theoretical performance evaluations of nmos double gate fets … kim... · 2009. 12. 2. ·...

Theoretical Performance Evaluations of

NMOS Double Gate FETs with

High Mobility Materials: Strained III-V, Ge and Si

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF

STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Donghyun Kim

Nov 2009

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/zg341nj2801

© 2010 by Donghyun Kim. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/zg341nj2801

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Krishna Saraswat, Primary Adviser


Philip Wong


Tejas Krishnamuhan

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

iv

Abstract

As Si CMOSFET technology scales down to nanometer scale, it becomes extremely

difficult to keep the same drive current due to limitations of channel mobility (µ), gate

oxide scaling and parasitics. Currently, strained-Si is the dominant technology for high

performance MOSFETs and increasing the strain provides a viable solution to scaling.

However, looking into future nanoscale MOSFETs, it becomes important to look at

novel channel materials such as strained SixGe(1-x) and strained III-Vs and new device

structures.

There are two fundamental problems with these candidates. First, the main advantage

of small transport mass to have high injection velocity may be undermined by low

density of states which reduces the inversion charge and hence reduces drive current.

Second, due to their small effective masses and small band gap combined with high

electric fields applied in nanometer-scale devices, large band to band tunneling (BTBT)

leakage current is a big concern.

To understand the physics and the performance limitation of nanoscale MOSFETs

with high mobility channel materials, we have developed first, a new BTBT model

which takes into account complete real and complex band structure, direct/phonon-

assisted tunneling, and quantum confinement effect, second, a quantum transport

simulator and third, a local pseudo-potential method (LPM) to calculate real and

complex band structures of strained semiconductors. While the BTBT models in

commercially available TCAD tools largely fail to predict accurate current even in

v

silicon, our model is well matched to experimental results of PIN diodes fabricated on

Si, Ge and GaAs, and heterostructure SiGe pMOSFETs.

We have investigated and benchmarked Double-Gate (DG) n-MOSFETs with

different channel materials (GaAs, InAs, InSb, Ge and Si). Our results show that with

gate oxide thickness of 0.7 nm, small density of states (DOS) of these materials does

not significantly limit the on-current (ION) and high µ materials still show higher ION

than Si. However, the high µ small bandgap materials like InAs, InSb and Ge, suffer

from excessive BTBT current and poor SCE, which limits their scalability. InP has

significantly higher ION and shorter intrinsic delay at a reduced IOFF,BTBT when

compared to Si.

Strained (uniaxial and biaxial) InxGa1-x As may have a very good tradeoff between the

excellent transport properties of InAs and the low leakage of GaAs. Full quantum

ballistic simulations with new BTBT models based on the bandstructures calculated by

LPM shows that at a l00nA/ µ m Ioff specification, 4% biaxial compressive strained

In0.75Ga0.25As (111) NMOS DGFET outperforms other InGaAs compositions because

of the excellent transport properties and reduced leakage current with strain

engineering.

vi

Acknowledgements

I wish to acknowledge the support of many people who have contributed to my

Ph.D. research and have made my life at Stanford very enjoyable.

First of all, I would like to express my heartfelt gratitude to my advisor

Professor Krishna Saraswat for meticulously guiding this dissertation, constant

encouragement and support and especially for his patience throughout the course of

my research. I am especially grateful for the time and the freedom he gave me to

choose a research topic and explore a large number of areas in novel device physics.

He has been undoubtedly one of the most kind and helpful people I have ever

interacted with at Stanford. I have been extremely fortunate to work with the finest

advisor that one could possibly hope for.

I would like to thank my associate advisor Professor Philip Wong for all his

help throughout my project especially with regards to nano-electronics and quantum

ballistic transport simulations. He has been an excellent role model and mentor. I have

learned a lot from his vast knowledge, his wide industry experience and strong

technical expertise.

My special thanks go to my mentor friend and once housemate, Prof. Tejas

Krishnamohan for all the wonderful discussions and guidance on a variety of topics

that forms large part of my dissertation. Tejas is one of the brightest scientists I ever

have interacted with. Throughout the hour-long discussions at office cubicle,

vii

restaurants, and at home I have learnt a lot. It was the most intellectual experience I

ever had.

I am also grateful to Professor David Miller for agreeing to serve as the chair

of my Ph.D. oral examination committee. Without his brilliant and insightful lectures

which formed the basis of my quantum mechanics and semiconductor physics, I would

not have been able to even start my Ph.D. project.

I would like to thank Dr. Lee Smith from Synopsys for being an excellent

mentor during my summer internships and for his support throughout my Ph. D. Initial

work on BTBT would not have been done without him letting me have an access to

the source code of TAURUSTM. I acknowledge the computational and technical help

and guidance offered by Professor Robert Dutton.

I am grateful to my undergraduate advisors – Professor Hyo Joon Eom,

Professor Kwyro Lee and Professor Kyunghoon Yang for initiating my interest in

research. I thank them for their guidance and support throughout my undergraduate

years.

I would also like to thank the administrative staffs - Diane, Gabrielle, Gail,

Irene, Miho and Natasha for being very helpful and supportive throughout my stay at

the Center for Integrated Systems.

My gratitude is expressed to the Defense Advanced Research Projects Agency

(DARPA), MARCO Materials, Structures and Devices (MSD) Center, Initiative for

Nanoscale Materials and Processes (INMP) and Samsung Scholarship Foundation for

their financial support.

viii

I owe a debt of appreciation to Professor Saraswat’s former and current group

members who have provided me with ideas, and have made my stay in CIS pleasant

and rewarding: Abhijit, Ali, Aneesh, Arunanshu, Chi On, Crystal, Duygu, Euijong,

Gunhan, Hoon, Hoyeol, Hyun-Yong, Jason, Jin-Hong, Ju Hyung, Jungyup, Kyung

Hoae, Pawan, Raja, Rohan, Rohit, Sarves, Shyam, Ting-Yen, WooShik and Yeul.

I am also grateful to my friends Abhi, Beomsoo, Bob, Byoung Du, Byoungil,

Christophe, Ed, Henry, Hopil, Hyejun, Hyunjoo, Insun, Jack, Karen, Kee-Bong,

Kyungryoon, Kyuwon, Jason, Jekwang, Jenny, Jeongha, Joon Seok, Joo Hyung,

Jungjoon, Kat, Katherine, Kyeongran, Kyungrok, Margie, Mathieu, Mehdi, Misung,

Moonjung, Paul, Rami, Rob, Saeroonter, Salman, Sangbum, Serene, Sergey,

Seunghwa, Soogine, Sora, Sooyoung, Sungwoo, Sunhae, Utkan, Wanki, Wonseok,

Yejin, Yeonjoo, Yeong Min, Youngjoon and Yoonyoung for their friendship and all

the fun that we have had together.

No words would ever do justice to express my deepest thanks and gratitude to

my most wonderful family – my parents and my sister. I will be eternally grateful to

them for being there for me at all times. Their continuous love, sacrifice, support and

encouragement have allowed me to pursue my ambitions.

This dissertation is dedicated to them.

ix

Table of Contents

Abstract .......................................................................................................................... iv

Acknowledgements ....................................................................................................... vi

Table of Contents .......................................................................................................... ix

List of Tables ............................................................................................................... xiv

List of Figures .............................................................................................................. xvi

Chapter 1 Introduction .................................................................................................... 1

1.1 Motivation ............................................................................................................ 1

1.2 Organization ......................................................................................................... 8

Chapter 2 Calculation of Bandstructure ......................................................................... 9

2.1 Introduction .......................................................................................................... 9

2.2 Principles of Pseudopotential ............................................................................. 10

2.3 Band structure calculation with plane wave basis .............................................. 12

2.4 Continuous q-space model potential .................................................................. 14

2.5 Strain dependence of the pseudopotentials ......................................................... 15

2.6 Kinetic-energy scaling ........................................................................................ 16

2.7 G-space smoothing function ............................................................................... 16

2.8 Spin orbit coupling ............................................................................................. 17

x

2.9 Atomic positions in strained materials ............................................................... 20

2.9.1 Strain tensor ................................................................................................. 20

2.9.2 Stress Tensor ............................................................................................... 22

2.9.3 Stiffness and Compliance Coefficients ....................................................... 24

2.9.4 Coordinate transformation ........................................................................... 24

2.9.5 Lattice vectors and reciprocal lattice vectors for a strained system ............ 26

2.9.6 Internal ionic displacement .......................................................................... 27

2.9.7 Uniaxial strain ............................................................................................. 28

2.9.8 Biaxial strain ................................................................................................ 29

2.10 Alloys material (virtual crystal approximation) ............................................... 32

2.11 Complex Bandstructure .................................................................................... 33

2.12 Summary ........................................................................................................... 34

Chapter 3 Modeling of Band to Band Tunneling ......................................................... 36

3.1 Introduction ........................................................................................................ 36

3.2 Direct-gap Band-to-Band Tunneling. ................................................................. 36

3.3 Phonon-Assisted Tunneling / Indirect-gap Tunneling ....................................... 46

3.4 2D Quantized System ......................................................................................... 59

3.5 Direct-gap Band-to-Band Tunneling (2D) ......................................................... 65

3.6 2-D Phonon-assisted Indirect-gap Band-to-Band Tunneling ............................. 71

3.7 Summary ............................................................................................................. 76

xi

Chapter 4 Phonon-Electron Interactions ...................................................................... 78

4.1 Introduction ........................................................................................................ 78

4.2 Electron-Phonon Interaction Element- Rigid Ion Model ................................... 79

4.3 Phonon Polarization Vectors .............................................................................. 80

4.4 Silicon. ................................................................................................................ 82

4.5 Germanium ......................................................................................................... 87

4.6 Summary ............................................................................................................. 91

Chapter 5 Application of band-to-band tunneling (BTBT) model ............................... 93

5.1 Introduction ........................................................................................................ 93

5.2 Comparison to Previous Models ........................................................................ 93

5.2.1 Direct Band .................................................................................................. 93

5.2.2 Indirect Band ............................................................................................... 95

5.2.3 Selection Rules on Phonon modes and Phonon-interaction Strength ......... 96

5.3 BTBT Generation Rates for Various Materials .................................................. 97

5.4 PIN Diodes ......................................................................................................... 99

5.5 Effect of Strain (Ge) ......................................................................................... 103

5.6 Quantum Confinement ..................................................................................... 106

5.7 Summary ........................................................................................................... 111

Chapter 6 Quantum-Ballistic Transport in 1-D Quantum Confined System ............. 113

6.1 Introduction ...................................................................................................... 113

xii

6.2 1-D Quantum Confined System ....................................................................... 114

6.3 1-D Schrodinger Equation with non-parabolic effective mass ......................... 116

6.4 Electron Wavefunction along the Channel ....................................................... 118

6.5 Non-parabolic Density of States ....................................................................... 120

6.6 Charge Density (with Non-parabolicity corrections) ....................................... 121

6.7 Quantum Ballistic Current (with Non-parabolicity corrections) ...................... 128

6.8 Band-to-Band Tunneling Current ..................................................................... 132

6.9 Summary ........................................................................................................... 139

Chapter 7 Properties and Trade-offs Of Compound Semiconductor MOSFETs ....... 141

7.1 Introduction ...................................................................................................... 141

7.2 Simulation Framework ..................................................................................... 145

7.2.1 Bandstructure calculation (Real and Complex) ......................................... 145

7.2.2 Band-to-Band Tunneling (Off-State Leakage) .......................................... 146

7.2.3 2D-Quantum Ballistic Current (On-State Drive Current) ......................... 149

7.2.4 Benchmark Test bed - Device Structure .................................................... 150

7.3 Power-Performance Tradeoffs in Binary III-V materials (GaAs, InAs, InP and

InSb) vs. Si and Ge. ................................................................................................ 151

7.3.1 Inversion charge and injection velocity ..................................................... 152

7.3.2 ION, IOFF,BTBT and Delay ............................................................................. 154

7.3.3 Effect of Scaling film thickness and VDD .................................................. 156

xiii

7.3.4 Power-Performance Tradeoff of Binary III-V Materials vs. Si and Ge .... 156

7.4 Power-Performance of Strained Ternary III-V Material (InxGa1-xAs) ............. 158

7.4.1 Strained InGaAs Band Structures .............................................................. 159

7.4.2 ION and IOFF,BTBT with Strain Engineering and Channel Orientation ......... 161

7.4.3 Power-Performance of Strained Ternary III-V Material (InGaAs) ........... 163

7.5 Strained III-V for p-MOSFETs ........................................................................ 165

7.5.1 Hole Mobility in Ternary III-V Materials (InGaAs vs. InGaSb) .............. 166

7.5.2 Hole Mobility Enhancement in III-Vs with Strain .................................... 167

7.6 Novel Device Structure and Parasitics ............................................................. 169

7.6.1 Quantum Well (QW) Strained Heterostructure III-V FETs ...................... 169

7.6.2 Parasitic Resistance ................................................................................... 171

7.6.3 Parasitic Capacitance ................................................................................. 172

7.7 Conclusion ........................................................................................................ 174

Chapter 8 Conclusion ................................................................................................. 176

References .................................................................................................................. 181

xiv

List of Tables

Chapter 4

Table 4.1 : Polarization vectors of major phonon modes in diamond lattice structure.

Since the diamond structure has a primitive cell formed by two-atom basis, each

phonon consists of two polarization vector. While longitudinal mode consists of one

LA and one LO polarizations, the transverse modes (TA and TO) are double

degenerate. .................................................................................................................... 81

Table 4.2 : The effective deformation potentials and phonon energies for intervalley

scattering between Γ25’ and ΔC ( 2 / (0,0,0.85)laπ ). Energy difference between two states is

1.12eV. Since there is degeneracy in Γ25’ states and transverse modes, the deformation

potentials were added to indicate the effective strength as follows: 2 2

,( ) ( )l

pp l

DK DK= ∑ ,

where p denotes the polarization of phonon and l is indices for degenerate valence bands. ... 83

Table 4.3 :The complex-k-space-intervalley-deformation potentials calculated for Band-to-

Band tunneling between Γ25’ and ΔC. Due to lack of theoretical work, previous BTBT models

(R,T,F,S) estimated the deformation potentials from dilation deformation potentials and

mobility deformation potentials. Experimental values are extracted from light absorption

coefficient study (Ge,Gp) and Esaki tunneling device (C). ................................................. 86

Table 4.4 :The effective deformation potentials and phonon energies for intervalley scattering

between Γ25’ and LC ( / (1,1,1)laπ ) (for LA phonon, between c2−Γ - c1L+ ). Energy difference

xv

between two states is 1.12eV. Since there is degeneracy in Γ25’ states and transverse modes,

the deformation potentials were added to indicate the effective strength as follows:

2 2

,( ) ( )l

pp l

DK DK= ∑ , where p denotes the polarization of phonon and l is indices for

degenerate valence bands. ................................................................................................ 89

Table 4.5 : The complex-k-space-intervalley-deformation potentials calculated for Band-to-

Band tunneling between Γ25’/ c2−Γ and c1L+ /ΔC. (100) at E=-4.12eV. ................................... 90

Chapter 5

Table 5.1 : Parameter F0 of Kane/Hurkx model AFσexp(-F0/F), extracted by our BTBT model

(by LEPM), Experiments and Kane(two band model for direct and effective mass

approximation (EPM) for indirect). The direction indicates tunnel direction ...................... 100

Chapter 7

Table 7.1 : Dominant leakage paths in nanoscale high mobility CMOS devices. High

mobility (low-bandgap) materials suffer from excessive BTBT leakage. ................. 144

xvi

List of Figures

Chapter 1

Fig. 1.1: Scaling the bulk MOSFET in accordance with Moore’s Law (source: Intel).. 1

Fig. 1.2: (a) Need to maintain an increase in the drive current (Ion) even at reduced

supply voltages (Vdd) in the future. (b) The increase in Ion with scaling is obtained at

the expense of an exponentially growing off-state leakage (Ioff) (source: Intel) ............ 2

Fig. 1.3: At the current rate, the exponentially growing static power dissipation will

cross the active power consumption (source: Intel). ...................................................... 3

Fig. 1.4: Carrier transport in ballistic limit. The current is proportional to the velocity

and the density of the carriers at the source, and the backscattering rate ...................... 5

Fig. 1.5: Leakage current components in the channel of highly scaled MOSFETs ....... 6

Fig. 1.6: Three major models to obtain I-V characteristics of highly scaled transistors 7

Chapter 2

Fig. 2.1: Pseudopotential approximation: The tightly bond inner electron shell can be

effectively replaced by the pseudopotential. ................................................................ 11

Fig. 2.2: of Ga and As in GaAs. This is the Fourier transform of the atomic

pseudopotential ............................................................................................................. 15

Fig. 2.3: Two- dimensional geometric deformation of an infinitesimal material

element. ...................................................................................................................... 21

xvii

Fig. 2.4: Components of stress in three dimensions ................................................... 23

Fig. 2.5: Crystal coordinate system ............................................................................ 25

Illustration of band-to-band tunneling paths. The electron can tunnel to indirect-gap

conduction band (L or ∆ valleys) through phonon interactions. Fig. 2.6: Location of an

atom relative to its four nearest neighbors in a crystal strained in the (111) direction.

For z=0 the location of the atom is deformed only by the homogeneous bulk strain,

while for z=1 all four bonds are of the same length. The vector u represents the

differential vector between two locations ..................................................................... 28

Chapter 3

Fig. 3.1:Crystal Potential and corresponding Wavefunctions ...................................... 37

Fig. 3.2: (a) 3D plot of complex bandstructure of germanium along the [111] direction.

The black lines depict the real bands and the red lines depict complex bands. (b) 2D

plot of complex bandstructure of germanium .............................................................. 39

Fig. 3.3: Illustration of electron tunneling process in the semiconductor with the strong

electric field. The electron in valence band tunnels in to the bandgap at the classical

turning point, , passes the branch point, and tunneling ends at . .................... 40

Fig. 3.4: Complex band structure of Ge obtained from local empirical pseudopotential

method (LEPM) along the [111] direction. Parabolic effective mass approximation (a)

fails to follow the complex dispersion in relaxed germanium. While the Kane’s two-

band model works well for relaxed materials (a), it breaks down in strained materials

xviii

(c). The proposed model well approximate the dispersion in both relaxed and strained

germanium (b, d). ......................................................................................................... 45

Fig. 3.5: Illustration of band-to-band tunneling paths. The electron can tunnel to

indirect-gap conduction band (L or ∆ valleys) through phonon interactions. ............. 46

Fig. 3.6: (a) Longitudinal and transverse waves in a 1D linear lattice. (b) Transverse

optical and transverse acoustical waves in a 1D diatomic lattice. (c) The ion is

displaced from equilibrium by an vector, . is the unit-length polarization vector of

the lattice vibration ....................................................................................................... 47

Fig. 3.7: Phonon dispersion relations in germanium. Generally at zone boundaries

such as 2 ⁄ 0.5,0.5,0.5 and 2 ⁄ 1,0,0 , the frequencies are

roughly constant ........................................................................................................... 50

Fig. 3.8: Complex band structure of germanium obtained from local empirical

pseudopotential method (LEPM) along the 110 direction. The valley located at k-

point 2 ⁄ 0.5,0.5,0.5 is projected to Γ point. The parabolic effective mass

approximation (a) completely fails to estimate the complex dispersion, since the

dispersion curve at Γ point deviates a lot from the parabolic curve. The suggested

model (b) successfully approximates the dispersion curve .......................................... 58

Fig. 3.9: Schematic view of the thin channel double-gate MOSFET (DGFET).

Thickness of the channel is . Current flows along x-direction and the channel is

quantized along the z-direction .................................................................................... 59

Fig. 3.10: Illustration of complex bandstructure of two-band two-subband system. The

left panel depicts the scenario where bands with same quantum number are connected

xix

and the right panel shows the scenario where bands with different quantum number are

connected ...................................................................................................................... 63

Fig. 3.11: Complex bandstructure of 14nm thick (001) germanium. The imaginary

values of the wavenumber k are drawn as red curves and the real values are shown in

black curves .................................................................................................................. 68

Fig. 3.12: Γ′ as a function of quantum confinement energy (Δ ) (eq. (3.83)) in

(001) germanium. Tunneling direction is along the (110) direction ........................... 70

Fig. 3.13: Γ , Γ , used in Fig. 3.12 and their fitting functions ..... 70

Fig. 3.14: L , ,′ as a function of quantum confinement energy (Δ ) (eq.

(3.104)) in (001) germanium. Tunneling direction is along the (110) direction .......... 75

Fig. 3.15: , Δ L and , Δ L used in Fig. 3.14 and their fitting

functions ....................................................................................................................... 76

Chapter 4

Fig. 4.1: Bandstructure of silicon calculated by EPM. Phonon-assisted transitions with

highest probability between band extrema are shown. In silicon, transitions between

Δ1-Γ25’ are most dominant. Due to selection rules imposed by symmetry of the two

points, TA and TO phonons are most effective in assisting the tunneling process ...... 82

Fig. 4.2: Comparison between the Lax’ Selection Rules and our work. Since most of

contribution in tunneling is coming at E=-4.67eV, the deformation potentials should

xx

be calculated between two points depicted in (b) not between two symmetry points in

(a). ................................................................................................................................. 85

Fig. 4.3: Bandstructure of germanium calculated by EPM. Phonon-assisted transitions

with highest probability between band extremum are shown. In germanium, transitions

between L1-Γ25’ are most dominant. Experimentally LA, TA and TO phonons are

most effective in assisting the tunneling process. Subindex c in Lc1 represents

conduction band while v represents valence band ....................................................... 88

Chapter 5

Fig. 5.1: (a) Kane’s 2-band model. (b) Full band model obtained by EPM. ................ 94

Fig. 5.2: ......................................................................................................................... 96

Fig. 5.3: (a) 3D plot of complex bandstructure of germanium along the [111] direction

(b) GaAs. ...................................................................................................................... 98

Fig. 5.4: BTBT generation rate vs. Electric Field ........................................................ 99

Fig. 5.5: Indirect tunneling process. Keldysh’s model does not capture full band-

information ................................................................................................................. 101

Fig. 5.6: I-V curves of GaAs PIN diodes. Solid lines with markers are actual I-V

characteristics of the diodes, solid lines are estimations from our BTBT model, and

dotted lines are currents calculated by Kane’s model. The length of I-region is varied

from 15nm to 55nm .................................................................................................... 102

xxi

Fig. 5.7: I-V curves of Ge PIN diodes. Solid lines with markers are actual I-V

characteristics of the diodes, solid lines are estimations from our BTBT model, and

dotted lines are currents calculated by Kane’s model ................................................ 103

Fig. 5.8: Calculated band shifts of germanium (001) as a function of in-plane biaxial

strain ( 0/ 1a a − ) ........................................................................................................ 104

Fig. 5.9: IOFF,MIN vs. biaxial strain. Compressive strain increases indirect BTBT while

it decreases direct BTBT. Optimum point at around x=0.6 ....................................... 105

Fig. 5.10: (a) A GaAs QW consists of 12 atom thick layer with 6 atom thick vacuum

layer. (b) Complex Bandstructure of the GaAs QW .................................................. 106

Fig. 5.11: Ultra-thin body and larger quantization increases the effective bandgap and

lowers the tunneling rate ............................................................................................ 107

Fig. 5.12: Iso-leakage (IOFF,MIN) Contours. These curves determine the maximum

allowable VDD to meet the ITRS off current requirement for a 15nm gate length, at a

given body thickness (constrained by technology). s-Si cannot be used for >0.5V

operation. Ge and InAs can be used for 0.9 operation when Tbody<5nm. For

Tbody<7nm s-Ge can be used for >1V operation. Si and GaAs comfortably meet the

off-current specification because of their large bandgap ........................................... 109

Fig. 5.13: Device structure and band diagrams of SiGe heterostructure p-MOSFET 110

Fig. 5.14: BTBT (GIDL) with varying TGe channel. Left plot is obtained from

experiments and right plot is drawn using simulation results .................................... 111

xxii

Chapter 6

Fig. 6.1: Schematic view of the thin channel double-gate MOSFET

(DGFET).Thickness of the channel is . Current flows along x-direction and the

channel is quantized along the z-direction ................................................................. 115

Fig. 6.2: Convergence of the first energy level. Effective mass is 0.069 and is

0.7eV . Channel thickness is 3nm. About in 5 iterations, it converges within 0.01eV

.................................................................................................................................... 118

Fig. 6.3: The error term Θ , becomes significantly large value when or are

large. (a) For 0.5, 4/ and 40, the error term accounts for more

than 10% of the correct integral. (b) For 2, 4/ and 40, the

error term accounts for more than 25% of the correct integral .................................. 124

Fig. 6.4: Relative error of the approximation in Eq. (6.34) vs. the integral in Eq. (6.24).

This approximation is accurate with an error below 0.3% ......................................... 126

Fig. 6.5: Relative error of the approximation of in ⁄ , Eq. (6.47) vs. the

integral in Eq.(6.44). This approximation is accurate in the range of error below 1.3%

.................................................................................................................................... 131

Fig. 6.6: Source to drain current ( DSI ) vs. gate voltage (Vg). When negatively large

Vg is applied, BTBT current generated at the edge between gate and drain grows

significantly ................................................................................................................ 133

Fig. 6.7: Illustration of the carrier flux components. Electron currents consist of the

one injected from source side, one from drain side and one generated by BTBT

xxiii

process. The BTB tunneling hole current is supplied to the channel cavity (around

). The accumulated holes can escape by thermionic emission from the cavity

................................................................................................................................... .134

Fig. 6.8: Sketchy of the band profile along the channel. Source Fermi level ( ),

Drain Fermi level ( ), Energy profile of the n-th state of valence band, hole quasi-

Fermi level ( ), hole barrier height ( ) are marked in the figure. Only the hole

with kinetic energy larger than can escape the cavity and reach the source side.. 136

Fig. 6.9: Flow chart of the carrier transport simulation of quantum-ballistic FETs ... 140

Chapter 7

Fig. 7.1: Dominant leakage paths in nanoscale high mobility CMOS devices. High

mobility (low-bandgap) materials suffer from excessive BTBT leakage .................. 142

Fig. 7.2: Tradeoffs between effective mass (m ), bandgap (EG), and dielectric constant

(κs) in semiconducting materials ................................................................................ 143

Fig. 7.3: (a) 3D plot of complex bandstructure of germanium along the [111] direction.

(b) 2D plot of complex bandstructure of InAs ........................................................... 146

Fig. 7.4: (a) Illustration of band-to-band tunneling paths. The electron can tunnel to

indirect-gap conduction band (L or ∆ valleys) through phonon interactions. (b) All

Possible (Direct and Indirect) BTBT Tunneling Paths, Γ-Γ, Γ-X and Γ-L, are captured

by the model ............................................................................................................... 148

xxiv

Fig. 7.5: Ultra-thin body and larger quantization increases the effective bandgap and

lowers the tunneling rate ............................................................................................ 149

Fig. 7.6: Test bed for benchmarking III-Vs as channel material. Gate Length LG=15nm,

equivalent gate oxide thickness (EOT) TOX=0.7nm, channel thickness TS=3-5nm and

VDD=0.3-0.7V ............................................................................................................. 151

Fig. 7.7: Inversion charge (Qi) at VGate=0.7V and Tox=0.7nm. Si channel stores 4 times

more electrons than InAs. As Ts scales, Qi becomes bigger due to reduced Vth and

nonparabolicity of band. ............................................................................................. 153

Fig. 7.8: Injection velocities (vinj) at VGate=0.7V and Tox=0.7nm. InAs and InSb have 6

times larger vinj than Si due to their small transport mass. As Ts scales, lowered

threshold voltage (vth) increases vinj, while nonparabolicity reduces vinj. .................. 154

Fig. 7.9: Drive currents (ION) for various (a) Ts and (b) VDD. LG=15nm, TOX=0.7nm.

.................................................................................................................................... 155

Fig. 7.10: (a) IOFF,BTBT is minimum achievable leakage current defined as a

intersecting point between BTBT current and Drive current. (b) IOFF,BTBT for various

Ts. VDD=0.5V. (c) IOFF,BTBT for various VDD .............................................................. 155

Fig. 7.11: Intrinsic delay vs. IOFF,BTBT trade-off in various materials. Lg=15nm,

Tox=0.7nm, Ts=5nm. VDD is varied from 0.3V to 0.7V. ............................................. 157

Fig. 7.12: Band shifts in biaxially strained (a) GaAs (001) and (b) In0.75Ga0.25As (111)

calculated by local empirical pseudopotential method. For GaAs(001), 4% biaxial

compressive (BiC) strain is applied and for In0.75Ga0.25As (111), 4% biaxial tensile

xxv

(BiT) strain is applied. Solid lines represent relaxed material and dotted lines represent

strained material. ........................................................................................................ 159

Fig. 7.13 (a) Quantized bandgap as a function of strain (e||) in 5nm film. Compressive

strain increases bandgap. Especially (111) orientation is strongly modified (b) Γ-L

Separations as a function of strain (e||) in 5nm film. Tensile strain widens the

separation. With tensile strain, ΔEΓL in GaAs can be over 0.3 (eV). ......................... 160

Fig. 7.14: (a) 001 (b) 111. Tunnel mass as a function of x (InxGa1-xAs) in 5nm film.

