simulationofthinsiliconlayers: impactof orientation

Master Thesis

Completed at Fraunhofer ENAS, Chemnitz

Simulation of Thin Silicon Layers: Impact ofOrientation, Confinement and Strain

Thomas JosephMatriculation number: 356637

Technische Universität ChemnitzFaculty of Electrical Engineering and Information Technology

January 17, 2018

Supervisor:

Fraunhofer ENAS: Dr. Jörg Schuster

Examiner:

Fraunhofer ENAS: Prof. Dr. Stefan E. Schulz

iii

AcknowledgmentsMy deepest gratitude to Dr. Jörg Schuster and Fraunhofer ENAS for providing the computationenvironment for this study. I am also much obliged to Florian Fuchs, without whom thisstudy would have been impossible. I also thank him for his insights and invaluable assistanceon this endeavour. Getting through this work required more than academic support and Ihave many people to thank for this. Especially my colleagues and Max Huber for sharing hisinsights on the study. Thanks to all the fellow members of the simulation team for their helpand support. A special thanks to Quantumwise and Synopsys for providing with numeroussample templates upon which much of this study is based. With deep feelings of respect andgratitude, I would like to extend my thanks to my family and friends for their whole-heartedsupport and encouragement.

List of Symbols v

List of Symbols

Symbol Description Unit

A Area m2

Cox Gate oxide capacitance per unit area F m−2

D Diffusion coefficent m2 s−1

Eelec Electronic energy JEg Band gap JE Energy of the system JF Electric field V m−1

Ids Drain to source current AIoff Subthreshold leakage current AM Nuclear mass (reduced mass) kgNc Conduction band density of states J−1 m−2

Nd Donor ion concentration in bulk silicon m−3

Nv Valence band density of states J−1 m−2

Qb Bulk charge per unit area C m−2

R Nuclear coordinate —T0 Transmission coefficient —T Temperature KVds Drain to source voltage VVfb Flatband voltage VVgs Gate to source voltage VVr Reverse bias voltage VVth Threshold voltage VV Potential energy JXdm Maximum depletion width under the gate mZ Nuclear charge C

Ψ Many particle wave function m−dN/2

`g Gate length of the FDSOI m`sp Spacer height m`tot Total length of the FDSOI mε0 Dielectric permittivity of vacuum F m−1

εsi Relative permittivity of silicon —η Minority carrier lifetime s~ Reduced Plank constant. Value: 6.626× 10−34 J sπ Mathematical constant pi. Approximated value: 3.14159 —e Charge of electron. Value: 1.602× 10−19 CkB Boltzmann constant. Value: 1.38× 10−23 J K−1

µ Carrier mobility m2 V−1 s−1

φsi Surface potential at onset of strong inversion Vφ Barrier height eV

vi List of Symbols

Symbol Description Unit

ψelec Electron wave function m−dN/2

ψnuc Nuclear wave function m−dN/2

ψ Wave function m−d/2

ρ Electron density m−3

τ Average scattering time sζ Space charge generation lifetime sf Fermi distribution —k Wave vector m−1

m∗ Effective mass —mr Reduced mass kgm∗conductivity Conductivity effective mass —m∗density Density of state effective mass —m∗l Longitudinal effective mass —m∗t Transverse effective mass —m∗z Quantization mass —m Electron mass (reduced mass) kgni Intrinsic carrier concentration m−3

q Electric charge Cr electronic coordinate —tbox Box thickness mtil Thickness of the intermediate layer mtox Thickness of the gate oxide mtsp Spacer width mt Thickness of the 2D material m

Acronyms vii

Acronyms

Acronym Description

ATK Atomistix TooKitDFT Density Functional TheoryDOS Density of StatesDZP Double Zeta PolarizedFCC Face Centered CubicFDSOI Fully Depleted Silicon On InsulatorFET Field Effect TransistorFFT Fast Fourier TransformFHI Fritz-Haber InstituteFINFET Fin Field Effect TransistorGGA Generalized Gradient ApproximationLBFGS Limited memory Broyden-Fletcher-Goldfarb-ShannoMGGA Meta Generalized Gradient ApproximationMOSFET Metal Oxide Field Effect TransistorPDSOI Partially Depleted Silicon On InsulatorSIESTA Spanish Initiative for Electronic Simulations with Thousands of AtomsSIMOX Seperation by Impantation of OxygenSOI Silicon On InsulatorSOIFET Silicon On Insulator Field Effect TransistorTCAD Technology Computer Aided DesignWKB Wentzel-Kramer-Brillouin

List of Figures ix

List of Figures

1.1 Key trends of the semiconductor industry . . . . . . . . . . . . . . . . . . . . 1

2.1 Schematic representation of MOSFET and SOIFET structures . . . . . . . . 32.2 Smart Cut process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Output characteristics of a SOIFET . . . . . . . . . . . . . . . . . . . . . . . 52.4 Mindmap describing applications of electronic structure theory in various domain 7

3.1 Multiscale modelling approach for electronic devices . . . . . . . . . . . . . . 153.2 Simulation models of silicon slabs . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Schematic illustration of the ATK simulation process . . . . . . . . . . . . . . 173.4 Schematic of ntype FDSOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Schematic illustration of the TCAD simulation process . . . . . . . . . . . . . 20

4.1 Electronic band structure of bulk silicon . . . . . . . . . . . . . . . . . . . . . 254.2 Brillouin zone of bulk silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Electronic band structure of strained bulk silicon . . . . . . . . . . . . . . . . 274.4 Constant energy surface of conduction bands in bulk silicon . . . . . . . . . . 284.5 Impact of strain on bandgap and conductivity effective mass of bulk silicon . 294.6 Impact of confinement on dispersion relation in silicon . . . . . . . . . . . . . 304.7 Bandstructure of 1 nm and 2 nm thick silcon slab with confinement in the 110

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.8 Plot of band gap as a function of thickness for 100, 110 and 111 cleaved

silicon slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.9 Energy gap from inplane and projected subbands as a function of slab thickness

for confinement in 100 and 110 direction . . . . . . . . . . . . . . . . . . 324.10 Impact of confinement on conductivity effective mass in silicon . . . . . . . . 334.11 Brillouin zones of various cleaved silicon slabs . . . . . . . . . . . . . . . . . . 344.12 Electronic band structure of strained 100 cleaved silicon slab with thickness

of 2 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.13 Electronic band structure of strained 110 cleaved silicon slab with thickness

of 2 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.14 Plot of band gap as a function of slab thickness and strain for 100 cleave and

110 cleave silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.15 Difference between direct and indirect band gap for 100 cleaved silicon slab

as a function of strain and confinement. . . . . . . . . . . . . . . . . . . . . . 394.16 Difference between direct and indirect band gap for 110 cleaved silicon slab

as a function of strain and confinement. . . . . . . . . . . . . . . . . . . . . . 404.17 Plot of conductivity effecitve mass as a function of slab thickness and strain

for 100 cleave and 110 cleave silicon . . . . . . . . . . . . . . . . . . . . . 41

5.1 Device characteristics of the n type FDSOI under study . . . . . . . . . . . . 435.2 Impact of SOI thickness of an n type FDSOI on mobility and threshold voltage 44

x List of Figures

5.3 Comparison of the device-off characteristics of the n type FDSOI simulatedusing TCAD’s inbuilt band gap and band gap extracted from DFT . . . . . . 45

5.4 Comparison of the device-on characteristics of the n type FDSOI simulatedusing TCAD’s inbuilt band gap and band gap extracted from DFT . . . . . . 46

5.5 Comparison of the device-off characteristics of the n type FDSOI simulatedusing TCAD’s default bulk parameters (Eg, m∗) and confined parameters (Eg,m∗) extracted from DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.6 Comparison of the device-on characteristics of the n type FDSOI simulatedusing TCAD’s default bulk parameters (Eg, m∗) and confined parameters (Eg,m∗) extracted from DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

List of Tables xi

List of Tables

3.1 Crystallographic description of k vectors . . . . . . . . . . . . . . . . . . . . . 183.2 Device dimensions of the n type FDSOI . . . . . . . . . . . . . . . . . . . . . 193.3 Meshing strategy for device simulation . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Electron and hole effective mass for bulk silicon in <100> direction . . . . . . 264.2 Bulk silicon conductivity and DOS effective mass for electron along <100>

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Quantization mass for projected and in-plane subband ladders of the silicon

slabs under study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Fit parameters for the emperical band gap model . . . . . . . . . . . . . . . . 324.5 Principle effective masses in cleaved silicon . . . . . . . . . . . . . . . . . . . . 35

Contents xiii

Contents

Acknowledgments iii

List of Symbols v

Acronyms vii

List of Figures ix

List of Tables xi

1 Introduction 1

2 Theoretical Background 32.1 Silicon on Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Smart Cut Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 SOIFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Electronic Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . 102.2.4 Hartree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.5 Hartree Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.6 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Multiscale Model 153.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Electronic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Model System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2.1 Effective Mass Evaluation . . . . . . . . . . . . . . . . . . . . 183.2.2.2 Electron Dispersion Evaluation . . . . . . . . . . . . . . . . . 18

3.3 Numerical Device Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.1 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.2 Model System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.2.1 Sentaurus Structure Editor . . . . . . . . . . . . . . . . . . . 203.3.2.2 Sentaurus Device . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Electronic Structure Simulation 254.1 3D Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Bulk Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Impact of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

xiv Contents

4.2 2D Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.1 Impact of Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Impact of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Numerical Device Simulation 435.1 Device Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Impact of Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Summary and Outlook 51

Bibliography 57

1

1 IntroductionThe Fig.(1.1) shows the key trends in the semiconductor industry. Microelectronic industry

is always fueled by the need for faster and smaller devices [1,2]. Gordon E. Moore, co-founderof Intel and Fairchild Semiconductor, made an empirical observation that the componentdensity would double approximately every two years [3]. For a long time geometric scalingtheory has served this purpose of doubling the components [4]. Over the years, from 2002to 2010, a steady trend is maintained by the Moore’s Law as shown in Fig.(1.1). However,from 2012 onwards the trend saturates. This represent the onset of the death of Moore’s Law.This is because the geometrical scaling has come to its limits as certain feature sizes havereached technologically possible smallest dimension. Moreover, as device dimensions enterthe deep-submicrometer regime, many parasitic effects like drain induced barrier lowering,velocity saturation, hot carrier generation, and subthreshold leakage limits further geometrical

*Forecast Source: Linley Group

2002

2004

2006

2008

2010

20122014* 2015*

180 nm

130 nm

90 nm

65 nm

40 nm

28 nm20 nm

16 nm

2.6 M4.4 M 7.3 M

11.2 M

16 M

20 M 20 M19 M

Figure 1.1: Key trends of the semiconductor industry. Note that the green blocks representchipset generation with the year of manufacture marked at the bottom. Bear inmind, that the technology node is marked above the year of manufacture. Alsothe total transistor bought per unit dollar in million is given on top of the chipset.Moreover, the trend of Moore’s law is marked by the grey plot.

2 CHAPTER 1. INTRODUCTION

scaling [5]. Therefore, the microelectronic community has come up with equivalent scalingtechnique which uses novel concepts for further scaling. Some of which include use of newdesign architecture, alternative materials, use of strain on the active device, metal gate andhigh-k dielectrics and low-k dielectrics for back end of line etc.One of the approach from design perspective of equivalent scaling technique is the use of

Fully Depleted Silicon On Insulator (FDSOI) technology [6]. FDSOI features a fully depletedbody which is isolated by an insulator box. This introduces better electrostatics, lower leakagecurrent and better channel control in comparison to bulk planar transistors [6, 7]. The deviceperformance is heavily influenced by the orientation, confinement, and strain in the ultra-thinbody [8]. The conventional body thickness of the FDSOI is in the range of 4 nm to 6 nm. Forsuch small body thicknesses, silicon loses its translational symmetry in confinement directionand the lattice becomes two dimensional [2].

Understanding the material at an atomistic level is essential, especially when size quantiza-tion plays a role. This has been demonstrated for quantum dots [9, 10] and nanowires [11, 12].Despite the high relevance in technology, there exists only a limited number of studies onthe electronic structure of two dimensional silicon [12–14]. Here, the study focuses on theimpact of orientation, confinement, and strain on the electronic properties of thin silicon slabs.Up to now, most of the literature only deals with the 100 cleaved slabs. However, thisstudy includes other low index orientations like 110 and 111 cleaved slabs. Moreover,the impact of strain on confined structures are also investigated. The influence on bandgap and type of band gap (i.e. indirect or direct) is studied systematically in this work.A detailed investigation of confinement and strain on the band gap indicates the possibleuse of silicon in optical applications [15, 16]. Besides, a methodical study of the impact oforientation, confinement and strain on the effective mass is also analysed. From comparingthe material properties of two dimensional silicon with bulk silicon, it is quite clear that amultiscale modelling approach is required for simulating electronic transport properties ofFDSOI structures. Therefore, the study carries out a multi-scale modelling of the n typeFDSOI by importing band gap and effective mass of confined two dimensional silicon into thenumerical simulator. The whole documentation is organised into the following chapters:

• Theoretical Background: This chapter briefs on the foundational theory of Silicon OnInsulator Field Effect Transistor (SOIFET). In addition, a basic build up on the electronicstructure theory especially on Density Functional Theory (DFT) is introduced.

• Multiscale Model: Here the electronic model and the numerical device model is discussedin detail. The slab structures and the associated simulation parameters for the electronicstructure study is summarized. Also a brief description of the numerical device model andthe associated model parameters for multiscale modelling is given.

• Electronic Structure Simulation: Illustrates and discusses the impact of orientation,confinement, and strain on the electronic structure of three dimensional and two dimensionalsilicon. Additionally, a systemic study on band gap and effective mass is also summarized.

• Numerical Device Simulation: Summarizes the electrical Characterization of the n typeFDSOI. Furthermore, a comparison of the electrical characteristics from the macroscopicmodel to the multiscale model is outlined.

• Summary and outlook: This chapter summarizes and concludes the electronic structurestudy of silicon and the numerical device simulation of n type FDSOI using a multiscalemodelling approach.

3

2 Theoretical Background2.1 Silicon on Insulator

2.1.1 Introduction

Monolithic integration, realized by Noyce and Kilby, revolutionized electronics and theworld around us [6]. The first Field Effect Transistor (FET) was patented by Lilienfeld in1930 [17]. Thirty years down the lane, Kahng and Attala of the Bell labs improvised on theFET design to come up with the Metal Oxide Field Effect Transistor (MOSFET) [18]. Sincethen, the MOSFET has been incorporated into integrated circuits and has played a key role inthe electronic industry. Fueled by the need for faster and smaller devices, the microelectronicindustry introduced geometrical scaling technique. However, due to technological limitationsin the recent years, the geometrical scaling has come to an end [5]. To circumvent this,the researchers popularized equivalent scaling technique. One of the corner concepts inthe scaling technique involves use of novel design architectures. Fin Field Effect Transistor(FINFET), nanowire transistor and SOIFET are some of the prime examples of the currentlymanufactured and researched transistor designs.

