Quantum Mechanical and Atomic Level ab initio Calculation of … · 2018-12-15 · The quantum mechanical effects are studied with a model, based on a self-consistent solution of

Quantum Mechanical and Atomic Level ab initio Calculation of Electron Transport through Ultrathin Gate Dielectrics of Metal-

Oxide-Semiconductor Field Effect Transistors

By the Faculty of Electrical Engineering and Information Technology at Chemnitz

University of Technology

approved

Dissertation

In fulfillment of the requirements for the degree

Doktor-Ingenieur

(Dr.-Ing.)

Submitted by

Dipl.-Ing. Ebrahim Nadimi

Born in 1.8.1972 in Tehran-Iran

Date of submission: 19.11.2007

Examiners: Prof. Dr. rer. nat. Christian Radehaus

Prof. Dr. -Ing. habil. John Thomas Horstmann

Dr. -Ing. Karsten Wieczorek

Date of defense: 16.04.2008

Quantenmechanische und atomistische ab initio Berechnung des Elektronentransports durch ultradünne Gatedielektrika in

MOSFETs

von der Fakultät für Elektrotechnik und Informationstechnik

der Technischen Universität Chemnitz

genehmigte

Dissertation

zur Erlangung des akademischen Grades

Doktor-Ingenieur

(Dr.-Ing.)

vorgelegt

von

Dipl.-Ing. Ebrahim Nadimi

geboren am 1.8.1972 in Teheran-Iran

eingereicht am: 19.11.2007

Gutachter: Prof. Dr. rer. nat. Christian Radehaus

Prof. Dr. -Ing. habil. John Thomas Horstmann

Dr. -Ing. Karsten Wieczorek

Tag der Verleihung: 16.04.2008

1

Abstract

The low dimensions of the state-of-the-art nanoscale transistors exhibit increasing

quantum mechanical effects, which are no longer negligible. Gate tunneling current is

one of such effects, that is responsible for high power consumption and high working

temperature in microprocessors. This in turn put limits on further down scaling of

devices. Therefore modeling and calculation of tunneling current is of a great interest.

This work provides a review of existing models for the calculation of the gate

tunneling current in MOSFETs. The quantum mechanical effects are studied with a

model, based on a self-consistent solution of the Schrödinger and Poisson equations

within the effective mass approximation. The calculation of the tunneling current is

focused on models based on the calculation of carrier’s lifetime on quasi-bound states

(QBSs). A new method for the determination of carrier’s lifetime is suggested and then

the tunneling current is calculated for different samples and compared to measurements.

The model is also applied to the extraction of the “tunneling effective mass” of electrons

in ultrathin oxynitride gate dielectrics.

Ultrathin gate dielectrics (tox<2 nm) consist of only few atomic layers. Therefore,

atomic scale deformations at interfaces and within the dielectric could have great

influences on the performance of the dielectric layer and consequently on the tunneling

current. On the other hand the specific material parameters would be changed due to

atomic level deformations at interfaces. A combination of DFT and NEGF formalisms

has been applied to the tunneling problem in the second part of this work. Such atomic

level ab initio models take atomic level distortions automatically into account. An atomic

scale model interface for the Si/SiO2 interface has been constructed and the tunneling

currents through Si/SiO2/Si stack structures are calculated. The influence of single and

double oxygen vacancies on the tunneling current is investigated. Atomic level

distortions caused by a tensile or compression strains on SiO2 layer as well as their

influence on the tunneling current are also investigated.

Keywords: Metal-Oxide-Semiconductor Field Effect Transistor (MOSFET),

Tunneling current, Effective Mass Approximation (EMA), oxynitride, Quasi-Bound State

(QBS), Lifetime, ab initio, Density functionally Theory (DFT), Non-Equilibrium Green’s

Function (NEGF), Oxygen Vacancy

2

Bibliografische Beschreibung

Quantenmechanische und atomistische ab initio Berechnung des Elektronentransports

durch ultradünne Gatedielektrika in MOSFETs

Nadimi, Ebrahim – 145 S., 52 Abb., 2 Tab., 144 Lit.

Stichworte

Metal-Oxide-Semiconductor Field Effect Transistor (MOSFET), Tunnelstrom, Effective

Mass Approximation (EMA), oxynitride, Quasi Bound States (QBS), Lebensdauer, ab

initio, Dichte Funktional Theorie (DFT), NonEquilibrium Green’s Function (NEGF),

Sauerstoff Leerstelle

Referat

Die vorliegende Arbeit beschäftigt sich mit der Berechnung von Tunnelströmen in

MOSFETs (Metal-Oxide-Semiconductor Field Effect Transistors). Zu diesem Zweck

wurde ein quantenmechanisches Modell, das auf der selbstkonsistenten Lösung der

Schrödinger- und Poisson-Gleichungen basiert, entwickelt. Die Gleichungen sind im

Rahmen der EMA gelöst worden. Die Lösung der Schrödinger-Gleichung unter offenen

Randbedingungen führt zur Berechnung von Ladungsverteilung und Lebensdauer der

Ladungsträger in den QBSs. Der Tunnelstrom wurde dann aus diesen Informationen

ermittelt. Der Tunnelstrom wurde in verschiedenen Proben mit unterschiedlichen

Oxynitrid Gatedielektrika berechnet und mit gemessenen Daten verglichen. Der

Vergleich zeigte, dass die effektive Masse sich sowohl mit der Schichtdicke als auch mit

dem Stickstoffgehalt ändert.

Im zweiten Teil der vorliegenden Arbeit wurde ein atomistisches Modell zur Berechnung

des Tunnelstroms verwendet, welche auf der DFT und NEGF basiert. Zuerst wurde ein

atomistisches Modell für ein Si/SiO2-Schichtsystem konstruiert. Dann wurde der

Tunnelstrom für verschiedene Si/SiO2/Si-Schichtsysteme berechnet. Das Modell

ermöglicht die Untersuchung atom-skaliger Verzerrungen und ihren Einfluss auf den

Tunnelstrom. Außerdem wurde der Einfluss einer einzelnen und zwei unterschiedlich

positionierter neutraler Sauerstoffleerstellen auf den Tunnelstrom berechnet. Zug- und

Druckspannungen auf SiO2 führen zur Deformationen in den chemischen Bindungen

3

und ändern den Tunnelstrom. Auch solche Einflüsse sind anhand des atomistischen

Modells berechnet worden.

4

Contents

CONTENTS ................................................................................................ 4

LIST OF SYMBOLS AND ABBREVIATIONS.................. ........................... 8

1 INTRODUCTION.................................................................................... 12

1.1 Motivation..................................... .......................................................................... 12

1.2 Quantum effects in nanoscale devices........... ..................................................... 14

1.3 Tunneling current models....................... .............................................................. 16

2 FULLY QUANTUM MECHANICAL MODEL OF TUNNELING CURRE NT BASED ON THE EFFECTIVE MASS APPROXIMATION AND SCHRÖDINGER-POISSON SOLVER ...................................................... 27

2.1 The Schrödinger-Poisson solver................. ......................................................... 28

2.2 Tunneling current model based on an one side op en boundary conditions ... 39

2.3 Results and discussions........................ ............................................................... 43

3 THE ELECTRON “TUNNELING EFFECTIVE MASS” IN ULTRA- THIN SILICON OXYNITRIDE GATE DIELECTRICS ................ ......................... 56

3.1 The “tunneling effective mass” vs. effective ma ss ............................................ 56

3.2 Oxynitride gate dielectric..................... ................................................................. 57

3.3 Extracting the “tunneling effective mass” from I-V measurements.................. 59

3.4 Discussing the results ......................... ................................................................. 62

4 ATOMIC LEVEL CALCULATIONS........................ ................................ 66

4.1 Density functional theory...................... ................................................................ 67

4.2 DFT implementation with localized numerical bas is set.................................... 75

4.3 Non-equilibrium Green's function method for tra nsport.................................... 77

5 SILICON/SILICON-DIOXIDE SYSTEM................... ............................... 87

5

5.1 Constructing the Si/Silicon dioxide model struc ture ......................................... 87

5.2 Interface stability ............................ ....................................................................... 90

5.3 Two probe configuration with intrinsic leads ... .................................................. 90

5.4 Transport under non-equilibrium condition...... .................................................. 96

6 THE INFLUENCE OF OXYGEN VACANCY DEFECTS IN SILICO N DIOXIDE ON THE TUNNELING CURRENT........................................... 100

6.1 Single oxygen vacancy .......................... ............................................................. 100

6.2 Double oxygen vacancies........................ ........................................................... 105

7 COMPRESSED AND STRAINED SILICON DIOXIDE.......... ............... 108

8 SUMMARY AND OUTLOOK .............................. ................................. 112

ZUSAMMENFASSUNG.................................... ...................................... 115

BIBLIOGRAPHY....................................... .............................................. 117

LIST OF FIGURES ................................................................................. 134

LIST OF TABLES..................................... .............................................. 138

ACKNOWLEDGEMENTS................................... .................................... 139

VERSICHERUNG ................................................................................... 141

THESES ................................................................................................. 142

CURRICULUM VITAE ................................... ......................................... 145

6

INHALTSVERZEICHNIS …………………………………………………….… 4

Liste der Symbole und Abkürzungen …………………………………… … 8

1 Einleitung ……………………………........................... ........................... 12

1.1 Motivation ………………………………………… ……………………………………... 12

1.2 Quanteneffekte in Nanobauelementen … …………………………………………... 14

1.3 Tunnelstrommodelle…………………………................... ....................................... 16

2 Quantenmechanisches Tunnelstrommodell basierend a uf der

effektiven Massenapproximation und der Lösung des S chrödinger-

Poisson Gleichungssystems …… ………………………………………… 27

2.1 Die Lösung des Schrödinger-Poisson Gleichungssy stems ……………………. 28

2.2 Tunnelstrommodell basierend auf einseitig offen er Randbedingung ………... 39

2.3 Ergebnisse und Diskussion …………… …………………………………………….. 43

3 Die effektive Masse in sehr dünnen Oxynitrid Gate dielektrika ……. 56

3.1 Die "Tunnel Effektive Masse" gegen die effektiv e Masse ………………………. 56

3.2 Die Berechnung der effektiven Masse aus der I-V Kennlinie …………………... 57

3.3 Oxynitrid Gatedielektrika ……………………………………………………… ……… 59

3.4 Ergebnisse und Diskussion …………… …………………………………………….. 62

4 Atomskalige Berechnungen ……………………………………………. 66

4.1 Dichtefunktionaltheorie…… ………………………………………………………... 67

4.2 Realisierung der DFT mit lokalen Basissätzen …… ……………………………… 75

4.3 Der Formalismus der Nichtgleichgewicht Greensch en-Funktion für das

Transportproblem…………………………......................... ........................................... 77

5 Das Silizium/Siliziumdioxid Schichtsystem.………………… ……….. 87

7

5.1 Konstruktion der Silizium/Siliziumdioxid Modell struktur ………………………. 87

5.2 Die Stabilität der Grenzfläche………………........... ............................................... 90

5.3 Zweiprobeanordnung … ……………………………………………………………… 90

5.4 Transport im Nichtgleichgewichtszustand… ………………………………………. 96

6 Der Einfluss von Sauerstoff-Leerstellen auf den T unnelstrom ….. 100

6.1 Einzelne Sauerstoffleerstelle………………............ ............................................100

6.2 Zwei Sauerstoffleerstellen…………………… …………………………………….. 105

7 Der Einfluss von Druck- und Zugspannungen auf den

Tunnelstrom…………………………… …………………………………….. 108

8 Zusammenfassung und Ausblick ……………………………………... 11 2

ZUSAMMENFASSUNG (auf Deutsch)…… ………………………………. 115

Bibliographie ………………………… ……………………………………... 117

Liste der Abbildungen ………………………… ………………………….. 134

Liste der Tabellen …………………… …………………………………….. 138

Danksagung …………………………… ……………………………………. 139

Versicherung………………………… ……………………………………… 141

Thesen……………… ………………......................................................... 142

Lebenslauf ……………………………… ……………………………………145

8

List of symbols and abbreviations A Spectral function

BTE Boltzmann Transport Equation

CMOS Complementary Metal Oxide Semiconductor

CPU Central Processor Unit

DBRTD Double-barrier resonant tunneling diode

DD Drift Diffusion

DFT Density functional theory

DG Density Gradient

DOS Density Of State

DT Direct Tunneling

EOT Effective Oxide Thickness

ε Dielectric constant

0ε Permittivity of free space

Cε Correlation energy

Xε Exchange energy

XCε Exchange correlation energy

E Energy

EF Fermi energy

Eϑ Valence band edge energy

f Impact frequency

F Electric field

FN Fowler-Nordheim tunneling

cφ Conduction band offset

quantφ Bohmian quantum potential

vφ Valence band offset

G Green’s function

Γ Energy broadening or imaginary part of energy

9

H Hamiltonian

HD Hydrodynamic

ℏ Reduced Planck’s constant

η Intrinsic impedance in a wave guide in transverse-resonant method

J Current density

k Wave number

Bk T Boltzmann constant multiplied by temperature

LDA Local Density Approximation

LDOS Local Density Of State

Lg Gate length

*lm Longitudinal electron effective mass in Si

*tm Transverse electron effective mass in Si

*lhm Light hole effective mass in Si

*hhm Heavy hole effective mass in Si

m*DT Direct tunneling electron effective mass

m*c Electron effective mass at the conduction band edge

ifM Matrix element

MOS Metal Oxide Semiconductor

MOSFET Metal Oxide Semiconductor Field Effect Transistor

µ Chemical potential

NEGF Non-Equilibrium Green’s Function

n Spatial distribution of Electron density

NA Acceptor doping concentration

Ni,j Density of total electrons on each subband

Nϑ Valence band effective density of states

Ω Volume

P Scattering Probability

PAO Pseudo Atomic Orbitals

PDOS Projected Density Of State

ρ Spatial total charge distribution

10

Ψ Wave function

q Electron charge

QBS Quasi-Bound State

QHD Quantum Hydrodynamic

Qi,j Total charge density on each subband

QTBM Quantum Transmission Boundary Method

( )r E Complex reflection coefficient

er Mean interelectronic distance

rc Cutoff radius

rs Second cutoff in double-ζ basis

Σ Self energy

S Perturbative input in Schrödinger equation

TEM Transmission Electron Microscopy

t time

tox Gate dielectric thickness

T Transmission probability

T Kinetic energy operator

τ Lifetime

( )R Eτ Reflection time

v Group velocity

V Voltage

Vbs Bulk source voltage

Vds Drain source voltage

Veff Effective potential

FBV Flat band potential

Vg Gate voltage

HV Hartree potential

WFTE Wigner Functions Transport Equation

WKB Wentzel-Kramers-Brillouin approximation

Wg Gate width

Wsep Work of separation

11

XPS X-Ray Photoemission Spectroscopy

lmY Spherical harmonic

Z

& Z

Terminal impedance in a wave guide model

12

1 Introduction

1.1 Motivation

For more than 30 years, MOS device technology has been improved at a

dramatic rate. A large part of the success of the MOS transistor was due to the fact that

it can be scaled to increasingly smaller dimensions, which results in a higher

performance. The ability to improve the performance while decreasing the power

consumption has made CMOS architecture the dominant technology for integrated

circuits. The scaling of the MOS transistor has been the primary factor driving

improvements in microprocessor performance. Transistor delay times have been

decreased by more than 30% per technology generation resulting in a doubling of

microprocessor performance every two years. Figure 1.1 shows the scaling scheme of a

MOS transistor.

Figure 1.1: Scaling scheme of a MOSFET transistor

However, down scaling of devices is accompanied by some unwanted effects

which generally originate from quantum mechanical effects in low dimensions. Such

effects lead to a degradation of the device performance as well as power consumption

and heat budget in nanoscale devices. One of the major problems, which have slowed

the downscaling trend, is the gate leakage current. The gate leakage current in

nanoscale devices with ultrathin gate dielectric is mainly due to the quantum mechanical

tox

p-type Si substrate

n+ n+

source drain Lg

Field oxide

Drain contact

source contact Gate

-Vbs

Vg Vds

Wg

λ < 1 scaling factor

Lg λ

tox λ

Vds λ

Junction depth λ

Doping concentration 1/λ

13

tunneling of carriers through a finite potential barrier. It increases exponentially as the

dielectric thickness decreases. The high leakage current in transistor leads to a high

power consumption of chips and high operating temperatures, which in turn require

improved cooling methods.

Therefore, the gate leakage current becomes one of the most important issues in

nanoscale MOS transistors. In this work we will try to understand the mechanism of

leakage current through ultrathin gate dielectrics and develop a new model for its

computation.

As mentioned above the leakage current is mainly due to the quantum

mechanical tunneling. Although extensions of classical models are applied to the

leakage current calculation, a thorough computation of it requires quantum mechanical

models. Such models have something in common with their classical counterparts; they

used to be fed with some material specified parameters such as band gaps, dielectric

constants, carrier effective masses etc.

The specified material parameters are generally calculated or measured for a

bulk material. However, the ultrathin gate dielectric with tox<2 nm are constituted of few

atomic layers in the direction perpendicular to the channel. As will be shown, the

material specified parameters in such thin layers could deviate from their values in bulk.

The two interfaces of the gate dielectric with channel and gate induce atomic level

distortions which could result in changes of material parameters.

An atomic level calculation could resolve this problem and automatically take the

atomic distortions at the interfaces into account. On the other hand, such an atomic level

model could be applied to mixed materials with unknown physical parameters. In the

second part of this work we apply an atomic level model to the calculation of tunneling

current through an ultrathin oxide layer. However, atomic level models are

computationally very demanding and the number of atoms and consequently the size of

the system under consideration are limited to few hundred atoms. This would limit the

application of such models to the real devices but could be very useful in studying

atomic level distortions and their effect on material parameters and device performance.

14

1.2 Quantum effects in nanoscale devices

As MOS gate dimensions have been reduced, the thickness of the gate oxide

must be approximately linearly scaled with channel length to maintain the same amount

of gate control over the channel to ensure good short channel behavior. In addition,

short channel behavior is governed by the ratio of channel depletion layer thickness to

channel length. The channel depletion layer is inversely proportional to the square root

of the channel doping concentration. Therefore, to suppress short channel effect it is

necessary to scale (increase) channel doping concentration as well. However,

increasing channel doping leads to a shift of the threshold voltage. Therefore, thinner

gate oxide is required to compensate this shift and regularize the threshold voltage in

short channel devices.

The result of high substrate doping is a narrow and deep potential well at the

substrate/gate-dielectric interface. This confining potential together with an ultrathin gate

oxide raise the importance of quantum mechanical effects such as energy discretization

of the conduction band, penetration and reflection of carrier wave function at the

channel/gate-dielectric interface, etc. The reflection of the wave function at the interface

causes the centroid of carrier density to move back from interface into the bulk, which in

turn leads to a change in gate capacitance which is not negligible in MOS structures with

ultra-thin dielectrics. As a result, it is obvious, that a quantum mechanical description of

down scaled MOS structures is required to take quantum effects into account. However,

increased computational burden of quantum mechanical calculations was the result of

emerging quantum corrections to classical models. These corrections are incorporated

in classical models to include quantum mechanical effects, while using further the

classical formalism.

Hänsch proposed a model that modifies the density of states to account for the

shifting of the charge centroid away from the interface [1]. A model originating from Van

Dort [2] captures both splitting of continuous energy bands and charge centroid shift

through an increase of the silicon band gap near the interface [3]. An alternative way to

include quantum mechanical effects is to use the effective potential approach that takes

into account the natural non-zero size of an electron wave packet in the quantized

system [4-6]. Ferry suggested an effective potential that is derived from a wave packet

15

description of particle motion. Within this formulation, the effective potential Veff is related

to the self-consistent Hartree potential V, obtained from the Poisson equation, through

an integral smoothing relation as follows:

Veff(x)=∫V(x+y)B(y,a0)dy (1-1)

where B is a Gaussian function with the standard deviation a0.

Another physically well-founded approach is the density gradient (DG), which

formalizes the quantum mechanical requirement that wave functions, and thus carrier

densities, can not change abruptly versus position. Mathematically, the DG model

modifies the continuity equation by adding a “quantum potential” to the electrostatic

potential [7]. The quantum potential acts like an additional quantum force term in the

particle simulation, similar in sprit to the Bohm interpretation [8] of quantum mechanics.

The Bohm quantum potential is proportional to the gradient of the charge density as

depicted in Eq. (1-2) and will vanish to zero if the Planck’s constant is zero.

2 2*

*6quant

n

m q nφ

∇=

ℏ (1-2)

The Bohm quantum potential appears in the quantum hydrodynamic (QHD)

model as well as in DG, which are corrections to their classical counterparts, the

hydrodynamic and drift-diffusion models. In fact, both the DG and QHD models can be

derived as simplifications of Wigner’s functions transport equation (WFTE) [7], in an

analogous manner to the derivations of DD and HD models from the Boltzmann

transport equations BTE [9].

Further it has been shown that the use of quantum corrections modifies the

classical models, so that some quantum phenomena may be captured. However

physical phenomena, such as tunneling, which will become much more important as the

transistor’s dimensions shrink, still cannot be captured with these corrections. There

were few attempts to apply physically well-founded DG correction to the tunneling

problem [10,11] however the main approach is the application of quantum mechanical

models to this problem. Based on the solution of the Schrödinger equation, as the heart

16

of quantum mechanics, different models have been introduced for the calculation of

tunneling current in MOS structures. Remarkably, the fabrication of nanoscale

transistors provides faster processors, which in turn make the expensive quantum

mechanical simulation of them manageable. In the next section a brief review of these

models will be presented.

1.3 Tunneling current models

Quantum mechanical tunneling of carriers through a potential barrier is one of the

interesting aspects of quantum mechanics, which has gained more attention due to its

application in electronic devices. The carrier tunneling occurs through a classically

forbidden region which has been sandwiched between two carrier reservoirs.

Fundamental parameters required for the calculation of the tunneling current are: density

of quantum states (DOS) in both initial and final reservoirs, carrier distribution function

on that states and the transparency of the barrier or in other words, the transmission

function. There exist various models of the tunneling current. They differ in methods and

approximations, which are applied to the calculation of above mentioned parameters. In

the following subsections some of these models are discussed.

1.3.1 Compact models

In a compact model the reservoirs are generally modeled by a continuum of

quantum states. The wave function of carriers on these states has a plane wave form.

The confinement of the electron gas to the interface and its influence on DOS are

generally ignored. The transparency of a barrier is calculated by applying the Wentzel-

Kramers-Brillouin (WKB) approximation.

Pioneering work for the calculation of tunneling current through a negative energy

or classically forbidden region has been done by Bardeen [12] and Harrison [13]. In

these approaches the probability of carrier transmission from an initial state to one of the

final states with the same energy in a continuum is calculated by using the Fermi Golden

rule:

17

22( ) (1 ( ))i f if f i fP M DOS E f f E

π→ = −

ℏ (1-3)

where i and f are the initial and final states with fi and ff are the occupation probabilities

of the corresponding states, DOSf(E) is the density of final states and Mif is the matrix

element of transition. In the particular case of tunneling, Mif is hard to define. Bardeen

suggested writing the transition matrix elements in terms of matrix elements of the

quantum current operator which is in turn calculated as a function of wave functions.

if ifM i J= − ℏ (1-4)

Where Jif is the current density operator evaluated in the tunneling region and expressed

as:

0

**

2ifz

d dJ

mi dz dz

Ψ Ψ = Ψ − Ψ

ℏ (1-5)

where m is the effective mass of carriers and Ψ is the wave function or the solution of the

Schrödinger equation.

This model has been applied to the calculation of tunneling current between two

metallic electrodes which are separated by an insulator. Harrison expands the wave

function of the system as a cosine function in the classical region and applies the WKB

approximation to determine the exponential decaying wave function in the negative

energy region. The resulting tunneling current expression reads:

2exp( 2 )( )

f

t i

z

z i fk z

qJ k dz f f dE

π

∞

−∞

= − −∑ ∫ ∫ℏ

(1-6)

where q is the electron charge, kt and kz are the transverse and perpendicular to the

interface wave numbers respectively and zi and zf are the barrier boundary points.

