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Strain Effects on Hole Mobility of Silicon and Germanium p-Type Metal-Oxide-Semiconductor Field-Effect-Transistors

Permanent Link: http://ufdc.ufl.edu/UFE0018960/00001

Material Information

Title: Strain Effects on Hole Mobility of Silicon and Germanium p-Type Metal-Oxide-Semiconductor Field-Effect-Transistors
Physical Description: 1 online resource (141 p.)
Language: english
Creator: Sun, Guangyu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: hole, mobility, mosfet, strain, stress
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My research explores the strain enhanced hole mobility in silicon (Si) and germanium (Ge) p-type metal-oxide-semiconductor field-effect-transistors (p-MOSFETs). The piezoresistance coefficients are calculated and measured via wafer bending experiments. With good agreement in the measured and calculated small stress piezoresistance coefficients, k.p calculations are used to give physical insights into hole mobility enhancement at large stress (3 GPa for Si and 6 GPa for Ge) for stresses of technological importance: in-plane biaxial and channel-direction uniaxial stress on (001) and (110)-surface oriented p-MOSFETs with h110i and h111i channels. The mathematical definition of strain and stress is introduced and the transformation between the strain and stress tensor is demonstrated. Self-consistent calculation of Schrodinger Equation and Poisson Equation is applied to study the potential and subband energy levels in the inversion layers. Subband structures, two-dimensional (2D) density-of-states (DOS), hole effective mass, phonon and surface roughness scattering rate are evaluated numerically and the hole mobility is obtained from a linearization of Boltzmann Equation. The results show that hole mobility saturates at large stress. Under biaxial tensile stress, the hole mobility is degraded at small stress due to the subtractive nature of the strain and quantum confinement effects. At large stress, hole mobility is improved via the suppression of the phonon scattering. Biaxial compressive stress improves hole mobility slightly. Uniaxial compressive stress enhances the hole mobility monotonically as the stress increases. In (001) surface oriented p-MOSFETs, the maximum enhancement factor is 350% for Si and 600% for Ge. The enhancement of (110) p-MOSFETs is smaller than (001) p-MOSFETs due to the strong quantum confinement and low DOS of the ground state subband. For (001) p-MOSFETs, the dominant factor to improve the hole mobility is the hole effective mass reduction at small stress and phonon scattering rate suppression at large stress. For (110) p-MOSFETs, the hole effective mass and phonon scattering rate are constant at large stress due to the saturation of the subband splitting and DOS caused by the strong confinement. Strain efects on non-classical devices (single-gate (SG) silicon-on-insulator (SOI) and double-gate (DG) p-MOSFETs) are also investigated. The calculation shows that the mobility enhancement for SG SOI and DG (001) p-MOSFETs is similar to traditional Si p-MOSFETs. Hole mobility enhancement in FinFETs is more than traditional (110) p-MOSFETs due to the subband modulation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Guangyu Sun.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Thompson, Scott.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0018960:00001

Permanent Link: http://ufdc.ufl.edu/UFE0018960/00001

Material Information

Title: Strain Effects on Hole Mobility of Silicon and Germanium p-Type Metal-Oxide-Semiconductor Field-Effect-Transistors
Physical Description: 1 online resource (141 p.)
Language: english
Creator: Sun, Guangyu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: hole, mobility, mosfet, strain, stress
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My research explores the strain enhanced hole mobility in silicon (Si) and germanium (Ge) p-type metal-oxide-semiconductor field-effect-transistors (p-MOSFETs). The piezoresistance coefficients are calculated and measured via wafer bending experiments. With good agreement in the measured and calculated small stress piezoresistance coefficients, k.p calculations are used to give physical insights into hole mobility enhancement at large stress (3 GPa for Si and 6 GPa for Ge) for stresses of technological importance: in-plane biaxial and channel-direction uniaxial stress on (001) and (110)-surface oriented p-MOSFETs with h110i and h111i channels. The mathematical definition of strain and stress is introduced and the transformation between the strain and stress tensor is demonstrated. Self-consistent calculation of Schrodinger Equation and Poisson Equation is applied to study the potential and subband energy levels in the inversion layers. Subband structures, two-dimensional (2D) density-of-states (DOS), hole effective mass, phonon and surface roughness scattering rate are evaluated numerically and the hole mobility is obtained from a linearization of Boltzmann Equation. The results show that hole mobility saturates at large stress. Under biaxial tensile stress, the hole mobility is degraded at small stress due to the subtractive nature of the strain and quantum confinement effects. At large stress, hole mobility is improved via the suppression of the phonon scattering. Biaxial compressive stress improves hole mobility slightly. Uniaxial compressive stress enhances the hole mobility monotonically as the stress increases. In (001) surface oriented p-MOSFETs, the maximum enhancement factor is 350% for Si and 600% for Ge. The enhancement of (110) p-MOSFETs is smaller than (001) p-MOSFETs due to the strong quantum confinement and low DOS of the ground state subband. For (001) p-MOSFETs, the dominant factor to improve the hole mobility is the hole effective mass reduction at small stress and phonon scattering rate suppression at large stress. For (110) p-MOSFETs, the hole effective mass and phonon scattering rate are constant at large stress due to the saturation of the subband splitting and DOS caused by the strong confinement. Strain efects on non-classical devices (single-gate (SG) silicon-on-insulator (SOI) and double-gate (DG) p-MOSFETs) are also investigated. The calculation shows that the mobility enhancement for SG SOI and DG (001) p-MOSFETs is similar to traditional Si p-MOSFETs. Hole mobility enhancement in FinFETs is more than traditional (110) p-MOSFETs due to the subband modulation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Guangyu Sun.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Thompson, Scott.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0018960:00001


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STRAIN EFFECTS ON HOLE MOBILITY OF SILICON AND GERMANIUM P-TYPE
METAL-OXIDE-SEMICONDUCTOR FIELD-EFFECT-TRANSISTORS



















By

GUANGYU SUN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































2007 Guangyu Sun






















To my dear wife Anita, and my parents









ACKNOWLEDGMENTS

I am grateful to all the people who made this dissertation possible and because of

whom my graduate experience has been one that I will cherish forever.

First and foremost I thank my advisor, Dr. Scott E. Thompson, for giving me an

invaluable opportunity to work on challenging and extremely interesting projects over the

past four years. He has ahli-- made himself available for help and advice and there has

never been an occasion when I have knocked on his door and he has not given me time.

He taught me how to solve a problem starting from a simple model, and how to develop it.

It has been a pleasure to work with and learn from such an extraordinary individual.

I thank Dr. Jerry G. Fossum, Dr. Huikai Xie, Dr. C'!i -I1!.ih. r Stanton, and Dr.

Jing Guo for agreeing to serve on my dissertation committee and for sparing their

invaluable time reviewing the manuscript. I also thank Dr. Toshi Nishida for a lot of

helpful discussions and kind help.

My colleagues have given me a lot of assistance in the course of my Ph.D. studies.

Dr. Yongke Sun helped me greatly to understand the physics model, and we ah--,v- had

fruitful discussions. Dr. Toshi Numata also gave me good advice and some insightful

ideas. I also thank Jisong Lim, Sagar Suthram, and all other group members who made

my life here more interesting.

I acknowledge help and support from some of the staff members, in particular,

Shannon Chillingworth, Teresa Stevens and Marcy Lee, who gave me much indispensable

assistance.

I owe my deepest thanks to my family. I thank my mother and father, and my wife,

Anita, who have alv--,v- stood by me. I thank them for all their love and support. Words

cannot express the gratitude I owe them.

It is impossible to remember all, and I apologize to those I have inadvertently left

out.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF FIGURES .................................... 7

LIST OF TABLES ....................... ............ 11

ABSTRACT ....................... ........... ...... 12

CHAPTER

1 INTRODUCTION AND OVERVIEW ................... .... 14

1.1 History of Strain in Semiconductors .......... ............ 15
1.2 Apply Strain to A Transistor ................... .... 17
1.3 Main Contributions of My Research .......... ............ 18
1.4 Brief Description of The Dissertation .................. ... 19

2 K P MODEL AND HOLE MOBILITY ............. .... .. 21

2.1 The k p Method ............... . . .... 21
2.1.1 Introduction to k p Method ............. .... . 21
2.1.2 Kane's M odel .................. ........... .. 25
2.1.3 Luttinger-Kohn's Hamiltonian ................... ... .. 28
2.2 Hole Mobility in Inversion L iv. rs ................ .... .. 32
2.2.1 Self-consistent Procedure .................. ... .. 32
2.2.2 Hole M obility .................. ........... 33
2.3 Scattering Mechanisms .................. ........ .. .. 34
2.3.1 Phonon Scattering. .................. ........ .. 34
2.3.2 Surface Roughness Scattering ............... . .. 36
2.4 Summ ary .................. ............... .. .. 38

3 STRAIN EFFECTS ON SILICON P-MOSFETS ................ .. 39

3.1 Piezoresistance Coefficients and Hole Mobility . . ..... 40
3.1.1 Piezoresistance Coefficients ................ .... .. 40
3.1.2 Hole Mobility vs Surface Orientation ................ .. 41
3.1.3 Hole Mobility and Vertical Electric Field . . ...... 42
3.1.4 Strain-enhanced Hole Mobility ................... ... .. 42
3.2 Bulk Silicon Valence Band Structure ............... . .. 48
3.2.1 Dispersion Relation .................. ........ .. 48
3.2.2 Hole Effective Masses .................. .. .... .. .. 49
3.2.3 Valence Band under Super Low Strain . . ...... 56
3.2.4 Energy Contours .................. ......... .. 56
3.3 Strain Effects on Silicon Inversion L rs .................. .. 60
3.3.1 Quantum Confinement and Subband Splitting . . ... 60









3.3.2 Confinement of (110) Si ...............
3.3.3 Strain-induced Hole Repopulation .. .......
3.3.4 Scattering Rate .. ...............
3.3.5 Mass and Scattering Rate Contribution ......
3.4 Sum m ary . . . . . . .

4 STRAIN EFFECTS ON NON-CLASSICAL DEVICES ....


4.1 Single
4.1.1
4.1.2


Gate SOI pMOS .
Hole Mobility vs Sili
Strain-enhanced Hol


4.2 Double-gate p-MOSFETs
4.2.1 (001) SDG pMOS
4.2.2 Strain Effect on FinI
4.3 Summary .. .......


con Thickness .........
e Mobility of SOI SG-pMOS .


ETs ...............


5 STRAIN EFFECTS ON GERMANIUM P-MOSFETS ....

5.1 Germanium Hole Mobility ...............
5.1.1 Biaxial Tensile Stress .. .............
5.1.2 Biaxial Compressive Stress ............
5.1.3 Uniaxial Compressive Stress ...........
5.2 Strain Altered Bulk Ge Valence Band Structure .....
5.2.1 E-k Diagram s .. ................
5.2.2 Effective M ass .. ................
5.2.3 Energy Contours .. ..............
5.3 Discussion Of Hole Mobility Enhancement ........
5.3.1 Strain-induced Subband Splitting .. .......
5.3.2 Biaxial Stress on (001) Ge ............
5.3.3 Uniaxial Compression on (001) Ge .........
5.3.4 Uniaxial Compression on (110) Ge .........
5.4 Sum m ary . . . . . . .

6 SUMMARY AND SUGGESTIONS TO FUTURE WORK .

6.1 Sum m ary . . . . . . .
6.2 Recommendations for Future Work .. ..........

APPENDIX

A STRESS AND STRAIN . ................

B PIEZORESISTANCE .. ..................

REFERENCES .......... ...............


. . 122


. . 133


BIOGRAPHICAL SKETCH .............................. .......









LIST OF FIGURES


Figure page

1-1 Schematic diagram of biaxial tensile stressed Si-MOSFET on relaxed Sil_-Ge1
1-iv-r . . ..... . . . . . .. .. 17

1-2 Uniaxial stressed Si-MOSFET with Sil_1Ge1 Source/Drain or highly stressed
capping 1- rv.r . . . . . . . . . .18

3-1 Hole mobility vs device surface orientation for relaxed silicon . . ... 41

3-2 Hole mobility vs inversion charge density for relaxed silicon. Both measurements
and simulation show larger mobility on (110) devices. . . 43

3-3 Hole mobility vs stress with inversion charge density 1 x 1013/cm2. ........ 44

3-4 Calculated strain induced hole mobility enhancement factor vs. experimental
data for (001)-oriented pMOS. .................. ..... 45

3-5 Hole mobility enhancement factor vs uniaxial stress for different channel doping. 45

3-6 Calculated strain induced hole mobility enhancement factor vs. stress for (001)
oriented pMOS with different inversion charge density. ............ ..47

3-7 E-k relation for silicon under (a) no stress; (b) 1GPa biaxial tensile stress; and
(c) 1GPa uniaxial compressive stress. .............. .... 50

3-8 Normalized E-k diagram of the top band under different amount of stress. Larger
stress warps more region of the band. The energy at F point for all curves is
set to zero only for comparison purpose. ................ ...... 51

3-9 C'!i i,,, I direction effective masses for bulk silicon under (a) biaxial tensile stress;
and (b) uniaxial compressive stress. ............... .... 52

3-10 Two-dimensional density-of-states effective masses for bulk silicon under (a) bi-
axial tensile stress; and (b) uniaxial compressive stress. ............. .53

3-11 Out-of-plane effective masses for bulk silicon under (a) biaxial tensile stress;
and (b) uniaxial compressive stress. ................ .... 54

3-12 Hole effective mass change under very small stress. The change in this stress
region explains the "d( I-. oi ii ,ly of the hole effective mass between the re-
laxed and highly stressed Si. ............... .......... .. 57

3-13 The 25meV energy contours for unstressed Si: (a) Heavy-hole; (b) Light-hole. .58

3-14 The 25meV energy contours for biaxial tensile stressed Si: (a) Top band; (b)
Bottom band . ............... ............... .. 59









3-15 The 25meV energy contours for uniaxially compressive stressed Si: (a) Top band;
(b) Bottom band ............... .............. .. 59

3-16 Quantum well and subbands energy levels under transverse electric field. . 61

3-17 Schematic plot of strain effect on subband splitting, the field effect is additive
to uniaxial compression and subtractive to biaxial tension. . . .. 64

3-18 Subband splitting between the top two subbands under different stress. . 65

3-19 Out-of-plane effective masses for (110) surface oriented bulk silicon under uni-
axial com pressive stress. .................. .......... ..66

3-20 The 2D energy contours (25, 50, 75, and 100 meV) for bulk (001)-Si. Uniaxial
compressive stress changes hole effective mass more significantly than biaxial
tensile stress .................... .................. .. 68

3-21 Confined 2D energy contours (25, 50, 75, and 100 meV) for (001)-Si. The con-
tours are identical to the bulk counterparts. ................ . 69

3-22 The 2D energy contours (25, 50, 75, and 100 meV) for bulk (110)-Si under (a)
no stress; (b) uniaxial stress along ( 110); and (c) uniaxial stress along (111). .70

3-23 Confined 2D energy contours (25, 50, 75, and 100 meV) for (110)-Si. The con-
fined contours are totally different from their bulk counterparts which -, '-.. -r
significant confinement effect. .................. ...... 71

3-24 Ground state subband hole population under different stress. . .... 72

3-25 Stress effect on the 2 dimensional density-of-states of the ground state subband. 74

3-26 Two dimensional density-of-states at E=4kT. .................. 75

3-27 Strain effect on (a) acoustic phonon, and (b) optical phonon scattering rate. .. 76

3-28 Strain effect on surface roughness scattering rate. .... . ... 77

3-29 Hole mobility gain contribution from (a) effective mass reduction; and (b) phonon
scattering rate suppression for p-MOSFETs under biaxial and uniaxial stress. 79

4-1 Hole mobility vs SOI thickness for single gate SOI pMOS. The mobility decreases
with the thickness due to structural confinement. ............... .. 83

4-2 Hole mobility for single gate SOI pMOS vs uniaxial stress at charge density p
1 x 1013/cm 2 . . . . . . . . . 84

4-3 Hole mobility enhancement factor of UTB SOI SG devices vs uniaxial compres-
sive stress at charge density p 1 x 1013/cm2. ................. 85

4-4 Subband splitting UTB SOI SG devices vs uniaxial compressive stress at charge
density p 1 x 1013/cm 2. .................. .. .......... 86









4-5 Comparison of the subband splitting of double gate and single gate MOSFETs. 87

4-6 Hole mobility of SDG devices under uniaxial compressive stress at charge den-
sity p = 1013/cm 2. ................................ 88

4-7 Hole mobility enhancement factor of SDG MOSFETs vs uniaxial compressive
stress at charge density p 1 x 1013/c2. ............ . 89

4-8 Hole mobility of FinFETs under uniaxial stress compared with bulk (110)-oriented
devices at charge density p 1 x 1013/cm2. ................ . 90

4-9 Hole mobility enhancement factor of FinFETs under uniaxial compressive stress
at charge density p 1 x 1013/cm2. ............... .... 91

4-10 Hole mobility gain contribution from effective mass and phonon scattering sup-
pression under uniaxial compression for (110)/(110) FinFETs compared with
SG (110)/(110) p-MOSFETs at charge density p 1 x 1013/c2. . ... 92

5-1 Germanium hole mobility vs effective electric field. .. . . ..... 97

5-2 Germanium and silicon hole mobility under biaxial tensile stress where the in-
version hole concentration is 1 x 1013/cm2. ............... .. .. 98

5-3 Germanium and silicon hole mobility under biaxial compressive stress where
the inversion hole concentration is 1 x 1013/cm2. ................ 99

5-4 Germanium and silicon hole mobility on (001)-oriented device under uniaxial
compressive stress where the inversion hole concentration is 1 x 1013/cm2. .. 100

5-5 Germanium and silicon hole mobility on (110)-oriented device under uniaxial
compressive stress where the inversion hole concentration is 1 x 1013/cm2. .. 101

5-6 E-k diagrams for Ge under (a) no stress; (b) 1 GPa biaxial tensile stress; (c) 1
GPa biaxial compressive stress; and (d) 1 GPa uniaxial compressive stress. .. 103

5-7 Conductivity effective mass vs biaxial tensile stress: (a) C('!h im, I direction (<110>)
and (b) out-of-plane direction. ............... ......... 104

5-8 Conductivity effective mass vs biaxial compressive stress: (a) C(! ,ii,, I direction
(<110>) and (b) out-of-plane direction. ............... ...... 105

5-9 Conductivity effective mass vs uniaxial compressive stress: (a) C(, ,ii,, I direc-
tion (<110>) and (b) out-of-plane direction. ................ . 106

5-10 25meV energy contours for unstressed Ge: (a) Heavy-hole; (b) Light-hole ..... 108

5-11 25meV energy contours for biaxial compressive stressed Ge: (a) Top band; (b)
Bottom band . ............... ............... .. 108









5-12 25meV energy contours for biaxial tensile stressed Ge: (a) Top band; (b) Bot-
tom band ................... ............ ...... 109

5-13 25meV energy contours for uniaxially compressive stressed Ge: (a) Top band;
(b) Bottom band .................. ................ .. 109

5-14 Ge subband splitting under different stress. ................... 110

5-15 Normalized ground state subband E-k diagram vs biaxial compressive stress. .. 112

5-16 Two dimensional density-of-states of the ground state subband for Si and Ge at
(a)E 5meV; (b)E 2kT 52meV under uniaxial compressive stress. ...... ..113

5-17 Phonon scattering rate vs uniaxial compressive stress: (a) Acoustic phonon,
and (b) optical phonon. .................. .......... 115

5-18 Surface roughness scattering rate vs uniaxial compressive stress for Ge and Si. 116

5-19 Mobility enhancement contribution from effective mass change (solid lines) and
phonon scattering rate change (dashed lines) for Si and Ge under uniaxial com-
pressive stress .............. ................. .. 117

5-20 Confined 2D energy contours for (001)-oriented Ge pMOS with uniaxial com-
pressive stress .................. ................. .. 117

5-21 Confined 2D energy contours for (H0)-oriented Ge pMOS with uniaxial com-
pressive stress .................. ................. .. 118

A-1 Stress distribution on crystals. .................. ....... 123









LIST OF TABLES


Table page

2-1 Luttinger-Kohn parameters, deformation potentials and split-off energy for sili-
con and germanium ................ ............. .. 31

3-1 Calculated and measured piezoresistance coefficients for Si pMOSFETs with
(001) or (110) surface orientation. The first value of each pair is from measure-
ments and the second is from calculation. .................. ..... 40

A-1 Elastic stiffnesses Cij in units of 101N/m2 and compliances Sij in units of 10-11m2/N126









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STRAIN EFFECTS ON HOLE MOBILITY OF SILICON AND GERMANIUM P-TYPE
METAL-OXIDE-SEMICONDUCTOR FIELD-EFFECT-TRANSISTORS

By

Guangyu Sun

August 2007

('C! i: Scott E. Thompson
Major: Electrical and Computer Engineering

My research explores the strain enhanced hole mobility in silicon (Si) and germanium

(Ge) p-type metal-oxide-semiconductor field-effect-transistors (p-MOSFETs). The piezore-

sistance coefficients are calculated and measured via wafer bending experiments. With

good agreement in the measured and calculated small stress piezoresistance coefficients,

k p calculations are used to give physical insights into hole mobility enhancement at

large stress (3 GPa for Si and 6 GPa for Ge) for stresses of technological importance:

in-plane biaxial and channel-direction uniaxial stress on (001) and (110)-surface oriented

p-MOSFETs with (110) and (111) channels.