On (111) Orientation, compressive strain increases mass, while on (001), it reduces

mass ............................................................................................................................ 161

Fig. 7.15: (a) ION as a function of strain (e||) in strained GaAs and In0.75Ga0.25As

NMOS DGFET. For comparison, IONs for 2% biaxially tensile-strained (BiT) Si and

unstrained Si are plotted as dotted lines. (b) IOFFBTBT as a function of strain in strained

GaAs and In0.75Ga0.25As. For comparison, IOFFBTBT for 2% biaxially tensile-strained Si

is drawn as dotted line. 4% (111) compressive strain reduces IOFF,BTBT by over 1000x

.................................................................................................................................... 163

Fig. 7.16: (a) ION vs. IOFFBTBT and (b) Delay vs. IOFFBTBT of the best performing n-

DGFETs with materials, biaxially tensile-strained GaAs (001), uniaxially

compressive-strained In0.25Ga0.75As (001) and biaxially compressive-strained

In0.75Ga0.25As (111). Strain levels are 0, 0.02 and 0.04. Values for biaxially tensile-

strained Si are given for comparison .......................................................................... 165

Fig. 7.17: Mobility for InXGa1-XSb and InXGa1-XAs. Sb’s have higher (~2X) hole

mobilities than As’s .................................................................................................... 167

xxvi

Fig. 7.18: Enhancement of mobility with respect to the unstrained case with biaxial

(left) uniaixal (right) strain. Enhancement with compression is always better than

tension. Upto ~4X / ~2X enhancement possible with 2% uniaxial /biaxial compression.

Enhancement with strain (especially for uniaxial) depends highly on the choice of

channel direction. ....................................................................................................... 168

Fig. 7.19: μh enhancement with biaxial stain; achievable by lattice mismatch between

channel and barrier layers during MBE growth. ........................................................ 169

Fig. 7.20: Conventional and QW double gate FET structures. .................................. 171

Fig. 7.21: Metal Source-Drain IIIV n-MOSFET. ....................................................... 172

Fig. 7.22: Delay in consideration of parasitic gate capacitance. Lg=15nm, Tox=0.7nm,

Ts=5nm. VDD =0.7V. .................................................................................................. 174

1

Chapter 1 Introduction

1.1 Motivation

The silcon-based Metal Oxide Semiconductor Field Effect Transistors (MOSFETs)

has been the most important building block of the Integrated Circuits (ICs), since they

were first demonstrated in 1960 [1]. Today's Integrated Circuits (ICs) use the

MOSFET as a basic switching element for digital logic applications and as an

amplifier for analog applications. Although the basic architecture of the device has

remained the same, the physical size has been continuously reduced by factor of two

every 2-3 years in accordance with Moore’s Law [2] (Fig. 2.1), resulting in faster and

more complex chips with lowered cost per transistor.

Figure 1.1 : Scaling the bulk MOSFET in accordance with Moore’s Law (source: Intel).

Scaling the digital switch.

2

This trend was the main driving force for electronics industry to put humongous

energy and budget into scaling device technologies. Currently, in the 32nm high-

performance processor technology, the physical gate length is scaled below 20nm,

with effective gate oxide thinner than 0.8 nm. This exponential scaling of the physical

feature size cannot continue forever, as beyond the 22nm node fundamental as well as

practical limitations start restricting the performance of the conventional Si MOSFETs.

As illustrated in Fig. 1.2 (a) and (b), after development of 90nm device technology,

enhanced drive currents were achieved with an exponential increase in the static, off-

state leakage of the device.

(a) (b)

Figure 1.2 : (a) Need to maintain an increase in the drive current (Ion) even at

reduced supply voltages (Vdd) in the future. (b) The increase in Ion with

scaling is obtained at the expense of an exponentially growing off-

state leakage (Ioff) (source: Intel).

3

Figure 1.3 depicts the evolution of static power dissipation (leakage) and active power

dissipation as the devices are scaled down. The static power density on the chip has

exponential increased at the much faster rate than the active power density, which

increases rather linearly with device scaling. The active power density arises due to

the dissipative flow of charges between the transistor gates and ground terminals

during switching of the device. The static power, also known as sub-threshold power,

or standby power, is dissipated even in the absence of any switching operation The

major source of the static power dissipation is the sub-threshold leakage in the

transistor because the MOSFET at below 50nm gate length scale does not behave as

an ideal switch and allows the leakage current. The static power dissipation was a

relatively insignificant component of the total chip power just a few generations back,

but it is now comparable in magnitude to the active power. Management and

Figure 1.3 : At the current rate, the exponentially growing static power

dissipation will cross the active power consumption (source: Intel).

4

suppression of static power is one of the major challenges to continued gate length

reduction for higher performance. Once the scaling of the conventional bulk Si

MOSFET starts slowing down, the insertion of performance boosters, like novel

materials and non-classical device structures, will become necessary to continue to

improve performance.

To achieve performance without sacrificing off state leakage power, strain technology

was adopted to boost the drive current. Mechanically stressed Silicon has higher

mobility than relaxed Silicon. However, for further scalability of the device, the

materials with even higher mobility than strained Si are needed. The materials such as

strained IIIV, strained SiGe or even more futuristic carbon nanotubes and graphene

have been actively investigated to study the possibility of substituting Si as a channel

material in future. While IIIV materials such as InAs and InSb exhibit 20 to 40 times

larger mobility than Silicon, high mobility generally comes along with large dielectric

constant and small bandgap which are the basic properties of semiconductor materials

which have weak bonding between atoms.

5

Figure 1.4 illustrates the carrier transport in ballistic limit. In transistors of which gate

length is smaller than 15nm, electrons move almost ballistic and back scattering is

negligible. In this limit, amount of current is multiplication of amount of charge in

source and injection velocity. High mobility materials show potential to improve the

drive current since they have high injection velocity due to its small mass compared to

Si. However, generally these materials suffer from the small density of states in source

side resulting in reduced number of carriers injected from source side. To answer the

question whether these high mobility materials outperform Si in terms of drive current

at the scale below 10nm gate length, more theoretical studies are needed.

Figure 1.4 : Carrier transport in ballistic limit. The current is proportional to the

velocity and the density of the carriers at the source, and the

backscattering rate [3].

Source

Drain

Channel

Injected (vinj)

Back scattered (r)Nsource

⎟⎠⎞

⎜⎝⎛

+−

×=rrvqNI inj

SourceSsat 1

1

Low m*transport High νinjLow r

Small mDOS Small Ns

Ballistic Limit

6

Figure 1.5 depicts possible paths of the leakage current in the channel of highly scaled

MOSFETs. Small mass and small bandgap of high mobility materials enhance source-

drain tunneling and band-to-band tunneling (BTBT) at the gate-drain edge. BTBT

leakage current is a major leakage component in III-V materials and this may hinder

scalability of transistor. To study applicability of III-V materials in highly scaled

MOSFETs, the investigation of off-state leakage current should be included in overall

benchmark.

Basic device performance can be evaluated by examining the Id-Vg characteristics

(Fig 1.6). The drain current has two main components, drive current and leakage

current (mostly BTBT current in III-Vs). Minimum off-state leakage is defined by the

intersection between leakage current line and the drive current line.

Figure 1.5 : Leakage current components in the channel of highly scaled

MOSFETs.

Source

Drain

Channel

Valence Band

Conduction Band S-D Tunneling

Band-to-Band Tunneling

7

To simulate the performance of transistor with novel materials, there are three major

steps. First, the energy band structure of the material should be calculated. Band

structure contains kinetic properties of electrons in semiconductors with periodic

lattice. To obtain energy band structures of semiconductors we have developed

empirical pseudopotential method (EPM). The energy band information in real

reciprocal space (k-space) obtained by the EPM method is fed into quantum ballistic

drive current model to calculate the drive current in the device. To calculate the BTBT

leakage in the device we information on real energy vs. wavenumber k is fed to

Figure 1.6 : Three major models to obtain I-V characteristics of highly scaled

transistors.

log (IDS)

Ioff,min

VGS

IBTBT

IDrive

-1 -0.5 0 0.5 1-12

-10

-8

-6

-4

-2

k ( 2π/A0)

E (e

V)

(111) (100)

Energy Bandstructure by Empirical

PseudopotentialMethod (EPM)

Quantum Ballistic Drive

Current Modeling

Complex E-k

Real E-k

Band-to-Band Tunneling Modeling

(BTBT)

8

quantum ballistic Drive Current Model. The complex band information is needed to

calculate BTBT leakage current.

1.2 Organization

The thesis is organized as follows:

Chapter 1: Introduction

Chapter 2: Calculation of Bandstructure

Chapter 3: Modeling of Band-to-Band Tunneling

Chapter 4: Phonon-Electron Interactions

Chapter 5: Application of BTBT model

Chapter 6: Quantum Ballistic Transport in 1-D Quantum Confined System

Chapter 7: Properties and Trade-offs Of Compound Semiconductor MOSFETs

Chapter 8: Conclusion

9

Chapter 2 Calculation of Bandstructure

2.1 Introduction

The complex movement of carriers in semiconductor can be simplified once we have

the bandstructure of the material. The bandstructure of a semiconductor describes

ranges of energy and crystal momentum (k) that an electron is “forbidden” or “allowed”

to have. It is due to the diffraction of the quantum mechanical electron waves in the

periodic crystal lattice. The bandstructure obtained in real 3-D crystal momentum (k)

space provides us with information on carrier’s density of states, quantum quantization

effective mass, transport effective mass. The band structure also can be expanded in

complex k space where k space is no longer real number rather complex number. Band

information in complex k-pace provides information on the decaying states coupling

two bands.

The pseudopotential method has been widely used to calculate the bandstructure of

semiconductors [4-9]. While ab-nitio methods are capable of calculating the

bandstructures of the wide range of materials and provide relatively accurate values in

entire k-space, they fail to predict observed bandgap and effective mass at the same

time. Local empirical pseudopotential method is a very powerful tool in that it requires

the least number of parameters and provides very accurate bandgaps (error <0.05eV)

and effective mass (error <5%) at the most important symmetry points (Γ, ∆, L).

10

2.2 Principles of Pseudopotential

The complete Hamiltonian for the periodic semiconductor structure is expressed as

follows [10]:

(2.1)

where

2e

(2.2)

and are the positions of the electrons and is the mass. The second term is:

2e

| | (2.3)

where are the positions, the atomic numbers and the masses of the nuclei.

The final term is:

e

,

(2.4)

In general, there are infinitely large numbers of electrons in a typical semiconductor,

and therefore it is almost impossible to track all these electrons. Furthermore, the

strong Coulomb potential of the nucleus terms tend to go to negative infinity as an

electron position gets close to the center of a nucleus, which requires very fine special

gridding in numerical calculation of this system.

However, there are several approximations [11] that allows one to achieve both

acceptably accurate bandstructure and to reduce the computation complexity.

11

i. Pseudopotential approximation (Fig.2.1): In semiconductor materials, the

electrons below the outer shell are tightly bound to the nucleus and the core

states remain almost unchanged by interaction from electrons in outer shell

(or valence bands). This permits us to replace the strong Coulomb potential

of the nucleus and the effects of the tightly bound core electron by an

effective ionic potential acting on the valence electrons.

ii. The independent electron approximation replace the complicated electron-

electron interactions with a time averaged potential.

With these approximations, this system can be treated as a one-electron problem and

the above Hamiltonian in Eq. (2.1) is simplified as:

2 (2.5)

Figure 2.1 : Pseudopotential approximation: The tightly bond inner electron shell can be effectively replaced by the pseudopotential.

0

V(r)

rqz eff

πε4−

Atomic Potential r

3s (valence electrons)

r

Ψ(r) 2s (non-valence)

Screened by core electrons (non-valence)

This Approximation reduces computational complexity

Pseudopotential

12

where is crystal potential that includes the interaction between the electrons and the

nuclei and the interaction between the electrons.

2.3 Band structure calculation with plane wave basis

The orthogonal plane wave (OPW) expansion [5] is the most general approach to

construct the Hamiltonian for periodic systems. Translational symmetry allows one to

build the complete set of plane waves:

,1

√Ωe (2.6)

where Ω is the volume of the lattice, G are the reciprocal lattice vectors which satisfy:

2 , where œ (2.7)

and are the Bravais lattice vectors.

Now we expand the Hamiltonian in Eq. (2.5) with the plane waves in eq. (2.6):

, , , 2| | , (2.8)

Here, we used the orthonormality property of the basis set:

, , , (2.9)

and is the potential given by:

1Ω

e e (2.10)

The crystal potential is expressed as a summation of potential contributions from

each atomic site, .

13

(2.11)

is assumed to be a local pseudopotential, dependent only upon . Although adding

non-local potentials is essential for photoluminescence study and provides better

fitting to wide range of electronic properties [4], it requires too many fitting

parameters. In this work since our interest is only around vicinity of a few symmetry

points (Γ, ∆, L) in valence band and conduction, treating only local terms is good

enough for accurate modeling of the range of interest and results in easier fitting

process. Using this Vc for eq. (2.10) gives:

1Ω

e d1Ω

e e d

ΩΩ

e1

Ωe d

(2.12)

where we write and Ω is the volume of single Bravais lattice point; in

diamond structure, that is:

Ω /4 (2.13)

where, 0A is the lattice constant.

The factor in eq.(2.12) :

1Ω

e d , (2.14)

is the Fourier transform of the atomic pseudopotential aV and is also called as

pseudopotential form factor. Therefore, the complete form for the matrix elements can

be written as:

14

, 2| | ,

ΩΩ

e , (2.15)

, can be obtained by self-consistent ab initio method or by fitting to empirical

data for the bulk constituents [12] (e.g., energy gaps, effective masses). In this work,

since the bulk constituents for the semiconductor materials in interest are well

measured and known, the local pseudopotential form factor will be empirically

fitted. This method is called the local empirical pseudopotential method [5] (LEPM).

2.4 Continuous q-space model potential

To derive the band-structures in relaxed IV or III-V semiconductor materials, only

information of on few numbers is required. In the standard derivation of the

EPM, the assumption is made that the atomic pseudopotentials are spherical with the

consequence that form factors, depend only on the magnitude of the reciprocal

lattice vectors , i.e., q where q | |. High frequency components in

the form factors are safely ignored and in general only 6 form factors are enough to

calculate acceptably accurate bandstructure. However for strained systems or

superlattice systems, the knowledge of the form factors on vicinity of non-reciprocal

lattice points is required. A variety of algebraic forms have been used in the past to fit

empirical pseudopotentials on continuous q-space [7-9, 12-14]. In this work, we take a

form suggested in [9], i.e.:

,1

Ω e 1 (2.16)

15

where are the fitting parameters. By using this form, where the value is

defined for q=0, fitting band-offsets and electron affinity is possible.

2.5 Strain dependence of the pseudopotentials

Under the hydrostatic strain, redistributed local charge affects the screened potentials

in (2.16) [15] . Since we solve this pseudo-potential non-self consistently, it is

necessary to introduce a new fitting parameter to accurately model the hydrostatic

deformation potentials [8, 15]. This approach is too add a strain dependent term is

analogous to the scaling law in the tight-binding method [16]. Our screened pseudo-

potential takes the form:

, , , , 0 1 Tr (2.17)

Figure 2.2 of Ga and As in GaAs. This is the Fourier transform of the atomic pseudopotential.

0 1 2 3 4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

q (2π/A)

V(q)

(ryd

berg

)

MP of Ga and As in GaAs at 300k

GaAs

16

The local hydrostatic strain Tr for a given atom at is basically a hydrostatic

volume change by the strain and it can be expressed as:

TrΩ Ω 0

Ω 0 (2.18)

2.6 Kinetic-energy scaling

While using the strain dependence term and continuous G-space form factors, we can

obtain acceptably accurate (error <0.01eV) bandgaps, it is not easy to get accurate

effective masses. It is due to neglect of non-local many-body potential. Cohen in [6]

showed that when the self-energy term is expanded to Taylor series, the lowest order

term can be represented as the kinetic energy. Thus, in this work we introduce the

kinetic energy scaling constant, , and the Hamiltonian matrix elements in eq. (2.15)

are now written as [9]:

, 2| | ,

ΩΩ

e , (2.19)

By introducing the constant , simultaneous fitting of both bandgaps and bulk

effective masses is achieved.

2.7 G-space smoothing function

The atomic pseudopotential evanesces quickly with the length of the reciprocal-lattice

vector G and the long G vectors have only negligible effect on the band-structure [5].

This property lets us use only small number of plane waves and in this work we use a

17

cutoff energy of 5 Ry, which generates about 60 plane waves. However, for strained

lattices or superlattices, if the reciprocal-lattice vectors are cut off sharply, then plane

waves at high frequency may be arbitrarily added or removed; this may result in

sudden change in bandstructure. To ease this problem, it is beneficial to introduce a

smoothing function to the atomic pseudopotentials [7], i.e. :

12

tanh5

0.31 (2.20)

where q are in atomic unit. After adding the smoothing function and hydrostatic strain

dependent constant, the atomic pseudopotential in eq. (2.17) now becomes:

, ,1

Ω e 11 Tr

12

tanh5

0.31 (2.21)

2.8 Spin orbit coupling

Although it is screened by core electron, extra positive nuclear charge creates an

electric field pointing radially outward. According to the special relativity theory of

Einstein, in the reference frame of an electron in valence band, it appears as if the

nucleus is moving and the moving electron experiences a magnetic field. Therefore

there is a spin-orbit interaction between the magnetic field associated with the orbital

motion of electrons in valence bands and the internal magnetic moment of the electron.

Based on the special relativity theory, the magnetic field (to first order) experienced

by electron moving with momentum in an electrostatic field is given by:

18

(2.22)

where is the electron mass and c is the speed of light. Since the electrostatic field

is roughly spherically symmetric, it can be expressed in spherical coordinates:

14

(2.23)

where e is an electron charge, is the dielectric constant of the material and Z is the

effective atomic number which partially takes into account the effect of screening by

core electrons on nuclear charge. Using eq. (2.23). and the relation , the

magnetic field can be expressed with the angular momentum of the electron , i.e.:

4 4 (2.24)

The magnetic dipole moment of electron spin angular momentum is given as:

g2

(2.25)

,where the factor g is the gyromagnetic ratio of the electron. in eq. (2.25) can be

expressed using Pauli spin matrices.

2 (2.26)

Pauli spin matrices are defined as:

(2.27)

In conjunction with eq. (2.24) and eq. (2.25), the spin-orbit interaction energy can be

written as:

19

g2 4 4 2

g2

(2.28)

In most of semiconductor materials, the outermost p-core states provide the biggest

contribution on the spin-orbit interaction. Inner core states and d-core states may be

neglected [17]. The resulted spin-orbit interaction terms from outermost p-core states

expanded with reciprocal lattice vectors and the result is as follows [4, 17-18]:

, , ,Ω2Ω

e , (2.29)

where we define:

(2.30)

is an adjustable parameters to be fitted to the band splitting. Reiger calculated

and found the analytic approximation form [19]. i.e:

1 ⁄ /51 ⁄ (2.31)

In this work, we cut off G at 5 Ry which corresponds to 4 . For this range of G:

⁄ 0.15 and1 ⁄ /51 ⁄ 1 (2.32)

With this approximation and using the ratio of the spin-orbit splitting of the free

atoms, we only need to find one fitting parameters of to get the band splitting.

Finally, the spin-orbit interaction term used in this work is:

, , ,Ω2Ω

e , (2.33)

Thus, the complete matrix elements including the spin-orbit interactions are:

20

, , ,

2

2 | |2, ,

ΩcΩ e , ,

Ωc2Ω e , (2.34)

The possible spin states combinations (s’,s)

are , , , , , and , .

2.9 Atomic positions in strained materials

2.9.1 Strain tensor

Under applied stress, a solid is strained resulting in a change of volume and shape.

Consider a 2-dimensional deformation of an infinitesimal rectangular material element

with dimensions dx by dy (Fig. 2.3). A vector describes the displacement as a result

of the stretching and it is a function of position. After deformation, it becomes a

rhombus. The distance between a and b, now can be expressed with dx and .:

2 2

1 22 2

(2.35)

For very small displacement gradients (i.e. 1), the distance is approximately, :

(2.36)

21

The normal strain in the x-direction of the rectangular element is defined by

(2.37)

This strain quantity measures the extension per unit length of the vector. Similarly,

the normal strain in y and z directions is:

, (2.38)

When shear strain is applied, the angle between two orthogonal material lines is

changed. The line , is more tilted to y-direction by amount of . The angle

between the line and the line is:

Figure 2.3 Two- dimensional geometric deformation of an infinitesimal materialelement.

22

tan (2.39)

The angle adopting the same derivation is .

For the rigid body to be strained by shear strain should not be zero. If the

addition is zero, the rigid body just rotates without being strained. Thus, the net shear

strain could be defined as:

12

12 (2.40)

Using the same analogy to define other strain tensor components, the 3-dimensional

strain tensor is constructed as:

12

12

12

12

12

12

(2.41)

From the relation above, we can discern:

(2.42)

2.9.2 Stress Tensor

When a solid is strained by external forces and deviates from its equilibrium condition,

the strain leads to the generation of internal forces. The measure of this internal forces

acting on the surface of an imaginary body is called stress. The stress tensor is

defined by:

23

,, ,

(2.43)

Where the are the components of the force vector acting on a small area dA which

can be represented by a vector perpendicular to the area element, facing outwards

and with length equal to the area of the element.

Note that equilibrium requires that the moment resulted by the stress must be zero.

Taking the moment of the forces above about any suitable point, it can be shown that

(2.44)

Figure 2.4 Components of stress in three dimensions

24

2.9.3 Stiffness and Compliance Coefficients

According to the generalized Hooke’s law, it is possible to define the proportionality

constants between stress and strain [20]. :

, , , , , , … , , (2.45)

where are the compliance coefficients and are the stiffness coefficients. Note

that the strain used in eq. (2.45) is called engineering strain. The shear strain of

engineering strain is twice bigger than the strain notation we used in previous

equations. We can construct a 6x6 matrix with either the compliance coefficients or

the stiffness coefficients. When the crystal-axis coordinate system is used for a cubic

crystal, the stiffness coefficients reduce to the following matrix:

c c c 0 0 0c c c 0 0 0c c c 0 0 0

0 0 0 c 0 0 0 0 0 0 c 00 0 0 0 0 c

(2.46)

A similar matrix can be constructed for the compliance coefficients.

2.9.4 Coordinate transformation

To find the components of the strain tensor and elastic stiffness tensor for any device

coordinates, it is necessary to perform a coordinate transformation.

25

The rotational transformation from the crystal axes to the arbitrary system , is

expressed as [20]:

, 1,2,3 (2.47)

The transformation operator can be defined as:

(2.48)

and the operation between the initial coordinate X to the transformed coordinate X’

can be written as:

and (2.49)

Using the result in eq. (2.49), the transformed strain tensor can be expressed with

the initial strain tensor and vice versa:

(2.50)

Figure 2.5: Crystal coordinate system

(001) (111)

(010)

(110)(100)a

26

Thus,

and (2.51)

The elastic stiffness tensor is transformed as follows:

, , , , ,

(2.52)

Similarly the compliance tensor can be transformed as:

, , , , ,

(2.53)

2.9.5 Lattice vectors and reciprocal lattice vectors for a strained

system

The homogeneous bulk strain deforms the atomic locations. The strained lattice

vector can be written in terms of the strain tensor and the relaxed lattice vector ,

i.e:

(2.54)

,where is the unit tensor, of which diagonal elements are one and off-diagonal

elements are set to zero.

The deformation of the lattice vectors leads to change in reciprocal lattice vectors as

well. The strained reciprocal lattice vectors, are calculated as:

1 (2.55)

27

2.9.6 Internal ionic displacement

For orthorhombic strain, all the locations of atoms in a strained crystal can be

expressed as in eq. (2.51). However, in the presence of shear strain, they are deformed

in a different way. Shear strain results in different length of four bonds [21]. As

depicted in Fig. 2.6, if the atoms are relocated only by bulk strain, the resulted bonds

to three circumcenter atoms b, c and d have larger energy than the bond to atom a.

This creates an internal strain in order to minimize the overall bonding energy. The

effect of internal strain can be represented by single proportionality parameter z. The

internal strain relocates the central atom somewhere between the position (z=0)

determined by the bulk strain tensor and the location (z=1) where all four bonds are of

same length. The finally the location of a strained central atom is:

z (2.56)

where is an internal ionic displacement vector depicted in Fig. 2.6. Since z is a weak

function of the level of strain and [22], in this work we use a single constant for each

material.

28

2.9.7 Uniaxial strain

If one-directional pressure is applied along a non-crystal coordinate axis, the applied

pressure generates off-diagonal stress components. The easiest way not to deal with

the off diagonal stress is to transform the coordinate so that its axis is aligned with the

applied pressure. Within that coordinate the applied pressure only induces normal

stress.

Let’s say the pressure is applied along the x-axis of coordinate system A, and the

crystal axis coordinate system is C. Two coordinate systems are connected through

transformation U.

(2.57)

The stress tensor in coordinate system A is:

Figure 2.6: Location of an atom relative to its four nearest neighbors in a crystal strained in the (111) direction. For z=0 the location of the atom is deformed only by the homogeneous bulk strain, while for z=1 all four bonds are of the same length. The vector u represents the differential vector between two locations.

z=1

z=0

a

b

cd

u

29

A0 0

0 0 00 0 0

or 00000

(2.58)

The strain tensor in coordinate system A is then given by:

A A A2⁄ 2⁄

2⁄ 2⁄2⁄ 2⁄

(2.59)

where the elements of compliance matrix can be obtained by applying eq. (2.53) on

. The factor ½ is put to transform the engineering strain notation.

Finally, the uniaxial strain tensor in crystal axis coordinate system C can be found by

performing the coordinate transformation on A as explained in eq. (2.51).:

C A (2.60)

2.9.8 Biaxial strain

A very thin epitaxial film A, of which relaxed lattice constant is , grown on a

substrate with lattice constant is biaxially strained in order to conform to the

substrate. The in-plane strain condition can be expressed as:

(2.61)

In the substrate coordinate system, in-plane strain represents and . Other

elements of the strain tensor can be calculated using two conditions that under biaxial

30

strain only in-plane stress and exists and two in-plane strain elements are of

same value. We first equate the strain tensor to the stress tensor using the compliance

matrix :

222

0000

(2.62)

By equating the first strain tensor element and the second one, and can be

expressed in terms of , i.e:

and (2.63)

Since :

31 32 (2.64)

by plugging eq. (2.63) into eq. (2.64) we have :

(2.65)

Shear strain elements are

2 (2.66)

2 (2.67)

2 (2.68)

31

The most commonly used substrates for semiconductor wafers are [001], [110] and

[111]. Three transformation operators for those substrates take the forms:

1 0 00 1 00 0 1

(2.69)

01

√21

√2

01

√21

√21 0 0

(2.70)

1√6

1√2

1√3

1√6

1√2

1√3

23

01

√3

(2.71)

Using the eq. (2.65)-(2.68) and transforming the strain tensors, the elements for the

strain tensors in the crystal axis coordinate system for these three substrates can be

obtained as:

[001] substrate:

,

2 12

11,

0. (2.72)

[110] substrate:

2 44 12

11 12 44,

,

11 2 12

11 12 44,

0. (2.73)

32

[111] substrate:

4 44

11 2 12 4 44,

11 2 12

11 2 12 4 44. (2.74)

, , and are the stiffness coefficients in the crystal axis coordinate.

2.10 Alloys material (virtual crystal approximation)

Mixing two binary or single 4 column element semiconductors to form an alloy is a

very widely used technique to yield a ternary or binary material with a desirable

bandstructure for specific applications. To calculate the bandstructure of alloy material,

the simplest way would be to put particular atoms at each lattice site. However this

method requires large basis especially if the composition ratio of two mixed materials

is not one to one. The easier method would be assuming that the properties would vary

linearly between the two constituents. This method is known as the virtual crystal

approximation.