18

Later on by Khairurrijal [14] the model is applied to the calculation of electron

tunneling form gate to the substrate in a MOS structure. The application of this model

leads to a compact analytical expression for the tunneling current in MOS structures.

One of the other widely used compact models for the tunneling in MOS structures

is the Fowler-Nordheim (FN) model [15-22]. This model is generally an analytical model

for the tunneling current of a 3D electron gas through a triangular barrier at applied

voltages higher than the barrier height. In the FN model the tunneling of electrons from

the conduction band of gate electrode to the conduction band of the gate oxide is

considered (figure 1.2). This is the main contribution to the tunneling in early devices,

which consisted of thick oxides (e.g. 5-10 nm) and worked under high supply voltages

(e.g. 5 V). The model applies a WKB approximation to the calculation of the

transmission coefficient as follows:

1/ 2

2

2 ( ( ) )( ) exp 2

x

s

z

ox zz

z

m q z ET E dz

ϕ − = − ∫

ℏ (1-7)

where Ez is the z component of the energy perpendicular to the interface, mox is the

effective mass of electrons in the barrier, zs and zx are the turning points of the classically

forbidden zone in the barrier, φ(z) is the electrostatic potential in the dielectric and ħ is

the reduced Planck’s constant. By summing over all energy states in the reservoir, the

FN current density can be expressed in the following general form:

2 exp( )FN ss

BJ AF

F= − (1-8)

where

3 20

2 216Si

ox ox c

q mA

m

επ ε φ

=ℏ

and 3/ 24 2

3ox c ox

Si

mB

q

ε φε

=ℏ

(1-9)

In the above expressions cφ is the barrier height, εSi and εox are the relative

dielectric constants of silicon and gate oxide respectively where Fs is the electric field at

the silicon surface.

19

Like the Harrison model and its extension by Khirrurijal, the FN model does not

account for quantum mechanical effects of the 2D electron gas at the interface.

Therefore it is mostly applicable to the tunneling of carriers from the gate to the

substrate; where due to the metallic property of the gate, a 3D approximation of electron

states is reasonable. However some extensions of this model are applied to the

calculation of inversion charge tunneling in MOS structures using triangular

approximation for the inversion potential well [22].

On the other hand, the WKB transmission probability used in this model is not

capable of taking the oscillation of the tunneling current due to the reflections of wave

functions at interfaces. An additional oscillatory term is proposed by Maserjian et al. [17]

to include the oscillatory nature of the tunneling current. This term is expressed using

the Airy functions as analytical solutions to the Schrödinger equation for a Triangular

barrier.

Figure 1.2: Fowler-Nordheim tunneling (left) and direct tunneling (right).

In the state-of-the-art MOSFET transistors with low working voltage and ultrathin

gate dielectric, the gate leakage current is dominated by direct tunneling (DT) in contrary

to early MOS generation where the FN tunneling had the major contribution. Figure 1.2

demonstrates the difference between FN and DT. However, the FN tunneling still has its

application in Non-volatile memory devices where the dielectric is still thick and the

applied gate voltage is high during writing and erasing cycles. However, in high

cφoxqV

e−

FN tunneling Direct tunneling

e−cφ

oxqV

20

performance processors the DT is the main contribution to the leakage current, which

leads to problems such as high static power consumption and processors thermal

budget management.

There have been some efforts to extend the FN formalism to the DT regime

[23,24]. This leads to an analytical expression of the tunneling current in DT regime

which has the same form as the FN current:

2 2

1

exp( )DT ss

BBAJ F

B F= − (1-10)

where A and B are given in (1-9) and B1 and B2 are:

2

31 21 1 , 1 (1 )Si s ox Si s ox

ox c ox c

F t F tB B

ε εε φ ε φ

= − − = − −

(1-11)

where tox is the barrier thickness.

However, the inherent shortcomings of FN, namely ignoring the 2D nature of

electron gas at the inversion layer and related quantum effects are still present in this

extended model. Some phenomenological corrections are added to the analytical

expression of Schuegraf et al., using some fitting parameters [25-28], in order to take the

2D nature of inversion layer into account and to obtain better agreement with measured

values of the tunneling current. However, more robust and complicated models are

required to include quantum mechanical phenomena for direct tunneling in nanoscale

MOS structures.

1.3.2 Bound and quasi-bound states

The analytical form of compact models and their low computational resource

requirement make them a proper choice for practical device simulation. That may be

necessary for a quick estimation of the dielectric thickness from I-V data or to predict the

impact of gate leakage on the performance of CMOS circuits [29-32]. However, more

21

detailed models are required to capture quantum mechanical effects due to carrier

confinement at the interface. Especially at the inversion regime of a MOS structure, the

assumption of 3D carriers in the reservoir is questionable. The quantum well at the

substrate/gate-dielectric interface, which is built due to band bending, confines the

carriers at the interface and leads to splitting of the continuous 3D states to energy

subbands or 2D states. The band bending profile and the position of resulting subbands

are important parameters, which can play an essential role in the tunneling current. On

the other hand, in compact models the transparency of barriers is generally calculated

within the WKB approximation and with the underlying assumption, that carrier’s wave

function has a plane wave form. However, this assumption is not valid for confined

carriers at the interface potential well. Some authors applied a triangular [33-40] or

exponential [41-43] approximation to the quantum well potential at the interface and

solved the Schrödinger equation analytically. However numerical self-consistent

solutions of the Schrödinger-Poisson equation system has been applied by others [44-

61] for more realistic description of the charge distribution at the interface quantum well.

The detailed procedure of self-consistent solution of Schrödinger-Poisson equation

system will be presented in the following section.

As mentioned above, the transmission coefficient is a well defined quantity for

continuous 3D states, where the traveling carriers impact the potential barrier and result

in a reflected and transmitted plane wave. However, this concept is not properly defined

for localized carrier states at the interface of an inverted or accumulated substrate.

These states are called bound state or more accurately, in the presence of leakage

current, quasi-bound states. Many Authors solved the Schrödinger equation applying

closed boundary conditions which force the wave function to vanish deep in substrate

and at either the substrate/dielectric or the dielectric/gate interface. Solving the

Schrödinger equation with closed boundary conditions leads to discrete bound states

with sharp energies but at the same time this implies that no current could be carried by

these states due to vanishing wave functions. The quasi-bound state (QBS) concept

resolves this paradox and explains the real nature of the 2D carriers at the interface. The

finite barrier height at the substrate/dielectric interface results in leaky states, which are

spatially localized near the interface but allows the carriers to leave these states and

tunnel to the gate after some time. The average time that a carrier stays in the QBS is

22

called the carrier lifetime in that state. The lifetime concept in a QBS replaces the

transmission coefficient of traveling states. Indeed, coupling between 2D states at the

interface and 3D states in the gate through a finite dielectric barrier, results in an energy

broadening of subbands (figure 1.3). The energy broadening is inverse proportional to

the lifetime of carriers in each subband. The transparency of the barrier is directly

reflected in the energy broadening and lifetime concepts. The transparency of a barrier

is connected to the barrier height and thickness. For a relatively transparent barrier, low

or thin, the energy broadening is high and the lifetime is short. In other words, the

carriers leave the QBS and tunnel with a rather fast rate. In contrary, in a thick or high

barrier the lifetime is long and energy broadening is negligible, which means that the

carriers stay longer on QBS and the tunneling rate is low.

The concept of lifetime is widely used in the calculation of tunneling current from

a 2D electron gas [34-40,51,54,55-60,62-67]. Different authors proposed different methods

for the evaluation of carrier’s lifetime in QBS.

Figure 1.3: Bound (closed boundary) and quasi-bound (open boundary) states at inversion layer of a

NMOS.

In a quasi-classical approximation, impact frequencies of carriers on the barrier

and transmission coefficient are used for the calculation of carrier’s lifetime

[34,35,37,38,40,51,65,67].

, ,,

1( ) ( )i j i j

i j

T E f Eτ

= (1-12)

∞ ∞

bound states

0∆ =E

t∆ → ∞

∞

∞

quasi-bound states

∆E finite

∆t finite

23

,i jτ is the carrier’s lifetime on the jth subband of ith valley with energy of Ei,j , f and

T are the impact frequency of carriers and transmission coefficient of the barrier,

respectively. The impact frequency of carriers indicates how many times a carrier hits

the barrier in unit time and the transmission coefficient is a ratio of successful impacts to

total number of impacts. The impact frequency is related to the kinetic energy and

localization length of carriers on subbands and applying a parabolic band structure

approximation the impact frequency could be expressed as follows:

1, , ,

,

12 ( ) 2 ( ( )

( )

s s

n n

z z

i j i j i j ci j z z

v z dz m E E z dzf E

−= = −∫ ∫ (1-13)

where vi,j is the group velocity of carriers in the subband, zs and zn are classically turning

points of wave function and Ec is the conduction band edge of the substrate.

The modified WKB approximation is widely used for the calculation of the

transmission probability. Transfer matrix methods as well as expressions in terms of Airy

functions are also applied to the calculation of the transmission probability. As

mentioned before, the transmission probability is well defined for 3D states assuming a

plane wave form for the incident, reflected and transmitted wave function but its

application for 2D states is questionable.

Other methods for the direct calculation of the lifetime without using transmission

probability are proposed. Lo et al. [54] suggested a method, based on the close analogy

between the confined electrons in a varying potential and electromagnetic waves in a

wave guide with varying reflection index. This analogy allows the utilization of the

transverse-resonant-method [68], which is often used for finding the eigenvalue equation

for inhomogeneously filled wave guides and dielectric resonators. To apply this method

the structure is divided to intervals of width d of a 1D grid along the direction

perpendicular to the Substrate/gate-dielectric interface. The transverse-resonant method

defines the intrinsic impedance *i i im kη = and iZ

and iZ

, terminal impedance of each

interval in which m* is the electron effective mass, k is the wave number and arrows

indicate the impedance looking to left or right. Consider an interval in the inversion layer

24

and applying the transmission-line transformation as is shown in equation 1-14

repeatedly, iZ

and iZ

could be expressed in terms of 1 2 1, , , iZ Z Z−

⋯ and 1 2 1, , ,i N NZ Z Z+ − −

⋯

respectively.

1

1

1 1 1 11

1 1 1 1

tan( )2,3, , .

tan( )

tan( )2, 3, , .

tan( )

m m m mm m

m m m m

m m m mm m

m m m m

Z j k dZ m i

jZ k d

Z j k dZ m N N i

jZ k d

ηηη

ηηη

−

−

+ + + ++

+ + + +

−= =−

−= = − −−

⋯

⋯

(1-14)

Under resonant conditions the input impedances looking to the left and right

should be equal, 0i iZ Z+ =

. This condition would be satisfied with a complex energy,

whose imaginary part Γ shows the energy broadening of leaky QBS, and the carrier’s

lifetime on each subband could be determined as follows:

2τ =

Γℏ

. (1-15)

Another method is borrowed from the theory of nuclear decay [69,70] and

proposed by Magnus et al. [56]. The potential is modeled by a piecewise constant profile

defined on a 1D mesh. An analytic ansatz for the wave function is suggested in each

interval. Applying the transfer matrix approach the solution of the Schrödinger equation

is obtained along with the density of available electronic states. The density of electronic

states has the characteristic of a resonant state with a narrow large peak around the

quasi-bound state energies. The lifetime then is calculated from the width of the

resonant peaks. Under the assumption that the resonance width or the energy

broadening is much smaller than their corresponding subband energy, the real wave

functions of bound states are extended to the complex plane. Putting a new wave

function with a complex energy in the Schrödinger equation and applying the quantum

current operator to a complex time dependent wave function, the time dependent current

density is obtained. This result corresponds to the situation in which the electron can

decay only once and is analogous to the Breit-Wigner theory of nuclear decay [71].

However, in contrary to the nuclear decay the electrons are continuously delivered to the

25

QSB of the inversion layer from drain and source junctions and randomized by

scattering mechanism. In a more precise model and under high tunneling regime, one

should take the electron delivery mechanism into account. However, for short channel

MOS structures with an equivalent oxide thickness tox>1 nm the drain and source

junctions refills the decaying states instantaneously. This instantaneous refilling allows

us to ignore the time dependent part of current operator and take the time independent

part as a steady state current of inversion layer electrons.

Quantum transmission boundary method (QTBM) is originally proposed to solve

the Schrödinger equation in a general quantum system with open boundaries [72]. As

mentioned before, closed boundary condition could not explain the current carrying

states. A scattering state or current carrying state is extended over space and is not

limited to the active region of device. The wave function of such a state is a solution of

the Schrödinger equation over an extended area which includes the active device

region. However, the QTBM introduces new boundary conditions which include the

effect of open boundaries; therefore the solution of the Schrödinger equation over an

extended area is reduced to the solution of the equation in the active region of the

device with proper boundary conditions. This method is applied to resonant superlattice

structures [73] as well as gate tunneling of MOS transistors [36,58-60,63]. In the MOS

structure the plane wave ansatz is used for the gate’s wave function. Using such a

complex wave function leads to a non-Hermitian Hamiltonian which has complex

eigenvalues. As expressed before, the imaginary part of the complex energy is the

energy broadening of QBS which is reverse proportional to the carrier’s lifetime.

Complex energy is characteristic for quantum systems with open boundary, which

exchange either particle or energy through their contacts. The imaginary part of the

energy shows how strong the system is correlated to its contacts. In a MOS transistor

strong correlation between inversion layer QBS and the continuum of states in the gate

would be a result of ultrathin or low potential barrier, which in turn leads to high tunneling

current.

An alternative method for the calculation of QBS lifetime is proposed within the

formalism of formal reflection delay time of wave packets [55,66], which is adopted from

a time independent approach used for resonance tunneling devices [74]. In this method

the lifetime of a QBS is identified with the reflection time of an electron wave packet on a

26

semi-infinite barrier. The phase ( )Eθ of the complex reflection coefficient ( )r E related to

each plane wave component with energy E is introduced as ( ) exp[ ( )]r E i Eθ= . Then the

reflection time ( )R Eτ is defined as

0

0

( )( )R

E

d EE

dE

θτ = ℏ (1-16)

If the energy E0 is far from any QBS energy the plane wave components of a

wave packet are weakly correlated to the QBS and experience almost the same

reflection time. Consequently the wave packet will be reflected with a small distortion. If

the energy width of the wave packet nearly coincides with one of the QBS some of the

component experience much longer reflection time than the other and the wave packet

will be reflected with a high distortion. The reflection time at the QBS energy is

interpreted as the lifetime of the QBS. Cassan et al. applied an energy scanning

procedure to obtain the accurate shape of ( )Eθ . The same energy scanning procedure

is applied to the direct calculation of the lifetime based on the wave function shape in the

next section. However, Govoreanu et al. use a perturbative schema and analytical wave

functions, which leads to an analytical evaluation of the reflection coefficient.

Before we present our new proposed method for the calculation of lifetime in a

QBS in the next section, we briefly mention a wave function based method which was

proposed by Oriols et al. [62] for the calculation of scattering states in a double-barrier

resonant tunneling diode (DBRTD). In this method two solutions of the Schrödinger

equation with different energies which satisfy the boundary condition at the closed and

open boundaries are considered and the difference between two energies is interpreted

as the energy broadening which leads to the calculation of lifetime. In the next section a

similar idea is applied to the QBS at the inversion layer of a MOS structure which leads

to a bunch of energies around each QBS energy, which satisfy the open boundary

condition.

27

2 Fully quantum mechanical model of tunneling curre nt based on the effective mass approximation and Schrö dinger-Poisson solver

In this section we present our tunneling current model, which is based on a self-

consistent solution of Schrödinger-Poisson equation system and the lifetime concept. In

the first part, the self-consistent procedure of solving Schrödinger-Poisson equation

system is described in detail. The method is applied to a NMOS structure without being

restricted to this structure. In the quantum mechanical description of particle dynamics

the effective mass approximation is applied, neglecting the discrete atomic structure of

materials. In the following part of this work we will consider the atomic structure of Si and

SiO2 and their interface.

Actually, the quantum mechanical dynamics of a particle in a MOSFET has

complex characteristics, requiring the solution of 3D Schrödinger-Poisson equation

system. The numerical solution of such a 3D problem involves computational

complexities such as functional boundary conditions on the surface surrounding the

system. However, it is possible to reduce the latter to a 1D form by neglecting minor

physical effects. In a real field-effect-device drain potential destroys the in-channel

homogeneity of particle distribution. A source supplied current carrier can accelerate due

to the in-channel potential and tunnels into the oxide around the drain. This idea

constitutes a basic ground for the hot carrier tunneling mechanism. The quantum

mechanical description of a particle dynamics in this mechanism needs a solution of at

least a 2D Schrödinger equation with a complicated potential. However, classical Monte

Carlo device simulation that accurately describes hot carrier phenomena has shown that

the maximum direct tunneling current for ultrashort MOSFETs with ultrathin gate

dielectric is obtained at zero drain voltages [75]. This justifies the extension of the

tunneling model from simple MOS structures to MOSFET transistors. As a result, we

simply can neglect inhomogeneity of the in-channel potential and reduce the problem to

the solution of a 1D effective mass Schrödinger-Poisson equation system. We also

neglect the surface states and bulk traps in the ultrathin dielectric, in the presence of

which the quantum mechanical problem would be, strictly speaking, a three dimensional

problem.

28

We also neglect the effect of image force at the Si/SiO2 interface. The barrier

lowering due to image force is still a hazy concept. Many authors neglect the effect [17,

34-36,51,54,55,63], while others take it into account [22,76-79]. Ludeke et. al. have shown

that ignoring the image force effect at large biases in the ballistic electron emission

microscopy experiments resulted in a small discrepancy [80]. On the other hand, the

uncertainty in the value of conduction band offset at the Si/SiO2 interface (2.9 – 3.2 eV)

and the effective mass of electrons in the oxide (0.3m0 – 0.86m0) have a larger effect on

the tunneling current as a small barrier lowering due to image force.

The self consistent solution of the 1D effective mass Schrödinger-Poisson

equation system gives the electrostatic potential at the interface, subband energies and

wave functions, electron density on subbands as well as the dielectric potential. In the

second part, a wave function based method is presented for the calculation of the

carriers’ lifetime on QBS. The tunneling current in different samples are then calculated

and compared to measurements, which shows excellent agreement.

2.1 The Schrödinger-Poisson solver

We consider a NMOS structure with a p-doped Si substrate grown in (100)

direction in inversion regime, where electrons are collected at the Si/SiO2 interface by a

positive voltage applied to the metallic gate. However, the procedure could also be

extended to the accumulation regime as well as other dielectric materials such as

oxynitride or high-k dielectrics. The band structure of the silicon conduction band is

constituted by six ellipsoidal valleys. In the (100) direction there are 2 fold degenerated

longitudinal and 4 fold degenerated transverse electrons with the effective masses of

*00.98lm m= and *

00.19tm m= , 0m being the free electron mass. The valence band consist

of light, *00.16lhm m= , and heavy, *

00.49hhm m= holes. Particularly for the NMOS structure

the hole tunneling can be neglected in comparison to the electron tunneling due to the

high valence band offset at the Si/SiO2 interface. In our computation we also neglect the

3D electrons at the inversion layer, assuming that the majority of electrons in the

inversion layer regime reside on quantized subbands. This is a reasonable assumption

at inversion regime due to the large distance of 3D states’ energy from the Fermi

29

energy. However, in accumulation regime the distance between 3D carriers’ energies

and Fermi energy is small, thus they should be taken into account.

The Fermi energy of electrons in p Si− substrate is defined from the charge-

balance equation written in equilibrium region deep into the substrate or far from the

Si/SiO2 interface. Assuming that electrons in the substrate obey the Fermi-Dirac

statistics and neglecting a small concentration of thermally excited electrons in the

conduction band, the charge balance equation is written for completely ionized

acceptors and gives:

1 2

2( )F AN F Nϑ η

π= , (2-1)

where F1/2 is the Fermi- Dirac integral of 1/2 order and is defined as follows

0

1( )

( 1) exp( ) 1

j

j

dF

j

ζ ζηζ η

∞

=Γ + − +∫ (2-2)

( )F F BE E k Tϑη = −

EF is the Fermi energy, 23 2 21.380 10 ( )Bk m kg s K−= × is the Boltzmann constant and T is

temperature. NA represents acceptor doping concentration and Nϑ is the effective

density of states at the valence band-edge and will be expressed as follows:

3 2

,

2

22 h Si Bm k T

Nhϑ

π ∗ =

. (2-3)

The value of the Fermi energy is calculated at room temperature by the numerical

evaluation of the Fermi integral and inverse solution of Eq. (2-1) and depicted in figure

2.1 for different doping concentrations. The conduction band edge of silicon substrate far

from the interface is taken as reference point for the energy. Such a choice is

reasonable, since states with negative energies become bounded, whereas electrons in

the conduction band acquire a positive energy. Figure 2.2 shows the Fermi energy and

30

the first two subbands energies together with the conduction band bending at the

Si/SiO2 interface due to the positive gate voltage applied to the structure.

The procedure of the self-consistent solution of the Schrödinger-Poisson equation

system starts with a triangular potential. The triangular start potential has no influence

on the converged results, however a properly chosen potential can speed up the

convergence. The MOS structure is enclosed in a quantum box with closed boundary,

which contains substrate and dielectric layer as depicted in figure 2.2.

1010101013131313

1010101014141414

1010101015151515

1010101016161616

1010101017171717

1010101018181818

1010101019191919

1010101020202020

-1.25-1.25-1.25-1.25-1.2-1.2-1.2-1.2

-1.15-1.15-1.15-1.15-1.1-1.1-1.1-1.1

-1.05-1.05-1.05-1.05-1-1-1-1

-0.95-0.95-0.95-0.95-0.9-0.9-0.9-0.9

-0.85-0.85-0.85-0.85-0.8-0.8-0.8-0.8

-0.75-0.75-0.75-0.75

NNNNAAAA (1/cm³) (1/cm³) (1/cm³) (1/cm³)

EE EEFF FF-E-E -E-E

cc cc (

eV

) (

eV

) (

eV

) (

eV

)

Figure 2.1: The position of the Fermi energy with respect to the conduction band edge of silicon, far from

interface, is depicted for a p-doped silicon substrate as a function of the doping concentration.

The 1D Schrödinger equation with closed boundary condition is solved for the

start triangular potential V(z) to obtain the subband energies Ei,j and wave functions

, ( )i j zΨ :

( ) ( ) ( )2 2

, , ,22 i j i j i ji

dV z z E z

m dz∗

− + Ψ = Ψ

ℏ (2-4)

i and j are the valley and subband indices respectively. For a (100) Si interface

according to degeneracy the Schrödinger equation should be solved for longitudinal and

transverse electrons. The first derivative of the wave functions has to be continuous at

31

the interface Si/SiO2 (z=zs). Thus the wave functions should satisfy the following

boundary conditions:

, ,(0) ( ) 0,i j i j LΨ = Ψ = and 2

, ,( ) ( )1 1

s s

i j i j

Si SiOz z z z

d z d z

m d z m d z− +

∗ ∗= =

Ψ Ψ= (2-5)

z=0 and z=L are the boundaries of the quantum box deep in the substrate and at the

dielectric/gate interface respectively. *Sim and

2

*SiOm are the electron effective masses in

the silicon substrate and in SiO2 in a direction perpendicular to the interface.

0000 5555 10101010 15151515 20202020 25252525 30303030 35353535 40404040-1.5-1.5-1.5-1.5

-1-1-1-1-0.5-0.5-0.5-0.5

00000.50.50.50.5

11111.51.51.51.5

2222

z (nm)z (nm)z (nm)z (nm)

EE EEcc cc (

eV)

(eV

) (

eV)

(eV

)

36363636 36.536.536.536.5 37373737 37.537.537.537.5 38383838 38.538.538.538.5 39393939 39.539.539.539.5-1.3-1.3-1.3-1.3

-1.25-1.25-1.25-1.25-1.2-1.2-1.2-1.2

-1.15-1.15-1.15-1.15-1.1-1.1-1.1-1.1

-1.05-1.05-1.05-1.05-1-1-1-1


EE EEcc cc (

eV)

(eV

) (

eV)

(eV

)

Figure 2.2: The band bending at the Si/SiO2 interface at 1 Volt applied gate voltage for a NMOS structure

with 1018 cm-3 doping concentration. The Fermi energy is Ef= -1.0579 eV and the first two subbands have

energies of E1,1= -1.0721 eV , E1,2= -0.9693 eV.