The mathematical definition of strain and stress is introduced and the transformation

between the strain and stress tensor is demonstrated. Self-consistent calculation of

Schrodinger Equation and Poisson Equation is applied to study the potential and subband

energy levels in the inversion 1vi--. Subband structures, two-dimensional (2D) density-

of-states (DOS), hole effective mass, phonon and surface roughness scattering rate are

evaluated numerically and the hole mobility is obtained from a linearization of Boltzmann

Equation.

The results show that hole mobility saturates at large stress. Under biaxial tensile

stress, the hole mobility is degraded at small stress due to the subtractive nature of the

strain and quantum confinement effects. At large stress, hole mobility is improved via the

suppression of the phonon scattering. Biaxial compressive stress improves hole mobility









slightly. Uniaxial compressive stress enhances the hole mobility monotonically as the stress

increases. In (001) surface oriented p-MOSFETs, the maximum enhancement factor is

35(0', for Si and 1,1111', for Ge. The enhancement of (110) p-MOSFETs is smaller than

(001) p-MOSFETs due to the strong quantum confinement and low DOS of the ground

state subband. For (001) p-MOSFETs, the dominant factor to improve the hole mobility

is the hole effective mass reduction at small stress and phonon scattering rate suppression

at large stress. For (110) p-MOSFETs, the hole effective mass and phonon scattering rate

are constant at large stress due to the saturation of the subband splitting and DOS caused

by the strong confinement.

Strain effects on non-classical devices (single-gate (SG) silicon-on-insulator (SOI)

and double-gate (DG) p-MOSFETs) are also investigated. The calculation shows that

the mobility enhancement for SG SOI and DG (001) p-MOSFETs is similar to traditional

Si p-MOSFETs. Hole mobility enhancement in FinFETs is more than traditional (110)

p-MOSFETs due to the subband modulation.









CHAPTER 1
INTRODUCTION AND OVERVIEW

Metal-oxide-semiconductor field-effect transistors (\ OSFETs) have been scaled

down .,-.--ressively to achieve density, speed and power improvement since 1960s [1]. As

the channel length is scaled to submicron even nanoscale level, the simple scaling of

complementary metal-oxide-semiconductor (C\ IOS) devices brings severe short-channel

effects (SCEs) such as threshold voltage roll-off, degraded subthreshold slope, and drain

induced barrier lowering (DIBL). Oxide thickness has to be reduced to sub-10 nm (about

1 nm in the state-of-the-art technology) and channel doping has to be increased up to

1019/cm3 in order to maintain good control of the channel [1]. The thin oxide and the

high channel doping result in high vertical electric field in the channel that severely

reduces the carrier mobility. Further scaling of the devices does not bring performance

improvement due to carrier mobility degradation.

With nothing to replace silicon C'\ OS devices in the near future and the need

to maintain performance improvements and Moore's law, feature enhanced Si C'\ IOS

technology has been recognized as the driver for the microelectronics industry. Strain is

one key feature to enhance the performance of Si MOSFETs. Biaxial tensile strain has

been investigated both experimentally and theoretically in C'\ IOS technology [2, 3, 4].

It improves the electron mobility [5], but degrades the hole mobility at low stress range

(< 500MPa) [3]. Recently, uniaxial stress has been applied to Intel's 90, 65, and 45-nm

technologies to improve the drive current without significantly increased manufacturing

complexity [5, 6].

The goal of this dissertation is to provide physical insights into the strain enhanced

hole mobility in Si and Ge p-MOSFETs. Before we investigate the hole mobility, the his-

tory of strain technology and the methods to apply strain to a transistor are is discussed

in this chapter. The organization of the dissertation is also introduced.









1.1 History of Strain in Semiconductors

The epitaxial growth of semiconductor lw.r-i is not new. The basis of 1i, I Iivs

experimental guide, piezoresistance [7, 8], and the theoretical approach to the strain effect,

i.e., deformation potential theory, can be traced back to the 1950s. But not until in the

early 1980's did scientists and engineers start to realize that strain could be a powerful

tool to modify the band structure of semiconductors in a beneficial and predictable way

[9, 10].

Deformation potential theory, which defines the concept of strain induced energy

shift of the semiconductor, was first developed to account for the coupling between

the acoustic waves and electrons in solids by Bardeen and Shockley [11], who stated

that the local shift of energy bands by the acoustic phonon would be produced by

an equivalent extrinsic strain, hence the energy shifts by both intrinsic and extrinsic

strain can be described in the same deformation potential framework. The deformation

potential theory was applied by Herring and Vogt [12] in 1955 in their transport studies of

semiconductor conduction bands. A set of symbols, E, was used to label the deformation

potentials. Herring and Vogt [12] also summarized the independent deformation potentials

constrained by symmetry at different conduction band valleys. At the F point, another set

of symbols are commonly used: ac, a,, b, and d, where a,, b, and d are three independent

valence band deformation potentials which have a correspondence to the Luttinger

parameters [13] employ, .1 in band calculations. The k p method we use in this work relies

on these three deformation potentials to account for the strain effects.

Smith measured the piezoresistance coefficients for n and p-type strained bulk

silicon and germanium in 1954 [7]. This was the first experimental work that studied

strain effects on semiconductor transport. Herring and Vogt used Shockley's band

model and ascribed the electron mobility change to two strain effects, "electron transfer

effect" and inter-valley scattering rate change caused by valley energy shift [12]. This









is essentially the same physics that explains the strain enhanced mobility in silicon

n-channel MOSFETs.

Piezoresistance coefficients are widely used in the industry due to its simplicity in

representing the semiconductor transport properties (mobility, resistance, and et al.)

under strain. It is defined as the relative resistance change with the stress applied on the

semiconductor. Piezoresistance coefficient (7r) can be expressed as


1 Ap-)
lAp (1 1)
a p

where a is the applied stress and p is the resistivity of the semiconductor.

In 1968, Colman [8] measured the piezoresistance coefficients in p-type inversion

l- ir-. This was the first time that strain effect on hole transport was investigated in the

inversion l-r-i~. The similarity and difference of the piezoresistance coefficients compared

with the bulk silicon was explained qualitatively in that work.

The first silicon n-channel MOSFET which used biaxial stress to improve the

electron mobility was demonstrated by Welser et al. [14] in 1992. The work showed that

the electron mobility was improved by 2.2 times. A biaxial stressed silicon p-channel

MOSFET was first reported by N i- I1: et al. [15] in 1993 where the hole mobility was

enhanced by 1.5 times. In 1995, Rim [16] showed the hole mobility enhancement in

silicon p-MOSFETs on top of Sil_-Gex substrate with different germanium components.

The idea of using longitudinal uniaxial stress to improve the performance of MOSFETs

was activated by Ito et al. [17] and Shimizu et al. [18] in the late 1990's through the

investigations of introducing high stress capping lI- r- deposited on MOSFETs to

induce channel stress. Gannavaram et al. [19] proposed Sil_-Ge1 in the source and drain

region for higher boron activation and reduced external resistance which also furnished

a technically convenient means to employ uniaxial channel stress. These studies opened

the gate to use strain as active factor in VLSI device design and resulted in extensive

industrial applications.









1.2 Apply Strain to A Transistor

Strain in the channel of Si and Ge MOSFETs is achieved by applying mechanical

stress to the wafer. The properties, and relations of strain and stress can be found in

the appendix. Here we first introduce how to apply biaxial and uniaxial stress in Si

MOSFETs.

For (001) wafer, biaxial tensile stress in Si MOSFETs is applied to the channel by

using the Sil-_Gex substrate. The lattice mismatch stretches silicon atoms in both (100)

and (010) directions which is illustrated in Figure 1-1. The percentage of germanium

content in the substrate determines the magnitude of the strain. This in-plane tensile

strain can also be achieved by applying uniaxial compressive stress from the out-of-plane

direction [20] with capping 1~v. r. The out-of-plane uniaxial compression is equivalent to

the in-plane biaxial tension in determining the transport properties of Si. The details are

shown in the appendix. For Ge MOSFETs, biaxial tensile stress is not applicable due to

its large lattice constant. Biaxial compressive stress is usually introduced by applying Si

or Sil-xGe substrate.


I train" "r !'a^ d_




Relaxed SiGe

Substrate



Figure 1-1. Schematic diagram of biaxial tensile stressed Si-MOSFET on relaxed Sil-zGez
l-,v,-r


Uniaxial stress can be applied from out-of-plane, in-plane longitudinal (parallel to

the channel), or in-plane transverse (perpendicular to the channel) direction. The in-plane

longitudinal stress is applied to the channel by either doping germanium to source and









drain or depositing compressive or tensile capping 1i,- r on top of the device which is

shown in Figure 1-2 [20].


45nm Gae High stress film












p-type MOSFET n-type MOSFET

Figure 1-2. Uniaxial stressed Si-MOSFET with Sil_,Ge, Source/Drain or highly stressed
capping li ,-r


Without further clarification, uniaxial stress in this work represents in-plane uniaxial

longitudinal stress. It is normally along (110) since it is the classical channel direction.

Biaxial stress means in-plane biaxial stress. For (O110)oriented wafer, biaxial stress

is employ, ,1 in both parallel and perpendicular direction to the channel (<110> and

<100> directions). The strain in those two directions are not as same as (001)-oriented

wafer (<100> and <010>-directions) due to the different Young's Modulus in <110>

and <100> directions.

1.3 Main Contributions of My Research

Strain enhanced hole mobility has been reported experimentally at small stress.

Little theoretical work has been done to provide physical insights into hole mobility

enhancement under large stress, especially uniaxial compressive stress. Strain effects on

hybrid ((ll0)-surface oriented), non-classical and Ge p-MOSFETs are not understood

either. In this work, piezoresistance coefficients are calculated and measured on (110) Si

p-MOSFETs. Physics of uniaxial stress enhanced hole mobility in (110) p-MOSFETs is









studied for the first time. The hole mobility dependence on device surface orientation

is calculated and the different quantum confinement effect is discussed. Strain-induced

changes in hole effective mass, subband structures, density-of-states (DOS), phonon and

surface roughness scattering rate are analyzed numerically. The results show that under

uniaxial stress, 350' and 1 1I 1'. mobility enhancement are achieved in (001) Si and Ge

p-MOSFETs, respectively. The more enhancement in Ge p-MOSFETs is due to smaller

hole effective mass of Ge under stress. In (110) Si and Ge p-MOSFETs, it is reported

for the first time that the maximum enhancement factor is only 10i1' due to the strong

quantum confinement undermining the strain effect.

Strain induced hole mobility enhancement is studied theoretically for the first time

in ultra-thin-body (UTB) non-classical p-MOSFETs, including single-gate (SG) silicon-

on-insulator (SOI), (001) symmetrical double-gate (SDG) p-MOSFETs, and (110) p-type

FinFETs. For SG SOI p-MOSFETs, the strain effects are as same as traditional Si

p-MOSFETs. For (001) SDG p-MOSFETs and (110) FinFETs, subband modulation

is found when the channel thickness is smaller than 20 nm. As the stress increases,

the mobility enhancement in (001) SDG p-MOSFETs is comparable to traditional

SG p-MOSFETs. For FinFETs, the form factors are much smaller than SG (110) p-

MOSFETs and the change with stress is larger which -, ii-.- -r more reduction of the

phonon scattering rate. Therefore, the strain-induced hole mobility enhancement (2111'.) is

larger than single gate (110) p-MOSFETs (1011' .).

1.4 Brief Description of The Dissertation

The main purpose of my research is to provide a simple but accurate physical

insight into strain effects on hole mobility in Si and Ge inversion l, -i~. We begin by

introducing the physics model. A six-band k p model with strain effects is derived

and finite difference method (FDM) is introduced briefly. Self-consistent calculation of

Schridinger Equation and Poisson Equation is discussed. The isotropic approximation of









scattering rate calculation is showed. In the calculation of the hole mobility, the Kubo-

Greenwood formula, which is from a linearization of Boltzmann Equation, is introduced.

Strain enhanced hole mobility in single-gate Si p-MOSFETs is then discussed. The

unstrained Si hole mobility versus device surface orientation and vertical electric field is

calculated. Hole mobility under biaxial and uniaxial stress in (001) and (110) p-MOSFETs

is showed. The band structure of bulk silicon under strain is discussed. In the Si inversion

1i.- -i~, the confined energy contours, subband splitting, hole population in ground state

subband, two-dimensional (2D) density-of-states (DOS), phonon and surface roughness

scattering rate are evaluated. The difference of strain induced hole mobility enhancement

in (001) and (110) p-MOSFETs under biaxial and uniaxial stress is explained.

Uniaxial strain-induced hole mobility enhancement is calculated for UTB non-

classical p-MOSFETs, including single-gate SOI, (001) SDG p-MOSFETs, and (110)

p-type FinFETs. The similarity and difference from the traditional Si p-MOSFETs are

discussed and physical insights are given.

Strain induced hole mobility enhancement in Ge p-MOSFETs is discussed. Un-

strained hole mobility in (001) and (110) Ge p-MOSFETs is calculated. Strain effect on

hole mobility in Sil_-Gex with arbitrary Ge components is evaluated. To understand the

physics, the bulk valence band structure and hole effective mass with strain effects are

calculated. In the inversion l. -,-i, the subband structure, 2D DOS and scattering rate are

calculated and their relation to hole mobility is analyzed.

We conclude with the results that we obtain in this dissertation and s-l::.- -1 possible

future research on strained Si and Ge.









CHAPTER 2
K P MODEL AND HOLE MOBILITY

Global descriptions of the dispersion relations of bulk materials can be obtained

via pseudo-potential or tight-binding methods [21]. However, such global solution over

the whole Brillouin zone is unnecessary for many aspects of semiconductor electronic

properties. What is needed is the knowledge of the dispersion relations over a small k

around the band extrema [21]. k p method is widely used in i i.-- I,1 vIs quantum well and

quantum dots calculation due to its simplicity and accuracy regarding the properties in

the vicinity of conduction band and valence band edges which govern most optical and

electronic phenomena.

To study the uniaxial or biaxial strain effect on hole mobility in the inversion 1~,-l. s,

a 6-band k p model, Luttinger-Kohn's Hamiltonian [13], is utilized in this work. In

this chapter, k p method and the derivation of the luttinger-Kohn's Hamiltonian is

introduced first. Then the procedure calculating the hole mobility is explained. Finally,

the evaluation of scattering mechanisms, mainly the phonon and surface roughness

II. 11. i i.- is discussed.

In the calculation of the hole mobility with strain effect in the inversion 1~,-l s,

Schr6dinger Equation and Poisson Equation are solved self-consistently to simulate the

potential energy in the channel. The subband structure and the two-dimensional density-

of-states (2D DOS) of each subband are calculated and the scattering relaxation time

is evaluated in k space. Finally, hole mobility is obtained from a linearization of the

Boltzmann equation.

2.1 The k p Method

2.1.1 Introduction to k p Method

The k p method [21, 22, 23] is essentially based on the perturbation theory and

was first introduced by Bardeen [24] and Seitz [25]. It is also referred to as effective mass

theory in literatures. The k p method is most useful for analyzing the band structure









near the extrema (ko) of the band. In the case of the band structure near the F point, i.e.

valence band edge of silicon and germanium, ko = 0.

For an electron in a periodic potential


V(r) V(r + R), (2-1)

where R = nal + n2a2 + n3a3, and al, a2, a3 are the lattice vectors, and ni, 72, and

n3 are integers, the electron wave function can be described by the Schr6dinger equation


H b(r) + V(r) (r) [2- V2 + V(r) b (r) E- (k)b (r) (2-2)
2mo I 2mo I
where p h= V/i is the momentum operator, mo is the free electron mass, and V(r)

represents the potential including the effective lattice periodic potential caused by the

nuclei, ions and core electrons or the potential due to the exchange correlation, impurities,

etc.

The solution of the Schr6dinger equation


H k(r) E k(r) (2-3)

satisfies the condition



bk(r + R) e= ikrk(r) (2-4)

k (r) = eik-r k(r) (2-5)

where

Uk(r +R) Uk(r), (2-6)

and k is the wave vector. Equations 2-4, 2-5 and 2-6 is the Bloch theorem, which gives

the properties of the wave function of an electron in a periodic potential V(r).

The eigenvalues for Equation 2-3 can be categorized into a series of bands E, n =

1, 2,... [26] due to the perturbation of the periodic potential at the Brillouin zone edge.









