] and [86]. In ZrO2

transport was assumed to be dominated by PF emission from defects. According to

the standard-PF model the current density j is proportional to[91]

j ∝ F · exp

− 1kBT

ED −√

√

√

√

e3F

πǫ0ǫopt

, (4.1)

where ED is the defect depth, F is the electric field, and ǫopt the optical permittivity

of the dielectric. However, good fits to experiments could only be achieved by using

unrealistic values for material parameters, especially for ǫopt. This is due to the fact

that this picture of charge transport is incomplete. Only charge carrier emission

from traps is described, while the injection from the electrodes into the traps is

neglected. A tunneling-assisted Poole-Frenkel (TAPF) conduction mechanism has

been suggested for HfO2[28] that incorporates the injection step. However, the

TAPF model only allows for homogeneous defect distributions and it does not take

4.1. Kinetic Monte Carlo (kMC) 65

into account that, by applying an electric field, the defect energy becomes a function

of the defect position. Furthermore, competing transport channels like detrapping of

electrons via quantum mechanical tunneling, cf. 3.2.1 and 3.2.3, are not considered.

In general, a closed model for charge transport in these high-κ films is missing.

These restrictions may be lifted by employing Monte Carlo techniques, which have

proven to be a versatile tool for transport simulations[92, 93, 94]. kMC provides a

rigorous stochastic framework[95, 96] to describe correctly the statistical dynamics

of the charge carriers in high-κ dielectrics, which are very complicated due to the

diversity of mutually interdependent transport processes, which play a role. Sim-

ulations take place at the level of single charge carriers, whose pathway through

the dielectric is tracked. In this way all relevant transport channels are accounted

for concurrently, whereas the well-defined relation between kMC simulation time

and real time, see 4.1.4, allows one to study the real-time dynamics of the charge

carriers. Local phenomena, as described above, are resolved and can be taken into

account in a straightforward manner, e.g. carrier-carrier interactions by coupling

the simulations to a Poisson solver.

4.1.2 History and Development of kMC

Figure 4.1: Sketch of a polymer

configuration.

The term "Monte Carlo" subsums a class of

computational algorithms that rely on the us-

age of random numbers. First algorithms of

this kind were developed in the late 1940’s at

the Los Alamos Scientific Laboratory to study

neutron diffusion in fissionable material[97].

These algorithms were developed to study

static or equilibrium properties of physical

model systems. For instance the conforma-

tion of polymers[98] was investigated using

the so-called Metropolis algorithm, which was

developed in 1953[99]. The idea behind this

is quite simple and sketched briefly in the fol-

lowing:

It shall be the aim of the simulation to find the

lowest energy configuration for a given poly-

mer. To this end a random walk in the con-

formation space of the polymer is performed, i.e. the polymer is transferred from

66 4. Methods for Transport Simulations

configuration to configuration, or from state to state, respectively, in order to find

the one with the lowest energy.

Let state A of the polymer, sketched in Fig. 4.1, be defined by the coordinates of

its N atoms, i.e. by the tuple

A = (r1, .. ri, .. rN ) . (4.2)

We now consider a transition from state A to state B, e.g. given by the tuple

B =(

r1, .. r′i, .. rN

)

, (ri′ 6= ri). (4.3)

The energies EA and EB for both states are computed and the change of the system

energy due to the transition ∆E = EB −EA is calculated. In order to direct the

random walk towards low energies, a transition A → B is only accepted with a

probability

P (A → B) =

1, if ∆E ≤ 0,

exp(

− ∆EkBT

)

, otherwise.(4.4)

This favors transitions for which ∆E ≤ 0, but stills allows for uphill steps so that

the system does not get stuck in local minima of the system’s total energy surface.

In this manner the polymer is iteratively driven into its state of minimum energy.

However, we do not learn anything about the system dynamics, since there is no

link between the probability P to accept a transition and the time needed for this

transition to happen.

A first algorithm, aiming at a simulation of system dynamics, appeared in the

1960’s with Beeler’s simulation of radiation damage annealing[100]. In the following

decades this development gained momentum and numerous applications of such al-

gorithms, studying surface processes, crystal growth etc., appeared. An informative

review of these developments can be found in ref. [101]. While this type of approach

originally was called "dynamic Monte Carlo", in the 1990’s, the term "kinetic Monte

Carlo" settled in. What makes it so attractive is the prospect that, in principle, it

delivers the exact dynamical behaviour of a system. Due to practical reasons, as

will be discussed below, this ideal is virtually never achieved.

For simulations of the dynamical behaviour of ensembles of atoms "molecular dy-

namics" (MD) were developed and, since then, have been the workhorse in this field.

In this simulation scheme, interatomic interactions and interactions with external

forces are described by a model potential, which is determined in advance and fed

4.1. Kinetic Monte Carlo (kMC) 67

into the simulations. Then the dynamics of the atoms emerge from forward prop-

agation in time of Newton’s classical equations of motion. No further assumptions

or user input are required.

While the first application in the late 1950’s assumed a hard sphere potential[102]

and was very successful in studying simple liquids, more accurate model poten-

tials appeared a few years later with the first use of Lennard-Jones potentials in

MD[103] for liquid argon. The development of faster computers enabled the us-

age of more sophisticated approaches, which are combined with high computational

costs. At the very core of these novel methods stands the idea that forces, acting

on atoms/molecules at a certain point in time, are calculated on-the-fly using the

results from ab initio electronic structure calculations. This type of MD is known

as Car-Parrinello MD or ab initio MD[104]. In contrast to the early approaches in

this case the treatment of the interatomic interactions adapts to the current situ-

ation, i.e. also systems in which the dynamics lead to a qualitative change of the

interatomic interactions during a simulation can be investigated.

Unfortunately, all MD simulations suffer from an intrinsic problem of this approach.

In order to accurately compute the atomic movements, the time steps ∆t, with which

the simulation advances, have to be chosen sufficiently short, i.e. ∆t < 10−15 s, in

order to resolve the vibrational motions of the atoms. Thus, typical MD simulations

only reach overall simulation times of less than a microsecond. The same applies to

the length scale. Unfavorable scaling of the computational effort with the number of

simulated atoms also sets strict limitations to the system size, which is accessible to

simulations. At present, one can simulate ensembles up to some hundred particles,

which translates to some hundred nanometers extension in every spatial dimension.

Often, however, processes one would like to study take place on much longer time

and length scales, e.g. macroscopic surface diffusion processes or phase transitions.

The ultimate goal is to unravel the link between macroscopic phenomena, which can

be directly observed, and the underlying mechanisms on the molecular scale. To this

end one has to bridge many orders of magnitude both in time and in space, setting

up large obstacles for computational approaches. This problem is known as time-

scale/length-scale problem and, in fact, was the major trigger for the development of

the kMC method. A lot of research has been dedicated to exactly this problem and

advances have been made. A good summary of the state-of-the-art can be found in

ref. [105]

A comprehensive model of such a system inevitably requires a multi-scale approach,

which closes the gap between the molecular scale, accessible to MD, and the macro-

68 4. Methods for Transport Simulations

Figure 4.2: Multi-scale modeling. kMC is thought to bridge the gap between ab

initio based approaches like ab initio MD and large-scale continuum descriptions.

scopic scale, treated via continuum approaches. kMC algorithms have been devised

for this purpose, see Fig. 4.2. Employing kMC brings along a boost in simulation

speed, which is achieved by describing the system dynamics in a coarse-grained fash-

ion, as will be explained in 4.1.3, neglecting in comparison to ab initio molecular

dynamics e.g. the atomic vibrations.

4.1.3 Basics of kinetic Monte Carlo

Before diving into the implementation of kMC for our specific task, the modeling of

electron transport through high-κ dielectrics, we will first give a general introduction

to the basic ideas of kMC and apply these on a simple example system.

We aim at an accurate description of the dynamics of a system. To this end the

system is regarded as an entity, which, for every instant of time, can be characterized

by its macroscopic state. Different system states are connected with each other via

state-to-state transitions, as sketched in Fig. 4.3 (a). All transitions are identified

and subsequently reduced to a set of relevant transitions, i.e. transitions which

are regarded as of minor importance for the overall system dynamics are neglected.

This coarse-graining of the system dynamics is responsible for the speed up of kMC

compared to MD. Obviously, one has to pay particular attention not to lose essential

parts of the dynamics. A transition rate R is associated with every transition,

whereas R is proportional to the probability of the transition. Usually, it is assumed

that the rate Rij for a transition from state i to state j does not depend one the

4.1. Kinetic Monte Carlo (kMC) 69

Figure 4.3: (a) Macroscopic states of a system interconnected via transitions. To

each transition i → j belongs a transition rate Rij , corresponding to the transition

probability. (b) Configuration space of a sample system. Each box corresponds

to a macroscopic state. States are connected by a complex network of transitions

(arrows).

system’s prehistory. This means that the system has no memory on how it reached

state i. If this is the case, we have to deal with so called Markov processes[106].

The time evolution of such a system is governed by the master equation

dPi(t)dt

= −∑

j 6=iRijPi(t)+∑

j 6=iRjiPj(t), (4.5)

whereas Pi(t) is the probability to find the system at time t in state i.

Essentially, kMC models the time evolution by solving the system’s master equation

in a stochastic framework. Often the number of possible states is huge and their

interrelation complex, as sketched in Fig. 4.3 (b), and an analytical solution of

equation (4.5) is not feasible. kMC offers a rigorous procedure to correctly propagate

the system from state to state. Such a trajectory (red in Fig. 4.3 (b)) is called

Markov chain. If one computes a large number of Markov chains and averages

over these, one obtains a numerical solution of equation (4.5) and thus the system

dynamics (for the reduced set of transitions).

The kMC algorithm itself, which will be presented in 4.2.1, is nothing but a stochas-

tic procedure. It is up to the researcher to incorporate the physics describing the

system under investigation via the transition rates.

70 4. Methods for Transport Simulations

Figure 4.4: (left) Trajectory of an atom adsorbed on a solid surface. Vibrational

motion of the atom around adsorption sites is interrupted by occasional hopping

to a neighbouring adsorption site. (right) Coarse-grained picture of the dynamics.

Vibrational motion is neglected.

Simple Showcase

Let us now regard a simple showcase: a single atom chemisorbed on a surface[105].

We want to study the atom’s diffusive motions between different adsorption sites,

drawn in Fig. 4.5 on the left side. In this case the macroscopic system (solid surface

+ adatom) state is defined by the location of the adatom. The surface atoms are

assumed to be immobile. Thus, the system dynamics reduce to the vibrational

motion of the adatom and to the hopping between adsorption sites. We further

reduce the dynamics by allowing only nearest-neighbour hopping, since we expect

this to be much more probable than longer jumps, and by neglecting the vibrational

motion of the adatom. After this coarse-graining procedure only the hopping events

are left, as depicted in Fig. 4.5 on the right.

How can we justify these simplifications? To this end we regard the potential energy

surface (PES) of our system, sketched in Fig. 4.5 (a). At a certain instant of time

the adatom is found at a certain adsorption site, corresponding to a local energy

minimum. In this basin it vibrates, whereas this vibrational motion is orders of

magnitude faster than the hopping events to another adsorption site, for which a

comparably large energy barrier has to be overcome. If we would carry out a MD

simulation, which would correspond to the pink trajectory in Fig. 4.5, even for such

a simple system, we would be stuck, as we would need a time step size of ∼ 1 fs in

order to resolve the vibrations of the adatom. This would mean that we would have

to carry out ∼ 109 MD iterations to observe one single hopping event. This is, at

4.1. Kinetic Monte Carlo (kMC) 71

Figure 4.5: (a) PES of the solid surface + adatom system. Each energy basin

corresponds to an adsorption site. The MD trajectory (pink) and the coarse-grained

kMC trajectory (green) are shown. (b) Lines of constant potential energy and

adatom trajectories are shown[101]

.

best, inefficient and often simply not feasible.

For the kMC calculation we neglect these vibrations saying that a vibrating atom

does not leave a certain state in the sense that, if, at a certain instant of time, we

would freeze all the atoms and reduce the system temperature to zero, the adatom

would relax to the bottom of the basin, it is currently found in. In short words, we

identify a macroscopic state with the energy basin on the PES, in which the adatom

is found, and we associate all vibrational motions in this basin to the very same

state. In this picture the dynamics are those of a so called infrequent-event system

(IES), where long periods of inactivity are interrupted by fast transitions. kMC is

predestined for such systems. Performing a kMC simulation for this simple showcase,

we arrive at the green trajectory in Fig. 4.5. Compared to a MD simulation we

generate the same sequence of adsorption positions, to which the atom moves. In

this sense the trajectories of MD and kMC simulations can be regarded as identical,

whereas the kMC simulation, in this case, would speed up the simulation by orders

of magnitude.

In general, of course, the investigated systems are of much higher complexity. While,

as said, for MD the system dynamics emerge naturally, in the case of a kMC sim-

ulation the researcher’s intelligence and his understanding of the system is asked

for. All possible events have to be foreseen. The kMC algorithm will carry out only

those transitions specified in advance. If processes that are important for the system

dynamics are overlooked, the simulation results will be completely incorrect. And

72 4. Methods for Transport Simulations

Figure 4.6: Sketch of the PES between two energy basins i and j. Barrier shapes

and heights are often extracted from ab initio calculations or they are estimated

from the experiment.

often the transitions in a system will not be so well-separated in terms of their rates,

i.e. there will be more or less a continuum of transition rates spanning many orders

of magnitude in time, which will decrease the efficiency gain of kMC. Actually, this

is one of the biggest problems of kMC, and it has to be addressed in an adequate

way that depends on the system under study. We will come back to this point in

4.2.2.

Back to the showcase. Via coarse-graining we have reduced the dynamics to the hop-

ping between neighbouring energy basins, corresponding to diffusion of the adatom

to a neighbouring adsorption site. In kMC the transition rates are user input, i.e.

they have to be supplied in advance. In the present case, sketched in Fig. 4.6, the

rate Rij for hopping from basin i to basin j depends on the shapes of the potential

basins i and j, and the shape of the ridge-top connecting them. It can be calculated

via ab initio methods[105] or it is derived from transition state theory[107, 108].

Then, it reads

Rij = ν0 · e−Eb

i→jkBT , (4.6)

with Ebi→j being the height of the barrier which separates the two basins. Accord-

ingly, one can calculate the rate for the back transition from j to i as

Rji = ν0 · e−Eb

j→ikBT . (4.7)

As can be seen in Fig. 4.6, one would obtain for the ratio of the two rates

4.1. Kinetic Monte Carlo (kMC) 73

Rij

Rji= e

− Ej−EikBT . (4.8)

Thus, the property of detailed balance is satisfied in this framework.

Figure 4.7: The adatom (red) has four neighbouring adsorption sites to which it

can hop.

In a next step we will conveniently assume that all transition rates in the system

are known. After the above-introduced reductions of the problem we are left with

the situation depicted in Fig. 4.7. An adatom located at a certain adsorption site

can carry out four different transitions. In order to correctly compute the reduced

dynamics, we have to answer two questions:

1. Which transition will be carried out?

2. When will the transition be carried out?

These questions were first answered by Bortz, Kalos, and Lebowitz (BKL) and Gille-

spie, who derived stochastic algorithms that randomly choose transitions and tran-

sition times under appropriate consideration of the transition probabilities/rates[96,

109]. Nowadays, these are known as BKL algorithm and Gillespie algorithm. In the

next section we will introduce these kMC techniques together with their rigorous

derivation.

4.1.4 A Simple kMC Algorithm

We summarize how, according to 4.1.3, a basic kMC algorithm should look like. A

flow chart is shown in Fig. 4.8. Every kMC simulation starts by defining the initial

macroscopic state of the system under investigation. In a next step a list of all

74 4. Methods for Transport Simulations

Figure 4.8: Flow chart containing the basic building blocks of a kMC algorithm.

transitions from this state to other system states is compiled, and the corresponding

transition rates are calculated according to the supplied models. This step also

encompasses coarse-graining of the dynamics, if desired. We have already studied

this on the showcase in subsection 4.1.3. The next step takes us back to the two

fundamental questions on page 73. The algorithm has to provide procedures to, in

a statistical sense, correctly choose the next transition and to compute the time ∆t,

after which the transition should be carried out. After this is done, the system is

propagated to the next state and the system time is incremented by ∆t. Then it is

checked, whether or not the kMC loop, framed blue in Fig. 4.8, should be continued.

If so, the next kMC iteration is started by recompiling the list of possible transitions.

This list may have changed due to the last transition, since the system is now in

another state. From this point one proceeds as described above until the user-

specified abortion conditions are fulfilled and the kMC simulation is terminated.

Choice of Transition and Time Step Size

One kMC iteration allows us to perform one single step in a Markovian random

walk. Via multiple iterations we obtain a Markov chain. As described above, the

calculation and subsequent averaging over a large number of Markov chains, sam-

pling the configuration space, is equivalent to a numerical solution of the master

4.1. Kinetic Monte Carlo (kMC) 75

equation (equation (4.5)). To this end we have to determine by which transition µ

and after which time τ the system will pass into another state. In the following, we

will closely follow the ideas of Gillespie[96] and adapt them to our needs[110].