The Schrödinger equation is discretized over a regular 1D mesh grid in the

direction perpendicular to the interface (figure 2.3).

1,2 0.9693E eV= −1.0579fE eV= −

1,1 1.0721E eV= −

sz L

32

Figure 2.3: The discretization scheme of the Schrödinger equation on a regular 1D mesh grid with

equivalent distance a in the direction perpendicular to the Si/SiO2 interface.

Employing the finite-difference method the Schrödinger equation as a partial

differential equation could be transformed to a difference equation system as follows:

1/ 2 1/ 2 22 21 1

, , , , ,2 1/ 2 2 1/ 2 1/ 2 2 1/ 2

( )

2 2 2ki i

i j i j i j i j i ji i i i

m mV E

a m a m m a m

− +− +

− − + +

+− Ψ + + Ψ − Ψ = Ψ

ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℏℏ ℏ (2-6)

1,2, , 1for N= −ℓ ⋯

Written as a matrix equation one gets:

1,1 1 1,21

2,1 2,2 2 2,3 2

, 1 , , 1

22, 2 1 2, 2 2 2, 2 1

11, 1 1 1, 1 1

ˆ ˆ 0 0 0 0

ˆ ˆ ˆ 0 0 0

0 0 0

ˆ ˆ ˆ

0 0

ˆ ˆ ˆ

ˆ ˆ0 0

NN N N N N N N

NN N N N N

T V T

T T V T

T T V T

T T V T

T T V

− +

−− − − − − − − − +

−− − − − − −

+ − Ψ − + − Ψ − + − Ψ− + −

Ψ − +

ℓ ℓ ℓ ℓ ℓ ℓ ℓ

⋯

⋯

⋮ ⋮ ⋮ ⋯ ⋮

⋮⋮ ⋮ ⋮ ⋮

⋮⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮

⋯ ⋯ ⋯

1

2

2

1

N

N

E

−

−

Ψ Ψ

=

Ψ Ψ

⋮

⋮

⋮

where the coefficients T are defined as follows:

1/ 2 1/ 2 22 2, 1 , , 1

2 1/ 2 2 1/ 2 1/ 2 2 1/ 2

( )ˆ ˆ ˆ, ,2 2 2

i i

i i i i

m mT T T

a m a m m a m

− +− +

− − + +

+= = =ℓ ℓ

ℓ ℓ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℏℏ ℏ (2-7)

The closed boundary condition implies that the Schrödinger equation has only

solution for specific energies, namely the eigenvalues of the matrix Eq. (2-7). Solution of

1/ 2im +ℓ1/ 2

im −ℓ a z

1−ℓ ℓ 1+ℓ

33

above matrix equation results in a set of eigenvectors and eigenvalues which

correspond to the wave functions and energies of carriers on different subbands.

To calculate the electron density at the inversion layer a summation over all

quantum states including spin degeneracy is required. All states are occupied according

to the Fermi-Dirac distribution function. In the directions parallel to the interface (x,y),

there is no confining potential. Thus, the solution of the Schrödinger equation results in

states with plane waves as wave functions and a continuous energy spectrum. On the

other hand, due to 1D potential the Schrödinger equation could be separated in two

parts; parallel and perpendicular to the interface:

, , || ,

||

( , , ) ( , ) ( ) ( )yx

x y z

ik yik xk k k i jx y z x y z e e z

E E E

−−⊥

⊥

Ψ = Ψ Ψ = Ψ

= + (2-8)

Now the summation over states could be separated into two parts, the sum over

subbands and the sum over parallel states with the same subband energy:

( )

2

2 ,

,, , ||

( , ) 1( , , ) 2 ( )

1 exp(( ) )x y

x y

k k

i ji j k k x y f B

x yn x y z z

L L E E E k T⊥

Ψ= Ψ

+ + −∑ ∑ (2-9)

Changing the summation into integration (, 2

x y

x yx y

k k

L Ldk dk

π⇒∑ ∫ ) for parallel continuous

states the electron density reads:

( )

2

2 ,

, 2, ||

( , )( , , ) 2 ( )

4 1 exp(( ) )x yk k x y

i ji j f B

x y dk dkn x y z z

E E E k Tπ

∞

−∞ ⊥

Ψ= Ψ

+ + −∑ ∫ ∫ (2-10)

Considering the parabolic dispersion relation and changing the variable into polar

coordinates we have:

34

222 2 22 2 2 2||

||, ||

( )

2 2 2 2x yz z

z x y z

kk kk kE E E

m m m m⊥

+= + = + = +

ℏℏℏ ℏ (2-11)

( )2

2 || ||,2

, 0 0 ||

1( , , ) ( )

2 1 exp(( ) )i j

i j f B

k dkn x y z z d

E E E k T

π

π

∞

⊥

= Ψ Θ+ + −∑ ∫ ∫ (2-12)

The integral over Θ provides a factor 2π . It should be mentioned that in the above

equation the dependence of charge distribution on x and y is eliminated due to the plane

wave form of the wave function in these two directions.

Changing the variable from ||k to ||E using following transformation gives:

|| || ||

|| || ||

||

2 2

2

m E mk d k dE

E= ⇒ =

ℏ ℏ (2-13)

( )2|| ||

,2, 0 ||

( ) ( )1 exp(( ) )

i ji j f B

m dEn z z

E E E k Tπ

∞

⊥

= Ψ+ + −∑ ∫

ℏ (2-14)

Applying the following relation:

[ ]ln 1 exp( ) .1 exp( )

dxx const

x= − + +

+∫ (2-15)

the integral could be exactly solved and reads:

( ) 2

,|| , ,2( ) ln 1 exp(( ) ) ( )B

i i f i j B i ji j

k Tn z g m E E k T z

π= + − Ψ∑ ∑ℏ

(2-16)

In the above equation the summation is evaluated over different valleys with

different degeneracy factors gi and density of state effective masses mi,||.

After calculating the electron density at the inversion layer, it is necessary to solve

the Poisson equation for the electrostatic potential ( )zϕ at the inversion layer. Then the

35

potential energy of electrons ( ) ( )V z q zϕ=− is obtained, which is required for the next step

of the self-consistent procedure. The Poisson equation is discretized and integrated on

the same regular 1D grid as the Schrödinger equation. The step by step integration is

performed by applying zero initial condition to the electrostatic potential and its first

derivative deep in the substrate (z=0). The integration continues into the dielectric region

by applying the following boundary condition at the Si/SiO2 interface:

2

20

( ) ( )d z z

dz

ϕ ρεε

= − (2-17)

( )0

( )0 0

z

d z

d z

ϕϕ=

= = and 2

( ) ( )

s s

Si SiO

z z z z

d z d z

d z d z

ϕ ϕε ε− += =

= (2-18)

In Eq. (2-17), ( )zρ is the total charge concentration in the silicon substrate, ε is

the relative dielectric constant being equal to 11.9 in the silicon substrate and 3.9 in the

silicon dioxide, and 0ε is the permittivity of free space. In p-type silicon substrates the

charge density in the inversion regime is constituted by inverted electrons of

concentration ( )n z and ionized donors of concentration NA, provided that doping atoms

are distributed uniformly in the substrate and completely ionized:

[ ]( ) ( ) Az q n z Nρ = − + (2-19)

Notice that in our study we neglect charged traps in the oxide, and assuming that

the contribution to the charge distribution in the oxide comes only from electrons

penetrated into the oxide.

After solving the Poisson equation and calculating the new electronic potential in

the current step Vp, its value in the next step is updated by mixing the new value and old

value by a mixing factor R, 0<R<1, as:

( )1 1 1p p p pV V R V V+ − −= − − (2-20)

36

The parameter R is chosen empirically to speed up convergence, and it may be,

generally speaking, a function of the space coordinate. Convergence in the

computations is reached when the relative difference between two subsequent values of

the potential over all 1D mesh points is smaller than a given critical value.

It should be noted that in addition to the boundary conditions given in Eq. (2-18)

the electrostatic potential should satisfy the following voltage balance condition under

the applied gate voltage Vg:

( ) g FB cL V Vϕ φ= − + (2-21)

where ( )Lϕ is the electrostatic potential at the dielectric/gate boundary, the cφ is the

conduction band offset of the substrate/dielectric interface and VFB is the flat-band

potential. Different Fermi energies of substrate and gate results in band bending or built

in potential on the structure in order to equalize the Fermi energies over the whole

system at equilibrium (Vg=0). The flat-band potential is determined according to Fermi

energies in the substrate and gate, which in turn are functions of doping concentrations.

The Poisson equation as a second order differential equation would be fully

determined in the substrate and in the dielectric region with the initial and boundary

conditions of (2-18). To fulfill the extra condition in Eq. (2-21) the charge density ( )zρ is

scaled with a scaling factor K. The ultimate solution is reached by altering the depletion

layer width L after each self-consistence step in order to reach a converged potential

which satisfies the potential balance equation (2-21) while K=1. The procedure of the

computation is schematically shown in figure 2.5.

The length of the depletion region strongly depends on doping concentrations in

the substrate. For substrate doping of NA=1017 cm-3, the maximum of the depletion layer

width is calculated as 115 nm. This value decreases drastically to almost 40 nm and 12

nm for NA=1018 cm-3 and for NA=1019 cm-3 respectively. On the other hand, before the

system enters the strong inversion regime, the depletion region increases continuously.

But, after the system reaches strong inversion the depletion layer will not increase

further. In figure 2.4 the changes of the depletion region width at different applied gate

voltages is depicted for a sample with NA=1018 cm-3 and tox=2 nm. As the figure shows,

37

the depletion width changes rapidly at low applied voltages and reaches its maximum

around Vg=0.8 V. Further increasing of gate voltage does not change the depletion

width.

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222232323232

33333333

34343434

35353535

36363636

37373737

38383838

39393939

40404040

VVVVgggg (V) (V) (V) (V)

De

ple

tion

Laye

r W

idth

(nm

)D

epl

etio

n La

yer

Wid

th (

nm)

De

ple

tion

Laye

r W

idth

(nm

)D

epl

etio

n La

yer

Wid

th (

nm)

Figure 2.4: The depletion layer width as a function of the gate applied voltage for a sample with NA=1018

cm-3 and tox=2 nm.

38

Figure 2.5: Flowchart diagram of the self-consistent solution of the Schrödinger-Poisson equation system.

Input parameters e.g. L, start potential, doping, etc.

Schrödinger solver Eq. (2-4)

Poisson solver Eq.(2-17)

Charge density calculation Eq. (2-16)

Check potential Eq. (2-21)

Scaling charge ρ(z)=K ρ(z)

Check potential Convergence

Mixing for new potential Eq. (2-20)

Check K

Increase L

Decrease L

End

No

Yes

No

Yes

K=1 K>1

K<1

39

2.2 Tunneling current model based on an one side op en boundary conditions

As mentioned before, the solution of the Schrödinger equation under closed

boundary condition results in stationary bound states, which are characterized by sharp

eigenenergies and vanishing wave function at the gate interface. These time-

independent states are not able to explain time-dependent phenomena such as

tunneling current. In order to explain time-dependent tunneling the solution of the time-

dependent Schrödinger equation is required. However, we take advantage of the

relation between time and energy in quantum mechanics through Heisenberg

uncertainty relation and obtain the required information of state’s time evolution without

solving the time-dependent Schrödinger equation. An energy scan procedure as

described in the following is used to extract the time evolution information of states,

which leads to the calculation of the tunneling current.

The first step is to relax the closed boundary condition and apply one side open

boundary conditions to the Schrödinger equation. The wave functions are considered to

vanish deep in substrate but allowed to penetrate the dielectric barrier and continue into

the gate. These boundary conditions indicate that localized electrons at the inversion

layer could escape only in the gate direction. We ignore also the phase information of

the localized electrons by applying zero boundary condition deep in substrate. The

electrons at the inversion layer originate from the substrate or source and drain regions.

In the steady state approximation they fill the inversion layer subbands according to

Fermi-Dirac distribution after experiencing different scattering processes. In other words,

the inelastic scattering processes can compensate the tunneling electrons in order to

maintain the quasi-equilibrium occupancy at the inversion layer.

Applying open boundary condition changes the Schrödinger equation from a

eigenvalue problem to a problem, where all energies are allowed. In other words, by

opening the boundary the energy levels at both sides of the barrier (discrete levels at the

inversion and continuous level in the gate) become common, resulting in a broadening

of the discrete levels in the substrate transforming the latter to quasi-bound states

(QBS). Nevertheless, the population weights of levels penetrating from the metallic side

to the substrate are much smaller than the host levels’ population weights.

40

The electronic potential profile obtained form the self-consistent solution of the

Schrödinger-Poisson system is used in this part and the Schrödinger equation is

integrated using Numerov’s algorithm [81] applying the following initial conditions:

,,

0

(0) 0, i ji j

z

darbitrary

dz =

ΨΨ = = (2-22)

An arbitrary initial derivative of the wave function has no effect on the position of

quasi-eigenvalues and shape of wave functions. It just scales the wave functions, which

can be regulated by normalizing the wave functions [49].

As mentioned above the open boundary condition implies that all energies are

allowed in the Schrödinger equation and the equation could be integrated for different

energies. We start from the bottom of inversion quantum well at the Si/SiO2 interface

and integrate the Schrödinger equation with gradually increasing energies. Many states

correspond to the metallic gate states. They are characterized by large wave function

amplitudes in the gate region and low amplitudes at the inversion quantum well. Just

around the energies of subbands the wave functions reach their maximum amplitude in

the inversion layer and decay rapidly into the gate. Figure 2.6 shows these two cases

corresponding to the gate and inversion layer states.

We are interested in the inversion layer states or the QBS which generally carry

the tunneling current. Each sign change of the wave function in the dielectric region

indicates that we just pass over a quasi-eigenenergy and the scanning should be

repeated on a finer energy grid to capture the QBS.

The energy interval around each QBS is found by utilizing the above sign change

property along with a bisection search algorithm. In order to exactly determine the

energy broadening of a QBS extra conditions are imposed to the wave functions:

• The transmission coefficient can not be greater than unity or the absolute value of

the wave function at the interface of Si/SiO2 should be larger than its value at the

interface of SiO2/gate.

41

26262626 28282828 30303030 32323232 34343434 36363636 38383838-2.5-2.5-2.5-2.5

-2-2-2-2-1.5-1.5-1.5-1.5

-1-1-1-1-0.5-0.5-0.5-0.5

00000.50.50.50.5

11111.51.51.51.5

22222.52.52.52.5

EE EEcc cc (

eV

) (

eV

) (

eV

) (

eV

)

z (nm)z (nm)z (nm)z (nm) Figure 2.6: Wave functions around the first QBS in a structure with NA=1018 cm-3 and tox=1 nm. The dotted

and dashed lines depict wave functions corresponding to the gate states, while the solid line corresponds

to the first QBS.

35.535.535.535.5 36363636 36.536.536.536.5

-0.875-0.875-0.875-0.875

-0.87-0.87-0.87-0.87

-0.865-0.865-0.865-0.865

-0.86-0.86-0.86-0.86

-0.855-0.855-0.855-0.855

-0.85-0.85-0.85-0.85

-0.845-0.845-0.845-0.845

-0.84-0.84-0.84-0.84

-0.835-0.835-0.835-0.835

-0.83-0.83-0.83-0.83


EE EEcc cc (

eV

) (

eV

) (

eV

) (

eV

)

25252525 30303030 35353535-2.5-2.5-2.5-2.5

-2-2-2-2

-1.5-1.5-1.5-1.5

-1-1-1-1

-0.5-0.5-0.5-0.5

0000

0.50.50.50.5

1111

1.51.51.51.5

2222

2.52.52.52.5


EE EEcc cc (

eV)

(eV

) (

eV)

(eV

)

Figure 2.7: A fine tune energy scan around the first QBS. In the right picture the wave functions with

dotted and dashed line violate the negative derivative and sign change condition in the dielectric region,

respectively. The solid line shows a valid wave function on the first QBS.

0.8464E eV= −

0.8486E eV= −

0.8489E eV= −

42

• The wave function can not change its sign within the classically forbidden region

(the gate dielectric region).

• The derivative of the wave function must be always negative in the classically

forbidden zone.

These three conditions ensure that the wave function belongs to an electron

which stays in a QBS at the inversion layer. The electrons stay in such states for a

relatively long time before they tunnel into the gate with a small probability. The last two

conditions guarantee that the wave functions exponentially decrease in the classically

forbidden region. Figure 2.7 shows three energies and the corresponding wave

functions, where only one of them fulfills the above mentioned conditions and two others

marginally violate them. The energy scan will end up after reaching an energy interval

where all the above mentioned conditions are satisfied. This interval ∆Ei,j defines the

energy broadening around each QBS. The lifetime of carriers on a QBS is then

determined as:

,,

i ji jE

τ =∆ℏ

(2-23)

The direct tunneling current from each subband is calculated as a multiplication of

total charge on a subband and the escaping rate of electrons from the subband, which is

equal to 1/τi,j. the total tunneling current is then calculated as a summation over all

subbands:

, ,

, ,, ,

i j i j

i j i ji j i j

N QJ q

τ τ= =∑ ∑ (2-24)

where ,i jN and ,i jQ are the number of electrons and charge density on j th subband of ith

valley, respectively.

43

2.3 Results and discussions

The computations are performed for MOS structures with different substrate

doping concentrations and dielectric thicknesses. The doping concentrations and the

dielectric thicknesses of stat-of-the-art MOSFETs used in high performance processors

are around NA≈1018 cm-3 and tox≈1.5 nm respectively. Therefore, doping concentrations of

1017, 5×1017, 1018 and 5×1018 cm-3 and dielectric thicknesses of 1,1.5 and 2 nm are used

in our calculation. The dielectric layer is chosen to be SiO2 with the relative dielectric

constant of ~3.9 and the band gap of ~ 8.9 eV. The conduction band offset at the Si/SiO2

interface is measured to be ~3.15 eV. Different values of the electron effective mass are

suggested by different authors. We use the value of 0.5m0 as the most popular value.

However, the issues of the electron effective mass and its thickness and material

dependencies are discussed in the next chapters.

As the gate potential increases the quantum well at the inversion layer becomes

narrower and deeper. The value of the surface potential (the potential which falls across

the silicon substrate) is a measure of the quantum well depth. Figure 2.8 shows the

surface potential at different applied gate voltages for a structure with NA=1018 cm-3 and

tox=2 nm. The first two subbands are also depicted in Fig. 2.8. An increase of

quantization effects is observed as the gate voltage increases. The distance between

the bottom of the quantum well and the first subband as well as the distance between

subbands are increases at higher gate voltages. Such quantization effects could result in

a degradation of the threshold voltage in the transistor.

The influence of the dielectric thickness on the surface potential and on the first two

subbands is depicted in figure 2.9. The structures in figure 2.9 have the same substrate

doping concentrations of 1018 cm-3 but different oxide thicknesses. Samples with thinner

oxide reach the strong inversion regime at lower gate voltages. This obviously shows the

reason for scaling the dielectric thickness along with the working potential of high

performance processors. The thinner oxide, with a higher gate capacitance, gives the

gate contact more control over the electrons at the inversion layer than a thicker one.

The total inversion charge density as depicted in the upper panel of figure 2.10 is also

confirming the above statement. However, it comes at its price, the higher leakage

current as will be shown soon.

44

Scaling the substrate doping concentration should result in a reduction of short

channel effects. However, a very high doping concentration could end up in a

degradation of the threshold voltage. The Fermi level of a p-type silicon can even cross

the valence band edge at doping concentrations higher than 5×1018 cm-3 (figure 2.1). The

large distance between the Fermi level and the conduction band edge at high doping

concentrations results in a reduction of electron density at the inversion layer, which in

turn leads to a shift of the threshold voltage to the higher values. This fact is clearly

shown in the lower panel of figure 2.10, where the total inversion charge is plotted for

samples with different substrate doping concentrations.

The inversion charges are distributed on different subbands. Figure 2.11

illustrates the inversion charge densities on the first four subbands and the total

inversion charges at the same picture. The first, third and fourth subbands belong to the

longitudinal electrons while the second subband is the lowest subband of the transverse

electrons. As the picture shows most contribution to the charge density comes from the

first subband and the other subbands make a small contribution to the total inversion

charge density.

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 2222-1.5-1.5-1.5-1.5

-1.4-1.4-1.4-1.4

-1.3-1.3-1.3-1.3

-1.2-1.2-1.2-1.2

-1.1-1.1-1.1-1.1

-1-1-1-1

-0.9-0.9-0.9-0.9

-0.8-0.8-0.8-0.8

-0.7-0.7-0.7-0.7

-0.6-0.6-0.6-0.6


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

Figure 2.8: The surface potential and the energy of the first two subbands as a function of applied gate

voltage for a structure with NA=1018 cm-3 and tox=2 nm. The solid line determines the Fermi level. The

triangles show the subband energies calculated under open boundary condition. The small filled circles

show the energies of subbands calculated under closed boundary condition.

First subband

Second subband

Surface potential

Fermi energy

45

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 2222-1.7-1.7-1.7-1.7-1.6-1.6-1.6-1.6-1.5-1.5-1.5-1.5-1.4-1.4-1.4-1.4-1.3-1.3-1.3-1.3-1.2-1.2-1.2-1.2-1.1-1.1-1.1-1.1

-1-1-1-1-0.9-0.9-0.9-0.9-0.8-0.8-0.8-0.8-0.7-0.7-0.7-0.7


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

Figure 2.9: The surface potential (open circles) and the first subband energy (triangles) as a function of

the gate voltage for structures with the same NA=1018 cm-3 and with different tox=1, 1.5 and 2 nm.

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-12-12-12-12

10101010-10-10-10-10

10101010-8-8-8-8

10101010-6-6-6-6

10101010-4-4-4-4


Q (

Col

/cm

²)Q

(C

ol/c

m²)

Q (

Col

/cm

²)Q

(C

ol/c

m²)

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-20-20-20-20

10101010-15-15-15-15

10101010-10-10-10-10

10101010-5-5-5-5

101010100000


Q (

Col

/cm

²)Q

(C

ol/c

m²)

Q (

Col

/cm

²)Q

(C

ol/c

m²)

data1data1data1data1data2data2data2data2data3data3data3data3

data1data1data1data1data2data2data2data2data3data3data3data3data4data4data4data4

Figure 2.10: The total inversion charge density for samples with the same substrate doping NA=1018 cm-3

and with different dielectric thicknesses (upper picture) and with the same dielectric thickness of tox=2 nm

but different substrate doping concentrations (lower picture).

tox=2 nm

tox=1.5 nm

tox=1 nm

tox=1 nm

tox=1.5 nm

tox=2 nm

NA=1017 cm-3

NA=5×1017 cm-3

NA=1018 cm-3

NA=5×1018 cm-3

46

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-10-10-10-10

10101010-9-9-9-9

10101010-8-8-8-8

10101010-7-7-7-7

10101010-6-6-6-6

10101010-5-5-5-5

10101010-4-4-4-4


Q (

Col

/cm

²)Q

(C

ol/c

m²)

Q (

Col

/cm

²)Q

(C

ol/c

m²)

d a ta 5d a ta 5d a ta 5d a ta 5 d a ta 2d a ta 2d a ta 2d a ta 2 d a ta 3d a ta 3d a ta 3d a ta 3 d a ta 4d a ta 4d a ta 4d a ta 4 d a ta 5d a ta 5d a ta 5d a ta 5

Figure 2.11: The Distribution of the inversion charge density on first four subbands for a sample with

NA=5×1017 cm-3 and tox=1 nm.