Consider the Schr6dinger equation in the nth band with a wave vector k,


P + V(r) blk(r) E- (k) .k(r). (2-7)
2mo I

Inserting the Bloch function Equation 2-5 into Equation 2-7, we have

[p2 h2k2 h 1
+ + k p + V(r) Uk(r) E(k)Unk(r). (2-8)
2mo 2mo mo

Including the spin-orbit interaction term


S(7 x VV) p (2-9)

in the Hamiltonian and simplifying the equation, Equation 2-8 becomes

p2 h 2k2 hk h h
+ + P + (x VV) + 2a x VV) p + V(r) Unk
2mo 2mo mo \ i 1,2 j 1 ',-,*

EE,(k)Unk(r). (2-10)



where c is the speed of light and a is the Pauli spin matrix. a has the components [22]


0 1 0 -i 1 0
(rx = y= z= (2-11)
1 0 i 0 0 -1

Rewriting the Hamiltonian in Equation 2.1.1, we have


[Ho + W(k)]unk = EnkUnk, (2-12)

where
p2 h
Ho =2 + (a x VV) p + V(r) (2-13)
2mo *1 -i,

and
hk ( h N h2k2
W(k) =- p+ ( x VV) + (2-14)
mo 1 ,,:2 2mo
Since only W(k) depends on wave vector k, Equation 2-13 can be used to evaluate

the band property at ko. If the Hamiltonian Ho has a complete set of orthonormal









eigenfunctions at k = 0, uo, i.e.,


Houno Enouno,


(2-15)


theoretically any function with lattice periodicity can be expanded using eigenfunctions

uno. Substituting the expression


(2-16)


Unk = ZCn(k)unmo
in


into Equation 2.1.1, multiplying from the left by u*0, integrating and using the orthonor-

mality of the basis functions, we have


(Eno Ek + )6nm Un0 + 1, 2 (7 x VV) U|no) c(k) 0.
m 2mo mo .2
(2-17)

Solving this matrix equation gives us both the exact eigenstates and eigenenergies. As

we mentioned earlier, only the dispersion relations over a small k range around the band

extrema are important describing the electronic properties of the semiconductor. Only

energetically .,ili i,:ent bands are normally considered when studying the k expansion of

one specific band for simpleness. To pursue acceptable solutions when k increases, one has

to increase the number of the basis states, or consider higher order perturbations, or even

both.

Neglecting the non-diagonal terms in Equation 2-17 for small k, the eigenfunction is

Unk Uno, and the corresponding eigenvalue is given by Enk = Eno + >- The solution

can be improved using the second order perturbation theory, i.e.

S2k2 UnO I H' umo) (umo H' uno)
Enk = Eno + + (2-18)
2mo mn Eno ~- Emo

where


H m' 1 i ,,, 2 ( x )


(2-19)









(uo (P + 4moc2(xvv) uno) = was applied in the calculation, which holds for a cubic
lattice periodic Hamiltonian due to the crystal symmetry. If we write


S= p + h ( x VV) (2-20)
1 t, ,,,,. 2

the second order eigenenergies can be written as

Sh2k2 h2 lrnm, k|2
Enk = Eno + m+ (2-21)
2mo0 'n mn n Emo

Equation 2-21 can also be expressed as


Enk = Eno+ ( + kk3, (2-22)
2 a,/3 m a/3

where
1 1 2 _- -_
m 6oa, + 2 E (2-23)
m* mo ',O mT, En0o Emo

m* in Equation 2-23 is the effective mass tensor, and a, = x, y, z. The effective mass

generally is anisotropic and k dependent. In the vicinity of the F point, sometimes m* can

be treated as k-independent, since at this level of approximation, the eigenenergies close to

the F point only depend quadratically on k [22, 23].

2.1.2 Kane's Model

Expanding in a complete set of orthonormal basis states in Equation 2-17 gives exact

solutions of both eigenenergies and eigenfunctions. In reality,it is almost impossible to

include a complete set of basis states, therefore only strongly coupled bands are included

in usual k p formalism, and the influence of the energetically distant bands is treated as

perturbation.

In Kane's model for Si, Ge, or III-V semiconductors, four bands are considered as

strongly couples bands-the conduction, heavy-hole (HH), light-hole (LH), and te spin-

orbit split-off (SO) bands are considered, which have double degeneracy with their spin

counterparts. The rest bands are treated as perturbation and can be analyzed with the

second order perturbation theory.









Our goal is to find the eigenvalue E of Equation 2.1.1 with eigenfunction


Unk(r) -= aUno(r)


(2-24)


The band edge functions uo(r) are

Conduction band: IS T), S 1) for eigenenergy E, (s-type),

Valence band: IX T), IY T), I T), IX 1), IY 1), Z 1) for eigenenergy E, (p-type).

Normally the following eight basis functions are chosen
)iS ), x-iY T Z), IX+-lY T)

and

TiS ), eigt bsisY ), Ists fr-Y )K
The eight basis states for Kane's model are


1
1 = 2'
3
U2= ,
2
3
2'
1
4 = | ,
24
1
5 '
3
6 = | ,
2
3
U7 = | ,
7 '
1
Us = ,
2


IS T) = S ),


IHH T) '|(X + iY) ),
v2
1 2
|LHT) |(X+iY) + |IZT),

ISO =) |\(X + iY)1) + |Z ),

= IS 1) = ),

= HH 1) = (X- iY) ),


= LH 1)

= SO )


v/z

l|(X iY) T) + Z ),

(X- Y) ) Z ).


(2-25)


This set of basis states is a unitary transformation of the basis functions and the eigen-

functions of the Hamiltonian 2-13. The eigenenergies for IS), IHH), ILH) and ISO) at

k = 0 are E,, 0, 0, -A, respectively, where E, is the band gap, and the energy of the top










of the valence band (HH and LH) is chosen to be 0. A is the split-off band energy, which

is 44meV for Si and 296meV for Ge.

At this level of approximation, the bands are still flat because the Hamiltonian 2-13

is k-independent. Including W(k) in Equation 2-14 into the Hamiltonian, and defining

Kane's parameter as


we obtain a matrix expression for


Eg + h 2
2mo
Pk_

- Pk+

- Pk+

0

0

- 3Pk,
- Pk
v1P3


Pk+
h2k2
2mo
0

0

0

0

0

0
0


'-Pk_

0
h2 k2
2TnO

0
o


P Pk

0

0

0


-ih
P (SI|Z),
mo

the Hamiltonian H =


-3Pk_

0

0

-A +
2mo
SPk,

0

0

0


0

0

- 3Pk,

Pk,

Eg + h2k2

Pk+

SPk_

3Pk_


(2-26)


Ho + W(k), i.e.,


0 Pk,

0 0

0 0

0 0

Pk_ Pk+
hk2 0
2m0O
0 h2k2
o mo
o o


Pkz

0

0

0
Pk+

0

0

A +


(2-27)

where k+ = k + iky, k_ = kx iky, and k,, ky, k, are the cartesian components of k.

The Hamiltonian 2-27 is easy to diagonalize to find the eigenenergies and eigenstates as

functions of k. We have eight eigenenergies, but due to spin degeneracy, there are only

four different eigenenergies listed below. For the conduction band,


1 1 4P2 2P2
S + + + A)
m mo+ 32E, 3+2(E, + A)


For the light hole and split-off bands,


k2k2/
Elh =
2mlh

A 2k2
Eso = -A -
2mso '


1
mlh

1
mso


1 4P2
p2
mo 3k2E,'

1 2P2
Smo 2+)
mo 3h (E9 + A)


h2k2
Ec = Eg + --
2m,c


(2-28)


(2-29)


(2-30)









For the heavy hole band we have


h2k2 1 1
Ehh =- (2-31)
2mhh mhh mo

These results are not complete since the effects of higher bands have not been included.

They will be taken into account next when discussing the Luttinger-Kohn model.

2.1.3 Luttinger-Kohn's Hamiltonian

For Si and Ge hole transport, we are only interested in the six valence bands (doubly

degenerate HH, LH, and SO). The coupling to the two conduction bands in Kane's

model is ignored due to the large band gap. It is convenient to use Lbwdin's perturbation

method [27] where the six valence bands are treated in class A and the rest bands are put

in class B.

We label class A with subscript n and class B with subscript 7. Wave function uk(r)

can be expanded as

A B
Uk(r) (k) (r) + a.(k) uo(r). (2-32)
n 7
C'! .... the eigenstates for class A, we have






3 ) -
3 1 -1
U3 I )= LH 1) = I(X+ iY) +|z
2'2 2,
311 2
U3 | ) ILH1)= (X iY) + Z{),
2'2 6 3
33 1
= I| ) = HH 1) = (X -Y) ),
11 1 1
U5 I ) ISO ) =t(X + iY) +) Z ),
2' 23
1 1 1 1
U6 |, ) ISO 1) = |(X -Y) T) |Z 1). (2-33)
2With Lwdin's method we only need to solve the eigenequation

With Liwdin's method we only need to solve the eigenequation











A
n(U
n


B H HH.
UA H,, + E -
i7nn 0 E


Hjn = (ujo IHIuo)


) 2k2
[Ej(0) + ] 6
2mo


H' = (ujo k In|uo)
Tmo


Let UjA = Dj, DjT can be expressed as



Dj = Ej(0)6j


where D"W is defined as


h2
n"D3 6j,5
in 2mo

(2-39)



To express DjT explicitly, we difine


hkO 38
So


af + D
a8


B
+E
,-7


mo(Eo -


Pj/P7n
- E7)


h2
Ao =
2mo
h2
Bo =-
2moo


h2 B x x
1,1' Eo E


1, 7 Eo E7
k2P TPT
+ ,, y G


h2 B pxp y py
Co -
i ,-, Eo E7


Then define the Luttinger parameters 71, 72, and 73 as


where


Ej,)a,(k) 0


(2-34)


B
Hj,+ E
Y3j,n


(2-35)


H7 H'
E E7



(j, ne A)


(2-36)


(j A,7 A)


(2-37)


(2-38)


(2-40)











h2 1
-h71 (ao + 2Bo)
2mo 3
h2 1
2 72 ( ao B)
2mo 6

2 73 C (2-41)
2mo 6

Finally we obtain the Luttinger-Kohn Hamiltonian




P +Q -S R 0 IS v2R 12
2 |3 1
-S+ P-Q 0 R -v2Q S

R+ 0 P-Q S S+ v2Q 22
H _) 3 1 (2-42)
T 3 3)
-' S+ Q 2s -VFR P+ 0o

2 1 1

where,


P_= 7(i (k2 1(k 2 + k 2),





Zmo 2
2mo 3[-(kx + 2I1 ,



S = 2 373(k iky)kz. (2-43)


When strain is present in the semiconductor, P, Q, R, and S in Equation 2-43 can be

resolved to two parts: k p terms (Pk, Qk, Rk, and Sk) and strain terms (Pe, Qe, Re, and

Se). They can be expressed as [13]












P = Pk + P Q2 = Qk + ,,

R = Rk + R S SSk+ S,,

Pk ( h2 71 (k 2+ k 2+ k ),

Qk 72(k + k 2k ),

Rk [72 (k2 k ) + 2i- .-, I ], (2-44)


2mo2
Pe = -a(6xzz + 6yy + 6zz),

Q = b(xx + ySy 2eZZ),
2
C v3 b(xx yy) id(xy

Se = -d(
where ij is the symmetric strain tensor as shown in C'! ipter 1; av, b, and d are the

Bir-Pikus deformation potentials for valence band; A is the spin-orbit split-off energy, and

the basis function Ij, m) denotes the Bloch wave function at the zone center. Energy zero

is taken to be the top of the unstrained valence band. Table 1 shows the parameters for

silicon and germanium [28].

Table 2-1. Luttinger-Kohn parameters, deformation potentials and split-off energy for
silicon and germanium.
71 72 73 av(eV) b(eV) d(eV) A(eV)
Si 4.22 0.39 1.44 2.46 -2.35 -5.3 0.044
Ge 13.35 4.25 5.69 2.09 -2.55 -5.3 0.296









2.2 Hole Mobility in Inversion Layers

2.2.1 Self-consistent Procedure

For Si or Ge pMOSFETs, holes are confined in the z-direction quantum well formed

by the Si/SiO2 interface and the valence band edge. Since the hole energy is not continu-

ous along z-direction kz should be replaced by -i- in Equation 2-45. Coordinate system

transformation is needed to calculate cases with other surface orientation.

Subband energy can be evaluated by solving Schr6dinger Equation,


[H(k, z) + V(z)]k(z) = E(k)k(z) (2-45)

where V(z) defines the potential energy in the quantum well. Triangular potential

approximation is widely used in simulations for simplicity. Stern [29] stated that it should

not be used when mobile charges are present. In order to accurately simulate the potential

in the quantum well, Schr6dinger Equation 2-45 is solved self-consistently with Poisson

Equation


d2 q2
dz2VH(z) = [p(z) n(z) + N(z)- N (z)] (2-46)

where p(z) and n(z) are mobile hole and electron density, ND+(z) and NA(z) are space

charge density.

To numerically evaluate Schr6dinger Equation and Poisson Equation, Finite-

Difference Method is utilized. The equations are evaluated on a z mesh of N. points

in the interval (0, zax) [3, 30, 31], where zax here is the sum of the thickness of silicon

1.i-r and oxide ly .-r. This yields a 6Nz x 6Nz eigenvalue problem of the tridiagonal block

form [3]. Schr6dinger Equation becomes












H_ H,_1 H+ 0 0 _

S0 H_ H H+ 0 =E(k) (2-47)

0 0 H_ Hii H '


where each ,' = (zi) is a six-component column-vector Qj(zi), the index j running

over the k p basis, and H_, Hi, H+ = H_ are 6 x 6 block-diagonal difference operators,

functions of the in-plane wave-vector k.

In principle, the potential V(z) results from three terms: an image-term, ', .(z);

an exchange and correlation potential, V,,(z); and the Hartree term, VH(z) [3, 30].

Fischetti [3] -,--.-. -1- that the image potential cancels the many-body corrections given by

the exchange and correlation term and the Hartree term is focused as the solution of the

self-consistent calculation of Schr6dinger Equation 2-45 and Poisson Equation 2-46.

2.2.2 Hole Mobility

The hole mobility in inversion l1.-.-is can be calculated from a linearization of the

Boltzmann equation. The xx component of the mobility tensor can be expressed as [3]



e 2T -E) K (E,)
4h2742kBTp, JO J f E,.
Dk| K,(E,w)

x (a KE,2 V) [K,(E, ), ] fo(E)[1 fo(E)] (2-48)
\0x K, (E,6)

where p8 = Ep, is the total hole concentration in the inversion -' --r, p, is the hole

density of subband v, Tf') (K, Q) is the x-component of the momentum relaxation time in

subband v, and



fo(E) = + (2-49)
1 exp )T









is the Fermi-Dirac distribution function.

The evaluation of density-of-states (DOS) and a term need further consideration.

In energy space a maximum kinetic energy Emax for each subband is selected in order to

account correctly for the thermal occupation of the top-most subband. In our calculation,

we assumed Emx = 120meV and divided the energy space to 1200 uniform parts, then

evaluated DOS and E in each part.

2.3 Scattering Mechanisms

Phonon i l 1. i ii- impurity scattering and surface roughness scattering are involved

in C'\ OS transistors. In the linear region of p-MOSFETs, neither charged-impurity nor

neutral-impurity scattering is important [3], hence they are neglected in our calculation.

Only phonon scattering and surface roughness scattering are investigated.

2.3.1 Phonon Scattering

Carriers migrate through the crystal with properties determined by the periodic

potential associated with the array of ions at the lattice points [32]. Vibration of the ions

about their equilibrium positions introduces interaction between electrons and the ions.

This interaction induces transitions between different states. And this process is called

phonon scattering.

Phonon scattering can be categorized to acoustic phonon scattering and optical

phonon scattering based on the phase of the vibration of the 2 different atoms in one

primitive cell. Both contribute to the momentum relaxation time. Acoustic phonon

energy is negligible compared with carrier energy, while optical phonon energy is about

61.3meV for silicon and 37meV for germanium at long wavelength limit. When strain is

applied to the i --I I1 the HH and LH degeneracy is lifted at F-point, as we mentioned

previously. Therefore, the inter-band optical phonon scattering will be limited due to

band splitting and mobility is enhanced. In fact, this is only significant when strain is

high and the band splitting is beyond the optical phonon energy. The reasoning will be

shown in the following section. One should also notice that the anisotropic nature of









silicon valence bands makes the modeling of scattering rate a complicated task. Since we

only need considering scattering in F valley for holes with the diamond crystal structure,

equipartition approximation [32] is used where we replace the anisotropic hole-phonon

matrix element with appropriate angle-averaged quantities.

First for acoustic phonon, relaxation time r can be expressed as [3]


1 2rkBT2~yy
2 e 7Fp,[E,(K)] (2 50)
Tc hpu1
where Eff = 7.18eV is the effective acoustic deformation potential of the valence

band, p, is the 2-dimensional density-of-states of subband v which is defined as


K(E, )
p0~ O[E -E)1E d E, (2-51)
oK K,(E,yL)
The two-dimensional carrier scattering rate for the phonon-assisted transitions of a

carrier from an initial state in the p-th subband and a final state in the v-th subband is

proportional to the form factor


1 1 r+foc
F, W = + II, (q,)2 dqz, (2-52)
27W,,, 27 -oo
where



I, (q) (z ) iqz (v) (z) dz. (2-53)

The form factor F,, illustrates the interaction between initial state and final state

due to the wave function overlapping, where "() (z) or < ")(z) is the envelope function

at k for subband p or v, respectively. z is the coordinate perpendicular to the Si/SiO2

interface, and q, is the change in the component perpenticular to the interfaces of the

carrier momentum in a transition from the p-th subband to the v-th subband.

Following Price's pioneering work, W,, can be expressed as










1 fZmax 2 2I I
S-270 dz k ''(z) ',k (z) (2-54)

If the final state is also p-th subband, W,, represents the effective quantum well

width for the p-th subband.

Since the acoustic phonon energy is small compared with subband splitting or

even the thermal energy kT, acoustic phonon scattering is an equal-energy scattering

process [32]. The scattering rate solely depends on the density-of-states of the final

states. Strain effect on acoustic phonon scattering is smaller than that on optical phonon

scattering which is shown in our simulation.

Second, the optical phonon scattering relaxation time is expressed as [3]



1 -D2 fo [E,(K) hu 1op 1
1 DZ p, V[E,(K) Fhw, p] x t nj p + (2-55)
T-o Pw~ op 1- fo[E,(K)1 2 2

For absorption and emission, respectively, where Dp = 13.24 x 108eV/cm is the

optical deformation potential constant of the valence band, huw = 61.3eV is the silicon

optical phonon energy. Optical phonon scattering is not significantly reduced for stress

< 1GPa since the subband splitting is less than the optical phonon energy.

2.3.2 Surface Roughness Scattering

In MOSFETs, carriers are confined close to the channel-oxide interface in strong

inversion region. Thermal movement of carriers also results in collision with the interface

and hence affects the carrier mobility. This interaction depends heavily on the roughness

of the interface. Therefore, this scattering mechanism is called surface roughness scat-

tering. Surface roughness scattering can be neglected when the transverse electric field

is small, since not many carriers are present and they are not strongly confined to the

channel-oxide interface. But when the electric field is high (carrier density over 1013/cm2),

surface roughness scattering must be taken into account in mobility calculation.









Unfortunately, people are still unable to model the roughness scattering accu-

rately [33, 3]. The early formulation by Prange and Nee, Saitoh, and Ando is still the best

model available [3]. Different roughness parameters are used in different references. Here,

we'll use Gamiz' model and corresponding parameters [34].