The problem we are dealing with may be outlined as follows: We are given a system

in a certain macroscopic state with a set of M possible transitions Tµ (1 ≤ µ≤M).

The set {Tµ} will usually comprise a multitude of, in part, very different physical

processes. We aim at simulating the time evolution of the system, knowing only the

initial state, the forms of possible transitions Tµ, and the values of the associated

transition rates Rµ.

We start by defining Rµdt as the probability, to first order in dt, that a particular

transition Tµ will take place in the next time interval dt. Here, Rµ is the transition

rate associated with transition Tµ. The time step dt is required to be infinitesimal

to ensure that the probability of another transition Tν , ν 6= µ, to occur in the time

interval dt is negligible.

In order to derive a rigorous algorithm for simulating the time evolution of the

system, we define the probability density function (PDF) P (τ,µ) as

P (τ,µ)dτ = probability at time t for the system to undergo a transition Tµ within

the differential time interval (t+ τ, t+ τ +dτ).

P (τ,µ) is a joint PDF on the space of the continuous variable τ (0 ≤ τ < ∞) and

the discrete variable µ (µ is a positive integer; 1 ≤ µ ≤ M). To correctly compute

the stochastic dynamics of the system, we have to randomly choose a pair (τ , µ)

according to P (τ,µ)dτ , thus specifying the next step in the Markov chain. To derive

an exact analytical expression for P (τ,µ), we calculate P (τ,µ)dτ as the product of

P0(τ), the probability at time t that no transition will occur in the time interval

(t, t+ τ) and Rµdτ :

P (τ,µ)dτ = P0(τ)Rµdτ. (4.9)

To calculate P0(τ), we devide the interval (t, t+ τ) into K subintervals of length

ǫ = τ/K. Then the probability that none of the transitions Tµ occurs in the first

subinterval (t, t+ ǫ) is, following the definition of Rµdt and the multiplication theo-

rem for probabilities,

M∏

µ=1

[1−Rµǫ] = 1−M∑

µ=1

(Rµǫ), (4.10)

whereas terms of higher order in ǫ have been neglected. This is also the subsequent

probability that no transition occurs in (t+ ǫ, t+2ǫ) and then in (t+2ǫ, t+3ǫ), etc.

76 4. Methods for Transport Simulations

Then P0(τ) reads

P0 (τ) =

1−M∑

µ=1

(Rµǫ)

K

(4.11)

=

1−M∑

µ=1

(Rµτ/K)

K

. (4.12)

In particular we are interested in the limit of infinitely large K. Thus,

P0 (τ) = limK→∞

1−∑M

µ=1 (Rµτ)

K

K

(4.13)

=exp

−

M∑

µ=1

Rµ

τ

= e−Rτ (4.14)

Here, R ≡ ∑Mµ=1Rµ is the cumulative transition rate. Equation (4.14) simply de-

scribes the exponential decay of the existing state through all available transition

channels. Inserting equation (4.14) into equation (4.9) delivers

P (τ,µ) =

Rµe−Rτ , if τ ∈ [0,∞) , 1 ≤ µ≤M,

0, otherwise.(4.15)

This PDF is properly normalized, as can be seen by evaluating

∫ ∞

0dτ

M∑

µ=1

P (τ,µ) =M∑

µ=1

Rµ

∫ ∞

0dτ × exp(−Rτ) (4.16)

=R[−1Rexp(−Rτ)

]∞

0(4.17)

=− (0−1) = 1 (4.18)

We continue by writing the two-variable PDF as product of two one-variable PDFs:

P (τ,µ) = P1 (τ)×P2 (µ|τ) , (4.19)

where P1 (τ)dτ is the probability that the next transition will occur between t+ τ

and t+τ+dτ , irrespective of which transition it might be. P2 (µ|τ) is the probability

that it will be a Tµ transition, given that the transition occurs between t+ τ and

t+ τ +dτ .

Using equations (4.15), (4.19) and the addition theorem for probabilities[96], we

obtain:

P1 (τ) =M∑

µ=1

P (τ,µ) =R · e−Rτ , (4.20)

4.1. Kinetic Monte Carlo (kMC) 77

P2 (µ|τ) =P (τ,µ)P1 (τ)

=Rµ · e−Rτ

R · e−Rτ=Rµ

R. (4.21)

The idea for the MC step is to first generate a random number value τ according to

the distributionP1 (τ), which determines the time after which the next transition will

occur, and then to generate a random integer µ according to the distribution P2 (µ|τ)

to select the transition. The resulting pair (τ,µ) will be distributed according to

P (τ,µ).

Inversion Method

The so called inversion method[96] allows us to use random numbers from the uni-

form distribution in the unit interval to construct random numbers distributed ac-

cording to any prescribed PDF. Suppose one wishes to generate a random number

according to the PDF P (x). Then P (x′)dx′ is the probability that x will lie between

x′ and x′ +dx′. One can define the function

F (x) =∫ x

−∞P(

x′)dx′. (4.22)

Then F (x0) is the probability that x < x0. Note that F (−∞) = 0 and F (∞) = 1.

F (x) rises monotonously, since P (x′)> 0 for all x′.

The inversion method can be described as follows. To generate x according to a given

PDF, one simply has to draw a random number r from the uniform distribution in

the unit interval and take for x that value which satisfies F (x) = r; in other words

take x = F−1 (r), where F−1 is the inverse of F , whose existence is ensured by the

monotonicity of F . To prove the correctness of this approach, we calculate the

probability that x lies between x′ and x′ +dx′. By construction this probability is

the same probability that r will lie between F (x′) and F (x′ +dx′), which is

F(

x′ +dx′)−F(

x′)= F ′ (x′)dx′ = P(

x′)dx′. (4.23)

Thus, x is indeed distributed according to P (x).

Determination of Time Step Size τ

We apply the inversion method to the desired PDF for the size of the time step τ ,

given in equation (4.20),

P1 (τ) =

R× exp(−Rτ) , 0 ≤ τ <∞,

0, else(4.24)

78 4. Methods for Transport Simulations

and obtain

F (τ) =∫ τ

−∞P1

(

τ ′)dτ ′ =R∫ τ

0exp

(

Rτ ′) (4.25)

=R

R

[

−exp(

−Rτ ′)]τ0

= [−exp(−Rτ)+1] (4.26)

⇔ F (τ) = 1− exp(−Rτ) != r (4.27)

⇔ τ =−ln(1− r)

R(4.28)

⇔ τ =−ln(r)R

. (4.29)

In the last step of the calculation (1−r) was replaced by the statistically equivalent

random variable r.

Choice of the Transition Tµ

First, we briefly demonstrate, how the inversion method can be applied to discrete

variables. The task shall be to generate a random integer number i distributed ac-

cording to P (i). P (i′) is the probability that i= i′. The corresponding distribution

function is

F (i) ≡i∑

i′=−∞P(

i′)

, (4.30)

and F (i0) is the probability that i ≤ i0. In this case, one has to draw a random

number r and take i, so that

F (i−1)< r ≤ F (i) . (4.31)

Proof:

F (i)−F (i−1) =i∑

i′=−∞P(

i′)

−i−1∑

i′=−∞P(

i′)

= P (i) . (4.32)

The desired PDF for the choice of the transition Tµ was given in equation (4.21),

P2 (µ|τ) =Rµ

R. (4.33)

Thus, we have to choose µ such that

µ−1∑

µ′=1

P2

(

µ′|τ)

< r ≤µ∑

µ′=1

P2

(

µ′|τ)

(4.34)

or

4.1. Kinetic Monte Carlo (kMC) 79

∑µ−1µ′=1Rµ′

R< r ≤

∑µµ′=1Rµ′

R(4.35)

⇔µ−1∑

µ′=1

wµ′ < r ≤µ∑

µ′=1

wµ′ . (4.36)

Here, we have introduced the transition probabilities wµ, which are easily obtained

by normalization of the corresponding transition rates Rµ to the cumulative transi-

tion rate. Of course, then∑M

µ=1wµ = 1.

Figure 4.9: Flow chart of the kMC algorithm, cf. Fig. 4.8.

In summary, one kMC step, i.e. the task of generating a random pair (τ,µ), dis-

tributed according to P (τ,µ), is cut down to the task of generating two random

numbers r1, r2 from the uniform distribution in the unit interval. τ and µ then

follow from eqns. (4.29) and (4.36). Accordingly, we adapt the flow scheme of the

kMC algorithm, see Fig. 4.9.

80 4. Methods for Transport Simulations

4.2 Leakage Current Simulations via kMC

As described in the previous section, kMC provides a rigorous stochastic framework[95,

96] to simulate the dynamical behaviour of a physical system, provided that valid

rate models for all relevant physical processes are supplied. The well-defined relation

between kMC simulation time and real time, for instance, allows one to study the

real-time dynamics of the charge carriers, whereas kMC captures the statistical in-

terplay between different intertwined microscopic processes. This makes it a perfect

candidate for our purposes. In the remainder of this section, we will closely follow

the general framework introduced in 4 and will shed light on its implementation for

our transport simulations.

Problem Definition

In the present work, transport through high-κ dielectrics was modeled in a MIM

capacitor configuration, whereas the high-κ film works as an insulating material in

between the two capacitor electrodes.

The problem, we are dealing with, may be outlined as follows. As simulation region

we choose a cuboid-shaped volume V of dielectric material with the two metal elec-

trodes attached to opposing faces. As said earlier, charge carriers are assumed to

flow preferentially across defect states in the band gap of the high-κ material. Thus,

by default, ND point-like defects are randomly allocated in V , with ND = nD ×V ,

where nD is the user-specified defect density. Each defect site can accommodate

at most one charge carrier, as long as we exclude multiple occupation. Within the

electrodes a continuous Fermi-Dirac distribution of the charge carriers is assumed.

Since our studies focus on the TiN/ZrO2 system, where hole transport is negligible

due to the large VB offset[111], we will limit our discussion to electrons. Nonethe-

less, the method is equally applicable to holes. We neglect the time electrons in

the CB need to reach the anode, as it is small compared to the time needed by

the tunneling processes[111]. Upon entering V , the electrons are discretized and

thenceforth treated as point-like particles with a defined energy until they leave the

dielectric to one of the electrodes. Their individual trajectories are simulated. The

number of electrons, N , in V may change upon injection from or emission into the

electrodes. The leakage current through the dielectric then simply equals the net

number of charge carriers that pass the dielectric in a given time interval.

4.2. Leakage Current Simulations via kMC 81

Figure 4.10: Schematic band diagram

of a MIM structure showing the trans-

port mechanisms, which are implemented

in the kMC simulator. Larger version de-

picted in Fig. 3.1.

Following the remarks of the previous

section, we first have to define what, in

this context, is to be understood as a

macroscopic state. We define the state

of the system at hand, i.e. the MIM

structure, via the electron distribution,

i.e. the positions of all N electrons in

the dielectric. We regard the movement

of a single electron as a transition of

the whole system from one macroscopic

state to another. Further following the

recipe presented in 4, we provide the full

set of possible transitions {Tµ}. They

are illustrated in Fig. 3.1 and, for conve-

nience, on the right in a smaller version

of the original figure. The list of possi-

ble transitions Tµ, which are taken into

account, reads:

(i) direct/FN tunneling

(ii) thermionic/SE

(iii) elastic/phonon-assisted tunneling into a defect

(iv) elastic or phonon-assisted tunneling out of a defect

(v) elastic or phonon-assisted tunneling between defects

(vi) PF emission out of a defect into the conduction band

(vii) defect relaxation upon electron occupation

Defect relaxation plays a special role, which will be revisited in 5.3. For now, we

will neglect it. Details on the rate models for the transitions (i) to (vi) are provided

in chapter 3.

In short, we now have defined our physical system and its macroscopic states, and

we have provided a set {Tµ} of transitions together with tested rate models. At this

point, the kMC procedure, introduced in 4.1.4, allows us to perform the single steps

in the Markovian random walk, thus mimicking the electron dynamics. Given an

electron distribution at time t, we can determine which of the N electrons will move

82 4. Methods for Transport Simulations

in the next kMC iteration (also injection from the electrodes is possible), where it

moves, i.e. also by which type of transition, and when it moves. The algorithm

determines by which transition Tµ and after which time τ the system will pass into

the next state.

4.2.1 Algorithm

Although some problems arise, to which we will dedicate special attention, in prin-

ciple the practical implementation of the kMC algorithm is straightforward. It can

be outlined as follows:

1) Set time variable t= 0 and define the initial state, i.e. the electron distribution,

by assigning N electrons to the ND defect states.

2) Solve Poisson’s equation on volume V with appropriate boundary conditions,

and calculate rates Rµ for all possible transitions. They collectively determine

the PDF. Specify sampling times at which the current flow is evaluated and a

stopping time tstop at which the simulation ends. Independence of the initial

conditions has to be assured.

3) Generate a pair of random numbers (r1, r2), and, using equations (4.29) and

(4.36), calculate the corresponding pair (τ,µ).

4) Advance t by τ , and carry out electron transition Tµ, i.e. adapt the electron

distribution accordingly.

5) If t has been advanced through a sampling point, read out current. Check,

if the current has reached a constant level. If t≥ tstop, terminate calculation,

otherwise return to step 2).

Let us briefly comment on the indivudual steps of the algorithm.

1) Setting up the system means that we first specify the simulation region, i.e. the

region on which e.g. Poisson’s equation has to be solved and in which electrons are

discretized. For a MIM structure, assuming continuous electron distributions within

the electrodes, the dimensions of the simulation volume are those of the dielectric

film. In transport direction these are given by the film thickness, i.e. of the order of

10 nm, and in the two lateral dimensions they are chosen as a compromise between

fast simulation runs, sufficient statistics, and negligible edge effects - typically some

ten nanometers. Defects are placed within the dielectric volume according to the

4.2. Leakage Current Simulations via kMC 83

desired distribution, e.g. homogeneously or with a concentration gradient from one

electrode to the other. All physical properties that are necessary to characterize the

dielectric material and the defects, such as permittivity, band gap, dispersion of the

tunneling mass, defect depth and charge etc., are specified.

2) Many of the rate models, introduced in chapter 3, to describe electron transitions

require precise knowledge of the electrostatic potential distribution within the di-

electric. Transmission coefficients for instance depend on the shape of the tunneling

barrier between initial and final electron position. Thus, we have to solve Poisson’s

equation and, doing so, take into account electron-electron interactions as well as

charged defects and image charge effects due to the presence of the electrodes.

By default, we assume ideally flat electrode/dielectric interfaces. If we want to study

electrode roughness effects, advanced techniques to generate the rough interfaces and

the mesh (adaptive mesh refinement, cf. 5.2.4), on which Poisson’s equation is solved

with a finite elements formalism, are applied. These allow us, e.g., to accurately

determine the electric field distribution in the vicinity of convex protrusions. More

details to that issue will be given in 5.2. In general, the boundary conditions for our

simulation volume are of mixed Dirichlet/von Neumann type. At the faces to which

the metal electrodes are attached the potential is given by the applied voltage, while

on the other faces the normal derivative of the electric field was assumed to vanish.

3) to 5) As discussed in 4.1.4, the task of propagating the investigated system from

one state to another can be reduced to picking two random numbers from the uniform

distribution in the unit interval. After having chosen a transition, it is carried out,

i.e. all necessary modifications to the system are made, mainly by manipulating

matrices in which information on the current system state is stored. As said earlier,

kMC is well-suited to study transient effects, e.g. charging of the dielectric by

applying defined voltage pulses. However, the focus in this work was laid on the

leakage current, i.e. the current which is observed after transient contributions have

faded away. Thus, simulation times have to be chosen significantly longer than the

slowest transient current contributions need to decay.