In this work, first, we determine the average lattice constant for the alloy. Since the

lattice constant of the alloy is not same as the ones for original materials, two

constituents should be strained to match it to the lattice of the alloy and then we

calculate the bandstructures of two strained constituents. The final bandstructure for

the alloy is obtained by mixing two strained bandstructures linearly in proportion from

components of each material.

33

This method has advantages over other commonly used methods where the form

factors are mixed. It allows using any pseudo-potential form factors without being

problematic for certain materials. In addition, it can accurately reproduce the values of

bowing parameters and hydrostatic deformation potentials measured in optical

experiments.

2.11 Complex Bandstructure

When a large electric-field is applied in semiconductor material, electron in valence

band can tunnel through the band gap and reach at conduction bands. This process is

called band-to –band tunneling (BTBT). During this process electron wavefunction

decays, in other words, it has imaginary wavenumber k. Thus, it is essential for precise

modeling of BTBT to obtain complex bandstructure of semiconductor material.

Normally the Hamiltonian is constructed and solved for eigen-energy for a given k. To

obtain complex band structure using the standard method, it is necessary to find a

specific complex k which ensures the Hamiltonian is still a Hermitian matrix so that

eigen-energy is still a real number. This process is very time consuming and

furthermore, it is quite difficult to find all of the correct complex k.

Instead, in this work, we transform the matrix into the equation finding for eigenvalue

k. Since the Hamiltonian matrix defined in eq. (2.34) is a Hermitian matrix, the eigen-

value of the matrix is also real:

, (2.75)

At the given real and energy , the Hamiltonian can be written as:

34

, , (2.76)

where . is a reciprocal lattice vector from which we

would like to find the complex k. is a complex number and is a unit

vector pointing the direction along which we would like to find a complex

bandstructure:

| | 1 (2.77)

We introduce a new vector , i.e.:

(2.78)

Using the new vector u, eq. (2.75) and eq. (2.76), the equation can be transformed into

the following form.

(2.79)

By diagonalizing a single matrix equation above, it is possible to obtain all of the

complex k solutions for fixed , and .

2.12 Summary

We have developed an empirical pseudo-potential method to calculate the

bandstructures of semiconductors in zinc-blende lattice structure. The atomic

dislocation induced by applied strain is carefully examined including the effect of

internal strain caused by shear strain based on the infinitesimal strain theory. Atomic

35

pseudo-potential is modeled in continuous reciprocal vector space as a function of

strain tensor and reciprocal vector. We included the spin-orbit coupling which causes

splitting in degenerate 3 valence bands. The concept of kinetic energy scaling is

adopted to take into account non-local many-body potential. For alloy materials,

virtual crystal approximation was used. Finally, the complex bandstructure was

obtained by modifying the Hamiltonian matrix such that crystal-momentum k-vector

is eigen-value.

36

Chapter 3 Modeling of Band to Band Tunneling

3.1 Introduction

High mobility materials are being investigated as channel materials in highly

scaled MOSFETs to enhance performance [23-34]. The materials such as GaAs, InAs,

Ge, strained Si and strained Ge have larger carrier mobility than silicon, but the

enhanced band to band tunneling (BTBT) because of their smaller bandgap or direct

band gap may limit their scalability [34]. Although it becomes important to predict the

band to band tunneling leakage of devices made with high-mobility materials, the

most commonly used BTBT models in commercial TCAD tools such as MINIMOSTM ,

MEDICITM, and SENTAURUSTM do not capture quantization effects and full band

information of nano-scale devices. In this work, we have developed a BTBT model,

which captures important quantum mechanical (QM) effects in highly scaled DGFETs.

3.2 Direct-gap Band-to-Band Tunneling.

Under slowly varying envelope function approximation, a wavefunction in lattice

structure can be written in the form:

Ψ , , (3.1)

where Ψ represents the actual electron (or hole) wave function in the band ‘b’ (either

conduction band or valence band). , is unit cell function and normalized in

37

space. , is an envelope function. Fig. 3.1 illustrates the unit cell function, the

envelope function and the crystal potential.

The Schrodinger equation for an electron in non-uniform external potential is:

2,

, ′ , ′

′,

(3.2)

,where xm is a transport effective mass, is an energy of the electron. The inter-

band coupling terms ′are defined as:

′ , ′ ′ , ′ (3.3)

In the above equation, is the force on the electron and it can be defined as,

. The quantity defined as eq. (3.4):

′ , ′ ′ , (3.4)

describes the amount of transition which occurs between the states per unit force.

These off-diagonal matrix elements connect between different bands and can be

Figure 3.1: Crystal Potential and corresponding Wavefunctions. Crystal Potential

Unit‐cell function

,

Envelope Function ,

38

interpreted as band-to-band tunneling (BTBT). If the inter-band terms in the eq. (3.2)

are ignored, the eigen-functions of the eq. (3.2) at the tunneling trails are (R.P.

Feynman, Revs. Modern Phys. 20, 367 (1948)) :

, 2

2 exp ′ ′ (3.5)

where is length of semiconductor along the tunneling direction and is an energy

of the band in the periodic lattice.

Note that ′ is a complex number inside the bandgap and the envelope wave function,

, becomes a decaying function toward the center of bandgap. Fig 3.2

illustrates the complex band structure of germanium along the [111] direction. The

black line depicts the real bands and the red line depicts the complex bands. The most

probable direct-gap tunneling occurs along the red line connecting Γ and Γ . When

electron at the top of the valence band starts tunneling toward a bandgap, ′ of the

tunneling wavefunction is given by values in the red line.

39

The tunneling probability per unit time between different bands is given by Fermi-

golden rule:

2′ (3.6)

and the perturbation potential connecting two bands b to b’, ′ is:

′ , ′ , ′ ⁄1

, ′ , ′(3.7)

where and are classical turning points as depicted in Fig. 3.3 and V is the

volume of the semiconductor. The density of state is given by inverse of energy

spacing between Wannier-Stark ladders (Δ :

(a) (b)

Figure 3.2: (a) 3D plot of complex bandstructure of germanium along the [111]direction. The black lines depict the real bands and the red lines depict complex bands. (b) 2D plot of complex bandstructure of germanium.

-0.2-0.15

-0.1-0.05

-0.200.20.40.60.8

‐3

‐3.5

‐4

‐4.5

‐5

‐5.5

00.8 0.400.10.2√ √

Energy (eV)

√√0 0.4 0.8 1‐0.2

Energy (eV)

‐3.5

‐4

‐4.5

‐5

‐5.5

40

1 Δ⁄2

(3.8)

,where and is a lattice constant.

Using equation (1.4), equation (1.5) is rewritten as:

′ (3.9)

By plugging eq. (3.5) into eq. (3.7), we have an expression:

exp ′ ′

.

(3.10)

After integrating over y and z dimensions first, the eq. (3.10) becomes:

Figure 3.3: Illustration of electron tunneling process in the semiconductor with thestrong electric field. The electron in valence band tunnels in to thebandgap at the classical turning point, , passes the branch point, and tunneling ends at .

E(k)

E(k)E(k)

Conduction BandValence Band

Energy

, ,

41

1

, ,

exp ′ ′

.

(3.11)

The terms , , represent the momentum conservation along the transverse

directions during the tunneling process. By using exchange of variables,

and , eq. (3.11) becomes as:

1

, ,

exp ′ ′

(3.12)

According to two band theory, has a pole around a branch point, . Around

the pole, is:

1

4

1

4

(3.13)

, where is a branch point in x-space. We rewrite the exponential term in eq. (3.12),

exp ′ ′ around the branch point ,

exp ′ ′ exp (3.14)

, where ⁄ and is:

′ ′,

′,

42

,,,

(3.15)

.With eq. (3.13) and eq. (3.14), perturbation energy element, becomes:

14 , , exp

(3.16)

The complex integration is performed in the vicinity of :

exp

exp

exp exp 3exp exp 3

, ∞

,

,

,

,

, ∞

2 6 exp43 exp

By using the complex integration result, off-diagonal perturbation energy elements is:

3 , , exp (3.17)

To compute the BTBT rate per unit volume, we integrate transition rate, in eq. (3.9)

over the entire k-space.

3 exp , , ,

2 3 exp 2 2 2

∞

∞

∞

∞

36 exp 2∞

∞

∞

∞

43

exp 2 1 ′ ′| ,

36′ ′

∞

∞

∞

∞

72

exp 2| ,

∂ ∂

(3.18)

, where is the imaginary part of complex number .To obtain the approximated

value of coefficient, / ∂ / ∂ , we adopt the Kane’s two band model.

1Γ Γ Γ⁄

Γ.

Γ.

4Γ

.

4 Γ 2 Γ 2 Γ

.

| ,

∂ ∂3 Γ

.Γ

.

16 Γ.

Γ.

(3.19)

where Γ is direct bandgap tunneling mass, Γ is relative density of states

(DOS) in y-direction and Γ is relative DOS in z-direction. These mass numbers

are to be determined by bandstructure calculation. Plugging eq. (3.19) into eq. (3.18)

results in total direct band-band tunneling rate per unit volume:

2 Γ.

Γ.

27 Γ.

Γ. exp 2 | , .

(3.20)

When electric field is relatively smaller than electric field in crystal momentum, the

path integration, in eq. (3.20) has pretty large number. Since the evaluation of

44

involves the path integral and is sensitive to changes in electric field during the

tunneling, we are allowed to compute only numerically. As pointed out by

Krieger [35], inaccurate evaluation of results in an error of several orders in

BTBT rate, due to the nature of exponential functions. In most of semiconductor

materials, where 4 valence bands are strongly interacting with each other, using the

Kane’s two band model inevitably leads to large amount of error. Furthermore, Kane’s

model is inherently unable to deal with strained materials.

Here we model as:

Γ′

1 Γ Γ Γ

Γ, 0 Γ

1 Γ Γ Γ Γ

Γ, Γ Γ

(3.21)

Γ Γ Γ Γ⁄ , Γ Γ Γ Γ , ΓΓ Γ

Γ Γ,

(3.22)

Γ′

(3.23)

where is the electron energy measured from the top of the valence band at point

x. Two fitting parameters Γ and Γ represent the BTBT tunneling mass for

valence band and conduction band respectively. Γ and Γ are extracted so that

Γ′ in eq. (3.21) matches with the complex band structure calculated by empirical

pseudo-potential method. We can see in Fig. 3.4 this suggested model can be precisely

fitted to bandstructure of semiconductors including strained materials. The parabolic

45

effective mass approximation (Fig. 3.4(a)) fails to follow the complex dispersion in

relaxed germanium. While the Kane’s two-band model works well for relaxed

materials (Fig. 3.4 (a)), it breaks down in strained materials (Fig. 3.4(c)). The

proposed model well approximates the dispersion in both relaxed and strained

germanium (Fig. 3.4( b, d)).

(a) (b)

(c) (d)

Figure 3.4: Complex band structure of Ge obtained from local empirical pseudopotential method (LEPM) along the [111] direction. Parabolic effective mass approximation (a) fails to follow the complex dispersion in relaxed germanium. While the Kane’s two-band model works well for relaxed materials (a), it breaks down in strained materials (c). The proposed model well approximate the dispersion in both relaxed and strained germanium (b, d).

0 0.04 0.080.020.06

‐3.4

‐4

‐4.6

‐5.2

Energy (eV)

√√

LEPM

Two‐Band

Effective mass

Relaxed Ge

0 0.04 0.080.020.06

‐3.4

‐4

‐4.6

‐5.2

Energy (eV)

√√

LEPM

This work

Relaxed Ge

0 0.04 0.080.020.06

‐3.4

‐4

‐4.6

‐5.2

Energy (eV)

LEPMTwo‐Band

Strained Ge on Si (001)

0 0.04 0.080.020.06

‐3.4

‐4

‐4.6

‐5.2

Energy (eV)

LEPM

This work

Strained Ge on Si (001)

46

3.3 Phonon-Assisted Tunneling / Indirect-gap Tunneling

In an indirect bandgap material, like silicon or aluminum arsenide, the smallest energy

separation for electrons in conduction band and holes in the valence band occurs for

very different crystal momentum. Band-to-Band tunneling is possible between these

states with the participation of a phonon to conserve momentum. Fig. 3.5 illustrates

the possible tunneling paths. Note that since this process requires both momentum and

energy to conserve, during the transition the electron either emits the phonon energy

or absorbs the phonon energy. Usually the energy of phonon is around tenths of meV.

The bandstructure of semiconductor is influenced by changes in lattice spacing. At

room temperature, vibrating atoms perturb lattice constant and bandstructure. For each

Figure 3.5: Illustration of band-to-band tunneling paths. The electron can tunnel to indirect-gap conduction band (L or ∆ valleys) through phonon interactions.

ConductionBand

Valence Band

Energy

Complex k‐+

-

-

Tunneling Process

Real k

Γc

Γv

-

-

L or ∆

Conduction Band

Real k

--

-

Phonon

Direct Tunneling (Γv Γc)

Indirect Tunneling : Phonon couples

Γv to Lc /∆c

47

wavelength of a vibration mode in a Zincblende crystal structure, there are six

branches, two transverse acoustic modes (TA), one longitudinal acoustic mode (LA),

two transverse optical modes (TO), and one longitudinal optical mode (LO) (Fig. 3.6

(a) and (b)).

Lattice displacement by phonon vibration of wave-vector can be expressed as:

12

12 2

exp (3.24)

where the top sign applies for phonon emission and the bottom for phonon absorption.

σ is phonon polarization vector (Fig. 3.6 (c)), is density of crystal, is phonon

frequency, is volume of crystal and is the number of the phonons in the mode.

The number of phonons is given by the Bose-Einstein factor as:

1exp ⁄ 1

. (3.25)

(a) (b) (c)

Figure 3.6: (a) Longitudinal and transverse waves in a 1D linear lattice. (b)Transverse optical and transverse acoustical waves in a 1D diatomiclattice. (c) The ion is displaced from equilibrium by an vector, . is the unit-length polarization vector of the lattice vibration.

Direction of Propagation

Displaced Atoms Atoms at equilibrium

Tran

sverse to Propagation Direction

Longitudinal mode

Transverse mode

Optical Mode

Acoustic Mode

Atom at equilibrium

Displaced Atom

48

According to the Rigid Ion Model [ref], the perturbation energy, δW caused by the

phonon mode is:

δW Nωβ12

12 2ρVωβ

exp (3.26)

where is the crystal potential.

We first establish the electron-phonon interaction matrix elements between two Bloch

states. Consider the Bloch functions in conduction band and valence band:

, exp i · and , exp i · . (3.27)

The electron-phonon interaction matrix elements H are:

H , exp · δW , exp · (3.28)

By inserting eq. (3.26) in eq. (3.28), we find:

H 12

12 2

, ,

exp ·

(3.29)

Since in the eq. (3.29) the unit-cell functions and the phonon perturbation terms are

varying fast in the unit cell, while the exponential term swings slowly, we calculate

the integration in the unit-cell first and then we have:

H12

12 2

ΩV

exp ·

, ,V

12

12

(3.30)

49

The delta functions in the matrix elements represent the conservation of momentum

and energy. The quantity

2 , ,V

(3.31)

describes the strength of interaction between electrons and phonons in the certain

phonon mode. These quantities have been measured for inner conduction bands or

inner valence bands but hardly measured for inter-bands. Several groups have

attempted to calculate by tight-binding or pseudo-potential method [36-38], but

most of the results are obtained between the symmetry points in real k-space. In the

next chapter we will elaborate on calculating proper the strength of interaction for

BTBT process and suggest values to be used in this BTBT model.

Now we work on the tunneling situation. Consider two tunneling states in different

bands defined as eq. (3.1) and eq. (3.5):

, exp i and ,

where is the momentum difference in the indirect bandgap. The electron-phonon

interaction matrix elements between these tunneling states are:

H , exp i δW , (3.32)

Since envelope functions and are slowly varying and stay nearly constant

in the unit cell, we can first perform the integrations within the each unit cell by

utilizing eq. (3.31).

50

H12

12 exp i

12

12

1exp i

(3.33)

Here we define as . Note that the integral part in eq. (3.33) have

significantly large amount only when is close to zero, or in other words, is close

to . This result naturally occurs to comply the momentum conservation rule.

Generally in indirect bandgap materials, valence band maxima is located at Γ point in

the Brillouin zone and conduction band minima at Δ, , or L points. Fig. 3.7 shows

phonon dispersion curves in germanium. For around these zone boundaries, can

be approximated to be constant and can be also assumed to be constant.

Figure 3.7: Phonon dispersion relations in germanium. Generally at zone boundaries such as 2 ⁄ 0.5,0.5,0.5 and 2 ⁄ 1,0,0 , the frequencies

are roughly constant.

The total generation rate of indirect-gap tunneling per unit time and per unit volume in

the semiconductor can be obtained by summing up transition rates for all the phonon

modes and carrier states in conduction band and valence band.

1012Hz8

6

4

2

TOLO

LA

TA

TO

LO

LA

TA

L (111) X (100)Γ

51

2, , ,

2H , 2

2, , , ,

2H , 2 2 2 2

(3.34)

In eq. (3.34), is conduction band wave-vector transverse to the tunneling direction

and is the one for valence band. represents the phonon modes such as

longitudinal acoustic mode (LA), longitudinal optical mode (LO), transverse acoustic

mode (TA) and transverse optical mode (TO). is the transition rate between two

states by phonon mode . is obtained by summing up all the transition rates over

the all the phonon states in the mode . Not all the phonon modes can participate in

tunneling process, since the selection rule, determined by symmetry of unit cell

functions and phonon modes, allows only specific types of phonon to couple between

bands. We first evaluate the first integral in eq. (3.34) :

H , 2

12

12 ,

1 ′ ′ ′ exp i ′

2 .(3.35)

Using the property of function:

exp i ′2

′ ,

we can rewrite eq. (3.35) as:

52

H , 2

12

12 , | | | | .

(3.36)

We can further evaluate the overlap integral in eq. (3.36) by substituting eq. (3.5) to

envelope functions:

| | | |

1

exp 2 ′ ′ .

(3.37)

The exponential term in eq. (3.37) has a maximum value at the branch point where the

first derivative of the exponent goes to zero, that is at , or in x-

space it is at . Existence of the maximum value and the second derivative of the

exponent let us use the method of steepest decent to compute the integral. After the

integration we have:

| | | |

|

exp 2

(3.38)

where

⁄ . (3.39)

53

Note that, since here we assume that the tunneling wavefunction in valence band

decays as x increases and the wavefunction in conduction band decays as x decreases,

both and are positive values. In eq. (3.39), has a correction term,

⁄ , since the phonon energy emission/absorption process effectively

increases/decreases the tunneling length.

Again, as we argued before, it is critical to accurately evaluate due to the nature of

exponential functions. However, the pre-factors in eq. (3.38) such as:

and

|

are safe to be obtained using an approximation. The parabolic band approximation let

us write:

2 Δ 2 Δ,

2.

(3.40)

In eq. (3.40), Δ is the indirect bandgap. Since the phonon energy is much

smaller than Δ , we ignored . We call the energy at the point , the

branch point energy . By taking the derivatives of eq. (3.40) the pre-factors in eq.

(3.38) are:

|

12 Δ

12 Δ

,

54

|

12 Δ

12

|

2 .Δ

. .

(3.41)

, where .

With the results of eq. (3.38) and eq. (3.41), we can rewrite eq. (3.37) as:

H , 2 Nωβ12

12 ,

. . .

2 . .Δ

. . exp 2ξ

(3.42)

The transition rate between two states by phonon mode , in eq. (3.34) is now

rewritten as:

2H , H , 2 exp 2ξ

(3.43)

where

. . .

2 . .Δ

. . Nωβ12

12 ,

(3.44)

By plugging eq. (3.43) into eq. (3.34), we are allowed to write:

55

∑ exp 2ξ

∑ exp 2ξ | , ∏ ⁄

| , , , ,

(3.45)

To evaluate the product term in eq. (3.45), we use the parabolic band approximation

again as described in eq. (3.40).

2 .

3

.Δ

.

,

2 .

3

.Δ

.

132 2 Δ

.

Δ.

2 . . (3.46)

Thus, the product term is:

/2

| ,, , ,

.

2 Δ .

(3.47)

Finally, by substituting eq. (3.47) into eq. (3.45), we have the generation rate of

indirect-gap tunneling per unit time per unit volume in the semiconductor, :

. .

2 . . .Δ

. ., , N , ,

12

12

exp 2 , | , ,

. .

2 . . .Δ

. ., , N , ,

12

12

exp 2 , | ,,

(3.48)

where denotes different conduction band valleys. In silicon, there are 6 conduction

band minima located at Δ points in the Brillouin zone. The efficiencies of tunneling in

56

these six valleys are different. Thus, to obtain the BTBT generation rate in silicon, it is

necessary to evaluate the tunneling rates in each of 6 valleys. In germanium or

aluminum arsenide, the conduction band minima are located at 4 L points in the

Brillouin zone and we need to repeat the process 4 times. In the materials like strained

SiGe, band minima of Γ, Δ, L valleys have similar energy levels. In such materials

both direct-gap tunneling and indirect-gap tunneling contribute significant amount to

tunneling current.

To calculate the path integral , in eq. (3.48), we need the precise model for E-k

dispersion relation in bandgap. For the model, we take the similar form as eq.(3.21),

(3.22) and (3.23).

Γ′

1 Γ Γ Γ

Γ, 0 Γ

1 Γ Γ Γ Γ

Γ, Γ Γ

(3.49)

Γ ΓΓ

Γ Γ, Γ Γ

Γ

Γ Γ

(3.50)

,′

1 , , , ,

,, α, ,

(3.51)

, Γ′

,′

, ⁄

(3.52)

57

In the model above, basically eq. (3.49) and eq. (3.50) are same as eq. (3.21) and

(3.22). Γ′ represents E-k dispersion in direct-gap and ,

′ represents E-k dispersion

in indirect-gap. Just like the direct-gap tunneling case, is the electron energy

measured from the top of the valence band at point x. Γ is the direct bandgap and

, is the indirect bandgap of the valley . Fitting parameters are Γ , Γ , ,

and , . Note that in all the semiconductors, the branch point of the indirect-gap,

, α, is located below the branch point of direct-gap, Γ . In all other non-

local BTBT generation models for TCADs, the E-k dispersion used in path integration,

, is modeled based on the parabolic effective mass approximation as in eq. (3.40).

Most of reported works on BTBT in germanium are done incorrectly based on the

same approximation as well [39-43]. As depicted in Fig.3.8 the parabolic effective

mass approximation completely breaks down especially in germanium where the

energy level of conduction band extremum at Γ point is close to the energy level of

bandedge of conduction band L valley.

58

(a) (b)

Figure 3.8: Complex band structure of germanium obtained from local empiricalpseudopotential method (LEPM) along the 110 direction. The valley located at k-point 2 ⁄ 0.5,0.5,0.5 is projected to Γ point.The parabolic effective mass approximation (a) completely fails toestimate the complex dispersion, since the dispersion curve at Γ pointdeviates a lot from the parabolic curve. The suggested model (b)successfully approximates the dispersion curve.

√√0.06 0.02 0.04 0.080

Energy (eV)

‐3.8

‐4

‐4.2

‐4.4

‐4.6

‐4.8

LEPMEffective mass

√

√0.06 0.02 0.04 0.080

Energy (eV)

‐3.8

‐4

‐4.2

‐4.4

‐4.6

‐4.8

LEPM

This work

59

3.4 2D Quantized System

In the thin channel, as seen in Fig. 3.9, states are quantized. When quantum

confinement is strong in the vertical direction (z), we can decouple the two-

dimensional Schrodinger equation in two one-dimensional equations in the vertical (z)

and longitudinal (x) directions. For certain valleys and orientations, vertical direction

and longitudinal direction are coupled due to nonalignment of the device coordinate

system with the principal axes of the constant energy conduction-band ellipsoids. If

the potential variations along the transport direction are slow enough, it is possible to

decouple two directions by adopting the Rahman’s decoupled effective-mass approach

[44]. In thin film channels with weak longitudinal voltage variations, wave function

can be decoupled using subband decomposition and it is,

Ψ , , , , ; , exp , , (3.53)

where n is the quantum number of quantized states and denotes bands (several

conduction bands valleys or a valence band). The wave-functions , ; are

Figure 3.9: Schematic view of the thin channel double-gate MOSFET (DGFET).Thickness of the channel is . Current flows along x-direction and the channel is quantized along the z-direction.

ChannelInsulator

Gate

Gate

xz

y

zx

z=0 z=tb

Eg

Insulator

Insulator

Energy

0

,

,

,

,

, ;

, ;

, ;

, ;

, ,

,

,

60

normalized in z direction and unit-cell functions , are normalized in the

space. 1D Schrodinger equation in the z confined direction provides eigen-functions

, ; and eigen-energies , .

21

,, ;

, , ; , , ; (3.54)

,where , is the bandedge energy level of band .

The wave functions , satisfy the following 1D Schrodinger equation in x

direction (or channel direction).

21 ,

′ ′,, , ′,

22 ′ ′ ′,

′, ,

0

(3.55)

where the coefficients, ′ , ′ and ′ , represent subband-to-subband

couplings and are given by,

′ , ′ ′ 1 , ′ , ′

, ; ′, ; ,

(3.56)

where

′ , , ′, , , (3.57)

, (3.58)

and

61

′ , ′ , ′ , ; ′, ; ,

′ , ′ , ′ , ; ′, ; . (3.59)

If channel thickness ( is much smaller than channel length ( ), , ; weakly

depends on x and the higher-order coupling terms, ′ , ′ , … can be safely

ignored. Since our semiconductor of interest is the slab where electron states are

confined, the solutions of Hamiltonian, sub-band Eigenfunctions , ; form

complete orthogonal set and the overlap integrals between them

( , ; , ; ) are zero as long as they stay in the same band ( ) .

Thus, the first order coupling terms ′ are zero for and the inner-band

couplings between two sub-bands are prohibited.

We can also prove that the inter-band couplings ( ) between two sub-bands with

different quantum number ( ) are prohibited. Consider the two-band system

with two sub-bands for each. The Hamiltonian for the system can be written as

Δ , 2⁄ ⁄ ⁄ ⁄

⁄ Δ , 2⁄ ⁄ ⁄⁄ ⁄ Δ , 2⁄ ⁄⁄ ⁄ ⁄ Δ , 2⁄

(1.60)

,where

62

, , , ′,

, , , ′, .

(3.61)

Δ , are the quantization energies of each sub-band and is the bandgap of the

semiconductor. From the orthogonal properties of , , we can set in eq.

(3.61) to be zero when and . The solutions of eq. (1.60) cannot be easily

put in analytical forms, but we cane roughly estimate them as

, Δ , 2

| |, Δ ,

| |, Δ ,

(3.62)

, Δ , 2

| |Δ , ,

| |Δ , ,

(3.63)

, Δ , 2| |

, Δ ,

| |Δ , ,

(3.64)

, Δ , 2| |

, Δ ,

| |Δ , ,

(3.65)

For the second sub-band in conduction band ( , ) to couple to the first sub-band in

valence band ( , ), there must be an intersection in energy between , and , at

the certain imaginary number k (Fig. 3.10).

63

Figure 3.10: Illustration of complex bandstructure of two-band two-subband system.

The left panel depicts the scenario where bands with same quantumnumber are connected and the right panel shows the scenario where bandswith different quantum number are connected.

By equating eq. (3.62) and (3.63), we have

Δ , Δ ,| |Δ , ,

| |, Δ ,

| |Δ , ,

| |, Δ ,

.

(3.66)

Using the eq. (3.61), eq. (3.69) can be rewritten as:

Δ , Δ ,| | , ,

Δ , ,

, ,

, Δ ,

, ,

Δ , ,

, ,

, Δ ,

(3.67)

Since Δ , Δ , and (k is a pure imaginary number) are both larger than

zero, to have a solution, the number inside the brackets in eq. (3.69) should be larger

than zero as well. Where the potential variation in z-direction is relatively small like in

Energy

Complex k

‐

Real k

Γc

Γv ΓvReal k

Intersection

(c,2) (v,2) (c,2) (v,1)

(c,1)

(c,2)

(v,1)

(v,2)

(c,1)

(c,2)

(v,1)

(v,2)

Γc

64

the thin-channel DGFETs or SOI FETs, , and , have similar symmetry and

generally the following inequality equation (3.68) holds:

, , > , , , , . (3.68)

Due to eq. (3.68) and the fact that Δ , Δ , , the number inside the brackets in

eq. (3.69) is smaller than zero and we can conclude that the there is no crossover in

energy between , and , at the certain imaginary number k and the inter-band

couplings ( ) between two sub-bands with different quantum number ( )

are prohibited.