As we know, the surface roughness scattering is caused by the roughness of the

surface and hence the abrupt potential change at Si/SiO2 interface. 2 assumptions are

needed in the simplification of the problem [34]. The first assumption is to consider

the interface between silicon and oxide is an abrupt boundary which randomly varies

according to a function A of the parallel coordinate, r, A(r). Another assumption is that

the potential V(z) close to the interface can be expressed by


8V(z)
V[z + A(r)] V(z) + A(r) () (2-56)
Oz
The scattering rate can be expressed as [34],



-EP[Ep(K)]2 )AV(z) ,(z)dz AL2
TsR (k) h Mv ,' (Z A,
ox2 d (2 57)
+ (I+ L22

In this equation -Av() is approximately equal to the effective electric field, which

means the scattering rate is proportional to the square of the electric field. Therefore,

surface roughness scattering becomes more significant when electric field reaches higher

level.

Different values for L and A are taken by different researchers to explain the exper-

imental data. Here, we use L = 20.4nm and A = 4nm [3] for silicon as -ir-'-i -I. 1 by

Fischetti. n = 0.5 [34] is chosen in this work.









2.4 Summary

The physics model used in the dissertation is reviewed in this chapter. The history

of k p method is introduced. The derivation of Kane's model and Luttinger-Kohn

Hamiltonian is showed. The calculation procedure of the hole mobility in inversion 1-i. r-

is introduced. Phonon and surface roughness scattering are taken into account as the main

scattering mechanisms. Form factors and their impact on scattering rate are discussed.

In this work, MATLAB and C codes are written to calculate the hole effective mass,

band and subband structures, and hole mobility in Si and Ge p-MOSFETs. To calculate

hole mobility dependence on device surface orientation, coordinate transformation is

performed to calculate hole mobility in (110), (111), and (112) oriented Si and Ge to

account for the different quantum confinement conditions. For different surfaces, different

surface roughness parameters are utilized to fit the interface roughness condition. DOS

and form factors are calculated in the whole k space.









CHAPTER 3
STRAIN EFFECTS ON SILICON P-MOSFETS

Hole transport in the inversion li. r of silicon p-MOSFETs under arbitrary stress and

device surface orientation is discussed in this chapter. Piezoresistance coefficients are cal-

culated and measured at stress up to 300 MPa via wafer-bending experiments for stresses

of technological importance: uniaxial compressive and biaxial tensile stress on (001)- and

(10O)-surface oriented devices. With good agreement in the measured vs calculated low

stress piezoresistance coefficients, k p calculation are used to give insight at high stress

(1-3 GPa). The results show that biaxial tensile stress degrades the hole mobility at low

stress due to the quantum confinement offsetting the strain effect. Uniaxial stress on

(001)/<110>, (110)/<110>, and (110)/<111> devices improves the hole mobility mono-

tonically. Unstressed (110)-oriented devices have superior mobility over (001)-oriented

devices due to the strong quantum confinement causing smaller conductivity effective mass

of the holes. When the stress is present, the confinement of (110)-oriented devices under-

mines the stress effect, hence the enhancement factor for (110)-oriented devices is less than

(001)-oriented devices. Hole mobility enhancement saturates as the stress increases. At

high stress, the maximum hole mobility for (001)/<110>, (110)/<110>, and (110)/<111>

devices is comparable.

Physical insights are given to explain the difference between biaxial and uniaxial

stress, and the difference of (110) and (001) p-MOSFETs. The bulk silicon valence band

structure under uniaxial compressive or biaxial tensile strain is shown and the difference

in effective mass change is calculated. The difference of the vertical electric field (quantum

confienment) effect on (001)- and (110)-oriented p-MOSFETs is explained. Subband

splitting, ground state subband hole population, and two dimensional (2D) density-of-

states (DOS) of subbands are calculated under stress. Scattering rate change with stress is

also discussed.









3.1 Piezoresistance Coefficients and Hole Mobility

Calculated and measured piezoresistance coefficients, and calculated hole mobility vs

stress and surface orientation of Si p-MOSFETs are covered in this section.

3.1.1 Piezoresistance Coefficients

Piezoresistance coefficients are widely used as an effective approach characterizing

the resistance change at low stress [7, 8]. Table 3-1 compared measured and calculated

piezoresistance coefficients. In the measurements, the stress is applied using 4-point

or concentric-ring bending of the wafers. The piezoresistance coefficients are obtained

through the linear regression of the measured resistance versus stress. The actual strain

in the devices is measured through the resistance change of a strain gauge mounted on

the sample, and via the laser-detected curvature change of bent wafer. In Table 3-1, 7rL,

TTT, and 7Biaxial represent longitudinal, transverse, and biaxial piezoresistance coefficients,

respectively.

Table 3-1. Calculated and measured piezoresistance coefficients for Si pMOSFETs with
(001) or (110) surface orientation. The first value of each pair is from
measurements and the second is from calculation.
Substrate (001) (110)
Channel <110> <110> <110>
7L 71.7 [6]/72.2 27.3/34 86 [35/79.1
T2 -33.8 [6]/ 45.8 -5.1/-6.6 -50 [35]/ 43
7Biaxial 40/35.7 35.7/28.7 15.1/10.2


Both measured and calculated results in table 3-1 show that under uniaxial longitu-

dinal stress, (110)/<111> devices have the largest piezoresistance coefficient, followed by

the (001)/<110> devices. The piezoresistance coefficient of (110)/<110> devices is the

lowest. Under uniaxial transverse stress, the piezoresistance coefficients are smaller than

longitudial stress for all p-MOSFETs. The table also shows that the biaxial tensile strain

increases the channel resistance and hence degrades the hole mobility at low stress.








3.1.2 Hole Mobility vs Surface Orientation
Surface and channel orientation dependence of electron and hole mobility has been
investigated experimentally since 1960's. Sato [36] reported that for p-type devices with
<110> channel, the mobility is the highest in (110)-oriented and lowest in (001)-oriented
p-MOSFETs. The hole mobility on a few surface orientations is simulated and compared
with Sato's experimental results [36, 37] in Figure 3-1. Good agreement is found between
the calculation and the experimental data. Two different surface roughness models [3, 34]
are used in the calculation. Both models are quite accurate and in the following results,
Gamiz' surface roughness model is utilized.

250

With Gamiz' surface
S200 roughness model *0
*) *

S150 0


S100A.


2 50 With Fischetti's surface
> 50 With Fischetti's surface
roughness model
0

(001) (112) (111) (110)

Surface Orientation
Figure 3-1. Hole mobility vs device surface orientation for relaxed silicon with <110>
channel. The hole mobility is highest on (110) and lowest on (001) devices.
Different surface roughness scattering models are used in the simulation(solid:
Gamiz 1999; dotted: Fischetti 2003).









3.1.3 Hole Mobility and Vertical Electric Field

The calculated hole mobility versus the effective electric field of unstressed (001)/<110>

and (110)/<110> Si p-MOSFETs are compared with experimental mobility curves [38,

39, 40] in Figure 3-2. The agreement between the calculation and the experimental results

-Ii--.- -I1 this work use reasonable scattering mechanisms. Normally (110)-Si has smoother

interface with the gate dielectric materials [41, 42], hence the surface roughness scattering

rate is lower than (001)-oriented devices. Lee [43] even s-l--.- -1. ,1 that the effective field in

(110)oriented devices is smaller than (001)-oriented devices, which also indicates smaller

surface roughness scattering rate considering that the scattering rate is inversely propor-

tional to the effective electric field [3, 34]. The smaller surface roughness scattering rate

is partly responsible for the higher hole mobility on unstressed (110)oriented deices than

that of the (001)-oriented devices. To fit the appropriate surface roughness condition, the

roughness parameters used are L = 2.6nm, A = 0.4nm for (001)-oriented p-MOSFETs

and L = 1.03nm, A 0.27nm for (110)oriented p-MOSFETs in this work. The same sur-

face roughness scattering model is utilized in the mobility calculation even when the strain

is present, assuming that the process-induced strain (uniaxial strain) does not change the

Si/SiO2 interface properties [3, 44].

3.1.4 Strain-enhanced Hole Mobility

Figure 3-3 shows the hole mobility versus (up to 3 GPa) stress at inversion charge

density pi, = 1 x 1013/cm2 and channel doping density ND = 1 x 1017/cm3) for

(001)/<110>, (110)/<110>, and (110)/<111> p-MOSFETs. Uniaxial compressive

stress improves the hole mobility monotonically as the stress increases. The hole mobility

enhancement saturates at large stress (3 GPa). Under uniaxial longitudinal compres-

sive stress, the maximum hole mobility enhancement factor is 350'. for (001)/<110>

p-MOSFETs, 150'-. for (110)/<111> p-MOSFETs, and 10l'. for (110)/<110> p-

MOSFETs. At 3 GPa uniaxial stress, (001) and (110) p-MOSFETs have comparable hole




















(11 0)/


SYang, 2003

,
izuno, 2003
Mizuno, 2003


Takagi, 19
Takagi, 19


Q2


(001)/<110>


0.3


0.6


Effective Electric Field / MV/cm


Figure 3-2.


Hole mobility vs inversion charge density for relaxed silicon. Both
measurements and simulation show larger mobility on (110) devices.


350

300

250

200

150

100


0
0)





0
E



.o


0.9


-..








mobility. Under biaxial tensile stress, the maximum hole mobility enhancement factor is
about 1(I0 '

400
(110)/<111> uniaxial

3 300 (110)/<110> uniaxial



o 200

.- y^ (001)/<110> uniaxial
o 100 _

(001)/<110> biaxial
0
0 1 2 3

Stress / GPa
Figure 3-3. Hole mobility vs stress with inversion charge density 1 x 1013/cm2. The
enhancement factor is the highest for (001)/<110> devices and lowest for
(110)/<110> devices. At high stress (3 GPa), three uniaxial stress cases have
similar hole mobility.


Calculated strain-induced hole mobility enhancement factor of (001)-oriented
pMOSFETs is shown in Figure 3-4 comparing with experimental data [45, 46, 47, 48, 49,
5, 50, 51]. Good agreement is found between the calculated and measured data.
In Figure 3-3 and 3-4, the channel doping density is set to be 1 x 1017/cm3 in the
calculation. The inversion charge density is 1 x 1013/cm2. In contemporary technology, the
actual channel doping is up to 1 x 1019/cm3. The mobility enhancement factor is calculated
with different channel doping density at inversion charge density of 1 x 1013/cm2 in
Figure 3-5. The enhancement factors are similar for all three doping levels. For simplicity,
the rest of the work will use channel doping 1 x 1017/cm3.

























0.01
Strain Exx=Eyy


Figure 3-4.


ci

E


(D

LU

>^


uniaxial



biaxial


A EJV
_^---D


Calculated strain induced hole mobility enhancement factor vs. experimental
data for (001)-oriented pMOS.


0L
0

U-
HL


1 2 3


Uniaxial Compressive Stress / GPa

Figure 3-5. Hole mobility enhancement factor vs uniaxial stress for different channel
doping.


Uniaxial Compression
A Lee 2005
o Rim 2003
Washington 2006
Thompson 2005
A Smith 2005
-Biaxial Tension
Wang 2004




0.02









Figure 3-6 compares the hole mobility enhancement factor for different inversion

charge density. The figure shows that the enhancement factor decreases as the inversion

charge density increases. This is because with more inversion charge, holes are populated

to the higher energy levels in the valence band, while the stress only affects the vicinities

of F point. This causes the average change of the hole effective mass decrease. More

inversion charges increases the electric field in the channel which undermines the strain

effect. The detail will be addressed later in this chapter.

With the strain-induced hole mobility change as we showed here, physical insights

of the difference of biaxial and uniaxial stress, and the difference between (001) and

(110)oriented pMOSFETs is given in the next sections. Strain-induced silicon valence

band structure change, subband structure caused by the transverse electric field, and the

hole effective mass and scattering rate change with the strain are analyzed.













E

co


0



0


1 2


Biaxial Tensile Stress / GPa
(a)


c 3

E





:5 0
a)
0^-2

0
o


0 0


0 1 2 3
Biaxial Tensile Stress / GPa
(b)

Figure 3-6. Calculated strain induced hole mobility enhancement factor vs. stress for
(001)-oriented pMOS with different inversion charge density.









3.2 Bulk Silicon Valence Band Structure

Carrier mobility is determined by the scattering rate and effective mass of the carrier

based on Drude's model:



P- = (31)

where r is the carrier momentum relaxation time that is inversely proportional to

scattering rate and m* is the carrier conductivity effective mass. In silicon inversion l v. rs,

carriers are confined in a potential such that their motion in one direction (perpendicular

to the silicon-oxide interface) is restricted and the electronic behavior of these carriers

is typically two-dimensional (2D). The mobility of the 2D hole gas is different from

the 3D holes in bulk silicon. But the simplicity of the bulk band structure calculation

can give us insights to how the effective masses of the holes change with the stress

and help understand how the quantum confinement modifies the subband position and

splitting which is important to 2D hole mobility. Therefore, bulk valence band structure is

discussed in this section before we move to the Si pMOSFETs.

3.2.1 Dispersion Relation

The E-k diagrams of unstressed, 1 GPa biaxial tensile stressed and 1 GPa uniaxial

compressive stressed silicon valence band are shown in figure 3-7. For the unstressed

silicon, the Heavy-hole (HH) and Light-hole (LH) bands are degenerate at Fpoint. This

is 4-fold degeneracy taking into account the spin. The Spin-orbital Split-off (SO) band is

44 meV below HH and LH bands. When stress is applied, the degeneracy of HH and LH

bands is lifted as shown in figure 3-7 (b) and (c). These two bands are also referred to as

the top and the second band indicating the split energy levels. The band splitting results

in the band warping which changes the effective mass of the holes. In the meantime,

the splitting causes the repopulation of the holes in the system. When the stress is large

and the splitting is high, most holes will locate in the top band based on Fermi-Dirac

distribution function as long as the density-of-states (DOS) of the topmost band is not









significantly less than that of the next bands. The repopulation of the holes alters the

average hole effective mass and phonon scattering change.

Figure 3-7 shows that the stress only affect the band property close to F point. The

figures show that away from the zone center, the band structure is almost identical to the

unstressed silicon. Figure 3-8 illustrates that more region around the zone center and more

carriers are affected by the stress when the stress increases. Therefore, the strain effect

cannot be explained only by the properties at the F point. Instead, the statistics of the

whole system should be considered. Figure 3-8 -i----.- -I; that as the stress increases from

500 MPa to 1.5 GPa, the band warping and effective mass at F point change very little.

The next subsection will also show this. In the process, more holes are affected by the

stress, therefore the average hole behaviors will still change. We showed in Figure 3-6 that

the mobility enhancement factor decreases as the amount of inversion charges increases.

This can be understood as follows. For devices with more inversion charges, more holes

occupy the higher energy states when the inversion charge density increases. At the same

stress, the average change induced by stress is smaller than the cases with fewer inversion

charges.

3.2.2 Hole Effective Masses

To better understand the stress effect on hole transport, the hole effective masses

at F-point of top and bottom bands under different stress are shown in figure 3-9, 3-10

and 3-11. Figure 3-9 shows the < 110 >-direction effective masses, figure 3-10 shows the

2-dimensional density-of-state effective masses, and the out-of-plane < 001 >-direction

effective masses are illustrated in figure 3-11.

Figure 3-9, 3-10 and 3-11 also -ii.;. -1 that with strain, the HH and LH bands are

no longer "pure" HH or LH anymore due to strong coupling of the wave functions. The

property of each band depends heavily on the <( i --I 1 orientation. A single band can

be HH-like along one direction, but LH-like along another. In general, if the < i i--1 I1

shows compressive strain along one direction, the top band is LH-like along this specific












































-0.2 0
Wave vector k / 2/a

(b)


-0.2 0
Wave vector k / 2/a


Figure 3-7. E-k relation for silicon under (a) no stress;
(c) 1GPa uniaxial compressive stress.


(b) 1GPa biaxial tensile stress; and














































E 0

S-0.05

1.0 GPa

-0.1 1.5 GPa



Uniaxial Compressive Stress
-0.15
-0.1 0 0.1
Wave Vector k / A
(b)

Figure 3-8. Normalized E-k diagram of the top band under different amount of stress.
Larger stress warps more region of the band. The energy at F point for all
curves is set to zero only for comparison purpose.













0.3



0
E
E
()










0.2
0


1 1.5 2
Biaxial Tensile Stress / GPa


0 0.5 1 1.5 2 2.5 3
Uniaxial Compressive Stress / GPa

(b)

Figure 3-9. C'I io,,, I direction effective masses for bulk silicon under (a) biaxial tensile
stress; and (b) uniaxial compressive stress.


STop Band












Bottom Band


-














0.3



0
E
E
()





uJ




0.2
0


0 0.5 1 1.5 2 2.5 3
Uniaxial Compressive Stress / GPa
(b)

Figure 3-10. Two-dimensional density-of-states effective masses for bulk silicon under (a)
biaxial tensile stress; and (b) uniaxial compressive stress.


0.5 1 1.5 2 2.5
Biaxial Tensile Stress / GPa
(a)


STop Band












Bottom Band


I















Bottom Band


1 1.5 2
Biaxial Tensile Stress / GPa


Top Band










Bottom Band





0 0.5 1 1.5 2 2.5
Uniaxial Compressive Stress / GPa


Figure 3-11. Out-of-plane effective masses for bulk silicon under (a) biaxial tensile stress;
and (b) uniaxial compressive stress.


0
E
;- 0.25
E





D 0.2
uJ


F


Top Band


L~L


n 0


n


0.3



0
E
E
()
()


4--

LuJ




0.2


F









direction; if the ( i--I I1 experiences tensile strain along a direction, the top band is HH-like

along this direction. For example, when in-plane biaxial tensile stress is applied to the

x-y plane of a silicon sample, in x-y plane, the sample experiences tensile strain, the

top band is HH-like in-plane, as shown in Figure 3-9. Along z-direction (out-of-plane),

the sample shows tensile strain as we shoed in ('!i lpter 1. The top band is LH-like along

this direction as shown in Figure 3-11. This is a very important issue for biaxial tensile

stress. As we will show in the following section, the transverse electric field effect offsets

the biaxial stress effect and causes the hole mobility degradation at low stress. Similar

analysis can be applied to uniaxial compressive stress. Under uniaxial compression, the

<110> channel direction experiences compressive strain, therefore the top band is LH-like

along the channel. At the same time, the out-of-plane direction experiences tensile strain,

the top band is HH-like out-of-plane.

The spin-orbital split-off (SO) band is also coupled with HH and LH band when

strain is present. This band is not as important due to the large energy separation from

HH and LH bands and hence very few holes locate in this band.

As stated previously that normal MOSFETs have < 110 > direction as the channel

direction, conductivity effective mass along this direction affects the hole mobility directly

according to Drude's model, a.k.a equation 3-1. Figure 3-9 tells us that compared with

the biaxial tensile stress, the uniaxial compressive stress induces much smaller top band

effective mass which -it--.- -1; greater hole mobility improvement is expected for uniaxial

compressive stress.