4.2.2 Time Scaling Problem

While the algorithm sketched in the preceding subsection is fairly straightforward,

in practice problems arise due to the fact that the transition rates range over many

orders of magnitude. It may happen that the major part of the computing time is

used for the simulation of the fast events, while the rare events, associated with slow

rates, are not captured by the simulation. Often these rare events are of particularly

84 4. Methods for Transport Simulations

high interest[112]. Thus, the algorithm should include a strategy to bridge the time

scales, allowing one to access all of them. Different approaches to tackle this problem

have been proposed, among these weighted kinetic Monte Carlo[112]. In our case,

a rate equation Monte Carlo approach turned out to be the best-suited. The basic

idea is to treat certain parts of the dynamics via rate equations and couple these to

the kMC algorithm as follows:

A simulation run starts with the full set of transitions {Tµ}, i.e. all possible tran-

sitions are allowed. After having executed a sufficient number of kMC iterations,

the algorithm identifies the most frequent events, i.e. transitions Tξ with Rξ larger

than a specified cutoff rate Rcutoff , and removes them from the simulation. Their

effect on both current flow and electron distribution in the dielectric is, in the follow-

ing, taken into account in the form of averaged values, extracted from the previous

kMC iterations. A simple example for such a situation is a defect state close to

an electrode. It will be frequently populated by electrons being injected from the

electrode, and it will be depopulated by the same electrons being reemitted to the

electrode (transitions of type (iii) and (iv) in Fig. 3.1). To save computation time

these individual transitions are removed from the set {Tµ}. Instead of resolving the

fast fluctuations of the defect occupation between 0 and 1, we calculate the occu-

pation probability f of the defect as ratio of time for which the defect has been

occupied to the time for which it has been unoccupied. Before each kMC iteration,

the defect occupation will be set either to 1 with a probability f , or it will be set

to 0 with a probability (1 − f). f is subsequently used as a weighting factor for

other transitions involving this defect. The size of the time step ∆t, which is cal-

culated according to equation (4.29), is mainly determined by the largest rates, as

these deliver the main contribution to the cumulative rate R. Now, following the

above-described scheme, ∆t can be increased stepwise by decreasing Rcutoff . The

inaccuracies resulting from this procedure were usually found to be negligible, as the

Coulomb interaction between the charge carriers is screened very efficiently in the

high-κ material. However, outmost care is necessary. Again, we are confronted with

the task to increase the simulation speed by coarse-graining the dynamics without

the loss of essential information which would deteriorate the real system dynamics.

Chapter 5

Leakage Current Simulations

On downscaling of the DRAM storage capacitor, as discussed in chapter 2, the

replacement of SiO2 with high-κ dielectrics became necessary to decrease flowing

leakage currents while achieving higher capacitance densities. Many different ma-

terials have been studied as potential candidates for the dielectric, e.g. Al2O3,

HfO2, ZrO2, TiO2, and SrTiO3[36, 113, 114]. Among these, Zr-based dielectrics are

currently favored[3, 115, 116], since ZrO2 in its tetragonal phase has a very high

permittivity of κ∼ 38 and a sufficiently large band gap of 5.5 eV, which suppresses

quantum-mechanical tunneling currents.

Since direct tunneling is not an issue, at least for the next chip generations, the focus

is now on the reduction of the intrinsically high defect density in the aforementioned

materials. The electronic states associated with these defects often lie within the

band gap and thus open up pathways for undesirable leakage currents. One strategy

to reduce the defect density that has been pursued is defect passivation via doping,

e.g. with F or La[13, 41, 42].

Defect-assisted tunneling has proven to be hard to model. Although the leakage

current in the aforementioned materials has been studied extensively, the analysis

has mostly remained on the level of simplified compact models[86, 90, 117]. These

typically account only for certain aspects of the observed transport and neglect other

essential parts. The widely used model for PF conduction, cf. equation (3) in ref.

[91], for example, describes the field-enhanced thermal emission of trapped electrons

from defect states into the CB. However, electron injection into the defects is not

considered, i.e., transport is implicitly assumed to be purely bulk limited. For HfO2,

a TAPF conduction mechanism has been proposed, which accounts for the injection

step but does not include the position-dependent alignment of defect levels with

respect to the electrode Fermi level upon an applied bias[28].

85

86 5. Leakage Current Simulations

In summary, existing models usually fail to provide a complete picture of transport.

Neither do they take into account that transport is mostly a multistep process, nor

do they include the competition between parallel conduction channels, e.g. different

transport mechanisms, leading to their mutual interdependence.

This motivated the development of our kMC simulator, which is described in chapter

4. It can handle tunneling transport within the band gap, including transport across

defect states, which is often predominant in high-κ dielectrics. Thus, our approach

promises to remove the aforementioned restrictions.

In the following we will present the results of our study of leakage currents through

DRAM storage capacitors with Zr-based high.κ dielectrics. By comparison of sim-

ulation and experiment transport models are derived. In a first step, models are

developed under the assumption of idealized conditions. These assumptions are

subsequently dropped stepwise. Electrode roughness is introduced, and its impact

on the electronic transport is evaluated. Finally, the role of structural relaxation

of defects upon charging is investigated with respect to its influence on the leakage

current. Following such a hierarchical appraoch, the transport models are iteratively

refined.

Identification of the most important transport channels is intended to support the

undergoing experimental efforts toward system optimization and to outline potential

for further scaling of the DRAM storage capacitor.

5.1 High-κ Thin Film TiN/ZrO2/TiN Capacitors

In the following, an elaborate evaluation of the TZT capacitor architecture with

respect to its electrical performance is provided. Measurements of TZT capacitors

are presented and subsequently compared to kMC simulation results. The transport

model derived from this analysis is presented and its implications for further down-

scaling of the DRAM capacitor are pondered. Temperature and thickness scaling

of the leakage current are discussed as well as contributions from different defect

positions to the overall leakage current. We analyze the mutual interdependence

of the competing conduction mechanisms and test our model with respect to its

sensitivity toward changes of the crucial model parameters. A concise overview of

our analysis is also given in refs. [111] and [110].

5.1. High-κ Thin Film TiN/ZrO2/TiN Capacitors 87

parameter symbol value

optical permittivity of ZrO2 ǫopt 5.6[31]

band gap of ZrO2 Eg 5.4 eV

Huang-Rhys factor S 1

effective phonon energy h̄ω 40 meV[31]

CB mass of ZrO2 mcb 1.2me[61]

VB mass of ZrO2 mvb 2.5me[61]

Table 5.1: Fixed model parameters used in the kMC simulations. In chapter 3

the sensitivity of the employed transport models with respect to changes of these

parameters is discussed.

5.1.1 Measurements

TZT capacitors with 9 nm (TZ9T) and 7 nm (TZ7T) thick ZrO2 layers were in-

vestigated. Measured κ-values of 38 for both capacitors evidence that the ZrO2 is

fully crystallized in the tetragonal phase. IV characteristics of both capacitors for

temperatures between 25◦C and 120◦C are shown in Fig. 5.1. Dielectric breakdown

of the samples occurred at voltages of 3.2 V and 2.6 V, respectively (not shown in

Fig. 5.1). While a strong temperature dependence of the current was observed for

voltages below 2.5 V and 2 V, respectively, above these values the leakage current

scales weaker with temperature. Especially for lower temperatures, the slope of

the curves increases. For voltages below ∼ 1 V constant, temperature-independent

leakage currents are observed.

5.1.2 kMC Simulations

kMC simulations for the TZ9T and TZ7T capacitors, whose leakage current curves

are shown in Fig. 5.1, were carried out using best-estimate physical parameters

from experiment and theory. The full set of employed model parameters is given in

Tab. 5.1 together with the corresponding literature sources. Throughout all kMC

simulations these values were kept fixed. To avoid ambiguities in the interpretation

of the experimental data, we have chosen the simplest model that describes the

essential features of the measured currents. A single defect level and a uniform

defect distribution in the dielectric were assumed. The model was calibrated on the

88 5. Leakage Current Simulations

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3 0 1 2 3

transient currents

(b) T7ZT

25°C 50°C 90°C 120°C

voltage U (V)le

akag

e cu

rrent

den

sity

j (

A/cm

2 )

simulation

(a) T9ZT

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

2.52.01.5

voltage U (V)

25°C 50°C 90°C 120°Csimulation

1.0

Figure 5.1: Leakage current data (symbols) of (a) TZ9T and (b) TZ7T capacitor

compared to kMC simulation results (lines). In the low-voltage region (exemplarily

depicted in (a)) no comparison between experiment and simulation is possible, as

here the measured current seems to be dominated by transient currents while the

kMC results describe the steady state. Dielectric breakdown was observed for U ≥3.2 V and U ≥ 2.6 V, respectively (not shown).

TZ9T capacitor. Only the CBO EB between TiN and ZrO2 and the defect depth

ED with respect to the ZrO2 CB were varied.

Very good agreement between simulation (lines) and experiment (symbols) was

achieved, especially in terms of the temperature dependence of the leakage current

(Fig. 5.1 (a)). The same parameter set was used to simulate the TZ7T capacitor.

As shown in Fig. 5.1 (b), thickness scaling of the leakage current is modeled very

accurately. The experimental data is closely fitted, and especially the more pro-

nounced increase of the slope of the current curves (here at ∼ 2 V), compared to

5.1. High-κ Thin Film TiN/ZrO2/TiN Capacitors 89

the TZ9T capacitor, is reproduced.

5.1.3 Transport Model

The transport model, as it emerges from kMC simulations, is sketched in Fig. 5.2.

Figure 5.2: Schematic band structure of a TZT capacitor for low (left side) and

high (right side) voltage. Arrows illustrate electron flow, whereas a dashed arrow

indicates that this electron emission path is slower than the other one. Barriers for

tunnel emission are highlighted with color.

For the TZ9T capacitor, for voltages below 2.5 V the leakage current was found to be

dominated by PF emission of electrons from positively charged defects with a depth

ED = 1.15 eV with respect to the ZrO2 CB and a density of nD = 3 ·1018 cm-3. Elec-

trons are injected from the TiN cathode into the defects via elastic and multiphonon-

assisted tunneling. Detrapping occurs in a next step via PF emission into the ZrO2

CB, where the electrons drift to the anode. Thermally activated detrapping of elec-

trons is the transport limiting step in this multistep process. Therefore, it governs

the temperature scaling of the current, which is found to be strong in this voltage

range.

As the kMC simulations, in contrast to standard treatments, account for the injec-

tion step, it is possible to extract information on the CBO EB between TiN and

ZrO2, which was found to be EB = 1.74 eV. According to the metal induced gap

states (MIGS) model[32], assuming a work function of 4.7 eV for TiN[118], an elec-

tron affinity of 2.5 eV and a charge neutrality level of 3.6 eV for ZrO2[32], a barrier

of 1.45 eV is predicted. Since the MIGS model tends to overestimate the barrier

reduction due to Fermi level pinning for wide gap oxides[119], the extracted value

seems reasonable (An upper limit for EB would be 2.2 eV in the Schottky limit, i.e.

without any Fermi level pinning.)

90 5. Leakage Current Simulations

The defect level lies well above the electrode Fermi level, suggesting a limitation

of the current flow due to the low occupation probability of the defect states. In

this sense, the current is electrode-limited, explaining the failure of the standard

PF formula in describing the leakage current[111], since the latter implies a bulk

limitation of transport. This point will be further discussed below.

For U > 2.5 V, at lower temperatures T ≤ 50◦C, an increase of the slope of the

current-voltage curves was observed together with a reduced temperature scaling.

Here, the dominant conduction mechanism changes from PF emission to tunnel

emission, also known as trap-assisted tunneling (TAT), i.e. electrons predominantly

escape from the defect states via tunneling through the potential barrier. At high

voltages, when the barrier shape changes from trapezoidal to triangular (cf. Fig.

5.2), the rate for this mechanism strongly increases with the applied voltage, so

that it finally surpasses the rate for PF emission. The stronger voltage dependence

of the only weakly temperature-dependent TAT explains the increased slope. For

T ≥ 90◦C the rate for PF emission is higher than the one for TAT over the whole

voltage range so that PF emission dominates the leakage current.

We assume that below ∼ 1.3 V transient currents, which arise due to the applied

voltage ramp (0.05 V/s), dominate over the steady-state leakage.

These transient currents are thought to consist of two contributions, (1) dielec-

tric relaxation currents and (2) charging/discharging phenomena like charge trap-

ping/detrapping involving defect states in the ZrO2 band gap.

In ZrSixOy films, as investigated in ref. [120], enormous transient currents, depend-

ing on the voltage ramp rate, were found. To get a rough feeling, one could compare

the measured currents at an applied electric field of 1 MV/cm. This corresponds to a

bias of 5 V for the 5 nm thick film investigated in ref. [120] compared to 0.9 V for the

9 nm film of our investigation. In ref. [120] currents of 2×10−7 A/cm2, 10−7 A/cm2,

4 × 10−8 A/cm2 and 3 × 10−8 A/cm2 were found for voltage ramp rates of 1 V/s,

0.33 V/s, 0.2 V/s and 0.14 V/s, respectively. We measured 10−9 A/cm2 for a voltage

ramp rate of 0.05 V/s. Thus, the order of magnitude of the measured currents seems

indeed to suggest a strong transient contribution. Additionally, these transient cur-

rents typically only weakly depend on the temperature[120], as it is observed here

(see Fig. 5.1 (a)).

However, the possibility cannot be excluded that a transport channel not captured

by our model is present, e.g. one involving a second defect type. The analysis

of transient leakage data, as suggested in ref. [121], should shed light on this in

the future. In general, the modeling of leakage current is a complex problem with

5.1. High-κ Thin Film TiN/ZrO2/TiN Capacitors 91

many unknowns, especially in case of DAT. Thus, further thorough experimental

investigation is needed in order to complement the present investigation and put

the transport model to the test, e.g. deep-level transient spectroscopy [122] to

study the spatial and energetic defect distribution.

5.1.4 Contributions from Different Defect Positions

In the presence of an electric field the energetic alignment of the defect states with

respect to the cathode Fermi level Ef,l, see Fig. 5.2, depends on their distance from

the electrode. Simulations show that, varying U , different defect positions deliver

the largest current contribution (see Fig. 5.3 and Fig. 5.4). In Fig. 5.3 the simulated

1 2 310-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

leak

age

curre

nt d

ensi

ty j

(A/

cm2 )

voltage U (V)

sum

2.5 nm2.0 nm1.5 nm

PF TAT

defectposition

3.0 nm3.5 nm4.0 nm

Figure 5.3: Leakage current of the TZ9T capacitor at 25◦C. Current contributions

of traps, lying at distances x = 1.5-4 nm of the cathode, are shown. Filled sym-

bols indicate transport via PF emission, empty symbols transport via trap-assisted

tunneling.

leakage current for the TZ9T capacitor at 25◦C is shown (continuous line). Con-

tributions of defects to the leakage current have been resolved with respect to their

position x. Additionally, Fig. 5.4 exemplarily depicts the occupation probability f

of defects at x= 2 nm and x= 4 nm as a function of U .

92 5. Leakage Current Simulations

1 2 310-16

10-12

10-8

10-4

100

10-10

10-8

10-6

10-4

10-2

100

1 2 3

10-10

10-8

10-6

10-4

10-2

100

defect at 2 nm

PF (f=1) TAT (f=1) PF TAT

leak

age

curre

nt d

ensi

ty j

(A/

cm2 )

voltage U (V)

occ

upat

ion

prob

abilit

y f

f

10-16

10-12

10-8

10-4

100

voltage U (V)

defect at 4 nm

(a)

(b)

Figure 5.4: Occupation probability vs. voltage of defects at (a) x= 2 nm and (b)

x= 4 nm is shown. Lines show upper limits for their current contributions, assuming

f = 1 for all voltages. Symbols indicate simulated currents.

For U ∼ 1 V defects deeper in the dielectric, with x= 3.5-4.0 nm, deliver the high-

est current contribution, as they are nearly level with Ef,l. Nonetheless, f stays

below 10−3, since electron injection is quite slow, in particular slower than electron

extraction via PF emission, which depopulates the defects. Contributions from de-

fects lying even deeper in the dielectric are suppressed due to the very slow electron

injection, while contributions from interface-near defects are low, since these lie well

above Ef,l. Tunneling into these defects is very fast, however, reemission into the

cathode is even faster, leading for defects at x= 2 nm to a low occupation f ∼ 10-6.

For U ∼ 2 V defects closer to the cathode are gradually aligned with Ef,l, thus, their

occupation rises, as can be seen in Fig. 5.4 (b). Here, defects at 2.5-3.0 nm give the

highest contribution to the current.

At 2.5 V the dominant transport mechanism changes from PF emission to TAT,

i.e. tunnel emission. In this case, defects between 2.0 and 2.5 nm, being nearly

fully occupied, deliver the highest current contribution. Figure 5.4 also shows the

theoretical maxima for the current contributions (lines) of the defects. Full occu-

pation is assumed, i.e. f = 1, independent of U . The real contributions (symbols),

5.1. High-κ Thin Film TiN/ZrO2/TiN Capacitors 93

as discussed, are roughly obtained by multiplying the theoretical maxima with f .

Especially for low voltages, the current, in this sense, is injection-limited.

Furthermore, Fig. 5.3 illustrates the mutual interdependence of the parallel trans-

port channels. At voltages where TAT dominates the leakage current, a drop of the

PF current can be observed, although the PF emission rate still increases. In this

voltage region, electrons are very efficiently extracted from the defects via tunneling

and thus are no longer available for PF emission. The defect occupation decreases

(see Fig. 5.4 (a)). Clearly the competition between the two transport channels alters

their individual characteristics.