35353535 35.535.535.535.5 36363636 36.536.536.536.5 37373737 37.537.537.537.5 38383838 38.538.538.538.5 39393939 39.539.539.539.5 404040400000

5555

10101010

15151515x 10x 10x 10x 10

19191919


Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

35353535 35.535.535.535.5 36363636 36.536.536.536.5 37373737 37.537.537.537.5 38383838 38.538.538.538.5 39393939 39.539.539.539.5 404040400000

5555

10101010

15151515x 10x 10x 10x 10

19191919


Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

Ele

ctro

n D

ens

ity (

1/c

m³)

data1data1data1data1




Figure 2.12: The total electron density at the Si/SiO2 interface for a sample with NA=1018 cm-3 and tox=2 nm

at different gate voltages (upper picture). The spatial distribution of charge on the first three subbands of

the same structure at Vg=2 V (lower picture).

,1lq ,2lq ,3lq,1tq totalq

nl,1 (z) su nl,2 (z)

nt,1 (z)

Vg =0.5V Vg =1.0V

Vg =1.5V

Vg =2.0V

SiO2

SiO2

47

The spatial distribution of electrons at the inversion layer, which has similar shape

as the wave function of the first subband, also indicates that the most of the electrons sit

on the first subband. Lower panel of figure 2.12 shows the spatial distribution of

electrons on the first three subbands at Vg=2. The upper panel shows the total inversion

charge at different gate voltages.

Figure 2.12 also shows that the inversion layer or the 2D electron gas at the

interface has a thickness of about 2 nm. The peak of the charge distribution, in contrary

to the classical model, is shifted back from the interface as a result of wave function

reflection at the interface. This shift is less than 1 nm but not negligible comparing to the

ultrathin gate oxide. Increasing of the gate potential pushes the 2D electron gas toward

the interface and reduces the shift of the inversion charges. The picture also shows that

the penetration of electrons into the dielectric increases rapidly as the gate voltage

increases. The tails of the inversion electron distribution at Vg=2 V are extended up to

0.2 nm into the dielectric region.

Before proceeding with the calculation of the lifetime in QBSs, we would like to

discuss the effect of open boundary on the subband energies. Many authors applied

closed boundary conditions for the calculation of subband energies and wave functions

and used these values in the calculation of the tunneling current. However, opening the

boundary at the dielectric/gate interface not only results in a broadening of energy states

but also causes a small shift in the position of subband energies. Although this

repositioning is small but it is not negligible. Due to the exponential relation between

subband energies and the electron density on each subband, tiny shifts could result

non-negligible changes in the electron density and consequently the tunneling current.

Figure 2.13 shows the energy shift in the first two subbands due to an open boundary

condition. The shifts in the first, second, third and fourth subbands of a sample with

NA=1017 cm-3 and tox=1 are 75.6, 31.8, 28.0 and 10.1 meV respectively. These values

agree well with the variational estimation reported in [66]. It indicates that the maximum

shift occurs on the first subband, which is the most critical one regarding the electron

density. The oxide thickness can cause changes in the energy shift. Increasing the oxide

thickness by 1 nm, results in a ~50% reduction of the energy shift. The energy shifts in

the first four subbands of a sample with tox=2 nm are 33.4, 13.8, 13.4 and 5.9 meV

48

respectively. This is an expected result because thicker oxides act similar to the closed

boundary condition and the effect of an open boundary could hardly transfer to the

inversion layer through a thick oxide. In contrary, in a thin oxide the inversion layer

electrons feel the open boundary more strongly and therefore the energy shift is higher.

As mentioned before, the open boundary condition results in a broadening of the

subband energies transforming sharp energy states to QBSs. The energy broadening,

which is inverse proportional to the lifetime of a QBS, is a measure of the tunneling

probability of an electron on each QBS. Thus, it should strongly depend on the oxide

thickness. Figure 2.14 shows the energy broadening of the first subband for three

otherwise identical samples with different oxide thicknesses. As expected the energy

broadening strongly increases as the dielectric thickness decreases. Decreasing the

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 2222

-1.2-1.2-1.2-1.2

-1.1-1.1-1.1-1.1

-1-1-1-1

-0.9-0.9-0.9-0.9

-0.8-0.8-0.8-0.8


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 2222

-1.2-1.2-1.2-1.2

-1.1-1.1-1.1-1.1

-1-1-1-1

-0.9-0.9-0.9-0.9

-0.8-0.8-0.8-0.8


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

Figure 2.13: The repositioning of subband energies due to the open boundary in samples with NA=1017 cm-

3 and different oxide thicknesses. Dashed lines are subband energies under the close boundary condition

where triangles (first subband) and open circles (second subband) depict the energies under the open

boundary condition.

tox=1 nm

tox=2 nm

49

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-14-14-14-14

10101010-13-13-13-13

10101010-12-12-12-12

10101010-11-11-11-11

10101010-10-10-10-10

10101010-9-9-9-9

10101010-8-8-8-8

10101010-7-7-7-7

10101010-6-6-6-6

10101010-5-5-5-5


Ene

rgy

Bro

aden

ing

(eV

)E

nerg

y B

road

enin

g (e

V)

Ene

rgy

Bro

aden

ing

(eV

)E

nerg

y B

road

enin

g (e

V)

Figure 2.14: The energy broadening of the first subband, for samples with NA=1017 cm-3.

oxide thickness from 2 nm to 1 nm causes a large increase (almost five orders of

magnitudes) in the energy broadening. The barrier lowering due to the applied gate

voltage also increases the energy broadening.

The energy broadening or the lifetime is the counterpart of the transmission

probability of free states in the case of QBSs. However, it does not behave exactly like

the transmission probability. The lifetime depends on the barrier transparency as well as

on the subband energy and the shape of localized wave functions. This is clearly shown

in the quasi-classical expressions of the lifetime in Eqs. (1-12) and (1-13). These

equations show that the particle energy and the classically turning points of the wave

function, which are functions of the potential profile at the inversion layer, are crucial

values in the determination of the lifetime. Although the transmission probability in Eq.

(1-12) is a monotonic function of the energy, the impact frequency could behave

differently. If the distances between classically turning points of the subbands grow

faster than their energies, then it could lead to a reduction of the impact frequency,

which in turns results in a reduction in the energy broadening at higher subbands. This

situation is presented in figures 2.15 and 2.16. Figure 2.15 shows the potential profile at

the interfaces of two samples with same oxide thicknesses but different doping

concentrations. As a result of a wider quantum well in the sample with lower doping

concentration, the distance between classically turning points grows rapidly. On the

tox=1 nm

tox=1.5 nm

tox=2 nm

50

other hand, the energy differences between subbands are smaller than highly doped

sample. These values are summarized in Table 2.1.

Table 2.1: The values of the subband energy, energy changes from subband to subband and the distance

between classically turning points on the first four longitudinal subband energies of samples with tox=2 nm

and NA=1017 and 1018 cm-3 respectively.

Subband Ei,j (eV) Energy change (meV) zs-zn (nm) Ei,j (eV) Energy change (meV) zs-zn (nm)

1 -1.1636 1.18 -1.1048 1.15

2 -1.0181 145.5 2.81 -0.9701 134.7 3.0

3 -0.9338 84.3 4.27 -0.9117 58.4 5.2

4 -0.8670 66.8 5.53 -0.8746 37.1 7.3

30303030 32323232 34343434 36363636 38383838 40404040

-1.4-1.4-1.4-1.4

-1.3-1.3-1.3-1.3

-1.2-1.2-1.2-1.2

-1.1-1.1-1.1-1.1

-1-1-1-1

-0.9-0.9-0.9-0.9

-0.8-0.8-0.8-0.8


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

106106106106 108108108108 110110110110 112112112112 114114114114 116116116116

-1.4-1.4-1.4-1.4

-1.3-1.3-1.3-1.3

-1.2-1.2-1.2-1.2

-1.1-1.1-1.1-1.1

-1-1-1-1

-0.9-0.9-0.9-0.9


Ene

rgy

(eV

)E

nerg

y (e

V)

Ene

rgy

(eV

)E

nerg

y (e

V)

Figure 2.15: The potential well and the first four subbands of longitudinal electrons of samples with tox=2

nm at Vg=2 V. The values of subband energies and the distance of classically turning points are listed in

table 2.1.

NA=1018 cm-3 tox=2 nm NA=1017 cm-3 tox=2 nm

El,1

El,2

El,3

El,4

El,1

El,2

El,3

El,4

NA=1017 cm-3 NA=1018 cm-3

zn zs

51

Above discussion could clarify the broadening behavior of subbands, which is

depicted in figure 2.16. In the sample with NA=1017 cm-3 the broadening decreases as we

move to higher subbands, due to a reduction in the impact frequency. However, in the

sample with NA=1018 cm-3 narrow potential well eliminates any reduction of the impact

frequency and the broadening increases on higher subbands. These results show the

difference between tunneling probability from 3D free electron states and the 2D

confined QBSs. In the 2D electron gas the shape of the potential well also plays a role in

the determination of the tunneling probability.

It should be noticed that the above discussions are valid for the subbands of the

same effective mass valley (e.g. longitudinal). Different effective masses of longitudinal

0000 0.50.50.50.5 1111 1.51.51.51.5 222210101010

-13-13-13-13

10101010-12-12-12-12

10101010-11-11-11-11

10101010-10-10-10-10


Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

0000 0.50.50.50.5 1111 1.51.51.51.5 222210101010

-13-13-13-13

10101010-12-12-12-12

10101010-11-11-11-11

10101010-10-10-10-10


Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)



Figure 2.16: The energy broadening on the first four longitudinal subbands of samples with tox=2 nm

El,1

El,2

El,3

El,4

El,1

El,2

El,3

El,4

NA=1018 cm-3 NA=1017 cm-3

52

0000 0.50.50.50.5 1111 1.51.51.51.5 222210101010

-13-13-13-13

10101010-12-12-12-12

10101010-11-11-11-11

10101010-10-10-10-10


Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

0000 0.50.50.50.5 1111 1.51.51.51.5 222210101010

-13-13-13-13

10101010-12-12-12-12

10101010-11-11-11-11

10101010-10-10-10-10


Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

Ene

rgy

Bro

ade

ning

(e

V)

data1data1data1data1data2data2data2data2

data1data1data1data1data2data2data2data2

Figure 2.17: The energy broadening of the first longitudinal and transverse subbands of the same samples

as in figure 2.16

and transverse electrons also affect the broadening. The light electrons (transverse),

according to Eq. (1-13), have a higher impact frequency which results in a higher energy

broadening (Figure 2.17). However, the above mentioned differences in the broadening

of QBSs are very tiny compared to the influence of the barrier thickness and height. The

broadening of the first four subbands, which contains almost all inversion electrons, are

close to each others.

The lifetime of the electrons on QBSs is calculated using the expression in Eq. (2-

23). The calculated lifetime for samples with tox=1, 1.5 and 2 nm at Vg=2 V are

τ=1.298×10-10, 2.889×10-8 and 6.182×10-6 s respectively. These values are in a very

good agreement with the values presented by other authors [82].

The direct tunneling current as a function of the gate voltage is depicted in Fig.

2.18 for different substrate doping concentrations and oxide thicknesses. The tunneling

current increases sharply at small voltages and is then slowed down at higher voltages.

Indeed, the substrate band edges bent approximately linearly at small gate voltages,

and the electron density at the inversion layer largely increases as the gate voltage

El,1 El,1

Et,1 Et,1

NA=1017 cm-3 NA=1018 cm-3

53

increases. Further increase in the gate voltage leads to the band bending in the oxide,

stabilizing the increasing of the tunneling current. The doping concentration plays a role

at low gate voltages but has no significant effect on the tunneling current in the strong

inversion regime. In contrary to the doping concentration, the oxide thickness has a

major influence on the tunneling current.

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-10-10-10-10

10101010-8-8-8-8

10101010-6-6-6-6

10101010-4-4-4-4

10101010-2-2-2-2

101010100000

101010102222

101010104444

101010106666


J (A

/cm

²)J

(A/c

m²)

J (A

/cm

²)J

(A/c

m²)

Figure 2.18: The tunneling current for samples with different oxide thicknesses and substrate doping:

NA=1017 cm-3 (circle), 5×1017 cm-3 (square) and 1018 cm-3 (triangle).

We compared our computation results for the tunneling current with the

experimental data measured in Ref. [35]. The direct tunneling current is measured on

MOS structures with gate oxide thicknesses of tox=1.46, 1.55, 1.79 and 2 nm, and a

substrate doping level of NA=5×1017 cm-3. The oxide thicknesses are extracted from C-V

analysis. Figure 2.19 demonstrates the measured and the modeled tunneling current.

Excellent agreement between the computed and the measured data is achieved in the

wide gate voltage values from 0.2 V to 2 V for all samples. Although the discrepancy

below 0.2 V is negligible small for the sample with thin oxide (1.46 nm), it increases in

samples with thicker oxides. We agree with the assumption argued in Ref. [35] that

small discrepancies at low gate voltages could be caused by interface states and

dielectric bulk traps.

The best fitting is achieved by using slightly different effective masses for

electrons in the oxide layer of different samples. The best fitting is achieved by using

tox=1 nm

tox=1.5 nm

tox=2 nm

54

electron effective masses of 0.64m0, 0.63m0, 0.58m0 and 0.55m0 for samples with

tox=1.46, 1.55, 1.79 and 2.0 nm respectively. In the next section we will discuss the issue

of the electron effective mass in ultrathin dielectric layers.

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-5-5-5-5

10101010-4-4-4-4

10101010-3-3-3-3

10101010-2-2-2-2

10101010-1-1-1-1

101010100000

101010101111

101010102222

101010103333

Vg (V)Vg (V)Vg (V)Vg (V)

J (A

/cm

-2)

J (A

/cm

-2)

J (A

/cm

-2)

J (A

/cm

-2)

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-5-5-5-5

10101010-4-4-4-4

10101010-3-3-3-3

10101010-2-2-2-2

10101010-1-1-1-1

101010100000

101010101111

101010102222


J (A

/cm

²)J

(A/c

m²)

J (A

/cm

²)J

(A/c

m²)

tox=1.46 nm m*=0.64m0

tox=1.55 nm m*=0.63m0

55

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-7-7-7-7

10101010-6-6-6-6

10101010-5-5-5-5

10101010-4-4-4-4

10101010-3-3-3-3

10101010-2-2-2-2

10101010-1-1-1-1

101010100000

101010101111


J (A

/cm

²)J

(A/c

m²)

J (A

/cm

²)J

(A/c

m²)

0000 0.20.20.20.2 0.40.40.40.4 0.60.60.60.6 0.80.80.80.8 1111 1.21.21.21.2 1.41.41.41.4 1.61.61.61.6 1.81.81.81.8 222210101010

-8-8-8-8

10101010-7-7-7-7

10101010-6-6-6-6

10101010-5-5-5-5

10101010-4-4-4-4

10101010-3-3-3-3

10101010-2-2-2-2

10101010-1-1-1-1

101010100000


J (A

/cm

²)J

(A/c

m²)

J (A

/cm

²)J

(A/c

m²)

Figure 2.19: The comparison between the measured (closed circles) and the calculated (open circles)

tunneling current in different samples.

tox=1.79 nm m*=0.58m0

tox=2.0 nm m*=0.55m0

56

3 The electron “tunneling effective mass” in ultrathin silicon oxynitride gate dielectrics

3.1 The “tunneling effective mass” vs. effective ma ss

Fundamental parameters of SiO2 such as the dielectric constant, the conduction

and valence band offset at the Si/SiO2 interface and the band gap, which are required

for the calculation of the tunneling current in MOSFETs are measured and known with

small discrepancies. However, the “tunneling effective mass” of carriers in the gate

dielectric of MOSFETs is a hazy parameter. It depends on the thickness of the dielectric

layer, as will be shown in this section. Different values for the electron “tunneling

effective mass” in SiO2 are reported, which are ranging from 0.3m0 [83] to 0.86m0 [18].

The determination of the “tunneling effective mass” becomes more complicated if the

SiO2 gate dielectric is doped with other atoms such as nitrogen, hafnium or zirconium.

The effective mass theory is generally adopted to approximate the kinetics of

carriers at the bottom/top of conduction/valence band. The effective mass of an

electron/hole lying at the bottom/top of conduction/valence band can be calculated from

the curvature of the band structure at these extremal points. Parameter-free density

functional theory (DFT) methods could be applied to extract the effective mass from the

calculated band structure of a periodic system [89]. However, the state-of-the-art

ultrathin gate dielectrics consist of only few atomic layers in the direction perpendicular

to the interface. Therefore, the periodicity of the system is almost destroyed in one

direction. In addition to that, the interfacial layers at both interfaces (Si/dielectric and

dielectric/gate) are deformed due to the mismatching of lattice constants which results in

a rather amorphous gate dielectric layer and even thinner bulk oxide. In such cases the

application of the band structure theory (which is defined for a periodic lattice) in the

ultrathin gate dielectrics is questionable. The other challenging issue is that in the DT

regime the electrons propagate neither at the bottom of the conduction band nor at the

top of the valence band of the dielectrics and therefore assigning the effective mass

extracted from the band structure at the bottom/top of conduction/valence band is

questionable.

57

All these facts challenge the application of the band structure theory in the

ultrathin gate dielectrics. In this case we use the term “tunneling effective mass” instead

of effective mass. The “tunneling effective mass” becomes rather a fitting parameter

which could include all the atomic scale deformations in ultrathin gate dielectric.

Fukuda et al. have addressed this issue [84] and suggested two different values,

m*DT =0.35m0 and m*c=0.60m0, for the electron effective mass in SiO2. m*DT is the

effective mass in the DT regime, or the “tunneling effective mass”, while m*c is the

effective mass of electrons which ballistically propagate at the bottom of the conduction

band. Indeed, in the Fowler-Nordheim tunneling regime, electrons propagate partly in

the conduction band of the dielectric, therefore using the conduction band effective mass

in this regime could be reasonable. But in the direct tunneling regime electrons never

enter the conduction band of the dielectric and tunnel directly to the other side of the

barrier. Considering the “tunneling effective mass” as a parameter, several authors have

introduced parametric dependent “tunneling effective masses” and reported for example

thickness dependent “tunneling effective masses” for electrons in ultrathin SiO2 layers

[14,95,96,85,86].

3.2 Oxynitride gate dielectric

Silicon oxynitride (SiOxNy) has been used in the last three years as an

intermediate solution toward high-k gate dielectric materials. Low concentrations of

interface defects and a higher dielectric constant in comparison to that of the silicon

oxide, as well as a higher resistance against boron penetration into the channel are the

most superior properties of the silicon oxynitride, which keep it as the first choice for the

11 Angstrom EOT technology node. Several theoretical [87-90] and experimental [91-93]

studies have been developed to investigate the characteristic parameters of silicon

oxynitride such as the band gap, the conduction and valence band offsets at the

Si/SiOxNy interface, the dielectric constant and the “tunneling effective mass” of carriers

in the oxynitride as a function of the nitrogen concentration. One of the most efficient

experimental approaches used in the determination of ultra-thin oxide and oxynitride

parameters is the extraction of the parameters by fitting the calculated gate leakage

58

currents to the measured one [18,94,95]. This method is widely applied to learn the

dependence of the “tunneling effective mass” on external parameters in ultra-thin gate

dielectrics [14,28,37,83,96,97]. However, different models and approximations in the

calculation of the gate tunneling current provide a wide range of values for the “tunneling

effective mass”.

As mentioned before, some authors proposed thickness dependent “tunneling

effective mass” in SiO2. However, in oxynitride the value of the “tunneling effective

mass” is affected by the nitrogen concentration too. Recently Mao et al. [89] and Ng et

al. [94,95] have addressed this issue using first principle calculations and gate leakage

current fitting, respectively. The authors of Ref. [89] used a supercell of α-quartz with 72

atoms and replaced O atoms by N atoms or vacancies. This replacement causes a

supercell, which contains dangling bonds. The strong band gap narrowing (≈2 eV at

N%=8.7), extra high electron effective mass at the bottom of the conduction band

(12.1m0 at N%=8.7) and a very high dielectric constant (≈25 at N%=8.7) are the results

of existing dangling bonds. They calculated the effective mass in two different ways.

First, they extract the electron effective mass at the bottom of the conduction band from

the calculated band structure. These values are extremely high and increase with

increasing nitrogen concentration. As it mentioned before using these values for the

“tunneling effective mass” in an ultrathin oxide layer is very questionable. In the second

approach, it is assumed that the tunneling current through both SiO2 and SiOxNy gate

dielectrics can be described by a simple analytical FN formula, but with different

coefficients. Then the “tunneling effective mass” can be obtained by using the band gap

values calculated within the DFT method. These values show a reduction in the electron

“tunneling effective mass” as the nitrogen concentration increases. However, the

calculated band gaps for dangling-bond-free structures [90] are in a better agreement

with the experimental values rather than values obtained for structures with dangling

bonds. In deed, thermal processes during the integration of MOS transistors anneal the

structures and lead to a strong reduction of dangling bonds; therefore the calculations

based on structures with dangling bonds are questionable.

Ng et al. [94,95] have determined the barrier height and the “tunneling effective

mass” by fitting current-voltage (I-V) measurements to the calculation. An analytical

expression for DT and FN regimes are used for the calculation in thin and thick

59

dielectrics, respectively. However, many approximations are required in order to express

the tunneling current analytically.

3.3 Extracting the “tunneling effective mass” from I-V measurements

To relax some of the approximations made in analytical models we use our fully

quantum mechanical model for the calculation of the tunneling current within the

effective mass approximation, which is extensively presented in the last section. The

accuracy of our self-consistent Schrödinger-Poisson method prevails over the accuracy

of quasi-classical expressions for DT and FN currents. The leakage currents measured

at AMD Saxony for NMOSFETs with SiOxNy gate dielectric of different thicknesses and

nitrogen concentrations are fitted to those calculated according to our model. The

“tunneling effective mass” of electrons is then extracted as a function of the dielectric

thickness and nitrogen concentration.

The main parameters required for the calculations are the conduction band offset

at the Si/SiOxNy interface, the dielectric constant of the SiOxNy layer, and the dielectric

physical thickness. The values of the conduction band offset and dielectric constant are

considered to vary linearly between the values of SiO2 and Si3N4. The linear behavior of

the barrier height at the conduction band and the dielectric constant have been verified

by experimental works [91,98] and used by other authors [97,99]. The conduction band

offsets of 3.15 eV and 2.1 eV and the relative dielectric constants of 3.9 and 7.5 have

been used for SiO2 and Si3N4 respectively.

The nitrogen concentrations and the physical thicknesses of the samples are

extracted by X-Ray photoemission spectroscopy (XPS). Although XPS is less precise for

thick layers, it is a reliable and precise method for ultrathin layers (tox<3 nm) [100]. In

general, ellipsometry is more precise than XPS for pure silicon dioxide layers, but it is

not suitable for oxynitride layers with unknown nitrogen concentrations, because the

results of the ellipsometry measurements depend on the nitrogen concentration in the

layer. On the other hand, using the XPS method, the thickness and the nitrogen

concentration may be extracted at the same time. These two parameters are extracted

using X-rays with the energy of 1486.6 eV.

60

We studied nine samples with different thicknesses and nitrogen concentrations.

The physical parameters of the samples are listed in Table 3.1. The sample thicknesses

range from 10.22 Å to 23.13 Å, while the nitrogen concentrations vary from 0.1% to

17.2%. The calculated gate currents are fitted to the experimental values at Vg=1 V. This

voltage corresponds to the working voltage of transistors in today’s CPUs, and this fitting

yields a good agreement between the experimental and the theoretical results for the

whole range of 0-2 V. Figure 3.1 shows the calculated current in comparison with the

experimental data for different samples. Very good agreement between measured and

calculated curves is obtained using values of electron “tunneling effective masses”,

which are listed in table 3.1.