Two-dimensional density-of-states effective masses as shown in Figure 3-10 gives a

qualitative estimation of the 2D density-of-states of the holes in each band. The 2D DOS

is not directly related to the bulk electronic properties of semiconductors. In the inversion

lI-.-is, large 2D DOS of the ground state subband -,t--.-i most holes locating in this

subband. This reduces inter-subband phonon scattering possibility. In the meantime,

if the ground state subband has very low conductivity effective mass, the large DOS









actually lowers the average hole conductivity effective mass in the system. 2D DOS will be

explained in a lot detail in the following section.

<001> out-of-plane effective mass is a important parameter defining the subband

energy levels in the inversion 1l-v- r as will explained in the following section.

3.2.3 Valence Band under Super Low Strain

If we compare the hole effective mass of unstrained bulk Si with Figure 3-9, 3-10

and 3-11, a significant discontinuity can be found at low strain (stress < 1 MPa). As

we mentioned before, HH band becomes LH-like along <110> direction under uniaxial

compression and along out-of-plane direction under biaxial tension. In the hole mobility

calculation, the discontinuity of the hole effective mass is also a confusing question,

although it is not important in industries since any single transistor would have much

larger strain in the channel in the process. To understand the "d- ... il oiily hole

effective mass at F point is calculated for super low stress [52] as shown in 3-12.

The figures show that under uniaxial compressive stress, the HH band is alvb--,-

HH-like and the LH band is albv--i LH-like out-of-plane. Along the <110> direction, the

effective mass curves cross over at about 3 kPa where HH band becomes LH-like and LH

band becomes HH-like. Biaxial tensile stress acts differently. The in-plane HH and LH

bands are still HH-like and LH-like, respectively. In the out-of-plane direction, the HH

band becomes LH-like and LH band becomes HH-like as the stress is greater than 1 kPa.

As the stress increases beyond 100 kPa, the conductivity effective mass does not

change at F point. The average effective mass change of the system comes from the fact

that more region of the bands is affected by the stress.

3.2.4 Energy Contours

Strain altered energy contours are straightforward describing the strain effect on

semiconductor band structures. The 25meV energy contours for I i, -i--hole and light-hole

bands are shown in figure 3-13 for unstressed bulk silicon. The anisotropic nature of the Si

valence band is clearly shown. Using the simple parabolic approximation E = *, where





























0.1'
0.01 0.1 1 10
Biaxial Tensile Stress / kPa
(a)


0.1 ...... ...... .. .. ......
0.01 0.1 1 10 100
Uniaxial Compressive Stress / kPa
(b)




Figure 3-12. Hole effective mass change under very small stress. The change in this stress
region explains the "dh- i...i fliii y of the hole effective mass between the
relaxed and highly stressed Si.










E stands for energy and m* is the effective mass, the thinner the contour is along one

direction, the smaller the effective mass is along that direction. The contours show that

the HH band has very large effective mass along <110> direction. When stress is applied,

the band structure is distorted as shown in figure 3-14 for 1GPa biaxial tensile stress and

figure 3-15 for 1GPa uniaxial compressive stress. The contours, as well as E-k relation

curves, show strain induces lower conductivity effective mass along <110> direction for

the top band. The effective masses for unstressed bulk silicon are 0.59mo for HH and

0.151,,, for LH band where mo is the free electron mass. Those two numbers become

0.28mo/0.22mo for 1GPa biaxial tensile stress and 0.11mo/0.2mo for 1GPa uniaxial

compressive stress. The bottom band effective masses do not show enhancement compared

with the LH band mass of the unstressed silicon. Again, for bulk electronic transport,

uniaxial compressive stress should enhance the hole mobility as stress increases, since the

top band is LH-like along <110> direction. Biaxial tensile stress does not have the mass

advantage since the top band is HH-like in-plane. The possible mobility enhancement

comes only from band splitting causing phonon scattering rate reduction. For holes in the

inversion liz-r's, the statement is still true as we will show next.



0.1 0.1




O 0 -



0.1 01
-01 0 -0.1 .0
-0.10 01 -01 01 0 0.1 -0.1

(a) (b)

Figure 3-13. The 25meV energy contours for unstressed Si: (a) Heavy-hole; (b) Light-hole.































0.1

0 0.1 -0.1


01
0 0.1- 0. 1
0 0.1 0.1


Figure 3-14.


The 25meV energy contours for biaxial tensile stressed Si: (a) Top band; (b)
Bottom band.


0.1


0


0.1

0 0.1 -0.1


01

0 0.1 -0.1


Figure 3-15. The 25meV energy contours for uniaxially compressive stressed Si: (a) Top
band; (b) Bottom band.









3.3 Strain Effects on Silicon Inversion Layers

As we mentioned before, in the silicon inversion 1.- ri, the carriers are confined in

the potential well formed by the Si/Si02 interface and the valence band edge of the

silicon. The motion of the holes is continuous in the horizontal x-y plane, but quantized

in z-direction [29]. The quantum confinement leaves a set of two dimensional subbands in

k-space (kx, ky). The subband structures are affected by both the stress and the transverse

electric field. In pMOSFETs, the topmost two subbands (4 counting the spin), the

ground state and the first excited state subbands, contain most of the holes and analyzing

those two subbands gives us qualitative understanding of the hole transport properties.

Therefore, those two subbands will be focused in the following discussions to explain the

strain effects, although up to 12 subbands are actually taken into account in the hole

mobility calculation.

In this section, we shall explain why the biaxial tensile stress and uniaxial com-

pressive stress affect the subband structure and the hole mobility differently under the

transverse electric field. The difference of (001) and (110)oriented devices under uniaxial

stress will also be studied.

3.3.1 Quantum Confinement and Subband Splitting

Carriers are confined in a potential well very close to the silicon surface in the

inversion l1-v-r of a MOSFET. The well is formed by the oxide barrier and the silicon

conduction band or valence band depending on electrons or holes as the carriers [1].

Taking holes (pMOS) as an example, the conduction and valence bands bend up (bend

down for nMOS) towards the surface due to the applied negative gate bias at strong

inversion region. This means hole motion in z-direction that is perpendicular to the silicon

surface is restricted and thus is quantized, leaving only a 2-dimensional momentum or k-

vector which characterizes motion in a plane normal to the confining potential. Therefore,

the inversion l-1v-r holes (or electrons) must be treated quantum mechanically as 2-

dimensional (2D). Figure 3-16 illustrates the quantum well and quantized subbands [51],









qualitatively. The band bending at the surface can be characterized as potential V(z).

Accurate modeling of V(z) requires numerically solving coupled Schrodinger's and

Poisson's Equations self-consistently. This is one of the main efforts of this work. The

details of the method can be found in Chapter 2.

SiO2/Si

STop of the well


Hol distribution of the ground state

/D 2/3
E. E(j=0) E [2hqE, + 3
) 4
a-40 -4 1 X
LU

S\ /E(j=1)
-60


Valence band edge

Hole energy level shift due to
quantization

Figure 3-16. Quantum well and subbands energy levels under transverse electric field.


The complex calculation procedure somehow prevents people understanding the

physics behind stress and electric field effect. To give the physical insights into the relation

between those two effects, triangular potential approximation is utilized to estimate the

subband energy levels. The triangular potential approximation states that the band

bending solely depends on depletion charges under subthreshold condition when the

mobile charge density is negligible. The potential V(z) is replaced by eEeffz, where Eeff

is the effective electric field in the depletion l1 .-r. Triangular potential approximation









is not a good approximation calculating accurate subband energies for strong inversion

region, but the physics can still be explained qualitatively.

Solving Schrodinger's equation,



[H(k, z) + V(z)lPk(z) = E(k)fk(z) (3-2)

one will get the subband energies. The energy of subband i can be expressed as [1],


Ei= 0i + i = 0, t, 2,... (3-3)
4 '-(u'2' 4
where h is plank constant, e is the electron charge, and m* is the out-of-plane hole

effective mass, also known as confinement effective mass. This effective field is defined as

the average electric field perpendicular to the Si -SiO2 interface experienced by the

carriers in the channel. It can be expressed in terms of the depletion and inversion charge

densities:


Es = (|Q| 11l + T Qi.|) (3-4)
where l = for electrons and 1 for holes [1, 53]. We focus on the inversion region of

MOSFETs where the effective field is over 0.5MV/cm throughout this work. This equation

for the effective electric field is an empirical equation. It may not be accurate to model

the carrier transport for devices with surface orientation other than (001) or other device

structures such as silicon-on-insulator (SOI) devices or double-gated (DG) devices.

Equation 3-3 shows that the subband energy of holes is inversely proportional to the

out-of-plane effective mass of the holes. With the transverse electric field, the subband

that is HH-like out-of-plane is shifted up (lower energy for holes) and the subband that

is LH-like out-of-plane is shifted down (higher energy). Figure 3-11 and 3-9 show that

in (001)-oriented devices, biaxial tensile strain shifts the out-of-plane LH-like band up

which is the in-plane HH-like band. The electric field effect offsets the biaxial tensile









strain effect. At low strain, this can be understood as follows. When the biaxial strain is

very small, i.e. 10 MPa, and the subband energy levels is dominated by the electric field

effect, the ground state subband is HH-like out-of-plane and LH-like along the channel.

As we increase the strain and keep the electric field constant, the energy splitting between

the ground state and the first excited state will decrease and at some stress level, the two

subbands will cross each other. The process is showed in Figure 3-17 schematically. If

the strain continues in, I iii.- the strain becomes dominant determining the subband

energies and structures. During the process, the average hole effective mass increases since

holes transfer from the in-plane LH-like subband to the HH-like subband. This increasing

effective mass is responsible to the initial mobility degradation under biaxial tensile strain

which is observed both in experiments and our calculation. The mobility enhancement

shown in Figure 3-3 comes from the suppressed inter-subband phonon scattering rate due

to the high subband splitting as will be shown later. Under uniaxial compressive strain,

the top band is HH-like out-of-plane and LH-like along the channel, which -i' -.- I- the

strain and the electric field effects are additive. Based on the similar an !1, i- both the

uniaxial compressive strain and the quantum confinement effects shift up the out-of-plane

HH-like band which is LH-like along the channel. Therefore the ground state subband is

alv-i -- LH-like along the channel and the average effective mass decreases monotonically

as the stress increases.

The calculated subband splitting between the ground state and the first excited

state is showed in Figure 3-18 for different stress and surface orientation. For biaxial

stress, the splitting is zero at 500 MPa which sir.-.- -I the crossing-over of the HH-like

and LH-like subbands. For all uniaxial stress cases, the subband splitting increases with

the stress. Like (001)/<110> devices, the ground state subband of both (110)/<110>

and (110)/<111> devices is HH-like out-of-plane and LH-like along the channel under

uniaxial compressive stress. The difference is tat the out-of-plane effective mass of the

ground state subband in (110)oriented devices is much larger 3-19 than that of the










Low Vertical Field High Vertical Field
Uniaxial Biaxial


Si02

...... E0
Second \ EtO

E.. Second Esecond

EV Ev

Band splitting due to strain Splitting (increases) / (decreases)

under confinement

Figure 3-17. Schematic plot of strain effect on subband splitting, the field effect is additive
to uniaxial compression and subtractive to biaxial tension.

(001)-oriented devices, which results in much larger subband splitting at low stress. The

splitting for (110)oriented devices does not change as much as (001)-oriented devices,

and the splitting saturates much faster with the stress compared with (001)-devices. This

is due to the strong quantum confinement underminging the strain effect, which is not

observed in (001)-oriented devices.

In general, in-plane compressive stress is desirable for pMOS, since it causes the

silicon top valence band to be HH-like out-of-plane and LH-like in-plane, which is additive

to the electric field effect. <110> uniaxial compressive stress is the best choice because it

gives very small conductivity effective mass.

3.3.2 Confinement of (110) Si

Figure 3-18 shows the difference of the subband splitting between (001)- and (110)

oriented devices. Figure 3-3 shows that the maximum enhancement factor at 3 GPa stress

for (001)-oriented devices under uniaxial stress is much larger than (110)oriented devices.








120
a) (001)/<110> uniaxial -
E 100


60
0 80
E 80 -------- ~--/ '-A "

cl) 60
0 "(110)/<110> uniaxial
40
0 ( (110)/<111> uniaxial
U0 20 0
S (001)/<110> biaxial
0 I ---------------
0 1 2 3
Stress / GPa
Figure 3-18. Subband splitting between the top two subbands under different stress.

To explain the physics, the bulk and confined 2D energy contours of the ground state
subband for (001) and (110)oriented Si are shown in Figure 3-20, 3-21, 3-22, and 3-23.
The figures show that for (001)/<110> devices, the ground state hole effective mass
decreases with uniaxial compressive stress (LH-like) along the channel), but the reduction
is not as notable under biaxial stress. Compared with the bulk Si energy contours, the
electric field does not modify the subband structure in kx ky plane for (001)-oriented
devices (it does affect the subband splitting though). The conductivity effective masses
along the channel direction are almost identical to those of bulk counterparts. The
confinement effect is much more significant on (110)oriented devices. The confined
effective mass of the ground state subband is very low along <110> and <111> direction
even for unstressed Si, which explains why unstressed (110)oriented devices have superior
hole mobility over (001)-oriented devices (the confinement effect is also significant in (111)
and (112) p-MOSFETs (3-1), though the hole effective mass is larger than that in (110)

























2.51


0 I lup DdlIU
E


S1.5

0)


tU

0.5

Bottom Band

0 0.5 1 1.5 2 2.5 3
Uniaxial Compressive Stress / GPa

Figure 3-19. Out-of-plane effective masses for (110) surface oriented bulk silicon under
uniaxial compressive stress.














66









p-MOSFETs). Furthermore, for (110)/<110> devices, stress shows very little effects on

the confined contours and the effective masses hardly change. For (110)/<111> devices,

the 2D contours are warped much more significantly and the effective mass decreases

more than (110)/<110> devices with uniaxial stress. This difference explains why the

hole mobility of (110)/<110> devices and (110)/<111> devices respond differently under

uniaxial stress.

3.3.3 Strain-induced Hole Repopulation

Strain induced hole population in the ground state subband is shown in Figure 3-24.

For (001) devices under uniaxial stress, the initial decrease of the hole population is due

to the decreased DOS near F point. As stress increases, the increasing subband splitting

causes the hole population increasing and the average conductivity effective mass keeps

decreasing since the ground state subband is LH-like along the channel under uniaxial

compressvie stress. For biaxial stress, the decrease of the hole population at low stress

again reflects the initial confinement effect lifting the in-plane LH-like subband and

reducing the subband splitting( 3-18). This in-plane LH-like subband is shifted down as

the stress increases and the in-plane HH-like subband is shifted up. After the crossing-over

of the two subbands, the ground state subband population starts increasing with the

stress. For (110)-oriented devices under uniaxial compressive stress, the ground state hole

population increases with the stress, but it saturates at much lower stress compared with

(001)-oriented devices which is consistent with the subband splitting change. The hole

population of (110)oriented devices is ah,--i- lower than (001)-oriented devices under

uniaxial compressive stress, although the subband splitting is much larger. The subband

splitting and hole population difference of (001)- and (110)oriented devices can be

explained by the ground state subband 2D DOS as shown in Figure 3-25. DOS difference

also -ii-.-. -; the different strain-induced mobility change. Both figures show that (001)

oriented devices have larger DOS than (110)oriented devices. For (001)/<110> devices

under uniaxial compressive stress, although the first excited state subband is HH-like

































0.15
<110>














1GPa Uniaxial Compression
-0.15 0 0.15
k
x
(b)

0.15
<110>















1 GPa Biaxial Tension
-0.15 0 0.15
k
x

(c)


Figure 3-20. The 2D energy contours (25, 50, 75, and 100 meV) for bulk (001)-Si.
Uniaxial compressive stress changes hole effective mass more significantly
than biaxial tensile stress.































0.15
0.15<110>













1 GPa Uniaxial Compression
-0.15 k0 0.15




















k

(b)
0.15---------------





















Figure 3-21. Confined 2D energy contours (25, 50, 75, and 100 meV) for (001)-Si. The
contours are identical to the bulk counterparts.

































0.15


1GPa Uniaxial Compression


0








-0.15 0 0.15
k
x
(b)

0.15
1 GPa Uniaxial Compression





111>
0








-0.15 0 0.15
k
x

(c)


Figure 3-22. The 2D energy contours (25, 50, 75, and 100 meV) for bulk (110)-Si under
(a) no stress; (b) uniaxial stress along ( 110); and (c) uniaxial stress along
(111).

70












Unstressed Si


<111>
0W








-0.15 0 0.15
k

(a)

0.15
1 GPa Uniaxial Compression



<110>



-0








-0.15 0 0.15
k

(b)

0.15
1 GPa Uniaxial Compression





<111>
0








-0.15 k0 0.15
x

(c)


Figure 3-23. Confined 2D energy contours (25, 50, 75, and 100 meV) for (110)-Si. The
confined contours are totally different from their bulk counterparts which
s i:--- -1- significant confinement effect.








along <110> channel, the subband splitting and the high 2D DOS of the ground state
subband (compared with (110)oriented devices) assures most holes populating to the
ground state subband as shown in Figure 3-24. The decreasing DOS in Figure 3-25 (b) for
both biaxial and uniaxial stress of (001)-oriented devices also -ii-.-- -i- that the phonon
scattering rate decreases with te stress. The DOS of (110)oriented devices does not
change with the stress especially at high stress region (1-3 GPa) which -ii-.-' -i the
phonon scattering rate should not change much.
1.0
(001)/<110> uniaxial


~0.8

o
O 0.6 0
-o / (110)/<110> uniaxial
0.4 C (110)/<111> uniaxial

c)
(001)/<110> biaxial

0.2
0 1 2 3
Stress / GPa

Figure 3-24. Ground state subband hole population under different stress.

As we mentioned in the previous section, the stress does not warp the band structure
evenly in the whole k-space. This can also be seen from the DOS change in Figure 3-25
(b) where the DOS at Energy E = 52meV (2kT where T = 300k) is shown. Taking
uniaxial stress on (001) devices as an example, when the stress is low, only a small region
close to F point is affected and becomes LH-like along <110> direction (still HH-like
along transverse and out-of-plane direction), while the rest of the band with higher energy
(including the energy level showed here) does not respond to the stress yet. As the stress









increases, more region is affected and becomes LH-like along the channel. The initial

constant DOS at low stress in Figure 3-25 (b) -,ii-.-- -1- when the stress is lower than about

500 MPa, the stress is too small to warp the band at this energy level. When the stress

increases, DOS starts decreasing because the stress starts warping the band at this energy

and the <110> direction becomes LH-like. The DOS curve becomes flat again when the

stress effect saturates for this energy level. For (001) p-MOSFETs under biaxial stress,

Figure 3-25 (b) does not show a DOS peak like Figure 3-25 (a) which means the position

crossing-over of the top two subbands only happens close to F point, and the HH-like

band is alv--,v- on top out of that region.