Fig.(2.1) illustrates a schematic representation of the planar bulk MOSFET and the SOIFET.The major difference is the presence of a buried oxide layer in the SOIFET structure. ForMOSFET integration, multiple transistor could be built into the same silicon substrate byisolating the adjacent device by reverse biased pn junction [6]. However, with rapid progressand from the associated scaling of the device, it has become increasingly clear that junctionisolation is not the best approach. It introduces extra capacitance and can have severe leakagecurrent at high ambient temperatures which can diminish the isolation [6]. In order to evadethese issues, the industry launched the Silicon On Insulator (SOI) based FETs. Here, thenorm of forming isolation from a pn junction is replaced by a dielectric. The SOI structure iscomposed of single crystalline silicon separated from the bulk substrate by a layer of silicondioxide. Therefore special processes like Seperation by Impantation of Oxygen (SIMOX) andwafer bonding process are used for manufacturing SOI. Smart Cut method and NanoCleaveare some of the popular techniques under wafer bonding process.

DrainSource

Gate

e

Si Substrate

STI

(a) MOSFET

DrainSource

Gate

eUltra Thin Buried Oxide

Si Substrate

STI

(b) SOIFET

Figure 2.1: Schematic representation of MOSFET and SOIFET structures.

4 CHAPTER 2. THEORETICAL BACKGROUND

2.1.2 Smart Cut Process

Wafer

Surface Oxidation

H Implant

Flip and Bond

Bubble Formation

Break

CMP and CutSOI Wafer

Handle Wafer

Figure 2.2: Smart Cut process.

Fig.(2.2) depicts an illustration of theSmart Cut process which revolutionized theSOI fabrication process. The fact is that itemploys blistering effect from ion implanta-tion to its advantage. High dose of hydrogenor inert gases implanted into the wafer canproduce swelling, flaking and exfoliation ofwafer [6, 19]. In silicon blistering can be in-duced in two ways. One of the process is byperforming high dose implantation. Blister-ing in this case becomes visible with dosesabove 2× 1017 cm−2. Another technique toinduce blistering is by annealing a medium-dose implanted wafer with dose in the rangeof 2× 1016 cm−2 to 2× 1017 cm−2. Here, atemperature budget of 400 C to 600 C isrequired for blistering [6, 19].

To form SOI wafers, the process comprisesof four main steps [6, 19]. The initial processinvolves cleaning of two wafers and oxida-tion of one of the wafers. The thickness ofthe buried oxide layer depends on the oxida-tion process and is usually of several nanome-ters thick [6, 19]. Following the oxidationprocess, hydrogen ions of dose greater than5× 1016 cm−2 are implanted into the waferthrough the oxide layer. In the second step,a cleaning process is done on both wafersin order to eliminate particles or contamina-tion and to make both surfaces hydrophyllic.After the cleaning process, the wafer withthe oxidized layer is bonded onto the secondwafer (handler wafer) via hydrophyllic waferbonding [6, 19]. The third step of the SOIfabrication technique is the annealing processwhich is carried out at 400 C to 600 C. Thisleads to microcavities due to segregation ofhydrogen molecules [6, 19]. Hence, the pres-sure builds up to a point of fracture resultingin cleaving of the bonded wafer above the oxide. The concept here is to suppress blisteringand redirect the pressure so to create a lateral cleave with the help of a handler wafer. Thefinal step involves, the polishing of the SOI wafer to eliminate the microstructures formedfrom the cleaving process [6, 19]. The primary advantage of the Smart Cut process is thatthe leftover wafers after the cleaving process can be reused as shown in Fig.(2.2). Anotherstrength is that it uses conventional manufacturing equipments. Also the process is highlyscalable and is suited for high volume manufacturing [6, 19].

2.1. SILICON ON INSULATOR 5

2.1.3 SOIFET

Lower leakage current, better electrostatics, lower short channel effects and possibility of aback bias are some of the major advantages of SOIFET over MOSFET. In addition, replacingthe pn junction isolation with the dielectric reduces the source drain capacitance. This inturn leads to faster transistor switching [6, 19].The SOIFET is classified into two types: Fully Depleted Silicon On Insulator (FDSOI)

and Partially Depleted Silicon On Insulator (PDSOI) [6, 19]. Fully depleted means thatthe depletion region covers the whole transistor body and does not extend with the gatebias. In MOSFET, the applied gate voltage is proportional to the sum of the bulk chargeand the inversion charge. However in an FDSOI since the body is depleted, the appliedvoltage is proportional to just the inversion charge. Therefore, excellent coupling betweenthe inversion charge and the gate bias produces improved subthreshold and drain currentcharacteristics [6, 19]. In partially depleted the depletion region does not cover the wholetransistor body. Hence, there exists a neutral region between the inversion layer and the buriedoxide layer. As a result, the advantage of coupling effect from gate voltage to inversion chargeas seen in FDSOI is absent in PDSOI devices. Moreover, the presence of the neutral regioncauses floating body effect to occur [6, 19]. Fig.(2.3) illustrates a side by side depiction of theoutput characteristics of the PDSOI and the FDSOI. Here, the impact of the floating bodyeffect which appears as a kink in the characteristics of the PDSOI is clearly marked. This isdue to the majority carrier, which is generated by impact ionization and gets collected in theneutral region. Hence, the body potential is raised and thereby lowering the threshold voltagewhich results in an increased drain to source current in the device. A similar feedback for draincurrent from floating body effect for weak inversion and high drain bias, can create hysteresis.When moving from weak inversion to strong inversion, majority carrier can get collected atthe neutral region. Now with reducing from strong inversion back to weak inversion, the drainto source current does not follow the same path. This is due to the positive feedback fromthe majority carrier resulting in a hysteresis. Floating body effect can also cause transientvariations of body potential, threshold voltage and current [6, 19].

0 Vds

Ids

(a) FDSOI0 Vds

Ids

(b) PDSOI

Figure 2.3: Output characteristics of a SOIFET. The red patch marks the kink effect in PDSOIarising from the floating body effect.


2.2 Electronic Structure Theory

2.2.1 Introduction

The concept of electronic structure theory originates in 1890 with the discovery of electron- a fundamental particle in matter. Later, the experiments from J.J. Thomson in 1897 atthe Cavendish Laboratory confirmed electrons to be negatively charged, with a charge tomass ratio similar to what was found by Lorentz and Zeeman [20, 21]. With the advent ofconclusions on atomic nulcei being positively charged, classical physics failed to describe theatom. This paved the path for the formulation of quantum mechanics which is centered onthe concept of wave-particle duality [20,21].

Although Bohr’s model was fundamentally incorrect, it paved path for the discovery of lawsof quantum mechanics especially works of de Broglie, Heisenberg, Schrödinger etc. Electronswere the testing ground for the new quantum theory. The famous Stern Gerlach experiment onthe deflection of electron in magnetic field validates the applicability [20, 21]. Simultaneously,Crompton based on his observations of convergence of xrays proposed that electrons havean intrinsic moment. Later Goudschmidt and Uhlenbeck, who noted Cromptons hypothesis,formulated a coupling between orbital angular momentum and electron spin. One of thesuccess of the quantum theory that came in 1925 was in explaining the periodic table ofelements based on Pauli’s exclusion principle [20,21]. Further in 1926, Fermi extended theprinciple to the general formula for the statistics of non-interacting particles. The generalprinciple is that wavefunction for many identical particles should be either symmetric orantisymmetric when two particles are exchanged [20,21]. This principle was first discussedby Heisenberg and Dirac in 1926. Further progress led to understanding the notions behindformation of chemical bonds and the laws of statistical mechanics. It was during these timesthat the Schrödinger equation, the corner stone of quantum mechanics, was published [20, 21].The first step towards understanding electrons in crystal come up with the formulation

of electronic bands in crystals explained by Felix Bloch. Bloch in his thesis formulated theconcepts of electron bands in crystals based upon what has come to be known as "Blochstheorm" [20, 21]. He discovered that the wavefunction in a perfect crystal is an eigen state ofthe crystal momentum. This theory resolved one of the key problem in Pauli-Sommerfieldtheory of conductivity of metals. It was only later that the consequence of band theory wasrecognized [20,21]. Based on band theory and Pauli’s exclusion principle, the allowed statesfor each spin can each hold one electron per unit cell of the crystal. With the work of A.H.Wilson, the foundation was laid for classification of all crystals into metal, semiconductorsand insulators [20,21].

The first quantitative calculations undertaken on multi-electron systems were for atoms byD.R Hartree and Hylleraas. Hartree’s work pioneered the self consistent field method whereone solves the equation numerically for each electron moving in a central potential formedby the nucleus and the other electrons [20, 21]. This technique serves as the backbone formany of the modern computational methods in use today. However, Hartree’s approach washeuristic and later Fock reworked the theory by introducing antisymmetrized determinantwavefunctions which later came to be known as Hartree-Fock method [20,21]. Many of thelater works based on perturbation theory originates from the work of Hylleraas, which providedaccurate solutions for the ground state of two electron systems as early as 1930 [20,21]. LaterKohn and Hohenberg came up with the DFT method which is several times scalable in termsof computational effort. The prime idea here was to self-consistently solve electron density andenergy unlike Hartree-Fock where wavefunction and energy is solved self-consistently [20, 21].

2.2. ELECTRONIC STRUCTURE THEORY 7

Electronic StructureTheory

Ground StateProperties

—

AtomicStructure

Forces ReactionBarriers

PhaseDiagrams

Excited StateProperties

— Spectroscopy

ResponseFunctions

ConductivityHeat

Capacity

First PrinciplesDescriptionof Matter

—

Super-conductivity

BondBreaking andFormation

StrongCorrelations

Figure 2.4: Mindmap describing applications of electronic structure theory in various domain


2.2.2 The Schrödinger Equation

The Schrödinger equation in quantum mechanics is analogous to Newton’s second law inclassical physics. That is it plays the role of Newton’s law and conservation of energy inclassical mechanics. Even though it predicts the behaviour of a dynamic system, the result ofa specific measurement on the wavefunction is uncertain. It is a partial differential equationwhich describes the wave function over time . This cornerstone expression from quantummechanics can describe particles by their wave function. The most general form is givenby [20,21]: [

− ~2

2mr∇2 + V (~r, t)

]ψ(~r, t) = ι~

∂ψ(~r, t)

∂t, (2.1)

where ~ is the reduced Plank constant, mr the reduced mass, V the potential energy, ~r andt are the position vector and time respectively and ψ the wave function. In Eq.(2.1), theHamiltonian H is defined as [20,21]:

H = − ~2

2mr∇2︸︷︷︸

kinetic energy

+

potential energy︷︸︸︷V (~r, t). (2.2)

The general form of the Schrödinger equation involves a second derivative in space and firstderivative in time. Factorizing, allows ψ to be written in terms of product of time and spacedependent functions:

ψ(~r, t) = ψ(~r)f(t). (2.3)

After substituting Eq.(2.3) in Eq.(2.1) and reorganizing the equation one gets:[− ~2

2mr∇2ψ(~r) + V (~r)ψ(~r)

ψ(~r)

]=ι~∂f(t)

∂t

f(t)= const = E. (2.4)

In Eq.(2.4), one side of the equation is a function of ~r and the other a function of t, bothsides can only be equal if they are a constant. Solving the R.H.S to a constant:

ι~∂f(t)

∂t= Ef(t). (2.5)

One gets a simple solution for f(t):

f(t) = Ce−ιEt

~ , (2.6)

where C is a constant. From Eq.(2.4), by equating L.H.S to the constant E, one can formulatethe time independent Schrödinger equation:[

− ~2

2mr∇2 + V (~r)

]ψ(~r) = Eψ(~r), (2.7)

Hψ = Eψ. (2.8)


A system of electrons and nuclei can be represented using the time independent form of theSchrödinger equation as:−∑

I

~2

2M I∇2

I︸︷︷︸kinetic: nuclei

kinetic: electrons︷︸︸︷−∑

i

~2

2mi∇2

i +1

2

∑I

∑J

I 6=J

1

4πε0

ZIZJ

|RI −RJ|︸︷︷︸potential: ion-ion interaction

+

potential: electron-electron interaction︷︸︸︷1

2

∑i

∑j

i 6=j

1

4πε0

e2

|ri − rj|+

∑i

∑J

1

4πε0

eZJ

|ri −RJ|︸︷︷︸potential: electron-ion interaction

Ψ = EΨ. (2.9)

Where m,M are the reduced mass of the electron and nuclei respectively, Z the nuclear charge,and r, R the electron and nuclear coordinates respectively. Analytically solving Eq.(2.9)is not possible. In practice, it is only possible to solve the Schrödinger equation for smallsystems like: particle in a box, harmonic oscillator and hydrogen atom. The electron-electroninteraction term alone is complex enough to deter solving the equation analytically. Sincethe Schrödinger equation is a partial differential equation, it may seem possible to solveit numerically on a grid. But this naive approach fails due to the high dimensionality ofΨ (Ψ(r1, r2, ...rNe ,R1,R2, ...RNn)). Over the years, research has led to various approaches tosolve the equation:

• Wave function based methods

– Quantum Chemistry: These methods involve successively improving the accuracy ofthe approximated wave function in a systematic manner. So far, these methods are mostlyemployed for finite systems like clusters, molecules etc. Hartree-Fock, Mjøller-Plessetperturbation theory, Coupled Cluster are some of the methods falling in this category.

– QuantumMonte Carlo: These methods rely on the stochastic solution of the Schrödingerequation. Both finite and periodic systems can be calculated using these methods. Wellknown methods from this category are Variational and Diffusion Monte Carlo. As thename suggests, the Variational Monte Carlo method employs the variational principleto solve the Schrödinger equation. On the other hand, Diffusion Monte Carlo employsGreen’s function to solve the Schrödinger equation.

• Density based methods: Density based methods reformulated the problem in terms ofthe electronic density. These methods can be employed for both finite and periodic systems.These methods have a wide range of functionals to chose from. The challenge is selectingan appropriate and systematically improvable approximations for the functional.

• Green’s function based method: The Many-body perturbation theory methods are oftenemployed in quantum chemistry and solid state physics and is improvable systematically.The popular methods from this class are GW, T-Matrix, BSE and FLEX.


2.2.3 The Born-Oppenheimer Approximation

The Born-Oppenheimer approximation or the clamped nuclei approach enables wave functionof the system, Ψ to be separated into electronic and nuclear components [22–24].

Ψ = ψelec · ψnuc, (2.10)

where ψelec is the electronic wave function and ψnuc the nuclear wave function. Thus, theapproximation simplifies the computation of wave function and energy of an average system.The approximation is justified by the fact that the nuclei are significantly heavier than electrons.Thus, nuclei react substantially slower than electrons to external perturbation. With theclamped nuclei approach the kinetic energy term of the nuclei becomes zero and potentialdue to ion-ion interaction results in a constant [23–25]. So applying the Born-Oppenheimerapproximation on Eq.(2.9) reduces the total hamiltonion to the so called electronic hamiltonian,Helec, expressed as [23,24]:

Helec = −∑

i

~2

2mi∇2

i︸︷︷︸kinetic: electrons

+

potential: electron-electron interaction︷︸︸︷1

2

∑i

∑j

i 6=j

1

4πε0

e2

|ri − rj|+

∑i

∑J

1

4πε0

eZJ

|ri −RJ|︸︷︷︸potential: electron-ion interaction

, (2.11)

Helecψelec = Eelecψelec. (2.12)

Where Eelec is the electron energy. This electronic energy contributes a potential term to themotion of the nuclei which is described by the nuclear wavefunction, ψnuc [24].