Table 3.1: The physical thickness, nitrogen concentration and extracted electron “tunneling effective

mass” of different samples.

sample tox (Å) N% m*/m0

(a) 10.22 0.1 0.888

(b) 13.15 0.1 0.838

(c ) 18.74 0.1 0.667

(d) 20.4 6.9 0.541

(e) 16.47 11.1 0.630

(f) 21.77 11.8 0.515

(g) 13.69 17.2 0.650

(h) 17.63 15.1 0.567

(i) 23.13 14.3 0.487

61

0 0.5 1 1.5 210

-4

10-2

100

102

104

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

100

102

Vg (V)

J (A

/cm

²)0 0.5 1 1.5 2

10-8

10-6

10-4

10-2

100

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

100

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-6

10-4

10-2

100

102

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

100

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-6

10-4

10-2

100

102

104

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

100

102

Vg (V)

J (A

/cm

²)

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

100

Vg (V)

J (A

/cm

²)

Figure 3.1: The measured (line) and the calculated (open circle) gate tunneling currents for samples (a),

(b), …, (i).

a b

c d

f e

g h

i

tox=1.022 nm N=0.1% m*=0.888m0

tox=1.315 nm N=0.1% m*=0.838m0

tox=1.874 nm N=0.1% m*=0.667m0

tox=2.040 nm N=6.9% m*=0.541m0

tox=1.647 nm N=11.1% m*=0.630m0

tox=2.177 nm N=11.8% m*=0.515m0

tox=1.369 nm N=17.2% m*=0.650m0

tox=1.763 nm N=15.1% m*=0.567m0

tox=2.313 nm N=14.3% m*=0.4871m0

62

3.4 Discussing the results

The silicon dioxide gates of samples (a), (b) and (c) contain a nitrogen

concentration of 0.1%, which is just at the limit of a contamination. A comparison of

these samples shows a strong correlation between the “tunneling effective mass” and

the layer thickness. As the thickness decreases, the “tunneling effective mass” increases

due to the compressive stress in the oxide layer at the SiO2/Si interface.

It has been reported that the oxide layer within ~ 1 nm from the Si/SiO2 interface

has a higher density than the bulk oxide layer, and consequently the Si-O-Si bonds are

compressively strained near the interface [101]. On the other hand, Eriguchi et al. [102]

have experimentally found that the tunneling current through an oxide layer with less

built-in compressive stress near the interface is higher than in the reverse case. This

result implies that the more compressed oxide shows a lower tunneling current or,

apparently a higher “tunneling effective mass”.

The same discussion can be applied to the samples with higher nitrogen

concentrations. For example, samples (e) and (f) with almost the same nitrogen

concentrations show the same trend. The “tunneling effective mass” increases as the

layer thickness decreases. Samples (h) and (i) display a similar trend at higher nitrogen

concentration.

As mentioned above, the electron “tunneling effective mass” is also a function of

the nitrogen concentration. Samples (b) and (g) have almost the same physical

thicknesses of 13.15 Å and 13.69 Å but different nitrogen contents of 0.1% and 17.2%,

respectively. Our calculations yield 0.838m0 for the “tunneling effective mass” of sample

(b), but 0.650m0 for sample (g). This difference shows that higher nitrogen

concentrations result in lower electron “tunneling effective masses”. Comparison of

samples (c) and (h) also shows the same behavior.

The “tunneling effective mass” values obtained in our calculations are slightly

higher than the values used in the previous section of this work and the values reported

e.g. by Simonetti et al. [96] but lower than the values reported by Mao et al. [89]. For

example, Simonetti et al. reported the value of ~0.72m0 for a 10 Å thick silicon oxide

layer. These discrepancies seem to be due to the different methods for the

measurement of the gate dielectric thickness. Simonetti [96] and Yang [35] et al.

63

extracted the oxide thickness with electrical methods such as fitting data to C-V

measurements. However, XPS measurement is known to result in smaller thicknesses

than the values extracted from ellipsometry, TEM or capacitive methods [103]. The

underestimation of the thickness results in slightly higher values for the “tunneling

effective mass” extracted in our case. However, it is obviously clear that the electron

“tunneling effective mass” tends to increase in thinner oxynitride layers and to decrease

as the nitrogen content increases.

Simonetti et al. proposed a linear relation between the “tunneling effective mass”

and the dielectric thickness. We also suggest a linear expression for the “tunneling

effective mass” as a function of two variables, thickness and nitrogen concentration, as

follows:

0* / %oxm m a b t c N= + × + × (3-1)

The values of the “tunneling effective mass” are interpolated to the above linear

function, targeting the lowest sum of squared absolute errors over all samples. The

values of the coefficients are calculated as: a = 1.1314, b = - 2.4171×10-2 and c = -

8.4536×10-3.

Next we add a correlation term, which correlates the thickness and the nitrogen

concentration as follows:

0* / % %ox oxm m a b t c N d t N= + × + × + × × (3-2)

Interpolation of the data with the same method as used in Eq. (3-1) results in

values of a = 1.2004, b = - 2.9034×10-2, c = - 1.7716×10-2, d = 5.9561×10-4 for the

coefficients. As it can be seen, the correlation coefficient “d” in Eq. 3-2 is relatively small,

which signals a weak correlation between the thickness and the nitrogen concentration.

The contour plots of the first and second interpolation function as well as the

surface plot of the first one are depicted in figure 3.2. The negative signs of coefficients

“b” and “c” in Eq. 3-1 and 3-2 indicate that an increase in the dielectric thickness or

nitrogen concentration leads to a decrease in the “tunneling effective mass”. On the

other hand the coefficient “d” has a positive sign which shows that the changes in the

64

“tunneling effective mass” according to the changes in thickness are less if the samples

have higher nitrogen concentration. For example, changing the thickness from 10 Å to

20 Å at N=0% causes ~31.9% reduction in the “tunneling effective mass”. However, the

same change at N=20% results in ~25.3% reduction. The same discussion is valid if we

exchange two parameters (tox, N%). Changing the nitrogen concentration from 0% to

20% at tox=10 Å leads to ~25.8% reduction in the “tunneling effective mass”, while the

same change at tox=20 Å results in ~19.0% reduction. These behaviors could be read

from right contour plot in figure 3-2.

Figure 3.2: The 3D surface (up) and contour (down-left) illustration of uncorrelated interpolation and the

contour plot of correlated (down-right) interpolation function.

65

The “tunneling effective mass” could also be considered as a function of the

effective oxide thickness (EOT), which includes the effect of nitrogen incorporation.

Figure 3.3 shows the “tunneling effective mass” as a function of the EOT. An almost

linear behavior of the “tunneling effective mass” could be observed and a linear

interpolation results in a following linear expression:

2

0* / 2.2872 10 1.0962m m EOT−= − × × +

By increasing nitrogen concentration the EOT increases, which again results in a

reduction of the “tunneling effective mass” as expected.

Figure 3.3: The electron “tunneling effective mass” as a function of the effective oxide thickness (EOT)

and its linear interpolation.

The above interpolation functions could be used to suggest proper values for the

“tunneling effective mass” of electrons in samples with specific thickness and nitrogen

concentration. The values would be valid at least in the range of thicknesses and

nitrogen concentrations of existing samples.

66

4 Atomic level calculations

In the previous part of this work a simplified model of MOS structure is used for

the calculation of the gate tunneling current. The simplifications reduce the problem to

the solution of a 1D effective mass Schrödinger equation. In the effective mass

approximation the problem of a moving carrier in a periodic atomic potential is replaced

with a free carrier moving with an effective mass tensor. However, state-of-the-art

ultrathin gate stacks consist of few atomic layers, and therefore the atomic structure of

gate dielectrics and their interface to the substrate and the gate could affect material

parameters and play an important role in the device operation. One example is the

“tunneling effective mass” of electrons through the dielectric layer as discussed in the

previous chapter. The atomic structure of an ultrathin oxide layer is dominated by

interfacial layers as the oxide layer becomes thinner and deviates from that of the bulk

material. This leads to a change in the electron “tunneling effective mass” as the

thickness decreases beyond 2 nm. On the other hand, the inclusion of atomic scale

distortions such as defects and vacancies in the 1D model is not straight forward,

because of the three dimensional character of such distortions. Atomic scale simulation

is therefore a proper tool to investigate such atomic level phenomena.

Quantum mechanic theory provides though a mysterious but a precise description

of systems consisting of atomic nuclei and electrons. Although, the interpretation of

quantum mechanics is still an area of debate, it has been extremely successful in

solving lots of problems in chemistry and physics. The total energy of a system,

consisting of ions and electrons, contains lots of information, which could be used in the

calculation of many physical properties of the system. The ability of quantum mechanics

to predict the total energy of a system of ions and electrons make it the proper choice for

the calculation of different properties of solids and molecules. For instance, the

equilibrium lattice constant of a crystal is determined with a set of total energy

calculations with different lattice constants. The value of lattice constant which minimizes

the total energy is then the equilibrium lattice constant. The interface of two solids as

well as the deformation in a crystal lattice due to defects or impurities could be

determined with the same scheme.

67

It should be noted that, in principle for the solution of quantum mechanical

equations in a solid at equilibrium only the specification of ions (their atomic number and

coordinates) is required. Such parameter free methods are generally called ab initio

methods. Many ab initio methods exist for more than a decade; however, their

application was limited to very small systems due to the high computational demands of

such calculations. Density functional Theory (DFT) proposed by Hohenberg and Kohn

[104] and formulated by Kohn and Sham [105] was a breakthrough in the ab initio

calculation, which extends the application of them to larger systems. In the next section

we briefly explore the DFT method and approximations which are essential to make the

theory tractable for moderately large systems.

4.1 Density functional theory

4.1.1 Born-Oppenheimer approximation

The total energy of a solid is a function of all degrees of freedom, namely the

electron and ion coordinates. However, the large difference between the mass of an

electron and a nuclei and the fact that the same forces act on both, suggest that the

response of electrons to the ionic motion is essentially instantaneous. Thus it would be a

reasonable approximation to treat the nuclei adiabatically and decouple the electronic

part of the many body wave functions. This is called the Born-Oppenheimer

approximation. The wave function of the system ( , )Ψ r R could be separated in

electronic ( ; )rΨ r R and ionic ( )RΨ R parts

( , ) ( ; ) ( )r RΨ = Ψ Ψr R r R R . (4-1)

The semicolon indicates a parametric dependence. Substituting the adiabatic wave

function into the Schrödinger equation with the following Hamiltonian:

ˆ ˆ( , ) ( ) ( ) ( , )H T T V= + +r R r R r R (4-2)

68

where T and V are kinetic and potential energy operators gives:

ˆ ˆ( ) ( ) ( ; ) ( ; ) ( ) ( ) ( , ) ( ; ) ( )

ˆ ˆ( ) ( ; ) ( ) ( ; ) ( ) ( ) ( ; ) ( )

R r r R r R

r R r R r R

T T V

T T E

Ψ Ψ + Ψ Ψ + Ψ Ψ +

+ Ψ Ψ − Ψ Ψ = Ψ Ψ

R r r R r R R R r R r R R

R r R R r R R R r R R (4-3)

Assuming that ˆ ˆ( ) ( ; ) ( ) ( ; ) ( ) ( ) 0r R r RT T Ψ Ψ − Ψ Ψ = R r R R r R R R , we can separate the

variables and get separate equations for the electronic and ionic subsystems

ˆ( ) ( , ) ( ; ) ( ) ( ; )r e rT V E + Ψ = Ψ r r R r R R r R (4-4)

ˆ( ) ( ) ( ) ( )e R RT E E + Ψ = Ψ R R R R (4-5)

Ee is the energy of the electronic subsystem when the ions are fixed in R. The total

energy of a system of electrons and fixed ions could be calculated adding the Coulomb

ion-ion interaction to the above energy.

4.1.2 Kohn-Sham energy functional and equations

Except the Born-Oppenheimer approximation, the DFT is an exact theory for the

ground state of a system. However, more approximations are required for a practical

realization of the DFT. The theory is based on two theorems introduced by Hohenberg

and Kohn. The first one states, that the Hamiltonian of the electronic subsystem and

subsequently the total energy are uniquely determined by the electron density. In other

words all physical properties of the system are determined as a functional of the electron

density. The second theorem assures that the global minimum of the energy functional

is the ground state energy of the system and the density which realizes this minimum is

the ground state electron density of the system. However, Hohenberg and Kohn

theorems provide neither the energy functional nor a procedure to minimize it. Kohn and

sham instead suggested a total energy functional and a method to find the ground state

electron density. The total energy functional contains the kinetic energy, the electron-

69

electron interactions, the ion-ion interaction as well as the electron-ion interaction and for

a system of doubly occupied electron states can be written as:

2 2

2 * 3 3 3 3( ) ( )( ) 2 ( ) ( )

2 2i i ion XC ioni

e n nE n d V n d d d E n E

mψ ψ ′ ′= − ∇ + + + + ′−

∑∫ ∫ ∫r r

r r r r r r r Rr r

ℏ

(4-6)

where 2

( ) 2 ( )ii

n ψ= ∑r r and (4-7)

Vion and Eion represent the electron-ion and ion-ion interactions respectively. The third

and forth terms are related to the electron-electron interaction. The third term is the

Coulomb interaction while the fourth term represents the exchange–correlation energy.

The fact that electrons as fermions obey the Pauli exclusion principle implies that the

many-body wave function of the electronic system should be antisymmetric under the

exchange of any two electrons. The antisymmetry restriction of the wave function leads

to an extra repulsion among electrons with the same spin and reduces the energy of the

electronic system. This reduction is called exchange energy. The inclusion of exchange

energy in the total energy functional is generally referred to as the Hatree-Fock

approximation. However, further spatial separation of electrons with different spins also

reduces the total energy. This further reduction is due to the difference between the

mean field approach of electron-electron interaction in the Hartree-Fock approximation

and the exact many-body approach and called correlation energy. The determination of

the correlation energy in a complex system is a very complicated task. Although some

quantum Monte Carlo simulations are applied to the calculation of electron gas

dynamics, these approaches are not tractable in total-energy calculations of complex

systems. In the next section we will discuss some approximations for this part of the

Hamiltonian.

It should be noted here that in the DFT formalism only the global minimum of the

energy and the corresponding electron density have a physical meaning as the ground

state energy and density.

70

Different methods are suggested for the minimization of the Kohn-Sham energy

functional. Kohn and Sham stated that the set of single particle wave functions which

minimize the total energy functional are the solution of the so called Kohn-Sham

equation [105].

22 ( ) ( ) ( ) ( ) ( )

2 ion H XC i i iV V Vm

ψ ε ψ − ∇ + + + =

r r r r rℏ

(4-8)

where

2 3( )H

nV e d

′ ′=′−∫

rr

r r (4-9)

[ ]( )

( )XC

XC

E nV

n

δδ

=r

r (4-10)

VH and VXC are the Hartree and the exchange-correlation potentials respectively

while εi are the Kohn-Sham eigenvalues.

The Kohn-sham equation reduces the problem of many interacting electrons into

a single electron problem moving in an effective potential due to all other electrons. As

mentioned above the Hamiltonian in (4-8) is a functional of the electron density. Thus

the solution of the above equation requires a self-consistent calculation, where the

resulting wave functions reconstruct the new Hamiltonian through the electron density.

The diagonalization of the Hamiltonian matrix is the standard method for the

solution of the Kohn-Sham equation. However, there are other methods for the

calculation of the ground state total energy. Two successfully applied methods for the

minimization of the energy functional are the Car-Parrinello molecular dynamic approach

and the direct minimization of the energy functional applying a conjugated gradient

algorithm [106].

4.1.3 Local density approximation for the Exchange- correlation energy

As mentioned before the calculation of the exact exchange-correlation functional

is a very complicated task, hence approximations are inevitable. On the other hand,

according to the Hohenberg-Kohn first theorem the exchange-correlation energy as part

71

of the total Hamiltonian should be a functional of the electron density. The simplest way

to express the exchange-correlation energy is to use a local density approximation

(LDA). This approximation is universally used as a standard method in DFT calculations.

In the LDA the exchange-correlation energy per electron at point r is related to the

electron density only at the same point and is equal to the exchange-correlation energy

of a homogenous electron gas, which has the same density as electronic system at point

r.

[ ] 3( ) ( ) ( )XC XCE n n dε= ∫r r r r (4-11)

This local approximation of the exchange-correlation energy further simplified the

solution of the Kohn-Sham equation.

Dirac [107] proposed an analytical expression for the exchange energy in a

homogeneous electronic system:

33 3 ( ) 0.458

4Xe

n

rε

π= − = −r

(4-12)

where re=(3/4πn(r))1/3 is the mean interelectronic distance expressed in atomic units (1

bohr=0.529177 Angstrom).

However, it is not possible to give an exact solution to the correlation part.

Several parameterizations exist for the correlation energy of a homogenous electron gas

[105,108-111], which result in very similar values of the total energies.

Predew and Zunger expressed the correlation energy of a homogeneous electron

gas analytical at high ( 1er ≤ ) and low ( 1er > ) densities:

1 2

ln ln , 1( ( ))

(1 ), 1

e e e e ePZC

e e e

A r B Cr r Dr rn

r r rε

γ β β+ + + ≤= + + >

r (4-13)

The coefficients are obtained by fitting to the quantum Monte Carlo results of Ceperly

and Alder for a homogeneous electron gas [112].

72

Although the LDA seems to be a very rough estimation for real systems, it work

exceptionally well in most solid-state systems and for the majority of their properties.

Part of this success is attributed to the small contribution of the exchange-correlation to

the total energy. Another reason for the success of the LDA is the fact that the

exchange-correlation hole describing the mutual electron repulsion is correctly described

within the LDA. It satisfies the sum rule expressing that the exchange-correlation hole

contains exactly one displaced electron. An improvement to the LDA could be achieved

by expanding the exchange-correlation to the higher order proportional to the gradient of

the density. However, such expansions show little improvement may be due to the

destruction of sum rule for the exchange-correlation hole, which is preserved in the LDA.

Structural properties such as the lattice constant and the bulk moduli and the

vibration frequencies are predicted with a very good accuracy within the LDA. But LDA

overestimates the binding energies of molecules and cohesive energies of solids while it

underestimates the band gap of insulators and semiconductors. However, these errors

are generally systematic; make trend predictions and comparative studies possible even

within the LDA. More exact calculations like GW using Green’s function to take a more

exact description of electron-electron interaction and exited states into account is

however computationally too expensive to be tractable for moderately large systems of

hundreds of atoms.

4.1.4 Pseudopotentials

It is well known that most physical properties of solids and molecules are

controlled by valence electrons and their interactions. On the other side, the core

electrons which occupy closed shells remain almost unchanged in different

environments. Therefore, it would be a big save in the calculation time if the Kohn-Sham

equations could be solved only for the valence electrons and the effect of core electrons

and positive nuclei are included in an effective potential called “pseudopotential”.

Additionally, the effective potential which contains screening effects of the core electrons

are much softer than the bare nuclei potential results in much smoother valence wave

functions. This is crucial in the plane wave as well as real space implementation of DFT;

because a large number of basis or a very fine mesh grid is required to mimic the wave

function of localized core electrons as well as the rapid changes in the wave function of

73

the valence electrons in the core region. Other advantage of the pseudopotential is that

the total energy of a valence electron system is orders of magnitudes smaller than that

of the all-electron system. Therefore, the accuracy required when comparing total

energies of two different ionic configurations is much smaller in the pseudopotential

calculation than what is required in the all-electron calculation.

In order to apply this simplification without losing accuracy, pseudopotentials and

pseudo wave functions should satisfy following general conditions. The radial part of the

pseudopotential and pseudo wave function should be similar to the true ionic potential

and valence wave function beyond a cutoff radius rc and the pseudo wave function

should have no node in the core region as depicted in figure 4.1. The similar values of

the true and pseudopotential beyond rc result in an identical scattering from both

potentials outside the core region.

Figure 4.1: A schematic illustration of the all-electron (dashed line) and the pseudo (solid line) wave

function and potential. Beyond the cutoff radius rc the pseudo and the true potential and wave function

become similar.

Additional constrains could be applied to the pseudopotential which lead to

different kinds of pseudopotentials. One of these conditions is the conservation of the

squared amplitude of the pseudo wave function outside the core region which results in

a norm conserving pseudopotential. As stated before the exchange-correlation energy is

74

a functional of the electron density. If this energy has to be calculated accurately it is

necessary to construct the pseudo wave function in the way not only to reconstruct the

radial dependence of the true valence wave function but also to reproduce the same

electron density outside the core region.

Starkloff and Joannopoulos [113] constructed one of the first norm conserving

pseudopotentials with a local form. However, the scattering from ion cores is best

described with a non-local psedopotential which uses different potentials for different

angular momentum components of the wave function [114-119]. The general procedure

for generating the pseudopotential is to fit the result of a calculation with a

parameterized pseudopotential to the result of an all-electron calculation of an isolated

atom. One should note that the same exchange-correlation functional is used in both

calculations.

4.1.5 Basis set

An efficient numeric solution of the Kohn-Sham equation is obtained if the

electronic wave functions are expanded over a proper basis set. The historically

standard basis sets in solids with periodic boundaries are plane waves. Periodic

boundary conditions in solids and orthogonality of plane waves result in a simple

representation of the Schrödinger equation in reciprocal space. Employing fast Fourier

transform algorithms the Schrödinger equation is computed easily and effectively. The

accuracy of the calculation could be controlled with a simple parameter, namely the

cutoff energy, which controls the number of basis functions. Floating basis sets like

plane waves are the natural choice to expand the electronic wave function in metallic

systems where the electrons are generally delocalized. Contrary to metallic systems, in

insulators and semiconductors electrons are generally localized and therefore the

expansion of the electronic wave function requires a large number of plane waves. The

natural choice for such systems is a localized atom-centered basis set. These bases

could be a set of analytical functions centered on each atom or numerical orbitals

adopted from isolated atomic calculations. The non-zero overlap of such basis however

complicates the construction of the Hamiltonian matrix, but a clever choice of strictly

localized numerical atomic orbitals and the expansion of the electronic wave function

over a linear combination of such atomic orbitals could considerably reduce the

75

computation time. Using strictly localized orbitals reduce the overlap of the basis sets

and results in sparse matrices which could facilitate the calculation by applying efficient

algorithms available for sparse matrices. In contrary to the plane waves, where the

accuracy is controlled systematically by a single cutoff energy, there is no systematic

way of improving the calculation accuracy in the flexible localized orbital basis. The

number of basis functions and their combination as well as their shape and cut off radius

are parameters which control the accuracy of the calculation with localized basis. It is

not an intrinsic lack of the localized basis set but more effort and care should be devoted

to the construction of an optimal basis set as it has bee done in the construction of

pseudopotentials.

4.2 DFT implementation with localized numerical bas is set

In the next sections of this work we want to address the problem of tunneling

current through the gate oxide of MOSFETs at the DFT level. The gate oxide is

sandwiched between two doped silicon electrodes. The tunneling current is then

calculated while voltage is applied between two electrodes. Such interface systems

contain a relatively large number of atoms which make the application of the efficient

localized basis preferable. On the other hand, in the leakage current calculations, which

will follow, zero interaction between two leads are desirable, which makes the

application of delocalized plane waves impossible.

The calculation will be carried out by applying a DFT implementation as realized

in the TranSIESTA [120] code, where numerical orbital basis sets and norm conserving

pseudopotentials are used. The DFT implementation of the code resembles the SIESTA

[121-125] code. An additional part for the calculation of transport properties is added in

TranSIESTA by applying the Green’s function formalism which will be explored in the

next section. In this section a short description of the localized basis and their

construction as well as their controlling parameters are presented. The solution of the

Kohn-Sham equation and the calculation of forces which are used in geometry

optimization are also briefly discussed.

76

The natural choice for localized orbitals would be the atomic orbitals themselves.

However, by using the pseudopotential approximation the bare nuclei is replaced by a

screened pseudo atom and atomic orbitals with pseudo atomic orbitals (PAO). The

numerical pseudo atomic orbitals are obtained by finding the eigenfunctions of the

isolated pseudo atoms confined within spherically symmetric potential wells. Different

shapes of such a confining potential are proposed in the literature [124,126-128]. SIESTA

uses the potential proposed by Junquera et al. [124], which keeps the orbitals restricted

while at the same time it takes care for the smoothness at the cutoff radius. Numerical

flexibility and confinement of the orbitals require a small number of non-overlapping

bases per electron, which in turn means more efficiency in the calculation. Within the

confined radius, the PAOs are defined as a product of a numerical radial function times

a spherical harmonic. For atom I located at RI,

( ) ( ) ( / ) ,Ilmn Iln I lm I I I Ir Y rφ φ= = −r r r r R (4-14)

The radial part of the basis function is the solution of the radial Schrödinger equation:

( )2

2 2

1 ( 1)( ) ( ) ( )

2 2 l l l l l

d l lr V r r r

r dr rφ ε δε φ +− + + = +

(4-15)

where lε are the eigenvalues of the pseudo atomic potential and Vl is the spherical

symmetric confining pseudopotential. The confinement causes a change of lδε at each

eigenvalue which is related to the cutoff radius rc. In other words lδε is determined by

the cutoff radius rc of the radial wave function. In order to achieve a well balanced basis

for different atoms and different angular momenta, it is better to fix the energy shift

rather than the cutoff radius. Therefore, the cutoff radius depends on the atomic species

and angular momentum.