As transport across interface-near defects is suppressed, for further scaling of these

ZrO2 films, the defect density deeper in the bulk has to be minimized. Concerning

electrode roughness, the leakage current is expected to be rather insensitive to the

local electric field enhancement near convex apertures. For extremely thin films,

however, the more important consequence of electrode roughness is local thinning

of the dielectric film. Then, TAT, depending exponentially on the tunneling barrier

thickness, is expected to strongly increase and to become the dominant transport

mechanism. Thus, also the sensitivity of the leakage current towards electrode

roughness is expected to increase. This will be discussed in detail in 5.2.

5.1.5 Sensitivity Analysis

The most crucial simulation parameters, which were attuned in order to reproduce

the leakage current data of the TZT capacitors, are the CBO EB between TiN

and ZrO2 and ED, the defect depth. They were varied systematically in order to

investigate the sensitivity of our model.

In Fig. 5.5 (a) simulation results for a fixed EB = 1.74 eV and varying ED =

0.95-1.25 eV are shown. For 25◦C the leakage current is maximum for ED = 1.05 eV,

while it is lower going to shallower or deeper defects. The shallower a defect is, the

higher the rate for PF emission of trapped electrons. The occupation probability,

however, is lower due to the higher energy difference to Ef,l. For deeper defects

the occupation probability is high, while the PF emission rate is low. Thus, the

existence of a maximum is expected. Temperature scaling of the leakage current is

found to be stronger for deeper defects, as expected for a thermally activated process

like PF emission. Increasing ED, the transition from PF conduction to TAT, visible

as change in the slope, occurs at lower voltages, for the depicted set of defect depths

best seen comparing ED = 1.25 eV and ED = 1.15 eV. Thus, the temperature scaling

of the leakage current and the change of the dominant conduction mechanism can

94 5. Leakage Current Simulations

10-1010-910-810-710-610-510-410-3

ED= 1.15 eV

EB= 1.74 eV

25°C 120°C

120°C

(b)

25°C 120°C

25°C

(a)

EB1.54 eV1.64 eV1.74 eV1.82 eV

120°C25°C ED0.95 eV1.05 eV1.15 eV1.25 eV

10-1010-910-810-710-610-510-410-3

voltage U (V)

leak

age

curre

nt d

ensi

ty j

(A/

cm2 )

1.0 1.5 2.0 2.5 3.0

3.02.52.01.51.0

Figure 5.5: Leakage current data of the TZ9T capacitor for 25◦C and 120◦C

(symbols). Simulated currents for (a) fixed EB = 1.74 eV and varying ED and (b)

fixed ED = 1.15 eV and varying EB are shown. As in Fig. 5.1, the low-voltage

region, dominated by transient currents, is not depicted.

be used to determine ED, given EB. In Fig. 5.5 (b) simulation results for a fixed

ED = 1.15 eV and varying EB = 1.54-1.84 eV are shown. For lower EB, especially

at lower voltages, where none of the defects are aligned with Ef,l, the current rises,

since the defects are pushed closer to Ef,l, and their occupation probability thus

rises. The effect is more pronounced at 120◦C, where the leakage current is mainly

limited by the low occupation of the defects, while at 25◦C it is limited by the slow

PF emission. Consequently, changing the CBO also changes the temperature scaling

of the leakage current.

Without further information onEB, ED or the defect density nD, the triplet (nD,EB,ED),

extracted from the kMC simulations is not completely unambiguous. However, as

soon as one of these values is fixed, the others can be determined with high accuracy.

We have chosen to fix nD to 3 · 1018 cm-3, resulting in EB = 1.74 eV, a reasonable

value for the CBO[111], and ED = 1.15 eV, a defect level that might be ascribed to

5.2. Electrode Roughness Effects 95

oxygen vacancies, cf. ref. [123].

5.1.6 Summary

Based on kMC simulations, a self-consistent leakage current model for TiN/ZrO2/TiN

capacitors was developed. kMC proved to be a useful technique for studying the

mutual interdependence of parallel transport channels. Most importantly, electron

injection into defect states is accounted for. In contrast to the pure bulk-limitation

of the transport suggested by the standard PF model, the CBO between TiN and

ZrO2 turns out to strongly limit the conduction. Bulk defects at a distance of

2-4 nm from the injecting electrode are found to deliver the main current contribu-

tion. These results strongly suggest an optimization of the electrode work function

and a reduction of the bulk defect density as a viable strategy for decreasing the

leakage currents.

5.2 Electrode Roughness Effects

By introducing high-κ materials like HfO2 and ZrO2 as dielectrics in the DRAM

storage capacitor[2, 3, 115, 116, 117] the undesired leakage currents could be re-

duced. However, the ITRS requires a downscaling of the storage capacitor to an

EOT of 0.4 nm within the next two years. The necessary thickness reduction of

the dielectric film will cause leakage currents to grow again. Sensitivity of these

thin-film capacitors on interface effects is expected to increase. Thus, in the face of

the stringent leakage current specifications for the DRAM storage capacitor, elec-

trode roughness (ER) is one of the major concerns for the realization of future chip

generations.

ER, leading to local electric field enhancement (LEFE) close to convex protrusions

of electrode material into the dielectric due to the aperture effect[124], was found to

cause an increase of the leakage currents in thin-film capacitors [125, 126] as well as in

the gate stack[127]. Standard one-dimensional models, usually employed to interpret

experimental leakage data[90, 128, 129], cannot describe these roughness effects. To

arrive at a comprehensive picture of transport in the TiN/ZrO2 material system,

it has turned out to be imperative to take ER effects into account. Such a model,

allowing to quantify the effect of ER on the leakage current of TZT capacitors, is

still missing.

So far, ER effects in thin-film capacitors have only been quantified based on a mean-

field point of view. Via a perturbation solution of Poisson’s equation Zhao et al.

96 5. Leakage Current Simulations

studied the effects of ER on the electric field and the leakage current[130]. Rough

electrode surfaces were found to increase the average electric field 〈F 〉 in the dielec-

tric. As many transport mechanisms are strongly field-dependent, such an increase

of the electric field is expected to cause an increase of the leakage current, as it has

been discussed for SE and PF emission. In the case of transport across defect states

the mean-field approach implies the assumption that all defects in the dielectric are

subject to an identical electric field 〈F 〉 and that they equally contribute to the

leakage current. Our investigations, cf. 5.1 and ref. [110], suggest that transport

in TZT capacitors occurs across defect states in the dielectric via PF conduction

for low and medium electric fields[111]. At high electric fields, and especially in

extremely thin films, TAT may become dominant. Thus, every transport channel

that can be used by charge carriers to cross the dielectric is linked to a point-like

defect.

The individual contribution of each channel to the conduction has been found to

depend on both the position of the defect and the electric field it is subjected to[110],

which, in presence of ER, can be strongly inhomogeneous. Thus, the mean-field

point of view is not applicable to TZT capacitors.

Instead, an approach that considers the local nature of the transport channels has

to be chosen. So does our kMC simulation technique and thus it seems to be well-

suited for this purpose. Here, tunneling transport of charge carriers in the band

gap of the dielectric, including DAT mechanisms, is treated fully self-consistently.

Transport is simulated at the level of single charge carriers, the trajectory of each

being computed as it crosses the dielectric. The simulator is coupled to a finite-

elements-based Poisson solver, working on an adaptively refined grid, cf. 5.2.4. In

this way, the dielectric, bounded by rough electrodes, can be modeled accurately,

and the local changes in the electric field, e.g. in the vicinity of surface features, are

resolved. This allows to correctly take the individual current contributions of the

transport channels into account.

In section 5.1 and in refs. [110, 111] a transport model for TZT capacitors was

developed by comparing kMC simulations to experimental leakage current data of

TZT capacitors with thicknesses d of the dielectric film between 10 nm and 7 nm,

corresponding to EOTs between 1.0 nm and 0.7 nm (assuming a permittivity of

40 for high-κ tetragonal ZrO2). Thickness scaling of the leakage current as well

as temperature scaling between 25◦C and 120◦C was accurately reproduced by the

simulations. Exemplary leakage current data of a 9 nm TZT capacitor (EOT of

0.9 nm) is shown in Fig. 5.1.

5.2. Electrode Roughness Effects 97

Therefore, based on the closed microscopic transport model for TZT thin-film capac-

itors, outlined in 5.1, as a step of further refinement, we investigated the influence

of ER on the leakage current. As will be seen, for thin dielectric films (<10 nm),

these roughnesses at the interfaces between electrode material and dielectric play

an important role. Besides leading to quantitative changes in the simulated leakage

currents, our simulations suggest that ER roughness also induces qualitative changes

in the transport for ever shrinking dielectric film thicknesses, as will be commented

on in 5.2.5.

We assessed the ultimate scaling potential of TZT capacitors in terms of EOT, film

smoothness, thickness fluctuations, defect density and distribution, and CBO in a

fictive scenario. Based on experimental data for an EOT of 0.9 nm, we extrapolated

via kMC simulations down to EOTs of 0.4 nm. ER was described in the framework

of fractal geometry, which will be introduced in 5.2.1. A concise overview of these

ER studies has been published in ref. [131].

5.2.1 Fractal Geometry

Aiming for a comparative study of ER effects, the ability to classify and describe

rough surfaces in quantitative terms is essential. Frequently, rough surfaces are

well-described by the concepts of fractal geometry[132], as they often exhibit self-

affinity[133]. Fractal geometry does not describe the roughness, but it allows to

describe the way roughness changes upon alteration of the observation scale. The

symmetry obeyed by self-affine objects on rescaling is used to classify them.

Often rough surfaces, as formed during a growth process, belong to the class of

self-affine random fractals, just like, e.g., a coastline, as shown in Fig. 5.6. Despite

their randomness they are self-affine in a statistical sense. A mathematical defini-

tion of self-affinity will be given below. A more descriptive explanation of what is

understood under "self-affinity" is that if you regard a self-affine surface and zoom

in/out, i.e. change the scale of observation, the surface will qualitatively still look

the same. Many approaches exist to characterize such systems. Ref. [135] provides

an excellent overview of these. Here, however, we will only introduce the approach,

which fits best to our needs, namely, for random rough surfaces, the height difference

correlation functions, defined as

Cq(∆r) =⟨

|h(x)−h(x+ r)|q⟩1/q

|r|=∆r. (5.1)

Here, h(x) is the height of the rough surface at position x above a smooth refer-

98 5. Leakage Current Simulations

Figure 5.6: Section of the coastline of South Britain (source: GoogleEarth). In

1967 Mandelbrot published his famous paper "How Long Is the Coast of Britain?

Statistical Self-Similarity and Fractional Dimension"[134], where he examined the

coastline paradox, i.e. the observation that the length of the coast of Britain depends

on the scale of measurement and theoretically is infinite.

Figure 5.7: One-dimensional random rough surface, parametrized by h(x), above

a flat reference plane. w measures the vertical surface fluctuations and will be used

below as a parameter to classify random rough surfaces.

5.2. Electrode Roughness Effects 99

ence surface, in general providing a coarse-grained, smooth approximation to the

rough surface, as shown in Fig. 5.7. 〈〉 stands for the average over all directions

in the reference plane. The problem of describing the surface morphology is then

transferred to describing the shapes of the correlation functions. The latter can be

advantageous. Often the correlation functions have the form

Cq(∆r) ∼ ∆rαq (5.2)

over a substantial range of length scales.

In most simple cases the exponent αq is independent of q, at least for q > 0 (If

αq depends on q, one speaks of multi-affine structures.). Then, the rough surface

is called a self-affine fractal with Hurst exponent α over the corresponding range

of length scales. The mean height difference ∆h ≡ C(∆r) scales with horizontal

distance ∆r as

∆h∼ ∆rα. (5.3)

Within the scaling regime the surface is invariant to a change of length scales by a

factor of λ in lateral directions combined with a change of length scales by λα in

vertical direction (cf. equation 5.7). The scaling regime is bounded by the upper

and lower correlation lengths ξ+ and ξ− in both the horizontal (||) and the vertical

(⊥) direction, i.e.

ξ−‖ <∆r < ξ+

‖ (5.4)

ξ−⊥ <∆h < ξ+

⊥ , (5.5)

related by

(ξ+⊥/ξ

−⊥) = (ξ+

‖ /ξ−‖ )−α. (5.6)

Mathematically, self-affinity then can be defined as follows: A surface is called self-

affine within a certain scaling regime, bounded by equations (5.4) and (5.5), if its

statistical properties are invariant to an anisotropical rescaling, i.e.

h(x) ∼ λ−αh(λx). (5.7)

Here, λ is the horizontal scaling factor, and α is the above-introduced Hurst expo-

nent, in the following referred to as surface smoothness (0<α< 1). Why this term is

appropriate can be understood regarding Fig. 5.8. There, self-affine random rough

100 5. Leakage Current Simulations

0 20 40 60 80 100-4

-2

0

2

4

6

8

vert

ical

coo

rdin

ate

h (n

m) 0.1

0.5 0.9

lateral coordinate x (nm)

Figure 5.8: Self-affine random rough surfaces generated with the algorithm in-

troduced on page 103 by using identical random seeds but different values for the

smoothness α. For better visibility the surfaces with α = 0.1 and α = 0.9 are offset

by -2 nm and 2 nm, respectively.

surfaces with different α, generated with the algorithm that will be introduced on

page 103 by using identical random seeds, are shown. A low smoothness α = 0.1

corresponds to a very jagged surface, for which the amplitude of vertical fluctuations

stays nearly constant down to smallest lateral length scales (short-range roughness).

α → 0 would mean ∆h = const., i.e. ∆h is completely independent of ∆r. The

higher α is, the smoother is the surface, since the amplitude of vertical fluctuations

decreases with decreasing lateral length scale. In other words, α describes, how fast

the size of vertical fluctuations changes upon a change of the horizontal length scale.

For a given surface α can be determined, e.g., from double-logarithmic plots of

C2(∆r),

C2(∆r) ∼ ∆rα, (5.8)

the height difference correlation function for q = 2, vs. ∆r. In the self-affine scaling

regime this should yield a linear curve with slope α.

5.2. Electrode Roughness Effects 101

5.2.2 Structural Characterization of the TiN Electrodes

The TiN electrodes, employed in the investigated TZT capacitor structures, were

investigated with respect to their self-affinity. Bare, planar TiN electrode surfaces,

characterized via atomic force microscopy (AFM), were found to exhibit self-affine

scaling up to ξ = 20 nm, as can be seen in Fig. 5.9.

1 10

fit C

2

C 2(nm)

20 nm

0.6

1

lateral separation r (nm)

0.1

Figure 5.9: Height difference correlation function of a bare TiN electrode, extracted

from AFM data, and fit according to equation (5.8).

This allows a systematic description of ER in the framework of fractal geometry.

Typically found TiN surface parameters are a smoothness of α= 0.6 and a thickness

fluctuation (or vertical surface width) of w ∼ 1.2 nm. w was introduced in Fig. 5.7.

According to the AFM data this can be translated to a root-mean-square roughness

of ∆r ∼ 0.3 nm via the empirical relation w ∼ 4×∆r.

5.2.3 Generation of Random Rough Electrodes

For simulation purposes self-affine random rough surface profiles h(x) - characterized

by the parameters α, ξ, and w, as introduced in 5.2.1 - were generated by adapting

a modified midpoint displacement algorithm[136]. Fractional Brownian motion is at

the very core of this algorithm, a famous special case of it being Brownian motion.

Let us start with the well-known Brownian motion. We consider a one-dimensional

102 5. Leakage Current Simulations

random walker, that start at the coordinate x = 0. It can perform steps xi in +x-

direction or in -x-direction. Then, its position after n steps is given by the sum

Xn = x1 +x2 + ...+xn. (5.9)

The steps are taken from a Gaussian distribution with 〈xi〉 = 0 and⟨

x2i

⟩

= σ2,

whereas 〈〉 denotes the average over the ensemble. We define the correlation function

as

F (p) ≡ 〈xixi+p〉 . (5.10)

If F (p) decays fast enough, such that∑∞

p=0F (p) < ∞, then, for large p, the mean

square displacement scales with p as[137]

⟨

x2p

⟩

∼ p. (5.11)

This is the case for the well-known ordinary Brownian motion. If the above sum

diverges, one arrives at

⟨

x2p

⟩

∼ p2α, (5.12)

where 0<α< 1 is the Hurst exponent. This type of random walk is called fractional

Brownian motion[138]. Obviously, then

⟨

x2p

⟩1/2 ∼ pα, (5.13)

reminding us of the height difference correlation functions, which were discussed

above, cf. equations (5.1) and (5.8). This property can be exploited to generate

fractal random rough surfaces with every desired smoothness α.

In general one can divide fractional Brownian motion into three categories: α < 12 ,

α = 12 , and α > 1

2 . As said, α = 12 is the case of ordinary Brownian motion, where

the increments are independent from each other, i.e. uncorrelated. For α > 12 the

increments are positively correlated, i.e., if xp0> 0, then the probability for xp0+1 > 0

is larger than 0.5. Drawing the curve, which connects the points (p,Xp), one will

obtain a rather smooth curve. For α< 12 the opposite is true. Subsequent increments

are negatively correlated, leading to a very jagged curve. This can immediately be

transferred to the generation of rough surfaces, as they were depicted for varying

α in Fig. 5.8. The simple algorithm used for the task of surface generation will be

presented in the following.

