Download ELECTRON MOBILITY CALCULATIONS IN SILICON, GERMANIUM

ELECTRON MOBILITY CALCULATIONS IN
SILICON, GERMANIUM, AND III-V SUBSTRATES
WITH HIGH-κ GATE DIELECTRICS
A Dissertation Presented
by
TERRANCE P. O’REGAN
Submitted to the Graduate School of the
University of Massachusetts Amherst in partial fulfillment
of the requirements for the degree of
DOCTOR OF PHILOSOPHY
May 2008
Electrical and Computer Engineering
3325153
Copyright 2008 by
O'Regan, Terrance P.
All rights reserved
2008
3325153
c Copyright by Terrance P. O’Regan 2008
All Rights Reserved
ELECTRON MOBILITY CALCULATIONS IN
SILICON, GERMANIUM, AND III-V SUBSTRATES
WITH HIGH-κ GATE DIELECTRICS
A Dissertation Presented
by
TERRANCE P. O’REGAN
Approved as to style and content by:
Massimo V. Fischetti, Chair
Neal G. Anderson, Member
Eric Polizzi, Member
Dimitrios Maroudas, Member
Christopher V. Hollot, Department Chair
Electrical and Computer Engineering
To my family.
ACKNOWLEDGMENTS
This work would not have been possible without the guidance and support of
my mentor and friend Professor Massimo Fischetti. Max is widely respected as a
leader in the field of solid state physics and its application to electronic transport in
semiconductor devices. I feel honored and quite frankly lucky to be Max’s first PhD
student.
I thank Dr. Seonghoon Jin, an energetic and very talented post doc that has
worked closely with me on much of the work in this dissertation. I thank Professor
Ting-wei Tang who was my Master’s Thesis advisor - without Dr. Tang’s support, I
would never have attended graduate school, never mind complete the PhD degree. I
thank Professors Neal Anderson, Eric Polizzi, and Dimitrios Maroudas for serving on
my dissertation committee and useful comments and criticism of my work.
I thank Dr. Bart Sorée, Dr. Wim Magnus, and Dr. Marc Meuris for their
interaction and collaboration. I thank them as well as IMEC for the opportunity
to intern at IMEC and for the chance to work with wonderfully smart scientists
and engineers. I especially want to thank Bart and Wim for their physical and
mathematical insight into III-V electron mobility and screening effects.
I thank my parents and family for their patience and support during the long road
through undergraduate and graduate school. Without their love and support I would
never have had the determination and resilience it takes to finish the PhD degree.
I thank Sang-Hyun Yim, Jseock Kim, Kevin Reibe, Jen Rose, Andy Phillips, Ilana
Shydlo, and Gary Pickering for their friendships that have helped me push ahead
through the ups and downs. I thank the IGERT program for the fellowship provided
me and Semiconductor Research Corporation for funding.
v
ABSTRACT
ELECTRON MOBILITY CALCULATIONS IN
SILICON, GERMANIUM, AND III-V SUBSTRATES
WITH HIGH-κ GATE DIELECTRICS
MAY 2008
TERRANCE P. O’REGAN
B.Sc., UNIVERSITY OF MASSACHUSETTS AMHERST
M.Sc., UNIVERSITY OF MASSACHUSETTS AMHERST
Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST
Directed by: Professor Massimo V. Fischetti
With the continued scaling down of MOSFET dimensions has come the introduction of high-κ gate insulators, high mobility substrates, and new device geometries.
The purpose of this work is to model the low-field electron mobility to evaluate the
performance of the various options to continue Moore’s Law. We model the electron
mobility in Si, Ge, and III-V inversion layers and quantum wells, including scattering with surface optical phonons associated with high-kappa gate insulators, bulk
phonons, and surface roughness scattering. We compare the low-field mobility results
with Monte Carlo simulations to understand the role of mobility in predicting device
performance in short-channel devices. For the first time, the theory describing surface
optical phonon scattering is extended to the symmetric double-gate structure with
a Si body - accounting for the coupling of the two interfaces and screening via the
substrate plasmon.
vi
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
1.2
1.3
1.4
Meeting Scaling Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Electron Mobility and Short-Channel Performance . . . . . . . . . . . . . . . . . . . . 2
III-V MOSFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
The Double-Gate MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. LOW-FIELD ELECTRON MOBILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1
2.2
Kubo-Greenwood Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Scattering Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1
2.2.2
2.2.3
2.2.4
2.3
2.4
Intravalley (acoustic) Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Intervalley (nonpolar optical) Phonons . . . . . . . . . . . . . . . . . . . . . . . . 9
Surface Roughness Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Surface Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Remote phonons and high-field transport (Monte Carlo) . . . . . . . . . . . . . . 13
Nonparabolic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. RESULTS FOR SI AND GE SUBSTRATES . . . . . . . . . . . . . . . . . . . . . . 20
3.1
Low-field mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1
Scattering-Limited Electron Mobility . . . . . . . . . . . . . . . . . . . . . . . . 21
vii
3.1.2
3.1.3
3.2
3.3
Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Metallic vs. PolySilicon Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
High-field transport (Monte Carlo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4. III-V SUBSTRATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1
4.2
4.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Longitudinal-Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Multi-subband Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.1
4.3.2
4.3.3
4.4
Dynamic Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Static Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Effective Screening Wavevector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Surface-Optical phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.1
4.4.2
4.4.3
4.4.4
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Plasmon and Phonon Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
SO Scattering Strength and Momentum Relaxation Rate . . . . . . . 50
5. RESULTS AND DISCUSSION FOR III-V SUBSTRATES . . . . . . . . 57
5.1
Valley Occupation and LO-Limited Mobility . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1
5.1.2
5.2
SO-Limited Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Comparison to Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6. ELECTRON MOBILITY:
THE SYMMETRIC DOUBLE GATE . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1
6.2
Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Surface Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.1
6.2.2
6.2.3
6.3
6.4
Secular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Even and Odd Dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Substrate Dielectric Function and 2D Plasmon . . . . . . . . . . . . . . . . 76
Total Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
viii
7. CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1
7.2
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2.1
7.2.2
III-V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Double Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
APPENDIX: MOBILITY PROGRAM MANUAL . . . . . . . . . . . . . . . . . . . 104
A.1 Example Set Up File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.2 Desciption of Important Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.3 Code Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
ix
LIST OF TABLES
Table
Page
2.1
Effective Masses and Parameters for Bulk Phonon Scattering. . . . . . . . . . 17
2.2
Dielectric parameters used in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1
Mobility and Transconductance Degradation for Si. . . . . . . . . . . . . . . . . . . 27
3.2
Mobility and Transconductance Degradation for Ge. . . . . . . . . . . . . . . . . . . 27
4.1
Parameters for GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2
Parameters for In0.53 Ga0.47 As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
x
LIST OF FIGURES
Figure
Page
2.1
The dielectric function in the insulator as a function of energy. . . . . . . . . . 18
2.2
The dielectric function in the Si substrate as a function of energy. . . . . . . 19
3.1
Total calculated mobility for Si and Ge substrates with four gate
dielectrics: SiO2 , HfO2 , Al2 O3 , and HfSiO4 . A substrate doping
concentration NA = 3 × 1017 cm−3 is used throughout this
work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2
Total calculated mobility for Si and Ge substrates with both SiO2
and HfO2 gate dielectrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3
SO-phonon limited mobility for Si substrate. . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4
Surface roughness limited mobility for Si substrate (using an
exponential distribution with step rms height ∆ = 0.1 nm and
step correlation length Λ = 2.5 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5
Same as in Fig. 3.3 but for Ge substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6
Same as in Fig. 3.4 but for Ge substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7
Total calculated mobility for Si substrates with both SiO2 and HfO2
gate dielectrics as the temperature is varied. . . . . . . . . . . . . . . . . . . . . . . 34
3.8
Same as in Fig. 3.7 but for a Ge substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.9
Total calculated mobility for Si and Ge substrates with HfO2 . A
comparison of metal and poly-Si gates as the equivalent oxide
thickness is varied for a sheet density of ns = 2 × 1011 cm−2 . . . . . . . . 35
3.10 Remote phonon limited mobility for Si and Ge substrates with HfO2 .
A comparison of metal and poly-Si gates as the electron sheet
density is varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
xi
3.11 Calculated current-voltage characteristics of Si MOSFETs with SiO2
(solid symbols) and HfO2 (open symbols) as gate insulators. The
dotted lines are a guide for the eyes to judge the
transconductance. Note that SiO2 and HfO2 devices exhibit a
different threshold voltage at the same equivalent thickness due to
fringing-field effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.12 Same as in Fig.3.11, but for Ge devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1
The real part, imaginary part, and magnitude of the (11,11) element
of the dielectric matrix for dynamic screening. . . . . . . . . . . . . . . . . . . . . 53
4.2
Dispersion for GaAs with SiO2 . This is the solution of Eq. 4.24
yielding four mixed solutions (substrate plasmon and 3
TO-modes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3
The relative phonon-3 content of each branch of the dispersion in
fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4
The substrate plasmon content of each branch of the dispersion in
fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1
Normalized valley occupation for a single-gate structure with HfO2
and GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2
Normalized valley occupation for a single-gate structure with HfO2
and InGaAs. The and L-valleys are assumed to be parabolic. . . . . . . . 63
5.3
Normalized valley occupation for a single-gate structure with HfO2
and InGaAs. The X- and L-valleys ar assujmed to be
nonparabolic with α = 0.5ev−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4
LO-phonon limited mobility for the unscreened, static screened,
dynamic screened and effective screened cases for HfO2 with
GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5
LO-phonon limited mobility for the unscreened, static screened,
dynamic screened and effective screened cases for HfO2 with
InGaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.6
SO-phonon limited mobility for a metallic gate with GaAs and
InGaAs substrates with SiO2 and HfO2 gate dielectrics as a
function of electron sheet density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xii
5.7
SO-phonon limited mobility for a metallic gate with GaAs and
InGaAs substrates with SiO2 and HfO2 gate dielectrics as a
function of temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.8
The scattering-limited and total mobility for InGaAs substrate with
HfO2 gate dielectic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.9
Comparison of the total calculated mobility for a Si, GaAs, and
InGaAs substrate with HfO2 gate dielectric. . . . . . . . . . . . . . . . . . . . . . . 69
6.1
The semi-infinte Double Gate structure studied in this chapter. . . . . . . . . 84
6.2
Solution to Eq. (6.21) and (6.22) when the substrate plasmon is
ignored. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3
The 3D and 2D plasmon frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4
The 3D and 2D plasmon frequency for different tsi . . . . . . . . . . . . . . . . . . . . 87
6.5
Solution to Eqs. (6.21) and (6.22) showing the even and odd sets of
solutions when the substrate plasmon is included. . . . . . . . . . . . . . . . . . 88
6.6
Same as Fig. 6.5 except the x-axis is in a log scale to illustrate the
large Q limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7
Solution to Eqs. (6.21) and (6.22) when the insulator phonon LO1 is
ignored. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.8
Solution to Eqs. (6.21) and (6.22) when the insulator phonon LO2 is
ignored. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.9
The normalized scattering potential as tsi is varied from 2 nm to 20
nm for Q = 8 × 106 cm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.10 The first four energy levels in the unprimed-ladder as a function of Si
thickness for ns = 2 × 1012 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.11 The relative electron occupation of the first two energy levels in the
unprimed-ladder as a function of Si thickness for ns = 2 × 1012
cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.12 The unscreened SO-limited mobility for the DG structure with HfO2
dielectrics as the Si body thickness is varied for ns = 4 × 1011
cm−2 . The SO-limited mobility converges to the SG limit in the
thick Si body limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
xiii
6.13 The unscreened SO-limited mobility for the DG structure with HfO2
dielectrics as the Si body thickness is varied for ns = 2 × 1012
cm−2 . The SO-limited mobility converges to the SG limit in the
thick Si body limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.14 A comparison of the SO-limited mobility for an HiO2 gate dielectric
with and without screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.15 The SO-limited mobility, including the substrate plasmon, as a
function of Si thickness for an SiO2 and HiO2 gate dielectric. . . . . . . . 97
6.16 The total and scattering-limited mobilities as a function of Si
thickness for an HfO2 gate dielectric at ns = 2 × 1012 cm−2 . . . . . . . . . 98
6.17 The total and scattering-limited mobilities as a function of Si
thickness for an HfO2 gate dielectric at ns = 1 × 1013 cm−2 . . . . . . . . . 99
6.18 The total and scattering-limited mobilities as a function of electron
sheet density for an HiO2 gate dielectric at tsi = 3 nm. . . . . . . . . . . . 100
A.1 Tabulation of the Prange-Nee term for surface roughness scattering in
single-gate bulk MOSFETs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 This block calulates the momentum relaxation rate for surface
roughness scattering (the actual rate is computed in
SUBROUTINE tauSR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
xiv
CHAPTER 1
INTRODUCTION
1.1
Meeting Scaling Demands
With the continued down-scaling of silicon transistors, gate dielectrics with an
equivalent oxide thickness (EOT) of less than 1 nm will become a requirement of the
very-large-scale integration (VLSI) technology within the next decade [1]. Since the
current standard, SiO2 , yields unacceptable gate leakage currents at 1 nm and below,
high-κ dielectrics, such as HfO2 and its silicate are actively being considered as a way
to limit leakage while maintaining EOT scaling as well as gate capacitance. However,
concerns have been raised by the theoretical observation that high-κ dielectrics introduce larger remote phonon scattering strengths and reduce the electron mobility
in the channel [2, 3, 4]. The use of high mobility semiconductors, such as Ge, combined with high-κ gate dielectrics is one possible solution to meet scaling demands.
Previous studies have focused on the mobility reduction in Si with high-κ gate dielectrics [2, 5, 6], but here we also present new results for Ge substrates with high-κ
gate dielectrics and study the importance of remote phonons in the off-equilibrium,
quasi-ballistic regime of short channel devices.
The use of metal gates may become necessary to overcome the loss of capacitance
due to the depletion of the polycrystalline-Si (poly-Si) gate, to limit the thermal budget – possibly deleterious to the compositional integrity of many high-κ materials
and leading to the growth of unwanted SiO2 interfacial layer – avoiding the dopant
activation anneal of poly-Si, as well as to limit power dissipation [7]. Besides these
advantages, metal gates have been argued to screen remote-phonon scattering and so
1
increase electron mobility in the channel[8, 9]. Here we confirm these observations
and quantify the increase of the mobility with decreasing EOT for SiO2 and HfO2 .
An increase of 5% for HfO2 with Si and 10% for HfO2 with Ge is seen at low sheet
densities and decreases as the EOT is increased. The mobility increase, with decreasing dielecrtric thickness, is found to be negligible for SiO2 and modest for high-κ
materials.
1.2
Electron Mobility and Short-Channel Performance
In this work, we also attempt to determine to what extent the remote-phonon
degradation of the electron-transport properties – as measured by the low-field mobility – is reflected in a degradation of the performance – as measured by transconductance – of short-channel Si and Ge devices driven in which electron transport
occurs in a strong off-equilibrium, high-field regime, situation quite different from
the near-equilibrium conditions implied by the concept of ‘mobility’. We find that as
the channel length is scaled aggressively into the near ballistic limit, the transport
is dominated by high field effects and the mobility plays less of a role, especially for
Ge. However, even in the smallest devices, the mobility plays an important role in
the linear regime of device operation.
1.3
III-V MOSFETs
III-V semiconductors are considered a possible replacement for conventional Si
substrate for nMOSFETs because of the expected increase in electron mobility. InGaAs single-gate (SG) MOSFETs with high-κ gate dielectrics have been fabricated
and show a peak mobility of 1100 cm2 /V-s compared to 500 cm2 /V-s for strained Si
and 330 cm2 /V-s for Ge.[10, 11, 12] The mobility enhancement may lead to overall
device performance improvement and faster switching speeds - making III-V’s, and
especially InGaAs, a possible solution to continue Moore’s law.[13].
2
1.4
The Double-Gate MOSFET
If Silicon is to lead us into the next decade of scaling, some form of a double
gate structure is likely to be needed [14, 15]. These structures vastly improve the
device electrostatices and limit short channel effects [16, 17, 18]. In this work, we extend the theory for scattering with surface-optical phonons in single-gate structures
to the symmetric double-gate structure. The coupling between the SO-phonons of
both dielectrics is included for the first time and the dispersion is separated into even
and odd modes via the seperation of the secular equation. For the unscreened case,
the convergence to the single-gate case, as the Si thickness is increased, is demonstrated. The substrate plasmon is derived within the random-phase-approximation
and included in the dispersion to account for screening of the surface-optical phonons.
3
CHAPTER 2
LOW-FIELD ELECTRON MOBILITY
This chapter explains the theoretical and mathematical formulation of the lowfield electron mobility and momentum relaxation rates. First, the Kubo-Greenwood
formula is presented for the calculation of the low-field mobility. Bulk Phonon, surface
roughness, and SO phonon momentum relaxation rates are presented for a Si and Ge
single-gate MOSFET structure. Then we discuss how we incorporated SO-phonon
scattering in the Monte Carlo simulation. The last section in this chapter discusses
nonparabolic corrections and presents the model used in this work.
Polar longitudinal-optical phonon scattering and surface-optical phonon scattering
in III-V single-gate structures is presented in a following chapter. SO phonon scattering and surface roughness scattering in double-gate structures is also presented
separately in a following chapter.
2.1
Kubo-Greenwood Formula
We can express the mobility tensor from the linearization of the two-dimensional
Boltzmann transport equation [19, 20, 21]:
∂f 1
e
τp,i vi
,
µij = −
h̄
∂Kj f
th
(2.1)
where i and j run over the two in-plane cartesian coordinates x and y, τp,i (K) is the
momentum relaxation rate, f (K) is the equilibrium distribution function, and < · >th
is the thermal average:
4
X gν Z dK
A(K)fν (K) ,
hAith =
nν
(2π)2
ν
(2.2)
where nν is the electron density and gν the degeneracy of subband ν.
The one dimensional Schrödinger equation:
h̄2 d2
+ V (z) ζν (z) = Eν ζν (z)
−
2mz dx2
(2.3)
is solved self-consistently with the one-dimensional Poisson equation:
d2
e
V
(z)
=
dz 2
ǫsc
p(z) − NA− − e
X
µ
nµ |ζν (z)|2
!
(2.4)
along a line perpendicular to the transport direction to yield the subband energy
minima, Eν , envelope wavefunctions, ζν (z), and potential energy profile, V (z). The
hole density is p(z), the acceptor doping is NA− , and the 2D parabolic density of states
for electrons is:
Eµ −EF
kB T X
kB T
nµ =
,
gµ md,µ ln 1 + e
πh̄2 µ
(2.5)
where ǫsc is the dielectric constant of the semiconductor, kB is the Boltzmann constant, gµ is the valley degeneracy for subband µ, m,µ is the density of states effective mass, ad EF is the Fermi level. In this work, the image term as well as the
exchange-correlation term is ignored in the potential energy. The transport direction
is arbitrarily aligned along the x-axis. For elliptical subbands, including nonparabolic
corrections as will be discussed in Section 2.4, we calculate the xx mobility tensor
component of subband ν as:
µ(ν)
xx
Z 2π
Z ∞
egν
= 2 2
dβmν (β)
dE cos2 β
2π mν,x kB T nν 0
Eν
f (E)[1 − f (E)]
,
× τp,x (K, β)
1 + 2α(E − Uµ )
5
(2.6)
and the total mobility as:
1 X
nν µ(ν)
xx ,
ns ν
µxx =
where
mν (β ′ ) =
cos2
(2.7)
1
,
β/mν,1 + sin2 β/mν,2
(2.8)
ns is the electron sheet density, nν is the electron sheet density in subband ν, K the
initial wavevector, E the initial energy (integration from each subband minimum up
to a maximum energy above the last subband considered), mν,x the conductivity mass
in subband ν, and gν the degeneracy including spin. The term 1+2α(E−Uµ ) accounts
for nonparabolic corrections and is derived in Section 2.3. The momentum relaxation
time, τp,x (K, β), is found by summing all momentum relaxation rates considered:
1
τp,x
=
1
τBP
+
1
τSR
+
1
τSO
,
(2.9)
where τBP , τSR , and τSO are the bulk phonon, surface roughness, and surface optical
phonons momentum relaxation times, respectively. These momentum relaxation rates
are discussed in the following subsections for the Si single-gate structure and following
chapters for III-V substrates and double-gate structures.
The following notation is used throughout this work. The three-dimensional transferred wavevector is q = k′ − k, where k and k′ are the initial and final wavevector
of the electron, respectively. From here on, all capital letters denoting vectors (and
magnitudes) represent two-dimensional in-plane vectors. For example, aligning the
quantization axis along the z-axis, we have q = Q + qz with Q2 = qx2 + qy2 . The
wavefunction of subband µ for wavevector K is:
1
ψµK (r) = ζµ (z) √ eiK·R ,
2π
(2.10)
where ζµ (z) is the solution of the Schrödinger equation with ζµ (z) vanishing at the
dielectric/substrate boundary.
6
We have chosen the standard orientation, the [100] field direction on the (011)
surface, for calculations with Si and III-Vs, and [111](112̄) for Ge. This particular
orientation was chosen for Ge because it yields the highest bulk-phonon limited mobility and we are interested in comparing ‘best possibe’ performance. The particular
orientation chosen will not affect the effective mass in the isotropic Γ-valley but will
change the various effective masses of the L and X-valleys and also the valley occupation. For Ge, GaAs, and InGaAs, the L, X, and Γ valleys are included in the
mobility calculations. See Table 2.1 for the transverse, mT , and longitudinal, mL ,
effective masses for each valley. Table 2.1 also lists the energy minimums of the three
valleys. The reference energy, the lowest minimum, is set to zero (e.g., EΓ = 0 for
GaAs and InGaAs). The nonparabolicity factor α is also listed in table 2.1 and 2.2.
The Γ-valley of InGaAs is strongly nonparabolic with α = 1.22 compared to α = 0.61
for GaAs. For the X and L-valleys, α = 0 because no reliable data exists in the
literature. Choosing α = 0 is the best case scenario, in terms of occupation, resulting
in the largest Γ-valley occupation. This is illustrated in the results and discussion
section.
7
2.2
2.2.1
Scattering Rates
Intravalley (acoustic) Phonons
Intravalley phonon scattering in the X and L-valleys (simplifies to isotropic in the
Γ-valley) is described by an anisotropic elastic model. The elastic approximation is
justified because the acoustic phonons that contribute to scattering at room temperature have energies that are smaller than kB T . Using the elastic and equipartition
approximation, the momentum relaxation rate is written as:
Z ∞
dqz
kB T
|Fµν (qz )|2
=
3 2 θ(E − Eν )
(i)
2πρh̄ ci
τµν (K, β)
−∞ 2π
Z 2π
K′
Ξ2i (ηQ )
1−
×
dφ 2 ′
cos φ ,
K
cos β /m1,ν + sin2 β ′ /m2,ν
0
1
(2.11)
using the deformation potential obtained by Herring and Vogt [22, 23]:
Ξi (ηQ ) =