−∑

I

~2

2M I∇2

I︸︷︷︸kinetic: nuclei

+Eelec +1

2

∑I

∑J

I 6=J

1

4πε0

ZIZJ

|RI −RJ|︸︷︷︸potential: ion-ion interaction

ψnuc = Eψnuc. (2.13)

2.2.4 Hartree Theory

In 1928, D.R. Hartree proposed a method to solve the wave function and energy of multiple-electron system, based on fundamental principle: the hartree method. This method involvessimplifying the interacting multiple-electron system to a non-interacting multiple-electronsystem. Thus, enabling the electronic wave function to be expressed as a product of oneelectron functions called the hartree product [23,26,27]. This can be better explained usingthe Helium system [28]:

Hψ(r1, r2) = Eelecψ(r1, r2), where (2.14)

H =−~2

2m1∇2

1 +1

4πε0

eZ1

|r1 −R1|︸︷︷︸H1

+−~2

2m2∇2

2 +1

4πε0

eZ1

|r2 −R1|︸︷︷︸H2

+

V 12︷︸︸︷1

4πε0

e2

|r1 − r2|. (2.15)


From Eq.(2.15), it can be understood that the V 12 (electron-electron potential) term preventsseparation of ψelec into product of one electron functions, ψ(r1) · ψ(r2). Now, ψelec can bemade separable by splitting V 12 into sum of effective potential: V 1 +V 2. With this, Eq.(2.14)transforms to:

[H1 + V 1 +H2 + V 2]ψ(r1, r2) = [E1 + E2]ψ(r1, r2), (2.16)

Hartree assumed that each electron moves in an averaged potential of the electrostaticiterations with surrounding electrons [28]. Typically, the effective potential or the hartreepotential, V H described with the mean field approximation for each electron is given as:

V H(ri) =∑

j

j6=i

∫e2|ψj(rj)|2

4πε0|rj − ri|dr. (2.17)

Since Eq.(2.16) are one index terms, following can be generalized:

ψelec = ψ1(r1) · ψ2(r2).....ψNe(rNe) and (2.18)Hiψi(ri) = Eiψi(ri). (2.19)

The one electron wave function ψi and energy, Ei are later called orbital and orbital energyrespectively. Now from Eq.(2.17), Eq.(2.19) which describes the single particle Hartreeequation can be transformed to:−~2

2mi∇2

i +∑

J

1

4πε0

eZJ

|ri −RJ|+∑

j

j 6=i

∫e2|ψj(rj)|2

4πε0|rj − ri|dr

ψi(ri) = Eiψi(ri). (2.20)

Now with the Hartree approximation, the many particle problem is broken into set of singleparticle equation. Even with this simplification, solving the problem is still complex. This isbecause the Hartree potential connects the solution of each of the single particle states. Inother words, the potential for particle i depends on the state of the other Ne − 1 particles. Soto solve this, Hartree incorporated the self-consistent field method. The approach is essentiallyto [28]:

1. Guess a set of single particle states, ψi.

2. Compute the Hartree potential, V H for each of the states using the single particle states.

3. Recomputing the single particle states using the calculated Hartree potential.

4. Comparison of states computed from step(3) and step(1).

5. If the states from step(3) are different from step(1), new states are computed.

6. Repeat until convergence.

2.2.5 Hartree Fock Theory

The Hartree approach to approximate the many body wave function is not a good approxi-mation as it does not incorporate exchange symmetry. In 1926, Heisenberg and Dirac proposed


the electron wave function to be antisymmetric to satisfy Pauli exclusion principle [20, 21].This introduces a new electron electron interaction in the Hartree equation arising from theantisymmetrization and is called the exchange interaction. Electrons being fermions, the manybody electron wave function needs to be antisymmetric with exchange of electrons [20,21]:

ψelec(r1, r2, ...rNe) = −ψelec(r2, r1, ...rNe) (2.21)

So a new approximate wavefunction which obeys the antisymmetry principle can be formulatedusing the Slater determinant [20,21]:

ψelec(r1, r2, ...rNe) =1√Ne!

∣∣∣∣∣∣∣∣ψ1(r1) ψ2(r1) · · · ψNe(r1)ψ1(r2) ψ2(r2) · · · ψNe(r2)· · · · · · · · · · · ·

ψ1(rNe) ψ2(rNe) · · · ψNe(rNe)

∣∣∣∣∣∣∣∣ (2.22)

The Hartree-Fock equation is formulated by using the variational principle with the Slaterdeterminant. All it does is introduce a new interaction term called the exchange in the Hartreeequation [20,21]:−~2

2mi∇2

i +∑

J

1

4πε0

eZJ

|ri −RJ|+∑

j

j 6=i

∫e2|ψj(rj)|2

4πε0|rj − ri|dr

ψi(ri)

−∑

j

j 6=i

[∫e2ψ∗j (rj)ψi(rj)dr

4πε0|rj − ri|

]ψj(ri) = Eiψi(ri). (2.23)

2.2.6 Density Functional Theory

The Hohenberg-Kohn theorem forms the base for the modern DFT. The first theoremstates that the many-body electron wavefunction can be replaced by the electron groundstate density without any loss of information. The second theorem is the equivalent of thevariational principle in quantum mechanics. Unlike in Hartree method where the wave functionis the central quantity, the DFT method employs electron density instead. For an Ne electronsystem, the electron density is defined as [20,21,28]:

ρ(r) =

Ne∑j=1

∫...

∫ψelec(r1, r2, ...rNe)δ(rj − r)ψ∗elec(r1, r2, ...rNe)dr1...drNe (2.24)

where ρ is the electron density and δ is the delta distribution. In DFT energy is a functionaldepending only on the electron density instead of the wavefunction. The Hohenberg-Kohntheorem relates to any system that containing electrons that are under the influence of externalpotential (electron-ion interaction). Stated as follows:Theorem 1. For any system of interacting particles in an external potential, the total

energy is determined uniquely by the ground state density [28].Theorem 2. A universal functional for the energy in terms of the electron density can be

defined, valid for any external potential. The exact ground state energy of the system is theglobal minimum of this functional and the density that minimizes the functional is the exact


ground state density [28]. The energy functional is expressed as:

EHK(ρ) = Ek(ρ) + Vint(ρ) +

∫Vext(r)ρ(r)dr (2.25)

where EHK is the total energy functional, Ek is the kinetic energy of the system, Vint is thepotential energy component from the electron electron interaction and Vext is the externalpotential.Although the Hohenberg-Kohn theorems are powerful, they offer no way in practiceto compute the ground state electron density. Recapitulating, it has been shown that densityfunctional theory provides a clear and mathematical exact framework for the use of the electrondensity as base variable. Nevertheless, nothing of what has been derived is of practical use. Orin other words, the Hohenberg-Kohn theorems, as important as they are, do not provide anyhelp for the calculation of atomistic properties and also don’t provide any information aboutapproximations for functionals like Ek and Vint. In the direct comparison to the variationalapproach of the Hartree Fock method, the variational principle introduced in the secondtheorem of Hohenberg and Kohn is even more tricky [20,21,28]. Whereas in wave functionbased approaches like Hartree Fock the obtained energy value provides information about thequality of the trial wave function, this is not the case in the variational principle based on theelectron density. More than that, it can even happen that some functionals provide energieslower than the actual ground state energy in certain calculations. Also important to mentionis that sometimes the electron density may fulfill the condition for ground state densities, butdo not correspond to a potential. Therefore another requirement on the electron density is itsrepresentability, i.e. it must correspond to some potential [20,21,28].Several years later Kohn and Sham devised a method for carrying out DFT calculations

which retain the exact nature of DFT. This formulation centers on transforming an interactingsystem with real potential to a fictitious non-interacting system. Whereby, the fictitiouselectrons move within an effective single-particle Kohn-Sham potential [20, 21,28]. Here, theauxiliary non-interacting system is assumed to have the same ground state density as the realinteracting system. Since, the electron electron interaction are accounted in the potentialterm, the 3-N dimensional Schrödinger equation translates to N 3-dimensional single particleequations for electrons moving in an effective potential [20, 21,28]:

Hφi =

[−1

2∇2 + VKS(r)

]φi = Eiφi (2.26)

where Ei are Lagrange multipliers corresponding to the orthonormality of the Ne single-particlestates φi. Here, the Kohn-Sham potential can be expanded as:

VKS(r) = Vext + VH + VXC (2.27)

= Vext(r) + VH +δEXC[ρ(r)]

δ[ρ(r)](2.28)

where Vext is the external potential, VH is the hartree potential and VXC is the exchangecorrelation. Solving the Kohn-Sham equation is similar to the Hartree method. The equationsrequires to be solved self-consistently just like in the case of Hartree. The major differencebetween the two methods is that, the Kohn-Sham method solves the energy and electrondensity self-consistently unlike in Hartree where energy of the system is coupled with thewavefunction [20,21,28].

15

3 Multiscale Model3.1 Introduction

Multiscale electronic circuit modelling involves an integrated approach to model the systemalong a range of scales from electronic model up to the circuit simulation. It is a necessityfor optimization of device and process design [29]. A schematic of the multiscale modellingmethodologies organized in the order of system size is shown in Fig.(3.1). The model follows asequential multiscale modelling, where a modelling methodology uses precomputed parametersfrom the previous methodology [29]. In Fig.(3.1), the highest in hierarchy is the circuitsimulator. Cadence is one good example for a circuit simulator. Circuit simulator uses coupledsystem behaviour from compact models to model the system. Below the circuit simulatoris the compact models, these are composed of analytical device models which may use fitparameters or derived equations based on the numerical device simulation. BSIM is one of thepopular compact models widely used for FET. Beneath the compact models is the numericaldevice simulators, which employs material parameters to model the respective devices understudy. Some of the popular software in this category are Synopys Technology ComputerAided Design (TCAD) and Global TCAD which are based on continuum mechanics. Thematerial parameters for numerical simulation is usually extracted from first principle studies orelectronic models. At the lowest level is the electronic structure simulators (Electronic Model).Quantumwise Atomistix TooKit (ATK) [30–32] and Quantum Espresso are couple of thepopular software from this class. The current study focuses on the lowest two methodologies.Here, FDSOI transistors are simulated using Synopsys TCAD with material parameters likeband gap and effective mass of confined SOI. The material paramters for confined SOI arecomputed using DFT from Quantumwise ATK [30–32].

System Size

Parameter

Circuit

Simulation

Compact

Model

System

Behavior

Parameter/

Derived Equations

Numerical

DeviceSimulation

Electronic

Model

Figure 3.1: Multiscale modelling approach for electronic devices.

16 CHAPTER 3. MULTISCALE MODEL

3.2 Electronic Model

3.2.1 Design Parameters

Fig.(3.2) depicts the system under study. For simulation, three silicon orientations areinvestigated 100, 110, and 111 respectively. The investigated structure is confined inone direction and periodic in the other two. These cleaved two dimensional structures areformed from geometrically optimized bulk silicon. With the chosen simulation parameters, thelattice constant of bulk silicon is optimized to 5.47 nm before cleaving. The confinement of thematerial is realized by vacuum padding of 2 nm in the confinement direction [31]. The cleavedstructures are hydrogen passivated forming dihydride Si-H with a bond length of 0.148 nm atthe surface. This suppresses surface states in the electronic structure calculation. After thecleave, an initial geometry relaxation is performed to optimize the atomic positions of theslab structure. The simulation is carried out for all three orientations of slab thicknesses. In

xyz[110][110]

[100]

(a) 100 cleave

xyz[001][110]

[110]

(b) 110 cleave

xyz[110][110]

[111]

(c) 111 cleave

Figure 3.2: Simulation models of silicon slabs with a thickness of 1 nm in a vacuumpadded bounding simulation cell. Note that the simulation box is notdrawn in scale to the slab structure. The black spheres denote hydrogenpassivation of the surface. Bottom figures represent the isometric viewof silicon cleaves while the figures on top show the respective top view ofthe slabs. The structures shown here are the thrice repeated simulationunit cell in x and y directions. The cells marked in red shows the unitcell used for simulation.

3.2. ELECTRONIC MODEL 17

addition, uniaxial and biaxial strain calculations for all three orientations are also performed.Strain is applied along x axis and y axis for biaxial case while for uniaxial scenario, strainalong x axis is only investigated.

3.2.2 Model System

Fig.(3.3) shows the schematic of the simulation process. The initial step involves relaxingof bulk silicon with an unconstrained unit cell. This is followed by the optimization of thecleaved structures, whose unit cell has been unconstrained in the confinement direction. Next,optimized cleaved structures are strained and re-optimized with the unit cell unconstrainedin the confinement direction. The final step involves, a band structure calculation on theoptimized strained structures.

The electronic structure simulation is performed using DFT [33] implemented in Quantum-wise Atomistix TooKit (ATK) 2015.1 [30–32]. For this study, a Double Zeta Polarized (DZP)basis set of Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA)type orbitals with norm-conserving Fritz-Haber Institute (FHI) pseudopotentials is used. It iswell known that DFT provides inaccurate band gap information. To circumvent this, a MetaGeneralized Gradient Approximation (MGGA) potential of Tran and Blaha is employed [34].This approach includes a empirical parameter, which can be used to fine-tune the band gap.It is set to 1.04774 to reproduce the experimental bulk silicon band gap of 1.12 eV. For theelectronic structure simulation, reciprocal space is sampled using 15x15x1 k-points. Thepoisson equation is solved using an Fast Fourier Transform (FFT) based method, where thereal space grid spacing is defined by an energy cutoff of 200Eh.

The simulation also involves geometry optimization of the structure after rendering cleaveand strain. The geometry optimization uses Limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm. For the optimization, Generalized Gradient Approximation(GGA) functional of Perdew, Burke and Ernzerhof is employed. This is because the simulationwith GGA over MGGA gives a lattice constant that is close to experimental result [35]. Thegeometry optimization is based on force minimization and stress minimization in the structure.As such, the simulation is carried out using a force tolerance of 0.0005 eVÅ−1 for the firsttwo optimization steps as shown in Fig.(3.3) and 0.01 eVÅ−1 for relaxing the structure afterstrain introduction. The stress tolerance is set to 0.01 eVÅ−3 for all optimization scenarios.

Optimization of bulk Si(x ,y ,z-unconstrained)

Optimization of cleaved Si(x ,y -constrained)

Optimization of strained Si(x ,y -constrained)Band structure calculation

Cleave

Strain

Figure 3.3: Schematic illustration of the ATK simulation process.


3.2.2.1 Effective Mass Evaluation

To begin, consider the case of an electron dispersion in a crystal where the energy momentumrelationship is given as [36,37]:

E =~2k2

2m∗+ constant (3.1)

where E is the energy, k is the wave vector and m∗ is the electron mass. The conductionband valleys can be effectively described by the parabolic band expression Eq.(3.1). So, bytaking the second derivative on Eq.(3.1) and rearranging the terms, the effective mass at theminima can be written as:

1

m∗=

1

~2

∂2E

∂k2(3.2)

The energy near the conduction band minima can be approximated by a parabola just likethat of a free particle [37]:

E = C1k2 + C2 (3.3)

where C1, C2 are constants. These constants can be evaluated by a polynomial fit of Eq.(3.3)on the dispersion relationship from the simulation. Now taking second derivative of E withrespect to k, Eq.(3.3) can be written as [37]:

∂2E

∂k2= 2C1 (3.4)

Now, substituting Eq.(3.4) in Eq.(3.2), the expression for effective mass can be rewrittenas [37]:

1

m∗=

2C1

~2(3.5)

3.2.2.2 Electron Dispersion Evaluation

Table 3.1: Crystallographic descrip-tion of k vectors for bulkand slab structures.