For (110) p-MOSFETs, the DOS is constant with the stress, which is due to the

strong quantum confinement effect. To discover the strain effect, 2D DOS at 4kT (102meV

at T 300K) is shown in Figure 3-26. For (001) p-MOSFETs, the curves have the similar

trend compared with the DOS curves at 2kT. The only difference is that the DOS starts

to decrease at higher stress. For (110) p-MOSFETs, DOS decreases at low stress and the

change is not as significantly as (001) p-MOSFETs. Figure 3-25 and 3-26 -i--.-. -1 that

the strain in (110) p-MOSFETs only warps the subband at high energy region due to the

strong quantum confinement. The strain induced mobility change should be less than

(001) p-MOSFETs, since smaller portion of holes locate at high energy compared with F

point.

3.3.4 Scattering Rate

Besides effective mass change, hole mobility is inversely proportional to the scattering

rate. Phonon scattering and surface roughness scattering are focused in this work, since

they are the predominant scattering mechanisms when the effective electric field in the

channel is over 0.5MV/cm [3, 1].

Figure 3-27 shows that for (001)-oriented devices, the phonon scattering rate does not

change much when the stress is lower than 500 MPa. This indicates that at low stress, the

hole mobility enhancement (or degradation) is almost purely caused by the effective mass










3x1014




2x1014




1x1014




0.0


S6x1014
0
O

a,




o
u 4xl 014





(r)
a,
Cz


> 2x1014


-o
70
C\ 0.0


1 2
Stress / GPa


Stress / GPa
(b)

Figure 3-25. Stress effect on the 2 dimensional density-of-states of the ground state
subband at (a) the top of the subband (E=0); (b) E=2kT. (110)devices
have much smaller 2D DOS which limits the ground state hole population
(larger inter-subband phonon scattering). Another observation is that DOS of
(110)devices does not change with stress.


(001)/<110> uniaxial


(1 0)/<1 uniaxial

(110)/<111> uniaxial


- ftr m m


(110)/<110> uniaxial


I








6x1014
E
o
(001)/<110> uniaxial
a(
3 4x1014 \ (001)/<110> biaxial




2x1014 -
a)
S((11 0)/<111> uniaxial >


^ 2x1014 -



o4 0.0
0 1 2 3

Stress / GPa

Figure 3-26. Two dimensional density-of-states at E=4kT.

change. When the stress increases from 500 MPa to 3 GPa, the phonon scattering rate
decreases by 5('. for both acoustic phonon and optical phonon scattering, the phonon
scattering rate reduction overweighs the effective mass change to become the main driving
force to improve the hole mobility in this stress range, especially for biaxial stress.
Unlike (001)-oriented devices, phonon scattering rate changes more at low stress
region rather than high stress region for (O110)oriented devices under uniaxial compressive
stress. This is consistent with Figure 3-18 and 3-24 that the subband splitting and the
ground state subband hole population only increase at low stress. The constant phonon
scattering rate at high stress explains why the hole mobility of (110)/<111> devices at 3
GPa is not significantly larger than (001)/<110> or (110)/<110> devices, regardless of
the largest piezoresistance coefficient at low stress.
Figure 3-28 shows that the surface roughness scattering rate increases with stress
for (001)-oriented devices. This is due to the increasing hole population in the ground








4x1012
(001)/<110> uniaxial

c 3x1012 (001)/<110> biaxial
0 )


0


S 110 (110)/<110>uniaxial
0
C/) (110)/<111> uniaxial
0 -
0 1 2 3
Stress / GPa
(a)
1x1013




c/) 000<
OD (001)/<110> uniaxial





( 6x1012 %mo


-0 24x1012


"_ (110)/<110> uniaxial
O (110)/<111> uniaxial
0
0 1 2 3
Stress / GPa
(b)

Figure 3-27. Strain effect on (a) acoustic phonon, and (b) optical phonon scattering rate.
Optical phonon scattering is the dominant scattering mechanism improving
the mobility. Phonon scattering rate changes mainly in high stress region for
(001)-devices and low stress region for (HO)-devices.









4x1012.


U)a
(D U) 3x1
a)
~cc
CzC
Cc 2x1
a)r
CZ a)

C/) 0
C/)


012


012[


012


Stress / GPa
Figure 3-28. Strain effect on surface roughness scattering rate of holes in the inversion
lwv-r. As stress increases, the scattering rate increases for (001)-devices due
to the increasing occupation in the ground state subband which brings the
centroids of the holes closer to the Si/SiO2 interface.

state subband which brings the centroids of the holes closer to the Si/SiO2 interface.

The magnitude of the surface roughness scattering rate is much smaller than the phonon

scattering rate and therefore the increasing surface roughness scattering does not affect

the hole mobility as much. The surface roughness scattering rate for (110)oriented

devices does not change much with the stress, which is consistent with the fact that the

ground state subband hole population is relatively constant with the stress.

3.3.5 Mass and Scattering Rate Contribution

Figure 3-29 illustrates the stress-induced hole mobility enhancement contribution

from hole effective mass and phonon scattering rate reduction, respectively. Under

uniaxial compression, (001)/(110) p-MOSFETs have the largest mobility improvement

from both aspects. Compared with (110)/(111) p-MOSFETs, (110)/(110) p-MOSFETs


(001)/<110> uniaxial

(001)/<110> biaxial









(110)/<111> uniaxial
(110)/<110> uniaxial









have smaller effective mass gain but larger phonon scattering rate gain. For (001) p-

MOSFETs under biaxial tension, the mobility enhancement is purely from the suppression

of the phonon scattering rate.

3.4 Summary

From the results of the self-consistent calculation of Schrodinger's Equation and

Poisson's Equation, we notice that the subband splitting between the ground state and

the first excited state decreases as the biaxial stress increases when the stress is smaller

than 600MPa, but the splitting increases with uniaxial compressive stress. The difference

is due to the subtractive or additive nature between the quantum confinement effect and

the stress effect which causes the increase or decrease of the average effective mass of the

holes in the inversion l~iv-r. As the stress keeps in' i i i:- the stress effect outweighs the

confinement effect for both stresses and the subband splitting increases so much that the

inter-subband phonon scattering rate reduces and hence the hole mobility increases.

Uniaxial stress on (110) devices improves the hole mobility too. But the improvement

is not as much as (001)-oriented devices. This is due to the strong confinement effect

on (110)-oriented devices undermining the stress effect. When no stress is present, the

confinement effect swaps the subband structure and reduces the hole effective mass around

the F-point. This effective mass advantage over the (001)-oriented unstressed pMOS

causes that the hole mobility is much larger. When the stress is applied, the effective mass

change is not as significant, neither does the subband splitting. Therefore, the mobility

enhancement with the stress is not supposed to be as much as the (001)-oriented pMOS.

It is also noticed that the subband splitting saturates when the stress reaches 2 or 3

GPa, so does the effective mass. This leads to the saturation of the stress enhanced hole

mobility.








1.5


a)
E
a)
o


LU
>-

0
Ill
3o


1.0



0.5



0.0


-0.5


1.5


ci
a)
E i

ai




O


1.0



0.5



0.0


-0.5


1 2
Stress / GPa
(a)


0 1 2
Stress / GPa
(b)


Figure 3-29. Hole mobility gain contribution from (a) effective mass reduction; and (b)
phonon scattering rate suppression for p-MOSFETs under biaxial and
uniaxial stress.

79









CHAPTER 4
STRAIN EFFECTS ON NON-CLASSICAL DEVICES

As the silicon C\ IOS technology is scaled to sub-100 nm, even sub-50 nm scale,

further simple scaling of the classical bulk devices is limited by the short channel effects

(SCEs) and does not bring performance improvement. The ultra-thin body (UTB) silicon-

on-insulator (SOI) transistor architecture [54, 55, 56, 57, 58] has been considered possible

replacement for the bulk MOSFETs. The basic idea of SOI Ci\OS fabrication [54, 56]

is to build traditional transistor structure on a very thin lv, -r of crystalline Si which is

separated from the substrate by a thick buried oxide li- -r (BOX). Compared with the

bulk C'\ OS, UTB SOI technology brings benefits such as reduced junction capacitance

which increases switching speed, no body effect since the body potential is not tied to

the ground or Vdd but can rise to the same potential as the source, low subsurface leakage

current, and et al..

SOI MOSFETs are often distinguished as partially depleted (PD) transistors that

the Si thickness is larger than the maximum depletion width and fully-depleted (FD)

SOI transistors that the Si is thinner than the maximum depletion width. FD SOI

technology [1] add additional performance enhancements over PD SOI including low

vertical electric field in the channel (higher mobility) due to the fact that most FD-SOI

transistors have undoped channel, further reduction of the junction capacitance, and

better scalability. Although FD SOI technology has better scalability than classical device

structures, it is still difficult to scale the device to sub-20 nm scale. In short-channel FD

SOI MOSFETs, the thick BOX acts like a wide gate depletion region and is vulnerable

to source-drain field penetration and results in severe short-channel effects [1, 59, 60]. To

better control the channel, double-gate (DG) transistors, especially FinFETs, have been

investigated theoretically and experimentally [61, 62, 63, 64]. DG-MOSFETs have better

scalability than single-gate (SG) SOI transistors and are considered promising candidates

for sub-20nm technologies [62]. Overall, SOI SG devices and DG devices have been shown









to increase circuit performance and reduce active power consumption. These non-classical

device structures are the future of the C'\ OS technology.

With the research of strain effects on bulk silicon devices, strained silicon UTB

FETs draw the attention of researchers as such devices may combine the strain induced

transport property enhancements with their scaling advantages. Stress enhanced hole

mobility in SOI-devices has been investigated experimentally in recent years [65, 66,

67, 68, 69, 70]. In 2003, Rim [45] reported the biaxial tensile stressed SOIpMOS hole

mobility with dependence of strain and inversion charge density. Zhang [71] showed

the hole mobility enhancement under low uniaxial longitudinal and transverse stress.

(10O)-surface SOI devices with strain effects are also investigated [72]. Those results are

consistent with the measured and calculated results for bulk Si devices that are showed in

the last chapter.

Strain research on double gate devices lags that on bulk devices and even single gate

SOI devices partly due to the difficulty employing stress to the channel without damaging

the properties of the channel and Si/SiO2 interfaces. Due to the better scalability and

higher hole mobility, more attention has been drawn to (110)oriented FinFETs over

planar DG FETs. Collaert [73] investigated strain effect on electron and hole mobility

enhancement on FinFETs. Shin [74] and his colleagues investigated multiple stress effects

on p-type FinFETs using wafer bending method. Verheyen [75] reported "-'. drive

current improvement of p-type multiple gate FET devices with germanium doped source

and drain. Although hole mobility enhancement is observed in those experiments, the

actual stress in the fin is unknown. Theoretically, strain effects on FinFETs are much less

understood. With the knowledge of stress enhancing hole mobility in bulk devices, it's

important to understand how that stress alters the hole mobility in FinFETs. Uniaxial

compressive stress will be focused in this work since it provides the greatest hole mobility

improvement than other stress on bulk devices. Another reason is that for (110)oriented

FinFETs, the stress in the channel is normally uniaxial longitudinal stress even if SiGe













S100-
0)~m ^ L --- ---- ---
UK p=6x1012/cm2
E 80


0 60-


4 / p=1.2x1013/cm2
40
I I I
0 5 10 15 20
SOI Thickness / nm

Figure 4-1. Hole mobility vs SOI thickness for single gate SOI pMOS. The mobility
decreases with the thickness due to structural confinement.


SOI thickness decreases. This does not bring smaller inter-subband scattering rate. The

rapidly increasing form factor actually keeps the scattering rate increasing.

Another issue related to the silicon thickness is subband modulation. Both measure-

ments and Monte-Carlo simulation show that the phonon-limited mobility increases at

very thin SOI thickness [67, 69, 77]. This issue only happens to nMOS. Uchida's mea-

surements show there is no such mobility peak in p-type UTB SOI FETs [67], which is

consistent with our calculation.

4.1.2 Strain-enhanced Hole Mobility of SOI SG-pMOS

Rim [45] reported that biaxial tensile strain improves (or degrades) the hole mobility

as same as it does to the bulk devices, which is supported by our calculation. Uniaxial

compressive strain is focused in this chapter due to its much larger mobility enhancement

factor than biaxial tensile strain.

Figure 4-2 shows the single-gate SOI pMOS hole mobility vs uniaxial compressive

stress comparing with bulk Si devices. Calculated curves for SOI thickness of 3 nm and 5








nm are shown in the figure. Simulation results for thicker SOI are not included because
they almost overlap with the bulk device curve.
400
o Conventional Si (001)/<110>

300 \ /
= E 1 tsoi = 5 nm

O-
200


2 100 .0-


Pinv= 1xi 013/cm2
0 0
0 1 2 3
Uniaxial Compressive Stress / GPa
Figure 4-2. Hole mobility for single gate SOI pMOS vs uniaxial stress at charge density
p 1 x 1013/cm2.

The hole mobility enhancement factor for SOI pMOS with SOI thickness of 3 nm
is shown in Figure 4-3. The enhancement factor for SOI devices is similar to the case of
bulk devices at low stress, but larger than bulk FETs at high stress. As we mentioned
in C'! lpter 3 that for (001)-oriented Si pMOS, the mobility is enhanced mainly due to
the decreased hole effective mass at low stress. At high stress, phonon scattering rate
reduction due to the increasing subband splitting is the main driving force to improve
the mobility. The overlapping curves at low stress sI:---- -1 the effective mass gain should
be similar for both cases. Calculation shows that the structure of each subband in SOI
pMOS is as same as the bulk counterpart which also -,.-':. -i the effective mass change for
both cases should be the same. Figure 4-4 shows the subband splitting of the ground state
and the first excited state subbands for SOI and bulk FETs. The larger splitting for SOI








devices -I, :. -- -; more inter-subband phonon scattering rate change, which is responsible
for the larger mobility enhancement.

5
tso = 3 nm


E t0so = 5 nm

|: S 3 y0-0

o /


.' Traditional Si (001)/<110
0

0 Pinv= 1xx1013/cm2
0 1 2 3

Uniaxial Compressive Stress / GPa
Figure 4-3. Hole mobility enhancement factor of UTB SOI SG devices vs uniaxial
compressive stress at charge density p 1 x 1013/cm2.

Uchida reported that as the SOI thickness reduces down to 2-3 nm, the fluctuation of
the Si/SiO2 interface is the main factor to limit the carrier mobility [67, 69]. Therefore,
the large hole mobility enhancement as shown in Figure 4-3 cannot be obtained in real
devices. A new surface roughness model is needed to solve this problem. In our discussion
of the double-gate devices including FinFETs later in this chapter, the smallest Si
thickness we consider would be 5 nm.
4.2 Double-gate p-MOSFETs
Due to the overwhelming research effort on FinFETs, FinFETs are focused in this
section. For (001)-oriented DG pMOS, only symmetrical-double-gate MOSFETs are
considered here. Unlike single gate devices, double gate MOSFETs have two surface
















140

0 120
Stsot = 3 nm
0 100 \ /

-. 80
/) 60

c 40 / Traditional Si (001)/<110>

= 20
20
U0nx Pinv = 1x1013/cm2
0 1 2 3
Uniaxial Compressive Stress / GPa
Figure 4-4. Subband splitting UTB SOI SG devices vs uniaxial compressive stress at
charge density p 1 x 1013/cm2.









channels. The wave functions of the two channels interact and one energy level splits to
two according to Pauli's exclusive principle (subband modulation). Schematic comparison
of the subband splitting for bulk and double gate devices is showed in Figure 4-5. The
subband splitting for SDG MOSFETs and FinFETs is very small when the Si thickness
is over 5 nm (5 meV when tsi = 5nm, 3 meV when tsi = 15nm). If the Si thickness is
below 5 nm, the strong interaction of the two surface channel causes the subband splitting
increasing drastically (i.e. 18 meV for tsi = 3nm).




EE
E V ... -top
Second

E Ethird


SG FET SDG FET
Figure 4-5. Comparison of the subband splitting of double gate and single gate
MOSFETs.


4.2.1 (001) SDG pMOS

The hole mobility and the mobility enhancement factor for SDG pMOSFETs are
shown in Figure 4-6 and 4-7, respectively. Double gate devices have higher mobility
than traditional bulk transistors mainly due to the undoped body, much smaller channel
effective electric field and bulk inversion [1]. Figure 4-6 shows that the hole mobility
decreases as the silicon thickness decreases. The reason is as same as single gate SOI
devices and has been explained in last section.
The mobility enhancement factor of SDG pMOS in Figure 4-7 is very similar to
the bulk case, but the mechanisms are a little different. The first excited subband (very
close to the ground state) provides smaller average effective mass to help the mobility
















S500
) 3tsi = 10 nm
> 400
N tsi = 5 nm

S3000

S200
Traditional Si (001)/<110>
0 100
I /
1 0 Pin, = x1 013/cm2
0 1 2 3
Uniaxial Compressive Stress / GPa
Figure 4-6. Hole mobility of SDG devices under uniaxial compressive stress at charge
density p 1 x 1013/c 2.










88








enhancement, but at the same time it also brings larger inter-subband phonon scattering
rate. Those two factors balance each other. Therefore the SDG devices show a little larger
mobility enhancement at low stress, but a little lower enhancement at high stress. The
difference is slim and the average effect is very similar to single-gate devices.

5 P 1Pin xlx 013/cm2


cI 4

S=L ts = 5 nm .





1
0
> CZ SG Si (001)/<110>

T5 1
o


0 1 2 3

Uniaxial Compressive Stress / GPa

Figure 4-7. Hole mobility enhancement factor of SDG MOSFETs vs uniaxial compressive
stress at charge density p 1 x 1013/cm2.

4.2.2 Strain Effect on FinFETs
The total hole mobility of the FinFET with respect to the stress is shown in Figure 4-
8, comparing with the single-gate (110)- and (001)-oriented p-type devices at the inversion
charge density of 1 x 1013/cm22. In the calculation of the single-gate devices, the doping
density is taken to be 1 x 1017/cm3. This is a low doping density compared with the
contemporary C'\ IOS technology. Even so, the FinFET shows significantly greater
mobility than the bulk devices. If larger doping density is applied, the mobility advantage
of the FinFET would be even larger. When 3 GPa uniaxial compressive stress is applied








to a FinFET, about 3:1i i'. enhancement of the mobility is expected, compared to only
2111 I' enhancement for a bulk (110)-oriented transistor as shown in Figure 4-9. Even
though the (001)-oriented pMOS shows greater relative enhancement (over !1111'.), the
absolute mobility is still lower than that of the FinFET due to its low mobility with no
stress.
700
Pinv= 1x1013/cm 2
( 600
) 600 FinFET

500

E 400 Bulk (001) FET

300

S200 ,k 0 F

0 100 Bulk (110) FET
I
0 1 1
0 1 2 3
Uniaxial Compressive Stress / GPa

Figure 4-8. Hole mobility of FinFETs under uniaxial stress compared with bulk
(10O)-oriented devices at charge density p 1 x 1013/cm2.