Finally, after taking out the forbidden transitions, the eq. (3.55) is simplified as:

21 ,

, , ′ ′,′

, (3.69)

where

′ , ′ ′ , ; ′, ; . (3.70)

If the inter-band terms in eq. (3.69) are ignored, the eigen-functions of the eq. (3.69) at

the tunneling trails have a form:

, 2 2 exp ′ ′ (3.71)

65

3.5 Direct-gap Band-to-Band Tunneling (2D)

The coefficients ′ include both inner-band and inter-band coupling. For

intervalley coupling Tunneling probability between two states in different conduction

band can be calculated using eq. As seen in eq. (3.72) the zero-th order coefficients

contain the overlap integral along the vertical direction (z). This overlap integral in

fact, the momentum conservation between conduction band sub-band state and

valence band sub-band state.

, , ⁄

,,

(3.72)

where

, ; , ; . (3.73)

The constant in eq. (3.73) denotes the thickness of the channel. We can notice that

in eq. (3.72) is exactly like the corresponding result for 3D carriers in eq. (3.17),

except that a form factor is used instead of the delta function , . The

term , represents the momentum conservation which is applicable only to the

infinite space. In quantized 2D carriers, due to the Heisenberg’s uncertainty principle,

66

crystal momentum has an uncertainty in its value, and form factor appears as a

result.

Using the results from eq. (3.12) to eq. (3.17), we can find:

3 ,exp

(3.74)

where

Γ′ ′.

(3.75)

The point is the branch point between nth valence subband and nth conduction

band subband. The point is where the electron in n-th valence subband starts to

enter the bandgap and the point is where the electron ends tunneling by reaching

the n-th conduction subband.

The tunneling probability per unit time between different bands is given as:

, , 9 exp 2 , (3.76)

Total generation rate per unit area by direct-gap band-to-band tunneling is obtained by

summing up tunneling rates in eq. (3.76) over the entire k-space.

, , ,

, , ,

29 exp 2 2 2

67

118 exp 2

/2⁄ .

(3.77)

Taking the similar step as in eq. (3.19), we have:

,

.

4 1 2 G,

.

and /2

⁄2 . . .

DOS.

3 . .G,. . .

(3.78)

Thus the total generation rate per unit time and unit area is

,1

18, exp 2

2 . .,

.DOS

.

3 . .G,. .

2 . .

DOS.

3 . . .,

.

G,. exp 2

(3.79)

Total generation rate per unit volume per unit time can be expressed by multiplying

square of sub-band eigen-functions, , ; . The electron generation rate is:

, , ,

2 . .DOS

.

3 . . . , .

G,. exp 2 , ,

(3.80)

and the hole generation rate is:

68

, , ,

2 . .DOS

.

3 . . . , .

G,. exp 2 , ,

(3.81)

where

Γ′ ′.

(3.82)

Again, the precise modeling of the wavenumber Γ′ is critical in accurate

prediction of BTBT generation rate. Fig. 3.11 shows the complex bandstructure in the

quantized thin slab. Quantum confinement strongly modifies complex bandstructure.

Each state in conduction band and valence band with same quantum number connect

each other through forbidden bandgap.

Figure 3.11: Complex bandstructure of 14nm thick (001) germanium. The imaginaryvalues of the wavenumber k are drawn as red curves and the real valuesare shown in black curves.

Since it is difficult to track and store all the imaginary k connecting between Γ and Γ

inside the bandgap, instead we propose a fitting model (eq. (3.83)) for the imaginary

Energy (eV)

√√

(110)(110)

00.5 0.50.3 0.30.1 0.1

‐3

‐4

‐5

69

part of k by using two variables Γ , and Γ , . These variables are

functions of the quantum confinement energy (Δ , ) of the tunneling states (eq.

(3.84)).

Γ′

1 , ,

,

, 0 ,

1 , , ,

,

, , ,

(3.83)

where

, , , , Δ , , , (3.84)

, , ⁄ , (3.85)

, , , , (3.86)

and 1

1 Δ ,⁄ 1 Δ ,⁄.

(3.87)

where , and , are the nth eigen-energies of the Γ point states of the slice

at x, in conduction band and valence band, respectively.

Fig. 3.12 illustrates the imaginary wavenumber Γ′ generated by LEPM and by our

model. It shows that our model successfully approximate the data given by LEPM.

70

The fitting function for the variables Γ , and Γ , are shown in Fig.

3.13.

Figure 3.12: Γ′ as a function of quantum confinement energy (Δ ) (eq. (3.83)) in

(001) germanium. Tunneling direction is along the (110) direction.

(a) (b)

Figure 3.13: Γ , Γ , used in Fig. 3.12 and their fitting functions.

√√

Energy (eV)

(110)(110)

‐3.4

‐3.8

‐4.2

‐4.6

0 0.040.040.08

=0.54 eV0.350.170.05

0 eVLEPMThis model

4

6

8

4

6

0 0.2 0.4 0.6 0.8 10.02

0.06

0.10

0.14

.

0 0.2 0.4 0.6 0.8 1

0.1

0.2. .

71

3.6 2-D Phonon-assisted Indirect-gap Band-to-Band

Tunneling

Consider two tunneling states in different bands defined as eq. (3.53) and eq. (3.71),

first, the wavefunction for the nth subband in conduction band:

Ψ , , , , ; , exp · , , , (3.88)

and the wavefunction for the mth subband in valence band:

Ψ , , , , ; , exp · , , (3.89)

where is the momentum difference in the indirect bandgap.

Using the eq. (3.33), the phonon-electron interaction off-matrix element can be written

as

H , , ,

12

12

.

,.

, ; , , ; , exp i

(3.90)

where we define as and is the phonon wave vector. The letter

denotes the phonon mode.

To evaluate the generation rate we need to add the square of perturbation energy in eq.

(3.90) over all the phonon states in the phonon mode j. By following the steps given in

eq. (3.35) and eq. (3.36), we have:

72

H , , 2 ,

12

12 ,

, , /

(3.91)

where

1 , ; , ; (3.92)

Just like the quantity obtained for 2D Direct-gap tunneling in eq. (3.73), the

quantity describes the effective extent of the interaction in the z-direction.

We further evaluate eq. (3.91) using equations (3.37)-(3.42) :

H , , 2 ,

12

12 ,

exp 2 . . .

2 . . . .

(3.93)

where

, . (3.94)

The point is the branch point and the energy is an energy gap between

mth valence subband and nth conduction band. The wavefunction , starts to

intrude the bandgap at point and , at the point . The driving force

is defined as:

12 (3.95)

73

and .

Now we can evaluate the transition rate between two subband states by phonon mode

, using eq. (3.93):

2H , , H , 2

exp 2 (3.96)

and in eq. (3.96) is defined as:

. . .

2 . . . . ,

12

12 ,

(3.97)

where the quantity is the volume of the semiconductor and it equals to .

The total generation rate per unit time and unit area in semiconductor can be obtained

by summing up the transition rates for all the phonons modes and carrier states in

conduction and valence subbands:

, 2 exp 2, , ,

2 2 2

2

2, , ,

exp 2 | , 2⁄

, ,.

(3.98)

where denotes different valleys in conduction band.

74

The product term in eq. (3.98) was already evaluated in eq. (3.46). By inserting eq.

(3.46) into eq. (3.98), , becomes:

,

2

2 exp 2 | ,

.,

2 . . ., , ,

.

2 . . . .,

12

12 ,

.,

.

.

exp 2 | ,

., , ,

(3.99)

or it can be rewritten using 2D-DOS mass of the plane:

, 2 . . .

,12

12 , ,

.

. .

exp 2 | ,

, , ,

.

(3.100)

Total generation rate per unit volume per unit time can be expressed by multiplying

square of sub-band eigen-functions, , ; . The electron generation rate is:

, , , 2 . . .

,12

12 , ,

.

. .

exp 2 | , , ,, ,,

(3.101)

and the hole generation rate per unit volume per unit time is:

, , , 2 . . .

,12

12 , ,

.

. .

exp 2 | , , ,, ,,

(3.102)

75

Now we model the path integral | , for indirect-gap tunneling as follows:

, ,⁄

(3.103)

where is defined in eq. (3.83) and

, 1 Δ Δ , Δ

Δ ,

, . (3.104)

In the eq. (3.104), , is the energy level of the m-th subband in valley

measured from the valence band edge. Both Δ and Δ are variables

to fit eq. (3.104) to complex bandstructure of quantized semiconductor film.

Figure 3.14: L , ,′ as a function of quantum confinement energy (Δ ) (eq.

(3.104)) in (001) germanium. Tunneling direction is along the (110)direction.

00.050.1

Energy (eV)

‐4

‐4.4

‐3.6

0.05 0.1√√

(110)

LEPMThis model

=0.37 eV

0.26

0.17

0.09

0eV

76

(a) (b)

Figure 3.15: , Δ L and , Δ L used in Fig. 3.14 and their fitting functions.

3.7 Summary

We have developed the BTBT model which takes into account full bandstructure,

direct and phonon assisted indirect tunneling, quantum confinement effect, non-

uniform electric field (non-local) and strain induced enhanced/suppressed tunneling in

semiconductors. In this model, the tail of electron wavefunction penetrating into the

bandgap is modeled and used to evaluate the interband matrix elements between

conduction band states and valence band states. The final BTBT carrier generation rate

(GBTBT) was calculated by adding up the transition rates obtained by Fermi-Golden

rule for all the possible transitions.

0 0.1 0.2 0.3 0.4 0.50.08

0.1

0.12

0.09

0.11

,

, . .

0 0.1 0.2 0.3 0.4 0.5

0.9

1

1.1

1.2

,

, . .

78

Chapter 4 Phonon-Electron Interactions

4.1 Introduction

During indirect tunneling process, only phonons with certain symmetry are allowed

to couple the conduction band to the valence band. Selection rules concerns if the

phonon-interaction strength, Mj,phV-1vanishes or not. Previous works on indirect

tunneling by Kane [41] and Keldysh [45] does not distinguish between different

phonon modes. Although Fischetti [39] tried to take into account various phonon

modes, he failed to consider selection rules and arbitrarily assumed all four phonon

modes (transverse acoustic (TA), transverse optic (TO), longitudinal acoustic (LA)

and longitudinal optic (LO)) contribute to BTBT current with almost equal strength.

Lax [46] elaborated selection rules based on group theory among symmetric points in

the Brillouin zone and explained the experimental results on electron-phonon

interactions – mobility and electron photoluminescence [38-39, 47-48]. However, the

selection rules fail to explain the results obtained by Chynoweth on Si and Ge Esaki

tunneling devices. In previous studies only phonon matrix elements between the

symmetry points in real bandstructure were obtained. In our work, for the first time we

use the electron orbital wavefunction in decaying states (or inside bandgap) in

calculating the interaction strengths and show that treating the phonon-electron

interaction in complex band structure successfully explains experiments. By adopting

the rigid-ion model [49] and using the empirical pseudo-potential method (EPM), we

suggest proper phonon-interaction strength during band-to-band tunneling process.

79

4.2 Electron-Phonon Interaction Element- Rigid Ion Model

In the previous chapter, we have shown that the matrix element describing the

strength of phonon interaction connecting two states can be written as:

, , 2 , ,V

(4.1)

, where l is the phonon mode, lσ is the polarization vector described in the chapter

on BTBT modeling.

This equation can be expressed in reciprocal space as [49]:

, , 2′

, ′,′, ,

′ exp ′ (4.2)

, where jτ = ( )( )/ 8 1,1,1la± are basis vector, la is the lattice constant andC kG is the

expansion coefficient of the pseudo-plane-wave representation of ku .

Conventionally, the phonon interaction strength ph,l,βM is written in the form of

intervalley deformation potentials DK, which is defined as:

, ,2

, , ⁄ (4.3)

Equation (4.3) can be readily adapted for use with diamond crystal structures. We

have evaluated the equation by using 118 pseudo-plane-wave expansions including

spin orbit coupling.

80

4.3 Phonon Polarization Vectors

The phonon polarization vectors, lσ are given by Lax [50] using group theory. The

polarization vectors for major phonons connecting symmetry points in conduction

bands (X or L) and valence band (Γ) of diamond lattice structure are listed in table 4.1.

In diamond structure, phonon modes are either pure longitudinal or transverse. The

phase difference between two atoms in primitive cell, gives the fraction of the mode

which is “acoustic” and “optic”. Actual displacement of the particles can be obtained

by multiplying the phase term, ( )exp li τβi to the polarization vectors.

81

Direction (β)

Mode ∆ = (0,0,q) Λ = (q,q,q)

LA (0,0,1), (0,0,1) (1, 1, 1)/ 3 , 5 (1, 1, 1)exp4 3

i π⎛ ⎞⎜ ⎟⎝ ⎠

LO (0,0,1), (0,0,-1) (1, 1, 1)/ 3 , (1, 1, 1)exp

4 3iπ⎛ ⎞

⎜ ⎟⎝ ⎠

TA

(1, -1, 0)/ 2 , ( , - , 0)/ 2i i (1, -1, 0)/ 2 , (1, -1, 0)exp

4 2iπ⎛ ⎞

⎜ ⎟⎝ ⎠

(1, 1, 0)/ 2 , (- , - , 0)/ 2i i (1, 1, -2)/ 6 , (1, 1, -2)exp

4 6iπ⎛ ⎞

⎜ ⎟⎝ ⎠

TO

(1, -1, 0)/ 2 , ( - , , 0)/ 2i i (1, -1, 0)/ 2 , 5 (1, -1, 0)exp4 2

i π⎛ ⎞⎜ ⎟⎝ ⎠

(1, 1, 0)/ 2 , ( , , 0)/ 2i i (1, 1, -2)/ 6 , 5 (1, 1, -2)exp4 6

i π⎛ ⎞⎜ ⎟⎝ ⎠

Table 4.1 : Polarization vectors of major phonon modes in diamond lattice structure. Since the diamond structure has a primitive cell formed by two-atom basis, each phonon consists of two polarization vector. While longitudinal mode consists of one LA and one LO polarizations, the transverse modes (TA and TO) are double degenerate.

82

4.4 Silicon.

The transition with highest probability during band-to-band tunneling process in

silicon occurs between ΔC-Γ25’, where the bandgap is 1.12eV. The second lowest point

of conduction band is located at L1 point. Although the transition from L1 to Γ25’ is

also possible, the large bandgap between the two points of 1.8eV, significantly lowers

the tunneling probability compared to the ΔC-Γ25’ transitions.

Although experimental and theoretical studies to estimate the intervalley deformation

potentials exist for inner-band transitions [36-38, 51] (conduction to conduction or

valence to valence), photoluminescence and photon absorptions [52] , the strength of

Figure 4.1: Bandstructure of silicon calculated by EPM. Phonon-assistedtransitions with highest probability between band extrema are shown.In silicon, transitions between Δ1-Γ25’ are most dominant. Due toselection rules imposed by symmetry of the two points, TA and TOphonons are most effective in assisting the tunneling process.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

Ener

gy (e

V)

1002aπ⎛ ⎞

⎜ ⎟⎝ ⎠

k 1113aπ⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

k

X1

L1

Γ15

Γ25’

Δ1

Δ5 Δ2’

Λ3

Λ1

L3’

X4

Δ2’ Λ1

TA/TO

83

phonon-electron interaction during band-to-band tunneling process is not well-studied

[39].

According to Lax [46], the selection rules allow only TA, TO and LO phonons to be

effective in assisting the intervalley transition between Δ1-Γ25’.

Γ′ Δ Δ Δ TA, TO LO (4.4)

The deformation potentials for intervalley scattering between two symmetry points,

Δ1-Γ25’ are calculated and listed in table 4.2.

Silicon

Mode (001) Phonon Energya (meV) Deformation Potentials (eV/Å)

LA 46.3 0.0

LO 54.3 2.6

TA 18.4 5.6

TO 57.6 2.9

Table 4.2 : The effective deformation potentials and phonon energies for intervalley scattering between Γ25’ and ΔC ( 2 / (0,0,0.85)laπ ). Energy difference between two states is 1.12eV. Since there is degeneracy in Γ25’ states and transverse modes, the deformation potentials were added to indicate the effective strength as follows: 2 2

,( ) ( )l

pp l

DK DK= ∑ , where p denotes the polarization of phonon and l

is indices for degenerate valence bands. a : Chynoweth [53]

84

The deformation potentials calculated using EPM, reconfirm the Lax’ selection rules

in that LA phonon does not participate in the scattering events, but TA, TO and LO

phonons can couple the two states. The values we obtained are also in reasonable

agreement with the values used in previous BTBT works. Rivas[54] guessed TA

deformation potential to be 2.45 eV/Å and Tanaka [55] used 6 eV/Å for overall

effective deformation potential. Although the selectrion rules and theoretical

calculation of deformation potentials between the symmetry points predict strong

participation of LO phonon in scattering process, experimental results showed

otherwise.

Chynoweth [53] and Holonyak [56] conducted experiments on silicon Esaki junction

and observed that only TA and TO phonons significantly contribute to a tunneling

current, but contrary to the selection rules, contribution from LO phonon contribution

is weak. The reason for the weak response of LO mode phonon has not been clearly

studied yet. The nearly equal contributions from TA and TO phonons imply that the

phonon interaction strengths for these phonon modes are roughly same.

The limitation of the previous theoretical studies lies in the fact that they assumed

that in the moment of scattering the state of electron is still same as the one where it

started to tunnel through the classical turning point. As we have shown in the chapter

on BTBT modeling, the most fraction of transition occurs at the point where

( )c vimag k k+ is minimized. In silicon, the transition point is at -4.67 eV (or 0.61 eV

above the valence band edge).

85

As seen in Fig.4.2 we have calculated the deformation potentials using the unit-cell

function ku at the transition point where crystal momentum k is imaginary, instead of

evaluating the phonon interaction between the symmetry in real bandstructure. The

results are listed in table 4.3.

(a) (b)

Figure 4.2: Comparison between the Lax’ Selection Rules and our work. Sincemost of contribution in tunneling is coming at E=-4.67eV, thedeformation potentials should be calculated between two pointsdepicted in (b) not between two symmetry points in (a).

Conduction Band

Valence Band

Energy

Complex k‐+

Real k

Γc

Γv -

∆

Conduction Band

Real k

-

Lax, Selection Rules

ConductionBand

Valence Band

Energy

Complex k‐+

Real k

Γc

Γv

-∆

Conduction Band

Real k

--

Phonon

This work, E=‐4.67eV

86

The complex-k-space-intervalley-deformation potentials in table 4.3 deviate from the

values obtained in real k space. They clearly show that, while the contribution from

both LA and LO modes are negligible, TA and TO modes are most dominant, which is

in very good agreement with experimental results by Chynoweth [53] and Holonyak

[56].

Due to lack of theoretical and experimental data, previous BTBT calculations in

electronic devices were based on “educational guess [39]” mostly estimated from

either dilation deformation potentials or mobility deformation potentials. While Rivas

[54] took into account only TA and TO phonon modes in his tunneling Esaki diode

simulations, Fischetti [39] added all four phonon modes without considering the

selection rules and assuming that they have almost equal deformation potentials.

Silicon (Γ-∆)

Mode (meV) This work (eV/Å) Previous works (eV/Å)

LA (46.3) 0.28 weakC

LO (54.3) 0.19 weakC

TA (18.4) 6.93 2.45R, 6T, 2.74S, 5.9F, strongC

TO (57.6) 4.42 5.6R, 5.23S, 4.06Ge, 4.96Gp,6.0F, strongC

Table 4.3 : The complex-k-space-intervalley-deformation potentials calculated for Band-to-Band tunneling between Γ25’ and ΔC. Due to lack of theoretical work, previous BTBT models (R,T,F,S) estimated the deformation potentials from dilation deformation potentials and mobility deformation potentials. Experimental values are extracted from light absorption coefficient study (Ge,Gp) and Esaki tunneling device (C). R: Rivas [54], T: Tanaka [55], F: Fischetti [39], S: Schenk [57] Ge: Glembocki (electron-phonon) [36], Gp: Glembocki (hole-phonon) [36]

87

Glembocki [36] experimentally measured phonon-assisted photon absorption length

and extracted the deformation potential for TO mode. The deformation potential

obtained by Glembocki is for electron 4.06 and for hole 4.96 eV/Å, which are in

astonishingly good agreement with our potentials 4.42 eV/Å.

4.5 Germanium

Germanium has direct bandgap slightly larger (0.14eV larger) than indirect bandgap

(Lc1-Γ25’). Since, direct BTBT process is more than an order efficient, phonon-assisted

tunneling is only significant for a small electric field. The transition with highest

probability during phonon-assisted-band-to-band tunneling process in Germanium

occurs between Lc1-Γ25’, where the bandgap is 0.66eV. The phonon-assisted

transitions between ∆c1-Γ25’ are also possible, they are negligible compared to the

strong direct tunneling transitions. Although for relaxed germanium the transitions

between ∆c1-Γ25’ are not strong, in strained germanium (ex. pseudomorphically grown

germanium on silicon) ∆c1-Γ25’ forms lowest bandgap and the transitions through this

path becomes most dominant process.

88

According to Lax [46], the selection rules allow only LA, TA and LO phonons to be

effective in assisting the intervalley transition between Lc1-Γ25’.

L Γ L L TA LO

L Γ L LA (4.5)

TA and LO phonons couple valence band to conduction band by first order

perturbation. LA phonon couples two bands by second order perturbation. Since Γ

valley is close to L valley, the electron tunneling from v25+Γ is mixed to c2

−Γ and then

LA assisted tunneling takes place from c2−Γ to c1L+ [42-43].

Figure 4.3: Bandstructure of germanium calculated by EPM. Phonon-assistedtransitions with highest probability between band extremum areshown. In germanium, transitions between L1-Γ25’ are most dominant.Experimentally LA, TA and TO phonons are most effective inassisting the tunneling process. Subindex c in Lc1 representsconduction band while v represents valence band.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

LA/TA/TO

1002aπ⎛ ⎞

⎜ ⎟⎝ ⎠

k1113aπ⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

k

L1

Λ3

Λ1L3’

Λ1 X1

X4

Energy (eV)

TA/TO Γ15

Γ25’

Δ1

Δ5

Δ2’

Δ2’

89

The deformation potentials for intervalley scattering between two symmetry points,

Lc1-Γ25’ (for LA phonon, between c2−Γ - c1L+ ) are calculated and listed in table 4.4.

Lax’ selection rules and the deformation potentials between symmetry points obtained

by EPM methods indicate that the contribution of LO phonons is comparable to LA or

TA phonons and the TO phonons do not participate in the tunneling process. However,

experiments on germanium Esaki diode performed by Chynoweth [53] show that,

while the transitions by LA and TA phonons are strongest, TO phonon interaction is

substantially large, the contribution from LO phonon is negligibly small.

Germanium

Mode (111) Phonon Energya (meV) Deformation Potentials (eV/Å)

LA 27.5 2.5

LO 31.6 4.5

TA 7.6 3.0

TO 36.0 0.0

Table 4.4 : The effective deformation potentials and phonon energies for intervalley scattering between Γ25’ and LC ( / (1,1,1)laπ ) (for LA phonon, between c2

−Γ - c1L+ ). Energy difference between two states is 1.12eV. Since there is degeneracy in Γ25’ states and transverse modes, the deformation potentials were added to indicate the effective strength as follows: 2 2

,( ) ( )l

pp l

DK DK= ∑ , where p denotes the

polarization of phonon and l is indices for degenerate valence bands. a : Chynoweth [53]

90

The discrepancy in TO and LO phonon interaction strength between the selection

rules and the experiments can be again explained by introducing complex-k space. As

we stated previously the most fraction of transition occurs at the point where

( )c vimag k k+ is minimized. In germanium, the transition point is around at -4.12 eV

Germanium

ModeC,W (meV) This work (eV/Å) Previous works (eV/Å)

LA (Γ-L) (27.5) 5.4 3.89K, 3.88G, 4.2Z, 4.5L, strongC

LO (Γ-L) (31.6) 0.22 very weakC

TA (Γ-L) (7.6) 1.6 1.42M, strongC

TO (Γ-L) (36.0) 1.8 mediumC

LA (Γ-∆) (25.6) 0.77 -

LO (Γ-∆) (31.7) 1.6 2.2L, 2.43K

TA (Γ-∆) (10.1) 0.15 -

TO (Γ-∆) (34.0) 0 -

Table 4.5 : The complex-k-space-intervalley-deformation potentials calculated for Band-to-Band tunneling between Γ25’/ c2

−Γ and c1L+ /ΔC. (100) at E=-4.12eV.

K: Krishnamurphy [51], ab initio tight-binding method (conduction band to conduction band scattering), G: Glembocki [37], rigid-ion model (conduction band to conduction band),

Z: Zhou [48], time-resolved photoluminescence method, L: Li [52], direct exciton optical absorption measurement, M: Kane[41], Macfarlane [47], optical absorption measurement,

C: Chynoweth [53], Esaki tunneling diode,

91

(or 0.54 eV above the valence band edge). The deformation potentials at the transition

point are listed in table 4.5.

The complex-k-space-intervalley-deformation potentials clearly shows prominent

LA(Γ-L) and TA(Γ-L) phonon interactions and vanishingly small contribution from

LO(Γ-L) phonon which is in very good agreement with Chynoweth [53]. Phonon

interaction strength of TO (Γ-L) phonon, TO( )LΓ−M is about ¼ of TA( )LΓ−M , which

successfully explain the reason for medium peak strength of TO (Γ-L) phonon [53].

The calculated deformation potentials for LA (Γ-L), TA (Γ-L) and LO (Γ-∆) are all

consistent with the values determined either by experiments[41, 47-48, 52] or

theory[37, 51].

4.6 Summary

The deformation potentials for phonon-assisted band-to-band tunneling have been

successfully calculated using rigid-ion model and empirical pseudo-potential method.

The previous theoretical calculations were all limited to inner-band scattering (ex.

conduction band to conduction band, valence band to valence band) or limited to

transitions between symmetric points in real k-space. By introducing complex-k space

in calculations of BTBT deformation potentials, we are able to explain the

discrepancies between previous theories to experimental results on tunneling diodes.

In relaxed silicon, only TA(Γ-∆) and TO(Γ-∆) phonons have significant contributions

92

on band-to-band tunneling and in relaxed germanium, LA(Γ-L), TA(Γ-L), TO(Γ-L)

phonons are participating tunneling process.

93

Chapter 5 Application of band-to-band

tunneling (BTBT) model

5.1 Introduction

It becomes important to predict the band-to-band tunneling leakage of devices made

with high-mobility materials. The most commonly used BTBT models in commercial

TCAD tools (MINIMOSTM, TAURUSTM, DESSISTM) do not capture quantization

effects, phonon-electron interactions and full band information of nano-scale devices.

The model developed in the last chapter captures important quantum mechanical (QM)

effects in highly scaled DGFETs. We will compare the previous models and our

model and discuss the applicability of this model for ultra thin body DGFETs with

high mobility channel.

5.2 Comparison to Previous Models

5.2.1 Direct Band

Mostly commonly used model is Kane’s model [40]. The model uses 2-band

approximation ( Fig. 5.1 (a) ) where it assumes only two bands exist (one conduction

band and one valence band) and interaction of bands is modeled using kÿp

approximation. This model works pretty well for relaxed small bandgap

semiconductors such as InSb. In InSb, the separation (∆so) between heavy hole (HH)

and split off bands is large enough compared to bandgap (Eg) – for InSb ∆so/Eg≈4.7.

Large separation ensures minimum interaction between these bands, which makes the

94

2-band based Kane’s model justifiable. However as we expands our interest of

material choice to large band-gap materials such as Si, InP and GaAs, this

approximation becomes inaccurate. Krieger [35, 59] expanded Kane’s model by using

Kane’s 4-band kÿp approximation, suggesting new analytical expression for effective

tunneling mass. This model also imposes limitation on applicability to semiconductors

such as Ge and Si, since Kane’s 4-band model assumes heavy hole band is isotropic.

In these semiconductors (Ge and Si), heavy hole band is severely warped, therefore

using analytic kÿp approximation leads to again inaccurate estimation of BTBT.

(a) (b)

Fig. 5.1: (a) Kane’s 2-band model. (b) Full band model obtained by EPM.