Route Bulk SlabΓ→ X [0.5, 0, 0] [0.5, 0, 0]Γ→ Y [0, 0.5, 0] [0, 0.5, 0]Γ→ Z [0, 0, 0.5] −Γ→ L [0.5, 0.5, 0] [0.5, 0.5, 0]Γ→ K [1/3, 1/3, 0] [1/3, 1/3, 0]Γ→ M [−0.5, 0.5, 0] [−0.5, 0.5, 0]Γ→ O [0.5, 0, 0.5] −Γ→ P [−0.5, 0, 0.5] −

The electron dispersion can be directly extractedfrom ATK after the electronic structure simulation un-like the effective mass evaluation. The inputs requiredfor the extraction are the points per segment and thepath along which the Energy versus momentum is tobe mapped. The points per segment represents thenumber of discrete points along a path that is to besampled. For the case study, 100 points per segmentis chosen. Now, the only concern is providing theroutes along which the dispersion relation has to begenerated. Some of the low symmetry directions re-quire to be mentioned explicitly. The complete routesemployed in the study is summarized in Tab.(3.1).Γ → K, Γ → M, Γ → O and Γ → P are the routesthat are to be specified explicitly.

3.3. NUMERICAL DEVICE MODEL 19

3.3 Numerical Device Model

3.3.1 Design Parameters

Doping [cm−3]

−2.0× 1018−5.4× 1015−1.5× 10136.0× 10122.2× 10158.2× 10173.0× 1020

tsp

t

Gate

`gDrainSource

`tot

tbox

tiltox

`sp

Figure 3.4: Schematic of the ntype FDSOI.

Tab.(3.2) and Fig.(3.4) togethersummarizes the device dimensionsof the 2D n type FDSOI understudy. The substrate material forthe n type FDSOI is composed ofsilicon. Here, the total width ofthe substrate, `tot is 74 nm. Fur-thermore, the silicon substrate isboron doped with a concentrationof 2.0 × 1018 cm−3. In design, sili-con dioxide on top of the substrateserves as the box material. For then type FDSOI, the thickness of thebox material, tbox is 20 nm. TheSOI layer with confinement in 100direction and channel oriented along [110] direction forms the active region. In simulation,the thickness of the SOI, t is varied systematically from 2 nm to 10 nm as given in Tab.(3.2).In addition, the SOI has a default boron doping of 5.0× 1015 cm−3. The source and drain aredoped with phosphorus for n type FDSOI and the doping profile is expressed as:

f(x) = 3.0× 1020 × 1

4× 10−3(1 + 10(x−0.019)

) × 1

1.5× 10−3(1 + 10(x−0.025)

) (3.6)

where x is the position along the channel. On the top central part of the active region apatterned hafnium oxide functions as the gate oxide. As shown in Tab.(3.2), the thickness ofthe gate oxide, tox is 1.8 nm. Since the gate stack is a high-k material, it requires an interfaciallayer between the high-k dielectric and the active silicon region. This buffer layer acts as abarrier against oxygen diffusion into the active silicon [38]. In this design, silicon oxynitrideserves as the interfacial layer with thickness til of 1 nm. Silicon nitride functions as the spacermaterial in the design. For easiness, the spacer is modelled as rectangular block with thickness,tsp of 6 nm. As shown in Tab.(3.2), the length of the spacer, `sp is 27 nm.

Table 3.2: Device dimensions of the ntype FDSOI.

Dimension Value [nm]`tot 74`g 24`sp 27tbox 20tsp 6til 1tox 1.8t 2...10

For simplicity, the contacts of the device is emulatedusing virtual pads placed directly on the contact points.Here, an external resistance of 100 Ω is added in seriesto the source contact. This is to mimic the influence ofmetallic and silicide contacts. For emulating the metalgate, a 50 Ω resistance is added to the virtual contact.In addition, a barrier of 4.385 eV is supplemented tothe virtual gate to account for the gate work function.To electrically characterize the n type FDSOI, theterminals are required to be biased. The gate to sourcevoltage, Vgs is biased in the regime of 0 V to 0.9 V.Similarly, the drain to source bias voltage, Vds in designtoo varies in the range of 0 V to 0.9 V.


3.3.2 Model System

Sentaurus Structure Editor• Structure• Doping• Meshing

Sentaurus Device• Output• Transfer

Sentaurus Visual• Analysis• Visualize

Figure 3.5: Schematic illustrationof the TCAD simula-tion process.

Synopsys TCAD is a software that can develop andoptimize semiconductor process technologies and devices.It uses finite element method for the characterization ofprocess and devices. The software fundamentally finds ap-proximate solution to boundary value problems consistingof partial differential equations like diffusion, transportetc [39, 40]. The technique subdivides structures into sim-pler units called finite elements using a meshing algorithm.These finite elements, each with a set of equations, aresolved to find the local state of the element. Later, globalassembly of these elements are performed to reflect thecomplete solution [41]. All device simulations for the twodimensional n type FDSOI is performed using SynopsysTCAD K-2015.06. Characterizing involves three stepsas shown in Fig.(3.5). The initial step is the modellingof the n type FDSOI structure using Sentaurus StuctureEditor [39]. This is followed by the simulation of electricalcharacteristics with Sentaurus Device [40]. The final stepis the analysis and visualization of results using SentaurusVisual [42].

3.3.2.1 Sentaurus Structure Editor

Sentaurus Structure Editor is an editor for two dimen-sional and three dimensional device structures. The script-ing language of Sentaurus Structure Editor is based onScheme, which is a LISP like programming language whichdiffers significantly from the conventional programminglanguages. The scripting language makes it very easy togenerate parameterized structures for varying design parameters [39]. Sentaurus Structure Ed-itor has three different modes: two dimensional structure editing, three dimensional structureediting and three dimensional process emulation. The process flow of the simulation can beclassified into the following steps: generating the device structure, binding of analytical dopingprofiles to respective regions, formation of virtual contacts and meshing of device for devicesimulation. The two dimensional device structures are formed geometrically using primitiveslike square, polygon etc. Complex structures can be generated by combining the availableprimitive elements [39]. The modelling of the structure is similar to making a drawing ofthe device; several basic geometrical elements are combined to form the structure. Each ofthese geometrical elements are also defined in terms of materials Eg. silicon, nitride, oxide,oxynitride etc. The Doping distributions are defined using analytical functions. The SentaurusStructure Editor supports binding of analytical functions or one dimensional profiles to arefinement window placed on predefined regions. With the virtual contact formation, a virtualpad is formed at the contact regions on the structure based on a placement window. Duringsource, drain contact formation, care must be undertaken so that the virtual contacts does nottouch the nitride spacer. If in case the virtual pads contact the spacer, it can cause erroneoussimulation results as the contact and spacer together can form a second gate [39]. The final


Table 3.3: Meshing strategy for the device simulation. Max and Min parameter indicates themaximum and minimum element size while the growth ratio represents the meshgrowth rate.

Region MaxX [nm]

MaxY [nm]

MinX [nm]

MinY [nm]

X GrowthRatio

Y GrowthRatio

Substrate `tot/4 50/8 0.005 0.005 - -Box `tot/4 tbox/4 0.005 0.005 - -SOI `tot/4 t/8 0.005 0.005 - -High-k `g/4 tox/4 0.005 tox/8 - -Interfacial `g/4 til/4 0.005 til/8 - -Channel `g/8 t/8 `g/10 0.0001 1.0 1.35

step in the process flow is the setting up of the meshing algorithm for device simulation.In Sentaurus Structure Editor the mesh generation occurs in two steps. The first step

defines the meshing strategy that is the mesh density. The second step binds the meshstrategy to a target which could be a region, material or a user defined window. Note thatdifferent regions have different roles in the device operation and as such requires differentmesh densities [39]. The meshes in Sentaurus Structure Editor is realised by the importedmesh engine Sentaurus Mesh. Sentaurus Mesh is a modular delaunay mesh generator whichgenerates high quality spatial discretizations for complex one dimensional, two dimensionaland three dimensional structures. It contains two mesh generation engines: an axis alignedmesh generator and a tensor product mesh generator [39]. The axis aligned mesh generatorproduces delaunay meshes compatible with Sentaurus Device. In one dimensional structure,the mesh contains segments. For two dimensional structure, the mesh comprises of triangleswhile for three dimensional structure the mesh contains tetrahedrons. The tensor productmesh generates meshes compatible with Sentaurus Device Electromagnetic wave solver andSentaurus Device Monte Carlo. The mesh contains rectangular elements in two dimensionalstructure and hexahedral elements for three dimensional structure [39].

The Tab.(3.3) details the meshing strategy for the device simulation. The Max parametersignify the maximum element size in a specific direction. Similarly, the Min parameterrepresents the minimum element size for a direction. The parameter which controls as tohow fast the mesh grows from minimum element size to maximum element size is given bythe Growth Ratios. In Tab.(3.3) the mesh strategy for all regions other than the channel,avoids explicitly specifying the growth ratios. In this scenario, the Sentaurus Mesh engine willautomatically choose the best possible growth ratios internally. Although a large growth ratiocan be specified, the internal algorithm can only increase the mesh spacing of adjacent cellsby a factor of two. Therefore, the structure might have sequences of cells with fixed spacingfollowed by a set of mesh cells with double-spacing, where the requested mesh spacing becomestwice the initial mesh spacing [39]. The sign of the ratio also determines the side at which thegrading starts at. A positive value for the ratio indicates the start of grading from left or topregion while a negative value means that the grading starts of at the right or the bottom [39]of the region. It is observed in Tab.(3.3), that different regions have different mesh spacings.The coarsest mesh is with substrate region where the impact from carrier is negligible. Onthe other hand, a finer mesh is required in the channel region which is essentially where thetransport of carriers takes place. The high-k gate oxide and the interfacial layers are the otherregions that require a finer mesh in the strategy.


3.3.2.2 Sentaurus Device

Sentaurus Device is a multidimensional, electrothermal, mixed-mode device and circuitsimulator. It incorporates advanced physical models and robust numerical methods forthe simulation. Terminal currents, voltages and charges are computed based on a set ofphysical device equations that describes the carrier distribution and conduction mechanisms.A real semiconductor device, such as a transistor, is represented in the simulator as a virtualdevice whose physical properties are discretized onto a non-uniform mesh of nodes [40, 43].Sentaurus Device simulation is based on semi-classical macroscopic transport models. Tosolve the transport using semi-classical theory, it requires three coupled equations: the poissonequation, the continuity equation and the transport equation. Sentaurus Device employsBoltzmann transport equation as the basic theory and uses the Monte Carlo method as awell established numerical technique to solve the Boltzmann transport equation [40,43]. Forrealistic structures the direct numerical solution of this equation by discretization of the phasespace is computationally too expensive. Thus, Sentaurus Device also has some simplifiedtransport models. These models are approximate solutions of the Boltzmann transportequation obtained by the method of moments. Using the first two moments of the Boltzmanntransport equation yields the drift diffusion equation. Hydrodynamic transport model on theother hand employs the first four moments and is a more accurate model than drift diffusiontransport model. These two are the major approximate transport models incorporated inSentaurus Device for low power devices [44].

Sentaurus Device solves the device equations, which are essentially a set of partial differentialequations, self-consistently on the discrete mesh in an iterative fashion. For each iteration,an error is calculated and the Sentaurus Device attempts to converge on a solution that hasan acceptably small error [40,43]. For this, a few settings for the numeric solver is required.This includes selection of solver type and a user defined convergence criterion. In this study,PARDISO is the chosen linear solver as it is the best 2D solver in Sentaurus Device [40,43].As for the convergence criterion, a relative error control is enabled with an error criteria of10−7. Moreover, the solver requires both the residue and the update error be small enoughto be considered as converged. The max criterion of the residue is set at 10−10, also theNewton iterations are limited by setting the Iteration factor to 24 with the NotDampedfeature set to 100. Iteration factor implies the number of Newton iterations allowed per biasstep. The NotDamped specifies the number of Newton iterations over which the residueis allowed to increase. With the default of 1, the error is allowed to increase for one steponly. It is recommended to set the NotDamped factor greater than Iterations so as to allowsimulations to continue despite the residue increasing. Here the Extrapolate feature is alsoenabled which causes the initial guess to be based on extrapolation from the previous twosteps. For most problems, Newton iterations converge best with full derivatives. Furthermore,for small-signal analysis or DC analysis using full derivatives is mandatory. For the simulationof the numerical model under study, the following models are used [40,43]:

In order to simulate the carrier transport, the Hydrodynamic model with Density Gradientcorrection is employed. The first term in the Hydrodynamic model takes into accountthe contribution from spatial variations of electrostatic potential, electron affinity and bandgap [40,43]. The other terms of the model represent the influence from gradient of concentration,the carrier temperature gradients and the spatial variation of the effective masses [40, 43]. Bydefault, the Boltzmann statistics is assumed for electrons and holes. So to enable a moreaccurate fermi statistics, the model has to be explicitly referred. The fermi statistics areimportant especially when the carrier concentration are above 1.0× 1019 in the active region


of the device [40,43]. The statistics results in better accuracy of the transport model over aslight overhead. Since the current FDSOI features has reached quantum mechanical lengthscales, the wave nature of holes and electrons cannot be neglected. The Density Gradientcorrection accounts for the quantization effects arising from the confinement [40, 43]. TheDensity Gradient model is a numerically robust and simple model in comparison to theother quantization correction models. To mimic the quantization effect in classical devicesimulation, Sentaurus Device includes a potential like correction factor to the classical densityformula [40,43]. Heavily doped regions in the semiconductor device can undergo band gapnarrowing effect. The band gap narrowing effect is ascribed to the emerging impurity bandfrom splitting of states due to high impurity concentration. This newly formed impurity bandscan overlap with the substrate bands whereby decreasing the band gap. This effect is enabledby using the Oldslotboom model with dopant concentration as the driving force [40,43].