In order to increase radial flexibility of the basis set an extra radial function could

be added for the same angular channel. This is conventionally called multiple-ζ. The

second-ζ function is constructed following the standard “split-valence” method in

quantum chemistry [129]. The first- and second-ζ functions have the same tails but the

77

second one changes to a simple polynomial shape ( 2( )ll lr a b r− ) inside a “split radius”

rs<r c. The coefficients a and b are chosen to ensure the continuity of the function and its

first derivative at rs. The radius rs is determined by fixing the norm of the first-ζ function in

r>r s. The value of 0.15 is empirically chosen in SIESTA as a reasonable value for the

“split norm”. In order to reduce the overlapping of the basis set further, a new second-ζ is

chosen as the difference between above expressed second-ζ and the first-ζ function.

This results in a second basis which is zero beyond rs and therefore reduces the number

of non-zero matrix elements.

A higher angular flexibility of a basis set is obtained by adding shells of higher

angular momentum, i.e. polarization orbitals, to the basis set. The polarization orbitals

account for the deformation induced by bond formations and leads to a faster

convergence. An efficient scheme for the generation of polarization orbitals has been

described in the literature [125]. The basic idea is to use the pseudo atomic orbital of

angular momentum l, such that there is no valence orbital with angular momentum of

l+1 . Applying a small electric filed and using first-order perturbation theory the polarized

orbital with angular momentum of l+1 is obtained and added to the basis set.

Atomic forces are obtained by direct differentiation of the total energy functional

with respect to atomic positions. The Pulay correction is required when an atomic

centered basis set is used. It includes the effect of moving atoms in a system described

by atomic centered basis set.

4.3 Non-equilibrium Green's function method for tra nsport

In the previous section an atomic level description of finite or periodic systems are

presented within the density functional theory. The DFT provides a strong formalism for

the ground state calculation of finite or extended systems with periodic boundary

conditions. Calculations beyond the ground state or of extended non-equilibrium

systems without periodic boundaries require some extension to the basic DFT. Non-

equilibrium Green’s function (NEGF) formalism is a strong tool, which has been applied

to the calculation of systems with open boundaries which break the periodicity of the

system. The problem of tunneling current through an insulator which is sandwiched

78

between two semi-infinite metallic leads is one of the problems which could be solved by

the application of the NEGF formalism.

The concept of Green’s function appears in many physical contexts including

circuit theory, electrostatics and electromagnetics. Whenever the response R is related

to the excitation S by a differential operator opD

opD R S= (4-16)

we can define a Green’s function and express the response in the form

1opR D S GS−= = where 1

opG D−≡ (4-17)

Our problem can be expressed in the form [132]

[ ]E H S− Ψ = (4-18)

where Ψ is the wave function and S is an equivalent excitation term due to a wave

incident from one of the leads. The corresponding Green’s function can be written as

[ ] 1( )G E E H

−= − or ( ) ( )E H G E I− = (4-19)

where H is the Hamiltonian operator of the system.

It is well-known that the response of a system to an arbitrary input could be

obtained if the impulse response of the system is known. The Green’s function is

actually the impulse response of the system and contains all information which is

required for the calculation of the system response to an arbitrary input.

A two probe system, which we will use as a simplified model in the atomic level

calculation of the tunneling current, is depicted in figure 4.2. As shown in the figure the

system can be divided into three interacting subsystems, left lead, central region and

right lead. Few atomic layers of each lead are also considered in the central region in

79

order to include the interface deformations into the central region. Beyond these

intermediate regions the atomic structure and the potential of the leads are obtained

from a bulk calculation.

Figure 4.2: The schematic diagram of a two probe system. C, L and R are the central region, left and right

leads respectively.

Introducing the indices L, C and R for the left lead, central region and right lead

the total self adjoint Hamiltonian H is represented as follows:

† †

0

0

L LC

LC C RC

RC R

H h

H h H h

h H

=

(4-20)

The interaction between two leads is assumed to be zero and hL(R)C stands for the

interaction between the left (right) lead and the central region. The application of

localized basis set assures the zero interaction between two leads if the central region

consists of more than a few atomic layers.

Using the above Hamiltonian and decomposing the Green’s function we have:

† †

0 1 0 0

0 1 0

0 0 0 1

L LC L CL RL

LC C RC LC C RC

RC R LR CR R

E H h G G G

h E H h G G G

h E H G G G

− − − − − = − −

(4-21)

The Green’s function of non-interacting subsystems is defined according to the non-

interacting Hamiltonian:

, , , ,( )L C R L C RE H g I− = (4-22)

L R C z

80

From equations (4-21) and (4-22) and after some algebra the total Green’s

function becomes:

† †

† †

† †

(1 )

(1 )

L LC c LC L L LC c L LC c RC R

c LC L c c RC R

R RC c LC L R RC c R RC c RC R

g h G h g g h G g h G h g

G G h g G G h g

g h G h g g h G g h G h g

+ = +

(4-23)

where:

† †( ( ) ) 1LC L LC C RC R RC Ch g h E H h g h G− + − − = (4-24)

The above equation states that GC is the Green’s function of the following effective

Hamiltonian:

( ) 1C L R CH G+ Σ + Σ = (4-25)

where: †, , , ,L R LC RC L R LC RCh g hΣ = are the so called “self-energies”.

The above formalism reduces the problem of solving an infinite Hamiltonian in Eq.

(4-21) to the solution of the effective Hamiltonian in Eq. (4-25) which is only extended in

the central region. The interactions of semi-finite leads are included through the self

energy terms. However, it does not yet simplify the problem, since to calculate the self

energies one has to obtain gL,R, which is a formidable task. The problem is simplified if

the coupling between the leads and the central region described by hLC and hRC is limited

only to the surface region of the leads. In this case it would be enough to calculate gL,R

only near the surface, since hLC and hRC are zero otherwise.

According to Eq. (4-17) the solution of the inhomogeneous Schrödinger equation

according to the excitation S is calculated as follows:

RG SΨ = − (4-26)

and since H is a self adjoint Hamiltonian another solution would be:

81

†AG SΨ = − (4-27)

These two wave functions are called retarded and advanced solutions of the

Schrödinger equation. The linearity of the Schrödinger equation however implies that the

difference of those solutions R LΨ − Ψ is a solution of the homogeneous Schrödinger

equation. It means for any input vector S ,

,A SΨ = where †( )A i G G= − (4-28)

solves the homogeneous equation and †( )A i G G= − is called the spectral function. On

the other hand, the solution of the homogeneous Schrödinger equation H EΨ = Ψ is

a set of eigenvalues kε and corresponding eigenfunctions kΨ . Considering the

representation of the Green’s function in the eigenbasis kΨ :

k k

k k

GE iε δΨ Ψ

=− +∑ (4-29)

the spectral function is expressed as:

1 1

k kk k k

A iE i E iε δ ε δ

= Ψ Ψ − − + − − ∑

2 20

lim( )k k

k kEδ

δε δ→

= Ψ Ψ− +∑ (4-30)

The spectral function is then an energy resolved density operator, which peaks at the

eigenvalues of the Hamiltonian. A Hermitian Hamiltonian results in real energy

eigenvlues and the spectral function consist of delta functions at each eigenvalue.

82

2 ( )k k kk

A Eπ δ ε= Ψ − Ψ∑ (4-31)

The Hamiltonian in Eq. (4-25) however is non-Hermitian due to the energy

dependent self energies. In this case the energy eigenvalues become complex and the

imaginary term δ in the Green’s function becomes finite. This imaginary part of the

energy has exactly the same sprite as discussed in the first part of this work, and

describes the hybridization and the broadening of the energy eigenstates due to the

interaction of the system with leads through its open boundaries. This broadening is

inverse proportional to the lifetime of electrons on each state, the time that electrons

spend in the localized states before they leave the state due to tunneling or scattering

mechanisms. The energy resolved density of state is then consists of a set of broadened

levels, occupied according to the Fermi distribution function and the broadening is

determined with the imaginary part of the self energies. A special notation is introduced

for this imaginary part in the Green’s function formalism as follows:

†( )iΓ = Σ − Σ . (4-32)

The spectral function can then be expressed in terms of Г:

† † 1 1 †

† † † † †

( ) ( )

( ) ( )L R L R L R

A i G G iG G G G

iG E H E H G G G G G

− −= − = −= − − Σ − Σ − + + Σ + Σ = Γ + Γ = Γ

(4-33)

As mentioned before the electron density plays an important role in the DFT

formalism, since the Hamiltonian is a functional of the electron density. In a two probe

system as depicted in figure 4.2 the electron density of the extended central region is

calculated in terms of Green’s function of the effective Hamiltonian. As in the general

DFT calculation a self-consistent procedure is required to find the converged electron

density and potential in equilibrium as well as under non-equilibrium conditions. Having

the spectral function or the energy resolved density of state the electron density is

simply calculated by integrating the spectral function which is multiplied by the Fermi

83

distribution function to take the occupation of the states into account. Applying (4-33) the

total electron density reads:

† †( , ) ( , )2L R C L C L C R C R

qn n n G G f E dE G G f E dEµ µ

π

∞ ∞

−∞ −∞

= + = Γ + Γ

∫ ∫ (4-34)

†

,

( , )2 C i C i

i L R

qn dE G G f Eµ

π

∞

=−∞

= Γ∑∫ (4-35)

Under the equilibrium condition where the Fermi energy of the leads are equal and by

using the alternative representation of the spectral function in (4-28) the charge density

takes the form:

Im ( ) ( , )C

qn dE G E i f Eδ µ

π

∞

−∞

= +∫ (4-36)

The numerical evaluation of the above integral is efficiently done by using a contour in

the complex plan as depicted in figure 4.3. The Green’s function changes sharply near

the real axis and therefore direct calculation of the integral in (4-36) requires a very fine

grid. However, the Green’s function behaves smoothly at the sufficient distance from the

real axis and therefore an accurate integration along the complex contour can be

achieved with a very small number of mesh points. In order to take all states and

therefore all electrons into account the contour is chosen in a way that includes all

eigenvlaues of the Hamiltonian. According to the residue theorem,

( ) ( ) 2 ( )p

B pz

dzG z f z ik T G zµ π− = − ∑∫ (4-37)

where zp are the poles of f(z) with a residues of -kBT. The integral in (4-36) is thus

evaluated as follows:

( ) ( ) ( ) ( ) 2 ( )p

B pzEB C L

dEG E i f E dzG z f z ik T G zδ µ µ π∞

+

+ − = − − − ∑∫ ∫ (4-38)

84

Figure 4.3: The contour in the complex plan. The numerical integration is evaluated along L(]∞+i∆,EF-λ+

i∆[ ) and C.

Under non-equilibrium condition where for example L Rµ µ> the electron density

could be separated in two parts: the equilibrium part which covers the states below Rµ

and the non-equilibrium part which consists of states in the energy window between two

Fermi energies.

†

,

( , )2

eqC i C R

i L R

qn dE G G f Eµ

π

∞

=−∞

= Γ∑∫ (4-39)

[ ]† ( , ) ( , )2

noneqC L C L R

qn dEG G f E f Eµ µ

π

∞

−∞

= Γ −∫ (4-40)

The calculation of the equilibrium part of the density is similar to the calculation of the

density for a system under equilibrium, using contour integration and the residue

theorem as described above. The integrand of the non-equilibrium part is not analytical,

and it is therefore not possible to extend the integral into the complex plane as used in

the equilibrium part. The integral in (4-40) is thus evaluated along the real axis, using a

dense set of integration points.

After calculating the charge density the self consistent loop as depicted in figure

4.4 is closed. The procedure in figure 4.4 is similar to that of the DFT except that the

charge density is calculated in terms if Green’s function which includes the effect of

open boundaries.

L C

EF EBL

∆

λ

δ

85

In the Landauer-Büttiker formalism [130] the current is calculated in terms of

electronic occupation and transmission probability. The current flow due to each state is

proportional to the occupation, charge and velocity of the electrons and the transmission

probability which gives the portion of the wave function transmitted through the system.

In a two probe system the scattering states in the left and right leads are propagating

with the wave vector k and –k’ respectively. The total net current flow from left to right is

then given by:

( ) ( )k k k L k k k Rk k

q qI v T f v T fε µ ε µ′ ′ ′− − −

′−

= − + −′Ω Ω∑ ∑ (4-41)

where Ω is the volume, k kv kε= ∂ ∂ℏ is the electron velocity and Tk is the transmission

probability. Transforming the sum into the integral ( 22k

dkπ

Ω→∑ ∫ ), including spin

degeneracy and using time reversal symmetry of the Hamiltonian (TL=TR=T) the current

is calculated as:

[ ]2( ) ( ) ( )L R

qI dE T E f E f E

hµ µ

∞

−∞

= − − −∫ (4-42)

In the framework of NEGF the transmission probability is expressed in terms of Green’s

function and self energies [131,132] as follows:

† †( ) Im ImL R L RT E Tr G G Tr G G = Σ Σ = Γ Γ (4-43)

It is clearly seen that the transmission probability is proportional to the energy

broadening. The higher broadening, which at the same time means higher coupling

between the central region and the leads, results in a higher transparency of the barrier.

This is exactly similar to the discussion about the quasi-bound states in the previous

part, where a higher broadening of an energy state results in a shorter lifetime and a

higher barrier transparency.

86

Figure 4.4: The flowchart shows the procedure of calculating the current through a two probe system

using NEGF formalism.

Set the atomic structure

Bulk calculation of leads

gL,gR, ΣL, ΣR

Guess n for the central region

Construct the effective

Hamiltonian

Heff=HC+ ΣL+ ΣR

Calculate the Green’s

function

G=[E-HC -ΣL -ΣR]-1

Calculate n’ from G

n=n’

Mix densities n=β n’+(1- β) n

Calculate the transmission probability and the current

Yes

No

87

5 Silicon/Silicon-dioxide system

In the previous section the theoretical background for the atomic level calculation

of the tunneling current is explored. Before we start with the atomic scale calculation of

the tunneling current, we should construct and optimize the atomic structure of the

system under consideration. In this section we begin with the construction and

optimization procedure of the Si/SiO2 interface system that will be used later for the

atomic level calculation of the tunneling current.

Before constructing the interface structure we specify input parameters used in

DFT calculations throughout this work. In all calculations we will use Norm conserving

non-local pseudopotentials of the Troullier-Martin type [119] for all atom species. The

localized numerical atomic orbital at the level of single-ζ plus polarization (SZP) with an

energy shift of 0.01 Ry and a cut off energy of 100 Ry are used in all calculations. The

exchange-correlation is approximated by the LDA using the Predew-Zang

parameterization [111] of Ceperly and Adler results [112]. The DFT total energy

calculation using the above parameters results in a lattice constant of aSi=5.5 Å for

crystalline Si. The band gap of Si is calculated to be 0.71 eV. As mentioned before the

underestimation of the band gap of semiconductors and insulators is a general accepted

error of the LDA approximation.

5.1 Constructing the Si/Silicon dioxide model struc ture

In most electronic structure calculations different phases of crystalline SiO2 have

so far been employed to generate model interfaces, mainly β-quartz [133-137], β-

cristobalite [133,136-139] and tridymite [134-138,140,141]. The electronic structure of

these model interfaces has been investigated successfully with the aim of understanding

effects such as Si core-level shifts [138], tunneling processes [134,141], and the carrier

mobility in the channel [137]. We start with a β-cristobalite crystalline SiO2 to model the

gate oxide. SiO2 β-cristobalite has a slightly larger lattice mismatch (about 6%) to the Si

(001) than tridymite, however, the cubic cristobalite has lots of polymorphs that are

almost energetically degenerate and differ only by relative rotations of the SiO4

tetrahedral units. Thus it can adjust easily to stress and local distortions. The mismatch

88

is minimized if the SiO2 crystal is rotated by 45° around its z axis. Thus, the lateral lattice

constant of SiO2 was fixed (expanded) to the value of √2×aSi, where aSi=5.5 Å is the

equilibrium lattice constant of Si, and the cristobalite cell was relaxed along its z

direction. The mismatch is small enough to obtain a stable interface but large enough to

introduce distortions in the SiO2 slab to mimic an amorphisation procedure. From the

“tetragonal” β-cristobalite bulk structure that was produced by this procedure, slabs

consisting of seven and eleven layers of SiO2 was composed and put on the Si

substrate consisting of nine layers of Si, as shown in figure 5.1. The number of atomic

layers ensures that the bulk properties are recovered in the respective slabs [133,139]

and that the interface does not interact with its periodic image. Note that the interface at

the right hand side of the supercell is rotated by 90° around the z axis of the supercell

(the direction perpendicular to the interface).

Figure 5.1: An interface model structure of the Si/SiO2 system.

To make up for the under coordination of Si atoms at the interface to SiO2, extra

bridging oxygen atoms were added, completing the coordination. This leads to a formal

oxidation state +2 of Si atoms at the interface, which is one of the intermediate oxidation

states that have been observed experimentally [142]. Different lateral arrangements of

the O atoms in the interfacial layer are possible, which would lead to equivalent

distributions of oxidation states. However, the details of the distribution do not affect

electronic properties of the interface such as the valence band offset [143].

89

-1-1-1-1 -0.5-0.5-0.5-0.5 0000 0.50.50.50.5 11110000

0.10.10.10.1

0.20.20.20.2

0.30.30.30.3

0.40.40.40.4

0.50.50.50.5

0.60.60.60.6

z-zo (Å)

Tot

al E

nerg

y D

iffe

renc

e (

eV

)T

ota

l Ene

rgy

Diff

ere

nce

(e

V)

Tot

al E

nerg

y D

iffe

renc

e (

eV

)T

ota

l Ene

rgy

Diff

ere

nce

(e

V)

-1-1-1-1 -0.5-0.5-0.5-0.5 0000 0.50.50.50.5 11110000

0.050.050.050.05

0.10.10.10.1

0.150.150.150.15

0.20.20.20.2

0.250.250.250.25

0.30.30.30.3

0.350.350.350.35

z-zo (Å)

Tot

al E

nerg

y D

iffe

renc

e (

eV

)T

ota

l Ene

rgy

Diff

ere

nce

(e

V)

Tot

al E

nerg

y D

iffe

renc

e (

eV

)T

ota

l Ene

rgy

Diff

ere

nce

(e

V)

Figure 5.2: The total energy as a function of the deviation from the optimal supercell size in the direction

perpendicular to the interface for a system with 7(left) and 11(right) atomic layers of SiO2.

The optimal distance between Si and SiO2 layers is obtained with several total

energy calculations. The cell size in the z direction is fixed and the total energy is then

calculated after relaxing all atomic coordinates. Atomic coordinates are relaxed until all

forces acting on atoms are smaller than 10-2 eV/Å. Figure 5.2 shows the total energy as

a function of deviation from optimal supercell size in the direction perpendicular to the

interface (zo) for two different oxide thicknesses. In the minimum energy configuration

the separation between the last Si layer in the bulk silicon and the first layer of Si in the

SiO2 is 2.32 Å. This means an expansion of almost 69% compared to the lattice spacing

along [001] in the bulk Si (1.375 Å), and of 28% compared to the separation of the Si

planes along [001] in the bulk SiO2 (1.76 Å). However, the Si-O bond lengths at the

interface are 1.67 Å, which shows only a slight extension of less than 2.5% compared to

the Si-O bond length of 1.63 Å in the SiO2 bulk region. This little distortion of the bond

length promises a nearly ideal interface. The larger distance of the Si layers at the

7 SiO2 atomic layers 11 SiO2 atomic layers

90

interface could be attributed to the extra interfacial oxygen atoms, which are added to

fully coordinate the Si atoms at the interface.

5.2 Interface stability

In order to study the stability of the constructed interface one has to compare the

total energy of the joined Si/SiO2 interface and the separated Si and SiO2 parts. The

difference between the total energies pro interface area is defined as the “work of

separation”, which is a measure of the interface stability for different interfaces.

( )( ) ( )

2tot tot

sepint

E joined E SeperatedW

A

−= − (5-1)

The factor 2 takes the two interfaces pro supercell into account, which exist due to the

periodic boundary conditions. It should be noted here that after separation the two

separated parts are relaxed again until the maximum force is smaller than 10-2 eV/Å. A

positive work of separation indicates a stable interface where a negative one signals an

unstable interface.

There are several scenarios for the separation of two subsystems, and for a

stable interface all should be results in a positive work of separation. Four different

separation scenarios for the Si/SiO2 system is depicted in figure 5.3 with corresponding

work of separations calculated for the system with a seven atomic layer oxide. The

positive values show that the model interface is stable. The (D) configuration, where the

Si surface is oxidized and the SiO2 has a Si terminated surface, seems to be the most

favorable configuration with the smallest work of separation.

5.3 Two probe configuration with intrinsic leads

After constructing the model interface and verifying its stability, the system should

be sandwiched between two leads in order to investigate the transparency of the SiO2

barrier layer and the tunneling current. The major difficulty in using an atomistic

91

Figure 5.3: Four different separation scenarios for the Si/SiO2 model interface(Si atoms are shown with

large blue balls where small red balls indicate O atoms) : (A): a clean Si surface, an O2 molecule and an

oxygen-terminated oxide surface, Wsep=10.39717 (J/m2) ; (B): a partially oxidized Si surface and an

oxygen- terminated oxide surface, Wsep=5.34923 (J/m2); (C): an oxidized Si surface and an oxygen –

deficient oxide surface, Wsep=3.97157 (J/m2); (D): an oxidized Si surface and a Si-terminated oxide surface,

Wsep=2.80924 (J/m2).

approach for the tunneling current simulation is the band bending at the Si/SiO2

interface. This band bending is extended over several tens of nanometers in Si, which

makes its inclusion in the atomic scale calculation impossible due to the large number of

atoms to be considered. To avoid this difficulty, and since we are interested in

microscopic aspects of the tunneling current, the structure is sandwiched between two

degenerated Si (and therefore metallic) leads. The leads are consisting of 8 Si atoms,

where one of them is replaced by P atom as n-type impurity. However, at zero applied

voltage there is no external electrostatic potential to be screened with the metallic leads

and therefore an intrinsic Si lead could be used in this case. The calculation of the

density of state (DOS), the projected density of state (PDOS) in the SiO2 region and the

92

transmission probability in the intrinsic system at zero bias provides information about

the Si and SiO2 band gaps as well as the valence and conduction band offsets at the

Si/SiO2 interface. However, the calculation of the current under applied voltage requires

highly doped metallic leads.

Figure 5.4: The atomic structure of the two probe Si/SiO2/Si system and the electron density on a slice

parallel to the (y,z) plane at x=0.

Figure 5.4 shows the atomic structure of a two probe Si/SiO2/Si system with

intrinsic leads and the electron density of the system on a cut parallel to the (y,z) plane.

A high density of localized electrons is observed around the oxygen atoms while a

93

periodic pattern of the electron density at left and right represents the two intrinsic leads

and the buffering Si layers. In the next picture (figure 5.5) the DOS of the whole system

as well as PDOS in the SiO2 region are depicted. The band gap of Si could be read from

the DOS plot and the PDOS plot shows the band gap of SiO2. However the interfacial

states are also presented in PDOS and DOS plots. In order to distinguish the interfacial

states from the bulk states and to determine the accurate band gap of Si and SiO2 the

projection of density of states on different Si atoms from the Si bulk into the oxide bulk is

plotted in figure 5.6. The picture shows that the interfacial states disappear in first three

atomic layers and the band gap reaches the values of the bulk SiO2. The projected

density of states also shows the band alignment of Si and SiO2 at the interface, which

depends on the atomic structure of the interface and the interface dipoles. The band gap

of Si is determined to be 0.71 eV where the band gap of SiO2 is 5.99 eV. The valence

and conduction band offsets at the Si/SiO2 interface is calculated to be 2.74 eV and 2.54

eV respectively. The discrepancy between these values and experimental values

( ,g SiE =1.12, 2,g SiOE =8.9, vφ =4.3 and cφ =3.1 (eV)) is due to the standard failure of the DFT-

LDA method in the determination of band gaps of semiconductors and insulators as

discussed earlier. However, obtained values are in a good agreement with other

calculations performed within the DFT-LDA [143,144].