5.2. Electrode Roughness Effects 103

Midpoint displacement method

The aim is to generate fractal random rough surfaces (with a variance σ), satisfying

the height difference correlation function

C2(∆r) = σ∆rα. (5.14)

Figure 5.10: Points (red crosses) generated by the midpoint displacement algo-

rithm after i iterations. Vertical displacements decrease according to equation (5.22),

here for α= 0.3.

Let us assume, we want to generate a surface h(x) on the interval 0 ≤ x ≤ L by

generating a set of points (x,h(x)), with the desired resolution on [0;L], which will

subsequently be connected by a continuous line (Straight lines are used in Fig. 5.10,

while a continuous connection of the points is realized in the kMC simulator via

cubic splines.). The midpoint displacement method is an iterative approach, from

which one can stepwise obtain random rough surfaces, as illustrated in Fig. 5.10.

We start by setting h(0) = 0 and selecting h(L) as a sample of a Gaussian random

variable with mean 0 and variance L2ασ2. Then,

var(h(L)−h(0)) = L2ασ2, (5.15)

and thus we expect for 0< x1 < x2 < L

104 5. Leakage Current Simulations

var(h(x2)−h(x1)) = (x2 −x1)2ασ2. (5.16)

We set h(L2 ) to be the average value of h(0) and h(L) plus a random Gaussian offset

D1 with mean 0 and variance ∆21. Then, we arrive at

h(L

2)−h(0) =

12

(h(L)−h(0))+D1. (5.17)

Consequently, h(L2 )−h(0) has a mean value 0 and so does h(L)−h(L

2 ). In order to

fulfill equation (5.16) we must require

var(h(L

2)−h(0)) =

14var(h(L)−h(0))+∆2

1!=(

L

2

)2α

σ2, (5.18)

which leads to

∆21 =

σ2L2α

22α

(

1−22α−2)

. (5.19)

For the next iteration of the algorithm, we apply the above idea to h(L4 ), which gives

var(h(L

4)−h(0)) =

14var(h(

L

2)−h(0))+∆2

2!=(

L

4

)2α

σ2, (5.20)

from which follows

∆22 =

σ2L2α

(22)2α

(

1−22α−2)

, (5.21)

and equally to h(3L4 ). Increasing the resolution via further iterations, we obtain

∆2n =

σ2L2α

(2n)2α

(

1−22α−2)

, (5.22)

as the variance of displacement Dn, corresponding to lateral distances L/2n.

Again, we see that the surface smoothness α, or the Hurst exponent, respectively,

describes, how vertical fluctuations scale upon a change of the horizontal scale. The

surface profiles generated in this way were subsequently fed into the kMC simulator.

By connecting several surface segments of length ξ in series, larger surfaces with an

approximate correlation length ξ and with the desired values for α and w can be

provided.

Throughout the following analysis ξ was fixed to 20 nm, the value extracted for the

TiN electrodes via AFM measurement, and α and w were varied. The influence of

the electrode topology on the electrical performance, i.e. the transport properties

of the TZT capacitor devices was investigated.

5.2. Electrode Roughness Effects 105

5.2.4 Mesh Generation and Solution of Poisson’s Equation

The task at hand is the following. Poisson’s equation has to be solved within the

simulation volume, in this case a cube with one or two rough faces, depending on

whether one wants to study one or two rough electrodes at the same time. Dis-

cretization of Poisson’s equation in the finite elements framework induces errors.

Via posteriori error estimation one has to gain control of these, whereas the com-

mon approach is to increase the number of degrees-of-freedom in regions, where the

initial finite elements model was not adequate. This is by nature an iterative pro-

cess. In a first step a mesh is generated, on which the differential equation is solved.

Then, errors are estimated, and, based on this estimate, the mesh has to be refined.

The brute force approach would be to globally refine the mesh, i.e. to subdivide

all the finite elements. However, to save computational resources, it is more conve-

nient to refine only in those regions with larger errors, i.e. adaptively. After error

estimation and mesh refinement, again, the differential equation has to be solved

and the errors have to be estimated. Convergency of the finite elements solution

to the correct results is monitored by suited error criteria, which are imposed, and

the refinement loop is to be repeated until the desired accuracy of the solution is

achieved.

Kelly et al.[139] developed a means to locally estimate the error of the finite elements

solution for every grid cell. This error estimator approximates the error per cell by

integration of the jump of the gradient of the solution along the faces of each cell.

It was specially designed in order to estimate the error for the generalized Poisson

equation

∇(κ(r)∇Φ(r)) = −ρ(r)ǫ0

, (5.23)

with either Dirichlet or generalized Neumann boundary conditions, i.e. exactly for

our needs. Here, κ(r) and Φ(r) denote the position-dependent permittivity and

electrostatic potential, respectively, and ρ(r) is the charge density. Implementation

of the error estimator is straighforward. The error ηK for a cell K is calculated from

η2K =

h

24

∫

∂Kκ(r)

[

∂Φ(r)∂n

]2

dS. (5.24)

The reason why we address this topic when discussing ER effects is that we expect

greater variations in the electrostatic potential close to roughness features. These

cause a LEFE, i.e. larger gradients of Φ, leading to higher errors of the finite elements

solution in these regions. The simulation grid has to be fine enough to resolve these

106 5. Leakage Current Simulations

features. Therefore, we adaptively refine the grid, wherever it is necessary. This

leaves us with grids, in which neighbouring cells may have a different degree of

refinement. As a constraint, we allow neighbouring cells to differ from each other in

their degree of refinement only by one. The framework for the generation of such

grids, taking advantage of the above-mentioned error estimation, is provided by the

open-source finite elements library "deal II", specially designed for the analysis of

differential equations[140].

A typical iterative generation process of locally refined grids, as used in our simula-

tions, can be seen in Fig. 5.11. First, for a given simulation volume, say a cube of

20 × 10 nm2 with one rough face, Poisson’s equation is solved on a globally refined

grid, as drawn in Fig. 5.11 (a), in order to identify regions of great variations in

the electrostatic potential, which reduce the quality of the finite elements solution.

Based on equation (5.24), the error for each grid cell is estimated and, e.g., the 30%

of all cells with the largest error or those with an error above a critical value, are

flagged. In a next step, the flagged cells, marking the critical regions, are refined,

see Fig. 5.11 (b) . Again, for the new grid the errors are estimate, and, if necessary,

subsequent refinement steps (Fig. 5.11 (c)-(d)) are carried out until one arrives at a

final mesh that provides a sufficiently accurate solution of Poisson’s equation in the

whole simulation volume. In Fig. 5.11 (e) a grid with a typical degree of refinement,

as used for our kMC simulations, is depicted, and in Fig. 5.11 (f) the electric field

distribution resulting from the solution of Poisson’s equation on this grid is shown.

For an applied voltage of 1 V across a distance of 10 nm (from top to bottom face)

one expects a mean electric field of 1 MV/cm. As can be seen, convex protrusions

lead to a local enhancement of the electric field by nearly a factor of two. In these

regions the electrostatic potential changes faster than in other regions. There, the

grid has to be refined to a very high degree.

5.2.5 Results and Discussion

In the preceding paragraphs we have introduced all modifications to our kMC sim-

ulator that are necessary, if one intends to study the effect of ER on the leakage

currents in TZT capacitor structures. We introduced fractal geometry as a means to

classify and describe self-affine random rough surfaces, such as as-grown TiN elec-

trode surfaces, and we presented the algorithm that allows to generate as-desired

random rough surfaces, which are fed into the kMC simulator. Issues concerning

grid generation, adaptive refinement, and the solution of Poisson’s equation in a

simulation volume with rough boundaries were discussed. The framework for the

5.2. Electrode Roughness Effects 107

Figure 5.11: Grids with different degree of refinement, generated by the kMC

simulator, for a simulation volume of 10×20 nm2 with one rough surface. (a) three

times refined globally, (b) three times refined globally and once adaptively(c) three

times refined globally and twice adaptively, (d) three times refined globally and

three times adaptively, (e) five times refined globally and three times adaptively. (f)

Electric field distribution for a bias of 1 V applied between top an bottom surface.

108 5. Leakage Current Simulations

kMC simulations is set.

For the sake of simplicity, in the following, we only investigate capacitor structures

with one rough electrode. The second electrode is assumed to be perfectly flat. In

reality of course both electrodes are rough. Often the large scale undulations of

the top electrode roughly follow those of the bottom electrode, since the dielectric

films in between the two electrodes, the substrate for the growth of the second

electrode, are grown via ALD, which guarantees very homogeneous growth. The

kMC simulator can handle a second rough electrode without further modifications.

However, while adding a second rough electrode may lead to quantitative changes

of the ER effects, it complicates the analysis, and it does not introduce additional

phenomena besides those discussed below.

Small thickness fluctuations

Figure 5.12: Electric field distribution in a capacitor with one rough (α = 0.5,

w= 2 nm) and one flat electrode. A voltage of 0.9 V was applied across the dielectric

with a mean thickness of d = 9 nm, leading to 〈F 〉 = 1 MV/cm. White lines mark

a distance of x = 2 nm and x = 3 nm from the rough electrode. In black, lines of

constant electric field are drawn for a field enhancement of 10-50 %. The inset shows

leakage current data of a 9 nm TZT capacitor at 120◦C and kMC simulation results

assuming flat and rough (w = 2 nm) electrodes for 0.3 ≤ α≤ 0.9.

Figure 5.12 shows the simulated electric field distribution in a TZT capacitor with

one rough (α = 0.5, w = 2 nm, ξ = 20 nm) and one flat electrode. The ER leads to

5.2. Electrode Roughness Effects 109

a LEFE near convex protrusions of the electrode material into the dielectric due to

the aperture effect. As can be seen in the inset of Fig. 5.12, the leakage current,

although being dominated by PF conduction, which is strongly field-dependent, re-

mains unaffected for small thickness fluctuations, up to w= 2.0 nm for the capacitor

with a 9 nm thick ZrO2 dielectric. As discussed in 5.1, transport in the 9 nm TZT

capacitor mainly occurs across defects in the interior of the film, 2-3 nm away from

the injecting electrode (This region is marked in Fig. 5.12). These defects are sub-

jected only to a weak LEFE, usually smaller than 10 %. The expected increase of

the leakage current due to ER can be roughly estimated as follows. The rate for PF

emission of electrons from defect states RP F is roughly given by equation (3.67)[85]

RP F = ν× exp

−ED −

√

e3F/(πǫ0ǫopt)

kBT

,

where F is the electric field at the defect, ǫopt is the optical permittivity and ν ∼1013 s−1 a typical phonon frequency. We define the field enhancement factor γ as

γ ≡ δF

〈F 〉 , (5.25)

where δF denotes a local change in the electric field with respect to the average field

〈F 〉. Then one can approximate (for δF << F ):

RP F (〈F 〉+γ 〈F 〉)RP F (〈F 〉) ≈ exp

√

e3 〈F 〉/(πǫ0ǫopt)

2kBT·γ

. (5.26)

For T = 393 K, 〈F 〉 = 1 MV/cm, and γ = 0.1 this yields

RP F (〈F 〉+γ 〈F 〉)/RP F (〈F 〉) ≈ 1.6, (5.27)

i.e. only a negligible increase in the emission rate and, in turn, in the current is

expected.

Defects located near the electrode, which are subject to the strongest LEFE, do not

significantly contribute to the transport because of their low electron occupation

probability (Fig. 5.4). These findings retrospectively justify the assumption of flat

electrodes made in 5.1 and in ref. [111], since for a 9 nm TZT capacitor, simulated

there, the thickness fluctuations of the employed electrodes (w ∼ 1.2 nm) do not

influence the leakage current.

We also observe that the leakage current is, at least for an EOT of 0.9 nm, insensi-

tive on changes of the smoothness α (see Fig. 5.13), in contrast to the predictions of

the mean-field approach[130]. In Fig. 5.14 the electric field distributions in two ca-

pacitors are compared, whose electrodes only differ in the smoothness α, governing

110 5. Leakage Current Simulations

1 2 310-10

10-8

10-6

10-4

10-2

100

(d)(c)

electric field F(MV/cm)

(b)(a)

= 0.3 = 0.5 = 0.7 = 0.9

flat electr. data

1 2 310-10

10-8

10-6

10-4

10-2

100

w = 3 nm

w = 6 nmw = 5 nm

w = 4 nm

leak

age

curre

nt d

ensi

ty j

(A/

cm2 )

= 0.3 = 0.5 = 0.7 = 0.9

flat electr. data

1 2 310-10

10-8

10-6

10-4

10-2

100

= 0.3 = 0.5 = 0.7 = 0.9 flat electr. data

1 2 310-10

10-8

10-6

10-4

10-2

100

= 0.3 = 0.5 = 0.7 = 0.9 flat electr. data

Figure 5.13: Leakage current data of a 9 nm TZT capacitor at 120◦C and kMC

simulation result assuming flat electrodes, taken from ref. [111]. Leakage currents

were simulated for 0.3 ≤ α≤ 0.9. Assumed thickness fluctuations w were (a) 3 nm,

(b) 4 nm, (c) 5 nm, (d) 6 nm.

the short-range roughness. Small surface features are more defined for the surface

with the lower α. On a larger length scale the surface topologies, i.e. the surface

undulations, are very similar. The field enhancement due to the short-range rough-

ness is highly localized near the surface. Deeper in the bulk the field distribution is

insensitive to such roughness. Here, the large-scale undulations determine the field

distribution which is consequently nearly identical for α = 0.3 and α = 0.7. As the

defects which deliver the highest current contributions lie in the bulk region, the

leakage current does not depend on α.

5.2. Electrode Roughness Effects 111

Figure 5.14: Electric field distributions in a capacitor with one flat and one rough

electrode (w = 3 nm). Identical surface profiles of the rough electrode were assumed

in (a) and (b), differing only in the smoothness, taken as (a) α = 0.3 and (b) α =

0.7, respectively. A voltage of 0.9 V was applied across the dielectric with a mean

thickness of 9 nm, leading to 〈F 〉 = 1 MV/cm. In black, lines of constant electric

field are drawn.

Large thickness fluctuations

For larger thickness fluctuations convex protrusions of electrode material extend

further into the dielectric. Consequently, besides a LEFE confined to the direct

surrounding of sharp surface features, a second effect becomes important: the local

reduction of the dielectric film thickness, e.g. from d to dmin in Fig. 5.14 (a). At such

constrictions a long-range increase γc of the electric field, which extends through the

whole film, is observed. Assuming dmin ≈ d−w/2, we can estimate γc in terms of

relative thickness fluctuations w/d as

γc(w/d) =Fc −〈F 〉

〈F 〉 =U/dmin

U/d−1 ≈ 1

2w/d −1

, (5.28)

where Fc is the average electric field in the region of the constriction. In this case,

those defects which deliver the dominant contributions to the leakage current are

subject to an average electric field enhancement γc, e.g. γc(3 nm/9 nm) = 20% for

the capacitor in Fig. 5.14 (a). Thus, the PF conduction slightly increases, as seen

in Fig 5.13 (a).

112 5. Leakage Current Simulations

However, up to w = 5 nm (Fig. 5.13 (b)-(d)) the 9 nm TZT capacitor fulfills the

leakage criterion. Experimentally achievable thickness fluctuations are far below

this value. Thus, for an EOT of 0.9 nm the effects of ER on the leakage current are

uncritical for the operation of the DRAM storage capacitor.

Downscaling of the capacitor

1 2 310-10

10-8

10-6

10-4

10-2

100

102

EOT = 0.6 nm

EOT = 0.5 nm EOT = 0.4 nm (d)(c)

electric field F(MV/cm)

(b)(a)

w = 0 nm w = 1 nm w = 2 nm w = 3 nm w = 4 nm

1 2 310-10

10-8

10-6

10-4

10-2

100

102

EOT = 0.9 nm

le

akag

e cu

rrent

den

sity

j (

A/cm

2 )

w = 3 nm w = 4 nm w = 5 nm w = 6 nm w = 7 nm

data w = 0 nm

1 2 310-10

10-8

10-6

10-4

10-2

100

102

w = 0 nm w = 1 nm w = 2 nm w = 3 nm

1 2 310-10

10-8

10-6

10-4

10-2

100

102

w = 0 nm w = 1 nm w = 2 nm w = 3 nm

Figure 5.15: Leakage current simulations for EOTs of (a) 0.9 nm, (b) 0.6 nm, (c)

0.5 nm, (d) 0.4 nm. As the leakage current does not depend on α, shown currents

are averaged over 0.3 ≤ α≤ 0.9.