Sentaurus Device incorporates a modular approach to describe the carrier mobilities. Themajor degradation effects are from the carrier-carrier scattering and the impurity scattering.The Thin Layer mobility model is a well-calibrated unified model that accounts for thethickness fluctuation scattering, surface phonon scattering and bulk phonon scattering [40,43].In the channel of a FDSOI, the high transverse electric field forces carriers to interact stronglywith the semiconductor-insulator interface. The Thin Layer mobility model specificallyapplies to devices with thin silicon layers as the active region. In such devices, the geometricconfinement leads to a mobility that cannot be expressed with a normal field dependentinterface degradation model [40,43]. It depends explicitly on the thickness of the active region.Thus, the Thin Layer mobility model is coupled with the Inversion and Accumulation Layermobility model to account for the proper interface degradation. This contributes influencefrom surface roughness scattering, 3D phonon scattering and coulomb scattering. If more thanone mobility model is activated for a carrier type, the different contributions are combinedusing the Mattheisens rule [40, 43]:

1

µ=

1

µ1+

1

µ2+ ...... (3.7)

where µ is the total mobility while µ1, µ2 etc are the individual contributions. In shortchannel devices, at high electric field the carrier drift velocity becomes no longer proportionalto the field, instead saturates. The contribution of this effect is included with the use of HighField Saturation model [40,43]. With this saturation model activated, the final mobility iscalculated in two steps. First the low field mobility according to Eq.(3.7) is determined. Inthe last step the final mobility is calculated as function of the low field mobility and thedriving force, which is carrier temperature for the hydrodynamic model [40].The material parameters extracted from ATK like band gap and effective mass can be

incorporated into Sentaurus Device through the parameter files. Parameter files are calibrationsheets for Sentaurus Device which contain material parameters used for simulation. Oncethe device script is setup, a companion parameter file also needs to be arranged wherein aregion or material of the device is bound to a set of material parameters specified in thefile [40, 43]. Though the band gap information can be incorporated directly, TCAD does nothave a built in feature which enables one to incorporate the effective mass information of theSOI. The best approach to work this out is through enabling strain models, especially thosethat recalculate the effective mass for mobility improvements from strain. Since the designis for unstrained n type FDSOI, switching on the strain models have slight to no impacton the device simulation. The strain based mobility model can be activated regionwise or


materialwise. Strain causes deformation of bands and shifting of energy levels. SentaurusDevice uses the Deformation Potential model to mimic the effect of strain induced changein the energy of carrier valleys or bands [40,43]. This is an apt model for devices under lowstrain. It considers change in the energy of carrier valleys or bands, caused by the deformationof the lattice, as a linear function of strain. Another means to model the effect of deformationof bands and valleys, is with the use of MultiValley model. This model accounts for the straineffect in the band structure by strain-induced change of the energy and effective masses ineach valley defined in the parameter file. Although the MultiValley model can work togetherwith any Deformation Potential model, it will still recompute the k · p valley energy shiftswith reference to the band edges defined by the deformation potential model [40,43]. Enablingthe MultiValley model will also enable the built in quantization model MLDA. Since DensityGradient model is already set up, enabling the MultiValley model can cause double countingof the correction factor. Thus, the carrier density computation from MLDA quantization ofthe MultiValley model has to be switched off explicitly [40,43]. Since the density calculationfrom MLDA is disabled, a separate strain-dependent conduction band and valence bandeffective Density of States (DOS) calculation is required and is accounted for by the DOSmodel. Once the Deformation Potential model, MultiValley model and the DOS model isset up the strain dependent mobility calculation can be enabled with the switching on theSubband model [40,43]. The Subband stress mobility model works as correction to the relaxedlow-field mobility. For the n type FDSOI under study, the low field mobility is calculated bythe Thin Layer mobility model which in turn acts as the base for the Subband stress mobilitymodel. This model has

The band gap extracted from ATK can be included in tcad simulation of the n type FDSOIby adding the confined band gap towards the end of the parameter file. Similarly, updatingeffective mass is also done through overwriting the band information in the parameter settingsof the strain based MultiValley model. Below is the parameter snippet for updating the bandgap and effective mass information for the 6 nm thick 100 cleaved silicon slab:

1 Bandgap2 Eg0 = 1.16055 3 MultiValley4 eValley"Projected"(1,0,0)(ml=0.2084, mt=0.2084, energy=0.580272e+00,

degeneracy=2, alpha=0.5)5 eValley"Inplane100"(1,0,0)(ml=0.923, mt=0.244, energy=0.611206e+00,

degeneracy=0, alpha=0.5)6 eValley"Inplane-100"(-1,0,0)(ml=0.923, mt=0.244, energy=0.611206e+00,

degeneracy=0, alpha=0.5)7 eValley"Inplane010"(1,0,0)(ml=0.923, mt=0.244, energy=0.611206e+00,

degeneracy=0, alpha=0.5)8 eValley"Inplane0-10"(0,-1,0)(ml=0.923, mt=0.244, energy=0.611206e+00,

degeneracy=0, alpha=0.5)9 hValley(0.16, 0.16, 0.16, 0.0000e+00, 1, 0.0000e+00)

10 hValley(0.49, 0.49, 0.49, 0.0000e+00, 1, 0.0000e+00)

The above definition represents the most general way of defining valleys in the MultiValleyband structure. For example, "Projected" defines a valley with the longitudinal effectivemasses ml = 0.2084 and transverse effective mass mt = 0.2084. The major axis of the band isdescribed to be oriented in the <100> direction with a nonparabolicity factor of α = 0.5 eVand a two fold degeneracy. The major difference between the parameters of the projected andinplane is mainly from the effective mass components and the degeneracy factor.

25

4 Electronic Structure Simulation4.1 3D Silicon

4.1.1 Bulk Silicon

W L Γ X W K

k

−3

−2

−1

0

1

2

3

E−

EF[eV]

Eg = 1.125 eV

C1

H2

H1L1

Figure 4.1: Electronic band structure of bulksilicon. H1, H2, L1 represents thetwo heavy hole bands and the lighthole band respectively while C1 rep-resents the lowest conduction band.

Fig.(4.1) shows the electronic structure forbulk silicon along low index orientations cal-culated using DFT. From simulation, thebulk band gap of silicon is found to be1.125 eV in comparison to the literature valueof 1.12 eV [45,46]. Similarly, the simulationmodel gives a lattice constant of 5.47 nmwhich is quite close to the literature valueof 5.43 nm [46]. In bulk silicon, the valenceband maxima lies at [000] (Γ) while the con-duction band minima is at 85% from [000]along <100> direction (Γ→ X). Thus, bulksilicon is an indirect band gap material andso shows weak optical properties [46,47]. Thecorresponding conduction band and valenceband valleys are marked in Fig.(4.1). Bulksilicon has three types of valence band val-leys heavy hole, light hole and spin orbitalcoupling respectively [45–47]. The spin orbital coupling is a relativistic effect and is notincluded in any of the DFT calculations in the study.

ky

kx

kz

[100][010]

[001]

Figure 4.2: Brillouin zone of bulk silicon. Theellipsoids are constant energy sur-faces indicating six equivalent con-duction band valleys.

Bulk silicon is depicted by two basisatoms and maybe thought of as two inter-penetrating Face Centered Cubic (FCC) lat-tice [47]. The Fig.(4.2) shows the brillouinzone for bulk silicon. The ellipsoids shown infigure represents the constant energy surfaces.This indicates the presences of six equivalentconduction band minima in bulk silicon inthe <100> direction (Γ → X) [45, 46]. Theeffective mass is a highly directional property.The effective mass calculated along the majoraxis of the ellipsoid is called the longitudinaleffective mass. Correspondingly, the trans-verse effective mass is the conduction bandeffective mass calculated along the minor axisof the ellipsoid. From the valence band per-spective, bulk silicon has two heavy hole andone light hole mass for the correspondingvalence bands valleys [45,46].

26 CHAPTER 4. ELECTRONIC STRUCTURE SIMULATION

Table 4.1: Electron and hole effective mass for bulk silicon in <100> direction.

m∗ [−]

Longitudinal (e) Transverse (e) Heavy hole (h) Light hole (h)Own work 0.920 0.208 0.32 0.21Literature [45,46] 0.98 0.19 0.49 0.16

Tab.(4.1) summarizes the effective mass extracted from simulation for bulk silicon. Inaddition, a comparison between literature values and simulation results is also depicted. FromTab.(4.1), the electron effective mass shows good conformity between literature and simulationvalues unlike hole effective mass. The longitudinal effective mass for electron predicted fromsimulation contains an error of −6 % in comparison to literature. Similarly, transverse effectivemass too from simulation carries an error of 9 % with respect to literature. In comparison toliterature, there is an error of −34 % in the simulated heavy hole effective mass. Likewise, thelight hole effective mass calculated from simulation also shows a significant error of 31 %.

Once the effective mass from the direction dependent band curvatures are determined, themasses are to be combined appropriately for different types of scenarios. The most prominentones are the conductivity effective mass and the DOS effective mass. The conductivity effectivemass is used in calculations where motion of carriers is involved. To get the conductivityeffective mass, the harmonic mean of the band curvature effective masses is calculated [46]:

m∗conductivity =3

1m∗

l+ 1

m∗t

+ 1m∗

t

(4.1)

where m∗conductivity is the conductivity effective mass, m∗l the longitudinal effective mass andm∗t the transverse effective mass. For cases which involve determining carrier concentration,the DOS effective mass is used. The DOS effective mass is formulated by taking the geometricmean of the band curvature effective masses involved [46]:

m∗density = 3

√g2m∗l m

∗tm∗t (4.2)

where m∗density is the DOS effective mass and g the degeneracy factor which in the caseof bulk silicon is six. Tab.(4.2) details the conductivity and DOS effective mass for elec-trons in bulk silicon. A comparison between the literature and values calculated fromsimulation is also given. Since the spin orbital coupling is not part of the simulation,it is not possible to calculate an accurate conductivity and density of states hole effec-tive mass. So, through out the material the study on hole effective mass is neglected.

Table 4.2: Bulk silicon conductivity and DOSeffective mass for electron along<100> direction.

Electron m∗conductivity m∗density

Own work 0.28 1.13Literature [45,46] 0.26 1.08

The conductivity effective mass for electronpredicted from simulation contains an errorof 7.6 % in comparison to literature. Simi-larly, DOS effective mass too from simula-tion carries an error of 4.6 % with respect toliterature. From these numerics, it can beconcluded that the electron effective massextracted from simulation agrees pretty wellwith the literature values.

4.1. 3D SILICON 27

4.1.2 Impact of Strain

Strain is defined as the relative change in the lattice constant of a material. Advantageousstrain can cause band warping and lifting of band degeneracy at electronic structure level.This is so because strain reduces crystal symmetry. Strain acting on any material can bedecomposed into hydrostatic strain and two types of shear strain. One type of shear strainrelates to change in length along the three axes while the other is related to the rotation ofthe axes of an infinitesimal cube [5, 48–51]. In cubic crystals like silicon hydrostatic straindoes not break symmetry and therefore only shifts the energy levels without impacting thedegeneracy. So large hydrostatic strain only results in band gap narrowing. It is the shearcomponent of the strain which causes subband splitting [5, 48–51].

M Γ L X Γ Z

k

−2

−1

0

1

2

E−

EF[eV]

−2 % biaxial ε

M Γ L X Γ Z

k

−2

−1

0

1

2

E−

EF[eV]

−2 % uniaxial ε

M Γ L X Γ Z

k

−2

−1

0

1

2

E−

EF[eV]

Unstrained

M Γ L X Γ Z

k

−2

−1

0

1

2

E−

EF[eV]

2 % biaxial ε

M Γ L X Γ Z

k

−2

−1

0

1

2

E−

EF[eV]

2 % uniaxial ε

Figure 4.3: Electronic band structure of strained bulk silicon. Both biaxial and uniaxial strainis applied along <110> direction. The red line denotes the energy of conductionband minima for unstrained bulk silicon. Note that negative values for strainindicate compressive strain while positive values stands for tensile strain.


Z

L

M

Γ

[010]

[001]

[100]

Figure 4.4: Constant energy surface of conduc-tion bands in bulk silicon. Har-monic mean of effective mass com-ponents perpendicular to the planesis taken to find the conductivity ef-fective mass along <100> direction.

Fig.(4.3) depicts the dispersion relation-ship for strained bulk silicon. The valleysalong Γ → L, Γ → M, Γ → Z represent theequivalent conduction band valleys in [100],[010] and [001] directions respectively. Nowwith the unstrained bulk silicon all the threevalleys are at the same energy level indicatingthe six degenerate conduction band valleys.The biaxial and uniaxial strain is appliedalong <110> direction. From Fig.(4.3), itcan be concluded that strain causes bothband warping and lifting of band degeneracy.With biaxial tensile strain, the subband val-leys along Γ → L and Γ → M shifts up inenergy while the valley along Γ → Z shiftsdown. In the case of biaxial compressivestrain, the shift of subbands is opposite tothat of the biaxial tensile case. That is, sub-band valleys along Γ→ L and Γ→ M shiftsdown in energy while the valley along Γ→ Zshifts up. Now with uniaxial strain, subbandvalleys along Γ → L and Γ → M shifts upin energy for tensile strain while the same

subband shifts down with compressive strain. As for the valley along Γ→ Z, for both uniaxialcompressive and tensile strain, the subband shifts down in energy.Fig.(4.5a) shows the impact of strain on band gap of bulk silicon. It can be concluded

that with both biaxial and tensile strain, the band gap narrows. This is due to the band gapnarrowing effect from strain as shown in Fig.(4.3). From the conduction band perspectivewith tensile strain, the Γ→ Z valley shifts down forming the new band minima. However forcompressive strain, it is the Γ→ L and Γ→ M valleys which shift down forming the new bandminimas. So for all scenarios, strain causes narrowing of band gap. Moreover, comparing theimpact from both biaxial and uniaxial strain, it is also observed that the band gap shows moreof a linear response to uniaxial strain. Fig.(4.5b) depicts the impact of biaxial and uniaxialstrain on conductivity effective mass of bulk silicon along <100> direction. From Fig.(4.5b) itis observed that conductivity effective mass decreases with tensile strain. As for compressivestrain, the conductivity effective mass increases. The reason for this effect is the lifting ofband degeneracy. The conductivity effective mass property depends not only on the effectivemass components but as well as on the weights of the respective components. Then againweights are a factor which depends on the carrier population. The effective mass componentsresulting in the conductivity effective mass along <100> are m∗l from Γ→ L valley, m∗t fromΓ→ Z and Γ→M valleys respectively. Now with tensile strain, Γ→ L and Γ→M valleysshift up in energy while Γ → Z valley shifts down in energy. This causes redistribution ofelectrons from Γ → L and Γ → M valleys to Γ → Z. Thus resulting in the decrease of theaverage effective mass. With compressive strain, it is the Γ→ Z valley which is at a higherenergy level than the other valleys. This causes repopulation of electrons from Γ→ Z valleyto Γ→ M and Γ→ L valleys. Thus increasing the weight of m∗l which in turn results in alarger average effective mass. Also Conductivity effective mass shows stronger response tobiaxial strain due to a larger degeneracy lift in comparison to uniaxial strain.

4.2. 2D SILICON 29

−3 −2 −1 0 1 2 3

Strain [%]

0.750.800.850.900.951.001.051.101.15

Eg[eV]

Biaxial εUniaxial ε

(a)

−3 −2 −1 0 1 2 3

Strain [%]

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

m∗ co

nd

uct

ivit

y

Biaxial εUniaxial ε

(b)

Figure 4.5: (a) Plot of band gap as a function of strain for bulk silicon (b) Plot of conductivityeffective mass along <100> as a function of strain for bulk silicon. Both biaxialand uniaxial strain is applied along <110> direction. Note that negative valuesfor strain indicate compressive strain while positive values stands for tensile strain.

4.2 2D Silicon

4.2.1 Impact of Confinement

Confining bulk silicon to thin silicon slab leads to strong size quantization resulting in twodimensional subband ladders. Several models are capable in explaining the quantization effectsfrom confinement. The parabolic band approximation is one of the simplest model, among theseveral, that explains the electron dispersion relationship at the extremas. For silicon slabs,using the potential well approximation, the dispersion relationship can be expressed as [36]:

E(kx, ky, t,n) ∝ ~2k2x

2m∗x+

~2k2y

2m∗y+

~2

2m∗z(nπ

t)2 (4.3)

where, t is the thickness of the slab, m∗z the quantization mass, m∗x, m∗y the effective massesin the x and y direction. kx, ky are the wave vectors in the x and y directions respectively andn = 0, 1, 2, ... is a positive integer which corresponds to the subband index. The quantizationmass of a valley is defined as the effective mass of the bulk material calculated along theconfinement direction. From Eq.(4.3) it can be concluded that the energy levels get quantizedin the confinement direction for a two dimensional slab. Also, the subband energy levels aredependent on the quantization mass. Thus, higher the quantization mass lower will be theenergy level of the subband.