-5 -4 -3 -2 -1 0 1 2 3 4 5E-E

f (eV)

Den

sity

of S

tate

DOSPDOS

Figure 5.5: The density of state and the projected density of state at the SiO2 layer of the sample with thin

oxide.

94

-5 -4 -3 -2 -1 0 1 2 3 4 5

E-Ef (eV)

PDOS

Figure 5.6: The projected density of states at different Si atoms in bulk Si, bulk SiO2 and their interface.

The band gaps and the band offsets are extracted from the projected density of states.

The spatial resolved density of states (see figure 5.7) also shows that the density

of states in the oxide layers for energies less than 2.8 eV is negligible but increase for

energies higher than that value, which indicates that the conduction band edge of SiO2

is around 2.8 eV.

Another method for the determination of the band gap and the band offset is to

extract them from the energy resolved transmission probability. The conduction and

valence band edge diagram of the Si/SiO2/Si system at zero bias is depicted in figure

5.8. As the diagram and the T(E) plot in figure 5.9 indicate the deep well in the T(E)

represents the Si band gap where the transmission probability has its lowest values. The

electrons with energies within the Si band gap should tunnel through the whole

structure, which results in a negligible small transmission probability. The electrons with

energies between the conduction or valence band edges of Si and SiO2 however,

experience a tunneling only in the SiO2 region and therefore have a higher transmission

probability comparing to the electrons within the Si band gap. Finally the electrons with

energies around the conduction or valence band edges of SiO2 are almost transmitted

without experiencing any tunneling and therefore have a transmission probability almost

equal to the unity.

Interfacial Si

Bulk Si

Bulk SiO2

0.71 eV 2.74 eV 2.54 eV

5.99 eV

95

0 5 10 15 20 25 30 35 40 45z (Å)

LDO

S

Figure 5.7: The spatial resolved density of states for energies between 2eV and 3eV.

Figure 5.8: The band edge diagram of the Si/SiO2/Si system. The transmitted wave function is mostly

attenuated at energies within the Si band gap. The transmission probability through SiO2 layer is also

small, while the incoming wave almost totally transmitted above the conduction band and below the

valence band of SiO2.

The band gap of Si and SiO2 are then obtained to be ,g SiE =0.71 eV and

2,g SiOE =6.10 eV where the band offset are determined to be vφ =2.75 eV cφ =2.61 eV.

These values are slightly different from those obtained from the projected density of

states. The discrepancies could be attributed to the interfacial states. As a result the

transmission probability is changing gradually around the SiO2 band edges, which make

the exact determination of the SiO2 band edges impossible. However, these deviations

are less than 3% and 0.5% in the conduction and valence band offsets respectively.

Si Si

SiO2

vφ

cφ

,g SiE

2,g SiOE

2 eV

2.2 eV

2.4 eV

2.6 eV

2.8 eV

3 eV Si Si SiO2

96

-5 -4 -3 -2 -1 0 1 2 3 4 510

-25

10-20

10-15

10-10

10-5

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

7 layers SiO2

11 layers SiO2

Figure 5.9: The transmission probability of the Si/SiO2/Si systems (thin and thick oxide).

The atomic model of the tunneling current takes the interface states and the

gradual change of the transmission probability automatically into account, in contrary to

the effective mass models presented in the first part of this work, where the band gap

and the band offsets are used as input parameters and the transition is approximated

with an abrupt jump in the conduction and valence band edges.

5.4 Transport under non-equilibrium condition

In order to extend the calculations to the non-equilibrium condition where a finite

voltage is applied along the structure, the leads are doped with P atoms. High

computation costs limit the supercell size or the number of atoms to be considered in the

calculation. Having small number of atoms in the lead, due to computational limits,

results in a very high doping level even if only one Si atom is replaced with P atom. This

high level doping shifts the Fermi energy into the conduction band which is not the case

in real devices with moderate doping levels. This can leads to higher tunneling currents

than that of in the real devices. However, we can comparatively study the effect of

atomic level distortions on the tunneling current using the same leads in all calculations.

97

Figure 5.10: The electron density of undoped (left) and doped (right) leads on a plane parallel to the (y,z)

plane at x=0.

Figure 5.11: The electrostatic difference potential across the Si/SiO2/Si structure on a cut parallel to the

(y,z) plane at 2 V applied potential between two leads.

98

Figure 5.10 shows the electron density in an intrinsic and a doped lead at the

same cut parallel to the (y,z) plane. The doped lead has a significantly higher electron

density around the doped atom due to the extra electron of the P atom. The metallic

character of the doped leads assures that the main part of the potential drops in the SiO2

region as it is shown in figure 5.11.

Using doped leads the tunneling current is then calculated for two samples with

two different SiO2 thicknesses. The thin sample has an oxide thickness of tox=14.93 (Å)

while in the thicker sample the oxide is tox=21.89 (Å). The oxide thicknesses are

measured as the distance between the two Si layers with +2 oxidation state at two

interfaces.

0 0.5 1 1.5 210

-13

10-11

10-9

10-7

10-5

10-3

Vg (V)

J g (µA

)

7 atomic layers siliconoxide11 atomic layers siliconoxide

Figure 5.12: The calculated tunneling current for two samples (tox=14.93 Å) and (tox=21.99 Å).

The tunneling current under applied voltages up to 2 V is depicted in figure 5.12

for two samples with different oxide thicknesses. The curves have similar shapes and

forms but a difference of more than three orders of magnitudes are observed, which is

due to the different oxide thicknesses. This is also reflected in the transmission

probability of the two samples which is depicted in figure 5.9. The difference of the

tunneling currents is generally consistent with the results of the first part of this work and

experimental measurements, where such changes in the oxide thickness results in

approximately similar variations in the tunneling current. This confirms again that,

although the DFT-LDA underestimates the band gaps and the band offsets in the

99

Si/SiO2/Si structure, we can still use such calculations for comparative studies of the

atomic scale deformations and their influences on the tunneling current. This can be

associated with the systematic behavior of the DFT-LDA error in the determination of the

band gap in insulators and semiconductors.

100

6 The influence of oxygen vacancy defects in Silico n dioxide on the tunneling current

6.1 Single oxygen vacancy

In the previous section the atomic level calculations of the tunneling current have

been applied to an almost ideal Si/SiO2/Si system. The main advantage of the atomic

level calculation is the ability to study atomic level distortions and their influences on the

tunneling current. In this section we will study the influence of neutral oxygen vacancies,

as a prevalent defect, in the SiO2 gate dielectric. Oxygen vacancies are implemented in

the system by removing one oxygen atom from SiO2 and relaxing all atomic coordinates

until all forces are smaller than 10-2 eV/Å. Oxygen atoms are removed from different

layers of the oxide in order to find the most energetically favorable position of an oxygen

vacancy and to study the correlation between the tunneling current and the position of a

vacancy. Figure 6.1 shows different selected positions for oxygen vacancies which are

used as single or in combination as double vacancies.

Figure 6.1: Different positions of implemented oxygen vacancies. Vac1-5 are used as single vacancies

where Vac3 is used in combination with Vac5/6/7/8 as double vacancies.

Vac: 1 2 3 4 5 6 7 8

101

Figure 6.2: The electron density on a cut parallel to the (y,z) plane for a system without (up) and with

(down) oxygen vacancy (Vac4). One of the red points which represent a high electron density around

oxygen atoms is disappeared.

It has been shown in the previous section that the real space electron density has

peaks around oxygen atoms. Therefore, after removing one oxygen atom one of the

peaks in the electron density will be disappeared as figure 6.2 shows for one of the

vacancies as an example.

Comparing the total energy of the systems with different vacancies shows that,

Vac1 and Vac2 are energetically favorable by amount of ~0.7 eV. This indicates that the

vacancies tend to sit at the interface of Si/SiO2 which leads to the mobility degradation in

channel of MOS transistors. In all vacancy sites the distance between two Si atoms after

removing the connecting oxygen and relaxing the atomic coordinates reduces by almost

83%. The two adjacent Si atoms form a bond, which introduce a bonding and an

antibonding defect levels in the SiO2 bad gap. The bonding level for the Vac1 (bridge)

appears at the valence band of Si as depicted in figure 6.3. In the Vac1 each bonded Si

atom has three other Si neighbors. However, other vacancies, which are within the SiO2

form a Si-Si bond where each Si has three O neighbors. The bonding level of these

102

vacancies are calculated to be placed few tenth of an electron volt above the top of the

SiO2 valence band (figure 6.3), in good agreement with other reported calculations [134].

There is no contribution from bonding levels in the tunneling current, because they are

far below the Fermi level, and out of the energy transmission window. In contrary, the

antibonding levels could have significant contribution to the tunneling. The antibondnig

levels are generally placed between the Si and SiO2 conduction band edges but the

exact position of them is less accurately known. In case of the bridge vacancy there is a

clear peak in the DOS around 0.67 eV above the Fermi level due to the antibonding level

(figure 6.3). Figure 6.4 shows the local density of states on each vacancy site before and

after creating the vacancy. The changes show the vacancy induced states, which

consist of bonding levels (near the SiO2 valence band edge) and antibonding levels

(between the conduction bands of Si and SiO2).

-4 -3 -2 -1 0 1 2 3 40

2000

4000

6000

E-Ef (eV)

DO

S

-4 -3 -2 -1 0 1 2 3 40

2000

4000

6000

E-Ef (eV)

PD

OS

without vac.

Vac1

without Vac. Vac2 Vac3 Vac4 Vac5

Figure 6.3: The bonding and antibonding defect levels for the Vac1 (up) and Vac2-5 (down).

103

-4 -3 -2 -1 0 1 2 3 4E-E

f (eV)

PD

OS

Figure 6.4: The local density of state on the vacancy sites before and after creating the vacancies.

After relaxing the atomic coordinates, the vacancy induced deformations in the

structure brings some changes in DOS and therefore in the transmission probability. At

each applied voltage the value of the transmission probability within the transmission

window around the Fermi energy is of essential importance for the tunneling current.

Figure 6.5 shows the transmission spectrum of samples without and with different

vacancies. The most contribution to the tunneling current comes from the conduction

band states around the Fermi energy and within the transmission window. The bridge

vacancy shows a higher transmission probability for energies below ~1 eV and almost

similar probabilities as the ideal system at higher energies. This is reflected in the

calculated tunneling current depicted in figure 6.6. The tunneling current through the

structure with a bridge vacancy is higher than the current of the ideal system at low

biases (Vg<1.2 V) but they become similar at the higher biases. All other vacancies result

in an increase of the transmission probability for energies above the leads conduction

band which causes a general escalation of the tunneling current at all applied potentials

( 0 2gV≤ ≤ ). The second vacancy, the so called arm vacancy that connects the last Si

layer in the Si lead to the first Si layer in SiO2, has the highest transmission probability at

E-Ef <1 eV and therefore the highest destructive influence on the tunneling current at low

biases. The bulk vacancies (3,4 and 5) show almost similar tunneling currents. The

Vac1

Vac2

Vac3

Vac4

Vac5 without Vac.

with Vac.

104

slightly higher tunneling current of Vac4 at Vg<1 V and Vac3 at Vg> 1 V could be

explained in the same way based on transmission probabilities. The Vac4 has a higher

transmission probability at E-Ef <1 eV while the Vac3 shows a higher transmission

probability at 1<E-Ef <2 eV .

-3 -2 -1 0 1 2 310

-20

10-10

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

-3 -2 -1 0 1 2 310

-20

10-10

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

Vac1

without Vac.

Vac2

without Vac.

Vac3

Vac4

Vac5

Figure 6.5: The comparison of the transmission probability between ideal system, bridge (Vac1) and arm

(Vac2) vacancies (up) as well as ideal system and Vac3, Vac4 and Vac5 (down).

105

0 0.5 1 1.5 210

-8

10-6

10-4

10-2

Vg (V)

J g ( µ

A)

without Vac.Vac 1

Vac 2

Vac 3

Vac 4Vac 5

Figure 6.6: The calculated tunneling current for the Si/SiO2/Si System without and with single vacancies.

6.2 Double oxygen vacancies

In the next step we built a second bulk vacancy in the presence of the Vac3. The

second site has been chosen with different distances to the Vac3 in order to study the

correlation between vacancy distances and their influence on the tunneling current. The

oxygen atoms which are numbered as 5,6,7 and 8 in figure 6.1 have been chosen and

removed as the second vacancy. After relaxing all atomic coordinates and calculating

the total energy, no significant correlation between distance and the total energy is

observed. The maximum deviation of the total energy is less than 0.08 eV.

Comparing the transmission probability and the tunneling current of systems with

double and single bulk vacancies (figure 6.7 and 6.8) shows that regardless of the

distance between vacancies the second vacancy deteriorates the insulating character of

the oxide and leads to a high barrier transparency and hence a high tunneling current. In

contrary to the total energy, the transmission probability shows some correlation with the

relative position of two vacancies. As the distance between vacancies reduces, the

transmission probability around the Fermi energy is increasing. The Vac3&5 shows the

highest transmission probability and therefore the highest tunneling current among the

systems with two vacancies. The tunneling current then reduces as the second vacancy

moves away from the first one.

106

-3 -2 -1 0 1 2 310

-20

10-10

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

-3 -2 -1 0 1 2 310

-20

10-10

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

Vac3&5

Vac3&6

Vac3&7

Vac3&8

without Vac.

Vac3

Vac2

Vac3&5

Figure 6.7: The transmission probability of samples with two bulk vacancies at the same time (up) and

comparing the Vac3&5 with the ideal system as well as with single vacancies (Vac2 and Vac3) (down).

These calculations may shed some lights on the degradation process of the gate

oxide due to vacancies and the reliability of the gate oxide. It shows that a path of

neighboring vacancies strongly deteriorates the barrier characteristic of the oxide and

could end in a breakdown of the gate oxide.

107

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Vg (V)

J g ( µ

A)

without Vac.

Vac3&5

Vac3&6

Vac3&7

Vac3&8

Vac3

Figure 6.8: The calculated tunneling current for systems containing two bulk vacancies compared with the

ideal system and the single vacancy Vac3.

In this section, it has been shown that even a single bulk neutral vacancy could

results in one order of magnitude enlargement of the tunneling current around 1 V

applied voltage. In the worst case, where two vacancies are lying only one atomic layer

away, the tunneling current enhancement reaches nearly two orders of magnitudes.

It should be noted that due to the limitation on the system size and the number of

atoms, the systems which considered here are corresponding to very high defect

concentrations. It is obvious that each vacancy interact with its images in the

neighboring cells which were repeated periodically. More accurate results for a real

single vacancy with negligible interaction with its images require a very large supercell

which in turn means a large number of atoms. The calculation of such a big supercell

under different non-equilibrium conditions is still beyond the available computation

power. However, calculation with smaller supercells could deliver some insight into the

defect problem and provide some useful results.

108

7 Compressed and strained Silicon dioxide

In the chapter 3 it has been shown that as the gate oxide thickness decreases the

“tunneling effective mass” which is required to fit the calculated tunneling current to the

measured one increases. Indeed, in the effective mass model, presented in chapter 2,

the only fitting parameter was the “tunneling effective mass” of electrons in the oxide and

all other parameters such as the dielectric constant and the barrier height are kept

constant. As it mentioned in chapter 3 the increase in the oxide “tunneling effective

mass”, which leads to a decrease in the tunneling current, might be due to the

compressive stress in the oxide at the oxide interface. This has been supported by some

experimental studies [101,102] which show that the tunneling current reduces in a

compressed oxide layer, while it increases in a decompressed oxide. In this chapter we

try to investigate this issue on the atomic level basis and study the influence of an

external compression as well as decompression on the barrier characteristic of the gate

oxide and the tunneling current.

Measuring the Si-O bond length in a relaxed Si/SiO2/Si structure shows a small

difference in the bond length at the interface and in the bulk oxide. The maximum

deviation of the bond length is less than 2.5%. However, the bond angle (Si-O-Si) is

strongly changed (~ 45%) as moving from the interface into the bulk. It means that the

distortion of the atomic structure at the interface is mainly absorbed in the bond angle

rather the bond length. The Si-O-Si bond angle is depicted in figure 7.1 at different

atomic layers for a thick (21.89 Å) oxide as well as a thin (14.92 Å) oxide. As the figure

shows in the thin oxide the interfacial layer is as thick as the bulk like layer, while in the

thicker sample the main part of the oxide has a bulk like bond angle. This shows how the

interfacial layer dominates the properties of an ultrathin (< 20 Å) gate oxide. The

unexpected changes in the barrier transparency, which is adjusted in the effective mass

model with a varying “tunneling effective mass”, might be due to the dominant interfacial

effects.

If a distortion in the bond angles could reduce the tunneling current it would be

interesting to study the impact of an external compression and decompression on the

gate oxide and their influence on the tunneling current. In order to model such strains in

109

the direction perpendicular to the interface, we make small changes in the supercell size

in that direction and relax all atomic coordinates again. The supercell size is increased

and decreased by 0.5 Å and 1 Å which correspond to ~2% and ~4%. Measuring the

average values of the bond lengths (Si-O) and bond angles (Si-O-Si) shows that when

the structure is compressed by 4% the bond lengths reduces only by ~0.7% while the

angles show ~3% change. In the case of a decompression ~1% change in bond lengths

and 2.5% in angles are observed. Figure 7.1 show the bond angles at different atomic

layers under compressed and decompressed for the thin oxide. The picture indicates

that applying a compressive pressure to the oxide in the direction perpendicular to the

interface reduces the bulk like area and could results in a less transparent barrier. On

the other hand, decompression of the oxide layer reduces the interfacial layer and

therefore leads to a more transparent barrier.

1 2 3 4 5 6 7 8 9 10 11 12120

130

140

150

160

170

180

Atomic Layer

Si-O

-Si A

ngle

1 2 3 4 5 6 7 8120

130

140

150

160

170

180

Atomic Layer

Si-O

-Si A

ngle

thick SiO2

thin SiO2 -1Å -0.5 Å +0.5 Å +1 Å

Figure 7.1: Si-O-Si bond angles in the thick (21.89 Å) (up) and thin (14.92 Å) oxide as well as compressed

and decompressed layers (down). The boxes show the bulk like part of the oxide where Si-O-Si bond

angles are near the corresponding bulk value.

110

The total energy of the supercell increases by ~90 meV and ~120 meV after

changing the supercell size by 2% for compressed and decompressed strain

respectively. In the case of 4% change in the supercell size, the total energy increases

by ~400 meV and ~450 meV corresponding to the compressed and decompressed case.

These values correspond to pressures of ~0.9, ~1.3, ~2.1, and ~2.5 GPa respectively,

which are far beyond the elastic modulus of silica ~ 70 GPa.

It is interesting to see that the similar behavior is also reflected in the transmission

probability of strained samples. Figure 7.2 shows the calculated transmission probability

for the unstrained and four compressed and decompressed samples.

-5 -4 -3 -2 -1 0 1 2 3 4 510

-20

10-15

10-10

10-5

100

E-Ef (eV)

Tra

nsm

issi

on P

roba

bilit

y

-1Å1 Å-0.5 Å0.5 Å0 Å

Figure 7.2: The transmission probability for the unstrained as well as four compressed and decompressed

gate oxides.

The above picture shows clearly the influence of the stress on the transmission

probability. Significant changes are observed at energies above the conduction band of

silicon where the most electrons are transmitted. As the compressive stress increases

the transmission probability in the transmission window reduces and the conduction

band is slightly shifted toward the higher energies. In other words, as the bond angles

reduce due to the compression, the conduction band edge moves into the higher

energies which could be interpreted as a higher barrier and therefore a lower tunneling

current. In contrary, decompression of the oxide layer causes the shrunk interfacial bond

111

angles to open up and leads to a higher transmission probability, which in turn means a

higher tunneling current.

The tunneling currents of all samples are depicted in figure 7.3 and shows that

the compressive stress on the oxide could result in a significant reduction in the

tunneling current where the decompression of the oxide layer increases the tunneling

current.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-8

10-7

10-6

10-5

10-4

10-3

10-2

Vg (V)

J g ( µA

)

0 Å-1 Å-0.5 Å0.5 Å1 Å

Figure 7.3: The calculated tunneling current for the unstrained oxide as well as compressed and

decompressed oxide layers.

112

8 Summary and outlook

Summary

The exponentially increasing gate tunneling current in nanoscale MOSFETs with

ultrathin gate dielectrics is a technological barrier against further performance

enhancement by means of dimensions downscaling. Therefore, understanding the

tunneling phenomena and calculating the tunneling current in such devices is of great

importance. In the first part of this work (chapter 1-3) we proposed a quantum

mechanical model for calculating the charge distributions on quantized states, potential

profiles and tunneling current at the inversion layer of a NMOS transistor. The model is

based on a self-consistent solution of the Schrödinger and Poisson equations within the

effective mass approximation. An open boundary condition at the dielectric/gate

interface is used for calculating the tunneling current. The open boundary results in a

broadening of quantized states at inversion layer and changes them to quasi-bound

states. Electrons on each QBS have a finite lifetime, which describes the tunneling rate

of electrons from that state. A new wave function based method is presented for the

lifetime calculation, which leads to the calculation of the tunneling current. It has also

been shown that open boundary not only causes a broadening of energy states but also

a tiny shift in their positions. The calculated tunneling currents for different samples with

ultrathin oxynitride gate dielectrics have been compared to the measured ones and

show a very good agreement. Like other authors, the best fitting has been achieved with

a varying “tunneling effective mass” in the dielectric layer. The “tunneling effective mass”

in the dielectric layer has shown to be a parameter of the chemical composition as well

as the thickness of the dielectric layer. It increases with decreasing the layer thickness

as well as the nitrogen concentration.

In the second part of this thesis (chapter 4-7) an atomic level model has been

studied and applied for the calculation of the tunneling current. The model is based on

the ab initio DFT method combined with the NEGF formalism. Very high computational

burdens of such models put limits on the size of systems under consideration. The

whole MOS structure with its extended depletion layer could not be calculated in such

113

models; however a real size SiO2 layer which is sandwiched between two small Si leads

could be calculated in a reasonable time. Atomic level distortions and their influence on

the tunneling current have been studied by means of this model. First of all a Si/SiO2

interface model has been constructed and optimized. Then the tunneling current has

been calculated for two samples with different oxide thicknesses. An increase in the

tunneling current due to a reduction of the oxide thickness has been observed as

expected. In the 5th chapter, the influence of oxygen vacancies on the atomic structure

and tunneling current has been studied. The arm vacancy seems to have the most

destructive effect among other single vacancies on the tunneling current at low bias. As

expected adding a second bulk vacancy to an oxide layer with one bulk vacancy

deteriorates the tunneling current. Two neighboring vacancies have shown to cause the

highest tunneling current.

In chapter 7 the influence of tensile and compressing strains on the atomic

structure of an ultrathin oxide and on the tunneling current through such layer is studied.

It has been shown that the atomic bonds in the first three layers at each interface are

deformed due to interface mismatches. Such deformations could be in principle

responsible for the variation of electron “tunneling effective mass”. It has been shown

that compressing the oxide layer results in a change of atomic structures which in turn

reduces the tunneling current. On the other hand, tensile strains work in the opposite

direction and increase the tunneling current.

Outlook

In the model developed in the first part of this work, the electron-electron

interactions at the inversion layer is calculated within the Hartree limit without the

inclusion of the exchange-correlation potential. Although, this part of electron interaction

is small comparing to the Hartree part, the inclusion of such potentials could results in

improved results for charge distribution at the inversion layer as well as the tunneling

current.