To study how ER effects on the leakage current behave on further downscaling of the

capacitor, we carried out leakage current simulations for capacitor structures with

film thicknesses of 6 nm, 5 nm, and 4 nm, as shown in Fig. 5.15 (b) - (d). For flat

electrodes (w = 0 nm) PF conduction remains the dominant transport mechanism

down to 4 nm. However, in comparison to the 9 nm film (Fig. 5.15 (a)), a drastically

5.2. Electrode Roughness Effects 113

10-10

10-8

10-6

10-4

10-2

7 6 5 4 3 2 1 0

EOT (nm)

cu

rrent

j a

t 1.3

MV/

cm (

A/cm

2 )

0.5 1.0

w(nm)

Figure 5.16: Simulated leakage currents at 〈F 〉=1.3 MV/cm for different EOTs and

thickness fluctuations w. Upon downscaling the sensitivity of the leakage current

toward ER drastically increases.

increased sensitivity of the leakage current on ER is observed, as illustrated in

Fig. 5.16.

In fact, when comparing equal thickness fluctuations w for different film thick-

nesses d, the relative thickness fluctuations, i.e. w/d, increase going to thinner

dielectrics. Thus, the electric field enhancement γc(w/d) at the dielectric constric-

tions is stronger. However, the resulting increase of PF conduction is not sufficient to

explain the large increase of the leakage current observed for EOTs ≤ 0.6 nm. Most

important for scaling, as it turns out, is the transformation of the dominant trans-

port mechanism from PF conduction to TAT, evidenced by the kMC simulations.

TAT scales much stronger with the film thickness than PF conduction, since it de-

pends exponentially on the thickness of the tunneling barriers. Thus, for extremely

thin films and large enough thickness fluctuations TAT across local constrictions of

the dielectric becomes the dominant transport mechanism and completely controls

the leakage current once w is further increased.

Accordingly, for the upcoming chip generations the requirements on the uniformity of

the fully 3D-integrated electrodes drastically increase, e.g. to w∼ 1 nm for an EOT

of 0.5 nm (see Fig. 5.16). The feasibility of an EOT of 0.4 nm seems questionable.

Strategies to achieve the smallest possible EOT values are needed. Based on the

114 5. Leakage Current Simulations

presented transport model, and considering ER effects, two technological options

exist.

The first is the reduction of the defect density nD. As discussed, the dominant

transport mechanisms PF conduction and TAT both involve bulk defects in the

ZrO2. Usually each single defect opens up one transport channel. Thus, the leakage

current is approximately proportional to nD, whereas only the contribution of bulk

defects in a certain distance from the electrodes, e.g. 2-3 nm for a 9 nm TZT

capacitor, should be considered. Both PF conduction and TAT would be affected

by a reduction of nD in the same way. In our simulations a homogeneous defect

density nD = 3 × 1018 cm−3 has been assumed. How much this defect density can

be reduced in the future is not clear. Further refinement of the transport model

by integration of structural relaxation of defects, discussed in the following section,

suggests the possibility of higher defect densities what would make this strategy

appear more promising.

The second option is to increase the CBO, e.g. by introducing an Al-doping at the

TiN/ZrO2 interface[36, 118, 141]. This option is a unique result of our model, as

standard leakage current models for TZT capacitors describe the current as purely

bulk-limited PF conduction and do not include the electron injection and thus the

influence of the electrode work function. Figure 5.17 depicts a possible scaling

scenario for TZT capacitors to realize an EOT of 0.4 nm. An increase of the CBO

by 0.1-0.2 eV with respect to 1.74 eV, extracted from kMC simulations, is sufficient

to fulfill the leakage current criterion for 0.4 nm EOT.

5.2.6 Summary

Performing 3D kMC simulations, we have carried out a quantitative analysis of the

influence of electrode roughness on the leakage current in TiN/ZrO2/TiN capacitors

for application in DRAM cells. While the leakage current is found to be insensitive

on the electrode smoothness (α), thickness fluctuations (w) turn out to have a

huge impact. We have found that, upon downscaling the EOT, the sensitivity of

the leakage current on thickness fluctuations drastically increases due to a change

of the dominant transport mechanism from PF conduction to TAT. For a feasible

thickness fluctuation of w ∼ 1 nm the leakage current criterion could not be fulfilled

for a desirable EOT of 0.4 nm. According to the new insights into the transport

mechanisms in TiN/ZrO2/TiN capacitors an increase of the CBO by 0.1-0.2 eV is

suggested in order to enable an EOT of 0.4 nm.

5.3. Structural Relaxation of Defects 115

1 2 3 410-10

10-8

10-6

10-4

10-2

100

EOT = 0.4 nm

leak

age

curre

nt d

ensi

ty j

(A/

cm2 )

electric field (MV/cm)

ED= 1.15 eV

EB1.64 eV1.74 eV1.84 eV1.94 eV

Figure 5.17: Leakage current simulations of a 4 nm TZT capacitor (EOT=0.4 nm)

at 120◦C, varying the CBO EB between TiN and ZrO2.

5.3 Structural Relaxation of Defects

So far, all studies of leakage currents in high-κ dielectrics did not take into account

structural relaxation of defects upon a change of their charge state (e.g. refs. [90,

129]). Also the transport model, developed in 5.1, was set up under the assumption

that structural relaxation is negligible. However, recently, M. Häufel carried out

accurate first principles calculations of defect energy levels for different charge states

in t-ZrO2[142], which, in contrast, demonstrate that structural relaxation is very

pronounced in t-ZrO2, leading to shifts of the defect energy levels larger than 1 eV.

Such large energy shifts suggest that in case of charge transport across defect states,

as it is envisaged for TZT capacitors, cf. 5.1 and 5.2, structural relaxation may play

a major role.

Consequently, a defect model was derived from Häufel’s first principles calculations

and subsequently coupled to our kMC charge transport simulations. ZAZ thin

films, which were shown to have slightly better insulating properties than pure

ZrO2 films[3], were taken as an exemplary system to assess the impact of structural

relaxation on the electron transport.

116 5. Leakage Current Simulations

5.3.1 Introduction

For decades, semiconductor bulk defects have been subject of intensive studies[143]

and it is common knowledge that the energy of defect levels depends on the de-

fect charge state[144]. Electrostatic interactions between charges localized at defect

sites and the surrounding lattice atoms lead to structural relaxation. The atoms in

the vicinity of the defect undergo a rearrangement in order to minimize the total

energy of the system. Remarkable shifts of atoms, and thus defect levels, can be

observed, a prominent example being the bistability of the oxygen vacancy in SiO2

(E′-center)[145].

Although structural relaxation of defects caused by a change of their charge state

has been known for a long time, its influence on charge transport across defect

states has been scarcely addressed[146]. Especially in modern nanoscale devices,

featuring film thicknesses below ten nanometers, unanticipated effects may arise.

Especially for the high-κ oxides this is to be expected. It is well-known that the

high-κ oxides, due to their large coordination, suffer from high defect densities[39].

As the main fields of application of these materials, usage as dielectrics in the state-

of-the-art transistor gate stack[147], in future carbon-nanotube transistors[148], and

in the storage capacitor of DRAMs[2], impose high requirements on their insulating

properties, numerous studies of defect levels in materials like HfO2, ZrO2, TiO2,

and SrTiO3[123, 149, 150, 151] were conducted, which incidentally demonstrated

the enormous effect of structural relaxation on the defect energy levels in these

materials.

As has been discussed in the previous sections, transport mechanisms involving

charge flow across defect levels in the oxide band gap dominate the conduction

in high-κ films currently employed as dielectrics[110, 111]. Obviously, as soon as

charges flow across defects, causing frequent changes of the defect charge states,

defect relaxation will strongly influence the electronic transport.

Considering the current stand of the semiconductor industry, which, only a few years

after successful integration of high-κ dielectrics, is pushing these materials to their

scaling limits[152], a better understanding of the charge transport is crucial. Envis-

aging the current downscaling speed, a value of 0.4 nm for the EOT will be required

within the next two years (according to the ITRS). Yet no feasible concept is known

that fulfills the strict leakage current criteria for the DRAM storage capacitor and,

at the same time, enables such low EOTs. Future material systems are investigated,

whereas the focus is laid on SrTiO3[153, 154, 155] and rutile TiO2[36] as dielectrics

in combination with Ru-based electrodes, since these materials provide permittivi-

5.3. Structural Relaxation of Defects 117

ties of more than 100. However, these material systems are far from being ripe for

integration into the large-scale production process.

Thus, other approaches have been suggested, under these laminate dielectric stacks,

such as trilaminate ZAZ dielectrics[3, 116, 89], which are used in the latest chip

generations. Here, two polycrystalline layers of t-ZrO2, featuring a permittivity

close to 40, surround a thin (∼ 0.5 nm) amorphous Al2O3 interlayer. It was found

that these ZAZ films have insulating properties slightly superior compared to pure

ZrO2 films[3], which were discussed in 5.1 and 5.2. The actual reason for this

improvement is still a subject of discussion. It was suggested that the interlayer

prevents ZrO2 grains from extending over the whole dielectric and causing excessive

surface roughness[156], which is known to have a detrimental effect on the capacitor

performance[131]. It was also argued qualitatively that the large band gap of Al2O3

sets up a barrier for charge flow[116]. However, a quantitatively predictive model for

the leakage currents in TZAZT capacitors is still missing. The fact that structural

relaxation of defects easily causes energetic shifts of the defect levels larger than one

electron volt[157] strongly suggests that relaxation processes have to be accounted

for in this context.

To take structural relaxation of defects into account, we augmented the established

kMC framework for transport simulations by coupling our kMC algorithm to a de-

fect model derived from first principles GW calculations. The impact of structural

relaxation of defects and the accompanying change of the defect energy levels on

charge carrier transport is investigated and a complete transport model for TZAZT

capacitors is developed. The first principles GW calculations, described in the fol-

lowing were carried out by Martin Häufel at the chair for Theoretical Semiconductor

Physics, Walter Schottky Institute, Technical University of Munich[142].

5.3.2 GW Calculations

As said, many elaborate studies of defect levels in the high-κ materials are available.

To save computation time, usually defect levels are calculated in the local density

approximation (LDA), i.e. the results suffer from the systematically underestimated

band gap. While precise rules for rescaling the band gap are known, due to the

rescaling process, the energy of the defect levels within the band gap is afflicted

with some uncertainty. For an unambiguous identification of structural defects which

might constitute the dominant leakage current channels a more accurate analysis is

needed. Alternative calculation schemes exist, which are on the one hand much more

costly, but on the other hand deliver very precise results for defect level energies.

118 5. Leakage Current Simulations

One of these methods is the GW approximation [158, 159], in which dynamical

screening effects are taken into account, leading to better results for the band gap

and excited states of many semiconducting and insulating materials [159, 160].

In the GW approach crystal electrons are viewed to be surrounded by a polarization

cloud. Thus, a screened interaction with the other electrons takes the place of bare

Coulomb interaction. The excitations of the many-particle system can therefore be

described as those of weakly interacting quasiparticles.

The GW approximation can formally be written as an expansion of the self-energy

Σ, defined as the energy difference between the quasiparticle and the bare electron,

in terms of the screened interaction W ,

Σ ≈GW, (5.29)

where G denotes the single-particle Green’s function. In Feynman diagram repre-

sentation this reads

(5.30)

The equation of motion is given by the quasiparticle equation

hΨi(r)+∫

d3r′ Σ(r,r′;Ei)Ψi(r′) = EiΨi(r), (5.31)

where h is the one-particle Hamiltonian in the Hartree approximation, and Ψi and

Ei are the quasiparticle wavefunction and energy, respectively. In this definition

the self-energy contains all the exchange and correlation contributions beyond the

Hartree approximation. Note that, for simplicity, spin degrees of freedom are ne-

glected in this presentation, whereas the actual calculations are carried out in a

spin-polarized way. Comparing the quasiparticle equation to the Kohn-Sham equa-

tions(

h+vxc(r))

ϕi(r) = εiϕi(r), (5.32)

with ϕi and εi denoting the Kohn-Sham orbitals and energies, respectively, one can

see that the self-energy term takes the place of the exchange-correlation potential

vxc.

As a starting point the Kohn-Sham energies and wavefunction were determined self-

consistently within the LDA. Then the quasiparticle effects enter the calculations

as energy corrections, obtained using first order perturbation theory to evaluate the

operator (Σ−vxc) at the Kohn-Sham energies and for the Kohn-Sham wavefunctions.

5.3. Structural Relaxation of Defects 119

All defect energy level calculations were carried out with the ABINIT [161, 162, 163,

164] software package. The density functional theory simulations were performed for

the crystal at zero temperature. Vanderbilt ultrasoft pseudopotentials (USPP) [165]

were used, as, according to our calculations, the band gap energy differs only by

0.1 eV to the value obtained with norm-conserving pseudopotentials. For the gener-

ation of the pseudopotentials the following reference configurations were used. For

oxygen all six electrons in the n = 2 shell (2s22p4) were included. For zirconium,

all electrons in the n = 4 shell plus the 5s electrons (4s24p64d25s2 = 12 electrons)

were treated as valence electrons. Exchange and correlation were described using

the Perdew-Zunger-Ceperley-Alder potential [166, 167]. Quasiparticle corrections

for the highest occupied valence band state and the lowest lying conduction band

state were calculated using the GW approximation. The one-shot G0W0 approxima-

tion was applied using the standard Plasmon-Pole model of Godby and Needs[168]

at two frequencies: the static limit (0 eV) and another frequency, imaginary, of the

order of the plasmon frequency (the peak in the electron energy loss spectrum). We

used the volume plasmon pole energy of 13.8 eV given in ref. [169].

First, the energy gap of t-ZrO2 was determined using the same 2×2×2 supercell

consisting of 96 atoms, that was used for the defect calculations. Note that the cell

(as well as all defect supercells used in the calculations below) was relaxed using a dif-

ferent ultrasoft pseudopotential (the volume per unit formula differs by ∼1%�). The

relative error caused by this is assumed to cancel in the final results, as we are only

interested in the positions of the defect states within the gap. The valence pseudo-

wavefunctions are expanded in a projector-augmented-wave (PAW) [170] basis set

up to a kinetic energy of 15 ha. In the screening calculation and in the calculation

of the self-energy matrix elements the summation was performed over 730 and 700

electronic bands, respectively. That is, in order to obtain converged GW results,

about 350 unoccupied buffer bands were included in the summation. The calculation

was carried out at the Γ-point only. All parameters were converged within 0.1 eV.

As a result a LDA energy gap of 3.52 eV and a GW gap of 5.52 eV were obtained,

which, in the following, are used as a reference for the defect levels.

5.3.3 Defect Model

Oxygen vacancies in t-ZrO2, a common defect in this material[171] and the most

probable candidate for causing the leakage currents in this dielectric, were studied.

Calculations were carried out using a 2 × 2 × 2 supercell consisting of 95 atoms, as

shown in Fig. 5.18 (top).

120 5. Leakage Current Simulations

Figure 5.18: (top) 95-atom supercell of t-ZrO2 used to perform the GW calcula-

tions. O atoms are shown in light red, Zr atoms in dark blue. An oxygen vacancy was

formed by removing one oxygen atom (green). Subsequently the atomic structure

was relaxed. (bottom) Zoom on the direct vicinity of the oxygen vacancy (missing

O atom colored green). The relaxed structure for the neutral vacancy V 0O is pic-

tured. Semitransparent, dark blue balls show the positions of the nearest-neighbor

Zr atoms for the positively charged vacancy. Upon removal of an electron, the Zr

atoms are pushed away from the vacancy by the localized positive charge. For better

visibility the displacement is upscaled by a factor of 5.

5.3. Structural Relaxation of Defects 121

charge state This thesis Theor.

LDA GW t-ZrO2[172] m-ZrO2[173] c-ZrO2[174]

-1 2.58 (0.94) 2.59 (2.94) 2.40 (4.20)

-1 3.49 (0.03) 5.00 (0.53) 4.40 (2.20)

neutral 0.96 (2.56) 3.06 (2.44) 3.35 (3.55) 2.20 3.45 (2.1)

+1 (α) 0.68 (2.84) 3.08 (2.43) 4.20 (2.70)

+1 (β) 3.41 (0.11) 4.41 (1.09) 6.10 (0.80)

+2 3.48 (0.04) 5.25 (0.27) 7.00 (-0.10)

Table 5.2: Defect energy levels of the neutral and of charged oxygen vacancies in

t-ZrO2 (in eV) in comparison to published theoretical results. In braces we give the

energy difference to the CB minimum.

In Tab. 5.2 the results of the defect level calculations are presented along with

published theoretical values.

According to defect formation energies, calculated in ref. [149] for the chemically

similar HfO2, the prevailing charge states for oxygen vacancies (VO) are neutral

(V 0O), positive (V +

O ), and twofold positive (V ++O ). The position of the corresponding

defect energy levels of the relaxed oxygen vacancy within the band gap of t-ZrO2 is

shown in Fig. 5.19.