We mentioned in the last chapter that 2D DOS of the topmost subband in (110)-
oriented devices is very small near F point no matter if the stress is present and the stress
does not warp the subbands much. Therefore the average effective mass does not change
as much as standard (001)-oriented devices when uniaxial stress is present. Regarding
FinFETs, strong subband modulation is observed where the topmost 2 subbands are close
to each other (like (001) SDG p-MOSFETs) as we illustrated in Figure 4-5. This extra
subband is so close to the ground state subband and it acts like increasing the DOS of the
ground state subband. More importantly, the band bending at the Si/SiO2 interface is






















E 3 -

S<] / FinFETs (110)/<110>

.co



S/ w--... w- SG (110)/<110>

0
0 1 2 3
Uniaxial Compressive Stress / GPa
Figure 4-9. Hole mobility enhancement factor of FinFETs under uniaxial compressive
stress at charge density p 1 x 1013/cm2.












91









very small in FinFETs when gate bias is applied and the ground state subband is much

closer to the Fermi-level than that of the single-gate FETs (in both cases, the ground

state subbands are on top of Fermi-level). With the same total amount of holes in both

systems, the lower ground state subband level keeps more holes close to F point that can

be affected by the strain and the electric field. Although the topmost two subbands in

FinFETs are close to each other, the form factors are extremely small (only about 1/6 of

single gate case) between these two subbands. This results in smaller phonon scattering

than single gate devices, and the change of the scattering rate with stress is larger than

SG p-MOSFETs.

2
0
4-
0

4- FinFETs (110)/<11 O0>









S 0 1 3
(110 \ 11

-1 ----------------

0 1 2 3

Uniaxial Stress / GPa

Figure 4-10. Hole mobility gain contribution from effective mass and phonon scattering
suppression under uniaxial compression for (110)/(110) FinFETs compared
with SG (110)/(110) p-MOSFETs at charge density p 1 x 1013/c2.


To understand the hole mobility difference between FinFETs and traditional single

gate (110)/(110), the hole mobility gain contribution from effective mass change and

phonon scattering rate change is shown in Figure 4-10. It shows that phonon scattering

rate change is the main factor to improve the hole mobility for both FinFETs and bulk









p-MOSFETs. Both the effective mass and phonon scattering rate for FinFETs change are

larger than single gate (110)/(110) devices, which leads to higher mobility enhancement.

Smaller surface roughness scattering rate due to small electric field in FinFETs also

contributes to the higher mobility enhancement.

The calculation also shows the enhancement is not a strong function of the silicon

thickness of the fin as the fin thickness is above 5 nm. If the fin is thinner than that, more

subband splitting is observed (about 18 meV for 3 nm of the fin thickness). Since the

splitting is still not too large, our analysis about the effective mass -I ,i-, true. Surface

roughness scattering rate is much larger and the hole mobility enhancement would not

be as large as that for thicker fin. An accurate surface roughness model for such devices

would be necessary to evaluate the mobility change numerically.

4.3 Summary

Strain effects on SOI MOSFETs, including planar symmetrical DG devices and

(110)oriented FinFETs are discussed in this chapter. For single gate SOI pMOS, the

mobility decreases as the SOI l-, r thickness decreases due to increasing phonon and

surface roughness scattering rate. The hole mobility enhancement under stress is similar

to that of bulk silicon devices unless when the SOI thickness is so small that the surface

roughness scattering out-dominates the phonon scattering.

For double gate devices, subband splitting is drastically smaller than the bulk devices

due to the interaction of the quantum states of the two surface channels. For (001)

oriented planar symmetrical DG pMOS, the structure of each subband is still identical to

the counterpart in the bulk devices. The extra effective mass gain is canceled by the inter-

subband phonon scattering and the total hole mobility enhancement is similar to the bulk

FETs at low stress. But when the stress is over 2 GPa, the effective mass gain is saturate.

The mobility gain is less than that of bulk FETs due to the larger inter-subband optical

phonon scattering. This effect is not that significant due to the smaller form factors.









As of the FinFETs, the extra subband provides much more effective mass gain while

the phonon scattering rate is similar to the bulk devices. This causes that the mobility

enhancement is higher than bulk FETs. Although the mobility enhancement factor for

FinFETs (3 times) is not as large as (001)-oriented bulk pMOS (>4 times), FinFETs still

have much higher mobility due to its high initial mobility without stress. Together with

the better scalability, FinFETs will be strong candidate for C'\ OS technology under 20

nm scale.









CHAPTER 5
STRAIN EFFECTS ON GERMANIUM P-MOSFETS

As short-channel-effects (SCEs) prevent the simple scaling of traditional Si MOSFETs

achieving historical performance improvement, new material, as well as feature enhanced

technology (strain technology), attract attention of the researchers. Germanium is one of

those new materials due to its large electron and hole mobility. With the strained silicon

technology in the industry, it's a interesting topic to discover how the strain affects the

electron and hole mobility in germanium MOSFETs.

Germanium has been of special interest in high speed C '\OS technology for years [78,

79]. The bulk germanium hole mobility is larger than that of other semiconductor

materials, and its electron and hole mobility are much less disparate than other materials.

In 1989, germanium hole mobility of 770cm2/V sec in a pMOSFET was exhibited by

Martin [80] and his co-workers using SiO2 as the gate insulator. Since then, more and

more work [81, 82] has been done on germanium or SiGe channel pMOS [83, 84, 85].

In order to reduce the surface roughness and limit the band-to-band tunneling issue,

silicon-germanium or Si-SiGe dual channel is also used in some applications. Different

gate dielectric materials [86, 87, 88] have been utilized to find the best material to

limit the surface roughness at the interface between gate insulator and germanium

channel. Due to the uncertainty in the surface roughness and the surface states, different

hole mobility values have been reported in those publications. In recent years, with

the strain technology applied to silicon C'\ OS, strain effect is also investigated on

germanium MOSFETs [87, 89, 90, 91, 92]. The strain is normally achieved by applying

SiGe substrate underneath the germanium or SiGe channel. But most of the work stays

only in experiments, the physical insights of the strain effect on germanium MOSFETs

have not been discussed carefully. The only available theoretical works are some Monte-

Carlo simulations [93, 94, 95]. The goal of this chapter is to give physical insights of strain

effects on germanium utilizing k p calculation.









In this chapter, strain-induced hole mobility change of Ge and Sil-zGez in pMOS

inversion lwir-i~ is investigated. The hole mobility vs electric field and surface orientation

is showed. Strain-enhanced hole mobility is calculated for different Ge concentration in

Si1lGex. To understand the difference between Ge and Si, hole effective mass, band and

subband splitting, and two-dimensional density-of-states are calculated and their effects on

hole mobility is analyzed. Phonon and surface roughness scattering is also evaluated under

strain.

5.1 Germanium Hole Mobility

Unstrained Ge hole mobility [86, 96] vs vertical electric field and device surface

orientation is shown in figure 5-1. Experimental works give a lot of different mobility

values ranging from 70cm2/V sec to over 1000cm2/V sec, depending on what the gate

dielectric materials are used [86, 87, 88] and if Si buffer is applied [97, 98] between the

Ge (or SiGe) and the gate oxide. With Si buffer, the device acts as a buried-Ge channel

transistor and normally shows large hole mobility due to the lack of confinement and

surface roughness scattering. Due to the bad scalability of buried-channel devices, only

surface channel Ge-pMOS is discussed here. Calculated Ge hole mobility matches the

measured data and the mobility is much larger mobility than silicon. (110)-oriented device

shows higher mobility than (001)-oriented device, which is consistent with the results

of Si. We shall show that the larger hole mobility of germanium mainly comes from the

smaller effective mass of the holes. The relative smaller inter-subband phonon scattering

rate due to the larger subband splitting (and smaller optical phonon energy) also improves

the germanium mobility.

5.1.1 Biaxial Tensile Stress

In silicon MOSFETs, biaxial tensile strain is obtained via applying Sil_-Ge, sub-

strate underneath the Si channel. Biaxial tension is not a popular stress type for germa-

nium devices due to the large lattice constant of germanium. For comparison purpose,








500
o Zimmerman, 06
S400 Ge (110)/<110>
S 400


E 300- El
0 0 Ge (001)/<110>

200
-g Chui, 02


0 Si (001)/<110>


0 0.2 0.4 0.6 0.8 1
Effective Electric Field / MV/cm

Figure 5-1. Germanium hole mobility vs effective electric field.

the biaxial tensile strain effect on germanium hole mobility is calculated and showed in
Figure 5-2.
Like silicon, the degradation of the hole mobility at low biaxial tensile stress is due
to the subtractive nature of strain effect and transverse electric field effect resulting in
the increase of the average effective mass, together with a little increased inter-subband
phonon scattering. At high stress, the mobility enhancement is obtained due to reduced
inter-subband optical phonon scattering.
5.1.2 Biaxial Compressive Stress
Biaxial compressive stress in germanium MOSFETs channel can be obtained by
germanium channel on top of Sil_,Ge, substrate. Silicon transistors can also have biaxial
compression with Sil_-C, substrate. This is not a favorable stress type for either case,
since it does not improve the hole mobility significantly as shown in Figure 5-3.



















300
in=1 xl 013/cm2
250
o (001) Ge
0)
200

E 150-
(001) Si
S100

S 50

0
0 2 4 6
Biaxial Compressive Stress / GPa
Figure 5-3. Germanium and silicon hole mobility under biaxial compressive stress where
the inversion hole concentration is 1 x 1013/cm2.












99









5.1.3 Uniaxial Compressive Stress

Uniaxial compressive stress on Si is has been applied to multiple technology nodes

because of the maximum mobility enhancement to hole mobility. The hole mobility

vs uniaxial compressive stress for Ge is shown in Figure 5-4 for (001)-oriented Ge and

Figure 5-5 for (O110)oriented Ge. For (001)-oriented devices, both Si and Ge show large

enhancement. One difference between the two curves is that the mobility enhancement for

Si saturates at about 3GPa, but it does not saturate until 6GPa of stress is applied to Ge.

1800
pinv=1x1013/cm2 Ge
a) 1500
0
> 1200 Si0.25Ge075


900 Si. .Geo .

S0.75 0.25
S 600
0


0~- ---------------------------
a) 300 S
0


0 2 4 6

Uniaxial Stress / GPa
Figure 5-4. Germanium and silicon hole mobility on (001)-oriented device under uniaxial
compressive stress where the inversion hole concentration is 1 x 1013/cm2.


5.2 Strain Altered Bulk Ge Valence Band Structure

To give the physical insights of the similarity and the difference of the hole mobility

enhancement under strain for Ge and Si, strain altered bulk germanium valence band

structure is discussed in this section. Strain brings band splitting and effective mass

change to semiconductor valence band. Here, we shall focus on the effective mass change

with strain and compare the difference between germanium and silicon. In next section,


















600
Pin=1 xl 013/cm2
S500 (110)Ge

S400

S300- (110) Si
>.
200

S 100

0 i-4
0 2 4 6
Uniaxial Compressive Stress / GPa
Figure 5-5. Germanium and silicon hole mobility on (1lO)-oriented device under uniaxial
compressive stress where the inversion hole concentration is 1 x 103/cm2.









Ge subband structure in inversion lv. -i~ will be discussed and the phonon scattering rate

will be calculated.

5.2.1 E-k Diagrams

Figure 5-6 shows the dispersion relation diagrams for (001)-Ge under different

stress. Like silicon, the heavy hole and light hole bands of relaxed Ge are degenerate at

F point as shown in Figure 5-6(a). The degeneracy is lifted when strain is applied. The

band splitting leads to band warping and the change of hole effective mass and phonon

scattering rate. The SO band energy is 296 meV lower than the HH and LH bands for

relaxed germanium which implies less coupling with HH and LH bands compared with

silicon. Under biaxial tensile strain, the top band is LH-like out-of-plane and HH-like

along (110). For both compressive strain in Figure 5-6(c) and (d), the top band is HH-like

out-of-plane and LH-like along (110). Uniaxial compressive strain brings the most warping

on the top valence band. The warping is the smallest under biaxial compressive strain,

which -ti-'-i -1- the least mobility enhancement as shown in Figure 5-3.

5.2.2 Effective Mass

Strain-induced <110> and out-of-plane effective mass change at F point are showed

in Figure 5-7 for biaxial tension, 5-8 for biaxial compression, and 5-9 for uniaxial compres-

sion. Compared with silicon, the effective mass for germanium is obviously much smaller

along both directions. This ti-'-. -1; larger hole mobility for germanium than silicon

according to Drude's model. One significant difference from Si effective mass is that the

hole effective mass change of Ge saturates with stress at much higher stress than silicon.

For some of the curves, i.e. "top" band of Figure 5-7(a) and 5-9(b), or "bottom band" of

Figure 5-8, the effective mass change does not saturate until the stress goes up to 7 GPa.

But for silicon, normally the effective mass change saturates at 2 or 3 GPa. This -Ii--.; -I-

higher stress for the mobility saturation.

The trend of the effective mass change with stress is similar for both silicon and

germanium. If we look at the channel direction ((110)) effective mass, the top band














































Figure 5-6. E-k diagrams for Ge under (a) no stress; (b) 1 GPa biaxial tensile stress; (c) 1
GPa biaxial compressive stress; and (d) 1 GPa uniaxial compressive stress.












































Bottom Band


Ton Band


1 2 3 4 5 6 7
(b)


Figure 5-7. Conductivity effective mass vs biaxial tensile stress: (a)
(<110>) and (b) out-of-plane direction.


('IC iin,, I direction


0.22-

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02-
0

























0.08


0.07


0.06 Top Band


0.05
0 1 2 3 4 5 6 7
(a)

0.22

0.2-

0.18- Top Band

0.16

0.14

0.12
bottom Band
0.1

0.08-

0.06

0.04 '''
0 1 2 3 4 5 6 7
(b)

Figure 5-8. Conductivity effective mass vs biaxial compressive stress: (a) C'!, ,iii, I
direction (<110>) and (b) out-of-plane direction.



















0.4

).35

0.3


0.251


0.21


0.15F


0.11


U-
0 1 2 3 4 5 6
(a)


Figure 5-9. Conductivity effective mass vs uniaxial compressive stress:
direction (<110>) and (b) out-of-plane direction.


(a) C'!i, ,ii, I


Bottom Band











Top Band









effective mass at F point under uniaxial compressive stress is only about 0.0 ,,,,, compared

with 0.;:,,,,, for relaxed germanium. The ratio of the change is 9.5 comparing with 5.4

of silicon (0.59mo to 0.11mo). This huge effective mass gain does not result in higher

mobility enhancement in Figure 5-4 because of the much smaller 2D DOS and initial large

subband splitting in the inversion Il-.-i which will be addressed in the next section.

Under biaxial tensile stress, the top band has higher channel direction effective mass

(increasing with stress) and lower out-of-plane effective mass which is similar to silicon.

This means the stress effect and the transverse electric field effect in the inversion 1-iv. r

should be subtractive and the hole mobility should be degraded at low stress as shown

in Figure 5-2. Under biaxial compressive stress, the top band has very low conductivity

effective mass at F point along (110). As we mentioned before, the band warping is not

significant and only happens very close to the F point, which -i-i.; -r; the average effective

mass of the system may not decrease much with the stress.

5.2.3 Energy Contours

The energy contours of the valence band provide a straightforward picture of the

conductivity effective mass and density-of-states of each band. The conductivity and

density-of-states effective mass change with strain can also be seen from the shape change

of the contours. The 25 meV contours of the unstressed Ge are shown in figure 5-10.

Contours (25 meV) under biaxial compressive and tensile stress are shown in figure 5-11

and 5-12. Figure 5-13 shows the contours under uniaxial compressive stress. The energy

contours are similar to those of silicon, but the shape of the contours changes more than

Si contours when the same amount of strain is present. Another difference is that under

uniaxial compressive stress, the 2D DOS of Ge looks much smaller than Si. From the

analysis of Si, lower F point DOS leads to smaller strain induced mobility improvement

due to fewer holes are affected by strain. This may explain why the mobility enhancement

factor for Ge is not larger than Si, although the effective mass change is much larger at F

point.

























0.05
0 005 -005


--- -i --<-....



0.05

0 0 05 -005


Figure 5-10. 25meV energy contours for unstressed Ge: (a) Heavy-hole; (b) Light-hole.


0 05 -


0.05

0 005 -0.05


0.05
0 005 -0.05


Figure 5-11. 25meV energy contours for biaxial compressive stressed Ge: (a) Top band;
(b) Bottom band.






























0.05
-005 0
-0.05
-0.0 005 -0.05

(a)


0.05
-0 05 -
0050 0 05 -0.05

(b)


Figure 5-12.


25meV energy
Bottom band.


contours for biaxial tensile stressed Ge: (a) Top band; (b)


0.05 -


0.05
0 0 05 -0.05
0 005 -0.05


0.05

0 0 05 -0.05


Figure 5-13. 25meV energy contours for uniaxially compressive stressed Ge: (a) Top band;
(b) Bottom band.









5.3 Discussion Of Hole Mobility Enhancement

5.3.1 Strain-induced Subband Splitting

Based on the triangular potential approximation, subbands with higher out-of-plane
effective mass tend to go closer to the top of the quantum well (lower energy for holes)

under vertical electric field. Figure 5-8 and 5-9 show that both biaxial compressive stress
and uniaxial compressive stress shift up the out-of-plane HH-like band. This effect is

clearly additive to the electric field effect. For biaxial tensile stress, the electric field effect
is subtractive to the strain effect and therefore if the electric field is fixed, the subband

splitting should decrease at low stress level and at some stress value, the top two subbands

would cross over each other just like Si. The subband splitting between the ground and
the first excited state of (001) Ge is illustrated in Figure 5-14.

140
140 (001) Ge, bi. tens. .. -

S120
E ,t
100 /





8 8
c 40 4
-5-


20 \ *
SCI

0
0 1 2 3 4 5 6

Stress / GPa

Figure 5-14. Ge subband splitting under different stress.


Figure 5-14 shows that the subband splitting for relaxed Ge p-MOSFETs is much

larger than that of the Si p-MOSFETs. The splitting is larger than the optical phonon









energy of Ge(37 meV), while for unstressed (001) Si, the splitting is smaller than optical

phonon energy. The difference, together with the fact that Ge has much smaller 2D DOS,

-,i--.- -i; that even without the stress, the inter-subband optical phonon scattering rate is

much smaller compared with silicon. Except biaxial tensile stress, the subband splitting

of all compressive stress cases increases with the stress. The amount of increase is much

smaller than that of the silicon cases, which indicates smaller phonon scattering rate

change with the stress for (001) Ge comparing with Si. This -ii--, -1' that the strain-

enhanced hole mobility is mainly because of the effective mass gain.

5.3.2 Biaxial Stress on (001) Ge

As we mentioned in C'! lpter 3, with more strain, more region in momentum space is

affected. The band which is out of the strain-affected region is normally warped. Figure 5-

15 shows the normalized in-plane E-k diagram under biaxial compressive stress. On the

one hand, the figure shows as the stress increases, more region is near the zone center is

warped and has lower DOS. On the other hand, out of the warped area, the band curves

up a little as the stress increases, which -ii--.- -I the increase of the DOS. The overall

effect is that the effective mass gain close to F point due to stress is compromised by

the heavier mass of the holes away from the F point. At low stress, the mass change,

together with the increasing subband splitting, enhances the hole mobility slightly. Under

higher stress, the enhancement is minimal. For Si pMOS under biaxial compressive stress,

the E-k diagram is similar to Ge. The difference is that Si has much larger DOS near F

point, therefore there is ahv-- effective mass gain. The different DOS results in the strain

enhanced hole mobility difference as in Figure 5-3.