Good agreement between calculated and measured data in chapter 2 and 3

confirms that the main leakage mechanism in ultrathin dielectric layers is the direct

tunneling. However, the application of thick high-K gate dielectric stacks, increase the

114

possibility of trap states in oxide and therefore trap assisted mechanism could contribute

to the leakage current. The inclusion of such mechanism could leads to better results for

the tunneling current through thick high-K gate stacks.

The atomic level model used in 5th -7th chapters shows a promising method for

the investigation of atomic level distortion and their influence on dielectric properties.

The investigation of neutral oxygen vacancies presented in chapter 5 could be extended

to the positive and negative charged vacancies as well as other defects in SiO2 such as

B atom contamination. The method could be also applied to the complex gates stacks

which consist of several layers or alloyed dielectrics such as SiOxNy, HfxSi1-xO2 or ZrxSi1-

xO2.

115

Zusammenfassung

Der exponentiell steigende Tunnelstrom stellt eine technologische Barriere gegen die

weitere Skalierung von MOSFETs dar. Daher sind die Untersuchung und Berechnung

des Tunnelstroms von großer Bedeutung. Im ersten Teil dieser Arbeit wurde ein

quantenmechanisches Modell zur Berechnung von Quantisierungseffekten,

Ladungsverteilung und Tunnelstrom in einem NMOS Transistor entwickelt und

eingesetzt. Das Modell basiert auf der selbstkonsistenten Lösung des Schrödinger-

Poisson-Gleichungssystems im Rahmen der effektive-Masse-Approximation. Die

endliche Höhe der dielektrischen Barriere wurde mit einer offenen Randbedingung an

der Gateseite modelliert. Diese offene Randbedingung wandelt die gebundenen

Subbandniveaus in der Inversionsschicht zu quasi-gebundenen Zuständen (QBS) um.

Im Gegensatz zu Elektronen auf gebundenen Zuständen haben die Elektronen auf

QBSs eine endliche Lebensdauer, was zur Energieverbreiterung der Subbandniveaus

führt. Die offene Randbedingung sorgt auch für eine kleine Änderung der

Energieniveaus. Eine auf Wellenfunktionen basierende Methode wurde zur Berechnung

der Elektronenlebensdauer und des Tunnelstromes entwickelt und eingesetzt. Der

Vergleich zwischen gemessenen und berechneten Tunnelströmen zeigt eine gute

Übereinstimmung. Die effektive Masse in der dielektrischen Schicht wurde wie bei

anderen Autoren als Anpassungsparameter variiert. Diese effektive Masse steigt in den

Proben mit dünneren Schichtdicken und sinkt in den Proben mit höherem

Stickstoffgehalt.

Im zweiten Teil dieser Arbeit wurde ein atomistisches Modell zur Berechnung des

Tunnelstroms eingesetzt. Dieses Modell basiert auf dem ab initio DFT- und NEGF-

Formalismus. Die Anwendung solcher atomistischen Modelle ist auf kleine Systeme mit

wenigen hundert Atomen beschränkt. Deswegen ist die Simulation des gesamten

Transistors nicht möglich. Jedoch ist die Simulation einer dünnen SiO2-Schicht, die

zwischen zwei Si-Elektroden eingebettet ist, handhabbar. Das atomistische Modell

ermöglicht die Untersuchung atomskaliger Versetzungen und Verzerrungen sowie deren

Einfluss auf den Tunnelstrom. Zuerst wurde ein Si/SiO2-Grenzflächenstrukturmodell

konstruiert und im Rahmen der LDA-DFT optimiert. Dann wurde der Tunnelstrom für

116

zwei unterschiedliche SiO2-Schichtdicken berechnet. Wie erwartet, fließt deutlich mehr

Tunnelstrom durch die dünnere SiO2-Schicht. Der Einfluss von Einzel- sowie

Doppelsauerstoffleerstellen wurde ebenfalls untersucht. Unter den Einzelleerstellen hat

die so genannte „Arm“ Leerstelle, die die letzte Si-Schicht im reinen Si mit der ersten Si

Schicht in SiO2 verbindet, die größte Wirkung auf die Vergrößerung des Tunnelstromes.

Das gilt insbesondere für kleine Gatespannungen. Eine zweite Sauerstoffleerstelle führt

zu noch größeren Tunnelströmen, vor allem wenn die beiden Leerstellen nahe

beieinander liegen.

Im siebenten Abschnitt wurde gezeigt, dass die ersten drei Atomschichten des SiO2 an

der Grenzfläche gespannt und deformiert sind. Sehr dünne Oxidschichten bestehen

daher hauptsächlich aus diesen gespannten Schichten. Dies kann eventuell die

steigende effektive Masse in den sehr dünnen Oxidschichten erklären. Eine zusätzliche

Druckspannung auf die Oxidschicht führt zu mehr Verformung und daher weniger

Tunnelstrom. Eine Zugspannung anderseits, verringert den Druck und erhöht damit den

Tunnelstrom.

117

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134

List of figures Figure 1.1: Scaling scheme of a MOSFET transistor..................................................... 12

Figure 1.2: Fowler-Nordheim tunneling (left) and direct tunneling (right). ...................... 19

Figure 1.3: Bound (closed boundary) and quasi-bound (open boundary) states at

inversion layer of a NMOS............................................................................................. 22

Figure 2.1: The position of the Fermi energy with respect to the conduction band edge of

silicon, far from interface, is depicted for a p-doped silicon substrate as a function of the

doping concentration. .................................................................................................... 30

Figure 2.2: The band bending at the Si/SiO2 interface at 1 Volt applied gate voltage for a

NMOS structure with 1018 cm-3 doping concentration. The Fermi energy is Ef= -1.0579

eV and the first two subbands have energies of E1,1= -1.0721 eV , E1,2= -0.9693 eV. .... 31

Figure 2.3: The discretization scheme of the Schrödinger equation on a regular 1D mesh

grid with equivalent distance a in the direction perpendicular to the Si/SiO2 interface. .. 32

Figure 2.4: The depletion layer width as a function of the gate applied voltage for a

sample with NA=1018 cm-3 and tox=2 nm. ........................................................................ 37

Figure 2.5: Flowchart diagram of the self-consistent solution of the Schrödinger-Poisson

equation system. ........................................................................................................... 38

Figure 2.6: Wave functions around the first QBS in a structure with NA=1018 cm-3 and

tox=1 nm. The dotted and dashed lines depict wave functions corresponding to the gate

states, while the solid line corresponds to the first QBS. ............................................... 41

Figure 2.7: A fine tune energy scan around the first QBS. In the right picture the wave

functions with dotted and dashed line violate the negative derivative and sign change

condition in the dielectric region, respectively. The solid line shows a valid wave function

on the first QBS. ............................................................................................................ 41

Figure 2.8: The surface potential and the energy of the first two subbands as a function

of applied gate voltage for a structure with NA=1018 cm-3 and tox=2 nm. The solid line

determines the Fermi level. The triangles show the subband energies calculated under

open boundary condition. The small filled circles show the energies of subbands

calculated under closed boundary condition.................................................................. 44

135

Figure 2.9: The surface potential (open circles) and the first subband energy (triangles)

as a function of the gate voltage for structures with the same NA=1018 cm-3 and with

different tox=1, 1.5 and 2 nm........................................................................................... 45

Figure 2.10: The total inversion charge density for samples with the same substrate

doping NA=1018 cm-3 and with different dielectric thicknesses (upper picture) and with the

same dielectric thickness of tox=2 nm but different substrate doping concentrations (lower

picture)........................................................................................................................... 45

Figure 2.11: The Distribution of the inversion charge density on first four subbands for a

sample with NA=5×1017 cm-3 and tox=1 nm. .................................................................... 46

Figure 2.12: The total electron density at the Si/SiO2 interface for a sample with NA=1018

cm-3 and tox=2 nm at different gate voltages (upper picture). The spatial distribution of

charge on the first three subbands of the same structure at Vg=2 V (lower picture)...... 46

Figure 2.13: The repositioning of subband energies due to the open boundary in

samples with NA=1017 cm-3 and different oxide thicknesses. Dashed lines are subband

energies under the close boundary condition where triangles (first subband) and open

circles (second subband) depict the energies under the open boundary condition. ...... 48

Figure 2.14: The energy broadening of the first subband, for samples with NA=1017 cm-3.

...................................................................................................................................... 49

Figure 2.15: The potential well and the first four subbands of longitudinal electrons of

samples with tox=2 nm at Vg=2 V. The values of subband energies and the distance of

classically turning points are listed in table 2.1. ............................................................. 50

Figure 2.16: The energy broadening on the first four longitudinal subbands of samples

with tox=2 nm.................................................................................................................. 51

Figure 2.17: The energy broadening of the first longitudinal and transverse subbands of

the same samples as in figure 2.16 ............................................................................... 52

Figure 2.18: The tunneling current for samples with different oxide thicknesses and

substrate doping: NA=1017 cm-3 (circle), 5×1017 cm-3 (square) and 1018 cm-3 (triangle)... 53

Figure 2.19: The comparison between the measured (closed circles) and the calculated

(open circles) tunneling current in different samples. .................................................... 55

Figure 3.1: The measured (line) and the calculated (open circle) gate tunneling currents

for samples (a), (b), …, (i).............................................................................................. 61

136

Figure 3.2: The 3D surface (up) and contour (down-left) illustration of uncorrelated

interpolation and the contour plot of correlated (down-right) interpolation function........ 64

Figure 3.3: The electron “tunneling effective mass” as a function of the effective oxide

thickness (EOT) and its linear interpolation. .................................................................. 65

Figure 4.1: A schematic illustration of the all-electron (dashed line) and the pseudo (solid

line) wave function and potential. Beyond the cutoff radius rc the pseudo and the true

potential and wave function become similar. ................................................................. 73

Figure 4.2: The schematic diagram of a two probe system. C, L and R are the central

region, left and right leads respectively. ........................................................................ 79

Figure 4.3: The contour in the complex plan. The numerical integration is evaluated

along L(]∞+i∆,EF-λ+ i∆[ ) and C. .................................................................................... 84

Figure 4.4: The flowchart shows the procedure of calculating the current through a two

probe system using NEGF formalism. ........................................................................... 86

Figure 5.1: An interface model structure of the Si/SiO2 system. .................................... 88

Figure 5.2: The total energy as a function of the deviation from the optimal supercell size

in the direction perpendicular to the interface for a system with 7(left) and 11(right)

atomic layers of SiO2. .................................................................................................... 89

Figure 5.3: Four different separation scenarios for the Si/SiO2 model interface(Si atoms

are shown with large blue balls where small red balls indicate O atoms) : (A): a clean Si

surface, an O2 molecule and an oxygen-terminated oxide surface, Wsep=10.39717 (J/m2)

; (B): a partially oxidized Si surface and an oxygen- terminated oxide surface,

Wsep=5.34923 (J/m2); (C): an oxidized Si surface and an oxygen –deficient oxide surface,

Wsep=3.97157 (J/m2); (D): an oxidized Si surface and a Si-terminated oxide surface,

Wsep=2.80924 (J/m2). ...................................................................................................... 91

Figure 5.4: The atomic structure of the two probe Si/SiO2/Si system and the electron

density on a slice parallel to the (y,z) plane at x=0. ....................................................... 92

Figure 5.5: The density of state and the projected density of state at the SiO2 layer of the

sample with thin oxide. .................................................................................................. 93

Figure 5.6: The projected density of states at different Si atoms in bulk Si, bulk SiO2 and

their interface. The band gaps and the band offsets are extracted from the projected

density of states............................................................................................................. 94

137

Figure 5.7: The spatial resolved density of states for energies between 2eV and 3eV. . 95

Figure 5.8: The band edge diagram of the Si/SiO2/Si system. The transmitted wave

function is mostly attenuated at energies within the Si band gap. The transmission

probability through SiO2 layer is also small, while the incoming wave almost totally

transmitted above the conduction band and below the valence band of SiO2. .............. 95

Figure 5.9: The transmission probability of the Si/SiO2/Si systems (thin and thick oxide).

...................................................................................................................................... 96

Figure 5.10: The electron density of undoped (left) and doped (right) leads on a plane

parallel to the (y,z) plane at x=0..................................................................................... 97

Figure 5.11: The electrostatic difference potential across the Si/SiO2/Si structure on a

cut parallel to the (y,z) plane at 2 V applied potential between two leads...................... 97

Figure 5.12: The calculated tunneling current for two samples (tox=14.93 Å) and

(tox=21.99 Å). ................................................................................................................. 98

Figure 6.1: Different positions of implemented oxygen vacancies. Vac1-5 are used as

single vacancies where Vac3 is used in combination with Vac5/6/7/8 as double

vacancies..................................................................................................................... 100

Figure 6.2: The electron density on a cut parallel to the (y,z) plane for a system without

(up) and with (down) oxygen vacancy (Vac4). One of the red points which represent a

high electron density around oxygen atoms is disappeared. ....................................... 101

Figure 6.3: The bonding and antibonding defect levels for the Vac1 (up) and Vac2-5

(down).......................................................................................................................... 102

Figure 6.4: The local density of state on the vacancy sites before and after creating the

vacancies..................................................................................................................... 103

Figure 6.5: The comparison of the transmission probability between ideal system, bridge

(Vac1) and arm (Vac2) vacancies (up) as well as ideal system and Vac3, Vac4 and

Vac5 (down). ............................................................................................................... 104

Figure 6.6: The calculated tunneling current for the Si/SiO2/Si System without and with

single vacancies. ......................................................................................................... 105

Figure 6.7: The transmission probability of samples with two bulk vacancies at the same

time (up) and comparing the Vac3&5 with the ideal system as well as with single

vacancies (Vac2 and Vac3) (down). ............................................................................ 106

138

Figure 6.8: The calculated tunneling current for systems containing two bulk vacancies

compared with the ideal system and the single vacancy Vac3. ................................... 107

Figure 7.1: Si-O-Si bond angles in the thick (21.89 Å) (up) and thin (14.92 Å) oxide as

well as compressed and decompressed layers (down). The boxes show the bulk like

part of the oxide where Si-O-Si bond angles are near the corresponding bulk value. . 109

Figure 7.2: The transmission probability for the unstrained as well as four compressed

and decompressed gate oxides. .................................................................................. 110

Figure 7.3: The calculated tunneling current for the unstrained oxide as well as

compressed and decompressed oxide layers.............................................................. 111

List of tables Table 2.1: The values of the subband energy, energy changes from subband to subband

and the distance between classically turning points on the first four longitudinal subband

energies of samples with tox=2 nm and NA=1017 and 1018 cm-3 respectively. ................. 50

Table 3.1: The physical thickness, nitrogen concentration and extracted electron

“tunneling effective mass” of different samples.............................................................. 60

139

Acknowledgements

This thesis arose out of years of research that has been done since I came to the

Professor C. Radehaus’s group at Technical University of Chemnitz. During that time I

took advantage of working with friendly and scientific qualified people, to whom I would

like to convey my gratitude.

In the first place I would like to record my gratitude to Prof. C. Radehaus for his

supervision, advice and guidance from the early stage of this research. I enjoyed many

scientific discussions with him around the subject of this thesis as well as several

discussions around God and world. For me, he has always been a perfect example of an

open minded scientist.

I gratefully acknowledge Dr. E. Nakmedov for his advice and supervision. As a

classical electrical engineer at the beginning I knew little about quantum mechanics. Dr.

Nakhmedov with his broad band knowledge in physics had always the proper answer to

my questions around quantum mechanics.

It was a pleasure to collaborate with several postdoctoral research fellows as well

as Ph.D. students. Many thanks go to our former colleagues Dr. A. Martinez-Lima, Dr. R.

Janisch, Dr. M. Bouhassoune and Dipl.-Phys. L. Mancera for very useful collaborations

and discussions. It is a pleasure to pay tribute also to the current member of our group

Dr. F. Thunecke, Dr. R. Öttking, Dr. F. Chiker and Dipl.-Phys. P. Plänitz. I also

appreciate other member of our group, whom I met regularly at the end of each year at

Christmas party. Many thanks in particular go to Elke Buchwaldt, our kind secretary, who

was every time ready to help me to understanding the German bureaucracy.

My special thanks go to Dr. Karsten Wieczorek for his strong supports and very

useful discussions from the beginning up to now. I also thank Dipl. Ing. Petra Hetzer and

Dr. Wolfgang Buchholtz for organizing very smooth cooperation between our group at

TU Chemnitz and AMD Saxony. Christian Golz, Martin Trentzsch, and L. Herrman

deserve a special thanks for the measurements conducted at AMD Saxony.

Furthermore, I would like to thank former and current members of AMD student

TaskForce. My special thanks go to Ivo Kabadshow, Carsten Schubert, Jan Rosam,

Markus Franke, Nico Mittenzwey and Mahyar Boostandoost for fixing computer

problems, compiling different codes, and testing the programs.

140

I also kindly appreciate DAAD (Deutscher Akademischer Austausch Dienst) for

financial support of the major part of my study. I convey special acknowledgment to Dr.

H. Liaghati and Prof. F. Rafipour for making me aware of DAAD scholarship and

encouraging me to continue my study in Germany.

My parents deserve special mentioned for their inseparable support and prayers

not only during this period of time but through my whole life. My father, Prof. Hadi

Nadimi, in the first place is the person who put the fundament of my learning character,

showing me the joy of intellectual pursuit ever since I was a child. My mother, Pari, is the

one who kindly raised me and provided me a calm atmosphere at home for learning

even in the hard time.

Words fail me to express my appreciation to my wife Mona whose patience and

love, has taken the load off my shoulder. Especially as my son Saleh and after two years

my daughter Sana joined us, she proved her ability to perfectly manage our family.

Finally, I would like to thank everybody who was important to the successful

realization of this work, as well as expressing my apology that I could not mention

personally one by one.

141

Versicherung

Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und

ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus

fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich

gemacht.

Bei der Auswahl und Auswertung des Materials sowie bei der Herstellung des

Manuskripts habe ich Unterstützungsleistungen von folgenden Personen erhalten:

KEINER

Weitere Personen waren an der Abfassung der vorliegenden Arbeit nicht beteiligt. Die

Hilfe eines Promotionsberaters habe ich nicht in Anspruch genommen. Weitere

Personen haben von mir keine geldwerten Leistungen für Arbeiten erhalten, die im

Zusammenhang mit dem Inhalt der vorgelegten Dissertation stehen.

Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher

Form einer anderen Prüfungsbehörde vorgelegt.

Chemnitz,

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

Ort, Datum Unterschrift

142

Theses

1. The scaling of metal-oxide-semiconductor transistors has been the primary factor

driving improvements in microprocessor performance. However, down scaling of

devices is accompanied by some unwanted effects which generally originate from

quantum mechanical effects in low dimensions.

2. One of the unwanted quantum mechanical effects is the gate leakage current. The

gate leakage current in devices with an ultrathin gate dielectric is mainly due to the

quantum mechanical tunneling through a finite potential barrier. It increases

exponentially as the dielectric thickness decreases. The high leakage current in

transistor leads to a high power consumption in chips and high operating

temperatures, which in turn limits further downscaling of devices.

3. In order to calculate the tunneling current first the potential profiles and the charge

distribution in device can be obtained with a self-consistent solution of the

Schrödinger-Poisson equation system within the effective mass approximation. The

results show that electrons in the inversion layer populate quantized states and the

peak of charge density as well as its centroid is pushed back from the interface due

to the reflection of electronic wave functions at the Si/SiO2 interface.

4. To model the finite barrier height at the dielectric/gate interface the Schrödinger

equation should be solved with an open boundary condition. The open boundary

transforms the quantized bound states to the quasi-bound states. In contrary to the

bound states electrons can escape from quasi-bound states with finite rate which is

identified with the electron lifetime or energy broadening on each subband.

5. The lifetime of electrons on each subband could be calculated by an energy scan

procedure, where the Schrödinger equation is solved for different energies. Only

energies which result in wave functions that satisfy specific conditions correspond to

allowed subband energies. The result shows an energy broadening for each

subband as well as tiny shift in the subband energy due to the open boundary.

6. The tunneling current is calculated as a result of number of electrons on each

subband and the lifetime of electrons on that subband. Small or thin barriers result in

143

a short lifetime and a high tunneling current while high and thick barriers yield a

longer lifetime and a lower tunneling current.

7. Comparing the calculated and measured tunneling current show a very good

agreement of model and experiment. The only fitting parameter used in our model is

the “tunneling effective mass” of the electron in the dielectric layer. It is expected that

some of the material specific parameters of an ultrathin dielectric experience some

changes mainly due to interfacial layers. All such deviations are absorbed within a

varying “tunneling effective mass”.

8. The effective mass is generally incorporates the effect of a periodic atomic potential.

However in ultrathin dielectric layers the periodicity of the system is destroyed in one

direction and therefore using the standard definition of the effective mass in that

direction is questionable. In this case we believe that the effective mass reduces to a

simple parameter, the “tunneling effective mass”, which could be used as fitting

parameter.

9. The electron “tunneling effective masses”, extracted from different samples with

ultrathin oxynitride gate dielectric show that the “tunneling effective mass” increases

as the layer thickness decreases. Also adding nitrogen atoms into the oxide reduces

the electron “tunneling effective mass”.

10. Ultrathin dielectrics consist of only few atomic layers. Thanks to several

improvements in methods and increasing computational power, even systems

containing hundreds of atoms are in the operating range of atomic scale modeling.

This encourages to apply such models for the calculation of the tunneling current

through an ultrathin oxide layer in order to understand the atomic level structure of

Si/SiO2 interface as well as the influence of atomic level distortions on the tunneling

current. To this end a combination of ab initio density functional theory and Non-

equilibrium Green’s function as implemented in TranSIESTA code has been utilized.

11. An interface model structure of Si/SiO2 has been constructed and optimized within

the local density approximation of the density functional theory. A β-cristobalite SiO2

is used in the construction of the supercell. The calculation of the work of separation

shows that all different separation scenarios result in a stable Si/SiO2 interface.

144

12. The band gaps of Si and SiO2 as well as the conduction and valence band offsets

can be extracted in two different ways. In the first method the local density of states

are used, where in the second method the values extracted from the energy resolved

transmission probability of the Si/SiO2/Si system. Both methods result in very similar

values which are in very good agreement with other calculation within the local

density approximation.

13. Two similar Si/SiO2/Si structures which only differ in the thickness of SiO2 layers

have been constructed. The tunneling current calculated in both system shows the

expected thickness dependence.

14. The total energy calculation of several structures with different positioned neutral

oxygen vacancies show that the oxygen vacancies tend to sit at the interface. The

calculation of the tunneling current shows that the arm vacancy, which connects the

last Si layer in the Si lead to the first Si layer in SiO2, has the most destructive effect

on the barrier transparency. The calculation of systems with two bulk oxide

vacancies show that the tunneling current increases as the distance between two

vacancies decreases.

15. Investigating the Si-O-Si bond angles show that the first three layers at the Si/SiO2

interface suffer from deformations. In an ultrathin gate dielectrics (tox<1.5 nm) the

main part of the layer consists of such deformed oxides which leads to changes in

material specific parameters of the thin layer. This may explain the thickness

dependency of the “tunneling effective mass” in ultrathin gate oxides.

16. Applying a compressive/tensile stress along the direction perpendicular to the

Si/SiO2 interface increase/decreases such deformations and leads to a

reduction/increase of the tunneling current.

145

Curriculum Vitae Name: Ebrahim Nadimi Date of birth: August 1, 1972 Place of birth: Tehran, Iran Nationality: Iranian Marital status: Married, two children

EDUCATION June 1990 High school diploma, at Nikan high school, Tehran - Iran. June 1995 Bs.c. in electrical engineering, electronics, at electrical engineering

instituted of university of Tehran. April 1999 Ms.c in electrical engineering, electronics, at electrical engineering

instituted of university of Tehran. Subject of thesis: “Investigation of hole dynamics in the valence band of

quantum wells”.

EMPLOYMENT Jun 1996 – Sep 2002 Developer engineer at HAD Co. Tehran - Iran Sep 2002 – Sep 2006 Awarded with DAAD scholarship. Since Apr 2003 Ph.D student and research assistant at opto- and solid state

electronics professorship of Professor Dr. rer. nat. C. Radehaus at Chemnitz University of Technology.

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