 Ξd + Ξu cos2 ηQ (i=LA)
(2.12)

 Ξu cos ηQ sin ηQ (i=TA),
where ηQ is the angle between the acoustic phonon and the longitudinal axis of the
valley’s equienergy surface. For example, for collisions involving intravalley transitions
within the primed valleys of Si on the (001) surface:
cos ηQ =
K cos β − K ′ cos β ′
.
(Q2 + qz2 )1/2
(2.13)
The uniaxial shear, Ξu , and the dilation, Ξd , deformation potentials as well as the
transverse, cT , and longitudinal, cL , sound velocities are listed in Table 2.1. The
transverse wave vector is Q = |K − K′ |, and the electronic form factor is a function
of the wave functions, ζi (z):
Fµν (qz ) =
Z
∞
dzζµ (z)eiqz z ζν (z).
tox
8
(2.14)
2.2.2
Intervalley (nonpolar optical) Phonons
Intervalley phonon scattering is approximated as an isotropic process:
(s)
(Dt K)siv gµν
1 1
iv
θ(E − Eν )
=
∓ + ns Fµν
2
(iv)
2
2
4πh̄ ρωs
τs;µν (K, β)
Z 2π
K′
1
1−
×
cos φ ,
dφ 2 ′
K
cos β /m1,ν + sin2 β ′/m2,ν
0
1
(2.15)
where the various processes (f- and g-scattering for Si) are labeled by the index s,
(Dt K)siv is the deformation potential for the s-th transition, ns is the Bose occupation
(s)
of phonon ωr , and gµν is the degeneracy of the final state. The form factor is:
iv
Fµν
=
Z
∞
dzζµ (z)2 ζν (z)2 .
(2.16)
tox
For the standard Si substrate orientation of (001)[110], m1,ν = mx,ν and m2,ν =
my,ν , and the density of states effective mass is:
md,ν =
cos2
β ′ /mx,ν
1
,
+ sin2 β ′ /my,ν
(2.17)
which simplifies Eq. (2.15):
(iv)
(s)
∆s md,ν gµν
=
θ(E − Eν )
(iv)
2h̄2 ρωs
τs;µν (K, β)
1
1 1
iv
∓ + ns Fµν
.
2 2
(2.18)
The parameters for intervalley phonon scattering are taken from the literature where
possible [24]. For Ge, where transitions include the Γ and L-valleys, the parameters
are taken from Ref.[25]
9
2.2.3
Surface Roughness Scattering
Scattering with surface surface roughness (SR) at the insulator-substrate interface
is accounted for following Ando’s approach and screened by computing the screening matrix with the appropriate Green’s function [26, 27]. The scattering rate is
proportional to:
∆2 Λ2
1
≈
τµν (K, β)
2h̄3
Z
0
2π
dβ ′|S[q(β ′)]|2 |Γµν [q(β ′)|2 ,
(2.19)
where an exponential-decaying model has been used for the spectral distribution of
steps at the interface:
|S[q(β ′)]|2 = (1 + q 2 Λ2 /2)−3/2 .
(2.20)
In this work, the rms height ∆, and the step distance autocorrelation Λ, are set to 0.48
nm and 1.3 nm respectively. [28] In general, these parameters are process dependent
and could vary between devices on the same wafer. We take the parameters of Ref. [28]
because the were correlated with experiment, keeping in mind that new technology
may have vastly smoother/rougher interfaces.
The other component of the SR scattering rate, Γµν [q(β ′ )], can be broken into two
(s)
(c)
(s)
matrix elements: Γµν and Γµν , where Γµν arises from the wavefunction shift at the
(c)
discontinuity of the interface or ’step’, and Γµν is the Coulomb contribution. The
first term is given by[27]:
Γ(s)
µν
(c)
h̄2 dζµ dζν .
=
2mz dz 0 dz 0
(2.21)
The Coulomb term Γµν includes three separate contributions:[27]
(c)
Γµν
=
Z
dzζµ (z) [γ1 (Q, z) + γ2 (Q, z) + γ3 (Q, z)] ζν (z)
(2.22)
The first matrix element is due to the shift of the charge of the free carriers caused
by the step:
10
e2 ǫ̃ 1
γ1 (Q, z) = −
ǫsc 2Q
Z
dz
′
) e−Q|z−z |
−Q|z+z ′ |
,
+e
∂z
ǫ̃
′ ∂n(z
′
(2.23)
0
∞
0
where ǫ̃ = (ǫ∞
s − ǫox )/(ǫs + ǫox ). The second term is caused by the dipole at the step:
γ2 (Q, z) =
e2 ǫ̃
(ns + Nd ) e−Qz ,
ǫsc
(2.24)
where ns and Nd are the electron an ionized impurity densities per unit area, respectively. The third term is the potential dipole induced by the polarization of the step
created by the carriers and their images:
e2 ǫ̃ Q2 K1 (QZ) ǫ̃
γ3 (Q, z) =
− K0 (QZ) ,
ǫsc 16π
Qz
2
(2.25)
where K0 (QZ) and K1 (QZ) are the modified Bessel functions.We leave the discussion of screening to section 4.3, where we discuss static and dynamic multi-subband
screening of surface roughness scattering and LO phonons.
2.2.4
Surface Optical Phonon Scattering
In this subsection we introduce the general formulation of SO phonons. Namely
we describe how the secular equation is derived by using the electrostatic boundary
conditions withe the general form of the scattering potential. We wait until the
chapter dealing with SO phonons with III-V semiconductors and high-κ insulators to
give a more detailed example because the III-V case is more general than the Si/Ge
case.
The LO phonons associated with high-κ insulators (and III-V semiconductors)
induce SO-phonons at the interface between the the insulator and the semiconductor.
The Fourier transform of the scattering potential associated with the SO-phonons is:
φ(R, z) =
X
Q
11
φQ (z)eiQ·R ,
(2.26)
where φQ (z), which extends into the semiconductor and acts a a scattering potential
for electrons, has the following basic form:
φQ (z) = Ae±Qz .
(2.27)
If we assume a poly-Si gate, and using the general form of the scattering potential combined with the electrostatic boundary boundary conditions at the insulator/semiconductors, we can write the secular equation as:[2]
ǫox (ω)2 + ǫox (ω)[ǫg (ω) + ǫs (Q, ω)]cotanh(Qtox )
(2.28)
+ ǫg (ω)ǫs (Q, ω) = 0.
Following the approach employed by Kotlyar et al. [6], metal gates are idealized
as perfect conductors unable to sustain any spatially varying potential, so that the
perturbing spatially-varying potential due to the interface excitations is assumed to
vanish at the gate-dielectric interface. With this approximation the secular equation
becomes:
ǫox (ω) cosh(Qtox ) + ǫs (ω) sinh(Qtox ) = 0.
(2.29)
with the insulator, substrate, and gate dielectric functions (ǫox , ǫs , and ǫg , respectively) as [2]:
ǫox (ω) = ǫ∞
ox
2
2
(ωLO1
− ω 2 )(ωLO2
− ω2)
,
2
2
(ωTO1
− ω 2 )(ωTO2
− ω2)
(2.30)
and
ǫs (Q, ω) =
ǫ∞
s
2
ωp,s
(Q)
1−
.
ω2
(2.31)
Figure 2.1 shows the gate insulator dielectric function as the energy is varied, assuming
HfO2 . This figure illustrates the various frequency and dielectric constants used in this
work. The LO-modes are the zeros of the dielectric function, whereas the poles are
the TO-modes. The intermediate dielectric constant, ǫi is used as a fitting parameter
12
for experimental data. [29] Figure 2.2 shows the Si dielectric function as the energy
and Q is varied. The zeros represent the plasmon energy as a function of Q. This
2
plot assumes the approximate form ωP;e
(Q) for the DG case, Eq. (6.37).
Table 2.2 lists the static permittivity, ǫ0ox , and transverse optical energies, ωT O1
and ωT O2 , associated with the four dielectrics considered here [2]. Notice that HfO2
has the largest permittivity and the lowest-energy modes.
Equation (2.28) yields four solutions representing the coupled gate-plasmon, substrateplasmon, and two SO-phonon modes. In the absence of the gate plasmon, as in the
case of idealized metallic gates, Eq. (2.29) yields the dispersions of three coupled
modes. Electrons in the channel scatter with the gate plasmon (poly gate) and two
SO phonons. Scattering with the substrate plasmon is not included since it does not
cause any momentum loss of the two-dimensional electron gas (2DEG) [2]. However,
the substrate plasmon is included in calculating the dispersion and it contributes
to screening the SO phonons. Landau damping is included approximately by ignoring the gate (or substrate) plasmon when the branch of the dispersion most like
the gate (or substrate) plasmon enters the single-particle continuum in the gate (or
substrate) [2].
2.3
Remote phonons and high-field transport (Monte Carlo)
In order to establish the importance of the electron-SO scattering in short-channel
devices, we have employed the DAMOCLES [30] program to perform the Monte Carlo
simulations. In attempting to include SO-scattering in the code, we had to face a
well-known complication. At large source-to-drain bias, electrons occupy low-energy
states in the subbands of the inversion layer only near the source end of the channel.
In this case, the scattering rate appropriate for a two-dimensional electron system
as used previously [2] could be used without complication when simulating electron
transport in inversion layers[24]. However, as electrons are accelerated by the electric
13
field towards the drain, they may be scattered into high-energy subbands. In this
case, a three-dimensional (bulk) continuum description is appropriate and the scattering rate between bulk electrons and interfacial modes is required. In turn, this
requires an expression for the wavefunction of the electrons. Employing plane waves,
as usually done for all other scattering processes in semiclassical Monte Carlo simulations, is not appropriate in this case since it would result in the non-physical result
of ignoring the distance, z0 , of the carrier form the interface and in a scattering rate
independent of z0 . Even more worrisome is the observation that this approach would
require the definition of some ‘normalization length’ (roughly, the ‘size of the electron’ along the direction, z, normal to the semiconductor-dielectric interface) which
cannot be determined unambiguously. On the contrary, assuming a wavefunction of
the form ∼ δ(z − z0 ) would correctly account for the distance of the particle from
the interface, but would result in a process in which the z-component of the crystal
momentum would not be conserved. Therefore, we have followed a phenomenological
approach, assuming that (crystal) momentum can only be exchanged on the plane of
the interface, since SO modes propagate on this plane. Moreover, the variation with
z of the potential associated with the SO modes, φQ (z) ∝ exp(−Qz), is treated as a
parameter, consistent with the semiclassical spirit of Monte Carlo simulations. Thus,
the scattering rate between an interface mode η with energy h̄ω (η) and a ‘bulk 3D’
electron, with wavevector k = (K, kz ) at a distance z0 from the dielectric-substrate
interface, is approximated by:
!
e2 ω (η)
1
1 1
1
1
=
nη + ±
− (η)
(η)
τ (η) (k, z0 )
8π 2
2 2
ǫhigh ǫlow
Z
e−2Qz0
× dQ
δ(Ek − Ek+Q ± h̄ω (η) ),
Q
(2.32)
(η)
(η)
where Q is the 2D wavevector of the interface mode and the term 1/ǫhigh − 1/ǫlow
in Eq. (2.32) represents the coupling constant, as in Ref. [2]. Here the energy mode,
14
ω (η) , is assumed to be independent of wavelength (corresponding to the infinitely-thick
insulator limit) and the coupling with substrate and gate plasmons is ignored. These
simplifications are necessary to limit excessive computational cost of computing the
dispersions of the interfacial modes, dispersions which would vary along the channel
as the concentration of electrons in the substrate and in the gate (and so the plasma
energies) also vary along the channel. As stated above, considering the electrons as
semiclassical bulk, three-dimensional, particles allows us to treat the z-dependence
of the scattering potential as a parameter. Only the components of the momentum
parallel to the dielectric-substrate interface are exchanged between the electrons and
the interfacial modes, Q being the momentum transfer. For a given magnitude Q,
the strength of the interaction is exponentially dependent on the distance z0 from the
interface.
Despite these simplifications, the evaluation of eq. (2.32) within a full-band context
is numerically challenging due to the energy conserving delta-function. A modification
to two dimensions of the Gilat-Raubenheimer algorithm, employed in ref. [31], is used
to evaluate the integral in 2.32.
2.4
Nonparabolic Corrections
The Γ-valley of InGaAs is isotropic and strongly nonparabolic, making it necessary
to include nonparabolicity to more accurately model the electron mobility. In 2D,
perturbation theory is used and the corrected energy minimum of subband µ is: [24]
Eµ ≈
Eµ(0)
−α
Z
0
∞
2
dz Eµ(0) − U(z) |ζν(0) (z)|2
(2.33)
where U(z) is the potential energy. The problem arises when the parabolic subband
(0)
energy, Eµ , becomes larger than 0.5 eV. This problem is encountered when first-order
perturbation fails - under strong quantization (double-gate, small mz ) and/or strong
15
nonparabolicity paramter α (0.5 eV−1 for Si compared to 1.22 eV−1 for InGaAs).
This problem is overcome phenomenologically in Ref. [32] with the total energy as
EµK ≈ Uµ +
−1 +
q
(0)
1 + 4α(γK + Eµ − Uµ )
2α
(2.34)
The corrected energy minimum of subband µ given as:
Eµ = Uµ +
q
(0)
−1 + 1 + 4α(Eµ − Uµ )
2α
,
(2.35)
with
Uµ =
Z
∞
dzU(z)|ζµ (z)|2 .
(2.36)
0
This phenomenological solution was chosen such that energy increases monotonically
and gives the exact solution in the infinite square well potential.
Up to this point, the momentum relaxation rates do not include nonparabolic
corrections. To include nonparabolic corrections in the momentum relaxtion rates,
[1 + 2α(E − Uµ )]
(2.37)
must be inserted in front of the integrals of all momentum relaxing rates and inside
the integral for the SO rate. The corrected 2D wavevector:
(0)
{2mν (φ)[E − Eµ + α(E − Uµ )2 ]}1/2
K(E, φ) =
h̄
with a similar expression for K ′ (E ′ , φ).
16
(2.38)
Table 2.1. Effective Masses and Parameters for Bulk Phonon Scattering.
Ge (Γ)
Ge (L)
Ge (X)
Si (X)
Emin
(eV)
0.135
0.000
0.173
0.000
mL
(mo )
0.062
1.454
1.353
0.916
mT
Ξd
(mo ) (eV)
0.062 2.5
0.112 2.5
0.288 2.5
0.190 1.1
Ξu
(eV)
4.5
4.5
4.5
10.5
cL
(cm/s)
5.31
5.31
5.31
9.0
cT
(cm/s)
3.61
3.61
3.61
5.4
Table 2.2. Dielectric parameters used in this work.
ǫ0ox (ǫ0 )
ǫiox (ǫ0 )
ǫ∞
ox (ǫ0 )
ωT O1 (meV)
ωT O1 (meV)
SiO2 Al2 O3
3.90
12.53
3.05
7.27
2.50
3.20
55.60 48.18
138.10 71.41
17
HfSiO4
11.75
9.73
4.20
38.62
116.00
HfO2
22.00
6.58
5.03
12.40
48.35
100
80
HfO2
εox (ε0)
60
40
20 ε0
ω(LO1)
εi
ε∞
ω(LO2)
0
-20
ω(TO1)
-40
0
10
ω(TO2)
20
30
40
50
ENERGY (meV)
60
70
80
Figure 2.1. The dielectric function in the insulator as a function of energy.
18
20
ns=2x1011 cm-2
ωp,s(Q)
Silicon
15
Q=1x107 m-1
εs (ε0)
10
Q=1x108
5
9
Q=1x10
0
Q=5x109
-5
-10
0
10
20
30
40
50
ENERGY (meV)
60
70
80
Figure 2.2. The dielectric function in the Si substrate as a function of energy.
19
CHAPTER 3
RESULTS FOR SI AND GE SUBSTRATES
3.1
Low-field mobility
Figure 3.1 shows the electron mobility versus electron sheet-density for Si and Ge
substrates with SiO2 , HfO2 , HfSiO4 , and Al2 O3 gate dielectrics. In this case EOT=1.0
nm and T=300 K. Ge [as seen in Fig. 3.1(b)] outperforms Si [shown in Fig. 3.1(a)]
for all four dielectrics. This is due to the large bulk phonon-limited mobility which,
in turn, is due to the smaller conductivity masses along the (111)[112̄] in Ge. For
both Si and Ge, HfO2 yields the lowest mobility while the other dielectrics yield
results on par with SiO2 . This is due to the low-energy phonons that give HfO2 the
largest dielectric constant of all four insulators considered [2]. Figure 3.2, illustrating
the same data shown in Fig. 3.1, emphasizes the mobility reduction introduced by
replacing SiO2 with a high-κ dielectric, in this case HfO2 . Dielectric screening by
carriers in the inversion layer reduces the strength of remote-phonon scattering at
large electron densities. Thus, the percentage mobility reduction, ∆µ = 38% for Si,
and ∆µ = 46% for Ge, is calculated at ns = 2 × 1011 cm−2 .
Of particular interest is the observation that the curves shown in the figures we
have just discussed exhibit a maximum. This ‘turn-around’ is usually attributed to
Coulomb scattering with the ionized dopants in the substrate. However, since this
process is not included in our mobility calculations (Coulomb scattering is included
in the Monte Carlo simulation), the turn-around appears to be due exclusively to
scattering with SO modes. The strength of this process depends on the overlap
20
between the SO-potential, φbf Q (z) ∼ e−Qz , and the initial, ζµ (z), and final, ζν (z),
wavefunctions,
Z
dz ζµ (z) e−Qz ζν (z) .
(3.1)
In calculations bypassing the self-consistency between Poisson and Schrödinger equations, as in Ref. [2], as the electron sheet density ns decreases, both the wavefunctions
ζ and the SO-potential, e−Qz , become less ‘localized’: The wavefunctions for obvious
reasons, the SO-potential since wavevectors close to the Fermi wavevector, KF , largely
control the mobility, so that the function e−KF z becomes less ‘squeezed’ against the
interface as ns decreases. Thus, the matrix element given by Eq. (3.1) above changes
monotonically (and relatively slowly) with ns and no ‘turn-around’ is observed. On
the contrary, when employing a self-consistent Poisson-Schrödinger approach with a
large density of dopants, nB , in the depletion layer, as the electron density is reduced,
the confinement of the wavefunction decreases very slowly at medium densities, since
the confinement is largely due to the bulk dopant charge, enB . Thus, the confinement
of the wavefunctions changes slowly. On the contrary, the SO-potential, dependent
only on ns via KF , becomes less squeezed against the interface as before as ns decreases. This causes the matrix element, Eq. (3.1), to grow with decreasing ns as
soon as nB controls the confinement, resulting in the turnaround. Eventually, at even
lower values of ns , also the dopant bulk charge enB will begin to drop, resulting in
an increasing mobility.
3.1.1
Scattering-Limited Electron Mobility
Figures 3.3 and 3.4 show the SO-limited and SR-limited mobility for Si on (001),
respectively. Figures 3.5 and 3.6 show the SO-limited and SR-limited mobility for Ge
on (111), respectively. SR-scattering is most important at high densities because the
wavefunctions are “squeezed” tightly to the dielectric/substrate interface. The main
factor determining the difference in SR-scattering among insulators is the permittivity
21
of the insulator and substrate, see Figs. 3.4 and 3.6. An important factor in the
expression for the SR-scattering rate, Eq. (2.22), is the effect of image charges and the
interfacial dipoles they induce [27]. This term is proportional to ǫs −ǫins , where ǫs and
ǫins are the permittivities of the substrate and insulator respectively. The magnitude
and even sign of this term is clearly dependent on the insulator and substrate materials
considered. Notably, in high-κ materials (ǫins > ǫs ), this term acts as a screening