Table 4.3: Quantization mass (effective mass along confinement direction in bulk) for projectedand in-plane subband ladders of the silicon slabs under study.

Subbandmz [−]

100 Cleave 110 Cleave 111 CleaveInplane Projected Inplane Projected Inplane Projected

Own 0.20 0.92 0.20 0.32 - 0.26Literature [8, 51] 0.19 0.98 0.19 0.318 - 0.259


Tab.(4.3) summarizes the quantization mass for the respective conduction band subbandvalleys for some of the popular confinement directions for silicon. Moreover, a comparisonbetween the simulation values and the literature values is also given. The explanation on theformation and impact of the subbands are explained in detail in the following paragraphs.For the 100 cleave scenario, simulation results of the projected valley has an error of −6 %while the inplane valleys has an error of 5 % when comparing to the quantization mass inliterature. In case of 110 cleave, the quantization mass calculated at the projected subbandhas an error of 6 % and the inplane subband has an error of −6 % in contrast to literature.Now with the 111 cleaved silicon slab, the quantization mass of the projected subbandscontain an error of 4 % in comparison to the literature.

Γ

[010]

[001]

[100] Γ

M Γ L X Γ Y L

k

−3

−2

−1

0

1

2

3

E−

EF[eV]

t = 2 nm

(a) 100 cleave

Γ

[010]

[001]

[100] Γ

Γ L X Γ Y L

k

−3

−2

−1

0

1

2

3

E−

EF[eV]

t = 2 nm

(b) 110 cleave

Γ

[010]

[001]

[100]Γ

M Γ K X Γ Y

k

−3

−2

−1

0

1

2

3

E−

EF[eV]

t = 2 nm

(c) 111 cleave

Figure 4.6: Impact of confinement on disper-sion relation in silicon. (subfig-ure: top left) The six equivalentconduction band valleys alongthree principal axes and the re-spective cleave planes. (subfig-ure: top right) Schematic of sub-band structure. (subfigure: bot-tom) Bandstrucutre for the re-spective cleaves with slab thick-ness of 2 nm.

4.2. 2D SILICON 31

Γ L X Γ Y L

k

−3

−2

−1

0

1

2

3E−

EF[eV]

t = 1 nm

Γ L X Γ Y L

k

−3

−2

−1

0

1

2

3

E−

EF[eV]

t = 2 nm

Figure 4.7: Bandstructure of 1 nm and 2 nm thick silicon slab with confinement in the 110direction.

The impact from confinement on the electronic structure with 100 cleave is illustrated inthe Fig.(4.6a). The top left subfigure in Fig.(4.6a), depicts the conduction band structure ofbulk silicon and the 100 cleave plane. The conduction band subbands are strongly affectedby confinement. With the 100 confinement the out of plane ellipsoids are projected ontothe cleave plane while the inplane ellipsoids are retained as such. The subband which lies onthe cleave plane, as shown in Fig.(4.6a), is called the inplane subbands. The projection of theout of plane subband onto the cleave plane, as marked in Fig.(4.6a), is called the projectedsubband. This phenomenon results in a dispersion relationship with four equivalent inplanesubband ladders along Γ→ L direction and a two-fold degenerate projected subband at the Γpoint [51, 52]. This is portrayed in the top right subfigure of Fig.(4.6a). Since the two folddegenerate inplane subband has a smaller quantization mass as shown in Tab.(4.3), it canbe inferred that the projected subband will be at a lower energy level in comparison to theinplane subbands, which follows from Eq.(4.3). Even though the conduction band subbandvalleys undergo transformations, the valence band ladder remains as such at the Gammapoint. Therefore, 100 cleave in silicon leads to direct band gap type as depicted in Fig.(4.6a)and shows strong optical properties. Thus, 100 cleave silicon slab has rich prospects in theopto-electronic industry [15,16].

Fig.(4.6b) shows the impact of confinement on electronic structure of 110 cleaved siliconslab. The conduction band structure of bulk silicon with 110 cleave plane is shown in thetop left subfigure in Fig.(4.6b). Now confinement in the 110 direction, leads to projectionof out of plane subbands onto the cleave plane forming the two equivalent two fold degenerateprojected subband along Γ → X direction. As for the subbands which lie on the cleaveplane, it is retained as such forming two equivalent inplane subband ladders along Γ → Ydirection. This is depicted in the top right subfigure of Fig.(4.6b). Just as in the 100 cleavescenario, valence band in the 110 cleave too retains itself at the Γ point. Unlike with100 cleave silicon, the 110 cleave results in an indirect band gap as there are no subbandladders at the Γ point as shown in Fig.(4.6b). From Tab.(4.3), it is seen that the projectedsubband ladder has a lower quantization mass than the inplane subbands. So in accordanceto Eq.(4.3), the projected subband will be at a lower energy level in comparison to othersubbands. Thus forming the conduction band minima at Γ→ X [51, 52]. Another effect ofconfinement observed with 110 cleave is when thinning the slab down to 1 nm. With theslab thickness down to 1 nm, the indirect band gap type transforms to a direct band gap typeas shown in Fig.(4.7).


1 2 3 4 5 6 7 8 9 10

t [nm]

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Eg[eV]

E Bulkg = 1.12 eV

100110111

FitDFT

Figure 4.8: Plot of band gap as a functionof thickness for 100, 110 and111 cleaved silicon slabs. Themarkers indicate the band gap val-ues from simulation while the solidlines represent the analytical modelwhich is based on the parabolic ap-proximation.

Table 4.4: Fit parameters for the empiricalband gap model.

Parameters 100 110 111a [-] 1.59 1.40 1.39b [nma] 0.73 0.47 0.51

The 111 cleave is an interesting configu-ration and is shown in Fig.(4.6c). The top leftsubfigure in Fig.(4.6c) shows the conductionband structure with the 111 cleave plane.Now with confinement in the 111 direction,there are only out of plane ellipsoids whichis projected onto the cleave plane formingthe projected subband ladder as indicated inFig.(4.6c). The configuration has six equiv-alent projected subband ladder along Γ→ Xdirection [51, 52]. As for the valence band,just like in the previous cases, the subband isretained at the Γ point. Thus, confinementin 111 direction results in an indirect bandgap type material.Fig.(4.8) shows the influence of confine-

ment on the band gap. From simulation itis observed that the band gap increases withdecreasing slab thickness. This is in agree-ment with Eq.(4.3). Another conclusion fromFig.(4.8) is that for the same slab thickness,100 cleave demonstrates a higher band gapfollowed by 111 cleave and 110. Also,it is recommended to use band gaps of con-fined material when studying transport inSOI FETs as the subthreshold current is influ-enced by the variation in band gap. Further,

we fit the Eg values to thickness dependent empirical model [7]:

Eg = EgBulk + b(1/t)a (4.4)

where Eg is the band gap of silicon slab, EgBulk the bulk band gap, t the slab thickness, a

and b the fitting parameters. This can be incorporated into numerical device simulator and

1 2 3 4 5 6 7 8 9 10

t [nm]

1.11.21.31.41.51.61.71.81.92.0

Eg[eV]

ProjectedInplane

(a) 100 cleave

1 2 3 4 5 6 7 8 9 10

t [nm]

1.1

1.2

1.3

1.4

1.5

1.6

Eg[eV]

ProjectedInplane

(b) 110 cleave

Figure 4.9: Energy gap of inplane and projected subbands as a function of slab thickness forconfinement in 100 and 110 direction.

4.2. 2D SILICON 33

for compact modelling for silicon devices so that atomistic calculations for arbitrary channelthicknesses can be avoided. The fit parameters for the investigated orientations are given inTab.(4.4). Fig.(4.9) shows the energy gap from projected and inplane subband ladders as afunction of thickness for silicon slabs with confinement in 100 and 110 direction. Here,the difference between the projected and the inplane increases with decreasing slab thicknessfor both slab orientations. This is due to the difference in the quantization mass between theinplane and the projected subband ladders. That is, the inplane subband ladder shift up inenergy faster in comparison to the projected subband as shown in Eq.(4.3). Moreover, thedifference between the inplane and the projected subbands are larger in the 100 cleave thanthe 110 cleave. This is due to the larger difference between quantization mass of inplaneand projected subband for 100 cleave in comparison to 110 cleave. It is also seen that theobserved trends are lost when thinning the slab from 2 nm to 1 nm.Fig.(4.10) shows the plot of conductivity effective mass as function of slab thickness for

100, 110 and 111 cleaved silicon slabs. The effective mass along different low symmetrytransport direction for each cleave is also depicted in Fig.(4.10). As aforementioned, theconductivity effective mass is calculated by taking the harmonic mean of the respectiveeffective mass components [46]. Here, harmonic mean for the first twenty bands of eachconduction band valley is taken to calculate the conductivity effective mass:

m∗conductivity =

20∑n=1

fn

20∑n=1

fn1m∗

n

(4.5)

2 3 4 5 6 7 8 9 10

t [nm]

0.210.220.230.240.250.260.270.280.290.30

m∗ co

nd

uct

ivit

y

Direction[100][110]

(a) 100 cleave

2 3 4 5 6 7 8 9 10

t [nm]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m∗ co

nd

uct

ivit

y

Direction[100] [010] [110]

(b) 110 cleave

2 3 4 5 6 7 8 9 10

t [nm]

0.200.220.240.260.280.300.320.340.360.38

m∗ co

nd

uct

ivit

y

Direction[100][110]

(c) 111 cleave

Figure 4.10: Plot of conductivity effectivemass as a function of slab thick-ness with confinement in 100,110 and 111 direction respec-tively.


Γ

L

X

Y

[100]

[110]

[010]

(a) 100 cleave

Γ

LX

Y

[100]

[010]

[110]

(b) 110 cleave

X

Y K

Γ[100]

[010] [110]

(c) 111 cleave

Figure 4.11: Brillouin zones of silicon slabs. Note that the grey lobes represent projectedsubbands while the navy blue ellipses denote the inplane subbands.

where m∗conductivity is the conductivity effective, f is the fermi distribution acting as the weightsand m∗ is the effective mass component along the transport direction.

The conductivity effective mass in 100 cleaved silicon slab decreases with decreasing slabthickness as seen in Fig(4.10a). This effect is primarily from the degeneracy lifting as shown inFig.(4.9a), which causes redistribution of electrons from the inplane subbands to the projectedsubband. From Tab.(4.5) and Fig.(4.11a), it is can be concluded that the projected subbandhas the lowest effective mass component in all of the low index orientations. Thus conductivityeffective mass being a weighted average, for any transport direction in the 100 cleavedsilicon, the conductivity effective mass decreases with decreasing slab thickness. Anotherconclusion from the Fig(4.10a) is that the conductivity effective mass along [110] is larger than[100] direction. The is because of the larger inplane conductivity effective mass componentalong [110] direction than [100] direction. Moreover, from excessive trimming of slab thickness,the degeneracy uplift causes the impact of inplane subbands on the conductivity effectivemass to decrease. Therefore, for thinner slabs the conductivity effective mass along [110] and[100] converges.In the 110 cleaved silicon slab, as shown in Fig(4.10b), the conductivity effective mass

along [010] direction is the lowest followed by [110] direction while [100] direction marksthe highest. Trimming the slab thickness causes degeneracy lifting as seen in Fig.(4.9b).This causes electron population of inplane subband to redistribute into projected subband.Thus, increasing the weights of projected subband while decreasing the weight of the inplanesubband. Now with transport in [100] direction, the contribution of effective mass componentof conductivity effective mass from the inplane subband is m∗t while for projected subband is(m∗l +m∗t ) /2 as given in Tab.(4.5)) and Fig.(4.11b). Hence, the conductivity effective massincreases with decreasing slab thickness. This trend is also observed in Fig(4.10b) for thicknessfrom 10 nm to 8 nm. In case of transport along [010] direction, the effective mass componentfrom inplane subband is m∗l while for projected subband is m∗t . Therefore, decreasing slabthickness causes conductivity effective mass to decrease. This is also observable in Fig(4.10b)for thickness ranging from 10 nm to 8 nm. As for transport along [110] direction, it can bepredicted that the effective mass component from projected subband is larger than the inplanesubband from referring to Tab.(4.5)) and Fig.(4.11b). Therefore, trimming slab thicknesscan increase conductivity effective mass along [110] direction. This trend is also observable

4.2. 2D SILICON 35

Table 4.5: Principle effective masses for each conduction band valley in 100, 110 and110 cleaves respectively [51]. This does not include impact of band warping fromconfinement.

Cleave Subband Longitudinal Transverse

100 Projected m∗t m∗tInplane m∗l m∗t

110 Projected (m∗l +m∗t ) /2 m∗tInplane m∗l m∗t

111 Projected (2m∗l +m∗t ) /3 m∗t

in Fig(4.10b) for thickness ranging from 10 nm to 8 nm. Also, in Fig.(4.10b) it is seen thatwhen trimming down the slab thickness from 4 nm to 2 nm, perceivable shifts in the trendsare observed. This is because of the lowering of the degeneracy uplift for thickness rangingfrom 4 nm to 2 nm as shown in Fig.(4.9b).The 111 cleaved silicon is a special case as seen in Fig.(4.11c). Silicon slabs with this

confinement direction only has projected subbands. Thus thinning the silicon slab producesno degeneracy uplift as all of the six equi-energy subbands shift equally. Thus, ideally theconductivity effective mass should be independent of slab thickness. However, in Fig.(4.10c)a decreasing trend of conductivity effective mass for decreasing slab thickness is seen. Thisarises from the impact of band warping from trimming of slab thicknesses. From Tab.(4.5)and Fig.(4.11b), it can be concluded that the conductivity effective mass along [100] directionis greater than [110] direction. The deciding factor in this case is the two dissimilar effectivemass component among the six valleys which are m∗t for [110] direction and m∗l for [100]direction respectively.A significant conclusion from Fig.(4.10) is regarding the impact on carrier transport. The

transmission coefficient for tunneling phenomenon can be described using the Wentzel-Kramer-Brillouin (WKB) approximation [53]:

T0 = exp−4√

2φ3m∗conductivity

3e~F(4.6)

where T0 is the transmission coefficient, φ is the barrier height and F is the electric field.So from Eq.(4.6) one can conclude that lower the conductivity effective mass, better thetransmission coefficient. Thus for 100 wafer confinement, the best channel orientation forballistic devices is along [100] direction. As for wafer with 110 confinement, the channelmust be oriented along [010] direction. Similarly, for 111 wafer surface the best channelconfiguration is along [110] direction.

4.2.2 Impact of Strain

As previously mentioned, strain causes band warping and shifting of bands in materials.Fig.(4.12) shows the impact of strain on the electron dispersion relationship of 2 nm thick100 cleaved silicon slab. Here, both biaxial and uniaxial strain is applied along <110>direction. The projected and inplane conduction subband ladder is marked in Fig.(4.12) as Pand I respectively. Now in the unstrained silicon slab, the projected subband is at a lowerenergy than the inplane subband. With biaxial tensile strain, the projected subband shiftsdown in energy while inplane subband shifts up. Here one may assume the band gap type


to be direct but the shift of valence band maxima from Γ point translates the material toindirect band gap type. Now with biaxial compressive strain the projected subband shifts upwhile the inplane subband shifts down transforming band gap type from direct to indirect.As for uniaxial strain, for both compressive and tensile strain the projected subband alwaysshifts down in energy. However, the inplane subband shifts up in energy with tensile strainand shifts down in energy with compressive strain. Another observation is that uniaxial straincan cause degeneracy lifting which is noticeable for both projected subband and the valenceband. This results in improved carrier mobility due to reduced intra-valley scattering [51].

M Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP −2 % biaxial ε

M Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP −2 % uniaxial ε

M Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP Unstrained

M Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP 2 % biaxial ε

M Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP 2 % uniaxial ε

Figure 4.12: Electronic band structure of strained 100 cleaved silicon slab with thickness of2 nm. Both biaxial and uniaxial strain is applied along <110> direction. Thered line denotes the energy of conduction band minima for unstrained 2 nm thick100 cleaved silicon. Note that negative values for strain indicate compressivestrain while positive values stands for tensile strain. The projected subband andthe inplane subband are marked as P and I respectively.

4.2. 2D SILICON 37

In the 110 cleaved silicon slab, the biaxial strain is applied along <110> and <100>direction while the uniaxial strain is administered only along <110> direction. Fig.(4.13)depicts the impact of strain on the electron dispersion relationship of 2 nm thick 110 cleavedsilicon slab. With Uniaxial strain, the projected subband shift up in energy for tensile strainwhile the subband shifts down in energy in the case of compressive strain. As for the inplanesubband, tensile strain causes the band to shift down while compressive strain shifts upthe band. Now in case of biaxial strain, both projected and inplane bands shift down with

Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP

−2 % biaxial ε

Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP

−2 % uniaxial ε

Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP

Unstrained

Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP

2 % biaxial ε

Γ L X Γ Y L

k

−2

−1

0

1

2

E−

EF[eV]

IP

2 % uniaxial ε

Figure 4.13: Electronic band structure of strained 110 cleaved silicon slab with thickness of2 nm. Here, biaxial strain is applied along <110> and <100> direction while theuniaxial strain is administered only along <110> direction. The red line denotesthe energy of conduction band minima for unstrained 2 nm thick 110 cleavedsilicon. Note that negative values for strain indicate compressive strain whilepositive values stands for tensile strain. The projected subband and the inplanesubband are marked as P and I respectively.


compressive strain while tensile strain shifts up both projected and inplane bands.Fig.(4.14a) shows the effect of strain in the range of −2% (compressive) to 2% (tensile)

on the band gap for 100 cleaved silicon slabs. This strain is within reasonable limits forsilicon as is shown in Ref.( [13,54]). With uniaxial strain, the band gap is found to decreasefor both compressive and tensile strain. As for biaxial compressive strain in slab thicknessranging from 4 nm and below, band gap increases for small strain values and then decreasesfor further increase in strain. With slab thicknesses above 4 nm, the band gap decreases forincreasing biaxial compressive strain. For biaxial tensile strain, the band gap responds witha decreasing trend for increasing strain. In the case of bulk silicon, the band gap shows adecreasing trend for both biaxial and uniaxial strain.

Fig.(4.14b) shows the band gap response for 110 cleaved silicon slabs to strain in the rangeof −2% (compressive) to 2% (tensile). With uniaxial tensile strain, the band gap increasesfor small strain values and then decreases for further increase in strain except for 1 nm thickslab. For the 1 nm case, the band gap shows a decreasing response to tensile strain. As forthe uniaxial compressive scenario, the band gap decreases for increasing strain except forthe 1 nm thick slab where the band gap increases for small strain values and then decreases

1.0 nm2.0 nm

4.0 nm6.0 nm

8.0 nmBulk

Direct Eg

Indirect Eg

−3 −2 −1 0 1 2 3

Biaxial ε [%]

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Eg[eV]

−3 −2 −1 0 1 2 3

Uniaxial ε [%]

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Eg[eV]

(a) 100 cleave

−3 −2 −1 0 1 2 3

Biaxial ε [%]

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Eg[eV]

−3 −2 −1 0 1 2 3

Uniaxial ε [%]

0.8

1.0

1.2

1.4

1.6

Eg[eV]

(b) 100 cleave

Figure 4.14: Plot of band gap as a function of slab thickness for 100 cleave and 110 cleavesilicon. Negative strain values indicate compression while positive values implytensile strain.

4.2. 2D SILICON 39

for further increase in strain. In the biaxial compressive case, the band gap decreases forincreasing compressive strain. As for biaxial tensile strain, the band gap shows a decreasingresponse to strain for thicker films above 4 nm. In case of 1 nm and 2 nm thin slabs, biaxialtensile strain shows an increasing trend of the band gap. In comparison, the bulk materialresponds with a decreasing trend of the band gap for both biaxial and uniaxial strain.The Fig.(4.15) and Fig.(4.16) summarizes the impact of strain and confinement on the

band gap type of silicon slabs. Since the graph depicts the difference of direct (Edirect -smallest direct band gap) and indirect band gap (Eindirect - smallest indirect band gap),positive values denote indirect band gap type material while others indicate direct band gaptype material. As aforementioned, unstrained 100 cleaved silicon is always direct bandgap type material. But with compressive strain the primed subband ladder can be loweredthus transforming from direct band gap type to indirect band gap type material as seen inFig.(4.12). An interesting trend observed here is that thinner films require larger compressivestrain to achieve a transition to indirect band gap material. For the thicknesses we studied,2% compressive strain is sufficient to transform a 1 nm thick silicon slab to an indirect bandgap material in the biaxial scenario. Similar behaviour is also observed in the uniaxial case,however a higher compressive strain is required to achieve the band gap transformation. Itcan be inferred from Fig.(4.15) that more than 2% compressive strain is required to transform1 nm and 2 nm thick slabs into an indirect band gap material. Fig.(4.15) also shows that forthe tensile biaxial strain, the silicon slab forms an indirect band gap for slab thickness in therange of 1 nm to 2 nm. This is due to the shift of valence band maxima from the [000] (Γ)due to band warping as seen in Fig.(4.12). Thus, for opto-electronic applications using 100cleaved silicon, it is best to use uniaxial tensile strain to engineer band gaps since the directband gap is always retained.In case of the 110 cleave, the material is mostly indirect band gap type except for the

1 nm thin slab. An interesting feature for 110 cleaved silicon is the transformation into adirect band gap type material with the application of tensile strain as shown in Fig.(4.16).This is due to band warping and lowering of primed subband ladder as seen in Fig.(4.13). Forboth biaxial and uniaxial strain scenarios, it is observed that thicker films require higher strainfor the transformation to direct band gap type material. From the slabs under investigation,

-2 -1 0 1 2Biaxial ε [%]

1

2

4

6

8

t[nm]

-2 -1 0 1 2Uniaxial ε [%]

0 10 20 30 40 50 60Edirect − Eindirect [meV]

Figure 4.15: Difference between direct and indirect band gap for 100 cleaved silicon slab asa function of strain and confinement.


-2 -1 0 1 2Biaxial strain [%]

1

2

4

6

8

t[nm]

-2 -1 0 1 2Uniaxial strain [%]

0 2 5 7 10 12 15Edirect − Eindirect [meV]

Figure 4.16: Difference between direct and indirect band gap for 110 cleaved silicon slab asa function of strain and confinement.

it is found that 2% tensile strain can transform slab thickness upto 4 nm into direct band gapmaterial. The study recommends use of tensile strain for band engineering in 110 cleavedsilicon slabs in the opto-electronic industry.The Fig.(4.17) shows a plot of conductivity effective mass as a function of slab thickness

and strain for 100 cleave and 110 cleave silicon. In the case of 100 cleave silicon,under uniaxial tensile strain the conductivity effective mass decreases with increasing strain.However, for compressive strain conductivity effective mass increases with increasing strain.Use of strained silicon in active devices is quite common to improve device characteristics.For devices with holes as the carrier compressive strain is advantageous while tensile strainimproves transport where electrons are the major carriers. So from Fig.(4.17a), it can beconcluded that the use of strain in active region can be applied only to unipolar devices. Withbiaxial strain, a common trend is only observed for low strain ranging from −1 % to 1 % strain.In this, regime with increasing tensile strain the conductivity effective mass decreases. On theother hand, for increasing compressive strain the conductivity effective mass is observed toincrease. Furthermore, for biaxial strain values more than 1 %, conductivity effective massseems to has only slight influence to strain. Moreover, thinner the slab thickness the impactof tensile strain on conductivity effective mass also decreases.The Fig.(4.17b) shows the impact of strain and thickness on the conductivity effective

mass of 110 cleaved silicon. With Biaxial tensile strain, the conductivity effective massdecreases with increasing strain. However, thicker the SOI larger is the strain required to lowerthe conductivity effective mass. So for 8 nm thickSOI, it appears as though a larger tensilestrain is required for the conductivity effective mass to respond. However, in the 1 nm case,conductivity effect mass seems to be independent of the applied strain as shown in Fig.(4.17b).With uniaxial strain on the 110 cleave, thicker films follow a similar trend a seen with the100. That is, with increasing tensile strain the conductivity effective mass is observed todecrease. Similarly, increasing compressive strain results in increasing conductivity effectivemass. The 1 nm case shows a completely different response in comparison to the thicker films.This could be because of the transitioning of the band gap type from indirect to direct due tolowering of the inplane subband.

4.2. 2D SILICON 41

t [nm]

1.0 2.0 4.0 6.0 8.0

−3 −2 −1 0 1 2 3

Biaxial ε [%]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m∗ co

nd

uct

ivit

y

−3 −2 −1 0 1 2 3

Uniaxial ε [%]

0.1

0.2

0.3

0.4

0.5

m∗ co

nd

uct

ivit

y

(a) 100 cleave with transport along [110]

−3 −2 −1 0 1 2 3

Biaxial ε [%]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

m∗ co

nd

uct

ivit

y

−3 −2 −1 0 1 2 3

Uniaxial ε [%]

0.15

0.20

0.25

0.30

0.35

0.40

0.45

m∗ co

nd

uct

ivit

y

(b) 110 cleave with transport along [110]

Figure 4.17: Plot of band gap as a function of slab thickness and for 100 cleave and 110cleave silicon. Negative strain values indicate compression while positive valuesimply tensile strain.

43

5 Numerical Device Simulation5.1 Device Characteristics

The Fig.(5.1) depicts the device characteristics of the n type FDSOI under study. FDSOIdevices with a grounded body can be represented with a simple analytical model [55]:

Ids ∝

µ((Vgs − Vth)Vds −

1

2Vds

2) when Vds < Vgs − Vth (5.1a)µ

2(Vgs − Vth)2 when Vds ≥ Vgs − Vth (5.1b)

where Ids is the drain to source current, µ is the mobility of carriers, Vgs is the gate to sourcevoltage, Vds is the drain to source voltage and Vth is the threshold voltage of the device. TheFig.(5.1a) shows the output characteristics of the n type FDSOI at gate to source voltage of0.9 V. Here with increasing Vds, a linear model is observed initially. This agrees well withEq.(5.1a) for low drain to source biasing especially when Vds << Vgs − Vth. For low drain tosource biasing, the 1

2Vds2 term can be neglected and thus Eq.(5.1a) becomes linear. Now with

increasing drain to source voltage, once Vds ≥ Vgs − Vth from Eq.(5.1b) the characteristicsshould saturate. This so because drain current becomes independent of Vds as shown inEq.(5.1b). However in Fig.(5.1a), a deviation from ideality due to parasitic affects is observed.The Fig.(5.1b) shows the transfer characteristics of the n type FDSOI at drain to sourcevoltage of 0.9 V. Here, initially for Vgs ≤ Vds−Vth the device is in saturation regime. Howeverthe characteristics fail to show the quadratic behaviour as given in Eq.(5.1b) due to theparasitic effects. Now with increasing gate to source voltage in the transfer characteristics, asVgs > Vds− Vth, the device moves into the linear regime. So at high gate to source voltage thetransfer characteristics show a linear behaviour. It is also seen in Fig.(5.1) that decreasing SOI

t [nm]

4.0 6.0 8.0 9.0 10.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vds [V]

0.0

0.5

1.0

1.5

2.0

I ds[m

A]

Vgs = 0.9 V

(a) Output characteristics

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vgs [V]

0.0

0.5

1.0

1.5

2.0

I ds[m

A]

Vds = 0.9 V

(b) Transfer characteristics

Figure 5.1: Device characteristics of the n type FDSOI under study. Note that the characteri-zation is done for various SOI thickness ranging from 4 nm to 10 nm.

44 CHAPTER 5. NUMERICAL DEVICE SIMULATION

2 3 4 5 6 7 8 9 10

t [nm]

0.150.200.250.300.350.400.450.500.55

Vth[V]

(a)

2 3 4 5 6 7 8 9 10

t [nm]

0

50

100

150

200

µ[cm

2V−1s]

(b)

Figure 5.2: Impact of SOI thickness of an n type FDSOI on mobility and threshold voltage.(a) Plot of threshold voltage as a function of SOI thickness (b) Plot of mobility asa function of SOI thickness.

thickness can improve device performance. This is due to improving mobility and decreasingthreshold voltage from trimming the SOI thickness.The Fig.(5.2a) shows the impact of SOI thickness on the threshold voltage. Here, the

threshold voltage decreases with increasing SOI thickness. So from Eq.(5.1a) and Eq.(5.1b),it be concluded that decreasing threshold voltage can improve drain current. The thresholdvoltage of the n type FDSOI can be defined as [55]:

Vth = Vfb + φsi −Qb(φsi)

Cox, here (5.2)

Qb ∝ qXdm (5.3)

where Vfb is the flat band voltage, φsi is the surface potential at the onset of strong inversion,Cox is the gate oxide capacitance per unit area, q is the elementary charge, Qb is the bulkcharge per unit area and Xdm is the maximum depletion width under the gate. So fromEq.(5.2), one can conclude that the threshold voltage will decrease with increasing bulk chargeper unit area. Then again, the bulk charge per unit area is directly proportional to themaximum depletion region under the gate as given in Eq.(5.3). Here, in the n type FDSOI,the thickness of the SOI is even smaller than the maximum depletion width. That is why, withincreasing SOI thickness the available depletion region increases thereby the threshold voltagelowers. However once the SOI thickness is far more than the maximum depletion width underthe gate, the threshold voltage will saturate and naturally the device will transform fromfully depleted to partially depleted. In the current design, with 10 nm SOI thickness, thedevice is still in fully depleted mode. This can be verified as there are no kinks in the outputcharacteristics in Fig(5.1a).

The Fig.(5.2b) depicts the impact of SOI thickness on the mobility factor. Here, the mobilityis observed to increase with increasing SOI thickness. Moreover, from Eq.(5.1a) and Eq.(5.1b),it be concluded that increasing mobility can improve the drain current. The mobility can beexpressed as [55]:

µ =qτ

m∗conductivity

(5.4)

5.2. IMPACT OF CONFINEMENT 45

where τ is the average scattering time. Since simulation is done using TCAD’s inbuiltparameter set the conductivity effective mass is a constant for all slab thickness. Hence,from Eq.(5.4) it may seem that the mobility should remain independent of SOI thickness.However, this is not true since the impact of SOI thickness on average scattering time isnot taken into account. Even though carrier-carrier scattering time remains independent ofSOI thinning, the phonon scattering time decreases with trimming of SOI thickness. So, theaverage scattering time decreases with thinning of SOI thickness. This causes mobility todecrease with decreasing slab thickness [56].