For V 0O, two degenerate, filled defect levels are found 2.44 eV below the ZrO2 CB,

corresponding to two electrons that are strongly localized at the vacancy site. Upon

removal of an electron (V +O ), the now unoccupied spin down state (β) moves up to

1.09 eV, while the occupied spin up state (α) remains energetically unaffected. For

the neutral oxygen vacancy, screened Coulomb repulsion between the two localized

electrons, as described by equation (5.33) below, leads to a potential energy con-

tribution of the order of 0.3-0.5 eV. Thus, upon removal of one electron one would

expect the defect energy level to shift downwards by this amount. However, upon

positive charging of the defect a large structural relaxation takes place. Figure 5.18

(bottom) shows a close up view of the relaxed atomic structure in the vicinity of a

neutral oxygen vacancy V 0O. If one electron is removed, the remaining positive charge

localized at the vacancy site forces the positive Zr ions to relax 0.11 Å away from

the vacancy (shown as semitransparent blue balls). Thus, the attractive potential

energy between the remaining electron and the Zr ions decreases, causing the (α)

122 5. Leakage Current Simulations

Figure 5.19: Energy level diagram showing the positions of the oxygen vacancy

states within the t-ZrO2 band gap. All defect energy levels were obtained using

the GW approximation; all values are given in electron volts (eV). For V +O the non-

degenerate spin up and down states are denoted (α) and (β), respectively. Open

(closed) circles denote unoccupied (occupied) states.

state to more or less remain energetically unchanged and the unoccupied (β) state

to shift 1.35 eV upwards. A similar value of 0.10 Å for the relaxation distance was

found in refs. [123] and [172]. However, in ref. [123] the exchange correlation was

described by the generalized gradient approximation leading to a defect level shift

of only 0.6 eV, while embedded cluster calculations in ref. [172], overestimating the

ZrO2 band gap, found a shift of 2.75 eV. By removing another electron (V ++O ), the

two unoccupied states move up close to the CB.

5.3.4 Kinetic Monte Carlo Framework

The defect model, derived from the GW calculations (Fig. 5.19), was incorporated

into our kMC algorithm to study the effect of the structural relaxation on the

electronic transport.

To get a correct picture of the electron transport and, especially, the statistical in-

terplay between different transport mechanisms, structural relaxation was added to

the list of state-to-state transitions, which have to be taken into account concur-

rently in the kMC algorithm. The full set of transitions is sketched in Fig. 5.20;

for comparison to the earlier model framework the reader should refer to chapter 3

5.3. Structural Relaxation of Defects 123

and Fig. 3.1 therein. These transitions encompass both, electrode-limited mecha-

nisms, such as (i) direct tunneling and (ii) Schottky emission, as well as bulk-limited

mechanisms, such as (iii) elastic/inelastic tunneling of electrons into and (iv) out of

defects, (v) field-enhanced thermal emission of electrons from defects (PF emission),

and (vi) tunneling between defects. In addition, we integrated self-consistently (vii)

structural relaxation of defects upon a change of their charge state, causing a shift

of the defect energy level.

Figure 5.20: Schematic band diagram of a MIM structure with applied voltage U .

Transport mechanisms, implemented in the kMC simulator: (i) direct/FN tunneling,

(ii) SE, (iii) elastic and inelastic tunneling (p= number of phonons with energy h̄ω)

into and (iv) out of defects, (v) PF emission, (vi) defect-defect tunneling, and (vii)

structural relaxation of a defect upon a change of its charge state (waved arrows).

The physical models for the different transport mechanisms are an essential input of

our kMC simulations. They have been introduced in chapter 3; a concise overview

124 5. Leakage Current Simulations

layer thicknesses (nm) layer permittivities

bott. top ZAZ bott. top ZAZ

samples ZrO2 Al2O3 ZrO2 stack ZrO2 Al2O3 ZrO2 stack

T7ZAZT 3.4 0.5 3.4 7.3 (7.3) 38 9 38 31.1 (31.4)

T9ZAZT 4.5 0.4 4.5 9.4 (9.4) 38 9 38 33.4 (35.2)

Table 5.3: Model parameters for the individual layers of the ZAZ dielectric used

in the kMC simulations. For comparison, measured values, only available for the

whole dielectric stack, are given in braces.

can be found in ref. [111].

In order to account for the structural relaxation of the defects due to a change of

their charge state the defect model is coupled to the kMC simulations by setting the

defect energies for different charge states to the GW values. It turns out that the

residence time of electrons in the defects is orders of magnitude larger than the time

needed for structural relaxation (in the picosecond regime[175]) around the defect

after a change of the charge state. Thus, we neglected the relaxation time in our

simulations, i.e. for transitions of type (vii) we assumed a rate Rrelax → ∞.

The electrostatic potential in the vicinity of charged defects was modeled as a

screened Coulomb potential. For defect i with electrical charge Ze, located at posi-

tion ~ri, it can be written as

V (~r) =Ze

4πǫoptǫ0 |~r− ~ri|. (5.33)

Here, Z is an integer number, giving the charge state of the defect, and ǫopt is the

optical permittivity of the surrounding dielectric material, taken as 5.6 in t-ZrO2[31].

In the above-described first principles kMC modeling framework charge transport

in planar TZAZT capacitors, fabricated as described in chapter 2.3, was simulated.

Total film thicknesses of the ZAZ stacks, determined via spectroscopic ellipsometry,

and their effective permittivities, extracted from capacitance measurements, given

in Tab. 5.3, are in excellent agreement with our model assumptions.

5.3.5 First-principles kMC Leakage Current Simulations

Figure 5.21 shows the leakage current data of the two capacitor structures for tem-

peratures between 25◦C and 150◦C as a function of the mean electric field 〈F 〉 =U/d,

5.3. Structural Relaxation of Defects 125

with the applied bias U and the total dielectric stack thickness d. In order to study

the effect of structural relaxation on electron transport in these thin film capaci-

tors by means of kMC some assumptions had to be made, whose justification will

be commented on later. For the sake of simplicity it was assumed that one single

type of defect, homogeneously distributed within the dielectric, causes the dominant

leakage current contributions.

Best estimate physical parameters, entering the kMC transition rates, were taken

to characterize the dielectric stack and kept fixed during the simulations. They are

listed in Tab. 5.1. Assumed thicknesses of the individual ZrO2 and Al2O3 layers,

given in Tab. 5.3, were chosen so as to reproduce the measured total film thicknesses

and permittivities of the ZAZ laminates. The only fitting parameters are the CBO

EB at the TiN/ZrO2 interface and the defect density.

In this model framework we found a single set of parameters that accurately re-

produces both the temperature and the thickness scaling of the leakage current, as

can be seen in Fig. 5.21 (a) and (b). The closest fit to the experimental data was

achieved for a barrier height of 1.9 eV, a value which compares well to the earlier

simulation result of 1.74 eV for TZT capacitors, studied in 5.1 and ref. [111], and

to theoretical estimates given on page 89.

For the best possible fit of the kMC simulations to the experimental data the defect

level energies for the different charge states were slightly changed with respect to

the GW values (given in braces) to E+d,(β) = 1.1 eV (1.09 eV) for the unoccupied (β)

state of the positively charged vacancy and E0d = 2.3 eV (2.44 eV) for the degenerate

states of the neutral defect with respect to the CB. The energies of the unoccupied

states of the twofold positively charged defect and the occupied (α) state of the

positively charged defect were left unchanged. These minor changes lie well within

the error margin of the GW results. This suggests that, indeed, oxygen vacancies

are responsible for the dominant leakage current contribution in TZAZT capacitors.

5.3.6 Trap-assisted Tunneling Across Oxygen Vacancies

From the kMC simulations the following picture of electron transport across the

dielectric, as shown in Fig. 5.22, emerges. An electron is injected from the TiN

electrode into the unoccupied electron state of a positive vacancy 1.1 eV below the

ZrO2 CB (Fig. 5.22 (a)) which subsequently relaxes by Erelax =E0d −E+

d,(β) = 1.2 eV

(Fig. 5.22 (b)). The trapped electron then moves by phonon-assisted tunneling, cf.

3.3.2, into a neighboring positively charged vacancy (Fig. 5.22 (c)) which again re-

laxes. In this way, percolating along a chain of oxygen vacancies, the electron crosses

126 5. Leakage Current Simulations

0 1 2 3 410-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

(a)

simdata

50°C

150°C100°C

leak

age

curr

ent

dens

ity j

(A/c

m2 )

mean electric field <F> (MV/cm)

25°C

relaxation currents

0 1 2 3 410-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

simdata

50°C

120°C 90°C

leak

age

curr

ent

dens

ity j

(A/c

m2 )

mean electric field <F> (MV/cm)

25°C

relaxation currents

(b)

Figure 5.21: Leakage current data (symbols) of TZAZT capacitors with dielectrics

of (a) 7 nm and (b) 9 nm thickness compared to kMC simulation results (lines).

In the low-field region the measured current is dominated by transient relaxation

currents, while the kMC results describe the steady state. Thus, no comparison is

possible[110]. Dielectric breakdown was observed for high fields.

5.3. Structural Relaxation of Defects 127

Figure 5.22: The different steps of an electron crossing the dielectric film are

sketched. (a) First the electron is injected into a positiv oxygen vacancy which (b)

subsequently undergoes structural relaxation causing the defect energy level to shift

downwards. (c) In a next step the electron moves via an inelastic, multiphonon-

assisted transition, cf. 3.3.2, to a neighbouring positively charged oxygen vacancy.

(d) By repeating the last two steps, the electron can percolate along a chain of defect

states until it is finally emitted to the counter-electrode. Here, p is the number of

phonons with energy h̄ω.

the dielectric (Fig. 5.22 (d)). The number of percolation sites varies between one

and three, depending on film thickness and applied bias (The thinner the dielectric

and the higher the applied bias, the fewer sites are needed.).

Earlier models which neglected structural relaxation effects came to the conclusion

that transport is dominated by PF conduction[90, 129, 110] via a defect level located,

e.g., 1.15 eV[110] below the CB. The analysis of the ab initio results shows that this

corresponds almost exactly to the relaxation energy Erelax of the oxygen vacancy.

Such coincidence explains why in both models a similar temperature scaling of the

leakage current is obtained. Clearly, only the model including structural relaxation

128 5. Leakage Current Simulations

effects is consistent with first principles defect level calculations, as shown here and

in refs. [123] and [172].

Best quantitative matching of the simulations with the experimental data was achieved

for a defect concentration of 1×1020 cm3. Such a high concentration might be due to

the presence of the high work function TiN electrodes, which are known to increase

the concentration of oxygen vacancies in such high-κ thin films[41]. Via soft x-ray

photoelectron spectroscopy defect densities in the range of 2×1019 to 1×1020 cm3

were estimated[176]. The investigation of (ZrO2)0.8(Al2O3)0.2 films via conductive

atomic force microscopy[177] suggests that transport predominantly occurs along

grain boundaries in the polycrystalline ZrO2. Since oxygen vacancies are known to

cluster at grain boundaries[176, 178], such experimental evidence strongly suggests

that they are responsible for opening the loss paths.

5.3.7 TiN/ZrO2/Al2O3/ZrO2/TiN Capacitors

In Fig. 5.23 (a) leakage current data of capacitors with pure ZrO2 dielectrics, dis-

cussed in 5.1, and of the present TZAZT capacitors is plotted as a function of 〈F 〉.As can be seen, introduction of the amorphous Al2O3 interlayer leads to a reduction

of the leakage currents. So far, the Al2O3 was thought to act as a barrier for elec-

tron flow due to its large band gap of around 8.9 eV. Additionally, it was suggested

that the interlayer interrupts the grain boundaries of the polycrystalline material.

While these effects may contribute to the reduction of the leakage current in ZAZ

films compared to pure ZrO2 films, see Fig. 5.23 (a), our simulations suggest that

their influence is negligible. Instead, the electron transport turns out to be mainly

controlled by FZAZ , the electric field in the ZrO2 layers of the ZAZ stack. Upon in-

troduction of the low-κ Al2O3 interlayer (κ= 9), the electric field in the ZrO2 layers

is reduced with respect to FZrO2 , the value it would have in a dielectric film of pure

ZrO2. Assuming that the electric displacement D= κ ·F is constant throughout the

dielectric film, FZAZ can be calculated from

FZAZ =U

d(ZrO2)+ κ(ZrO2)κ(Al2O3)d(Al2O3)

. (5.34)

Here, U is the applied voltage, κ(Al2O3) and κ(ZrO2) are the permittivities in

Al2O3 and ZrO2, respectively, and d(Al2O3) and d(ZrO2) are the total thicknesses

of the materials.

In Fig. 5.23 (b) we replot the data from Fig. 5.23 (a) as a function of FZAZ for the

ZAZ films and FZrO2 for the ZrO2 films, so as to eliminate the field reduction effect

5.3. Structural Relaxation of Defects 129

of the Al2O3 interlayer. Apparently, the major effect of the interlayer is the reduction

of the electric field in the ZrO2 layers. While this might also be advantageous in

terms of reliability[89], it comes with the cost of an increased EOT. The fact that,

for an identical FZAZ , the current does not scale with the film thickness, at least in

the considered thickness range, is consistent with our transport model. As electrons

pass the dielectric in a multistep process, the rate for this to happen is governed by

the slowest single step, i.e. phonon-assisted tunneling of an electron between two

defects. Thus, relevant for the scaling of the current is not the total film thickness

but the defect-defect distance, i.e. the defect concentration.

5.3.8 Summary

In summary, multistep, inelastic, phonon-assisted tunneling across oxygen vacancies

has been identified as the dominant conduction mechanism in TZAZT capacitors.

A closed model, taking structural relaxation of the involved defects upon a change

of their charge state into account, was presented. Consideration of defect relaxation

leads to a qualitative modification of our picture of electron transport. Instead of

PF conduction, TAT across oxygen vacancies is identified as the dominant transport

mechanism.

130 5. Leakage Current Simulations

0 1 2 3 410-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

9 nm ZAZ1

9 nm ZrO2

7 nm ZAZ1

7 nm ZrO2

leak

age

curr

ent

dens

ity j

(A/c

m2 )

<F> (MV/cm)

(a)

0 1 2 3 410-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

ZAZ ZrO2

(b)

9 nm ZAZ1

9 nm ZrO2

7 nm ZAZ1

7 nm ZrO2

leak

age

curr

ent

dens

ity j

(A/c

m2 )

F , F (MV/cm)

Figure 5.23: Experimental leakage current data of pure ZrO2 dielectrics and of

ZAZ stacks for a temperature of 120◦C is compared. In (a) the current density j is

plotted vs. 〈F 〉, the mean electric field in the dielectric and in (b) vs. the field in

the ZrO2 layers, i.e. FZAZ for the ZAZ films and FZrO2 for the pure ZrO2 films,

thus accounting for the field reduction in the ZAZ stack caused by the low-κ Al2O3

interlayer.

Conclusions and Outlook

The common practice in analyzing leakage currents in thin-film capacitor structures

is to fit experimental results with simplified compact models. Their big advantage

is their low complexity and their speed, as they usually provide analytical expres-

sions, reducing the task of fitting to the adjustment of a small set of parameters.

However, often these models oversimplify the situation, i.e. conclusions, which can

be drawn based on the fitting of experimental data, are limited or, worse, might

even be misleading. In general, an in-depth description of electronic transport in

such structures is a very ambitious task, facing alone the large number of unknown

quantities in the typical systems of interest.

It was the goal of the present work to go beyond the so far existing approaches

and to establish a higher-level simulation framework for leakage current studies.

Therefore, a modeling environment was set up, where microscopic models for ev-

ery relevant physical transport process were united in a comprehensive simulation

framework with a kinetic Monte Carlo algorithm at its heart. The unique prop-

erties of the kinetic Monte Carlo formalism were exploited, such as modeling at

the microscopic scale, e.g. to resolve single charge carrier motions, straightforward

coupling to finite elements based differential equation solvers allowing to take into

account, e.g., carrier-carrier interactions, the study of phenomena spatially localized

in all three dimensions (electrode roughness effects, structural relaxation of defects,

defect-assisted transport), and the investigation of the statistical interplay of dif-

ferent microscopic processes, e.g. the concurrent simulation of multiple transport

mechanisms which allows to investigate their mutual interdependence.

Having such a modeling environment at hand, we carried out elaborate studies

on the TiN/ZrO2 material system, which is of outmost importance for the semi-

conductor industry, where it is employed in the DRAM storage capacitor. Closed

transport models were developed, which shed light on the microscopic transport

in high-performance TiN/ZrO2/TiN capacitors. Again, making use of the kinetic

Monte Carlo framework and its modular architecture, which allows to augment

131

132 5. Leakage Current Simulations

transport models with only little effort, we iteratively improved our transport mod-

els by taking more and more physical effects into account. At the end, we could

provide a self-consistent transport model for the TiN/ZrO2 material system. The

detailed modeling on the microscopic scale, in contrast to the compact models, en-

abled us to propose target-oriented strategies how to optimize the performance of

the capacitor structures. Problems which may arise in the future due to further

downscaling of the capacitors could be identified in advance, and viable approaches

how to tackle these could be provided.