component which reduces the strength of SR-scattering (see Figs. 3.4 and 3.6). SRscattering depends on the quality of the fabricated interface. Information about real
interfaces, such as can be seen using an atomic force microscope [33], may make SR
models more realistic.
High-κ effects are more noticeable at lower densities, and hence for small values of
the Fermi wavevector kF , since carriers close to the Fermi surface contribute most to
the mobility, and since the SO-scattering strength behaves as 1/q at small q. Instead,
at large densities screening by the channel electrons results in an enhanced mobility.
So far, the mobility in fabricated Ge MOSFETs has been disappointing. [12, 34]
These disappointing results have been attributed to the inability of Hydrogen to
passivate the Ge dangling bonds (as is done for Si), resulting in non ideal interfaces
with high-κ gate dielectrics and reduced electron mobility. [35, 36] Hence our results
set an upper limit on the mobility and comparison to fabricated Ge devices is not yet
meaningful.
3.1.2
Temperature Dependence
Figures 3.7 and 3.8 show the mobility, respectively, for Si and Ge for T = 77, 300,
and 375 K. The left figure, (a), is for SiO2 , and the right, (b), for HfO2 . In both
figures, EOT = 1.0 nm, and the sheet density is varied from 2 × 1011 to 2 × 1013
cm−2 . For all cases, the mobility increases as the temperature is increased. This is
mostly due to the decrease in bulk phonon scattering with decreasing temperature.
22
As the temperature decreases, the population of bulk phonons decreases in accordance
with Bose statistics. At ns = 2 × 1012 the total mobility decreases approximately as
T −1.5 for SiO2 with Si and Ge. For HfO2 , the dependence is closer to T −1 because
the low energy SO phonons are easier to excite at low temperatures [5]. For lower
temperatures, the mobility curves exhibit a large positive slope for smaller sheet
densities. This is a consequence of stronger screening of SO phonons by electrons in
the channel and a stronger contribution of SO phonons to the overall scattering rate at
lower densities. Notice that Si (also Ge) with HfO2 can exhibit a mobility larger than
Si (also Ge) with SiO2 for large electron sheet densities. This is due to the image
term in the SR relaxation rate. For high-κ dielectrics this term can be negative increasing the total mobility. This is even more noticeable at low temperatures, when
the SR limited mobility plays a larger role (see Figs. 3.7(a) and 3.7(b) at ns > 5×1012
cm−2 ).
3.1.3
Metallic vs. PolySilicon Gate
Figure 3.9 shows the mobility as the EOT is varied form 1 to 5 nm at T=300 K
and ns = 2 × 1011 cm−2 . The higher set of curves is for Ge and the lower for Si. The
gate dielectric is HfO2 with a metal gate (solid symbols with solid line) and a poly-Si
gate (open symbols with dashed line). In all cases the mobility is weakly dependent
on the EOT. As the EOT is decreased the mobility increases due to screening by the
gate. The metal gate is expected to yield higher mobilities compared to the polySi gate because the scattering potential associated with the remote phonons in the
dielectric is forced to zero at the metal dielectric interface. For the poly-Si gate, the
potential can decay into the gate and the overall magnitude is higher. This is more
noticeable for the Ge substrate: as the EOT is decreased the mobility associated with
the metal gate increases faster as compared to the poly gate. Overall, an increase of
23
5% for HfO2 with Si and 10% for HfO2 with Ge is seen at low sheet densities for the
thinnest EOT, 0.5 nm.
In hindsight, the noticeable but still modest improvement observed when replacing the poly-Si gate with an ideal metallic gate could have been expected for the
following reasons: The effect of dielectric screening by the electrons in the gate is
most pronounced when the gate is “sufficiently close” to the substrate. (Here “close”
and “far” must be interpreted as relative to the characteristics length-scale of the
two-dimensional electron gas, its Fermi wavelength, since the f (E)[1 − f (E)] factor
in (2.6) shows that electrons with energy close to the Fermi energy yield the largest
contribution to the mobility.) Thus, the gate appears “sufficiently close” at low electron sheet densities. In this case poly-Si gates are not depleted and their screening
effects are noticeable (as shown in Ref. [2]). Metal gates show a beneficial effect, but
only a modest one as shown in Fig. 3.10. In the opposite limit of large sheet electron
densities, poly-Si gates are depleted (because of the large gate bias) and metallic gate
should be vastly superior. However, in this case the gate appears “far”, as the Fermi
wavelength is very short, and the channel becomes insensitive to the presence of the
gate electrons.
3.2
High-field transport (Monte Carlo)
In Figs. 3.11 (Si) and 3.12 (Ge) we show the current-voltage (drain-to-source
current, IDS versus drain-to-source voltage, VDS ) characteristics of the MOSFETs
obtained from the DAMOCLES simulation. The dotted lines are just visual aids to
facilitate a qualitative valuation of the threshold voltage and of the transconductance.
Note that for a given EOT, SiO2 and HfO2 exhibit slightly different threshold voltages
do to slightly different fringing-field effects.
Figure 3.11(a) shows that for Si devices operating in the linear region HfO2 introduces a significant relative degradation to the linear transconductance for long
24
channel lengths (namely, ∆gm = 50% for the 60 nm device), and a decreasing degradation for shorter channel lengths (∆gm = 30% for the 15 nm device). For the Si
devices, this result corresponds to the mobility reduction seen in Fig. 3.2. On the
contrary, in the saturated region of operation, illustrated in Fig. 3.11(b), the mobility
plays a smaller role due to large electric fields driving the carriers away from thermal
equilibrium, away from the band minima, and the reduction in transconductance is
lower.
Moving to the smallest Ge device (15 nm), Fig. 3.12 illustrates the fact that the
linear and saturated transconductance is negligibly affected by the presence of the
high-κ insulator. For the largest device (60 nm), remote phonon scattering may still
play a role in the linear region, ∆gm = 30% compared to a relative mobility reduction
∆µ = 46%. The saturated transconductance for the 60 nm device is reduced by less
than 10%. The likely cause of this behavior lies in the fact that Ge has a large
bulk phonon-limited mobility and is more severely affected by the strong Coulomb
scattering (which is included in the Monte Carlo simulation but not in the mobility
calculations) with the large density of dopants in the channel. This ‘hides’ the effect
of scattering with high-κ phonons, at least as compared to the more noticeable remote
phonon scattering in Si.
The results are summarized in Table 3.1 for Si and Table 3.2 for Ge. The mobility
degradation is calculated in the long-channel limit and is reported in the table for
reference. In general, the transconductance degradation improves as the channel
length is shortened with the exception of Si devices in the saturated region. The
transconductance degradation more closely follows the mobility degradation for Si
devices in the linear region as compared to the Ge devices.
25
3.3
Conclusions
We have computed the electron mobilities for Si and Ge inversion layers including
bulk phonons, surface roughness and SO-phonon scattering. Ge outperforms Si but
is significantly affected by the introduction of high-κ insulators. Decreasing oxide
thickness does not significantly increase remote phonon scattering. HfO2 yields the
lowest mobilities and other materials such as HfSiO2 or Al2 O3 , or AlN should be
considered over HfO2 .
The low-field electron mobility reduction, due to surface-optical modes associated
with high-κ dielectrics, plays less of a role in determining device performance as the
gate length is scaled from 60 nm to 15 nm. This is especially true for devices operating
in saturation and Ge MOSFETs. HfO2 , the dielectric with the largest dielectric constant and lowest energy SO modes, yields the lowest mobility. Other materials, such
as HfSiO4 and Al2 O3 , should be considered a good compromise, yielding mobilities
close to SiO2 while having a larger dielectric constant. A metal gate reduces phonon
scattering when compared to a poly-Si gate. The performance gain is expected to be
on the order of 10%.
26
Table 3.1. Mobility and Transconductance Degradation for Si.
Gate length
(nm)
60
30
15
EOT ∆µ
(nm) (%)
2.8
38
1.4
38
0.7
38
Linear ∆gm
(%)
50
44
30
Saturated ∆gm
(%)
10
19
17
Table 3.2. Mobility and Transconductance Degradation for Ge.
Gate length
(nm)
60
30
15
EOT ∆µ
(nm) (%)
2.8
46
1.4
46
0.7
46
Linear ∆gm
(%)
27
16
9
27
Saturated ∆gm
(%)
8
3
1
EOT = 1.0 nm
Si
SiO2
HfO2
HfSiO4
Al2O3
103
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
ELECTRON MOBILITY (cm2/Vs)
ELECTRON MOBILITY (cm2/Vs)
104
(a) Si
104
SiO2
HfO2
HfSiO4
Al2O3
103
EOT = 1.0 nm
Ge
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
(b) Ge
Figure 3.1. Total calculated mobility for Si and Ge substrates with four gate
dielectrics: SiO2 , HfO2 , Al2 O3 , and HfSiO4 . A substrate doping concentration
NA = 3 × 1017 cm−3 is used throughout this work.
28
ELECTRON MOBILITY (cm2/Vs)
EOT = 1.0 nm
T = 300K
10
3
Ge ∆µ = 46%
Si ∆µ = 38%
102 11
10
SiO2
HfO2
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.2. Total calculated mobility for Si and Ge substrates with both SiO2 and
HfO2 gate dielectrics.
29
SO-LIMITED MOBILITY (cm2/Vs)
105
104
103
SiO2
HfSiO4
Al2O3
HfO2
Si
teq=1 nm
102 11
10
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.3. SO-phonon limited mobility for Si substrate.
30
SR-LIMITED MOBILITY (cm2/Vs)
104
SiO2
HfSiO4
Al2O3
HfO2
103
Si
teq=1 nm
102 11
10
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.4. Surface roughness limited mobility for Si substrate (using an exponential
distribution with step rms height ∆ = 0.1 nm and step correlation length Λ = 2.5
nm).
31
ELECTRON MOBILITY (cm2/Vs)
105
104
SiO2
HfSiO4
Al2O3
HfO2
Ge
teq=1 nm
103 11
10
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.5. Same as in Fig. 3.3 but for Ge substrate.
32
ELECTRON MOBILITY (cm2/Vs)
105
SiO2
HfSiO4
Al2O3
HfO2
104
103
102 11
10
Ge
teq=1 nm
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.6. Same as in Fig. 3.4 but for Ge substrate.
33
EOT = 1.0 nm
Si/SiO2
77K
300K
375K
104
103
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
ELECTRON MOBILITY (cm2/Vs)
ELECTRON MOBILITY (cm2/Vs)
105
105
77K
300K
375K
EOT = 1.0 nm
Si/HfO2
104
103
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
(a) SiO2
(b) HfO2
105
EOT = 1.0 nm
Ge/SiO2
77K
300K
375K
104
103
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
ELECTRON MOBILITY (cm2/Vs)
ELECTRON MOBILITY (cm2/Vs)
Figure 3.7. Total calculated mobility for Si substrates with both SiO2 and HfO2
gate dielectrics as the temperature is varied.
(a) SiO2
105
77K
300K
375K
EOT = 1.0 nm
Ge/HfO2
104
103
102
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
(b) HfO2
Figure 3.8. Same as in Fig. 3.7 but for a Ge substrate.
34
ELECTRON MOBILITY (cm2/Vs)
1100
1000
900
Ge
800
700
Metal Gate
Poly-Si Gate
600
T = 300K
HfO2
500
Si
400
300
0
1
2
3
4
5
EQUIVALENT OXIDE THICKNESS (nm)
Figure 3.9. Total calculated mobility for Si and Ge substrates with HfO2 . A comparison of metal and poly-Si gates as the equivalent oxide thickness is varied for a
sheet density of ns = 2 × 1011 cm−2 .
35
6
ELECTRON MOBILITY (cm2/Vs)
104
Ge
Si
103
102 11
10
EOT = 1.0 nm
T = 300K
HfO2
Metal Gate
Poly-Si Gate
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 3.10. Remote phonon limited mobility for Si and Ge substrates with HfO2 .
A comparison of metal and poly-Si gates as the electron sheet density is varied.
36
5
6
Si
HfO2
SiO2
3
IDS (103 µA/µm)
IDS (103 µA/µm)
4
Si
VDS = 0.2 V
15 nm
∆gm = 30%
2
30 nm
44%
60 nm
50%
1
VDS = 1.0 V
HfO2
SiO2
15 nm
∆gm = 17%
4
30 nm
∆gm = 19%
2
60 nm
10%
0
–0.2 –0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
VGS–VT,lin (V)
(a) Linear Region
0
–0.2 –0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
VGS–VT,sat (V)
(b) Saturated Region
Figure 3.11. Calculated current-voltage characteristics of Si MOSFETs with SiO2
(solid symbols) and HfO2 (open symbols) as gate insulators. The dotted lines are a
guide for the eyes to judge the transconductance. Note that SiO2 and HfO2 devices
exhibit a different threshold voltage at the same equivalent thickness due to fringingfield effects.
37
5
6
Ge
HfO2
SiO2
15 nm
∆gm = 9%
3
IDS (103 µA/µm)
IDS (103 µA/µm)
4
Ge VDS = 1.0 V
HfO2
SiO2
VDS = 0.2 V
30 nm
16%
2
15 nm
∆gm = 1%
4
30 nm
∆gm = 3%
2
60 nm
1
60 nm
∆gm = 8%
27%
0
–0.2 –0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
VGS–VT,lin (V)
(a) Linear Region
0
–0.2 –0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
VGS–VT,sat (V)
(b) Saturated Region
Figure 3.12. Same as in Fig.3.11, but for Ge devices.
38
CHAPTER 4
III-V SUBSTRATES
4.1
Introduction
The continuation of Moore’s law past the 22nm node challenges the conventional scaling methods practiced over the last few decades.[37] Many options exist
to continue scaling within Si technology; such as different device geometries (DoubleGates, Tri-gates, FinFets, nanowires, multi-bridge FETs), fully depleted SOI (FDSOI), strain, high-κ dielectrics, and orientation.[38, 39, 40, 41, 11, 42, 43] Besides
Si technology, another possible solution is to use high mobility substrates, such as
Ge and III-V semiconductors, to increase device performance. The use of electron
mobility, as the ballistic limit is approached, as marker for device performance is less
clear and gives us only part of the picture.[44, 45, 46] Our results show that as the
gate length is reduced, the electron mobility becomes less correlated with the device
performance measured as transconductance. This brings into question the validity
of the ‘ballistic mobility’ as a meaningful concept other than a fitting parameter for
drift-diffusion simulation [47, 48, 49]. Therefore we focus on electron mobility calculations for III-V materials while recognizing mobility as ‘one indicator among many’
of device performance.
In this work, scattering with longitudinal-optical (LO) phonons is included in the
mobility calculations for III-V substrates. Here we include dynamic multi-subband
screening within the random-phase approximation - with the dynamic screening parameter accounting for the inelastic process with energy transfer determined by the
energy of the bulk LO-mode. Static screening is compared to dynamic screening and
39
the results deviate by less than 5%. Previous works have dealt mainly with dynamic
screening in III-V heterostructures [50, 51]. Other work has either ignored screening,
or included screening through the use of a reciprocal screening length.[52, 53] An effective screening wavevector (reciprocal screening length) is defined in this work and
compared to the dynamic and static screening approximations. The effective screening is found to qualitatively reproduce the dynamic screening case in low and high
sheet densities while decreasing computation time considerably.
The LO phonon mode associated with III-V substrates induces a Surface-Optical
(SO) phonon at the dielectric/III-V interface. This mode, as well as the modes
induced from the dielectric LO-modes and coupled to interface plasmons, generates a
scattering potential in the substrate and reduces the electron mobility of the 2DEG.
In this work we extend the existing model in Ref. [2] to account for the substrate
mode and calculate the electron mobility for an ideal metallic gate. To calculate
the dispersion, we present the secular equation in terms of dielectric function in the
insulator and III-V substrate. The plasmon and phonon contents of each branch of
the dispersion are defined, then the scattering strengths and finally the momentum
relaxation rate is given.
We assume In0.53 Ga0.47 As throughout this chapter because it is nearly latticed
match to InP, a material that has recently been considered as a barrier layer in III-V
nMOSFET design. The Schrödinger and Poisson equations are solved self-consistently
and the mobility is calculated using the Kubo-Greenwood formula with nonparabolic
corrections. Scattering with LO phonons, SO phonons, bulk phonons (intra- and
inter-valley phonons) and surface roughness is included. We focus on the LO and
SO-limited mobilities for the III-V substrates and include bulk phonons (intra- and
inter-valley phonons) and surface roughness when comparing to Si and Ge.
40
4.2
Longitudinal-Optical Phonons
Scattering with polar LO-phonons is an additional scattering mechanism that
is not present in nonpolar semiconductors such as Si and Ge. Polar LO-phonons
are present in compound semiconductors where the vibration of different (Ga and
As for example) atoms results in an LO phonon with a specific energy. For IIIV semiconductors, LO-phonon scattering is stronger than nonpolar intra-valley and
inter-valley phonon scattering. The potential energy associated with the LO-phonon
scattering is given by Fröhlich:[54]
"
h̄ωLO
VLO (q) = e
2q 2
1
1
− 0
∞
ǫs
ǫs
!#1/2
eiq·r ,
(4.1)
0
where ǫ∞
s and ǫs are the optical and static permittivity of the substrate, respectively,
e the electron charge, and h̄ωLO the energy of the substrate LO phonon mode (≈ 35
meV for GaAs and ≈ 32.5 meV for In0.53 Ga0.47As). [55] The squared magnitude of
the matrix element between an initial state in subband µ and a final state in subband
ν, is:
!
2 e2h̄ωLO 1
1
ψ µK |VLO (q)|ψνK′ =
− 0
2q 2
ǫ∞
ǫs
s




 nLO 
|Fµν (qz )|2 δ(K − K′ − Q),
×


 1 + nLO 
(4.2)
where nLO is the Bose occupation of the LO phonons with the upper value for absorption and the lower value for emission. Applying Fermi’s Golden Rule, the scattering
rate for an electron in state |ψµK i scattering to state |ψνK′ i is:
Z
Z
2
dq
2π
dK′ 1
=
ψµK |HLO |ψνK′ 3
2
τµν (K)
h̄
(2π)
(2π)
× δ Eµ (K) − Eν (K′ ) ± h̄ωLO ,
41
(4.3)
with the upper sign for absorption and the lower sign for emission. Switching to polar
coordinates, and a change of variable form K ′ to E ′ , we obtain:

!
1 
e2 ωLO
1
=
τµν (K)
4πh̄2
×
Z
1
− 0
ǫ∞
ǫs
s
2π
dβ ′ mµ (β ′)
0
nLO




 1 + nLO 

∞
Z
−∞
(4.4)
dqz (ex) 2
ϕ
,
2π Q,µν
where the unscreened or ‘external’ potential is all terms with Q or qz dependence:
(ex)
ϕQ,µν = p
Fνµ (qz )
qz2 + Q2 (β ′ )
,
(4.5)
The integration over qz in Eq.4.4 can be carried out to obtain:

!
1 
1
e2 ωLO
=
τµν (K)
8πh̄2
×
Z
1
− 0
ǫ∞
ǫs
s
2π
dβ ′ mµ (β ′)
0
nLO




 1 + nLO 

(4.6)
′
Hµν [Q(β )]
,
Q(β ′ )
with
Q2 (β ′ ) = K 2 + K ′2 − 2KK ′ cos φ,
(4.7)
where β ′ is the angle between K′ and the direction of electron transport (we choose
kx arbitrarily), θ = β − β ′ is the angle between K and K′ , and the ‘form factor’ is:
[56]
′
Hµν [Q(β )] =
Z
∞
dz
0
Z
∞
dz ′ ζν (z)ζν (z ′ )
0
(4.8)
−Q|z−z ′ |
′
× ζµ (z )ζµ (z)e
42
.
We have thus far derived the LO-scattering rate. To turn the scattering rate into
a momentum relaxation rate, so that we can calculate the electron mobility, we need
to insert an additional term inside the integrals above:
#
"
x,tot x
τνK
v
1 − fν (E ′ )
′
′
,
1 − x,tot νK
x
1 − fµ (E)
τµK vµK
(4.9)
x,tot
where τµK
is the total momentum relaxation time including all scattering mechax
nisms, vνK
is the velocity along the x-direction, f (E) is the Fermi-Dirac distribution
function, and E ′ = E ± h̄ωLO is the final energy after absorption (+) or emission
(−) of the bulk LO phonon, h̄ωLO . For isotropic and inelastic scattering, Eq. (4.9)
simplifies to [1 − (K ′ /K) cos φ], but in the case of LO-phonons (and surface roughness scattering as well) this simplification is not strictly correct. However, we make
the simplification, Eq. (4.9) → [1 − cos φ], so that we do not have to solve the full
integral equation self-consistently and to avoid unphysical negative mobilities when
using [1 − (K ′ /K) cos φ].[21]
In the next section we will discuss screening and show results for the mobility
when the LO-phonons are unscreened and screened.
4.3
Multi-subband Screening
In the last section, the screening of LO-phonons was neglected. Here we describe
a general method of dynamic multisubband screening through the dielectric function
and complex dynamic screening parameter. For comparison, the elastic limit is taken
to recover static screening. In the static limit, the imaginary part of the dynamic
screening parameter and the dielectric function vanishes. Finally an effective screening wavevector is defined in terms of the magnitude squared of the dynamic screening
parameter.
43
4.3.1
Dynamic Screening
(ex)
In general, an external potential ϕQ (z) (in this case the LO-phonon scattering
(ind)
potential) will induce a charge δρQ (z ′ ) which will in turn induce a potential ϕQ
(z)
such that the net potential is:
(ex)
(ind)
ϕQ (z) = ϕQ (z) + ϕQ
(z),
(4.10)
where
(ind)
ϕQ (z)
=
Z
dz ′ GQ (z, z ′ )
δρQ (z ′ )
,
ǫs
(4.11)
where ρQ (z ′ ) is the charge density and the Green’s function GQ (z, z ′ ) is the solution
to:
!
d2
− Q2 GQ (z, z ′ ) = δ(z − z ′ ),
dz 2
(4.12)
which depends on the geometry of the problem. For the case of two semi-infinite
layers:[24]
"
#
ǫs − ǫox −Q|z+z ′|
1
−Q|z−z ′ |
e
+
.
e
GQ (z, z ) = −
2Q
ǫs + ǫox
′
(4.13)
Following Ref. [24], the net potential for intrasubband scattering can be found by
inverting the linear problem:
X
(ex)
ǫµµ,λλ (Q, ω)ϕQ,λλ = ϕQ,µµ ,
(4.14)
λ
where the dielectric matrix is:
(2D)
ǫµν,λλ (Q, ω) = δµν δλλ +
βλλ
ΩQ,µν,λλ .
Q
(4.15)
Ω is another form factor:
ΩQ,µν,λλ = 2
Z
dz
Z
dz ′ G̃Q (z, z ′ )ζµ (z)ζν (z)ζλ (z ′ )ζλ (z ′ ),
44
(4.16)
and G̃ = 2QG is the reduced Green’s function. Using the intrasubband net scattering
potentials, the intersubband scattering potentials are obtained directly:[24]
(ex)
ϕQ,µν = ϕQ,µν −
X
ǫµν,λλ ϕQ,λλ.
(4.17)
λ
This method speeds up the computation time and saves memory by ignoring the off
diagonal terms of the dielectric matrix, thereby ignoring inter-subband
The two-dimensional screening wavevector in the high temperature limit is:[24, 57]
( "
!1/2
!#
i
h
1/2
m
ω
h̄Q
π
µ
(2D)
Φ
+
= βDH
Re βλλ
Qlµ
2kB T
Q 2mµ
"
!1/2
!#)
mµ
h̄Q
ω
−Φ
,
−
2kB T
Q 2mµ
(4.18)
and
i
h
πh̄ω
(2D)
= βDH
Im βλλ
Qlµ kB T
h̄2 Q2
−mµ ω 2
−
2kB T Q2 8mµ kB T
× exp
!
sinh[h̄ω/(2kB T )]
,
h̄ω/(2kB T )
(4.19)
where βDH = e2 nµ /(2ǫs kB T ) is the two-dimensional Debye-Hückel limit of the static
screening wavevector, lµ = [2πh̄2 /(mµ kB T )]1/2 is the thermal wavelength of electrons
in subband µ, and Φ(y) = 2e−y
2
Ry
o
dtet
2
is the plasma dispersion function. For dynamic
screening, we set ω = ωLO in Eqs. (4.18) and (4.19).
4.3.2
Static Screening
For static screening, as is strictly correct for screeing surface roughness scattering,
we set ω = 0 to obtain:
"
#
i
h
1/2
2π
Ql
µ
(2D)
= βDH
Φ
Re βλλ
Qlµ
4π 1/2
45
(4.20)
and
i
h
(2D)
= 0.
Im βλλ
(4.21)
Including screening adds extra computational cost associated with calculating the
screened potential (inverting the dielectric matrix). For the unscreened case, Eq. (4.8)
can be tabulated as a function of Q for each sheet density and used with Eq. (4.6).
However, when including screening, the screened scattering potential must be used
with Eq. (4.4) where the explicit integration over qz slows down the computation.
4.3.3
Effective Screening Wavevector
A less computationally expensive method, which comes from assuming a scalar
dielectric “matrix”, to include screening is to define an effective screening wavevector:
qs =
X
Re(βQ,ωLO ;λλ )
(4.22)
λ
and modify the scattering potential:
(eff)
ϕQ,λλ′ = p
Fνµ (qz )
,
qz2 + Q2 + qs2
(4.23)
where qs is also referred to as the reciprocal screening length [53]. The effective
screening parameter, qs , increases with sheet density through the βDH term and the
scattering potential is subsequently decreased - leading to stronger screening and an
increase in mobility at high sheet densities.
4.4
4.4.1
Surface-Optical phonons
Dispersion
In this work we assume an MOS system with a metallic gate, high-κ dielectric, and
III-V substrate. The secular equation for this system is the solution of the Laplace
46
equation with the appropriate electrical boundary conditions at the interface:[6]
ǫox (ω)cosh(Qtox ) + ǫs (ω)sinh(Qtox ) = 0,
(4.24)
where tox is the physical oxide thickness. The SO mode originating from the substrate
is included in the dielectric function of the substrate as:
ǫs (Q, ω) =
ǫ∞
s
2
ωp,s
(Q)
1−
ω2
+ (ǫ0s − ǫ∞
s )
2
ωTO3
.
2
ωTO3
− ω2
(4.25)
where the first term in Eq. (4.25) is the substrate plasmon contribution and the second
term is due to the polar-optical phonon in the substrate. The plasmon frequency of
P
1/2
the two-dimensional electron gas (2DEG) is ωp,s (Q) = [ ν e2 nν Q/(ǫ∞
, where
s mν )]
nν and mν are the electron density and conductivity mass of subband ν, respectively,
and h̄ωT O3 the energy of the substrate transverse-optical (TO) phonon mode. The
notation ‘TO3’ is used to distinguish the substrate mode from the two insulator
modes. The dielectric function in the insulator is:
ǫox (ω) = ǫ∞
ox
2
2
(ωLO1
− ω 2 )(ωLO2
− ω2)
,
2
2
(ωTO1
− ω 2 )(ωTO2
− ω2)
(4.26)
where ǫ∞
ox is the optical permittivity of insulator and h̄ωTO1 and h̄ωTO2 are the energies
of the two insulator LO phonon modes.
For InGaAs, Eq. (4.25) could be extended as:
ǫs (ω) = ǫ∞
s
2
2
(ωLO3
− ω 2)(ωLO4
− ω2)
,
2
2
(ωTO3
− ω 2)(ωTO4
− ω2)
(4.27)
to account for two separate substrates LO-modes as measured experimentally [58].
However, to avoid complication, we lump the two closely spaced LO-modes into one
mode and use Eq. (4.25).
47
The solution of Eq. (4.24) yields four solutions with each dispersion constituting
a mixture of the substrate plasmon, the SO phonon originating in the substrate, and
two SO modes from the insulator. An example dispersion, a numerical solution of
Eq. (4.24) for SiO2 with GaAs at a sheet electron density ns = 2 × 1012 cm−2 , is
shown in fig. 4.2. In the next section we will show how to calculate the plasmon and
phonon contents of each branch of the dispersion.
4.4.2
Landau damping
Landau damping is approximately included by switching off the substrate plasmon
2
ωTO3
2]
TO3 −ω
0
∞
[i.e., setting ǫs (Q, ω) = ǫ∞
s + (ǫs − ǫS ) ω 2
when ωp,s ≤ ωLDs , where ωLDs =
[h̄Q/(2mt )](Q + 2Kf ) is the ‘boundary of the single-particle continuum of the 2DEG
in the extreme quantum limit’, Kf = (πns )1/2 , and mt is the transverse mass.[2] When
Landau damping is included in this way, Eq. (4.24) admits three solutions instead of
four - basically three unscreened SO-phonons.
4.4.3
Plasmon and Phonon Content
To define separate phonon contents of each mode, we consider three solutions
(−T O3,α)
ωQ
(α = 1, 2, 3) obtained from the secular equation, Eq. (4.24), by ignoring the
2
2
response of phonon 3, that is ǫs (Q, ω) → ǫ∞
s (1 − ωp,s (Q)/ω ). Extending the theory of
Ref. [2], the relative phonon-3 (the TO mode of the substrate) content of each branch
i (i = 1, 4) is given by:
(i)
RTO3 (ωQ )
(i)2
(ω − ω (−TO3,1)2 )(ω (i)2 − ω (−TO3,2)2 )(ω (i)2 − ω (−TO3,3)2 ) Q
Q
Q
Q
Q
Q
≈
2
2
2
2
2
2
(l)
(i)
(k)
(i)
(j)
(i)
(ω − ω )(ω − ω )(ω − ω )
Q
Q
Q
Q
and the TO-phonon-3 content of mode i is:
48
Q
Q
(4.28)
(i)
(TO3)
Φ
(i)
(ωQ )
≈
RTO3 (ωQ )
(i)
(i)
(i)
RTO1 (ωQ ) + RTO2 (ωQ ) + RTO3 (ωQ )
(4.29)
(i)
× [1 − Π(s) (ωQ )],
where the substrate plasmon content of mode i is:
(i)
Π(s) (ωQ )
(i)2
(ω − ω (−s,1)2 )(ω (i)2 − ω (−s,2)2 )(ω (i)2 − ω (−s,3)2 ) Q
Q
Q
Q
Q
Q
≈
,
2
2
2
2
2
2
(l)
(i)
(k)
(i)
(j)
(i)
(ωQ − ωQ )(ωQ − ωQ )(ωQ − ωQ )
(−s,α)
and i, j, k, l are cyclical. The three solutions ωQ
(4.30)
(α = 1, 2, 3) are obtained from
the secular equation, Eq. (4.24), by ignoring the substrate plasmon response, that is
0
∞
ǫs (Q, ω) → ǫ∞
s + (ǫs − ǫs )
2
ωTO3
.
2
ωTO3
− ω2
(4.31)
(i)
Following the the same logic, similar expressions are obtained for Φ(TO1) (ωQ ) and
(i)
Φ(TO2) (ωQ ).
Fig. 4.3 shows the relative phonon-3 content for the dispersion in Fig. 4.2. For
small Q, ω (3) is basically the TO3 mode, while for larger Q there is a crossover
with ω (4) gradually becoming the TO3 mode. Fig. 4.4 shows the substrate plasmon
content for the dispersion in Fig. 4.2. The substrate plasmon is “mixed” with all
three SO phonons across all four branches of the dispersion. For example, focusing
on the ω (4) curves in Figs. 4.2 and 4.4, as the substrate plasmon content decreases
the phonon 3 content increases in proportion. Including the substrate plasmon in the
dielectric function of the substrate inherently incorporates the screening effects of the
2D electron gas. However, scattering with the substrate plasmon is ignored because
there is no net effect on the momentum of the 2D electron gas.
49
4.4.4
SO Scattering Strength and Momentum Relaxation Rate
Following the arguments of Ref. [2], the “total” effective dielectric function for
the metallic gate/high-κ stack is: [6]
ǫTOT (Q, ω) = ǫsubstrate − ǫsub
ǫinsulator
ǫox
(4.32)
where ǫsubstrate and ǫinsulator are chosen depending on which response of the system,
ionic or electronic, we are considering.[2]
The scattering strength for optical modes 1 and 2 is obtained as follows. For mode
1 (and similarly for mode 2): setting ǫsubstrate (Q, ω) = ǫs (Q, ω), and ǫinsulator (ω) =
2
2
2
2
ǫ∞
ox (ωLO2 −ω )/(ωTO2 −ω ) (phonon 1 does not respond) results in the effective dielec(TO1)
tric function ǫTOT,high . Again setting ǫsubstrate (Q, ω) = ǫs (Q, ω), and now ǫinsulator (ω) =
2
2
2
2
2
ǫ∞
ox (ωLO2 − ω )/(ωTO2 − ω )(ωLO1 /ωTO1 ) (phonon 1 fully responds) results in the ef(TO1)
fective dielectric function ǫTOT,low . Then the scattering strength is:
(i) "
#
h̄ω
1
1
Q
(i) (TO1)
Φ
(ω
)
|AQ |2 = −
Q ,
(TO1)
2Q ǫ(TO1)
ǫ
TOT,high
TOT,low
(4.33)
and a similar equation for mode 2.
For the TO mode originating in the bulk: setting ǫinsulator (ω) = ǫox (ω), and
2
2
ǫsubstrate (Q, ω) = ǫ∞
s (1 − ωp /ω ) (phonon does not respond) results in the effective di(TO3)
electric function ǫTOT,high . Again setting ǫinsulator (ω) = ǫox (ω), and now ǫsubstrate (Q, ω) =
2
2
0
∞
ǫ∞
s (1 − ωp /ω ) + (ǫs − ǫs ) (phonon fully responds) results in the effective dielectric
(TO3)
function ǫTOT,low . Then the scattering strength is:
(i) "
#
h̄ω
1
1
Q
(i) (TO3)
Φ
(ω
)
−
|AQ |2 = Q .
(TO3)
2Q ǫ(TO3)
ǫ
TOT,high
TOT,low
(4.34)
Once we have the scattering strength, we are ready to calculate the momentum
relaxation rate:[2]
50
e2
1
=
τµν (K, β)
2πh̄3
Z
2π
dφ
0