As described in the previous chapter, we obtain the E-k dispersion in bandgap

(complex band structure) by empirical pseudopotential method (EPM), where entire

128 bands are included (Fig. 5.1 (b)). We found that using Kane’s 2-band model leads

Kane : 2-band Approximation

Conduction Band

One Valence Band

Forbidden Bandgap

Complex kReal k

Conduction Band

Three Valence Bands

HH

LHSO

Entire 128-bands obtained by EPM

Γc

Complex k

95

to 10 times smaller BTBT rate for 1MV/cm electric field and 300 times smaller BTBT

for 0.5MV/cm electric field.

5.2.2 Indirect Band

Most commonly used indirect tunneling model is Keldysh’s model [45]. In his

formulation, the author used band-edge effective mass and assumed this mass does not

change throughout the tunneling process. This assumption is justifiable for Si, where

transition energies between conduction band and valence band for Γ point and for Δ

points are pretty big (>3eV). In Si, since the energy transition an electron experiences

during tunneling process is relatively small compared to these transition energies,

using simple effective mass’ at band minima and maxima is a good approximation.

However, in Ge, proximately located Γc interacts strongly with Γv, bending the

dispersion curve steeper, thus substantially lowering tunneling mass [60].

96

Again as in direct tunneling case, precise modeling of band-dispersion is necessary.

We used EPM to numerically calculate the dispersion as discussed in the previous

chapters.

5.2.3 Selection Rules on Phonon modes and Phononinteraction

Strength

During indirect tunneling process, only phonons with certain symmetry are allowed

to couple the conduction band to the valence band. Selection rules concerns if the

phonon-interaction strength, Mj,phV-1 vanishes or not. Previous works on indirect

tunneling by Kane [41] and Keldysh [45] does not distinguish between different

phonon modes. Although Fischetti [39] tries to take into account various phonon

Figure 5.2: Indirect tunneling process. Keldysh’s model does not capture full band-information.

Valence Band

Energy

Complex k-+

Tunneling Process

Real k Γv

L or ∆

Conduction Band

Real k

--

-

Phononunknown-

Effective Mass Approx.

Γc band is ignored

Effective Mass Approx.

97

modes, he failed to consider selection rules and arbitrarily assumed all four phonon

modes (transverse acoustic (TA), transverse optic (TO), longitudinal acoustic (LA)

and longitudinal optic (LO)) contribute to BTBT current with almost equal strength. In

our work we use the electron orbital wavefunction in decaying states (or inside

bandgap) in calculating the interaction strengths, unlike all the previous studies where

phonon elements were calculated between the symmetry points in real bandstructure,

and show that treating the phonon-electron interaction in complex band structure

successfully explain experiments. As explained in the chapter on phonon By adopting

the rigid-ion model [49]and using the empirical pseudo-potential method, for the first

time we accurately treated the electron-phonon interactions during band-to-band

tunneling process.

5.3 BTBT Generation Rates for Various Materials

Since BTBT process requires the movement of electron in the bandgap, it is

important to predict the E-k relationship in the bandgap where k vector becomes

complex number. Local empirical pseudopotential method (LEPM) is used to obtain

complex bandstructure. We used simplex search method to fit a model potential to

experimental band parameters and exact effective mass at bandedge. Fig. 5.3 shows

the complex bandgap structures of Ge and GaAs.

98

(a) (b)

Fig. 5.3: (a) 3D plot of complex bandstructure of germanium along the [111] direction

(b) GaAs.

BTBT generation rate vs. electric field is calculated for various materials and

the results are shown in Fig. 5.4. Generally, the wider bandgap materials show smaller

BTBT current. Direct tunneling is dominant in Ge due to the proximity of the Γ-valley

to the L-valley. Si exhibits very low generation rate, due to its indirect bandgap.

Although the direct bandgap material, GaAs has a bigger bandgap than Si, the BTBT

rate of GaAs is two orders of magnitude higher than Si. Since InAs has smaller

bandgap and mass than Ge, BTBT generation rate for InAs is the highest among four

materials.

-0.2-0.15

-0.1 -0.05

-0.200.20.40.60.8

‐3

‐3.5

‐4

‐4.5

‐5

‐5.5

00.8 0.4

Energy (eV)

0.1 00.2

0

‐3

‐6

‐4

‐5

0.5 0

0.4

Energy (eV)

10 0.2

99

Fig. 5.4: BTBT generation rate vs. Electric Field

5.4 PIN Diodes

To verify our model, we compared our simulated current for PIN diode to fabricated

PIN diode’s IV data available on literature.

To validate our model, we collected existing experimental data on tunnel diodes and

compared it to the theoretically calculated values. Kane [40-41] and Hurkx [61]

showed that BTBTG can be expressed as 0exp( / )AF F Fσ − . A and 0F are constants

related to bandstructure and phonon-interactions of the materials, σ is 2 for direct

tunneling, 2.5 for phonon-assisted tunneling and F is an average electric field across

tunneling path. Although most of existing data on BTBT is very out dated and hard to

interpret, extracting F0 is possible. Table 1 shows the theoretically calculated 0F from

0 0.5 1 1.5 21010

1015

1020

1025

1030

Electric Field (MV/cm)

BTB

T G

ener

atio

n R

ate

(cm

-3s-1

)

SiGaAsGeInAs

100

our BTBT model explained in the chapter on BTBT, Kane’s two band model and

experimentally extracted values. Due to nonparabolicity of effective mass in bandgap,

0F predicted by our model is lower than the ones by Kane’s model and is closer to the

experimental values.

The experimental values of 0F in Si vary from 19 to 35 MV/cm. When applied bias

is low in a tunnel diode, the local E-field based model inevitably gives wrong results,

since it doesn’t account for the non-uniform E-field effect.

There are a few literature works on fabricated PIN structure where the accurate

information on doping profile is also available. Tyagi [62, 69] and Agarwal [70]

fabricated Si PIN diodes and measured the I-V characteristics in reverse bias region.

In Fig. 5.5, the I-V curves calculated using our BTBT model and Hurkx model [61]

are depicted together with experimental data.

Fo Mat.

Calc. (MV/cm) Exp. (MV/cm) Kane (MV/cm)

Si [100] 21.6 19~35[59, 61-64] 24

GaAs [111] 16.6 14.6[65] 17.5

Ge [111] Direct 5.3 5.7[66] 5.9

Indirect 4.6 4.6[67] 6.1

InAs[111] 1.3 1.3[68] 1.3

Table 5.1 : Parameter F0 of Kane/Hurkx model AFσexp(-F0/F), extracted by our BTBT model (by LEPM), Experiments and Kane(two band model fordirect and effective mass approximation (EPM) for indirect). Thedirection indicates tunnel direction.

101

Since Hurkx fitted his model to experimental data for high voltage bias, his model

matches with experiments pretty well at high bias region, but at low bias it largely

overestimates currents. At very low bias, defect assisted tunneling becomes more

dominant than BTBT [57], our model starts to underestimate the current in this region

of bias. In actual device, as tunneling distance increases, the trailing wavefunction

starts to scatter and lose quantum coherence. As we assume complete quantum

coherence throughout tunneling process, our model results in slight overestimation in

lightly doped diodes.

Gaul et al. [65] fabricated very abrupt junction GaAs PIN diode by MBE and their

data matches very well with our prediction. Fig 5.6 shows the I-V of PIN diode and

theoretical values given by our model and Kane’s BTBT model.

Figure 5.5 I-V curves of silicon PIN diodes. Solid lines with markers are actualI-V characteristics of the diodes, solid lines are estimations from ourBTBT model, and dotted lines are currents calculated by Hurkx’model.

0.5 1 1.5 2 2.5 3 3.5 4 4.5

10 -5

10 0

R e v e rs e B ia s (V )

J (A

cm-2

)

T h is w o rkH u rkxE xp .

Ty ag i N o1

Ty ag i N o2

A garw a l 33nm

102

Our calculated values closely match with experimental data while kane’s model

always underestimates by big amount due to inaccurate estimation of tunneling mass

using two-band model. The only deviation happens in device where I-region is 55nm

long. As in the case of Si, tunneling electrons in 55nm device are scattered and lose

quantum coherence during tunneling process.

Tyagi [66] and Tokuyama [71] fabricated abrupt junction PN diodes on germanium

wafer and measured the reverse bias breakdown currents. The estimations by our

model and Kane’s model are drawn in Fig. 5.7 and are compared with experimental I-

V characteristics. As in Si and GaAs, our model accurately predicts I-V characteristics

of Ge PN diode.

Figure 5.6 I-V curves of GaAs PIN diodes. Solid lines with markers are actual I-V characteristics of the diodes, solid lines are estimations from ourBTBT model, and dotted lines are currents calculated by Kane’smodel. The length of I-region is varied from 15nm to 55nm.

1 2 3 4 5 6

10-5

100

Reverse Bias (V)

J (A

cm-2

)

This workKaneExp.

15nm

20nm

55nm

103

5.5 Effect of Strain (Ge)

In Fig. 5.8, we show the energy shifts at symmetry points obtained from our LEPM

calculations in biaxially strained germanium (001). The strain is indicated by the ratio

0/ 1a a − . Since all three important symmetry points ( CΓ , 100Δ , CL ) form conduction

band minima at similar energy levels, strain can effectively deform and change the

electronic properties of germanium. The results show that compressive strain lifts up

CΓ valley and two-fold 001Δ valley, while it pushes down four-fold conduction band

100/010Δ valleys. When strong enough compressive strain is applied ( 0/ 1 0.02a a − < − ),

energy level of 100Δ valley comes down below CL valley leading to forming indirect

bandgap at 100Δ point. Tensile strain pushed CΓ further down and around at

Figure 5.7 I-V curves of Ge PIN diodes. Solid lines with markers are actual I-Vcharacteristics of the diodes, solid lines are estimations from ourBTBT model, and dotted lines are currents calculated by Kane’smodel.

1 1.5 2 2.5 3 3.5 4 4.5 510-6

10-4

10-2

100

102

Reverse Bias (V)

J (A

cm-2

)

This workKaneExp.

TokuyamaNo.12

TyagiNo.2

TyagiNo.1

104

0/ 1 0.025a a − = the strained germanium becomes direct bandgap materials. The

large separation between 1vΓ and 2vΓ is also observed due to breaking of symmetry

and degeneracy in diamond lattice structure. For both tensile strain and compressive

strain, the combined effect of the band shifts in conduction band and valence band

leads to shrinkage in bandgap

This change in bandgap and energy levels of symmetry points alters band-to-band

tunneling by large amount. In relaxed germanium where high electric field is applied,

major band-to- band tunneling component is direct tunneling through vΓ to CΓ . When

the small amount of biaxial-compressive strain is applied, band-to-band tunneling rate

goes down due to increased direct band-gap. Appling further strain reverses the trend

Figure 5.8 Calculated band shifts of germanium (001) as a function of in-plane

biaxial strain ( 0/ 1a a − ).

-0.04 -0.02 0 0.02 0.04-5

-4.8

-4.6

-4.4

-4.2

-4

-3.8

-3.6

-3.4

Strain (all/a0-1)

Ener

gy (e

V)

ΓC

ΓV2 ΓV1

Δ100

Δ001

L6,C

105

and brings up the band-to-band tunneling rate again, since the indirect tunneling

component starts to increase and become bigger than direct tunneling component. This

trend implies that current strain technology to boost a hole current in germanium

MOSFET would reach the barrier imposed by the enhanced leakage current.

Since our BTBT model is based on EPM calculations of complex bandstructures, it

can take into account any strain effects, while previous model which is solely based on

simple two band model fail to account for the effects of strains on BTBT. After

obtaining complex band structure information and phonon-electron matrix elements of

each strain level, we have simulated the I-V characteristics of strained germanium

double gate MOSFET device of which gate length is 15nm and oxide thickness is

0.9nm. As seen in figure 5.9, we found that 60% (out of max. 0.0410 strain) biaxially

compressive-strained Ge has the lowest band-to-band tunneling leakage.

Figure 5.9: IOFF,MIN vs. biaxial strain. Compressive strain increases indirect BTBTwhile it decreases direct BTBT. Optimum point at around x=0.6.

0 0.4 0.6 0.8 1 0.210

-9

10-8

10-7

xI O

FF,M

IN (A

/um

)Lg=50nm, Tox=0.9nm, Tb = 5nm

ss--Ge grown on SiGe grown on Si11--xxGeGexx

Direct TunnelDirect TunnelIndirect Indirect TunnelTunnel

Lowest BTBTLowest BTBT

s-Ge

106

5.6 Quantum Confinement

In highly scaled MOSFETs built on ultra thin bodies such as DGFETs or ultra-thin

silicon on insulator (UT-SOI), the bandstructure is also heavily altered by quantum

confinement effect. Figure 5.10 illustrates the complex bandstructure of GaAs (001)

quantum well (QW) which consists of 12 atomic layers (atm) of GaAs sandwiched by

two 6 atomic layers of vacuum. The bandstructure of QWs shows projections of bands

with out of plane k-component fall onto in-plane k-space. As each subband exhibits

different tunneling mass in QWs, for proper BTBT modeling it is required to calculate

the curvature of complex bandstructure of each subband.

Using the complex bandstructure of QWs, we have shown for the first time that

quantum confinement effect suppresses band to band tunneling. As seen in figure 5.11,

the tunneling wavefunctions in ultra thin body, experience exponentially bigger

(a) (b)

Figure 5.10: (a) A GaAs QW consists of 12 atom thick layer with 6 atom thickvacuum layer. (b) Complex Bandstructure of the GaAs QW.

GaAs (001)

Vacuum

Vacuum

12 atoms

6 atoms

6 atoms

Im(k) Γ Re(k) L Im(k)

ΓC

ΓV

LC

107

attenuations due to increased bandgap and tunneling mass which leads to reduced

band-to-band tunneling. To take into account the quantization effect caused by ultra-

thin body during device simulations, the wavefunctions and the energy levels of

quantized subband states were obtained by solving 1-D Schrödinger equation along z-

direction for both electrons and holes. By plugging the subband information into the 2-

D quantized BTBT model developed in the chapter on BTBT, it is possible to

calculate BTBT leakage currents in thin body devices [23-33, 72-74].

We have applied our model to calculate the BTBT leakage current in ultra thin film

(3~10nm) DGFETs with high mobility channel materials such as Ge, 4% biaxially

Figure 5.11:Ultra-thin body and larger quantization increases the effective bandgap and lowers the tunneling rate.

ΔETb

S

D

GBTBT

Small Tunneling Rate

ΨeΨh

Eg

Large Tunneling BarrierStrong Quantization

S

D

GBTBT

Large Tunneling Rate

ΨeΨh

Eg

Eg

Thick BodyDGFET

Thin Body DGFET

Tb

ΔE

Channel | OxideOxide |

108

compressive-strained Ge, Si, 4% biaxially compressive-strained Si, InAs, GaAs.

Through these simulations we found that reducing body thickness is very effective in

reducing the BTBT current. However it is not easy to manufacture thin body DGFETs

because of process variations, technological complexity and increased cost. Larger

VDD increases the drive current but results in a worse IOFF,MIN. To evaluate the

practicability of various materials based on BTBT leakage current at the 15nm

technology node, we plot the maximum VDD versus Tb curve (Fig. 5.12). The figure

shows the maximum allowable VDD at a given Tb for a constant IOFF,MIN of 0.1uA/um

(iso-leakage contours). The combination of VDD and Tb below the curve satisfies the

ITRS HP requirement for the 15nm node Off-current specification. The result shows

that 100% s-Si cannot be used unless it is operated under 0.5V, while Ge and InAs can

be operated at 0.9V as long as body is thinner than 5nm. Si and GaAs can be operated

over 1.5V due to their large bandgap. Although s-Ge has a smaller bandgap than Ge,

s-Ge shows a much lower BTBT leakage current than Ge due to its large quantization

and indirect bandgap.

109

The effect of quantum confinement on BTBT leakage current in MOSFET devices

predicted by theory is proved by experiments [32]. Strained-SiGe MOSFETs having

ultra-thin (UT) (TGe<3nm), very high germanium fraction (~80%) channel and Si cap

(TSi cap<3nm) have been successfully fabricated on relaxed Si substrates (Fig. 5.13). In

this device, big band-offset (>0.7eV) between germanium and silicon layer creates

quantum well for holes. The quantum confined holes in germanium layer are

quantized and the energy levels are shifted up.

Figure 5.12: Iso-leakage (IOFF,MIN) Contours. These curves determine the

maximum allowable VDD to meet the ITRS off current requirement for

a 15nm gate length, at a given body thickness (constrained by

technology). s-Si cannot be used for >0.5V operation. Ge and InAs

can be used for 0.9 operation when Tbody<5nm. For Tbody<7nm s-

Ge can be used for >1V operation. Si and GaAs comfortably meet the

off-current specification because of their large bandgap.

3 5 7 100

0.3

0.6

0.9

1.2

1.5

1.8

Tbody (nm)

VDD

(V)

SiGaAss-GeGeInAss-Si

Towards Decreasing

Ioff,MIN

110

Although predicting the BTBT current in complicated device structure is very

challenging work due to lack of accurate knowledge on dopant profile, it is still

possible to predict the trend in change of I-V characteristics as we vary design specs.

In this experiment, thickness of germanium layer is varied from 6nm to 2nm and we

observed over 10x BTBT reduction in 2nm strained Ge over 4nm strained Ge

(Fig.5.14). As illustrated in figure 5.14 this trend is also well predicted by simulations

using our BTBT model. Reducing the thickness of germanium layer imposes three-

fold effects on BTBT current. First, reducing the thickness lowers the BTBT current

linearly with respect to reduced cross-section area. Second, parallel electric field

applied across the germanium layer is increased, which results in higher BTBT current.

Third, increased quantum confinement effect widens up effective bandgap accordingly

the BTBT current is also reduced. Our simulations support the experimental results in

that despite of the enhanced parallel electric field, quantum confinement effect is

Figure 5.13: Device structure and band diagrams of SiGe heterostructure p-MOSFET.

Cross-sectionDevice geometry

Quantum Confinement

111

strong enough to bring down the BTBT current level when germanium layer is thinned

down below 2nm.

5.7 Summary

We have shown the limitations of previous BTBT tunneling models for TCAD

simulators. Since previous BTBT tunneling models are based on the two-band model

in relaxed bulk semiconductor system, it fails to predict BTBT current for highly

strained ultra thin film devices. Moreover the phonon-electron interactions were not

properly modeled in any of previous works. We have explained in this chapter the

difference in modeling direct tunneling (zero-phonon assisted) and indirect tunneling

(phonon-assisted). Depending on transition paths we have carefully chosen phonon

modes according to selection rules and deformation potentials calculated by applying

Figure 5.14:BTBT (GIDL) with varying TGe channel. Left plot is obtained from experiments and right plot is drawn using simulation results.

Experimental Id-Vg10x Reduction (reducing Tge)

Simulation Id-Vg (DGFET)

112

rigid-ion model on electron orbital in complex-k space. Our model was verified using

various experimental data on fabricated PIN structures on silicon, germanium and

GaAs wafers. We have shown for the first time that biaxial compressive-strain reduces

the direct tunneling component, but the increases in-direct tunneling component in

leakage current in germanium devices. With 2.4% biaxial compressive-strain applied

across germanium devices, the BTBT leakage current reaches the minimum point. By

applying our BTBT model in DGFET simulations we found that reducing the

thickness of channel layer is very effective in suppressing the BTBT leakage current.

This quantum confinement effect has been verified by comparing the simulation

results to the I-V characteristics of fabricated thin-film strained germanium MOSFET

devices.

.

113

Chapter 6 Quantum-Ballistic Transport in 1-D

Quantum Confined System

6.1 Introduction

According to the 2008 update of ITRS [75], MOSFETs with effective channel length

of 20 nm will enter large scale production in 2012, and MOSFETs with effective

channel length of 8.1 nm in 2022, at the end of the Roadmap. Such devices exhibits

several characteristics in nanometer scale: quantum confinement in the channel,

source-to-drain tunnel current, band-to-band tunnel current, and non-equilibrium

transport. Already in MOSFETs fabricated by the current technology node, the most

of fraction of electrons contributing to the drain current (in an n-MOSFET) go through

the channel without suffering from inelastic scattering. The ballistic fraction is

increasing with further scaling down, and it is predicted by some authors to be

predominant over the fraction of electrons experiencing inelastic scattering in devices

with channel length shorter than 20 nm [76]. Natori [77] has developed the first

analytical ballistic model for MOSFETs and Monte Carlo tools [76] have been

developed and used to treat carrier transports in nano-scale devices.

Nanoscale MOSFETs exhibits a significantly large quantum confinement effect in the

channel, due to the high vertical electric field and the thin channel thickness. A

quantum simulation is therefore necessary to take into account the quantized

subbands, the quantum gate capacitance, 2-D density of states and degeneracy lift in

114

conduction band of semiconductors [34]. Although various authors have shown works

on developing quantum transport simulations [78-79], they mostly ignore the non-

parabolicity of conduction bands. In very highly scaled devices, the quantum

quantized subband energies are located far above the band minima where the effective

masses obtained at the minima points are not valid anymore. In this chapter, we will

develop a quantum ballistic transport model in 1-D quantum confined double gate

FET (DGFETs) structures which takes in to account non-parabolicity of conduction

bands.

6.2 1-D Quantum Confined System

In the BTBT chapter, we have shown that if quantum confinement is strong only in

one direction (z direction) the states are written as a sum of 2-dimensional subbands,

the edges of which are obtained by solving the Schrodinger equation in the z-direction

for each mesh point along the x axis. For materials with degenerate minima in the

conduction band, the states can be computed with the directionally decoupled effective

mass approximation [44] for each minimum, taking into account mass anisotropy and

non-parabolicity. The same procedure is applied to the valence band maxima.

115

We recall the equations derived in the BTBT chapter. (Eq. 1.53 to 1.69). Carrier

wavefunctions are decoupled using subband decomposition and take the form:

Ψ , , ; exp , (6.1)

where , n and represent degenerate valleys, quantum number for subbands,

thickness of the channel, respectively. The wave-functions , ; are normalized

in z direction and unit-cell functions , are normalized in the space. 1D

Schrodinger equation in the z confined direction provides eigen-functions , ;

and eigen-energies , .

Figure 6.1: Schematic view of the thin channel double-gate MOSFET (DGFET).Thickness of the channel is . Current flows along x-direction and the channel is quantized along the z-direction.

ChannelInsulator

Gate

Gate

xz

y

xz=0 z=tb

Eg

Insulator

Insulator

Energy

0

,

,

,

,

, ;

, ;

, ;

, ;

, ,

,

,

116

21

,, ;

, , ; , , ; (6.2)

where , is the bandedge energy level of band . Under the assumptions that

subband wavefunctions , ; have very weak dependency on x and inter-subband

couplings are also very small due to the strong quantum confinement along z-direction,

the wave functions , satisfy the following 1D Schrodinger equation in x

direction (or channel direction).

21

′ ′

′ ,′

(6.3)

where coefficients ′ denote the inter-band couplings which represent the

BTBT.

6.3 1-D Schrodinger Equation with non-parabolic effective

mass

The electron levels in 1-D quantized Double Gate FETs with small effective mass like

in IIIV materials can be several tenths of an electron volt above the bulk-conduction

band edge. For that range of quantization, without considering non-parabolicity of

bandstructure, errors in eigen-energies obtained by solving eq. (6.2) could be

significantly large. The E-k dispersion relationship for the two-band Kane model [80]

can be expressed in the simple form:

12

(6.4)

117

where m* is the effective mass and a coefficient contains nonparabolicity

corrections. This simple relation is accurate enough in cases of our interest. In this

work we extract along the quantization direction (here z-direction) from the data

obtained by the empirical pseudo-potential method.

Although several different approaches have been used to take into account in solving

Schrodinger equations [81-82], we take a simpler approach utilizing energy dependent

effective mass and iterations. The E-k dispersion relationship in eq. (6.4) can be

interpreted as variant effective mass dependent on energy.

, 1 , ; (6.5)

At each iteration, solving the eq. (6.2) to get energy levels of quantized states, ,

are updated from eq. (6.5). To avoid oscillation and slow convergence we used a

damping method to prevent an abrupt change in , and the Hamiltonian. As

depicted in fig. (6.2), after about 5 iterations, energy levels converge within 0.01eV.

For effective mass of 0.069 , of 0.7eV and channel thickness of 3nm, the error

in using a parabolic band is over 0.15eV.

118

6.4 Electron Wavefunction along the Channel

Since in thin film DGFETs vertical quantum confinement is particularly strong and

wavefunctions along the vertical direction (z-direction) vary slowly along the transport

direction (x-direction), we have shown in the BTBT chapter that in this case the inter-

subband couplings can be ignored and the transport equation in eq. (6.3) can be

decoupled from other subbands. The only significant coupling is from inter-band

BTBT due to the breaking of orthogonality between unit-cell functions in different

bands by the applied electric field. Without BTBT coupling, the solution of eq. (6.3)

can be approximated as plane waves, i.e.:

Conduction band wavefunction from source-side

exp ,′ ′ (6.6)

Figure 6.2: Convergence of the first energy level. Effective mass is 0.069 and is 0.7eV . Channel thickness is 3nm. About in 5 iterations, it

converges within 0.01eV.

0 2 4 6 8 10 120.4

0.45

0.5

0.55

0.6

E(eV)

# of iteration

119

0 (6.7)

| | | | 2 | | (6.8)

Conduction band wavefunction from drain-side

exp ,′ ′ (6.9)

0 (6.10)

| | 2 | | (6.11)

where the non-parabolic imaginary part of ,′ was given in the BTBT chapter. The

term 2 | | represents the component traveling in opposite direction

to the injected flux. As a result, the magnitude of the wavefunction is 1 when all the

injected flux passes through the channel without any reflection and its 2 when all is

reflected back by the potential barrier. The term is added to ensure the flux

conservation. is given as:

| || | (6.12)

In tunneling problems, the necessity of dealing with complex k and negative kinetic

energy naturally arises. We followed the method by suggested by Buttiker [83-85] to

treat the velocity of tunneling and just took the absolute value of velocity.

120

6.5 Non-parabolic Density of States

To sum up total charges in the channel, non-parabolic density of states (DOS) is

needed to be obtained. Let us approximately write the dispersion relation between

and as:

1 Δ (6.13)

, where Δ is the quantization energy of the n-th state in the valley and it can be

written as:

Δ , ; (6.14)

One-dimensional (1D) DOS along the y direction is given by,

, 21

2 (6.15)

By taking a derivative of eq. (6.13) with respect to , we have

2 ΔExz

ΔExz 2 , ΔExz 0 (6.16)

, where

ΔE 1 Δ (6.17)

By plugging in the eq. (6.16) into (6.15), we obtain the explicit form of 1D DOS along

the y direction:

,Δ 2

2 ΔE

(6.18)

The 1D DOS along the x direction can be obtained by the following procedure:

121

, ,1

2| |

12

| | 2 | |

12

| | 2 | | (6.19)

,where represents the contact where the carriers are injected and represents the

contact where the carriers are absorbed. We assume here that the x-directional electron

density is limited by the electron flux from the contacts, thus it follows the electron

density at the contact. By utilizing the derivative obtained in Eq. (6.18) and introducing

the term | ; | | | 2 | | , the x-directional 1D DOS

can be accordingly rewritten as:

, ,Δ

2 2| | ΔE| ; | (6.20)

Note that for 1D DOS in x direction we do not multiply it with the directional factor 2

(Eq.(6.15)), since we already took into account the reflected wavefunctions when we

obtained them in eq. (6.6)-(6.11)

6.6 Charge Density (with Non-parabolicity corrections)

The source and drain contacts are assumed to be ideal reservoirs in thermal

equilibrium with the Fermi energies and . They supply carriers to the channel

and absorb carriers from the channel without reflections. The Fermi distribution

functions at the contacts are given as:

122

,1

1 exp

(6.21)

,where k is the Boltzmann constant and T is the temperature.

If the total energy of the subband E is , where nzE is the eigen-

energy of n-th quantized states in z-direction, the 3D DOS in the channel can be

expressed as follows:

2 , , ;

Δ

2 | |; ;

Δ 2ΔE

(6.22)

The total charge density injected from one-side of the contact is given as:

| |2

; ;

| |

Δ Δ 2

ΔE 1 exp

(6.23)

Eq. (6.23) requires double numerical integration and this is computationally very

intensive work to be done on every point. To relieve the burden, it is tempting to carry

out the integration with respect to first and find the analytical form. When is

sufficiently small, the integration can be rewritten as:

Δ Δ 2

ΔE 1 exp(6.24)

123

ΔE

1. 2 .

.

1

1 exp

ΔE

1. 2 . .

1 expΘ ,

ΔE Δ .. 1 Δ .

.