In general, the flexibility of the developed modeling framework makes it easily trans-

ferable allowing to study electronic transport in a variety of other, even very differ-

ent, systems, as will be commented on in the outlook.

5.3. Structural Relaxation of Defects 133

While in this work TiN/ZrO2, the currently employed material system for the

DRAM storage capacitor, was in the focus of our investigations, it will, in the

medium term be replaced by other materials, such as StTiO3 and TiO2, which

feature even higher permittivities. These, unfortunately are weak insulators; their

band gaps are around 3.5 eV. Currently ongoing investigations, thus, not unsurpris-

ingly, measure very high leakage currents for capacitor structures employing these

dielectrics. The conduction band offsets provided by these materials are roughly

0.5 eV too low to surpress intrinsic leakage currents due to, mostly, Schottky emis-

sion. Strategies to increase these offsets via acceptor doping are on pursued. How-

ever, these systems are far from being ripe for introduction into the DRAM produc-

tion process. The simulation framework proposed in the present work will, in the

future, be used to the study of these systems in order to provide theoretical sup-

port for the experimental efforts by enraveling the dominant microscopic transport

mechanisms in these systems.

But, besides DRAM applications, the developed simulation tool, due to the variety

of the implemented models and the versatility of the kinetic Monte Carlo algorithm

and the program architecture itself, is well-suited to investigate transport in other

metal-insulator-metal structures. For example, tunneling diodes for the rectification

of high-frequency ac currents, which consist of an aluminum and a gold electrode

separated by a ultrathin (∼ 3.6 nm) Al2O3 film, were investigated[179, 180]. Cur-

rents were found to be dominated by direct and Fowler-Nordheim tunneling and

electrode roughness, for such ultrathin dielectric films, was suggested to play an

important role[181, 182]. Further optimization of the fabrication process of these

devices is on its way and will be supported by transport simulations.

Another field of rising interest, for which kinetic Monte Carlo transport simulations

seem to be a very promising tool for theoretical studies, is the fabrication and op-

timization of organic solar cells, especially those based on the bulk-heterojunction

concept[183]. The simulation framework established in the present work has been

adapted in order to simulate the complex microscopic processes involved in the

photocurrent generation in these devices, namely exciton generation, difussion, sep-

aration, decay and charge carrier injection, transport, recombination, extraction etc.

The current functionality of the simulator corresponds to the state-of-the-art in this

field, subsumed in refs. [184, 185, 186, 187, 188]. Thus, the foundations are laid to

study experimental data of such devices in the future.

134 5. Leakage Current Simulations

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bel, G. Scarpa, W. Porod, and P. Lugli, “Energy harvesting using nano an-

tenna arrays,” in 11th International Conference on Nanotechnology, NANO,

2011.

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tovoltaic cells using small-molecular-weight organic thin films,” Nature, vol.

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[184] M. Casalegno, G. Raos, and R. Po, “Methodological assessment of kinetic

Monte Carlo simulations of organic photovoltaic devices: The treatment of

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[185] F. Yang, K. Sun, and S. R. Forrest, “Efficient solar cells using all-organic

nanocrystalline networks,” Advanced Materials, vol. 19, no. 23, pp. 4166–4171,

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[186] F. Yang and S. R. Forrest, “Photocurrent generation in nanostructured organic

solar cells,” Nano, vol. 2, p. 1022, 2008.

[187] R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the

behavior of nanostructured photovoltaic devices,” Journal of Applied Physics,

vol. 101, p. 083509, 2007.

[188] P. K. Watkins, A. B. Walker, and G. L. B. Verschoor, “Dynamical monte

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154 List of Own Publications and Contributions

List of Own Publications and Contributions

G. Jegert, M. J. Haeufel, D. Popescu, W. Weinreich, A. Kersch, P. Lugli,

The role of defect relaxation for trap-assisted tunneling in high-k thin films: A first

principles kinetic Monte Carlo study,

Physical Review B (under review)

G. Jegert, A. Kersch, W. Weinreich, P. Lugli,

Monte Carlo Simulation of Leakage Currents in TiN/ZrO2/TiN Capacitors,

IEEE Transactions on Electron Devices 58, 327 (2011)

G. Jegert, A. Kersch, W. Weinreich, P. Lugli,

Ultimate scaling of TiN/ZrO2/TiN capacitors: Leakage currents and limitations due

to electrode roughness,

Journal of Applied Physics 10, 014504 (2011)

M. Bareiss, B. Tiwari, A. Hochmeister, G. Jegert, U. Zschieschang, H. Klauk, B.

Fabel, G. Scarpa, G. Koblmüller, G. Bernstein, W. Porod, P. Lugli,

Nano Antenna Array for Terahertz Detection,

IEEE Transactions on Microwave Theory and Techniques, accepted for publication

(2011)

M. Bareiss, A. Hochmeister, G. Jegert, G. Koblmüller, U. Zschieschang, H. Klauk,

B. Fabel, G. Scarpa, W. Porod, P. Lugli,

Energy Harvesting using Nano Antenna Arrays,

11th International Conference on Nanotechnology, NANO (2011)

M. Bareiss, A. Hochmeister, G. Jegert, U. Zschieschang, H. Klauk, B. Fabel, G.

Scarpa, W. Porod, P. Lugli,

Quantum carrier dynamics in ultra-thin MIM tunneling diodes 17th International

Conference on Electron Dynamics in Semiconductors,

Optoelectronics and Nanostructures, EDISON 17 (2011)

List of Own Publications and Contributions 155

M. Bareiss, A. Hochmeister, G. Jegert, U. Zschieschang, H. Klauk, R. Huber, D.

Grundler, W. Porod, B. Fabel, G. Scarpa, P. Lugli,

Printed Array of Thin-Dielectric MOM Tunneling Diodes,

Jour. Appl. Phys., 110, 044316 (2011).

D. Zhou, U. Schroeder, J. Xu, J. Heitmann, G. Jegert, W. Weinreich, M. Kerber, S.

Knebel, E. Erben, T. Mikolajick,

Reliability of Al2O3-doped ZrO2 high-k dielectrics in three-dimensional stacked metal-

insulator-metal capacitors,

Journal of Applied Physics 108, 124104 (2010)

G. Jegert, A. Kersch, W. Weinreich, U. Schroeder, P. Lugli,

Modeling of leakage currents in high-k dielectrics: Three-dimensional approach via

kinetic Monte Carlo,

Applied Physics Letters 96, 062113 (2010)

D. Zhou, U. Schroeder, G. Jegert, M. Kerber, S. Uppal, R. Agaiby, M. Reinicke, J.

Heitmann, L. Oberbeck,

Time dependent dielectric breakdown of amorphous ZrAlxOy high-k dielectric used

in dynamic random access memory metal-insulator-metal capacitor,

Journal of Applied Physics 106, 044104 (2009)

W. Weinreich, R. Reiche, M. Lemberger, G. Jegert, J. Müller, L. Wilde, S. Teichert,

J. Heitmann, E. Erben, L. Oberbeck, U. Schroeder, A.J. Bauer, H. Ryssel,

Impact of interface variations on J-V and C-V polarity asymmetry of MIM capaci-

tors with amorphous and crystalline Zr(1−x)AlxO2 films, Microelectronic Engineer-

ing 86, 1826 (2009)

U. Schroeder, W. Weinreich, E. Erben, J. Mueller, L. Wilde, J. Heitmann, R. Agaiby,

D. Zhou, G. Jegert, A. Kersch,

Detailed Correlation of Electrical and Breakdown Characteristics to the Structural

Properties of ALD Grown HfO2- and ZrO2-based Capacitor Dielectrics,

The Electrochemical Society Transactions 25, 357 (2009)

156 List of Abbreviations

List of Abbreviations

3D Three-Dimensional

ALD Atomic Layer Deposition

BKL Bortz Kalos Lebowitz

BL Bitline

CB Conduction Band

CBO Conduction Band Offset

DAT Defect-Assisted Transport

DRAM Dynamic Random Access Memory

EOT Equivalent Oxide Thickness

ER Electrode Roughness

FN Fowler-Nordheim

IBS Imaginary Band Structure

IC Integrated Circuit

IES Infrequent Event System

ITRS International Technology Roadmap for Semiconductors

IV Current-Voltage

JEDEC Joint Electron Device Engineering Council

kMC kinetic Monte Carlo

LDA Local Density Approximation

LEFE Local Electric Field Enhancement

MA Miller-Abrahams

MD Molecular Dynamics

MIGS Metal-Induced Gap States

MIM Metal-Insulator-Metal

PAW Projector-Augmented Wave

pb parabolic

PDF Probability Density Function

PES Potential Energy Surface

157

158 List of Abbreviations

PF Poole-Frenkel

SE Schottky Emission

SEM Scanning Electron Microscopy

t-ZrO2 tetragonal ZrO2

TAPF Tunneling-Assisted Poole-Frenkel

TAT Trap-Assisted Tunneling

TB Tight Binding

TxZAZT TZAZT capacitor with x nm thick dielectric

TxZT TZT capacitor with x nm thick dielectric

TZAZT TiN/ZrO2/Al2O3/ZrO2/TiN

TZT TiN/ZrO2/TiN

USPP Ultra-Soft Pseudopotentials

VB Valence Band

WKB Wentzel Kramers Brillouin

WL Wordline

Acknowledgements 159

Acknowledgements

At this point, I would like to take the opportunity to mention all those people I

owe a debt of gratitude for their support, their participation and their contributions

which helped me to complete this work:

First, and most of all, I would like to thank my supervisor Prof. Dr. Alfred Kersch

for his unlimited support and for sharing his vast knowledge in the field of the

semiconductor device simulations and many other fields in countless discussions. He

has been the best possible guide and advisor on this long journey.

I thank Prof. Dr. Paolo Lugli, for his support, for his huge efforts which allowed me

to complete this work after the bankruptcy of Qimonda, for many fruitful discussions

and for providing me an excellent work environment.

The present work would not have been possible without several scientific collab-

orations which I enjoyed very much. I would like to thank all the people which

were involved in these. In particular, I am indebted to Wenke Weinreich, Dr. Uwe

Schröder, Dr. Dayu Zhou, Mario Bareißand Martin Johannes Häufel.

Moreover, I thank all my colleagues at the Institute for Nanoelectronics and the

members of the Qimonda simulation group for scientic support as well as for creating

an excellent atmosphere in which it was a pleasure to work.

I owe special thanks to Dr. Wolfgang Hönlein for being a fantastic group leader

during my time at Qimonda and for his outstanding support during the turbulent

times following the bankruptcy of Qimonda.

Last, but most importantly, I thank Angela Link for the unconditioned and loving

support throughout the thesis, which helped me to endure also the stressful phases

during the last years.

In der Schriftenreihe des Walter Schottky Instituts der Technischen Universität München sind bisher folgende Bände erschienen: Vol. 1 Cornelia Engel Si/SiGe basierende Phototransistoren 131 Seiten ISBN 3-932749-01-4

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Vol. 102 Vol. 109 Andreas Florian Härtl Sebastian Strobel Novel concepts for biosensors using Nanoscale contacts to organic molecules diamond-based field effect transistors based on layered semiconductor 255 Seiten substrates ISBN 978-3-941650-02-2 140 Seiten ISBN 978-3-941650-09-1 Vol. 103 Felix Florian Georg Hofbauer Realization of electrically tunable single Vol. 110 quantum dot nanocavities Ying Xiang 160 Seiten Semiconductor nanowires and ISBN 978-3-941650-03-9 templates for electronic applications 152 Seiten Vol. 104 ISBN 978-3-941650-10-7 Dominic F. Dorfner Novel photonic biosensing based on Vol. 111 silicon nanostructures Michael Kaniber 169 Seiten Non-classical light generation in ISBN 978-3-941650-04-6 photonic crystal nanostructures 177 Seiten Vol. 105 ISBN 978-3-941650-11-4 Till Andlauer Optoelectronic and spin-related properties Vol. 112 of semiconductor nanostructures in Martin Hermann magnetic fields Epitaktische AlN-Schichten auf 157 Seiten Saphir und Diamant ISBN 978-3-941650-05-3 216 Seiten ISBN 978-3-941650-12-1 Vol. 106 Christoph Bihler Vol. 113 Magnetic semiconductors Dominik Heiss 190 Seiten Spin storage in quantum dot ensembles ISBN 978-3-941650-06-0 and single quantum dots 196 Seiten Vol. 107 ISBN 978-3-941650-13-8 Michael Huber Tunnel-Spektroskopie im Vol. 114 Quanten-Hall-Regime Tillmann Christoph Kubis 164 Seiten Quantum transport in semiconductor ISBN 978-3-941650-07-7 nanostructures 253 Seiten Vol. 108 ISBN 978-3-941650-14-5 Philipp Achatz Metal-insulator transition and super- Conductivity in heavily boron-doped diamond and related materials 151 Seiten ISBN 978-3-941650-08-4

Vol. 115 Vol. 122 Lucia Steinke Ilaria Zardo Magnetotransport of coupled quantum Growth and raman spectroscopy Hall edges in a bent quantum well studies of gold-free catalyzed semi- 194 Seiten conductor nanowires ISBN 978-3-941650-15-2 184 Seiten ISBN 978-3-941650-22-0 Vol. 116 Christian Lauer Vol. 123 Antimonid-basierte Vertikalresonator- Andre Rainer Stegner Laserdioden für Wellenlängen oberhalb 2 µm Shallow dopants in nanostructured and 180 Seiten in isotopically engineered silicon ISBN 978-3-941650-16-9 185 Seiten ISBN 978-3-941650-23-7 Vol. 117 Simone Maria Kaniber Vol. 124 Optoelektronische Phänomene in hybriden Andreas J. Huber Schaltkreisen aus Kohlenstoffnanoröhren und Nanoscale surface-polariton spectros- dem Photosystem I copy by mid- and far-infrared near- 136 Seiten field microscopy ISBN 978-3-941650-17-6 144 Seiten ISBN 978-3-941650-24-4 Vol. 118 Martin Heiß Vol. 125 Growth and properties of low-dimensional Marco Andreas Höb III-V semiconductor nanowire hetero- Funktionalisierung von Gruppe structures IV-Halbleitern 172 Seiten 186 Seiten ISBN 978-3-941650-18-3 ISBN 978-3-941650-25-1 Vol. 119 Vol. 126 Sandro Francesco Tedde Daniel Claudio Pedone Design, fabrication and characterization of Nanopore analytics – electro-optical organic photodiodes for industrial and studies on single molecules medical applications 114 Seiten 277 Seiten ISBN 978-3-941650-26-8 ISBN 978-3-941650-19-0 Vol. 120 Vol. 127 Danche Spirkoska Jovanov Casimir Richard Simeon Katz Fundamental properties of self-catalyzed Multi-alloy structures for injectorless GaAs nanowires and related heterostructures Quantum Cascade Lasers 200 Seiten 131 Seiten ISBN 978-3-941650-20-6 ISBN 978-3-941650-27-5 Vol. 121 Vol. 128 Jürgen Sailer Barbara Annemarie Kathrin Baur Materials and devices for quantum Functionalization of group III-nitrides Information processing in Si/SiGe for biosensor applications 158 Seiten 215 Seiten ISBN 978-3-941650-21-3 ISBN 978-3-941650-28-2

Vol. 129 Vol. 136 Arne Laucht Christian Claus Jäger Semiconductor quantum optics with Polycrystalline Silicon Thin Films for tailored photonic nanostructures Electronic Applications 232 Seiten 202 Seiten ISBN 978-3-941650-29-9 ISBN 978-3-941650-36-7 Vol. 130 Vol. 137 Jia Chen Gunther Christian Jegert Compact laser-spectroscopic gas sensors Modeling of Leakage Currents in using Vertical-Cavity Surface-Emitting High-к Dielectrics Lasers 170 Seiten 150 Seiten ISBN 978-3-941650-37-4 ISBN 978-3-941650-30-5 Vol. 131 Hans Leonhard Prechtel Ultrafast Photocurrents and Terahertz Radiation in Gallium Arsenide and Carbon Based Nanostructures 160 Seiten ISBN 978-3-941650-31-2 Vol. 132 Roland Enzmann Technologieentwicklung für eine Einzelphotonenquelle 142 Seiten ISBN 978-3-941650-32-9 Vol. 133 Alexander Bachmann Antimonide-Based Vertical-Cavity Surface-Emitting Lasers 144 Seiten ISBN 978-3-941650-33-6 Vol. 134 Markus Andreas Mangold Two-Dimensional Gold Nanoparticle Arrays – A Platform for Molecular Optoelectronics 140 Seiten ISBN 978-3-941650-34-3 Vol. 135 Stefan Birner Modeling of semiconductor nanostructures and semiconductor–electrolyte interfaces 227 Seiten ISBN 978-3-941650-35-0