nQ




 1 + nQ 

1 − f (E ′ )
(SO) 2
×
|AQ |2 |Fµν
|
1 − f (E)
K′
cos φ θ(E ′ − Eν ),
× 1−
K
(4.35)
where
(SO)
Fµν
(qz )
=
Z
∞
dzζµ (z)e−Qz ζν (z),
(4.36)
0
is the form factor, f (E) is the Fermi-Dirac distribution function, θ(x) is the step
function, and E ′ = E ± h̄ωQ is the final energy after absorption (+) or emission (−)
of an SO phonon of energy h̄ωQ .
51
Table 4.1. Parameters for GaAs
Emin Egap
(eV) (eV)
Γ 0.000 1.42
L 0.323 —–
X 0.447 —–
mL
mT
α
(mo ) (mo ) nonparabolicity factor
0.067 0.067
0.61
1.537 0.127
0.0
1.987 0.229
0.0
Table 4.2. Parameters for In0.53 Ga0.47 As
Emin Egap
mL
mT
α
(eV) (eV) (mo ) (mo ) nonparabolicity factor
Γ 0.000 0.728 0.048 0.048
1.22
L 0.767 —– 1.565 0.232
0.0
X 1.250 —– 2.258 0.253
0.0
52
DIELECTRIC MATRIX, ε11,11
2.5
2
1.5
Real Part
Imaginary Part
Magnitude
Static
1
0.5
0
-0.5
0.01
ns = 2 x 1012 cm-2
0.1
1
10
IN-PLANE WAVEVECTOR, Q (108 m-1)
100
Figure 4.1. The real part, imaginary part, and magnitude of the (11,11) element of
the dielectric matrix for dynamic screening.
53
160
ω(1)
DISPERSION (meV)
140
120
100
80
ω(2)
60
ω(3)
40
ω(4)
20
0
0
1
2
3
4
5
IN-PLANE WAVEVECTOR (108 m-1)
Figure 4.2. Dispersion for GaAs with SiO2 . This is the solution of Eq. 4.24 yielding
four mixed solutions (substrate plasmon and 3 TO-modes).
54
PHONON-3 CONTENT
1
0.8
ω1
ω2
ω3
ω4
0.6
0.4
0.2
0
0
1
2
3
4
5
IN-PLANE WAVEVECTOR (108 m-1)
Figure 4.3. The relative phonon-3 content of each branch of the dispersion in fig. 4.2.
55
SUBSTRATE PLASMON CONTENT Π(s)
1
0.8
ω(1)
ω(2)
ω(3)
ω(4)
0.6
0.4
0.2
0
0
1
2
3
4
5
IN-PLANE WAVEVECTOR Q (108 m-1)
Figure 4.4. The substrate plasmon content of each branch of the dispersion in
fig. 4.2.
56
CHAPTER 5
RESULTS AND DISCUSSION FOR III-V SUBSTRATES
5.1
Valley Occupation and LO-Limited Mobility
For GaAs, we see from Fig. 5.1 that the X-valley becomes more occupied than
the Γ-valley at sheet densities above 1013 cm−2 with the L-valley accounting for nearly
15% of the electron occupation. In table 4.1 we see that the X and L-valley minima
are raised above the Γ-valley in terms of energy. At low sheet densities nearly all
of the inversion electrons sit in the Γ-valley. However, at larger densities, strong
quantization pushes the energy levels of the small-mass Γ-valleys higher in energy
compared to the heavier X and L-valleys. This strong quantization, along with the
higher density of states, leads to a significant occupation of the X and L-valleys. If
nonparabolic corrections in the X and L-valleys were to be included, one would see a
larger occupation of the X and L-valleys at and at smaller ns .
For InGaAs, we see from Fig. 5.2 that the L-valley becomes significantly populated
compared to the Γ-valley at large electron sheet densities. Similarly to GaAs, this is
due to strong quantization in the Γ-valley compared to the L-valley which is the result
of the small quantization mass in the Γ-valley (0.048m0 compared to 0.32m0 in the
L-valley), which is similar to the result for the double-gate case.[59] The X-valleys do
not become significantly occupied due to the large energy offset between the minimum
of the X and Γ-valley. When the X and/or L-valley becomes significantly populated,
the mobility in InGaAs and GaAs will be determined by transport in the X and/or
L-valleys instead of the low-mass Γ-valley.
57
In Fig. 5.2 the X and L-valleys were assumed to be parabolic. Fig. 5.3 shows
the valley occupation if we now assume the X and L-valleys are nonparabolic with
α = 0.5ev−1 . In this case the L-valley becomes significantly occupied at a smaller
density than in Fig. 5.1. Because we do not have reliable data for the nonparabolicity
parameters, all the following results are computed assuming the case of Fig. 5.1. This
is considered a best case scenario as the Γ-valley is more strongly occupied.
Figure 5.4 compares the LO-phonon limited mobility for GaAs with an HfO2 gate
dielectric for the unscreened, static screened, dynamically screened, and effectively
screened cases. For the unscreened case, we observe a mobility decrease in correlation
with the increased occupation of the ’heavier’ X-valley. However, screening is stronger
in the X-valleys (large mass means cooler electrons and stronger screening) resulting
in a larger percentage mobility increase at large ns . Dynamic and static screening
show similar results because of the large range of Q for which the 11,11 element of
static dielectric matrix coincides with the real part (and magnitude) of the 11,11
element of the dynamic dielectric matrix (see Fig. 4.1). The effective screening case
approaches the dynamic screening result at small and large ns .
Figure 5.5 compares the LO-phonon limited mobility for InGaAs with an HfO2
gate dielectric for the unscreened, static screened, dynamically screened, and effectively screened cases. As expected, the LO-phonon-limited mobility for an InGaAs
substrate is larger than for a GaAs substrate - corresponding to the lighter effective
mass in the Γ-valley of InGaAs. The LO-limited mobility peaks at ns = 1013 cm−2
and then drops in correlation with the increasing L-valley occupation seen in Fig. 5.2.
In this case, the effective-screening result predicts a mobility significantly higher than
the mobilities calculated using the static and dynamic screening models.
58
5.1.1
SO-Limited Mobility
Figure 5.6 shows the SO-phonon-limited mobility for a GaAs and InGaAs substrate with HfO2 and SiO2 gate dielectrics. For both GaAs, and InGaAs, the mobility is suppressed from 30% to 50% when SiO2 is replaced by HfO2 . The low energy
phonons associated with HfO2 as compared to SiO2 are ’easier’ to excite giving a
larger Bose occupation and stronger scattering strength. Again, as in Figs 5.4 and
5.5, the mobility decreases with increasing occupation of the L-valley for InGaAs and
the X and L-valley for GaAs.
Figure 5.7 shows the SO-phonon-limited mobility for a GaAs and InGaAs substrate with HfO2 and SiO2 gate dielectrics as the temperature is varied. For temperatures between 200K and 500K, with an HfO2 dielectric and GaAs or InGaAs,
the mobility remains roughly constant. For SiO2 with GaAs or InGaAs the mobility
increases. As mentioned above, HfO2 has lower energy phonons which are more easily excitable even as the temperature approaches 200K. Below 200K, HfO2 and SiO2
exhibit roughly the same temperature dependence. These results are qualitatively
similar to experimental data for GaAs-GaAlAs heterostructures. [60]
5.1.2
Comparison to Silicon
Figure 5.8 shows the SO-limited, SR limited, bulk phonon (BP)-limited, and total
calculated mobility for an InGaAs substrate with a HfO2 gate dielectric. The BPlimited mobility includes scattering with screened LO-phonons, intravalley phonons,
and intervalley phonons with LO-scattering dominating. Ando’s model is used to
include SR-scattering with static multisubband screening included.[24] We see that
SR-scattering dominates at large ns when the confinement enhances the scattering
with the rough interface. At small ns , SO-phonon scattering is dominant because of
the low-energy phonons associated with HfO2 . LO-scattering is a significant contri-
59
bution to the total mobility at small to mid-range values of ns . Screening at large ns
diminishes the importance of both LO and SO phonons and SR-scattering dominates.
Figure 5.9 compares the total calculated mobility for a Si, GaAs, and InGaAs
substrate. For each substrate material, a HfO2 gate dielectric is assumed. InGaAs
has an order of magnitude improvement in electron mobility over Si and roughly a
factor of two improvement over GaAs. The promise of the large mobility of InGaAs
based devices is offset by the difficulty in integrating InGaAs in to Si processing as
well as the difficulty in fabricating high-κ gate dielectrics on III-V substrates.
60
5.2
Conclusion
The LO-limited electron mobility was calculated accounting for multisubband
dynamic screening. We have shown that including screening is necessary to accurately
calculate the LO-limited mobility. Dynamic and static screening yield nearly identical
results with dynamic screening adding minimal computational cost over the static
case. Defining an effective screening parameter captures the qualitative aspects of
dynamic screening with the advantage of significant computational savings.
The SO-limited mobility was calculated for GaAs an InGaAs substrates with SiO2
and HfO2 gate dielectrics. The dispersion was calculated with the extra substrate
LO-mode accounted for in the substrate dielectric function. The existing model was
extended to account for scattering with the SO-mode originating from the substrate
LO-mode. The SO-limited mobility was also plotted as a function of temperature.
A comparison of InGaAs and GaAs to Si was made, with both InGaAs and GaAs
showing a significant moblty increase over Si.
61
1
0.9
Valley Occupation
0.8
0.7
0.6
0.5
Γ-valley
X-valley
L-valley
0.4
0.3
0.2
0.1
0
1011
1012
1013
ELECTRON SHEET DENSITY IN THE CHANNEL (cm-2)
Figure 5.1. Normalized valley occupation for a single-gate structure with HfO2 and
GaAs.
62
1
0.9
Valley Occupation
0.8
0.7
0.6
0.5
Γ-valley
X-valley
L-valley
0.4
0.3
0.2
0.1
0
11
12
13
10
10
10
-2
ELECTRON SHEET DENSITY IN THE CHANNEL (cm )
Figure 5.2. Normalized valley occupation for a single-gate structure with HfO2 and
InGaAs. The and L-valleys are assumed to be parabolic.
1
0.9
Valley Occupation
0.8
0.7
0.6
0.5
Γ-valley
X-valley
L-valley
0.4
0.3
0.2
0.1
0
11
12
13
10
10
10
-2
ELECTRON SHEET DENSITY IN THE CHANNEL (cm )
Figure 5.3. Normalized valley occupation for a single-gate structure with HfO2 and
InGaAs. The X- and L-valleys ar assujmed to be nonparabolic with α = 0.5ev−1
63
LO-PHONON LIMITED MOBILITY (cm2/Vs)
dynamic screening
static screening
effective screening
uncreened
T = 300K
GaAs
HfO2
104
1011
1012
1013
ELECTRON SHEET DENSITY IN THE CHANNEL (cm-2)
Figure 5.4. LO-phonon limited mobility for the unscreened, static screened, dynamic
screened and effective screened cases for HfO2 with GaAs.
64
LO-PHONON LIMITED MOBILITY (cm2/Vs)
T = 300K
InGaAs
HfO2
104
dynamic screening
static screening
effective screening
uncreened
1011
1012
1013
ELECTRON SHEET DENSITY IN THE CHANNEL (cm-2)
Figure 5.5. LO-phonon limited mobility for the unscreened, static screened, dynamic
screened and effective screened cases for HfO2 with InGaAs.
65
SO-LIMITED MOBILITY (cm2/Vs)
10
6
InGaAs SiO2
InGaAs HfO2
GaAs SiO2
GaAs HfO2
105
104
103
1011
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 5.6. SO-phonon limited mobility for a metallic gate with GaAs and InGaAs
substrates with SiO2 and HfO2 gate dielectrics as a function of electron sheet density.
66
SO-LIMITED MOBILITY (cm2/Vs)
10
5
InGaAs SiO2
InGaAs HfO2
GaAs SiO2
GaAs HfO2
104
103
102
103
TEMPERATURE (K)
Figure 5.7. SO-phonon limited mobility for a metallic gate with GaAs and InGaAs
substrates with SiO2 and HfO2 gate dielectrics as a function of temperature.
67
ELECTRON MOBILITY (cm2/Vs)
10
5
104
103 11
10
µSO
µBP
µSR
µtotal
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 5.8. The scattering-limited and total mobility for InGaAs substrate with
HfO2 gate dielectic.
68
ELECTRON MOBILITY (cm2/Vs)
10
4
InGaAs
GaAs
Si
103
102 11
10
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 5.9. Comparison of the total calculated mobility for a Si, GaAs, and InGaAs
substrate with HfO2 gate dielectric.
69
CHAPTER 6
ELECTRON MOBILITY:
THE SYMMETRIC DOUBLE GATE
There are many published investigations of semiconductor thickness effects and
volume inversion on electronic transport in DG structures [61, 62, 63, 64]. The decrease in electron mobility with decreasing film thickness is observed and attributed to
various scattering mechanisms. One important mechanism, scattering with surfaceoptical phonons, has either been neglected, or the interfaces have been uncoupled and
screening neglected [65]. Previous studies have examined SO-phonon scattering in SG
MOSFET structures with poly-Si and metallic gates [2, 5, 21, 45, 66, 67]. Here, we
advance the theory to deal with the coupling of SO-phonons from two insulator/Si
interfaces. Screening is included by derivation of the substrate plasmon frequency
within the RPA approximation. Similar work has been published dealing with SOphonons in III-V quantum wells [68, 69, 70, 71]. That work has dealt mostly with
optical properties of III-V quantum wells and ignored screening.
The treatment of surface roughness scattering in DG geometries has recently been
published in Ref. [72]. We utilize that theory to make a more complete study of the
electron mobility in thin films. Scattering with bulk phonons is also included and the
resulting electron mobility in the Si channel is investigated as the channel thickness,
tsi , and electron sheet density, ns , is varied for SiO2 and HfO2 gate dielectrics.
The gate insulators are approximated as semi-infinite slabs, sketched in Fig. 6.1,
with the Si channel defined between 0 < z ≤ tsi and the insulators defined for z ≤ 0
and z > tsi . In this work we assume a symmetric DG structure with identical SiO2
or HfO2 gate dielectrics.
70
This chapter is organized as follows. First we briefly review the DG Surface
Roughness model of Ref. [72]. Next we derive the full secular equation through the
use of the electric boundary conditions. Then we show how the problem can be
simplified by separating the SO-scattering potential into even and odd parts. Next
we derive the even and odd ‘scattering strengths’ by deriving the total energy of the
system and defining even and odd ‘total dielectric functions’. The substrate dielectric
function is then discussed in terms and the substrate plasmon frequency is derived
within the RPA approximation. Next results and discussion are presented and finally
conclusions.
6.1
Surface Roughness
Ando’s surface roughness model for SG MOSFETs, Refs. [73, 74, 27], was recently
generalized to handle DG SOI structures with parabolic bands [72]. The matrix
element for surface roughness scattering can be written as follows:
GPN
SR
+ ΓQ
VµKνK
′ = Γµν
µν ∆Q ,
(6.1)
where ΓGPN
is the generalized Prang-Nee term (GPN), and ΓQ
µν
µν includes the Coulomb
scattering terms. The power spectrum representing the surface roughness is given by
the exponetnial model [28]:
−3/2
|∆Q |2 = π∆2 Λ2 1 + Q2 Λ2 /2
,
(6.2)
where ∆ = 0.48 nm is the RMS step height and Λ = 0.13 nm is the correlation length.
Because we assume the roughness of the top and bottom interface is uncorrelated, and
beacuse we assume a symmetric DG, we need only to calculate the matrix elements
at the top interface and multiply the momentum relaxation rate by a factor of two.
71
The GPN term, which acounts for the change in kinetic energy with thickness
fluctuations due to the surface roughness is given by:
ΓGPN
µν
z ∂ζν
= (Eµ − Eν ) dzζµ 1 −
tsi ∂z
2
Z
h̄ ∂ 1 ∂
z
∂V
+ dzζµ −
1−
ζν .
+
tsi ∂z mz ∂z
∂z
tsi
Z
(6.3)
In our case we assume the wavefunction does not penetrate the oxide and allows us
to write Eq. (6.3) as:
ΓGPN
µν
z ∂ζν
= (Eµ − Eν ) dzζµ 1 −
tsi ∂z
Z
2 Z
h̄
∂ζµ ∂ζν
z
∂V
dz
1−
ζν .
+
+ ζµ dz
tsi mz
∂z ∂z
∂z
tsi
Z
(6.4)
The Coulomb term is broken into two terms, where the the third term due to the
change in image potential is ignored because the image potential was not included in
the self-consitent Schrödinger-Poisson solver.
ΓQ
µν
=
Z
dzζµ (z) VQµ (z) + VQσ (z) ζµ (z)
(6.5)
The first term is due to the redistribution of charge along the z-direction:
VQµ (z)
e2
=−
ǫsi
Z
′
′
dz GQ (z, z )
z′
∂n(z ′ ) n(z ′ )
,
−1
+
tsi
∂z ′
tsi
(6.6)
where the Green’s function in the Si body can be written as [75]:
1
1
GQ (z, z ′ ) = −
2
2Q 1 − ǫ̃ e−2Qtsi
i
h
−Q|z−z′ |
−Q|z+z′ |
−Q(2tsi −|z+z′ |)
2 −Q(2tsi −|z−z′ |)
.
× e
+ ǫ̃e
+ ǫ̃e
+ ǫ̃ e
72
(6.7)
The second term is due to the interface polarization at the Si/High-κ interface caused
by the mismatch in dielectric constants:
VQσ (z) = eǫ̃Fs
e−Qz + ǫ̃eQ(z−2tsi )
1 − ǫ̃2 e−2Qtsi
(6.8)
where Fs is the electrical field at the Si surface.
Up to this point the bands have been assume to be parabolic. Following Ref. [32],
nonparabolic corrections leave the coulomb pieces unchanged but alter the GPN as
follows. The intra-subband terms are given by:
ΓGPN
µ,µ
1
=q
0)
1 + 4α(Eµ − Uµ )
2 (0)
(E − Uµ ) +
tsi µ
Z
∂V
ζµ dz
∂z
z
1−
ζν ,
tsi
(6.9)
and the inter-subband terms are approximated as:
ΓGPN
µ,ν
∂ζµ ∂ζν
= sign
∂z ∂z
GPN
ΓGPN
µ,µ Γν,ν
z=0
1/2
,
(6.10)
(0)
where Eµ are the parabolic subband energy minima, and Uµ is the expectation value
of the electrostatic potential.
Degenerate screening is included as in Ref. [24] with the degenerate screening
parameter[72]:
T
βQ,µµ
"
#
p
Z z
1 − x/z
e2 ρµ
=
1 + tanh(y) −
dx
,
2ǫsi
cosh2 (x − y)
0
(6.11)
where ρµ = gµ mµ /πh̄2 is the density of states effective mass in the µth subband, mµ
is the density of states effective mass, y = (EF − E)/(2kB T ), z = h̄2 Q2 /(16mv kB T ),
and mv is the conductivity effective mass.
73
6.2
6.2.1
Surface Optical Phonons
Secular Equation
Starting from the general solution of the scattering potential, Eq. 2.27, the general
form for the scattering potential in the DG structure is:



aeQz
(z ≤ 0)



φQ (z) =
beQz + ce−Qz (0 < z ≤ tsi )




 de−Qz
(z > tsi )
(6.12)
The parallel, E|| , and transverse, E⊥ , fields associated with the electrostatic potential
are:
E|| = −∇R φ(R, z) =
E⊥ = −
X
(−iQ)φQ (z)eiQ·R ,
(6.13)
Q
X dφQ (z)
∂
φ(R, z) = −
eiQ·R .
∂z
dz
Q
(6.14)
The electrostatic boundary conditions on the electric field give four equations with
four unknowns, a, b, c, d:
E|| (0+ ) = E|| (0− )
→ a=b+c
D⊥ (0+ ) = D⊥ (0− ) → ǫs (b − c) = aǫox
D⊥ (t+ ) = D⊥ (t− )
(6.15)
→ ǫs (beQtsi − ce−Qtsi ) = −de−Qtsi ǫox
Setting the determinant to zero of the matrix defined by the system of equations
above, results in the secular equation:
ǫ2s (Q, ω) + ǫ2ox (ω) + 2ǫs (Q, ω)ǫox (ω) coth(Qtsi ) = 0,
(6.16)
where the dielectric function in the insulator is:
ǫox (ω) = ǫ∞
ox
2
2
(ωLO1
− ω 2 )(ωLO2
− ω2)
,
2
2
(ωTO1
− ω 2 )(ωTO2
− ω2)
74
(6.17)
and the dielectric function in the substrate is:
ǫs (Q, ω) =
ǫ∞
s
2
ωp,s
(Q)
1−
.
ω2
(6.18)
ǫ∞
ox is the optical permittivity of insulator and h̄ωTO1 and h̄ωTO2 are the energies of
2
the two insulator LO phonon modes. The plasmon frequency, ωp,s
(Q), in Eq. (6.18)
accounts for screening of the SO-phonons and is discussed below.
6.2.2
Even and Odd Dispersions
The secular equation, Eq. (6.16), can be factored, resulting in even and odd solutions. Using the continuity of the potentials at 0 and tsi , the even, (e), and odd, (o),
scattering potentials can be written in terms of the overall normalization factors ae
and ao :
and



ae eQz
(z ≤ 0)



eQz +e−Q(z−tsi )
φeQ (z) =
(0 < z ≤ tsi )
a
e
1+eQtsi




 a e−Q(z−tsi )
(z > tsi )
e



ao eQz
(z ≤ 0)