Δ Θ ,

,where is the complete Fermi-Dirac integral of j order:

1 exp (6.25)

and

(6.26)

The analytical approximations of the complete Fermi-Dirac integrals of the orders, 1.5

and 0.5 are suggested in [86] with error less than 0.7%. We extended this work to treat

the integral of order -0.5, 1 and 2. The detail of approximations for the Fermi-Dirac

integrals is covered in Appendix.

In Eq. (6.24), we also introduced new variables , and ΔE , which are defined as:

/ΔE (6.27)

ΔE (6.28)

and

124

ΔE 1 (6.29)

We introduced the error terms Θ ; resulted from the Taylor series expansion of

.

1

1 exp

.

1 expΘ ,

(6.30)

As seen in Fig. 6.3, this approximation becomes inaccurate and leaves significantly

large Θ ; when and become large and we cannot just ignore it. To find

the analytical form for Θ ; , the simplest approximation would be to consider the

asymptotic behavior of the integral.

(a) (b)

Figure 6.3: The error term Θ , becomes significantly large value when

or are large. (a) For 0.5, 4/ and 40, the error term accounts for more than 10% of the correct integral. (b) For

2, 4/ and 40, the error term accounts for more than 25% of the correct integral.

-40 -20 0 20 400

2

4

6

8

10

12

EF0/kT

Θ (

E F0, β

) /In

tegr

al (E

rror

%)

by=1/eV

by=2/eV

by=4/eV

bxEx=0.5

-40 -20 0 20 400

5

10

15

20

25

30

Θ (

E F0, β

) /In

tegr

al (E

rror

%)

EF0/kT

by=1/eV

by=2/eV

by=4/eV

bxEx=2

125

In the limit of ∞ , the asymptotic behavior of the integration of

. in Eq. (6.24) is:

lim.

11

1 exp

.

1

2 . . atan . ..

. Θ , (6.31)

For this limit, the analytical form of the error term can be found by subtracting the

integral with the one without non-parabolicity ( 0).

Θ , 2 . . atan . . 23

. Θ , 0

, (6.32)

For the limit of ∞, the asymptotic behavior is approximated as:

Θ , .. , (6.33)

Utilizing the Eq. (6.32) and (6.33), now we have the analytical approximated form of

Θ ⁄ , , which is

Θ ⁄ ,

.. , 0

2 atan23

0 , 0

(6.34)

126

The relative error of the expression in Eq. (6.34) is plotted in Fig. 6.4. The maximum

error for of 4/eV is below 1.3%. Normally, the semiconductor materials we are

interested in have lower than 4/eV. For example, the in the Γ valley of GaAs is

just 0.69/eV and that of InAs is about 2/eV.

Now, we go back to the DGFET case which forms two contacts to source and drain as

depicted in Fig. (6.1). The total electron charge density supplied from source and drain

sides can be obtained utilizing the previous results.

| |2

;(6.35)

(a) (b)

Figure 6.4: Relative error of the approximation in Eq. (6.34) vs. the integral in Eq. (6.24). This approximation is accurate with an error below 0.3%.

-40 -20 0 20 40-0.1

-0.05

0

0.05

0.1

0.15

E'F/kT

by=1/eV

by=2/eV

by=4/eV

bxEx=0.5

Error (%)

-40 -20 0 20 40-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

EF0/kT

by=1/eV

by=2/eV

by=4/eV

bxEx=2

Error (%)

127

| ; | Θ ,

| ; | Θ ,

ΔExzΔ

| |

,where the non-parabolicity correction term Θ ⁄ , is

Θ ⁄ , 1 Δ .. Δ Θ ,

(6.36)

and ΔE 1 Δ , E

, E

, ΔE 1 ,

and as already defined in Eq. (6.17),

(6.27) , (6.28) , (6.29) and (6.26). Θ , is defined in Eq. (6.34).

These results correspond to the charge density in ballistic DGFET formulated by

Natori ([77]). We can easily show that his formula is the special case of the quantum-

ballistic DGFET with parabolic bandstructure. By taking the conditions 0,

Δ 1, 0, ; 0 ; 0 1, we have:

| |2

; (6.37)

Using the relation

ln 1 exp (6.38)

Eq. (6.37) can be written as:

128

| |2

;

ln 1 exp 1 exp (6.39)

, which is same as what Natori derived.

6.7 Quantum Ballistic Current (with Non-parabolicity

corrections)

The quantum-ballistic current is given by the Landauer’s formula [87] where it is the

sum of the current component flowing in each sub-channel. Each current component is

expressed as the product of the unit charge, the number of carriers flowing through the

sub-channel per unit time and the transmission coefficient.

2;

,,

(6.40)

where W is the width (y) of the device, is the carrier group velocity, is the

transmission coefficient and is the thickness of the channel. Let’s take a look at the

flux from source to drain.

,

,| ; |

| | , ,∞

0

∞

∞

(6.41)

Using the definitions in Eq. (6.12), (6.15), (6.18), (6.19) and (6.20) to substitute terms in

Eq. (6.41) with, we rewrite Eq. (6.41) as

129

,

1

2 | |

Δ 2

ΔE

1

1 exp

∞

0

∞

∞

(6.42)

,where .

Eq. (6.42) requires double numerical integration and this is computationally very

intensive work to be done on every point. As we did for calculating charge density, we

will carry out the integration with respect to first and find the analytical form.

When is sufficiently small, the integration can be rewritten as:

Δ 2

ΔE

1

1 exp

ΔE1 exp

, ΔE ,

(6.43)

We introduced the error term ; resulted from the Taylor series expansion of

E

E

+Higher order terms.

To find the analytic solution for ; , we first take a look at the asymptotic

behavior of the integral in Eq. (6.43) in the limit of ∞.

limΔ 2

ΔE

1

1 exp

(6.44)

130

Δ 2

ΔE

ΔE 2 . ΔE ,

For this limit, the analytical form of the error term can be found by subtracting the

integral with the one without non-parabolicity ( 0 ).

, 2 . 2 . , 0 , (6.45)

In the limit of ∞, by expanding E

with Taylor series expansion,

we can easily find that:

,32

. , (6.46)

By properly connecting two limits of , at 0, we can write , as

⁄ ,

32

. , 0

2 232

. 0 , 0

(6.47)

Fig. 6.5 depicts the accuracy of this approximation. Even with 4/ , the error is

less than 1.3%.

131

After the integration with respect to , using the analytic form of , given in

Eq. (6.47), Eq. (6.41) can be rewritten as

2 | | ΔE

.. ⁄ ,

(6.48)

Plugging Eq. (6.48) into Eq. (6.40), we can derive the equation to evaluate the current

flowing from source to drain in the device.

2 ;

| | ΔExz

12

12 0.5 ⁄ ,

| | ΔExz

12

12 0.5 ⁄ ,

(6.49)

Figure 6.5: Relative error of the approximation of in ⁄ , Eq. (6.47)vs. the integral in Eq.(6.44). This approximation is accurate in the range of error below 1.3%.

-40 -30 -20 -10 0 10 20 30 40-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

EF0/kT

Error (%)

by=1/eV

by=2/eV

by=4/eV

132

ΔE 1 Δ , E

, and

as already defined in Eq. (6.17), (6.28) and (6.26). , is defined in

Eq. (6.47).

6.8 Band-to-Band Tunneling Current

In the last section, we have developed the current formula for the majority carriers (in

this case, electrons). In devices made of high mobility materials, due to their small

band gaps and effective mass, the current generated by band-to-band tunneling (BTBT)

becomes increasingly important.

As shown in Fig. 6.6, BTBT current increases as we try to turn off the by applying

negative gate bias (Vg). This behavior imposes a minimum value of the leakage

current. If the crossover between the quantum-ballistic current and the BTBT current

occurs at relatively large Vg, it is not possible to reduce the leakage current any

further by just lowering gate bias, even if we can engineer the device structure to

achieve steep sub-threshold slope. In MOSFETs, the BTBT currents are generated at

the gate-drain edge (Fig. 6.7). Low gate bias and high drain bias create large electric

field enough to pull out the electrons in valence band to conduction band and as a

result, electron-hole pairs are generated and flow to opposite direction until they are

collected by contacts. Thus, the stronger gate field to lift up the channel potential,

more BTBT current is generated.

133

The approach to treat the BTBT current is slightly different from the modeling of the

current by majority electron carriers. In the previous section, when an electron travels

from one contact to the other, we assumed that it flows through complete quantum

coherent transport and that no-scattering events are allowed, since the channel length

is shorter than the relaxation length. However, unless the drain Fermi level ( ) is

lower than the source Fermi level ( ) by more than the band-gap ( ),

the holes generated by BTBT cannot overcome the potential barrier created by valence

band at source-side, consequently they are unable to reach the source contact (See Fig.

6.7). Thus, generated holes are reflected back to the cavity created between two

potential barriers and accumulated in the channel. As trapped holes are reflected back

and forth, they start to scatter and eventually they are thermalized. Thermalization

Figure 6.6: Source to drain current ( DSI ) vs. gate voltage (Vg). When negatively large Vg is applied, BTBT current generated at the edge between gateand drain grows significantly.

-0.2 0 0.2 0.4 0.610

-10

10-8

10-6

10-4

10-2

Vg (V)

Ids (A/um) Lg=15nm

Quantum-Ballistic

BTBT

Total Current

134

allows a certain fraction of lucky holes get enough energy to overcome the barriers.

Since the potential barrier along the source side is lower than the barrier toward the

drain contact, most of thermionic emissions occur toward the source. This

accumulation process continues until the thermionic emissions are balanced by the

incoming holes supplied by BTBT process.

As the first step to model the hole transport in the channel, we assume that a hole

quasi-Fermi level ( ) can be defined in the hole cavity. This assumption that holes

roughly exhibit the Fermi-Dirac distribution is not irrational since in the off-state the

barrier height is large enough (>0.1eV or 5KT) that the most of holes in the channel

are trapped inside the cavity, while only a small fraction of holes can flow to reach

source side.

Figure 6.7: Illustration of the carrier flux components. Electron currents consistof the one injected from source side, one from drain side and onegenerated by BTBT process. The BTB tunneling hole current issupplied to the channel cavity (around ). The accumulated holes can escape by thermionic emission from the cavity.

IHole,Therm

IHole,BTBT

xEmax

Q-BallisticIe,Source

Q-BallisticIe, Drain

BTBT

Ie,BTBT

EFS

EFH

EFD

135

The procedure to find the hole BTBT current in equilibrium state can be roughly

divided into three steps. The first step involves calculation of BTBT current. The

second step is to evaluate the thermionic emission current and the final step is to find

the hole quasi-Fermi level such that it equates the thermionic current and the BTBT

current.

In the BTBT chapter, we derived the expressions for the BTBT tunneling electron-

hole pair generation rates in 2-D quantized system. The model is useful for the particle

based device simulator (ex. Monte Carlo or drift-diffusion), since it provides with the

carrier generation rate at each point. One way to find the BTBT current is to calculate

the generation rate through the path searching and then summing up the generation

rates of each point over the space. When the BTBT process rate is very large, the hole

quasi-Fermi level goes down below the valence band level. Low hole quasi-Fermi

level not only increases the increase the thermionic emission, but also reduces BTBT

rate. When the states in valence band are depleted (in other terms occupied with holes),

it limits the BTBT process. To take into account the depletion effect, we add the

Fermi-Dirac distribution to the generation rate, which is as follows:

,,,

1

1 exp

1

1 exp

(6.50)

, where is the Fermi level of the drain contact and is the hole quasi-Fermi

level in the channel (Fig. 6.7). and are the energy levels of states in valence

136

band and conduction band, respectively. Since the path finding algorithm searches for

the point of which energy level is same as the original points, and are

related such that:

, For Direct Tunneling

, For Phonon assisted Tunneling (6.51)

Now we evaluate the hole thermionic current injected from the channel to the source.

The energy levels governing the thermionic emission are illustrated in Fig.6.8. is

the barrier height that the hole in n-th state experiences.

We will use Eq. (6.49) to derive the expression for the thermionic emission from the

cavity. Let’s first take a look at in Eq. (6.47). BTBT current becomes significantly

large only when the drive current becomes few orders lowers than its maximum (Fig.

Figure 6.8: Sketchy of the band profile along the channel. Source Fermi level( ), Drain Fermi level ( ), Energy profile of the n-th state of valence band, hole quasi-Fermi level ( ), hole barrier height ( ) are marked in the figure. Only the hole with kinetic energy larger than

can escape the cavity and reach the source side.

xEmaxxS

EFS

EFH

EFD

Ev

Enz| EB |

137

6.6). Thus, we can assume that hole barrier , which is roughly same as electron

barrier height, is at least bigger than 0.1eV when BTBT current has any meaningful

number. This condition can be expressed as:

0.1 0.1 (6.52)

Note that the sign of has changed from the electron case. Under this condition,

can be approximated as

,32

34

exp , 0.1 (6.53)

With the same condition, the Fermi-Dirac Integral of -0.5 order, . in Eq. (6.49)

can be approximated as

. exp , 0.1 (6.54)

The | | for holes have the following values.

| | 1 0 (6.55)

Utilizing Eq. (6.53) , (6.54), and (6.55), Eq. (6.49) can be simplified:

2 ΔExz 1

34

exp (6.56)

, where

ΔE 1 E E (6.57)

and

/ΔE (6.58)

138

The integration over can be approximately carried out since the term exp

decreases very fast as increases, while the term ΔE 1 stays at

relatively same values. After the integration Eq. (6.56) becomes

2 ΔEB 1

34

exp(6.59)

where

ΔEB ΔE 1 E E EB (6.60)

Δ (6.61)

E EFH EB (6.62)

Physically, there should not be any current flowing between two thermal equilibrium

points when the Fermi levels of the points are same. To ensure the current obtained by

Eq. (6.59) goes to zero as EFH EFS , we slightly modify the Eq. (6.59), and that

becomes:

expEFS EFH 1 (6.63)

,where

2 ΔEB 1

34

expE EFS EB

(6.64)

Finally, by equating the thermionic emission current (Eq. (6.63)) and the BTBT current

(Eq. (6.50)), we can find the expression for the hole Fermi level EFH:

EFH EFS log 1 (6.65)

139

Once we have the hole Fermi level, hole density can be also obtained. The expression

for hole density is similar to the electron density in Eq. (6.35).

| | ;

Θ ,ΔExzΔ

| | (6.66)

,where the non-parabolicity correction term Θ ⁄ , is

Θ ⁄ , 1 Δ .. Δ Θ ,

(6.67)

and ΔE 1 Δ , E

, E

, ΔE 1 , and

. Θ , is defined in Eq. (6.34).

6.9 Summary

We have a developed quantum ballistic transport model to accurately model the carrier

transport in highly scaled DGFETs. This model takes into account the effects of non-

parabolicity of conduction bands on transport mass and 1-D density of states.

Asymptotic approximation helped to simplify model. The BTBT leakage current is

also included in the ballistic transport model. The flow chart of the entire device

simulation using this transport model is given in Fig. 6.9.

140

Figure 6.9: Flow chart of the carrier transport simulation of quantum-ballistic

FETs.

InitializeQe and Qh

Solve Poisson V

Solve 1-D Schrodinger Ea,n

Evaluate Qe , Qh , EFH , Ie and IBTBT

If Q and I areconverged

Set New Bias on Electrodes

No

Yes

141

Chapter 7 Properties and Trade-offs Of

Compound Semiconductor MOSFETs

7.1 Introduction

As we continue to aggressively scale transistors in accordance with Moore’s Law to

sub-20-nm dimensions, it becomes increasingly difficult to maintain the required

device performance. Currently, the increase in drive currents for faster switching

speeds at lower supply voltages is largely at the expense of an exponentially growing

leakage current, which leads to a large standby power dissipation. There is an

important need to explore novel channel materials and device structures that would

provide us with energy-efficient nanoscale MOSFETs. Due to their significant

transport advantage, high mobility materials are very actively being researched as

channel materials for future highly scaled CMOS [27, 31-32, 88-94]. However, most

high mobility materials also have a significantly smaller bandgap compared to Si,

leading to very high band-to-band tunneling (BTBT) leakage currents, which may

ultimately limit their scalability.

142

The different significant components of channel leakage are shown in Fig. 7.1.

(Assuming an ideal high-k gate dielectric technology, gate leakage has not been

addressed in the present study.) Conventionally, the exponentially rising diffusion (or

subthreshold) current due to lowered threshold voltages has been the dominant leakage

mechanism in Si MOSFETs. However, with continued device scaling into the

nanometer regime and the increasing E-fields in the channel, there is also an

exponential growth in the tunneling components (BTBT and direct source/drain (S–D)

tunneling) of leakage. Fig. 7.2 (a) and (b) shows the material parameters of low

effective mass (high mobility) semiconducting channel materials. A universal trend is

observed with respect to the bandgap and the dielectric constant of low effective mass

Figure 7.1 : Dominant leakage paths in nanoscale high mobility CMOS devices.

High mobility (low-bandgap) materials suffer from excessive BTBT

leakage.

IHole,Therm

IHole,BTBT

xEmax

Q-BallisticIe,Source

Q-BallisticIe, Drain

BTBT

Ie,BTBT

EFS

EFH

EFD

143

semiconductors. As we push toward lower effective mass (high mobility) channel

materials, the bandgap rapidly drops and the dielectric constant increases.

As seen in Table 7.1, III-V materials have significantly smaller effective mass and

higher electron mobility compared to Si and Ge. However, most high mobility

materials like Ge, InAs, and InSb also have a significantly lower bandgap compared to

Si. Due to the increasing E-fields in the channel and the smaller bandgap in these high

mobility materials, the BTBT leakage current can become excessive and can

ultimately limit the scalability of high mobility channel materials. Another point to

(a)

(b)

Figure 7.2 : Tradeoffs between effective mass (m ), bandgap (EG), and dielectric

constant (κs) in semiconducting materials.

144

note is that since most high mobility materials also typically have a higher permittivity

(κs), they also suffer from worse short cahnnel effects (SCEs).

At an initial glance, due to their extremely small transport mass leading to high

injection velocity (vinj), III-V materials appear to be very attractive candidates as

channel materials for highly scaled n-MOSFETs [26, 88, 95]. However, III-V

materials have many significant and fundamental issues such as small density of states

(DOS), bigger dielectric constant and large BTBT leakage current, which may prove

to be severe bottlenecks to their implementation. These tradeoffs need to be

systematically and thoroughly investigated from a theoretical standpoint in order to

help in providing useful guidelines, which can efficiently streamline the resources

invested by several research groups throughout the world in their experimental

development.

Table 7.1 : Dominant leakage paths in nanoscale high mobility CMOS devices.

High mobility (low-bandgap) materials suffer from excessive BTBT

leakage.

17.717.714.812.41611.8εr

0.170.170.360.361.420.661.12EG (eV)

77,00077,00040,00040,000920039001600μn(cm2/Vs)

0.0140.0140.0230.0230.0670.080.19meff*

InSbInSbInAsInAsGaAsGaAsGeGeSiSiMaterial/PMaterial/Propertyroperty

17.717.714.812.41611.8εr

0.170.170.360.361.420.661.12EG (eV)

77,00077,00040,00040,000920039001600μn(cm2/Vs)

0.0140.0140.0230.0230.0670.080.19meff*

InSbInSbInAsInAsGaAsGaAsGeGeSiSiMaterial/PMaterial/Propertyroperty

145

In this chapter, we will look at novel III-V channel materials for n- and p- MOSFETs,

and the application of various strain along different surface/channel orientations to

improve their performance, in terms of increased drive current and lower leakage. We

will also discuss novel device structures, and approaches to address the problem of

parasitics, which can enable energy-efficient nanoscale III-V n- and p- MOSFETs.

7.2 Simulation Framework

7.2.1 Bandstructure calculation (Real and Complex)

The transport of electric charge carriers (electrons and holes) in crystal

semiconductors is governed inherently by their bandstructures. Real part of

bandstructure provides information on how the carriers are distributed in energy-k

space (density of states and valleys) and how the carriers react to the external electric

fields (effective masses). The imaginary part of bandstructure helps us to understand

the tunneling process of carriers between different bands.

The Plane-wave based empirical pseudopotential method (EPM) [5] is one of the

useful ways to accurately estimate the details of the bandstructure. While the tight-

binding method or kÿp theory requires many fitting parameters, EPM only requires a

146

few fitting parameters and provides more accurate bandstructures, thus EPM is more

suitable when it is needed to study the entire first Brillouin zone and the effect of the

strain.

Fig. 7.3 shows the result of the bandstructure calculation done by EPM for Ge and

InAs described in the Chapter 2. Using this method, both complex part and real part of

bandstructure can be obtained.

7.2.2 BandtoBand Tunneling (OffState Leakage)

In semiconductors, most of states in valence bands are filled with electrons. Under a

strong applied external electric field, electrons in valence bands can tunnel through the

forbidden bandgap and when they reach the conduction bands, electron-hole pairs are

(a) (b)

Figure 7.3 : (a) 3D plot of complex bandstructure of germanium along the [111]

direction. (b) 2D plot of complex bandstructure of InAs.

147

created (Fig. 7.4) [96]. In in-direct band-gap materials such as Si, since the minimum

of the conduction band is located at ∆ point away from the top of valence band which

is located at Γv, the momentum conservation law prohibits the tunneling between two

bands (Fig. 7.4). Phonons created by thermal lattice vibration can provide the

momentum difference and allow two bands to be coupled. This tunneling process

incorporating phonons is called in-direct BTBT or phonon assisted BTBT, which is

the dominant BTBT process in Si [73].

We evaluated the tunneling rate interband matrix elements between quantized states

for both direct and indirect tunneling, as described in the Chapter 3. The final BTBT

carrier generation rate was calculated by adding up the transition rates for all the

possible transitions [72-73]. We included all the possible transition from valence band

to conduction bands such as ΓV-ΓC, ΓV-L and ΓV-X (Fig. 7.4 (b)). While counting, for

direct tunneling the momentum conservation selection rule was applied, while for the

indirect tunneling the selection rule was relieved.

148

The efficiency of BTBT process is altered by large amounts in few nm thick slabs

compared to bulk semiconductors [73]. An ultra-thin body has a larger effective band

gap and smaller DOS than bulk, due to strong quantization. With larger effective band

gap, the wave function decays faster inside the forbidden gap and lowers BTBT rate

(Fig. 7.5). To take into account the quantization effect caused by ultra-thin body, the

wavefunctions and the energy levels of quantized subband states were obtained by

solving 1-D Schrödinger equation along z-direction for both electrons and holes [97].

(a) (b)

Figure 7.4 : (a) Illustration of band-to-band tunneling paths. The electron can

tunnel to indirect-gap conduction band (L or ∆ valleys) through

phonon interactions. (b) All Possible (Direct and Indirect) BTBT

Tunneling Paths, Γ-Γ, Γ-X and Γ-L, are captured by the model.

149

7.2.3 2DQuantum Ballistic Current (OnState Drive Current)

For high mobility channel FETs with gate length shorter than 20nm, the majority

carriers injected from source travel through channel without significantly hindered by

scattering [98-99]. In [98] Wang and Lundstrom explained the validity of ballistic

transport and in [77] Natori formulated the ballistic transport in thin oxide MOSFETs

where states are quantized by strong vertical gate electric field. However, in highly

scaled DG MOSFETs where channel is only few nm thick, quantum confinement

Figure 7.5 : Ultra-thin body and larger quantization increases the effective

bandgap and lowers the tunneling rate.

150

effect is strong enough to lift quantized energy states far above the band-edge. In this

case, E-k dispersion significantly deviates from an ideal parabolic curve, thus adopting

Natori’s formula would result in over-estimations on quantized energy level and

under-estimations on the density of states. Moreover, in high mobility channel, its

small effective mass’ significantly enlarges the source-to-drain (S/D) tunneling which

makes sub-threshold slope lower and increases leakage current. As described in the

Chapter 6 , we extended applicability of Natori’s model so that the effects of non-

parabolicity and S/D tunneling are included.

7.2.4 Benchmark Test bed Device Structure

Since the purpose of this chapter is on exploring the suitability of high mobility III-Vs

as channel materials, we need to focus solely on carrier transport characteristics

(Ion/Ioff), while the structural variability and optimization should be excluded. The

structure, which suits best to the purpose and still closely resembles the structure

manufacturable in a next few generations, is a double gate FET structure (Fig. 7.6).

We assumed source and drain are perfect carrier absorbers, undoped channel and

abrupt doping profiles with a 1.3nm/dec. roll-off.

151

7.3 Power-Performance Tradeoffs in Binary III-V materials

(GaAs, InAs, InP and InSb) vs. Si and Ge.

Although their small transport mass leads to high injection velocity (vinj), III-V

materials have a low density of states (DOS) in the Γ-valley, tending to reduce the

inversion charge (Qinv) and hence reduce drive current [34, 99-102]. Furthermore, the

small direct band gaps of Ge and III-V materials inherently give rise to very large

Figure 7.6 : Test bed for benchmarking III-Vs as channel material. Gate Length

LG=15nm, equivalent gate oxide thickness (EOT) TOX=0.7nm,

channel thickness TS=3-5nm and VDD=0.3-0.7V.

152

band to band tunneling (BTBT) leakage current compared to Si [73-74]. They also

have a high permittivity and hence are more prone to short channel effects (SCE).

Quantum confinement in these ultra-thin nanoscale DGFETs plays a very important

role. In III-V materials, large quantum confinement makes electrons populate and

conduct in heavier L- or X-valleys. Furthermore, there is a significant increase in the

conductivity mass in quantized sub-bands due to large non-parabolicity of Γ-valley

[23, 34]. Quantum confinement effects also reduce BTBT leakage in an ultra thin

channel [72-74]. In this section, we will discuss Double Gate (DG) n-MOSFETs with

Ge and IIIV materials (GaAs, InP, InxGa(1-x)As, InAs and InSb) and compared to Si

[23-24, 33, 72].

7.3.1 Inversion charge and injection velocity

Generally, low DOS in III-V materials reduces the effective gate capacitance (CGeff),

resulting in lower inversion charge at the given gate voltage (Fig. 7.7). For example,

InAs can store only one third of electrons that Si can store. Fig. 7.8 shows injection

velocities (vinj) at VGate of 0.7V. Due to their small transport effective mass, III-V

materials and Ge have several times larger injection velocity than Si. Among all,

transport effective masses of InAs and InSb are only 0.023 m0 and 0.014 m0,

respectively and they exhibit 6 times larger vinj than Si. As we scale channel thickness

(TS), the Γ-valley energies in III-V materials rise up rapidly due to very low mz

resulting in most of the Qinv in thin films moving to the heavier L-valley. Once the L-

153

valleys are resided by electrons, III-V materials start to behave like Ge – large CGeff

and large Qinv. The L-valleys start to contribute for GaAs at TS=7nm and for InSb at

3nm. Scaling TS improves subthreshold slopes and DIBL effects, permitting larger

Qinv or higher Fermi level at the given gate voltages; thus carriers are injected at a

higher vinj from source to drain in thin body (Fig. 7.8). However, in small bandgap

materials such as InAs and InSb, due to strong non-parabolicity in Γ-valley, high

Fermi levels and lifted-up quantized energy states in thin body make electrons heavier

and injection velocity is reduced.

Fig. 7.7: Inversion charge (Qi) at VGate=0.7V and Tox=0.7nm. Si channel stores 4 times

more electrons than InAs. As Ts scales, Qi becomes bigger due to reduced Vth and

nonparabolicity of band.

S i G a A s I n P G e I n A s I n S b0

5

1 0

1 5

Qi (

1012

#/cm

2 )

1 0 n m5 n m3 n m

154

Fig. 7.8: Injection velocities (vinj) at VGate=0.7V and Tox=0.7nm. InAs and InSb have 6

times larger vinj than Si due to their small transport mass. As Ts scales, lowered

threshold voltage (vth) increases vinj, while nonparabolicity reduces vinj.

7.3.2 ION, IOFF,BTBT and Delay

Device simulation results for LG=15nm, TOX=0.7nm, TS=3-5nm and VDD=0.3-0.7V

are shown in Fig. 7.9 (ION) and Fig 7.10 (IOFF,BTBT). There have been concerns that III-

V materials would underperform than Si because of their low Qinv [23-24, 33, 99-100,

103]. However our study shows that despite of low Qinv, thanks to their large vinj, III-V

materials like InAs, InSb and InP can flow up to 80% larger drive current than Si.

Although L valleys of conduction bands in Ge has large density of states and small

transport effective mass, the drive current in Ge channel is lessened by worst SCE. In

3nm thick channel, SCE can be minimized, therefore Ge allows highest drive current.