5.2 Impact of Confinement

In this section, comparison of the on-off device characteristics simulated with default bulkparameters and confined material parameters for the respective SOI thickness is analyzed.Here, the discussions are done for two sets of data: characterization with band gap extractedfrom DFT and lastly a simulation where both conductivity effective mass and band gapextracted from DFT is used.

The Fig.(5.3a) shows the device-off characteristics of an n type FDSOI with SOI thicknessof 2 nm. Here, the simulation is done using both in-built TCAD parameters as well as withband gap extracted from the DFT simulation. From Fig.(5.3a), it is seen that the devicecharacteristics simulated with the bulk parameter exceeds the one simulated using band gapextracted from DFT. Moreover from Fig.(5.3b), this discrepancy is observed to increase withdecreasing SOI thickness. The Fig.(5.3b) is a plot of relative shift of device-off current fromthe default characteristics when simulated with band gap extracted from DFT for confinedSOI as a function of SOI thickness. From Fig.(5.3b), it is seen that the relative error whenusing bulk band gap rises from 3 % to 30 % when thinning the slab from 10 nm to 2 nm. Asdepicted in Fig.(5.3b), it is advisable to use the exact band gap value of confined SOI insteadof the bulk silicon band gap when simulation FDSOI with SOI thickness below 6 nm. Thisdiscrepancy arises due to the difference in the intrinsic carrier concentration in the active

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vds [V]

0

2

4

6

8

10

12

14

16

I off[pA]

t = 2 nm

TCAD (Eg )DFT (Eg )

(a)

2 3 4 5 6 7 8 9 10

t [nm]

0

5

10

15

20

25

30

∆I o

ff[%

]

Vds = 0.9 V

(b)

Figure 5.3: Comparison of the device-off characteristics of the n type FDSOI simulated usingTCAD’s inbuilt band gap and band gap extracted from DFT. (a) Off characteristicsfor SOI thickness of 2 nm (b) Plot of relative shift of device-off current from thedefault characteristics, when simulated with band gap extracted from DFT forconfined SOI, as a function of SOI thickness. The red dotted line marks thethickness below which confined parameters are required for device simulation.


region of the FDSOI. Besides, the intrinsic carrier concentration in the active region is heavilyinfluenced by the band gap of the SOI. The off current in the device is a leakage current andcan be expressed as [57]:

Ioff = qA

[√Dp

ηp

ni2

Nd+ni

ζ

√2ε0εsiVr

qNd

](5.5)

ni =√NcNv exp

−Eg

2kBT(5.6)

where Ioff is the device-off current, A is the device area, Dp is the minority carrier diffusioncoefficient, ηp is the minority carrier lifetime, ni is the intrinsic carrier concentration, Nd isthe donor doping concentration in the source/drain region, εsi is the relative permittivityof silicon, Vr is the reverse bias voltage, ζ is the space charge generation lifetime, Nc isthe conduction band density of states, Nv is the valence band density of states, kB is theBoltzmann constant and T is the ambient temperature. From Eq.(5.5) and Eq.(5.6), it can

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vds [V]

0102030405060708090

I on[µA]

t = 2 nm

TCAD (Eg )DFT (Eg )

(a)

2 3 4 5 6 7 8 9 10

t [nm]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

∆I o

n[%

]

Vds = 0.9 V

(b)

2 3 4 5 6 7 8 9 10

t [nm]

0.6

0.8

1.0

1.2

1.4

∆V

th[%

]

(c)

Figure 5.4: Comparison of the device-on characteristics of the n type FDSOI simulated usingTCAD’s inbuilt band gap and band gap extracted from DFT. (a) On characteristicsfor SOI thickness of 2 nm (b) Plot of relative shift of device-on current from thedefault characteristics, when simulated with band gap extracted from DFT forconfined SOI, as a function of SOI thickness (c) Plot of relative shift of thresholdvoltage from the default value, when simulated with band gap extracted fromDFT for confined SOI, as a function of SOI thickness.


be inferred that increasing band gap decreases intrinsic carrier concentration which in turndecreases the off current in the device. Now from Fig.(4.8), it is seen that confinement opensup band gap. Thus, trimming SOI thickness can lead to lower off current as shown in Fig.(5.4)when using true band gap of SOI extracted from DFT for device simulation.

The Fig.(5.4a) shows the device-on characteristics of an n type FDSOI with SOI thicknessof 2 nm. Here, a comparison on the device-on characteristic simulated using in-built TCADparameters as well as with band gap extracted from the DFT simulation is shown. In Fig.(5.4a),it is seen that the device-on characteristics simulated with the bulk parameter exceeds theone simulated using band gap extracted from DFT only by a small factor. Moreover fromFig.(5.4b), this discrepancy is observed to increase with decreasing SOI thickness. Since theimpact of confinement on the device-on characteristics is negligible, it is tolerable to use thebulk band gap parameter for device-on simulation. The trend observed in Fig.(5.4b) is due tothe impact of confinement on the intrinsic carrier concentration. This can be explained fromthe threshold voltage expression [55]:

Vth = Vfb + φsi −√

2ε0εsiq(Nd + ni)φsi

Cox(5.7)

Since intrinsic carrier concentration decreases with trimming SOI thickness, from Eq.(5.7)and Eq.(5.1) it can be inferred that decreasing intrinsic carrier concentration results in lowerdevice-on current. However, the discrepancy when using bulk band gap instead of confinedband gap for simulation is negligible as ni <<< Nd.

The Fig.(5.5a) depicts the device-off characteristics of an n type FDSOI with SOI thicknessof 2 nm. Moreover, a comparison of the device-off characteristics simulated using in-builtTCAD default parameters (Eg, m∗) and confined parameters (Eg, m∗) extracted from theDFT simulation is also shown in Fig(5.5a). Here, it is seen that the device characteristicssimulated with the bulk parameter exceeds the characteristics simulated using the confinedparameters extracted from DFT. The Fig.(5.5b) shows a plot of relative shift of device-off

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vds [V]

0

10

20

30

40

50

60

70

I off[pA]

t = 2 nm

TCAD (Eg, m∗)DFT (Eg, m∗)

(a)

2 3 4 5 6 7 8 9 10

t [nm]

102030405060708090

100

∆I o

ff[%

]

Vds = 0.9 V

(b)

Figure 5.5: Comparison of the device-off characteristics of the n type FDSOI simulated usingTCAD’s default bulk parameters (Eg, m∗) and confined parameters (Eg, m∗)extracted from DFT. (a) Off characteristics for SOI thickness of 2 nm (b) Plot ofrelative shift of device-off current from the default characteristics, when simulatedwith confined parameters (Eg, m∗) extracted from DFT for SOI, as a functionof SOI thickness. The red dotted line marks the thickness below which confinedparameters are required for device simulation.


current from the default characteristics, when simulated with the band gap and effective massparameters extracted from DFT for confined SOI, as a function of SOI thickness. Furthermore,this discrepancy is observed to increase with decreasing SOI thickness. The relative errorwhen using bulk band gap rises from 18 % to 92 % when thinning the slab from 10 nm to 2 nm.As depicted in Fig.(5.3b), it is advisable to use the exact band gap and effective mass values ofconfined SOI instead of TCAD’s default bulk silicon band gap and effective mass parameterswhen simulating FDSOI with SOI thickness below 6 nm. From comparing Fig.(5.5b) andFig.(5.3b), it can be inferred that simulating with exact effective mass has a larger impact ondevice-off current characteristics than when simulating with exact band gap. The impact ofeffective mass on device-off current characteristics can be explained from the density of statesexpression [55]:

Nc = 2

(kBTm

∗density(n)

2π~2

)3/2

(5.8)

Nv = 2

(kBTm

∗density(p)

2π~2

)3/2

(5.9)

where Nc is the conduction band density of states, Nv is the valence band density of states,m∗density(n) is the conduction band density of states effective mass, m∗density(p) is the valenceband density of states effective mass, kB is the Boltzmann constant and T is the ambienttemperature. From Fig.(4.10a), it is known that the conductivity effective mass decreaseswith decreasing silicon slab thickness. Since the density of states effective mass is a gometricmean, it will be always less than the conductivity effective mass which is an arithemetic meanof effective masses. Thus, density of states effective mass will also decrease for decreasing slabthickness. Now, from Eq.(5.8) and Eq.(5.9), it can be concluded that decreasing slab thicknesswill lead to decreasing density of states for both valence and conduction band. Thus, fromEq.(5.6) and Eq.(5.5) the device-off current should decrease with trimming of SOI thickness.This agrees well with the findings in Fig.(5.5).

The device-on characteristics of an n type FDSOI with SOI thickness of 2 nm is shown inFig.(5.6a). Here, a comparison on the device-on characteristic simulated using TCAD’s defaultbulk parameters (Eg, m∗) and confined parameters (Eg, m∗) extracted from DFT is alsodepicted. In Fig.(5.6a), it is seen that the device-on characteristics simulated with the bulkparameters (Eg, m∗) exceeds the one simulated using confined parameters (Eg, m∗) extractedfrom DFT. The fig.(5.6b) depicts a plot of relative shift of device-on current from the defaultcharacteristics, when simulated with confined parameters (Eg, m∗) extracted from DFT forSOI, as a function of SOI thickness. Here, the relative shift in off current is observed to increasewith decreasing SOI thickness. The relative error when using TCAD’s in-built bulk parameterrises from 15 % to 42 % when thinning the slab from 10 nm to 2 nm. As depicted in Fig.(5.6b),it is advisable to use the exact band gap and effective mass values of confined SOI instead ofTCAD’s default bulk silicon band gap and effective mass parameters when simulating FDSOIwith SOI thickness below 6 nm. Furthermore, the impact on device-on current characteristicsare less severe than the impact on the device-off current. From comparing Fig.(5.6b) andFig.(5.4b), it can be inferred that simulating with exact effective mass has a larger impact ondevice-on current characteristics than when simulating with exact band gap. The device-offcurrent simulated with confined parameter gives a slightly lower value than when usingTCAD’s in-built bulk parameters. This discrepancy is due to the variation on carrier mobility


bought in by using the true effective mass data from DFT for the SOI as given by Eq.(5.1)and Eq.(5.4). The mobility calculated from using the confined parameters are slightly lowerthan when using bulk parameters. Moreover, this error is observed to increase with decreasingslab thickness as shown in Fig(5.6c). The mobility is a factor which depends on conductivityeffective mass and the average scattering time as given in Eq.(5.4). Since the conductivityeffective mass decreases with decreasing SOI thickness as shown in Fig(4.10a), one expects themobility to improve. However, the degradation from scattering time exceeds the improvementfrom conductivity effective mass resulting in a larger mobility shift from when using bulkparameters with trimming the SOI thickness. Here, for all SOI thicknesses the impurityconcentration is the same. That means, for all SOI thicknesses the carrier scattering centersare the same. So, the impact on average scattering time is from the change in the mass ofthe carrier. The average scattering time is controlled by the inelastic collisions, which meansheavier the mass greater the numbers of collisions to transfer its kinetic energy to the targetedpoint. That means heavy carriers require much longer time due to multiple collisions [55, 56].

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Vds [V]

0

10

20

30

40

50

60

70

80

I on[µA]

t = 2 nm

TCAD (Eg, m∗)DFT (Eg, m∗)

(a)

2 3 4 5 6 7 8 9 10

t [nm]

10

15

20

25

30

35

40

45∆

I on[%

]

Vds = 0.9 V

(b)

2 3 4 5 6 7 8 9 10

t [nm]

15

20

25

30

35

40

45

50

55

∆µ[%

]

(c)

Figure 5.6: Comparison of the device-on characteristics of the n type FDSOI simulated usingTCAD’s default bulk parameters (Eg, m∗) and confined parameters (Eg, m∗)extracted from DFT. (a) On characteristics for SOI thickness of 2 nm (b) Plot ofrelative shift of device-on current from the default characteristics, when simulatedwith confined parameters (Eg, m∗) extracted from DFT for confined SOI, as afunction of SOI thickness (c) Plot of relative shift of mobility from the defaultvalue, when simulated with confined parameters (Eg, m∗) extracted from DFT forSOI, as a function of SOI thickness.

51

6 Summary and OutlookIn this study, we carry out a multiscale modelling of an n type FDSOI. This was done as to

analyze the impact of confinement on the transport properties of the FDSOI. The multiscalemodelling approach used is a two step process. The first step involves the electronic structuresimulation of thin silicon slabs. The effect of orientation, confinement, and strain on theelectronic structure of silicon was investigated systematically by using DFT. Especially, theinfluence on subband ladders, band gap, band gap type and effective masses. In the secondstep of the multiscale modelling, a device simulation is performed using TCAD. Moreover, acomparison of the FDSOI characteristics when simulated using TCAD’s in-built parameterand parameter extracted from DFT is also analyzed and discussed.From the electronic structure simulation of free standing silicon slabs, it is observed that

confinement can change the band gap type of silicon. With confinement of bulk silicon inthe 100 direction, it is seen that the band gap type can transition from indirect to direct.However for the 110 and 111 cleaved slabs the band gap type is always indirect, exceptfor the 1 nm thick silicon slab with confinement in 110 direction. In addition, varying thethickness of silicon slabs is shown to be an effective means to engineer the band gap. Strain isalso found to be an excellent technology parameter to engineer the band gap and the band gaptype. Our study shows that both uniaxial and biaxial compressive strain can transform thedirect band gap type to an indirect one in 100 cleaved silicon. Biaxial tensile strain for smallthicknesses also shows a similar transformation to indirect band gap type in 100 cleavedsilicon. On the other hand, the material is always direct band gap type for uniaxial tensilestrain. Thus, it is recommended to use uniaxial tensile strain when band gap engineering100 cleaved silicon for opto-electronic applications. In the case of 110 cleaved silicon slab,tensile strain is found to transform the band gap type from indirect to direct. From the 110cleaved slabs under study, it is found that 2% tensile strain can transform band gap type fromindirect to direct for slab thickness up to 4 nm. Here the impact on conductivity effectivemass is also studied methodically. Based on the simulation results, it is found that for the bestdevice characteristics the channel of ballistic transistors on 100, 110 and 111 wafersshould be oriented along [100], [010] and [110] directions respectively.From the device simulation of the n type FDSOI, it is found that using TCAD in-built

bulk parameters for the device simulation results in overestimated device characteristics. Thisseverity is strongly influenced by the SOI thickness. It is seen that effective mass has a largerimpact than band gap on the device transport characteristics. The influence of band gapfrom confinement on the device-on characteristics generates a shift of less than 4 % for 2 nmthick SOI. On the other hand, the shift in the device-off characteristics can be as high as30 % for the same thickness. Similarly, simulating the device with band gap and effectivemass extracted from DFT shifts the device-off current by 92 % for SOI thickness of 2 nm. Therelative shift in device-on characteristics is as hight as 42 % for SOI thickness of 2 nm

The study indicates prospects in opto-electronic applications for 100 and strained 110silicon slabs [15,16]. From the analysis of the multiscale model it is found that it is better touse the confined parameters for ultra-thin body instead of bulk values when studying transportin FDSOI. Especially for SOI structures below 6 nm. This is of relevance for device on-offcharacteristics for thinner SOI, as they are heavily influenced by material parameters. [55].

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simulationofthinsiliconlayers: impactof orientation

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