Qz
−Q(z−t
)
si
e −e
φoQ (z) =
(0 < z ≤ tsi )
ao
1−eQtsi




 −a e−Q(z−tsi )
(z > tsi )
o
(6.19)
(6.20)
using the boundary conditions in Eq. (6.15), we can write the secular equation for
the even and odd scattering potentials:
ǫs + χe ǫox = 0
(6.21)
ǫs + χo ǫox = 0
(6.22)
and
75
where χe = coth(Qtsi /2) and χo = tanh(Qtsi /2). Eqs. (6.21) and (6.22) are cubic
(i)
(i)
equations in ω 2 resulting in three even and three odd solutions, ωe and ωo where
i = 1, 2, and 3.
Before we include the substrate plasmon, it is useful to first ignore screening effects
in order to understand the coupling of the two interfaces as a function of Si thickness.
Figure 6.2 shows the unscreened dispersion, which is found by solving the secular
equation with ǫs = ǫ∞
s . There are two doubly degenerate SO-modes at large Q. As
Q → 0 the degeneracy is broken and the SO-modes transition into two ‘LO-like’ and
‘TO-like’ modes for which ωLO1 > ωTO1 > ωTO1 > ωTO2 . As in Ref. [68], we refer to
these modes as ‘LO-like’ and ‘TO-like’ interface phonons because they aproach the
bulk phonon frequencies. The LO1 and TO1 modes are denoted ωe−s,i and similarly
the LO2 and TO2 modes ωo−s,i with i = 1, 2.
6.2.3
Substrate Dielectric Function and 2D Plasmon
We write the screened potential as:
φQ,ω (z)
(0)
=φQ,ω (z)
×
X
+
Z
∞
0
(0)
dz ′ GQ (z, z ′ )
βµ,µ′ (Q, ω)φQ,ω;µµ′ ζµ (z ′ )ζµ′ (z ′ ).
(6.23)
µ,µ′
where the “free” Green’s function is:
′
(0)
GQ (z, z ′ )
e−Q|z−z |
.
=−
2Q
(6.24)
We will use the the boundary condition on the perpendicular component on the Dfield across the boundary at z = 0. Before we use the boundary condition, we first
derive the D-field in the Si body using
Dz,Q,ω (z = 0+ ) = −ǫ∞
s
76
dφQω (z = 0+ )
.
dz
(6.25)
Inserting Eq. (6.23) into Eq. (6.25), for the even (top sign) and odd (bottom sign)
potential, Eq. (6.41) and (6.42), we obtain
ae Q
(1 ∓ eQtsi )
1 ± eQtsi
X
ae
(0)
βµµ′ Ψe,o
+ ǫ∞
s
Qµµ′ , ΦQ,µµ′
Qt
si
2 (1 ± e ) µµ′
Dz,Q,ω (z = 0+ ) = − ǫ∞
s
(6.26)
where the form factor is:
(0)
ΦQ,µµ′
=
Z
tsi
′
dz ′ e−Qz ζµ (z ′ )ζµ′ (z ′ ),
(6.27)
0
and we have approximated the screened matrix element as the matrix element of the
unscreened potential:
ae
Ψe,o ′
1 ± eQtsi Qµµ
Z tsi
≡
dz eQz ± e−Q(z−tsi ) ζµ (z)ζµ′ (z).
φQµµ′ →
Ψe,o
Qµµ′
(6.28)
(6.29)
o
Equation (6.26) can be manipulated and written as:
Dz,Q,ω (z = 0+ ) = −ǫ̃s
ae Q
Qtsi
1
∓
e
1 ± eQtsi
(6.30)
with
ǫ̃s = ǫ∞
s
Noticing that:
#
X βµµ′ e,o
±1
(0)
.
Ψ ′Φ
1 + Qt
′
(e si ∓ 1) ′ 2Q Qµµ Q,µµ
"
(6.31)
µµ
Dz,Q,ω (z = 0− ) = −ǫox (ω)ae Q,
(6.32)
we can use the third boundary condition in Eq. (6.15) to write:
ǫox = ǫ̃s
1 ∓ eQtsi
,
1 ± eQtsi
77
(6.33)
or for the top sign
ǫ̃s + ǫox coth(Qtsi /2) = 0,
(6.34)
ǫ̃s + ǫox tanh(Qtsi /2) = 0,
(6.35)
and the bottom sign
which correspond precisely to Eqs. (6.21) and (6.21), the secular equations for the
even and odd scattering potentials respectively.
Ignoring inter-subband plasmons and taking the long wavelength limit allows us
to write Eq. (6.31) as:
"
X e2 nµ Q
±1
(0)
ǫ̃s = ǫ∞
1
−
Ψe,o
s
Qµµ ΦQ,µµ
∞
Qt
2
si
∓ 1) µ 2ǫsi mµ ω
(e
#
"
2
ω
e (Q)
P;
o
,
= ǫ∞
1−
s
ω2
#
(6.36)
where we have defined the 3D-like substrate plasmon for the even modes:
2
ωP;e
(Q) =
1
(eQtsi
X e2 nµ Q
(0)
ΨeQµµ ΦQ,µµ
− 1) µ 2ǫ∞
m
µ
si
(6.37)
and the 3D-like substrate plasmon for the odd modes:
2
ωP;o
(Q)
X e2 nµ Q
−1
(0)
ΨoQµµ ΦQ,µµ = 0.
= Qt
∞
si
(e
+ 1) µ 2ǫsi mµ
(6.38)
2
We notice that ωP;o
(Q) = 0 because ΨoQµµ is an integration over an odd potential and
an even function, ζµ (z)2 . This means the odd scattering potentials remain unscreened
because we have only considered the intra-suband plasmons. Equation (6.37) is a 3Dlike plasmon because it was derived in the absence of the second substrate-dielectric
interface. If we now plug this expression into the secular equation for the even modes
78
and set ǫox (ω) = ǫ0ox (the plasmon generally responds a lower frequencies than the
phonons) we obtain the 2D-like plasmon:
ωP;e(Q)
ωP2D;e (Q) = q
0
1 + χe ǫǫox
∞
(6.39)
s
A similar derivation for plasmon in the SG case gives:
2
ωP;SG
(Q) =
X e2 nµ Q (0) 2
ΦQ,µµ
2ǫ∞
si mµ
µ
(6.40)
Figure 6.3 shows the 3D plasmon, Eq. (6.37), the 2D plasmon, Eq. (6.39), and
the boundary of the single-electron phase space ωLDs . The 2D plasmon goes to zero
at small Q because it can not respond at long wavelengths whereas the 3D plasmon
does not have this restriction. This is seen more clearly in Fig. 6.4 where the 3D and
2D plasmon frequencies are plotted for tsi from 1 to 10 nm. For smaller tsi , the 2D
plasmon increases at Q = 0, while the 2D plasmon always goes to 0, regardless of tsi .
Figure 6.5 shows the full dispersion, three even solutions and two odd solutions.
(i)
First, let us focus on the ωe set of solutions in the Qtsi → 0 limit. This limit implies
that χe → 0 which occurs when either, ǫox → 0 or ǫs → ∞. From Eq. (6.17), ǫox → 0
(1)
(1)
when ωe → ωLO1 and ωe → ωLO2 . From Eq. (6.18), ǫs → ∞ when ω → 0.
(i)
Now, let us focus on the ωo set of solutions in the Qtsi → 0 limit. This limit
implies that χo → ∞ which occurs when either, ǫox → ∞ or ǫs → 0. From Eq. (6.17),
(1)
(1)
ǫox → ∞ when ωo → ωTO1 and ωo → ωTO2 . From Eq. (6.18), ǫs → 0 when ω → ωp;o
but ωp;o = 0 so there are only two odd solutions because the opposite charges at the
two interfaces cancel in the Qtsi → 0 limit.
In the Qtsi → ∞ limit, as is seen in Fig. 6.6, χe → 1 and χo → 1 which collapses
the five solutions to two degenerate solutions that correspond to what is found in
the unscreened single interface limit. The fifth solution goes to zero because the
79
plasmon frequency goes to zero (see Fig. 6.3). The plasmon frequency goes to zero
because it was derived using an expansion in Q where only the first order terms were
retained. This unphysical solution is luckily not an issue because when we include
Landau damping the response of the substrate plasmon is ignored at Q’s generally
much smaller than where the plasmon frequency begins to decrease.
Figures 6.7 and 6.8 show the dispersion when the LO1 and LO2 modes are ignored,
respectively. In both cases, the first even mode goes to the LO-mode of the remaining
phonon, while the odd mode approaches the TO-mode as Q → 0. The second even
mode goes to zero for the same reasons discussed above.
6.3
Total Dielectric Function
The even and odd scattering potentials felt by electrons in the substrate (0 < z ≤
tsi ) are given by Eq. (6.19) and (6.20) respectively:
eQz + e−Q(z−tsi )
φeQ (z) = ae
1 + eQtsi
(6.41)
eQz − e−Q(z−tsi )
φoQ (z) = ao
1 − eQtsi
(6.42)
and
The potentials above are defined up to an overall factor, ae for the even potential and
ao for the odd potential. The purpose of this section is to calculate in a semiclassical
way the ‘scattering strengths’, ae and ao , by calculating the total energy of the system
and defining even and odd ‘total dielectric functions’.
To calculate the total energy of the system, we first calculate the polarization
charge resulting from the even potential, using the boundary condition of the D-field
across the interface z = 0:
ρQ,ω (z) = D⊥ (z = 0− ) − D⊥ (z = 0+ )
80
(6.43)
1
(6.44)
ρQ,ω (z) = −δ(z)Qae ǫinsulator + ǫsubstrate
χe
E
D
(i)
The total energy associated with the scattering field, WQ , is twice the time averE
D
(i)
aged potential energy, UQ .
D
(i)
WQ
E
E
D
(i)
= 2 UQ
Z
Z ∞
2
e
=
dR
dzφQ (R, z, t)ρQ,ω (R, z, t) ,
Ω
Ω
−∞
(6.45)
where h·i denotes time average, and Ω is a normalization area. Using Eq. (6.19) and
Eq. (6.45), we obtain
E
D
1
(i)
2
.
WQ = 2ae Q ǫinsulator + ǫsubstrate
χe
(6.46)
(i)
The total energy is set equal to the zero-point energy, h̄ωQ /2. This determines the
scattering strengths, ae and similarly ao , respectively:
(i)
(i)
h̄ωe
ae =
2QǫeTOT
h̄ωo
ao =
2QǫoTOT
1/2
,
(6.47)
1/2
.
(6.48)
The total effective dielectric function for the even potential is:
ǫeTOT (Q, ωe(i) )
=2
1
ǫsubstrate + ǫinsulator ,
χe
(6.49)
and the total effective dielectric function for the odd potential is:
ǫoTOT (Q, ωo(i) )
=2
1
ǫsubstrate + ǫinsulator
χo
(6.50)
Figure 6.9 shows the even and odd nature of the SO-scattering potential. Depending on the subband, and thus the even or odd nature of the wavefunction, there is a
81
cancelation effect when calculating the matrix element. For instance, the ground state
(i)
wavefunction is an even function and will couple strongly to the even ωe modes and
(i)
weakly to the odd ωo modes. On the other hand, the wavefunction of the first excited
state is an odd function and will couple strongly to the odd scattering potential.
6.4
Results and Discussion
Figure 6.10 shows the first four energy levels in the unprimed valley as a function
of tsi for ns = 2 × 1012 cm−2 . For long tsi the four energy levels converge to two
degenerate levels. At shorter tsi the quantization effect breaks the degeneracy. This
corresponds to two separate triangular wells converging to a single square well as tsi is
shortened. Figure 6.11 shows the relative electron occupation of the first two energy
levels of Fig. 6.10. At ts i = 2 nm, nearly 100% of the inversion electrons occupy the
ground state. This illustrates the strong quantization effect in thin DG structures
as well as the smooth transition from the square quantum well to two uncoupled
triangular wells.
Figures 6.12 and 6.13 show the unscreened SO-limited mobility as a function of
tsi for ns = 4 × 1011 cm−2 and ns = 2 × 1012 cm−2 , respectively. In both Figures
the SO-limited mobility coincides with the SG limit at large tsi . The mobility for
the SG case is computed at half the electron density of the DG case because the
electron density in the DG case is divided between the two triangular wells at large
tsi . For both Figures, the SO-limited mobility is strongly reduced at short tsi because
of the increasing scattering potential in Fig. 6.9 as well as the strong overlap of the
wavefunctions in the matrix element.
Figure 6.14 a comparison of the SO-limited mobility for an HiO2 gate dielectric
with and without screening. For both sheet densities, including screening (substrate
plasmon) is vital as the screened SO-limited mobility is about an order of magnitude
larger that the unscreened SO-limited mobility.
82
Figure 6.15 shows the SO-limited mobility, including the substrate plasmon, as a
function of Si thickness for an SiO2 and HiO2 gate dielectric. For both sheet densities,
ns = 2×1012 cm−2 and ns = 1×1013 cm−2 , the SiO2 dilectric yields significantly higher
(≈ 2 orders of magnitude) SO-limited mobilities. As is the case for SG structures, the
energies of LO-modes of the SiO2 insulator are higher than for the HfO2 insulator.
This means that when SiO2 is used for DG structures, SO-phonon scattering can
be ignored as the mobility will be dominated by surface roughnes and bulk phonon
scattering. We notice the mobility increases for the larger sheet density for the SiO2
case but decreases for the HfO2 case. This is due to the complex interplay of the
substrate plasmon and the confinement effect.
Figures 6.16 and 6.17 show the total and scattering-limited mobilities as a function
of Si thickness for an HfO2 gate dielectric at ns = 2×1012 cm−2 and ns = 1×1013 cm−2 ,
respectively. In both cases, surface roughness scattering is the dominant scattering
mechanism below 2 nm. At 3 nm, SO-phonon scattering becomes as strong as surface
scattering. This is an importnant result, as DG MOSFETs will likely be fabricated
in the 3-5 nm Si thicknes range to achieve a gate length in the range of 12-15 nm.
Figure 6.18 shows the total and scattering-limited mobilities as a function of electron sheet density for an HiO2 gate dielectric at tsi = 3 nm. The bulk phonons are the
dominate scattering mechanism with SO and SR-limit mobilities 2-3 times higher. At
higher sheet densies we see that SO-phonons and SR are comparable in importance.
83
Figure 6.1. The semi-infinte Double Gate structure studied in this chapter.
84
60
ENERGY (meV)
50
ωe
(-s,1)
ωo
(-s,1)
ωSO1
40
ns=2x1012 cm-2
30
(-s,2)
20
ωe
10
ωo
(-s,2)
ωSO2
tsi=3 nm
HfO2
0
0
2
4
6
8
10
12
14
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.2. Solution to Eq. (6.21) and (6.22) when the substrate plasmon is ignored.
85
100
ωP;e
ω2DP;e
ωLDs
ENERGY (meV)
80
60
40
tsi=3 nm
20
ns=2x1012 cm-2
0
0
2
4
6
8
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.3. The 3D and 2D plasmon frequency.
86
10
tsi=1 nm
ENERGY (meV)
100
tsi=10
tsi=1
10
tsi=10 nm
ns=2x10
12
ωP;e
-2
ω2DP;e
cm
1
0
0.5
1
1.5
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.4. The 3D and 2D plasmon frequency for different tsi .
87
2
70
(i)
ωe
(i)
ωo
(1)
ωe
ENERGY (meV)
60
50
40
ωo
(1)
ωe
(2)
30
20
ωo
(2)
10
ωe
(3)
ns=2x1012 cm-2
0
1
2
3
4
5
6
tsi=3 nm
HfO2
7
9
8
10
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.5. Solution to Eqs. (6.21) and (6.22) showing the even and odd sets of
solutions when the substrate plasmon is included.
88
70
ωe
(1)
ωo
(1)
ωe
(2)
(i)
ωe
(i)
ωo
ENERGY (meV)
60
50
40
tsi=3 nm
HfO2
30
ns=2x1012 cm-2
20
10
ωo
(2)
ωe
(3)
0
0.1
1
10
100
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.6. Same as Fig. 6.5 except the x-axis is in a log scale to illustrate the large
Q limit.
89
70
ωe
(-TO1,1)
ωo
(-TO1,1)
ωe
(-TO1,2)
ENERGY (meV)
60
50
40
30
20
ns=2x1012 cm-2
tsi=3 nm
10
HfO2
0
0
2
4
6
8
10
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.7. Solution to Eqs. (6.21) and (6.22) when the insulator phonon LO1 is
ignored.
90
60
ENERGY (meV)
50
ωe
40
(-TO2,1)
ns=2x1012 cm-2
tsi=3 nm
30
HfO2
20
10
ωo
(-TO2,1)
ωe
(-TO2,2)
0
0
2
4
6
8
10
IN-PLANE WAVEVECTOR (108 m-1)
Figure 6.8. Solution to Eqs. (6.21) and (6.22) when the insulator phonon LO2 is
ignored.
91
NORMALIZED SCATTERING POTENTIAL
1.5
Q=8x106 cm-1
ns=2x1012 cm-2
1
0.5
0
-0.5
-1
φe/ae
φo/ao
-1.5
0
5
10
15
Si BODY THICKNESS (nm)
20
Figure 6.9. The normalized scattering potential as tsi is varied from 2 nm to 20 nm
for Q = 8 × 106 cm−1 .
92
ENERGY (eV)
100
10-1
10-2
1
10
SILICON THICKNESS (nm)
100
RELATIVE OCCUPATION
Figure 6.10. The first four energy levels in the unprimed-ladder as a function of Si
thickness for ns = 2 × 1012 cm−3 .
100
10-1
10
-2
10-3
1
10
SILICON THICKNESS (nm)
100
Figure 6.11. The relative electron occupation of the first two energy levels in the
unprimed-ladder as a function of Si thickness for ns = 2 × 1012 cm−3 .
93
SO-LIMITED ELECTRON MOBILITY (cm2/V-s)
104
DG, ns=4x1011 cm-2
SG limit, ns=2x1011 cm-2
103
102
0
20
40
60
80
100
120
SILICON BODY THICKNESS (nm)
140
160
Figure 6.12. The unscreened SO-limited mobility for the DG structure with HfO2
dielectrics as the Si body thickness is varied for ns = 4 × 1011 cm−2 . The SO-limited
mobility converges to the SG limit in the thick Si body limit.
94
SO-LIMITED ELECTRON MOBILITY (cm2/V-s)
104
DG, ns=2x1012 cm-2
SG limit, ns=1x1012 cm-2
103
102
0
20
40
60
80
100
120
SILICON BODY THICKNESS (nm)
140
160
Figure 6.13. The unscreened SO-limited mobility for the DG structure with HfO2
dielectrics as the Si body thickness is varied for ns = 2 × 1012 cm−2 . The SO-limited
mobility converges to the SG limit in the thick Si body limit.
95
SO-LIMITED ELECTRON MOBILITY (cm2/V-s)
105
symbols: ns=1x1013 cm-2
lines: ns=2x1012 cm-2
104
screened
103
102
unscreened
HfO2
101
0
2
4
6
8
SILICON BODY THICKNESS (nm)
10
Figure 6.14. A comparison of the SO-limited mobility for an HiO2 gate dielectric
with and without screening.
96
SO-LIMITED ELECTRON MOBILITY (cm2/V-s)
106
SiO2
105
104
HfO2
103
symbols: ns=1x1013 cm-2
no symbols: ns=2x1012 cm-2
102
0
1
2
3
4
SILICON BODY THICKNESS (nm)
5
Figure 6.15. The SO-limited mobility, including the substrate plasmon, as a function
of Si thickness for an SiO2 and HiO2 gate dielectric.
97
ELECTRON MOBILITY (cm2/V-s)
105
104
103
102
101
HfO2
ns=2x1012 cm-2
100
1
2
3
µSO
µSR
µBP
µtot
4
5
6
7
8
SUBSTRATE THICKNESS (nm)
9
10
Figure 6.16. The total and scattering-limited mobilities as a function of Si thickness
for an HfO2 gate dielectric at ns = 2 × 1012 cm−2 .
98
ELECTRON MOBILITY (cm2/V-s)
105
104
103
102
101
HfO2
ns=1x1013 cm-2
100
1
2
3
µSO
µSR
µBP
µtot
4
5
6
7
8
SUBSTRATE THICKNESS (nm)
9
10
Figure 6.17. The total and scattering-limited mobilities as a function of Si thickness
for an HfO2 gate dielectric at ns = 1 × 1013 cm−2 .
99
104
ELECTRON MOBILITY (cm2/V-s)
HfO2
tsi=3 nm
103
102 11
10
µSR
µBP
µSO
µtot
1012
1013
ELECTRON DENSITY IN THE CHANNEL (cm-2)
Figure 6.18. The total and scattering-limited mobilities as a function of electron
sheet density for an HiO2 gate dielectric at tsi = 3 nm.
100
CHAPTER 7
CONCLUSIONS AND FUTURE WORK
7.1
Conclusions
We have computed the electron mobilities for Si and Ge inversion layers including
bulk phonons, surface roughness and remote phonon scattering. Ge outperforms
Si but is significantly affected by the introduction of high-κ insulators. Decreasing
oxide thickness does not significantly increase remote phonon scattering. HfO2 yields
the lowest mobilities and other materials such as ZrSiO2 , Al2 O3 , or AlN should be
considered over HfO2 . Better models for surface roughness that include real interface
data and account for fabrication process are needed.
The low-field electron mobility reduction, due to surface-optical modes associated
with high-κ dielectrics, plays less of a role in determining device performance as the
gate length is scaled from 60 nm to 15 nm. This is especially true for devices operating
in saturation and Ge MOSFETs. HfO2 , the dielectric with the largest dielectric constant and lowest energy SO modes, yields the lowest mobility. Other materials, such
as HfSiO4 and Al2 O3 , should be considered a good compromise, yielding mobilities
close to SiO2 while having a larger dielectric constant. A metal gate reduces phonon