S i G a A s I n P G e I n A s I n S b0

2

4

6

8

v inj

(107 cm

/s)

1 0 n m5 n m3 n m

155

(a) (b)

Fig. 7.9: Drive currents (ION) for various (a) Ts and (b) VDD. LG=15nm, TOX=0.7nm.

The minimum achievable standby leakage by BTBT (IOFF,BTBT) is at the intersection of

the BTBT leakage with the sub-threshold leakage. DGFET cannot have lower leakage

current than IOFF,BTBT by adjusting threshold voltage. Small bandgap materials such as

InAs, InSb and Ge have extremely large IOFF,BTBT higher than 0.1µA/µm. However, if

the technology allows to thin channel thickness to 3nm, it is possible to limit IOFF,BTBT

below 0.1µA/µm. Based on our study, larger than 1000x reduction is possible in

GaAs, InP and Ge by scaling Ts from 10nm to 3nm.

Si GaAs InP Ge InAs InSb0

1

2

3

4

5

6

ION (m

A/ μ

m)

10nm5nm3nm

Si GaAs InP Ge InAs InSb0

1

2

3

4

5

ION (m

A/ μ

m)

0.7V0.5V0.3V

(a) (b) (c)

Figure 7.10 : (a) IOFF,BTBT is minimum achievable leakage current defined as a

intersecting point between BTBT current and Drive current. (b)

IOFF,BTBT for various Ts. VDD=0.5V. (c) IOFF,BTBT for various VDD.

log (IDS)

IOFF,BTBT

VGS

IBTBT

IDrive

Si GaAs InP Ge InAs InSb

IOFF

,BT

BT

(A/ μ

m)

10- 1

310

-910

- 710

-510

- 310

-11

7nm5nm3nm

10nm

Si GaAs InP Ge InAs InSb

IOFF

,BT

BT

(A/ μ

m)

10-1

310

-910

-710

-510

-310

-11

0.7V0.5V0.3V

156

7.3.3 Effect of Scaling film thickness and VDD

Scaling TS enhances the device performance by eliminating the SCE effect and

reducing BTBT leakage. IOFF,BTBT in Ge, InAs, GaAs and InSb can be reduced by over

1000x by scaling Ts to 3nm (Fig. 7.10), due to enlarged bandgap by quantization of

sub-bands. As Ts scales, ION is increased by 2.5 times (Fig. 7.9), but concurrently

electrons become heavier, resulting in losing the advantage of III-Vs over Si. Scaling

VDD is desirable since it reduces BTBT leakage (Fig. 7.10) by relieving maximum

electric field applied in the device and it decreases switching energy. III-Vs and Ge

with VDD of 0.5V provide with higher drive current than Si does with VDD of 0.7V

(Fig. 7.9 (b)), which means a transistor made with III-Vs and Ge can consume only ½

of active energy consumed by a transistor with Si. However large threshold voltages

of III-Vs prevent VDD from scaling below 0.5V where III-Vs start to drive smaller

currents than Si. InP exhibits good performance (high ION, short delay and low

IOFF,BTBT) in any scaling scenarios.

7.3.4 PowerPerformance Tradeoff of Binary IIIV Materials vs.

Si and Ge

157

Relationships between delay and IOFF,BTBT for different materials are depicted in Fig.

7.11. Subthreshold leakage is fixed to be 0.1µA/µm and VDD is varied from 0.3V to

0.7V. The best material should show short delay and low IOFF,BTBT at the same time. Si

exhibits slowest switching and lowest IOFF,BTBT. Although delays for InAs and InSb are

small, they suffer extremely large IOFF,BTBT. InP exhibit fast switching time, large ION

at the reduced IOFF,BTBT.

Fig. 7.11: Intrinsic delay vs. IOFF,BTBT trade-off in various materials. Lg=15nm,

Tox=0.7nm, Ts=5nm. VDD is varied from 0.3V to 0.7V.

Our results show that with oxide thickness of 0.7 nm, small density of states (DOS) of

these materials does not significantly limit the on-current (ION) and high µ materials

still show higher ION than Si. However, the high µ small bandgap materials like InAs,

InSb and Ge, suffer from excessive BTBT current and poor SCE, which can limit their

scalability. Ge has highest ION due to its large DOS and small transport effective mass,

158

but it also suffers from large BTBT leakage. Scaling TS enhances the device

performance by eliminating the SCE effect and reducing BTBT leakage. Overall,

among all the unstrained IIIV materials In0.25Ga0.75As and InP exhibit the best

performance - high ION, low delay and low IOFF,BTBT compared to Si.

7.4 Power-Performance of Strained Ternary III-V Material

(InxGa1-xAs)

InxGa1-xAs is a very promising candidate for future NFETs because it allows for a very

good tradeoff between the excellent transport properties of InAs and the low leakage

of GaAs [23, 34, 102]. Just like in the case of Si or Ge, strain engineering can be used

to further enhance the performance of III-V materials in terms of both, increasing the

drive current and reducing the off-state leakage. By varying strain conditions and

orientations for the different compositions, the best performing strained InxGa(1-x)As

was identified. Unlike p-MOS [28], in the case of n-MOS uniaxial strain does not

significantly improve performance [25].

In this section, we will discuss power-performance tradeoffs in nanoscale DG NFETs

with channel materials composed of InxGa1-xAs under various biaxial strain conditions

with different orientations.

159

7.4.1 Strained InGaAs Band Structures

Local empirical pseudopotential method (LEPM) [8] is used to obtain the

bandstructure of strained InGaAs. Fig. 7.12 shows the band shifts of GaAs (001) and

In0.75Ga0.25As (111) by applying a biaxial strain.

(a) (b)

Fig. 7.12: Band shifts in biaxially strained (a) GaAs (001) and (b) In0.75Ga0.25As (111)

calculated by local empirical pseudopotential method. For GaAs(001), 4% biaxial

compressive (BiC) strain is applied and for In0.75Ga0.25As (111), 4% biaxial tensile

(BiT) strain is applied. Solid lines represent relaxed material and dotted lines represent

strained material.

Application of strain can strongly modify the bandgap (Eg) of material. Fig. 7.13 (a)

shows Eg of the different 5nm thin InxGa1-xAs as a function of biaxial

(compressive/tensile) strain and orientation. On quantization, due the small mass of

the electrons in the Γ-valley, the carriers start occupying the high L- and X- valleys.

1 0.8 0.4 0 0.4 0.8

-6

-5

-4

-3 eV

Relaxed

Strained

BiT 4%

LX1 0.8 0.4 0 0.4 0.8

-6

-5

-4

-3 eV

Relaxed

StrainedBiC 4%

LX

160

Hence, in III-V materials, apart from the Eg, the separation between the Γ- and L-

valley (ΔEΓL) is a very important parameter in determining the transport. Fig. 7.13 (b)

shows ΔEΓL under different strain/orientation conditions. Due to strong non-

parabolicity in the Γ-valley, transport effective mass is bigger in thin layers than in

bulk material.

(a) (b)

Fig. 7.13 (a) Quantized bandgap as a function of strain (e||) in 5nm film. Compressive

strain increases bandgap. Especially (111) orientation is strongly modified (b) Γ-L

Separations as a function of strain (e||) in 5nm film. Tensile strain widens the

separation. With tensile strain, ΔEΓL in GaAs can be over 0.3 (eV).

Tunnel mass (mTunnel) determines how well electron and hole penetrate into bandgap

and cause BTBT leakage. Tunnel mass is also strongly modified with application of

strain [25]. Unlike Eg and ΔEΓL, mTunnel shows a strong dependency on wafer

orientation (Fig. 7.14). In (111) wafer, compressive strain increases mTunnel, while in

(001) wafer, tensile strain increases mTunnel.

Ega

p (e

V)

-0.04 -0.02 0 0.02 0.040.4

0.8

1.2

1.6

2

2.4

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Ts=5nm

Ega

p (e

V)

-0.04 -0.02 0 0.02 0.040.4

0.8

1.2

1.6

2

2.4

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Ega

p (e

V)

-0.04 -0.02 0 0.02 0.040.4

0.8

1.2

1.6

2

2.4

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Ts=5nm

ΔE Γ

L (e

V)

-0.04 -0.02 0 0.02 0.04

-0.4

0

0.4

0.8

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Ts=5nm

ΔE Γ

L (e

V)

-0.04 -0.02 0 0.02 0.04

-0.4

0

0.4

0.8

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Ts=5nm

161

7.4.2 ION and IOFF,BTBT with Strain Engineering and Channel

Orientation

Device simulation results for LG=15nm, TOX=0.7nm, TS=5nm, VDD=0.7V (IOFF=10-

7A/um) are shown in 7.15 (a) (ION) and 7.15 (b) (IOFF). In relaxed GaAs, ΔEΓL is not

large enough to prevent the population of electrons in L-valleys. In spite of the

occupation in L-valleys providing large channel charge, relaxed GaAs suffers from

low injection velocity (Fig. 7.8), which results in overall low drive current. This

implies that with 0.7nm-thick-gate oxide, having small mass and thus, fast injection

velocity is more effective for larger ION than having large mass and large DOS. Indeed,

with the application of tensile-biaxial strain, which increases ΔEΓL (Fig. 7.13(b)), the

drive current of GaAs can be boosted up to over 4mA/μm and the delay is shortened to

(a) (b)

Figure 7.14 : (a) 001 (b) 111. Tunnel mass as a function of x (InxGa1-xAs) in 5nm

film. On (111) Orientation, compressive strain increases mass, while

on (001), it reduces mass.

162

55fs [25]. The drive current of In0.75Ga0.25As is only slightly affected by strain, as

large ΔEΓL ensures that the majority of carriers reside in Γ-valley. Therefore,

regardless of strain levels, In0.75Ga0.25As exhibits 30% higher ION (4.3mA/μm) than 2%

biaxial tensile strained Si. Although both (111) and (001) orientation tensile strains

enhance the ION of GaAs, biaxial-tensile strained GaAs (111) suffers large IOFF,BTBT

due to reduction in bandgap and decrease in tunnel mass. Relaxed In0.75Ga0.25As have

small bandgap, leading to too high IOFF,BTBT (Fig. 7.15(b)). (111) compressive strain

effectively reduces leakage current in In0.75Ga0.25As by concurrently increasing

bandgap and tunneling mass. Due to the enlarged bandgap and tunneling mass, the

leakage current for 4% biaxially compressive-strained In0.75Ga0.25As (111) is below

the leakage specification of 0.1μmA/μm. However, (001) compressive strain does not

notably reduce the BTBT leakage current, since despite the increased bandgap, it

simultaneously reduces the tunneling mass, leading to no significant reduction in

BTBT.

163

7.4.3 PowerPerformance of Strained Ternary IIIV Material

(InGaAs)

InxGa1-xAs is a very promising candidate for future NFETs. Fig. 7.16 shows the best

channel materials and how these materials can be improved by strain engineering.

Based on our simulation results, GaAs (001), In0.25Ga0.75As (001) and In0.75Ga0.25As

(111) are selected as the best channel materials. Although GaAs (001) exhibits the

lowest IOFF,BTBT due to its large bandgap (>1.4eV), the drive current of GaAs (001) is

higher than strained Si. By applying biaxial tensile-strain, GaAs (001) can be

(a) (b)

Figure 7.15 : (a) ION as a function of strain (e||) in strained GaAs and In0.75Ga0.25As

NMOS DGFET. For comparison, IONs for 2% biaxially tensile-

strained (BiT) Si and unstrained Si are plotted as dotted lines. (b)

IOFFBTBT as a function of strain in strained GaAs and In0.75Ga0.25As. For

comparison, IOFFBTBT for 2% biaxially tensile-strained Si is drawn as

dotted line. 4% (111) compressive strain reduces IOFF,BTBT by over

1000x.

I ON

(mA

/ μm

)

-0.04 -0.02 0 0.02 0.04

2

3

4

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Si

2% Strained Si (BiT)I O

N (m

A/ μ

m)

-0.04 -0.02 0 0.02 0.04

2

3

4

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

Si

2% Strained Si (BiT)

I OFF

,BT

BT

(A/ μ

m)

10-9

-0.04 -0.02 0 0.02 0.04

1

10-1

110

-710

-510

-3


100nA/μm

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)I OFF

,BT

BT

(A/ μ

m)

10-9

-0.04 -0.02 0 0.02 0.04

1

10-1

110

-710

-510

-3

I OFF

,BT

BT

(A/ μ

m)

10-9

-0.04 -0.02 0 0.02 0.04

1

10-1

110

-710

-510

-3


100nA/μm

GaAs (001)GaAs (111)In0.75Ga0.25As (001)

In0.75Ga0.25As (111)

164

improved to have ION as high as In0.25Ga0.75As, with marginal increase in IOFF,BTBT.

In0.25Ga0.75As (001) has both good ION and low IOFF,BTBT because of its large bandgap

(>1eV) and large ΔEΓ-L. 4% uniaxial compressive strain can further reduce the leakage

in In0.25Ga0.75As (001), without significant reduction in ION. In0.75Ga0.25As (111) has

excellent carrier transport properties, but it suffers large IOFF,BTBT. Biaxial compressive

strain can reduce the leakage in In0.75Ga0.25As (111) below 0.1μA/μm. Considering the

continuation in device scaling, In0.75Ga0.25As (111) would be the better material choice

over other InxGa(1-x)As compositions. As the gate length scales down close to 10nm,

further scaling in channel thickness would be necessary to control the short channel

effects. Larger quantization effect in thinner body will further increase the bandgap of

In0.75Ga0.25As (111), leading to even smaller leakage current [25]. By contrast, the

scalability of devices with the channel materials of GaAs (001) and In0.25Ga0.75As (111)

would be limited, since the strong quantization effect in Γ-valley will diminish ION due

to reduced ΔEΓ-L.

165

7.5 Strained III-V for p-MOSFETs

As seen in the previous sections, III-V compounds are one of the most promising

materials for future high-speed, low-power n-MOSFETs due to their high electron

mobility. Recently, high performance III-V n-FETs have been demonstrated. However,

for CMOS logic, there is a significant challenge of identifying high mobility III-V p-

FET candidates [104-106]. Strain in silicon has been successful in significantly

enhancing the p-MOS performance and is now employed ubiquitously in the industry.

(a) (b)

Figure 7.16 : (a) ION vs. IOFFBTBT and (b) Delay vs. IOFFBTBT of the best

performing n-DGFETs with materials, biaxially tensile-strained GaAs

(001), uniaxially compressive-strained In0.25Ga0.75As (001) and

biaxially compressive-strained In0.75Ga0.25As (111). Strain levels are 0,

0.02 and 0.04. Values for biaxially tensile-strained Si are given for

comparison.

166

Use of strain to reduce hole effective masses by splitting the heavy-hole (hh) and

light-hole (lh) valence bands was first demonstrated in p-channel InGaAs/ (Al)GaAs

[107-108]. More recently, the technique has been applied to strained InSb [109], GaSb

[110] and InGaSb [111] based channels for improving hole nobility (μh). Given the

many choices available for materials, stoichiometry, strain, channel orientation, a

modeling effort is necessary to evaluate different options and narrow down the choice

for experimentation.

7.5.1 Hole Mobility in Ternary IIIV Materials (InGaAs vs. InGaSb)

Unlike for electrons in III-V where polar scattering is the dominant scattering

mechanism (non polar optical phonons do not interact due to s-like spherical

symmetry of conduction band in III-V’s [12])) both deformation potential and polar

scattering mechanisms are important for hole mobility. Varying group III element

while keeping the group V element same (i.e. InAs → GaAs) μh does not change

appreciably while a large change is observed when the group V element is varied (i.e.

InP → InAs → InSb). This is a consequence of the fact that the valence bands of III-V

mostly derive from the p-orbitals of the anion [112]. Fig. 7.17 plots the low field μh of

InxGa1-xAs and InxGa1-xSb (Sheet Charge (NS) =1012/cm2). Sb’s have significantly

higher mobilities than As’s.

167

7.5.2 Hole Mobility Enhancement in IIIVs with Strain

Next we look at the effect of biaxial and uniaxial strain on μh enhancement in these

III-V materials. Fig. 7.18 shows the μh enhancement with respect to the unstrained

case (for uniaxial case the strain is always applied along the channel direction). In all

cases compression is better than tension for μh enhancement. In the case of biaxial

strain, the enhancement in (100) is isotropic and maximum enhancement with 2%

biaxial strain is ~2X. Much higher μh enhancement is achieved with same % of

uniaxial strain (~4X with 2% uniaxial compression). Also, in the case of uniaxial

strain, the enhancement is highly anisotropic. Also note that ~2 times lower stress is

Figure 7.17 : Mobility for InXGa1-XSb and InXGa1-XAs. Sb’s have higher (~2X)

hole mobilities than As’s [104].

168

required to produce the same amount of strain in InSb/GaSb as compared to Si, due to

their lower elasticity constants. Fig. 7.19 shows that a hole mobility greater than

1600cm2/Vs can be obtained by application of biaxial strain in InGaSb (achievable

through lattice mismatch between channel and barrier layers during MBE growth).

Fig. 7.18: Enhancement of mobility with respect to the unstrained case with biaxial

(left) uniaixal (right) strain. Enhancement with compression is always better than

tension. Upto ~4X / ~2X enhancement possible with 2% uniaxial /biaxial compression.

Enhancement with strain (especially for uniaxial) depends highly on the choice of

channel direction [104].

m/m

unst

rain

ed

BiaxialNs=1e12cm‐2, T=300K

Uniaxial

169

Fig. 7.19: μh enhancement with biaxial stain; achievable by lattice mismatch between

channel and barrier layers during MBE growth.

In summary, our comprehensive analysis of the hole mobility enhancement in ternary

III-V materials shows that Sb’s are significantly better than As’s. Further, application

of strain significantly enhances the mobility in (In/Ga/InXGa1-X)Sb’s. A hole mobility

greater than 1600cm2/Vs can be obtained by application of biaxial strain in InGaSb.

Further enhancement should be possible using uniaxial strain. Strained Sb’s are

extremely promising candidates for nanoscale III-V p-MOSFETs.

7.6 Novel Device Structure and Parasitics

7.6.1 Quantum Well (QW) Strained Heterostructure IIIV FETs

170

Strain in general results in reduction in the EG and hence enhanced off-state leakage

(IBTBT). Confinement on the other hand results in increased EG as shown in Fig. 7.5,

and hence reduces IBTBT. The heterostructure QW-FET of Fig. 7.20 proposes a unique

and novel device structure to combine strain and quantum mechanical confinement to

obtain desired transport properties with reduced off-state leakage [29-32]. In these

structures the transport can be confined to the center of the channel in a high mobility

material flanked by a high EG material. The capping layer can be chosen such that it

helps in providing a very good high-k gate dielectric interface and reduced mobility

degradation due to interface states. The mobility is further enhanced due to strain,

reduced electric field in the center of the double gate structure due to symmetry and

the channel being away from the dielectric interface. The bandgap of the center

channel can be increased due confinement by keeping it very thin. However, QW

FETs are not completely free from their tradeoffs. QW FETs are found to have

dramatic mobility degradation at QW thickness of less than ~4nm due to strong

quantum confinement effects [32, 113-114]. To take complete advantage of high

mobility III-V materials, heterostructure quantum-well FETs, which can

simultaneously achieve high drive currents and low off-state leakage should be

investigated, similar to the case of Ge [30, 32].

171

Fig. 7.20: Conventional and QW double gate FET structures.

7.6.2 Parasitic Resistance

Parasitic resistance in the S/D regions of the conventional MOSFET has been

identified as one of the primary problems of the non-scaling of drive currents in

transistor scaling. This problem can be significantly worse in III-V n-MOSFETs due

to their low value of the maximum achievable n-type dopant concentration. Replacing

the S/D regions of transistors with metals has been suggested as one of the techniques,

which might help address this problem (Fig. 7.21). Use of the metal S/D offers

additional advantages of low-temperature processing for S/D formation, elimination of

172

the parasitic bipolar action and inherent physical scalability of the gate lengths due to

the abrupt metal-semiconductor interface. However, for the ION of the metal S/D to be

better than diffused S/D the Schottky barrier to channel needs to be very small. The

free-electron wavefunction penetrates from the metal into the semiconductor, which

generates metal-induced gap-states (MIGS) and pins the Fermi level near the charge

neutrality level (ECNL) [115]. In the case of Ge we have found that the Fermi level at

metal-Ge Schottky barriers is pinned near the valence band of Ge for a variety of

metals, including, Ni, Co and Ti [116-117]. In the case of III-Vs such as InAs, the

Fermi level pins very close to conduction band [118]. This could provide a very small

barrier to the electrons in the channel in a IIIV n-MOSFETs. Due to the small barrier

height to electrons coupled with the high inversion electron mobility, Schottky S/D

IIIV n-MOS transistors are very attractive future device candidates.

Fig. 7.21: Metal Source-Drain IIIV n-MOSFET.

7.6.3 Parasitic Capacitance

173

It has been shown that scaling MOSFET below 32nm may increase parasitic gate

capacitance and it may significantly limit the performance [119-120]. Parasitic gate

capacitance can undermine the advantage in delay of III-V materials over silicon. Fig.

7.22 shows how much the delays worsen by adding parasitic gate capacitance. In this

analysis, transit time is time required for electron to pass through the channel and

defined as, Lg/vinj. Gate delay is ΔQ/Ieff. To take into account parasitic effect, we

assumed parasitic gate capacitance to be half of gate capacitance of Si and added the

parasitic gate capacitance to gate capacitance of all the materials as:

Delay = (ΔQChannel + ΔQParasitic) /Ieff

, where ΔQParasitic= 1/2 ΔQSi and Ieff =1/2×[ID(VDD,VDD/2) + ID(VDD/2,VDD)];

ID(VG,VD) [121].

Although parasitic gate capacitance reduces the gap between high mobility materials

and Si, III-V materials and Ge still outperform Si in delay.

174

Fig. 7.22: Delay in consideration of parasitic gate capacitance. Lg=15nm, Tox=0.7nm, Ts=5nm. VDD =0.7V.

7.7 Conclusion

MOSFETs utilizing high mobility III-V channel materials can take us to the sub-20

nm regime. Ternary III-V materials, such as strained InxGa(1-x)As could be suitable for

n-MOSFETs and strained InGaSb for p-MOSFETs. However, to take full advantage of

high mobility/small bandgap channel materials, novel device structures, such as

heterostructure quantum well (QW) FETs along with strain engineering will be needed,

in order to achieve high drive currents while maintaining low off-state leakage. For

III-V materials to become mainstream, important challenges, such as, good surface

passivation, low parasitic resistance/capacitance, and heterogeneous integration on Si

platform must be overcome.

176

Chapter 8 Conclusion

Due to several fundamental and manufacturing limits, it is becoming clear that new

materials will be needed to supplement or even replace the conventional bulk

MOSFET in future technology generations. In this thesis, we have developed models

to calculate the bandstructure, the quantum ballistic carrier transport and band-to-band

tunneling and have used these models to benchmark various high mobility materials.

In the chapter 2, an empirical pseudo-potential method is developed to calculate the

bandstructures of semiconductors in zinc-blende lattice structure. The atomic

dislocation induced by applied strain is carefully examined including the effect of

internal strain caused by shear strain based on the infinitesimal strain theory. Atomic

pseudo-potential is modeled in continuous reciprocal vector space as a function of

strain tensor and reciprocal vector. We included the spin-orbit coupling which causes

splitting in degenerate 3 valence bands. The concept of kinetic energy scaling is

adopted to take into account non-local many-body potential. The complex

bandstructure was obtained as well by modifying the Hamiltonian matrix such that

crystal-momentum k-vector is eigen-value.

In the chapter 3, new BTBT model is developed to takes into account full

bandstructure, direct and phonon assisted indirect tunneling, quantum confinement

effect, non-uniform electric field (non-local) and strain induced enhanced/suppressed

177

tunneling in semiconductors. In this model, the tail of electron wavefunction

penetrating into the bandgap is modeled and used to evaluate the interband matrix

elements between conduction band states and valence band states. The final BTBT

carrier generation rate (GBTBT) was calculated by adding up the transition rates

obtained by Fermi-Golden rule for all the possible transitions

In the chapter 4, the deformation potentials for phonon-assisted band-to-band

tunneling have been successfully calculated using rigid-ion model and empirical

pseudo-potential method. The previous theoretical calculations were all limited to

inner-band scattering (ex. conduction band to conduction band, valence band to

valence band) or limited to transitions between symmetric points in real k-space. By

introducing complex-k space in calculations of BTBT deformation potentials, we are

able to explain the discrepancies between previous theories to experimental results on

tunneling diodes. In relaxed silicon, only TA(Γ-∆) and TO(Γ-∆) phonons have

significant contributions on band-to-band tunneling and in relaxed germanium, LA(Γ-

L), TA(Γ-L), TO(Γ-L) phonons are participating tunneling process.

In the chapter 5, we have shown the limitations of previous BTBT tunneling models

for TCAD simulators. Since previous BTBT tunneling models are based on the two-

band model in relaxed bulk semiconductor system, it fails to predict BTBT current for

highly strained ultra thin film devices. Moreover the phonon-electron interactions

were not properly modeled in any of previous works. Depending on transition paths

178

we have carefully chosen phonon modes according to selection rules and deformation

potentials calculated by applying rigid-ion model on electron orbital in complex-k

space. Our model was verified using various experimental data on fabricated PIN

structures on silicon, germanium and GaAs wafers. We have shown for the first time

that biaxial compressive-strain reduces the direct tunneling component, but the

increases in-direct tunneling component in leakage current in germanium devices.

With 2.4% biaxial compressive-strain applied across germanium devices, the BTBT

leakage current reaches the minimum point. By applying our BTBT model in DGFET

simulations we found that reducing the thickness of channel layer is very effective in

suppressing the BTBT leakage current. This quantum confinement effect has been

verified by comparing the simulation results to the I-V characteristics of fabricated

thin-film strained germanium MOSFET devices.

.

In chapter 6, the quantum ballistic transport model is developed to accurately model

the carrier transport in highly scaled DGFETs. This model takes into account the

effects of non-parabolicity of conduction bands on transport mass and 1-D density of

states. Asymptotic approximation helped to simplify model. The BTBT leakage

current is also included in the ballistic transport model.

In the chapter 7, we have benchmarked strained III-V and strained SiGe to study the

scalability of DGFETs with these materials as a channel material.

179

Among the relaxed materials, Si exhibits the slowest switching and the lowest

IOFF,BTBT. Although delays for InAs and InSb are small, they suffer extremely large

IOFF,BTBT. InP exhibit fast switching time, large ION at the reduced IOFF,BTBT. Our results

show that with oxide thickness of 0.7 nm, small density of states (DOS) of these

materials does not significantly limit the on-current (ION) and high µ materials still

show higher ION than Si. However, the high µ small bandgap materials like InAs,

InSb and Ge, suffer from excessive BTBT current and poor SCE, which can limit their

scalability. Ge has highest ION due to its large DOS and small transport effective mass,

but it also suffers from large BTBT leakage. Scaling TS enhances the device

performance by eliminating the SCE effect and reducing BTBT leakage. Overall,

among all the unstrained IIIV materials In0.25Ga0.75As and InP exhibit the best

performance - high ION, low delay and low IOFF,BTBT compared to Si.

Strained InxGa1-xAs is a very promising candidate for future NFETs. Based on our

simulation results, GaAs (001), In0.25Ga0.75As (001) and In0.75Ga0.25As (111) are

selected as the best channel materials. Although GaAs (001) exhibits the lowest

IOFF,BTBT due to its large bandgap (>1.4eV), the drive current of GaAs (001) is higher

than strained Si. By applying biaxial tensile-strain, GaAs (001) can be improved to

have ION as high as In0.25Ga0.75As, with marginal increase in IOFF,BTBT. In0.25Ga0.75As

(001) has both good ION and low IOFF,BTBT because of its large bandgap (>1eV) and

large ΔEΓ-L. 4% uniaxial compressive strain can further reduce the leakage in

In0.25Ga0.75As (001), without significant reduction in ION. In0.75Ga0.25As (111) has

excellent carrier transport properties, but it suffers large IOFF,BTBT. Biaxial compressive

180

strain can reduce the leakage in In0.75Ga0.25As (111) below 0.1μA/μm. Considering the

continuation in device scaling, In0.75Ga0.25As (111) would be the better material choice

over other InxGa(1-x)As compositions. As the gate length scales down close to 10nm,

further scaling in channel thickness would be necessary to control the short channel

effects. Larger quantization effect in thinner body will further increase the bandgap of

In0.75Ga0.25As (111), leading to even smaller leakage current. By contrast, the

scalability of devices with the channel materials of GaAs (001) and In0.25Ga0.75As (111)

would be limited, since the strong quantization effect in Γ-valley will diminish ION due

to reduced ΔEΓ-L.