scattering when compared to a poly-Si gate. The performance gain is expected to be
on the order of 10%.
The LO-limited electron mobility was calculated for III-V semiconductors, accounting for multisubband dynamic screening. We have shown that including screening is necessary to accurately calculate the LO-limited mobility. Dynamic and static
screening yield nearly identical results with dynamic screening adding minimal com101
putational cost over the static case. Defining an effective screening parameter captures the qualitative aspects of dynamic screening with the advantage of significant
computational savings.
The SO-limited mobility was calculated for GaAs and InGaAs substrates with
SiO2 and HfO2 gate dielectrics. The dispersion was calculated with the extra substrate
LO-mode accounted for in the substrate dielectric function. The existing model was
extended to account for scattering with the SO-mode originating from the substrate
LO-mode. The SO-limited mobility was also plotted as a function of temperature.
A comparison of InGaAs and GaAs to Si was made, with both InGaAs and GaAs
showing a significant mobility increase over Si.
The existing SO-phonon model was extended to handle the symmetric double-gate
within the semi-infinite insulator approximation. At the smallest Si body thickness,
tsi = 1 nm, surface roughnes is the dominant scattering mechanism, while SO-phonon
scattering is stonger than surface rougness scattering for tsi > 3 nm. Surface roughness scattering could in principle be eliminated by better fabrication techniques. On
the other hand, SO-phonons are an intrinsic property of high-κ/semiconductor interfaces that cannot be eliminated by better processing techniques. The use of a SiO2
interface layer would mitigate the scattering with SO-phonons but at the cost of a
thicker EOT and stronger short channel effects. This may be acceptable for double
gate MOSFETs as the short channel effects are limited by the improved electrostatics
of the double gate structure.
7.2
7.2.1
Future Work
III-V
Other III-V semiconductors of interest could be added to the exisitng code. This
could be easily accomplished by gathering the necessary material parameters (either
from experimental, or numerical data, or both) and inserting them into the code.
102
Within the framework of the III-V quantum well, delta doping could be included in
the barrier layer (InAlAs layer), as suggested by Prof. Mark Rodwell (UCSB), to
limit the occupation of the X and L-valleys.
7.2.2
Double Gate
So far we have a assumed a symmetric double gate stucture and semi-infinite
insulators. The next step would be consider mettallic gates with finite insulator
thickness. This is the most straightforward step as the equations (secular equation,
total dielectric function, etc.) are easily modified within the framework of the semiinfinite approximation. The next setp would be to the assymetric problem within
the semi-infinte approximation (or possibly the mettallic gates). This is a much more
complicated extension because the different gate insulators must be accounted for
within the secular equation and proper book keeping is required. Extension to a polySi gate (within even the semi-infinte insulator, symmetric double gate) adds serious
complications to first the algebra and second to the number dispersions. Thus, the
poly-Si gate should only be attempted after all the previous structures are analyzed.
The relevence of low-field electron mobility for short channel highly off equilibrium
devices was assesed. It would be interesting to see the same analysis for double gate
MOSFETs, perhaps as a function of Si body thickness and gate length.
103
APPENDIX
MOBILITY PROGRAM MANUAL
This appendix gives a brief overview of the Fortran90 code written for this dissertation. Instructions on how to set up and run the program are given as well as
descriptions of various subroutines that will help future students understand and
possibly extend the program.
A.1
Example Set Up File
The following source code shows an example of the MODULE Set job which is in
the file set job.f90. This module is where the device structure is chosen, as well as
doping, temperature, dilectric material, etc. In MODULE Set job, and throughout
the program, all units are in Meters-Kilograms-Seconds, and then converted to more
conventional units on output (such as cm−3 ). The only exception is energy, which is
kept in eV and is multplied by the electron charge when needed.
The acceptor doping is Na and To is the temperature (default is 300K, when
varying the temperature see main.f90). The start and finally sheet densities are
define by nstart and nfinish with the defaults shown. The Scrödinger-Poisson solver
will converge to the sheet densities chosen. In double gate mode, lstart and lfinish
set the range of semiconductor body thicknesses. In DG mode, the nstart is generally
set to some value and the semiconductor thickness is varied while maintaining the
nstart sheet density. For III-V heterostructures, Qstart and Qfinish set the range of
quantum well thicknesses.
104
The range of insulator thicknesess is set by teq start and teq finish. The insultor
thicknes is usually varied for the SO-limited mobility. The number of density points
that the mobility is calculated on is denf+1. In the example the electron sheet density
is varied between 2 × 1017 m−3 and 2 × 1015 m−3 with the mobilty calculated at 16
densities. The mobility is usually plotted on a logarithmic grid, so the spacing of
these 16 points is logarithmic so that they appear evenly spaced when plotting the
miobility. The number of semiconductor body thickness points is tf+1 for the double
gate case. For the single gate case, set tf=0 and the substratte will default to 100 nm
thick. Similarly to tf, Qf+1 sets the number of quantum well thickness to calculate
the mobility on. Leave Qf=0 unless III V and InGaAs InAlAs are set to true. The
number of insulator thickness points is teqf+1. If teqf=0 then the default insultor
thickness is 1 nm which can be changed in constants.f90 by changing the variable teq.
The scattering mechanisms are chosen. Schred-Pois should always be set to true
unless debugging the code. To plot electron occupation or things like subband energy minima as a function of electron density (body thickness for the double gate)
set mobility=.false. to avoid computing the mobility. To include bulk phonons,
set phonons=.true., intravalley=.true.
for intravalley phonons, intervalley=.true.
for intervalley phonons. There is an anisotropic and isotropic model for intravelly
phonons. In this example the isotropic model is used. When calculating the mobility of III-V semiconductors (LO-phonons are automatically ignored for Si and Ge),
set pol-opt=.true. to include polar LO-phonons. Set screen pol=.true. to include
dynamic screening of the LO-phonons. Set high k=.true. to include SO-phonons
and sub plasma=.true. to include screening of the SO-phonons. Set rough=.true.
to include surface roughness scattering and screenSR=.true. to statically screen the
surface rougness matrix elements. Set coul=.true. to included the coulomb pieces
of the surface roughness scattering. For degenerate screening set sunghoon=.true.
otherwise the screenig is non-degenerate.
105
The substrate (or body for double gate case) and insulator material is chosen.
Make sure to choose only one semiconductor and one insulator. For III-V, set IIIV=.true. and choose one III-V option below. For the quantum well structure choose
InGaAs InAlAs which is the only heterostructure implemented at the time of writing
this dissertation. The delt doped option is not functioning and should always be set
to false.
A mettallic gate, polysilicon gate, or the infinite oxide approximation can be chosen for the SO-phonons. For a mettalic gate, set metal ate=.true. and infox=.false.
For a polysilicon gate, set metal ate=.false. and infox=.false. For the infinite oxide
approximation, set metal ate=.false. and infox=.true. Set landau=.true. to include
landau damping in the SO-phonon dispersions (generally Landau damping should be
included). Leave image=.false. as this option is not functioning at the time of writing this dissertation. To include nonparabolic corrections, set nonparab=.true. and
choose between ‘fischett’ and ‘jin’ [24, 32]. For double gates ‘jin’ must be chosen.
The crystallographic surface and electron transport direction are chosen. It is up
to the user to make sure these two vectors are orthogonal. The usual orientaion for
Si is [100](011). Finally the single gate or double gate structure is set with ‘SG’ or
‘DG’. ‘SOI’ is not an available option at this time.
To compile the program, at the command line type: ./compile and press enter.
Remember to change -o /home/a.out in the executable file ‘compile’ to whatever
directory you want the executable to be in. If new subroutines are added, there
names must be added to the list in the ‘compile’ file. To run the program, go to the
folder where the exectuble is (this location is chosen in the compile file) and type
./run and press enter. Then the program will run in the backround and data will
be output in the same folder. The names of the output files can be changed in the
executable file ‘run’. Various runtime output is stored in the file output.data.
106
!Copyright 2008 Terrance O’Regan, University of Massachusetts Amherst
MODULE Set_job
IMPLICIT NONE
! set temperature (Kelvin), acceptor doping (m^3)
DOUBLE PRECISION, PARAMETER :: Na=3.d23
DOUBLE PRECISION :: To
! start and final sheet densities
DOUBLE PRECISION, PARAMETER ::
nstart=2.d17, nsfinish=2.d15
! start and final substrate thickness, Si DG
DOUBLE PRECISION, PARAMETER :: lstart=10.d-9, lfinish=1.d-9
! start and final Quantum Well thickness for III-V heterostructure
DOUBLE PRECISION, PARAMETER :: Qstart=15.d-9, Qfinish=2.d-9
! start and final equivalent insulator thickness
DOUBLE PRECISION, PARAMETER :: teq_start=2.5d-9, teq_finish=0.5d-9
! denf+1 is the number of density points for the mobility curve
INTEGER, PARAMETER :: &
denf = 14
! tf+1 is the number of substrate thickness points for the mobility curve
107
! default for single gate is tf=0 with tsi = 100nm
INTEGER, PARAMETER :: &
tf = 0
! Qf+1 is the number of Quantum Well thickness points for the mobility curve
! default is Qf=0
INTEGER, PARAMETER :: &
Qf = 0
! teqf+1 is the number of insulator thickness points for the mobility curve
! default for single gate is tf=0 with teq = 1nm, set to 6 otherwise
INTEGER, PARAMETER :: &
teqf = 0
logical*4 :: &
Scrhed_Pois = .true., &
mobility =
.true., &
! select scattering
phonons
= .true., &
intravalley = .true., &
intervalley = .true., &
anisotropic = .false., &
isotropic
= .true., &
pol_opt
= .false., &
screen_pol
= .false., &
high_k
= .true., &
sub_plasma
= .true., &
108
rough
= .true., &
screenSR
= .true., &
coul
= .true., &
sunghoon
= .true.
! select substrate semiconductor
logical*4 :: &
Si =
.false., &
Ge =
.true., &
III_V
= .false., &
GaAs
= .false., &
InP
= .false., &
InGaAs = .false., &
InGaAs_InAlAs = .false., &
delta_doped = .false.
! select insulator material
logical*4 :: &
SiO2 =
.false., &
ZrO2 =
.false., &
Al2O3 =
.true., &
AlN =
.false., &
HfO2 =
.false., &
ZrSiO4 = .false.
109
! metal (.true.) or poly-gate (.false.)
logical*4 :: &
metal_gate = .true.
!infox (.true.) or poly-gate (.false.)
logical*4 :: &
infox = .false.
! Landau Damping? (for High-k scattering)
logical*4 :: &
landau
= .true.
! Include Image and Exchange-Correlation term in S-P solver?
logical*4 :: &
image
= .false.
! nonparabolic corrections? (this is strictly false for double-gate)
logical*4 :: &
nonparab
= .true., &
fischett
= .false., &
jin
= .true.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!WARNING MAKE SURE THE FOLLOWING VECTORS ARE ORTHOGONAL!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
110
! choose crystal surface (h k l) miller indices
Integer, Parameter :: &
hh = 1, &
kk = 1, &
ll = 1
! choose direction of the field (n1 n2 n3) (the kx axis)
Integer, Parameter :: &
n1 = 1, &
n2 = 1, &
n3 = -2
! select MOS structure (single or double gate or SOI)
logical*4 :: &
SG
= .true. , &
DG
= .false. , &
SOI = .false.
END MODULE Set_job
111
A.2
Desciption of Important Subroutines
Most material parameters, such as effective masses, phonon parameters, dielectric
constants, etc., are set in the file material constants.f90. If a new semiconductor
or dielectric material is needed, then add it to material constatns.f90 and make a
new flag in set job.f90. SUBROUTINE qmass in file qmass.f90 caculates the various
masses which depend on the crystallogrhapic surface and field orientation chosen
in set job.f90. The main program is found in main.f90. This is where most major
subroutines are called from, such as the Schrödinger and Poisson solvers, tabulation,
subroutines, scattering rate subroutines, etc. Much of the output from the program
is at the end of main.f90.
The number of grid points for various grids, such as energy, angle, and real space,
are set at the top of constants.f90. Also in constants.f90 are various global arrays,
such as the electron density, and various arrays for tabulation. Also in constants.f90
are global subroutines and functions: FUNCTION trapzd is an extended trapezoidal rule for numercial integration; FUNCTION ctrapzd is the same as FUNCTION
trapzd but handles complex numbers; FUNCTION screening parameter computes the
degernate screening parameter; SUBROUTINEs LUDCMP and LUBKSB compute
the LU decomposition and solve Ax=B, respectively. SUBROUTINEs CLUDCMP
and CLUBKSB are the same as SUBROUTINEs LUDCMP and LUBKSB but handle
complex numbers.
File SP.f90 contians MODULE some routines with the following SUBROUTINES
andd FUNCTIONS: various Poisson solvers; SUBROUTINE tri solve a tridagonal
solver (for 1D Poisson equation); SUBROUTINEs nfermi calc and pfermi calc calculates the Fermi integral for electron and hole cacluations; SUBROUTINE schrodinger
solves the Schrödinger equation, normalizes the wavefunctions and created the subband ladders; SUBROUTINE Ef bisection computes the Fermi level and quantum
mechanical electron density; SUBROUTINEs bessk0 and bessk1 compute Bessel func-
112
tions for the Surface Rougness scattering; SUBROUTINE fetter calculates the plasma
dispersio function for the nondegenerate screening parameter; SUBROUTINE grid
sets up the various grids based on the input form constatnts.f90.
SUBROUTINE tabulate in the file tabulate.f90 tabulates form factors for isotropic
and anisotropic intravalley phonons ad well as the various matrix elements for surface roughness scattering. Screening of the surface roughness matrix elements is also
carried out in SUBROUTINE tabulate. SUBROUTINE tabulate pol opt in the file
tabulate pol opt.f90 tabulates the matrix element (and dynamically screens the matrix element) for LO-phonon scattering in III-V semiconductors.
File tau rates.f90 contains the various subroutines that caclulate the various momentum relaxation rates. These subroutines are called ‘on the fly’ from main.f90.
The various dispersions (i.e. metal gate, poly gate, infox approximation) for computing the SO-phonon momentum relaxation rates are found in files ending with
dispersion.f90. The file max energy.f90 determines the last subband to include in the
mobility calculation. The last subband is determined by the minimum subband electron density and is set in SUBROUTINE max energy.f90 (for example, if a subband
has less than 10−4 relative occupation it is not included int the cacluation).
A.3
Code Structure
The purpose of this section is to help the graduate students that follow me to
extend the source code - possibly following the suggestions in the ‘Future Work’
section. A couple typical examples of blocks of code will be explained. You will find
it helpful to understand the examples below because they are typical blocks of code
that are found frequently throught the source code
113
Figure A.1 is a piece of code tabulates the Prange-Nee term for surface roughness
scattering in single-gate bulk MOSFETs. The first DO-loop is over the valleys, where
vi=1 for Γ-valley, vi=2 for L-valley, vi=3 for X-valley, and numvalleys is always 3. For
Ge and III-V, gi=1 meaning that all three valleys are included. For Si, gi=3, meaning
only the X-valley is included. The next DO-loop is over the number of ladders in each
valley. In general, moblad(1)=1 for the isotropc Gamma-valley, moblad(2)=4 for the
8 halves in the L-valley, and moblad(4)=3 for the 3 sets of equivalent ladders in the
X-valley. The next two DO-loops are are over the subbands µ and ν in the specific
ladder of the specific valley. The next DO-loop is over the real space coordinate and
sets up the integrand of the matrix element. Finally, the matrix element is computed
and stored in gam(vi,s,u,vv,1)for later use in the momentum relaxation scattering
rate. The energy levels are stored in e1(vi,s,vv) the wavefunctions are stored in
psi(vi,s,u,i) the first derivate of the potential energy is stored in dV(i) and the
first derivates of the wavefunctions are stored in dpsi(vi,s,vv,i)
The tabulation is over the valleys, ladders, and various subbands. The last index
in gam(vi,s,u,vv,1) is an index over the Q grid. In this unscreened case the matrix
element does not depend on Q - but after screening it will depend on Q so we need
the extra index. This example illustrates a very common construction in the code,
but I warn you the index ordering is consistent across the program, but sometimes
other integers are used in the DO-loops.
Figure A.2 is founf in main.f90 and computes the momentum relaxation rate for