MOSFETs utilizing high mobility III-V channel materials can take us to the sub-20

nm regime. Ternary III-V materials, such as strained InxGa(1-x)As could be suitable for

n-MOSFETs and strained InGaSb for p-MOSFETs. However, to take full advantage of

high mobility/small bandgap channel materials, novel device structures, such as

heterostructure quantum well (QW) FETs along with strain engineering will be needed,

in order to achieve high drive currents while maintaining low off-state leakage. For

III-V materials to become mainstream, important challenges, such as, good surface

passivation, low parasitic resistance/capacitance, and heterogeneous integration on Si

platform must be overcome.

181

References [1] D. Kahng, and M. M. Atalla, “Silicon-silicon dioxide field induced surface

devices,” in IRE-AIEEE Solid-State Device Research Conference, Carmegie

Institute of Technology, Pittsburgh, PA, 1960.

[2] G. E. Moore, “Cramming more components onto integrated circuits (Reprinted

from Electronics, pg 114-117, April 19, 1965),” Proceedings of the Ieee, vol.

86, no. 1, pp. 82-85, Jan, 1998.

[3] K. Natori, “Ballistic metal-oxide-semiconductor field effect transistor,”

Journal of Applied Physics, vol. 76, no. 8, pp. 4879, 15 Oct. 1994, 1994.

[4] J. R. Chelikowsky, and M. L. Cohen, “Nonlocal pseudopotential calculations

for the electronic structure of eleven diamond and zinc-blende semiconductors,”

Physcial Review B, vol. 14, no. 2, pp. 556, 1976.

[5] M. L. Cohen, and T. K. Bergstresser, “Band Structures and Pseudopotential

Form Factors for Fourteen Semiconductors of the Diamond and Zinc-blende

Structures,” Physical Review, vol. 141, no. 2, pp. 789, January 1966, 1966.

[6] Author ed.^eds., “The fitting of pseudopotentials to experimental data and their

subsequetn application,” Solid State Physics, Advances in Research and

Applications, New York: Academic Press, 1970, p.^pp. Pages.

[7] P. Friedel, M. S. Hybertsen, and M. Schluter, “Local empirical pseudopotential

approach to the optical properties of Si/Ge superlattices,” Physcial Review B,

vol. 39, no. 11, pp. 7974, 1989.

182

[8] K. Kim, P. R. C. Kent, and A. Zunger, “Atomistic description of the electronic

structure of InxGa1-xAs alloys and InAs/GaAs superlattices,” Physcial Review

B, vol. 66, pp. 045208, 2002.

[9] L.-W. Wang, S.-H. Wei, T. Mattila et al., “Multiband coupling and electronic

structure of (InAs)n/(GaSb)n superlattices,” Physical Review B, vol. 60, no. 8,

pp. 5590, 15 August 1999, 1999.

[10] P. Harrison, Quantum Wells, Wires and Dots, Second ed.: Wiley, 2005.

[11] R. M. Martin, Electronic Structure: Basic Theory and Practical Methods,

Cambridge: Cambridge University Press, 2004.

[12] M. J. Shaw, “Microscoptic theory of scattering in imperfect strained

antimonide-based heterostructures,” Physical Review B, vol. 61, no. 8, pp.

5431, 15 February 2000, 2000.

[13] J.-B. Xia, “Theoretical analysis of electronic structures of short-period

superlattices (GaAs)m/(AlAs)n and corresponding alloys

Aln/(m+n)Gam/(m+n)As,” Physcial Review B, vol. 38, no. 12, pp. 8358, 1988.

[14] I. Morrison, M. Jaros, and K. B. Wong, “Strain-induced confinement in

Si0.75Ge0.25 (Si/Si0.5Ge0.5) (001) superlattice systems,” Phsycial Review B,

vol. 35, no. 18, pp. 9693, 1987.

[15] L.-W. Wang, J. Kim, and A. Zunger, “Electronic structures of [110]-faceted

self-assembled pyramidal InAs/GaAs quantum dots,” Physcial Review B, vol.

59, no. 8, pp. 5679, 1999.

183

[16] G. Grosso, and C. Piermarocchi, “Tight-binding model and interactions scaling

laws for silicon and germanium,” Physcial Review B, vol. 51, no. 23, pp.

16772, 1995.

[17] J. P. Walter, and M. L. Cohen, “Calulated and Measured Reflectivity of ZnTe

and ZnSe,” Physcial Review B, vol. 1, no. 6, pp. 2661, 1970.

[18] G. Weisz, “Band Structure and Fermi Surface of White Tin,” Physical Review,

vol. 149, no. 2, pp. 504, 1966.

[19] M. M. Rieger, and P. Vogl, “Electronic-band parameters in strained Si1-xGex

alloys on Si1-yGey substrates,” Physcial Review B, vol. 48, no. 19, pp. 14276,

1993.

[20] W. J. Wortman, and R. A. Evans, “Young's Modulus, Shear Modulus, and

Poisson's Ratio in Silicon and Germanium,” Journal of Applied Physics, vol.

36, no. 1, pp. 153, 1965.

[21] L. Kleinman, “Deformation Potentials in Silicon. I. Uniaxial Strain,” Physical

Review, vol. 128, no. 6, pp. 2614, 1962.

[22] O. H. Nielsen, and R. M. Martin, “Stressees in semiconductors: Ab initio

calculations on Si, Ge, and GaAs,” Physcial Review B, vol. 32, no. 6, pp. 3792,

1985.

[23] D. Kim, T. Krishnamohan, and K. C. Saraswat, "Performance evaluation of III-

V double-gate n-MOSFETs." pp. 67-8.

184

[24] D. Kim, T. Krishnamohan, and K. C. Saraswat, “Performance evaluation of

15nm gate length double-gate n-MOSFETs with high mobility channels: IIIV,

Ge and Si ” ECS Transactions, vol. 16, no. 10, pp. 47-55, 2008.

[25] D. Kim, T. Krishnamohan, and K. C. Saraswat, “Performance evaluation of

uniaxial- and biaxial-strained In(x)Ga(1-x)As NMOS DGFETs,” International

Conference on Simulation of Semiconductor Processes and Devices (SISPAD),

pp. 101-104, 2008.

[26] D.-H. Kim, J. d. Alamo, J.-H. Lee et al., “Logic suitability of 50-nm

In0.7Ga0.3AsHEMTs for Beyond-CMOS applications,” IEEE Transactions on

Electron Devices, vol. 54, no. 10, pp. 2606-13, Oct. 2007, 2007.

[27] T. Krishnamohan, C. Jungemann, and K. C. Saraswat, “A novel, very high

performance, sub-20 nm depletion-mode double-gate (DMDG),” Technical

Digest - International Electron Devices Meeting, pp. 687-90, 2003.

[28] T. Krishnamohan, D. Kim, T. V. Dinh et al., "Comparison of (001), (110) and

(111) uniaxial- And biaxial- strained-Ge and strained-Si PMOS DGFETs for

all channel orientations: Mobility enhancement, drive current, delay and off-

state leakage."

[29] T. Krishnamohan, D. Kim, C. Jungemann et al., “Strained-Si, relaxed-Ge or

strained-(Si)Ge for future nanoscale p-MOSFETs?,” Digest of Technical

Papers - 2006 Symposium on VLSI Technology, pp. 144-5, 2006.

[30] T. Krishnamohan, D. Kim, C. Nguyen et al., “High-mobility low band-to-

band-tunneling strained-germanium double-gate heterostructure FETs:

185

Simulations,” IEEE Transactions on Electron Devices, vol. 53, no. 5, pp.

1000-9, May 2006, 2006.

[31] T. Krishnamohan, Z. Krivokapic, K. Uchida et al., “Low defect ultrathin fully

strained Ge MOSFET on relaxed Si with high mobility and low Band-To-

Band-Tunneling (BTBT) ” Digest of Technical Papers - Symposium on VLSI

Technology, pp. 82-3, 2005.

[32] T. Krishnamohan, Z. Krivokapic, K. Uchida et al., “High-mobility ultrathin

strained Ge MOSFETs on bulk and SOI with low band-to-band tunneling

leakage: Experiments ” IEEE Transactions on Electron Devices, vol. 53, no. 5,

pp. 990-9, May 2006, 2006.

[33] T. Krishnamohan, and K. C. Saraswat, “High mobility Ge and III-V materials

and novel device structures for high performance nanoscale MOSFETS,”

ESSDERC 2008, pp. 38-46, 2008.

[34] A. Pethe, T. Krishnamohan, D. Kim et al., “Investigation of the performance

limits of III-V double-gate n-MOSFETs,” Technical Digest - International

Electron Devices Meeting, pp. 605-608, 2005.

[35] J. B. Krieger, “Theory of Electron Tunneling in Semiconductors with

Degenerate Band Structure,” Annals of Physics, vol. 36, no. 1, pp. 1-60, 1966.

[36] O. J. Glembocki, and F. H. Pollak, “Calculation of the Gamma-Delta-Electron-

Phonon and Hole-Phonon Scattering Matrix-Elements in Silicon,” Physical

Review Letters, vol. 48, no. 6, pp. 413-416, 1982.

186

[37] O. J. Glembocki, and F. H. Pollak, “Calculation of the Gamma-L Electron-

Phonon and Hole-Phonon Scattering Matrix-Elements in Germanium,”

Physical Review B, vol. 25, no. 12, pp. 7863-7866, 1982.

[38] O. J. Glembocki, and F. H. Pollak, “Intervalley Electron-Phonon and Hole-

Phonon Scattering Matrix-Elements in Germanium,” Physica B & C, vol. 117,

no. Mar, pp. 546-548, 1983.

[39] M. V. Fischetti, T. P. O'Regan, S. Narayanan et al., “Theoretical study of some

physical aspects of electronic transport in nMOSFETs at the 10-nm gate-

length,” Ieee Transactions on Electron Devices, vol. 54, no. 9, pp. 2116-2136,

Sep, 2007.

[40] E. O. Kane, “Zener Tunneling in Semiconductors,” Journal of Physics and

Chemistry of Solids, vol. 12, no. 2, pp. 181-188, 1959.

[41] E. O. Kane, “Theory of Tunneling,” Journal of Applied Physics, vol. 32, no. 1,

pp. 83-91, 1961.

[42] J. J. Tiemann, and Fritzsch.H, “Theory of Indirect Interband Tunneling in

Semiconductors,” Physical Review, vol. 1, no. 6A, pp. 1910-&, 1965.

[43] J. J. Tiemann, and H. Fritzsche, “Temperature Dependence of Indirect

Interband Tunneling in Germanium,” Physical Review, vol. 132, no. 6, pp.

2506-&, 1963.

[44] A. Rahman, M. S. Lundstrom, and A. W. Ghosh, “Generalized effective-mass

approach for n-type metal-oxide-semiconductor field-effect transistors on

187

arbitrarily oriented wafers,” Journal of Applied Physics, vol. 97, no. 5, pp. -,

Mar 1, 2005.

[45] L. Keldysh, “Influence of the Lattice Vibrations of a Crystal on the Production

of Electron-Hole Pairs in a Strong Electrical Field,” Soviet Physics JETP -

USSR, vol. 7, no. 4, pp. 665-668, 1958.

[46] M. Lax, and J. J. Hopfield, “Selection Rules Connecting Different Points in

Brillouin Zone,” Physical Review, vol. 1, no. 1, pp. 115-123, 1961.

[47] G. G. Macfarlane, T. P. Mclean, J. E. Quarrington et al., “Fine Structure in the

Absorption-Edge Spectrum of Ge,” Physical Review, vol. 108, no. 6, pp. 1377-

1383, 1957.

[48] X. Q. Zhou, H. M. Vandriel, and G. Mak, “Femtosecond Kinetics of

Photoexcited Carriers in Germanium,” Physical Review B, vol. 50, no. 8, pp.

5226-5230, Aug 15, 1994.

[49] M. Schluter, G. Martinez, and M. L. Cohen, “Pressure and Temperature-

Dependence of Electronic-Energy Levels in Pbse and Pbte,” Physical Review B,

vol. 12, no. 2, pp. 650-658, 1975.

[50] M. J. Lax, Symmetry principles in solid state and molecular physics, New

York: Wiley, 1974.

[51] S. Krishnamurthy, and M. Cardona, “Self-Consistent Calculation of Intervalley

Deformation Potentials in Gaas and Ge,” Journal of Applied Physics, vol. 74,

no. 3, pp. 2117-2119, Aug 1, 1993.

188

[52] G. H. Li, A. R. Goni, K. Syassen et al., “Intervalley Scattering Potentials of Ge

from Direct Exciton Absorption under Pressure,” Physical Review B, vol. 49,

no. 12, pp. 8017-8023, Mar 15, 1994.

[53] A. G. Chynoweth, D. E. Thomas, and R. A. Logan, “Phonon-Assisted

Tunneling in Silicon and Germanium Esaki Junctions,” Physical Review, vol.

125, no. 3, pp. 877-881, 1962.

[54] C. Rivas, R. Lake, G. Klimeck et al., “Full-band simulation of indirect phonon

assisted tunneling in a silicon tunnel diode with delta-doped contacts,” Applied

Physics Letters, vol. 78, no. 6, pp. 814-816, Feb 5, 2001.

[55] S. Tanaka, “Theory of the Drain Leakage Current in Silicon Mosfets,” Solid-

State Electronics, vol. 38, no. 3, pp. 683-691, Mar, 1995.

[56] N. Holonyak, I. A. Lesk, R. N. Hall et al., “Direct Observation of Phonons

during Tunneling in Narrow Junction Diodes,” Physical Review Letters, vol. 3,

no. 4, pp. 167-168, 1959.

[57] A. Schenk, “Rigorous Theory and Simplified Model of the Band-to-Band

Tunneling in Silicon,” Solid-State Electronics, vol. 36, no. 1, pp. 19-34, Jan,

1993.

[58] W. Weber, “Adiabatic Bond Charge Model for Phonons in Diamond, Si, Ge,

and Alpha-Sn,” Physical Review B, vol. 15, no. 10, pp. 4789-4803, 1977.

[59] P. M. Solomon, J. Jopling, D. J. Frank et al., “Universal tunneling behavior in

technologically relevant P/N junction diodes,” Journal of Applied Physics, vol.

95, no. 10, pp. 5800-5812, May 15, 2004.

189

[60] M. I. Nathan, “Current Voltage Characteristics of Germanium Tunnel Diodes,”

Journal of Applied Physics, vol. 33, no. 4, pp. 1460-1469, 1962.

[61] G. A. M. Hurkx, “On the Modeling of Tunnelling Currents in Reverse-Biased

P-N-Junctions,” Solid-State Electronics, vol. 32, no. 8, pp. 665-668, Aug, 1989.

[62] M. S. Tyagi, “Zener and Avalanche Breakdown in Silicon Alloyed P-N

Junctions .I. Analysis of Reverse Characteristics,” Solid-State Electronics, vol.

11, no. 1, pp. 99-115, 1968.

[63] R. A. Logan, and A. G. Chynoweth, “Effect of Degenerate Semiconductor

Band Structure on Current-Voltage Characteristics of Silicon Tunnel Diodes,”

Physical Review, vol. 131, no. 1, pp. 89-95, 1963.

[64] R. H. Haitz, “Mechanisms Contributing to Noise Pulse Rate of Avalanche

Diodes,” Journal of Applied Physics, vol. 36, no. 10, pp. 3123-3131, 1965.

[65] L. Gaul, S. Huber, J. Freyer et al., “Determination of Tunnel-Generation Rate

from Gaas Pin-Structures,” Solid-State Electronics, vol. 34, no. 7, pp. 723-726,

Jul, 1991.

[66] M. S. Tyagi, “Determination of Effective Mass and Pair Production Energy for

Electrons in Germanium from Zener Diode Characteristics,” Japanese Journal

of Applied Physics, vol. 12, no. 1, pp. 106-108, 1973.

[67] P. N. Butcher, K. F. Hulme, and J. R. Morgan, “Dependence of Peak Current

Density on Acceptor Concentration in Germanium Tunnel Diodes,” Solid-State

Electronics, vol. 5, no. Sep-Oct, pp. 358-360, 1962.

190

[68] H. P. Kleinknecht, “Indium Arsenide Tunnel Diodes,” Solid-State Electronics,

vol. 2, no. 2-3, pp. 133-140, 1961.

[69] M. S. Tyagi, “Zener and Avalanche Breakdown in Silicon Alloyed P-N

Junctions .2. Effect of Temperature on Reverse Characteristics and Criteria for

Distinguishing between 2 Breakdown Mechanisms,” Solid-State Electronics,

vol. 11, no. 1, pp. 117-&, 1968.

[70] P. Agarwal, M. J. Goossens, V. Zieren et al., “Impact ionization in thin silicon

diodes,” Ieee Electron Device Letters, vol. 25, no. 12, pp. 807-809, Dec, 2004.

[71] T. Tokuyama, “Zener Breakdown in Alloyed Germanium P+-N Junctions,”

Solid-State Electronics, vol. 5, no. May-J, pp. 161-&, 1962.

[72] D. Kim, “Theoretical Performance Evaluations of NMOS Double Gate FETs

with High Mobility Materials : Strained III-V, Ge and Si,” Ph. D. DIssertation,

Electrical Engineering, Stanford University, Stanford, 2009.

[73] D. Kim, T. Krishnamohan, Y. Nishi et al., "Band to band tunneling limited off

state current in ultra-thin body double gate FETs with high mobility materials:

III-V, Ge and strained Si/Ge." pp. 389-92.

[74] D. Kim, T. Krishnamohan, L. Smith et al., “Band to band tunneling study in

high mobility materials: III-V, Si, Ge and strained SiGe,” 2007 65th DRC

Device Research Conference, pp. 57-8, 2007.

[75] "International Technology Roadmap for Semiconduction," http://public.itrs.net.

191

[76] S. E. Laux, “A simulation study of the switching times of 22-and 17-nm gate-

length SOI nFETs on high mobility substrates and Si,” Ieee Transactions on

Electron Devices, vol. 54, no. 9, pp. 2304-2320, Sep, 2007.

[77] K. Natori, “Ballistic metal-oxide-semiconductor field effect transistor,”

Journal of Applied Physics, vol. 76, no. 8, pp. 4897, 1994.

[78] S. Scaldaferri, G. Curatola, and G. Iannaccone, “Direct solution of the

boltzmann transport equation and poisson-schrodinger equation for nanoscale

MOSFETs,” Ieee Transactions on Electron Devices, vol. 54, no. 11, pp. 2901-

2909, Nov, 2007.

[79] L. Worschech, A. Schliemann, H. Hsin et al., “Ballistic transport in nanoscale

field effect transistors revealed by four-terminal DC characterization,”

Superlattices and Microstructures, vol. 34, no. 3-6, pp. 271-275, Sep-Dec,

2003.

[80] G. Bastard, “Theoretical investigations of superlattice band structure in the

envelope-function approximation,” Physcial Review B, vol. 25, no. 12, pp.

7584, 1982.

[81] D. F. Nelson, R. C. Miller, and D. A. Kleinman, “Band nonparabolicity effects

in semiconductor quantum wells,” Physcial Review B, vol. 35, no. 14, pp. 7770,

1987.

[82] A. Godoy, Z. Y. a. U. Ravaioli, and F. Gamiz, “Effects of nonparabolic bands

in quantum wires,” Journal of Applied Physics, vol. 98, pp. 013702, 2005.

192

[83] M. Buttiker, “Larmor precession and the traversal time for tunneling,” Physical

Review B, vol. 27, no. 10, pp. 6178, 1983.

[84] M. Buttiker, and R. Landauer, “Traversal Time for Tunneling,” IBM Journal of

Reserach and Development, vol. 30, no. 5, pp. 451, September, 1986.

[85] M. Buttiker, and R. Landauer, “Traversal Time for Tunneling,” Physical

Review Letters, vol. 49, no. 23, pp. 1739, 1982.

[86] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, “An Analytical

Approximation for the Fermi-Dirac Integral F3/2(eta),” Solid State Electronics,

vol. 24, no. 10, pp. 981, 1981.

[87] R. Landuer, “Electrical Resistance of Disordered One-dimensional Lattices,”

Philosophical Magazine, vol. 21, no. 172, pp. 863, 1970.

[88] R. Chau, “Benchmarking nanotechnology for high performance and low-power

logic transistor applications,” IEEE Transactions on Nanotechnology, vol. 4,

no. 2, pp. 153-8, Mar. 2005, 2005.

[89] S. Datta, G. Dewey, M. Doczy et al., “High mobility Si/SiGe strained channel

MOS transistors with HfO2/TiN gate stack,” Technical Digest - International

Electron Devices Meeting pp. 653-6, 2003.

[90] T. Tezuka, N. Sugiyama, T. Mizuno et al., “Ultrathin body SiGe-on-insulator

pMOSFETs with high-mobility SiGe surface channels,” IEEE Transactions on

Electron Devices, vol. 50, no. 5, pp. 1328-33, 2003.

193

[91] H. Shang, K.-L. Lee, P. Kozlowski et al., “Epitaxial silicon and germanium on

buried insulator heterostructures and devices,” Applied Physics Letters, vol. 83,

no. 26, pp. 5443-5, 29 Dec 2003, 2003.

[92] H. Shang, J. O. Cho, X. Wang et al., “Channel design and mobility

enhancement in strained germanium buried channel MOSFETs ” Digest of

Technical Papers - Symposium on VLSI Technology, pp. 204-205, 2004.

[93] A. Ritenour, S. Yu, M. L. Lee et al., “Epitaxial strained germanium p-

MOSFETs with HfO<sub>2</sub> gate dielectric and TaN gate electrode ”

Technical Digest - International Electron Devices Meeting, pp. 433-6, 2003.

[94] M. Lee, E. Fitzgerald, M. Bulsara et al., “Strained Si, SiGe and Ge channels

for high mobility metal oxide semiconductor field effect transistors,” Journal

of Applied Physics, vol. 97, no. 1, pp. 11101-1-27, 2005.

[95] S. Datta, T. Ashley, J. Brask et al., “85nm gate length enhancement and

depletion mode InSb quantum well transistors for ultra high speed and very

low power digital logic applications ” Technical Digest - International

Electron Devices Meeting, pp. 763-6, Dec. 2005, 2005.

[96] E. O. Kane, “Zener Tunneling in Semiconductors,” Journal of Physics and

Chemistry of Solids, vol. 12, no. 2, pp. 181-8, 1959.

[97] A. Rahman, M. S. Lundstrom, and A. W. Ghosh, “Generalized effective-mass

approach for n -type metal-oxide-semiconductor field-effect transistors on

arbitrarily oriented wafers,” Journal of Applied Physics, vol. 97, no. 5, pp. 1-12,

2005.

194

[98] J. Wang, and M. Lundstrom, “Ballistic Transport in High Electron Mobility

Transistors,” IEEE Transactions on Electron Devices, vol. 5, no. 7, pp. 1604-9,

2003.

[99] S. E. Laux, “A simulation study of the switching times of 22- and 17-nm gate-

length SOI nFETs on high mobility substrates and Si,” IEEE Transactions on

Electron Devices, vol. 54, no. 9, pp. 2304-20, 2007.

[100] M. V. Fischetti, and S. E. LAUX, “Monte Carlo simulation of transport in

technologically significant semiconductors of the diamond and zinc-blende

structures--II: Submicrometer MOSFET's ” IEEE Transactions on Electron

Devices, vol. 38, no. 3, pp. 650-60, Mar. 1991, 1991.

[101] A. Asenov, K. Kalna, I. Thayne et al., “Simulation of implant free III-V

MOSFETs for high performance low power Nano-CMOS applications ”

Microelectronic Engineering, vol. 84, no. 9-10, pp. 2398-403, Sept. - Oct.

2007, 2007.

[102] M. D. Michielis, D. Esseni, and F. Driussi, “Analytical models for the insight

into the use of alternative channel materials in ballistic nano-MOSFETs,”

IEEE Transactions on Electron Devices, vol. 54, no. 1, pp. 115-23, 2007.

[103] M. V. Fischetti, T. P. O'Regan, S. Narayanan et al., “Theoretical study of some

physical aspects of electronic transport in nMOSFETs at the 10-nm gate-

length,” IEEE Transactions on Electron Devices, vol. 54, no. 9, pp. 2116-36,

2007.

195

[104] A. Nainani, D. Kim, T. Krishnamohan et al., "Hole Mobility and its

Enhancement with Strain for Technologically Relevant III-V Semiconductors."

[105] P. P. Ruden, M. Shur, D. K. Arch et al., “Quantum-well p-channel

AlGaAs/InGaAs/GaAs heterostructure insulated-gate field-effect transistors,”

IEEE Transactions on Electron Devices, vol. 36, no. 11, pp. 2371-9, 1989.

[106] A. Nainani, S. Raghunathan, D. Witte et al., "Engineering of Strained III-V

Heterostructures for High Hole Mobility."

[107] G. C. Osbourn, “Electron and hole effective masses for two-dimensional

transport in strained-layer superlattices,” Superlattices and Microstructures,

vol. 1, no. 3, pp. 223-6, 1985.

[108] M. Jaffe, J. E. Oh, J. Pamulapati et al., “In-plane hole effective masses in

InxGa1-xAs/Al0.15Ga0.85As modulation-doped heterostructures ” Applied

Physics Letters, vol. 54, no. 23, pp. 2345-6, 1989.

[109] M. Radosavljevic, T. Ashley, A. Andreev et al., “High-performance 40nm gate

length InSb p-channel compressively strained quantum well field effect

transistors for low-power (VCC=0.5V) logic applications,” Technical Digest -

2008 IEEE International Electron Devices (IEDM), pp. 727-30, 2008.

[110] B. R. Bennett, M. G. Ancona, J. B. Boos et al., “Strained GaSb/AlAsSb

quantum wells for p-channel field-effect transistors,” Journal of Crystal

Growth, vol. 311, no. 1, pp. 47-53, 15 Dec 2008, 2008.

196

[111] B. R. Bennett, M. G. Ancona, J. B. Boos et al., “Mobility enhancement in

strained p-InGaSb quantum wells,” Applied Physics Letters, vol. 91, no. 4, pp.

042104, 2007.

[112] G. Bastard, Wave Mechanics applied to semiconductor heterostructures, New

York: Halsted Press, Wiley, 1988.

[113] F. A. T. M, and A. V. K., “Quantum engineering of nanoelectronic devices: the

role of quantum confinement on mobility degradation,” Microelectronics

Journal, vol. 32, no. 8, pp. 679-86, Aug. 2001, 2001.

[114] K. Uchida, and S.-i. Takagi, “Carrier scattering induced by thickness

fluctuation of silicon-on-insulator film in ultrathin-body metal–oxide–

semiconductor field-effect transistors,” Applied Physics Letters, vol. 82, no. 17,

pp. 2916, 2003.

[115] J. Tersoff, “Schottky Barriers and the Continuum of Gap States,” Physical

Review Letters, vol. 52, no. 6, pp. 465-8, 1984.

[116] A. Pethe, and K. C. Saraswat, “High-mobility, low parasitic resistance Si/Ge/Si

heterostructure channel Schottky source/drain PMOSFETs,” 65th Device

Research Conference, pp. 55-6, 2007.

[117] A. Dimoulas, P. Tsipas, A. Sotiropoulos et al., “Fermi-level pinning and

charge neutrality level in germanium,” Applied Physics Letters, vol. 89, no. 25,

pp. 252110-1-3, 18 Dec. 2006, 2006.

197

[118] H. H. Wieder, “Surface and interface barriers of InxGa1-xAs binary and

ternary alloys,” Journal of Vacuum Science and Technology B, vol. 21, no. 4,

pp. 1915-9, 2003.

[119] A. Khakifirooz, and D. A. Antoniadis, “MOSFET performance scaling. Part I.

Historical trends,” IEEE Transactions on Electron Devices, vol. 55, no. 6, pp.

1391-400, 2008.

[120] A. Khakifirooz, and D. A. Antoniadis, “MOSFET performance scaling-Part II:

future directions ” IEEE Transactions on Electron Devices, vol. 55, no. 6, pp.

1401-8, 2008.

[121] D. A. Antoniadis, I. Aberg, C. Ni Chleirigh et al., “Continuous MOSFET

performance increase with device scaling: The role of strain and channel

material innovations,” IBM Journal of Research and Development, vol. 50, no.

4-5, pp. 363-76, 2006.

theoretical performance evaluations of nmos double gate fets … kim... · 2009. 12. 2. ·...

Documents