surface roughness scattering. The DO-loop is over all final subbands where onlu
intravalley transitions are considered. The final subband energy is +Ev and the
nonparabolic correction ofthe final subband is +Ev. The if(E-Ev>0.d0) statement
makes sure of energy conservation. The momentum relaxation rate is calculated in
tausr which is founf in tau.f90. The nonparabolic correction comes next with either
Fischetti’s or Jin’s model. Not shown here are the outer loops that caclulate the
114
Kubo-Greenwood formula. These outside loops are over intial energy, intital angle,
and over all initila subbands. The momemtum relation rate (tsr is a rate, so 1/tsr is
the relaxtion time) is summed over all the final energies, where const3 (and all similar
constants) is defeined in material constatns.f90. A similar block of code is repeated
for each scattering meachanism, with all the rates computed in tau rates.f90.
115
DO vi=gi,numvalleys
DO s=1,moblad(vi)
DO u=1,nue(vi,s)
DO vv=1,nue(vi,s)
DO i=1,N
ff(i) = psi(vi,s,u,i)*q*dV(i)*psi(vi,s,vv,i) &
+ q*e1(vi,s,vv)*dpsi(vi,s,u,i)*psi(vi,s,vv,i) &
+ q*e1(vi,s,u)*dpsi(vi,s,vv,i)*psi(vi,s,u,i)
END DO
gam(vi,s,u,vv,1) = trapzd(ff,xh,N)
END DO
END DO
END DO
END DO
Figure A.1. Tabulation of the Prange-Nee term for surface roughness scattering in
single-gate bulk MOSFETs.
! !!!!!!!!!!!!!!!!! SURFACE ROUGHNESS !!!!!!!!!!!!!!!!
IF( rough ) then
DO g=1,nue(vi,t)
Ev = e1(vi,t,g)
ev1 = np(vi,t,1,g)
if(E-Ev>0.d0) then
CALL tauSR(Isr, btat, E, Eu, Ev, eu1, ev1, s, g, t, vi)
IF(fischett) then
kv = 1.d0 - 2.d0*alpha(vi)*(E-Ev)-2.d0*alpha(vi)*ev1
ELSEIF(jin) then
kv = 1 - 2.d0*alpha(vi)*(E-vnp(vi,t,g))
END IF
tsr = tsr + const3*kv*Isr
end if
END DO
END IF
Figure A.2. This block calulates the momentum relaxation rate for surface roughness scattering (the actual rate is computed in SUBROUTINE tauSR).
116
BIBLIOGRAPHY
[1] International Technology Roadmap for Semiconductors 2006 Update. Semiconductor Industry Association, 2006. [Online]. Available: http://public.itrs.net/
[2] M. V. Fischetti, D. A. Neumayer, and E. A. Cartier, “Effective electron mobility
in Si inversion layers in metal-oxide-semiconductor systems with a high-κ insulator: The role of remote phonon scattering,” J. Appl. Phys., vol. 90, no. 9, pp.
4587–4608, Nov. 2001.
[3] D. Esseni, “On the modeling of surface roughness limited mobility in SOI MOSFETs and its correlation to the transistor effective field,” IEEE Trans. Electron
Devices, vol. 51, no. 3, pp. 394–401, Mar. 2004.
[4] T. Zhu, W.J.; Ma, “Temperature dependence of channel mobility in hfo/sub 2/gated nmosfets,” Electron Device Letters, IEEE, vol. 25, no. 2, pp. 89–91, Feb.
2004.
[5] Z. Ren, M. Fischetti, E. Gusev, E. Cartier, and M. Chudzik, “Inversion channel
mobility in high-/spl kappa/ high performance mosfets,” Electron Devices Meeting, 2003. IEDM ’03 Technical Digest. IEEE International, pp. 33.2.1–33.2.4,
8-10 Dec. 2003.
[6] R. Kotlyar, M. Giles, P. Matagne, B. Obradovic, L. Shifren, M. Stettler, and
E. Wang, “Inversion mobility and gate leakage in high-k/metal gate MOSFETs,”
Electron Devices Meeting, 2004. IEDM Technical Digest. IEEE International, pp.
391–394, 2004.
[7] E. Gusev, V. Narayanan, and M. Frank, “Advanced High-κ Dielectric Stacks
with PolySi and Metal Gates: Recent Progress and Current Challenges,” IBM
J. Res. & Dev, pp. 387–410, 2006.
[8] R. Kotlyar, B. Obradovic, P. Matagne, M. Stettler, and M. D. Giles, “Assessment
of room-temperature phonon-limited mobility in gated silicon nanowires,” Appl.
Phys. Lett., vol. 84, no. 25, pp. 5270–5272, Jan. 2004.
[9] R. Chau, S. Datta, M. Doczy, B. Doyle, J. Kavalieros, and M. Metz, “High/spl kappa//metal-gate stack and its MOSFET characteristics,” Electron Device
Letters, IEEE, vol. 25, no. 6, pp. 408–410, 2004.
117
[10] Y. Xuan, Y. Wu, H. Lin, T. Shen, and P. Ye, “Submicrometer InversionType Enhancement-Mode InGaAs MOSFET With Atomic-Layer-Deposited$
hbox {Al} {2} hbox {O} {3}$ as Gate Dielectric,” Electron Device Letters,
IEEE, vol. 28, no. 11, pp. 935–938, 2007.
[11] K. Rim, S. Koester, M. Hargrove, J. Chu, P. Mooney, J. Ott, T. Kanarsky,
P. Ronsheim, M. Ieong, A. Grill et al., “Strained Si NMOSFETs for high performance CMOS technology,” VLSI Technology, 2001. Digest of Technical Papers.
2001 Symposium on, pp. 59–60, 2001.
[12] W. Bai, N. Lu, A. Ritenour, M. Lee, D. Antoniadis, and D.-L. Kwong, “Ge
n-mosfets on lightly doped substrates with high-/spl kappa/ dielectric and tan
gate,” Electron Device Letters, IEEE, vol. 27, no. 3, pp. 175–178, March 2006.
[13] S. Laux, “A Simulation Study of the Switching Times of 22-and 17-nm GateLength SOI nFETs on High Mobility Substrates and Si,” Electron Devices, IEEE
Transactions on, vol. 54, no. 9, pp. 2304–2320, 2007.
[14] D. Frank, R. Dennard, E. Nowak, P. Solomon, Y. Taur, H. Wong et al., “Device
scaling limits of Si MOSFETs and their application dependencies,” Proc. IEEE,
vol. 89, no. 3, pp. 259–288, 2001.
[15] B. Yu, L. Chang, S. Ahmed, H. Wang, S. Bell, C. Yang, C. Tabery, C. Ho,
Q. Xiang, T. King et al., “FinFET scaling to 10 nm gate length,” Electron
Devices Meeting, 2002. IEDM’02. Digest. International, pp. 251–254, 2002.
[16] D. Hisamoto, W. Lee, J. Kedzierski, H. Takeuchi, K. Asano, C. Kuo, E. Anderson, T. King, J. Bokor, and C. Hu, “FinFET-a self-aligned double-gate MOSFET
scalable to 20 nm,” Electron Devices, IEEE Transactions on, vol. 47, no. 12, pp.
2320–2325, 2000.
[17] D. Frank, S. Laux, and M. Fischetti, “Monte Carlo simulation of a 30 nm dualgate MOSFET: how short canSi go?” Electron Devices Meeting, 1992. Technical
Digest., International, pp. 553–556, 1992.
[18] K. Suzuki, T. Tanaka, Y. Tosaka, H. Horie, and Y. Arimoto, “Scaling theory for
double-gate SOI MOSFET’s,” Electron Devices, IEEE Transactions on, vol. 40,
no. 12, pp. 2326–2329, 1993.
[19] R. Kubo, “Statistical-mechanical theory of irreversible
J. Phys. Soc. Jpn., vol. 12, no. 6, pp. 570–586, Jun. 1957.
process.
I.”
[20] D. A. Greenwood, “The Boltzmann equation in the theory of electrical conduction in metals,” Proc. Phys. Soc. London, vol. 71, pp. 585–596, 1958.
[21] M. V. Fischetti, “Long-range Coulomb interactions in small Si devices. Part II.
effective electron mobility in thin-oxide structures,” J. Appl. Phys., vol. 89, no. 2,
pp. 1232–1250, Jan. 2001.
118
[22] C. Herring and E. Vogt, “Transport and deformation-potential theory for manyvalley semiconductors with anisotropic scattering,” Phys. Rev., vol. 101, no. 3,
pp. 944–961, Feb 1956.
[23] M. V. Fischetti, F. Gámiz, and W. Hänsch, “On the enhanced electron mobility in
strained-silicon inversion layers,” Journal of Applied Physics, vol. 92, no. 12, pp.
7320–7324, 2002. [Online]. Available: http://link.aip.org/link/?JAP/92/7320/1
[24] M. V. Fischetti and S. E. Laux, “Monte carlo study of electron transport in
silicon inversion layers,” Phys. Rev. B, vol. 48, no. 4, pp. 2244–2274, Sep. 1993.
[25] C. Jacoboni and L. Reggiani, “The Monte Carlo method for the solution of charge
transport in semiconductors with applications to covalent materials,” Rev. Mod.
Phys., vol. 55, no. 3, pp. 645–705, Jul. 1983.
[26] D. A. Dahl and L. J. Sham, “Electrodynamics of quasi-two-dimensional electrons,” Phys. Rev. B, vol. 16, no. 2, pp. 651–661, Jul 1977.
[27] T. Ando, A. B. Fowler, and F. Stern, “Electronic properties of two-dimensional
systems,” Rev. Mod. Phys., vol. 54, no. 2, pp. 437–672, Apr. 1982.
[28] S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy, and
O. L. Krivanek, “Surface roughness at the Si(100)-SiO2 interface,” Phys. Rev. B,
vol. 32, no. 12, pp. 8171–8186, Dec. 1985.
[29] M. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surface Science Reports, vol. 41, no. 1-8, pp. 1–416, 2001.
[30] M. V. Fischetti and S. E. Laux, “Monte Carlo analysis of electron transport in
small semiconductor devices including band-structure and space charge effects,”
Phys. Rev. B, vol. 38, no. 14, pp. 9721–9745, Nov. 1988.
[31] G. Gilat and L. Raubenheimer, “Accurate Numerical Method for Calculating
Frequency-Distribution Functions in Solids,” Physical Review, vol. 144, no. 2,
pp. 390–395, 1966.
[32] S. Jin, M. V. Fischetti, and T.-w. Tang, “Modeling of electron mobility in gated
silicon nanowires at room temperature: Surface roughness scattering, dielectric
screening, and band nonparabolicity,” J. Appl. Phys., vol. 102, no. 8, p. 083715,
Oct. 2007.
[33] J. Nasrullah, G. Tyler, and Y. Nishi, “An atomic force microscope study of
surface roughness of thin silicon films deposited on SiO/sub 2,” Nanotechnology,
IEEE Transactions on, vol. 4, no. 3, pp. 303–311, 2005.
[34] M. Frank, H. Shang, S. Rivillon, F. Amy, C. Hsueh, V. Paruchuri, R. Mo,
M. Copel, E. Gusev, M. Gribelyuk et al., “High-κ Gate Dielectrics on Silicon
and Germanium: Impact of Surface Preparation,” Solid State Phenom, vol. 103,
no. 104, pp. 3–6, 2005.
119
[35] C. O. Chui, S. Ramanathan, B. Triplett, P. McIntyre, and K. Saraswat, “Germanium mos capacitors incorporating ultrathin high-/spl kappa/ gate dielectric,”
Electron Device Letters, IEEE, vol. 23, no. 8, pp. 473–475, Aug 2002.
[36] N. Lu, W. Bai, A. Ramirez, C. Mouli, A. Ritenour, M. L. Lee, D. Antoniadis,
and D. L. Kwong, “Ge diffusion in ge metal oxide semiconductor with chemical
vapor deposition hfo[sub 2] dielectric,” Applied Physics Letters, vol. 87, no. 5, p.
051922, 2005. [Online]. Available: http://link.aip.org/link/?APL/87/051922/1
[37] W. Haensch, E. Nowak, R. Dennard, P. Solomon, A. Bryant, O. Dokumaci,
A. Kumar, X. Wang, J. Johnson, and M. Fischetti, “Silicon CMOS devices beyond scaling,” IBM J. Res. & Dev, vol. 50, no. 4/5, pp. 339–361, 2006.
[38] B. Doyle, S. Datta, M. Doczy, S. Hareland, B. Jin, J. Kavalieros, T. Linton,
A. Murthy, R. Rios, and R. Chau, “High performance fully-depleted tri-gate
CMOS transistors,” Electron Device Letters, IEEE, vol. 24, no. 4, pp. 263–265,
2003.
[39] E. Nowak, I. Aller, T. Ludwig, K. Kim, R. Joshi, C. Chuang, K. Bernstein, and
R. Puri, “Turning silicon on its edge- double gate CMOS/FinFET technology,”
IEEE Circuits and Devices Magazine, vol. 20, no. 1, pp. 20–31, 2004.
[40] C. Black, “Self-aligned self assembly of multi-nanowire silicon field effect transistors,” Applied Physics Letters, vol. 87, p. 163116, 2005.
[41] Z. Krivokapic, V. Moroz, W. Maszara, and M. Lin, “Locally strained ultrathin channel 25nm narrow FDSOI devices with metal gate and mesa isolation,”
Electron Devices Meeting, 2003. IEDM’03 Technical Digest. IEEE International,
pp. 18–5, 2003.
[42] D. Wang, Q. Wang, A. Javey, R. Tu, H. Dai, H. Kim, P. McIntyre, T. Krishnamohan, and K. Saraswat, “Germanium nanowire field-effect transistors with
SiO 2 and high-k HfO 2 gate dielectrics,” Appl. Phys. Lett, vol. 83, no. 12, pp.
2432–2434, 2003.
[43] M. Yang, M. Ieong, L. Shi, K. Chan, V. Chan, A. Chou, E. Gusev, K. Jenkins,
D. Boyd, Y. Ninomiya, D. Pendleton, Y. Surpris, D. Heenan, J. Ott, K. Guarini, C. D’Emic, M. Cobb, P. Mooney, B. To, N. Rovedo, J. Benedict, R. Mo,
and H. Ng, “High performance cmos fabricated on hybrid substrate with different crystal orientations,” Electron Devices Meeting, 2003. IEDM ’03 Technical
Digest. IEEE International, pp. 18.7.1–18.7.4, 8-10 Dec. 2003.
[44] M. Fischetti, T. O’Regan, S. Narayanan, C. Sachs, S. Jin, J. Kim, and Y. Zhang,
“Theoretical Study of Some Physical Aspects of Electronic Transport in nMOSFETs at the 10-nm Gate-Length,” Electron Devices, IEEE Transactions on,
vol. 54, no. 9, pp. 2116–2136, 2007.
120
[45] T. O’Regan and M. Fischetti, “Remote Phonon Scattering in Si and Ge with SiO
2 and HfO 2 Insulators: Does the Electron Mobility Determine Short Channel
Performance?” Japanese Journal of Applied Physics, vol. 46, no. 5B, pp. 3265–
3272, 2007.
[46] P. Solomon and S. Laux, “The ballistic FET: design, capacitance and speed
limit,” Electron Devices Meeting, 2001. IEDM Technical Digest. International,
pp. 5–1, 2001.
[47] J. Wang and M. Lundstrom, “Ballistic transport in high electron mobility transistors,” Electron Devices, IEEE Transactions on, vol. 50, no. 7, pp. 1604–1609,
2003.
[48] J. Lusakowski, W. Knap, Y. Meziani, J. Cesso, A. El Fatimy, R. Tauk,
N. Dyakonova, G. Ghibaudo, F. Boeuf, and T. Skotnicki, “Ballistic and pocket
limitations of mobility in nanometer Si metal-oxide semiconductor field-effect
transistors,” Applied Physics Letters, vol. 87, p. 053507, 2005.
[49] Z. Ren, R. Venugopal, S. Datta, M. Lundstrom, D. Jovanovic, and J. Fossum,
“The ballistic nanotransistor: a simulation study,” Electron Devices Meeting,
2000. IEDM Technical Digest. International, pp. 715–718, 2000.
[50] X. Lei, “Dynamical screening and carrier mobility in GaAs-GaAlAs heterostructures,” Journal of Physics C: Solid State Physics, vol. 18, no. 20, pp. L593–L597,
1985.
[51] ——, “Dynamical screening and electron self-energy in the presence of polar
interaction in quasi-2D heterostructures,” Journal of Physics C: Solid State
Physics, vol. 18, no. 24, pp. L731–L735, 1985.
[52] D. Anderson, N. Zakhleniuk, M. Babiker, B. Ridley, and C. Bennett, “Polaroptical phonon-limited transport in degenerate GaN-based quantum wells,”
Physical Review B, vol. 63, no. 24, p. 245313, 2001.
[53] F. Riddoch and B. Ridley, “On the scattering of electrons by polar optical
phonons in quasi-2D quantum wells,” Journal of Physics C: Solid State Physics,
vol. 16, no. 36, pp. 6971–6982, 1983.
[54] M. A. Stroscio, “Interaction between longitudinal-optical-phonon modes of a
rectangular quantum wire and charge carriers of a one-dimensional electron gas,”
Phys. Rev. B, vol. 40, no. 9, pp. 6428–6431, Sep 1989.
[55] J. Blakemore, “Semiconducting and other major properties of gallium arsenide,”
Journal of applied phyics, vol. 53, p. R123, 1982.
[56] K. Yokoyama and K. Hess, “Monte Carlo study of electronic transport in Al {1x} Ga {x} As/GaAs single-well heterostructures,” Physical Review B, vol. 33,
no. 8, pp. 5595–5606, 1986.
121
[57] A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems. New
York: McGRAW-HILL, 1971.
[58] T. Mozume and J. Kasai, “Micro-Raman scattering study of InGaAs/(AlAs)/AlAsSb quantum wells grown by molecular beam epitaxy,”
Journal of Crystal Growth, vol. 278, no. 1-4, pp. 178–182, 2005.
[59] A. Pethe, T. Krishnamohan, D. Kim, S. Oh, H. Wong, and K. Saraswat, “Investigation of the Performance Limits of III-V Double-Gate n-MOSFETs,” University/Government/Industry Microelectronics Symposium, 2006 16th Biennial,
pp. 47–50, 2006.
[60] E. Mendez, P. Price, and M. Heiblum, “Temperature dependence of the electron
mobility in GaAs-GaAlAs heterostructures,” Applied Physics Letters, vol. 45, p.
294, 1984.
[61] T. Ernst, S. Cristoloveanu, G. Ghibaudo, T. Ouisse, S. Horiguchi, Y. Ono,
Y. Takahashi, and K. Murase, “Ultimately thin double-gate SOI MOSFETs,”
Electron Devices, IEEE Transactions on, vol. 50, no. 3, pp. 830–838, 2003.
[62] F. Gámiz and M. V. Fischetti, “Monte Carlo simulation of double-gate silicon-oninsulator inversion layers: The role of volume inversion,” J. Appl. Phys., vol. 89,
no. 10, pp. 5478–5487, May 2001.
[63] B. Majkusiak, T. Janik, and J. Walczak, “Semiconductor thickness effects in the
double-gate SOI MOSFET,” Electron Devices, IEEE Transactions on, vol. 45,
no. 5, pp. 1127–1134, 1998.
[64] D. Esseni, M. Mastrapasqua, G. Celler, C. Fiegna, L. Selmi, and E. Sangiorgi,
“An experimental study of mobility enhancement in ultrathin SOI transistors
operated in double-gate mode,” Electron Devices, IEEE Transactions on, vol. 50,
no. 3, pp. 802–808, 2003.
[65] D. Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically based modeling of
low field electron mobility in ultrathin single- and double-gate SOI n-MOSFETs,”
IEEE Trans. Electron Devices, vol. 50, no. 12, pp. 2445–2455, Dec. 2003.
[66] K. Hess, “Remote Polar Phonon Scattering in Silicon Inversion Layers,” Solid
State Communications, vol. 30, no. 12, pp. 807–809, 1979.
[67] B. Laikhtman and M. Solomon, “Remote phonon scattering in field-effect transistors with a high kappa insulating layer ,” Journal of Applied Physics, vol. 103,
no. 1, p. 014501, 2008.
[68] R. Chen, D. Lin, and T. George, “Optical-phonon modes in a double heterostructure of polar crystals,” Physical Review B, vol. 41, no. 3, pp. 1435–1442, 1990.
122
[69] M. Stroscio, G. Iafrate, K. Kim, M. Littlejohn, H. Goronkin, and G. Maracas, “Transition from longitudinal-optical phonon scattering to surface-optical
phonon scattering in polar semiconductor superlattices,” Applied Physics Letters, vol. 59, p. 1093, 1991.
[70] K. Kim et al., “Electron-optical-phonon scattering rates in a rectangular semiconductor quantum wire,” Journal of Applied Physics, vol. 70, no. 1, p. 319,
1991.
[71] S. Yu, “Transfer matrix method for interface optical-phonon modes in multipleinterface heterostructure systems,” Journal of Applied Physics, vol. 82, no. 7, p.
3363, 1997.
[72] S. Jin, M. V. Fischetti, and T.-W. Tang, “Modeling of surface roughness scattering in ultrathin-body SOI MOSFETs,” IEEE Trans. Electron Devices, vol. 54,
no. 9, pp. 2191–2203, Sep. 2007.
[73] T. Ando, “Screening effect and quantum transport in a silicon inversion layer in
strong magnetic field,” J. Phys. Soc. Japan, vol. 43, no. 5, pp. 1616–1626, Nov.
1977.
[74] S. Mori and T. Ando, “Intersubband scattering effect on the mobility of a Si
(100) inversion layer at low temperatures,” Phys. Rev. B, vol. 19, no. 12, pp.
6433–6441, Jun. 1979.
[75] M. V. Fischetti, Z. Ren, P. M. Solomon, M. Yang, and K. Rim, “Six-band k · p
calculation of the hole mobility in silicon inversion layers: Dependence on surface
orientation, strain, and silicon thickness,” J. Appl. Phys., vol. 94, no. 2, pp. 1079–
1095, Jul. 2003.
123