Biaxial tensile stress affects the Ge hole mobility similar to Si devices: the subtractive

nature of the strain and transverse electric field effects degrades the hole mobility at low

stress, and the decrease of the phonon scattering rate enhances the mobility at high stress.

Uniaxial compressive stress on (001)-oriented Ge is focused next because it provides the






























-0.1 -0.05 0 0.05 0.1
k
X

Figure 5-15. Normalized ground state subband E-k diagram vs biaxial compressive stress.


most mobility enhancement and the mobility enhancement mechanism is a little different

from Si.

5.3.3 Uniaxial Compression on (001) Ge

Ground state 2D DOS of Si and Ge are shown in Figure 5-16 at different energies.

DOS of Ge is much lower than Si. The trend of the DOS change with stress is similar for

Si and Ge. Figure 5-16(a) shows that the DOS of Ge saturates with stress at higher stress

than Si, since the effective mass changes with stress at higher stress. This is consistent

with the mobility saturation curves in Figure 5-4.

Phonon and surface roughness scattering rate change vs uniaxial stress is shown in

Figure 5-17 and 5-18. For both Si and Ge, phonon scattering rate does not change much

at low stress, and at high stress the phonon scattering rate decreases as the uniaxial stress

increases. Ge has lower scattering rate than Si due to the smaller DOS of Ge. For Si, both

acoustic phonon and optical phonon scattering rate decreases by 50' when the stress









2.5 014
2.5
2D DOS at Energy=5 meV

2.0 ,' Si
(u)
cz 1.5
(U)
I
0
> 1.0
.i-
U)

-> 0.5
"o Ge
C)
0.0
0 2 4 6
Uniaxial Compressive Stress / GPa
(a)
x1014
6

5


)4
0)

3 Si
-* *-- - *
0
22
(1 N- Ge
a)
TS 1------------

C\j0

0 2 4 6

Uniaxial Compressive Stress / GPa
(b)

Figure 5-16. Two dimensional density-of-states of the ground state subband for Si and Ge
at (a)E5mneV; (b)E 2kT 52meV under uniaxial compressive stress.


113









increases from zero to 6 GPa. For Ge, the phonon scattering rate only decreases ;: .' The

surface roughness scattering rate increases with the stress for both Si and Ge due to the

hole repopulation under stress as we explained previously.

Figure 5-19 shows the mobility enhancement contribution from effective mass (solid

lines) and phonon scattering rate (dashed lines) for Si and Ge. For Si, effective mass gain

is the main driving force of the mobility enhancement at low stress, and the scattering

rate change is dominant at high stress range (1 GPa-3 GPa). From unstressed case to 3

GPa of stress, effective mass gain and phonon scattering rate decrease have comparable

enhancement to the hole mobility. For Ge, the phonon scattering only contribute 1.5 times

of the enhancement. The effective mass change is dominant in the whole stress range. As

we mentioned before, this is because the effective mass change ratio is large under stress

(0.;:,,,,, to smaller than 0.0 1,,,). Another observation of the effective mass is that as

the stress is over 1 GPa, increasing the stress does not change the hole effective mass for

Si, but the effective mass of Ge continue to decrease as the stress increases. This extra

effective mass gain contribute to the hole mobility enhancement for Ge at very high stress.

5.3.4 Uniaxial Compression on (110) Ge

The confined 2D energy contours are showed in Figure 5-20 for (001)-oriented

MOSFETs and Figure 5-21 for (110)-oriented p-MOSFETs. For (110) Ge p-MOSFETs

under uniaxial stress, the strain effect is similar to (110) Si p-MOSFETs. The strong

quantum confinement warps the subband structure and results in small hole effective

mass, which explains the higher unstrained hole mobility than (001) Ge p-MOSFETs. As

the uniaxial compressive stress is applied, the strain effect is undermined by the strong

quantum confinement and only warps the high energy region of each subband. As a result,

the hole mobility is not enhanced as significantly as (001)-oriented p-MOSFETs.

5.4 Summary

Germanium hole mobility improvement under biaxial tensile, biaxial compressive

and uniaxial compressive stress is analyzed and compared with silicon. The trend of









x1012






4 Si

r iw ** *


0 2 4

Uniaxial Compressive Stress/ GPa
(a)

x1012







\ Si





Ge
I


Uniaxial Compressive Stress / GPa
(b)


Figure 5-17. Phonon scattering rate vs uniaxial compressive stress:
and (b) optical phonon.


(a) Acoustic phonon,


0.0


9.0


0)


c-
0
CO
0
0
O
r-



a5-
0
O


6.0




3.0


0.0




















x1012
4.5


) (D *

0. 0
a) to ,
I t



a)


C/) 0 Ge

0.0
0 2 4 6

Uniaxial Compressive Stress / GPa

Figure 5-18. Surface roughness scattering rate vs uniaxial compressive stress for Ge and
Si.


































1 2

Uniaxial Stress / GPa


Figure 5-19.






0.15r


Mobility enhancement contribution from effective mass change (solid lines)
and phonon scattering rate change (dashed lines) for Si and Ge under
uniaxial compressive stress.


-0.15


0.15







0


-0.15


0.15


Figure 5-20. Confined 2D energy contours for (001)-oriented Ge pMOS with uniaxial
compressive stress.


Unstressed Ge


1GPa Uniaxial Compression


,


- 0[











1GPa Uniaxial Compression


-0.15 0 0.15 -0.15 0 0.15
k k
x x
(a) (b)

Figure 5-21. Confined 2D energy contours for (1O0)-oriented Ge pMOS with uniaxial
compressive stress.


each stress type for both germanium and silicon is similar-uniaxial compressive stress

on (001)-oriented transistors has the most hole mobility improvement mainly from the

reduced hole conductivity effective mass. Uniaxial compressive stress on (110)-oriented

devices does not provide as much improvement due to the strong quantum confinement

undermining the strain effect. Hole mobility is degraded under low biaxial tensile stress

due to the subtractive nature of the strain and vertical electric field effects and hence the

increase of the average effective mass. The mobility is enhanced at high stress because

of the reduction of the inter-subband scattering rate. Biaxial compressive stress does not

improve the hole mobility much due to the small DOS after band/subband warping and

not much effective mass gain.


Unstressed (110) Ge









CHAPTER 6
SUMMARY AND SUGGESTIONS TO FUTURE WORK

6.1 Summary

In this work, uniaxial stress-induced hole mobility enhancement in (001)-oriented

Si p-MOSFETs is calculated at high stress (up to 3 GPa) and large enhancement factor

(4.5x) is obtained. For the first time, coordinates system transformation of Luttinger-

Kohn's Hamiltonian and Kubo-Greenwood Equation is performed to investigate the hole

mobility in Si and Ge p-MOSFETs with surface orientations other than (001). The strong

quantum confinement in (110), (111), and (112)-oriented p-MOSFETs is reported for the

first time. The results show that, unlike (001) p-MOSFETs, the subband structures of Si

and Ge in (110), (111), and (112)-oriented p-MOSFETs are warped by the confinement.

The strong confinement causes smaller hole effective mass and lower phonon scattering

rate due to larger subband splitting, which explains the higher hole mobility in those

p-MOSFETs. To analyze the difference of the stress-induced phonon scattering rate for

(001) and (110) p-MOSFETs, two-dimensional density-of-states (2D DOS) are evaluated

at arbitrary energy in the subbands. Comparing with (001) p-MOSFETs, (110) p-

MOSFETs have smaller DOS and DOS does not vary much as the uniaxial stress increases

due to the stronger quantum confinement. Under uniaxial stress, the phonon scattering

rate for (110) p-MOSFETs does not change as much as (001) p-MOSFETs. 2D energy

contours of the subbands in (001) and (110) p-MOSFETs under stress are investigated

and smaller effective mass change with stress for (110) p-MOSFETs is found which is

again due to the stronger quantum confinement. The smaller change of effective mass and

phonon scattering rate results in lower mobility enhancement in (110) p-MOSFETs. As

a result, at high uniaxial stress (3 GPa), (001)/<110>, (110)/<110>, and (110)/<111>

p-MOSFETs have similar hole mobility.









Strain induced hole mobility enhancement is studied theoretically for the first time

in ultra-thin-body (UTB) non-classical p-MOSFETs, including single-gate (SG) silicon-

on-insulator (SOI), (001) symmetrical double-gate (SDG) p-MOSFETs, and (110) p-type

FinFETs. For SG SOI p-MOSFETs, the strain effects are as same as traditional Si p-

MOSFETs. For (001) SDG p-MOSFETs and (110) FinFETs, subband modulation is

found when the channel thickness is smaller than 20 nm. Due to the interaction of the

two surface channels, the subband splitting between the ground state and the first excited

state is small (about 3 to 5 meV) as the body thickness is larger than 5 nm. This splitting

does not change as the stress increases. Compared with the single gate p-MOSFETs,

this small splitting is similar to increasing the DOS of the ground state subband. As

the stress increases, the average effective mass change is larger than that in single gate

p-MOSFETs. The low form factors due to the symmetrical structure and low electric

field in the channel -ii--.- -1 the phonon scattering rate in double gate pMOSFETs is lower

than single gate p-MOSFETs, regardless the small subband splitting. For (001) SDG

p-MOSFETs, the phonon scattering rate change is a little smaller than single gate p-

MOSFETs as the stress increases. The larger effective mass change and smaller scattering

rate change result in similar hole mobility enhancement factor compared with single gate

p-MOSFETs. For FinFETs, the form factors are much smaller than single gate (110)

p-MOSFETs and the change with stress is larger which -i.-.-.- -i- larger scattering rate

change. Therefore, the strain-induced hole mobility enhancement (3x) is larger than single

gate (110) p-MOSFETs (2x).

Strain effect on hole mobility improvement in (001) and (110) Ge and Sil_,Gex

p-MOSFETs is calculated for the first time. The mobility enhancement at low stress

is similar to Si. At high stress, the maximum mobility enhancement factor for (001)

Ge is larger than Si due to the greater effective mass change, especially at high stress.

The phonon scattering rate change for Ge p-MOSFETs is a little smaller than Si. For

(110) Ge p-MOSFETs, strong quantum confinement is found and the strain induced









mobility enhancement is smaller than (001) Ge. Biaxial compressive stress effect on Ge

p-MOSFETs is also calculated, and very small enhancement is found.

6.2 Recommendations for Future Work

The ...r'essive scaling of silicon C'L\ OS technology has pushed the channel length to

nanometer regime. Strain, especially uniaxial compressive strain, can improve the hole

mobility of pMOSFETs dramatically and hence enhance the device performance. To

further improve the performance of C' \!OS technology, other feature-enhanced technology

and even new material will be a must have. Non-classical devices have been seen as

possible replacement for simple planar layout single gate bulk silicon devices and have

the potential to be scaled down further in the roadmap. Although theoretical calculation

shows the performance could be improved by strain, the question still exists how strain

can be applied to these devices, especially FinFETs.

Germanium is one new material that has been considered to replace silicon in C'\ OS

technology. Uniaxial strain even has higher enhancement on germanium pMOS. But the

experimental work is still lack for germanium. People are still trying to find out the best

layout, proper dielectric and gate materials. It will be a long way but definitely worth

working on.

How about after all of this? There will be an ultimate limit for the scaling that

ballistic transport will take place and the mobility concept will not be valid. Will strain

still be useful at that stage? The answer is probably yes, since the strain can reduce the

effective mass of the carriers and this will still help the transport. That being said, serious

calculation will be necessary to further explain this.









APPENDIX A
STRESS AND STRAIN

Stress a is defined as the force F applied on unit area A.

F
S= lim (A-l)
A-O A

Any stress on an isotropic solid body in a cartesian coordinate system can be

expressed as a stress matrix a [13, 99],


O-xx T xy T xz

Tyx yy c Tyz (A-2)
TZX Tzy -zz

where
Fi
aii lim
Ai-O Ai

is called the normal stress on the i-face in the i-direction and

F.
T, = lim
Ai->O Ai

is the shear stress on the i-face in the j-direction [100] as shown in Figure A-i. This

stress matrix completely characterizes the state of stress at ( i --I i-

For stress S along < 100 >-direction, the matrix can be written as


1 0 0

a= S 0 0 0 (A-3)



For stress S along both <100> and <010>-direction (biaxial stress),


1 (A4)0 0

a=S 0 1 0 (A-4)
o o o




























Figure A-i. Stress distribution on ( i-- I i-


Stress S along <110>-direction is a little complicated. The stress is applied on both

(100) and (010) planes. If we resolve each component along x and y axes to get both
normal and shear terms, each term has the same magnitude of S/2. The stress tensor can

be expressed as,



S= 1 1 0 (A-5)

0 0 0
For stress S along <111>-direction, based on the similar analysis, the stress is

actually acted on (100), (010), and (001) planes. Each component can be resolved along x,
y, and z axes and the stress along each direction is S/3. Therefore the stress tensor is,



S= 1 1 1 (A-6)
3 1 1










The stress matrix is symmetric where ij = ji, and only 6 components are necessary

to represent the stress. Therefore, the 3 x 3 matrix can also be written as a 6 x 1 stress

vector.





-yy
-zz


az (A-7)


RTz



Strain is defined as the distortion of a structure caused by stress. Normal strain is

defined as the relative lattice constant change [13, 99],



a a (A8)
ao

where ao and a are lattice constant before and after the strain.

However, the deformation of the ( i- -I I1 cannot be fully represented with the normal

strain. It also has shear terms that are defined as change in the interior angles of the unit

element. Like stress, strain can also be expressed with a symmetric 3 x 3 tensor or 6 x 1

vector e [100].







ezz
C 6 yx g YY y or, C (A-9)
2cyz
,zx zy tzz
2 xz

2cxy









For most materials the stress is a linear function of strain. The transformation

between stress and strain is through a 6 x 6 stiffness matrix C or compliance matrix

S [99].



S= C. (A 10)



zxx C11 C12 C13 C14 C15 C16 6xx

jyy C21 C22 C23 C24 C25 C26 EyY

0zz C31 C32 C33 C34 C35 C36 Czz
(A-11)
TyZ C41 C42 C43 C44 C45 C46 2cy

TXz C51 C52 C53 C54 C55 C56 2ez

Txy C61 C62 C63 C64 C65 C66 2ecy
or,


c S'y



Cxz S11 S12 S13 S14 S15 S16 x-x

yy S21 S22 S23 S24 S25 S26 yy

Czz S31 S32 S33 S34 S35 S36 zz(
(A 12)
2yz S41 S42 S43 544 S45 S46 Tyz

2ez S51l S52 S53 S54 S55 S56 Tz

2cy S61 S62 S63 S64 S65 S66 I Ty



For diamond or zinc-blende-type (i -1I I stiffness matrix and compliance matrix can

be simplified as [99]










Table A-i. Elastic stiffnesses Ci in units of 1011N/m2 and compliances Sij in units of
10-11m2/N


C11
Si 1.657
Ge 1.292


S11
0.768
0.964


S12
-0.214
-0.260


C12
0.639
0.479


C44
0.7956
0.670




C12

C12

C11

0

0

0


S12

S12

S11

0

0

0


S44
1.26
1.49


6xx





2cyz

2c22

2cy
2(Exz

o-xx

o-yy





Tx

-XY


(A-13)












(A-14)


The stiffness and compliance coefficients for

following table.


silicon and germanium are listed in the


Let's go back to the strain tensor. Each strain can be decomposed to two compo-

nents: hydrostatic term and shear term. The shear term can be further decomposed to

shear-100 term which only has diagonal elements and shear-111 term which only contains

non-diagonal elements.




6 hydrostatic + 6shear-100 + 6shear-111


0 0 0

0 0 0

0 0 0

744 0 0

0 C44 0

0 0 C44


6xx

cyy

6zz

2cyz



2cy


C12

C11

C12

0

0

0


S12

S11

S12

0

0

0


0 0

0 0

0 0

S44 0

0 S44

0 0












0 0
2cxx + (Cyy +czz 0 0
1








0 0 2czz (Cx + ) / -







The hydrostatic term in the strain tensor shifts the energy of all the bands in

semiconductors by the same amount simultaneously but does not cause band splitting,

since it is actually a constant and in the calculation of the band energy it only acts like

adding an additional potential term to the hamiltonian. The semiconductor transport

property is independent on the hydrostatic strain term. For two different stress, as long
as the shear terms of their strain tensors are equal, their impact to the carrier mobility

should be identical.

Stress can be applied to semiconductors from any direction. For a silicon MOS-

FET, only in-plane biaxial stress or channel direction uniaxial stress has technological

importance. The common silicon wafers that are used in industry are (001)-oriented, and

normally the channel of the MOSFET is along <110>direction. Biaxial stress here means

that the stress is applied in both <100> and <010>-directions of the wafer with the

same magnitude. Uniaxial stress represents the stress along the <110> channel direc-

tion. This stress is also called uniaxial longitudinal stress. In the same manner, uniaxial

transverse stress normally means the uniaxial stress applied perpendicular to the channel

direction. Both of those stresses are applied in the plane of the wafer, therefore they are

also "in-pl! .ii, stresses. Another kind of uniaxial stress is called "out-of-pl! i., uniaxial









stress which means the stress is applied in the direction perpendicular to the surface of the
wafer.

For the out-of-plane uniaxial stress and the in-plane biaxial stress on (001) wafer,

the strain matrices only have diagonal terms and all non-diagonal terms are zero. The

question is, how do these two stresses differ from each other? Let's assume we have out-

of-plane uniaxial stress -a on one sample and in-plane biaxial stress a on another sample.
For case 1, based on (1.4) and (1.15), the strain tensor can be expressed as, in the form of

(1.16),


S12
Cu 0

0

S11 + 2S

3 0
0

Sii S12
0
3
0
? S -
+I o


12 0 0
S11 + 2S12 0

0 S11 + 2S12
S/ hydrosatic
0 0

S11 S12 0
0 2(S12 Sn)
8]hear


For in-plane biaxial stress,


S11 + S12
0

0


0

Sn1 + S12
0


(A16)


Cb "


0

0

2S12









2(Sil + 2S12)

3 0
0


0

2(SIl + 2S12)

0


0

0

2(S + 2S2) hydrostatic
/ hydrosatic


SI S12 0 0

+ 0 S11 S12 0 (A-17)

0 0 2(S12 S1l)

(1.17) and (1.18) show that the hydrostatic terms of those two strain tensors are

different, but the shear terms are identical. This tells us that the biaxial tensile (com-

pressive) stress should have the same effect as the out-of-plane uniaxial compressive (or

tensile) stress in determining the transport property of the holes.









APPENDIX B
PIEZORESISTANCE

The piezoresistance, or piezoresistive effect, describes the electrical resistance change

of materials caused by applied mechanical stress. The first measurement of piezoresistance

was performed by Bridgman in 1925 and extensive study on this topic was done ever since.

In 1954, Smith measured the piezoresistance effect on Si and Ge [7]. This effect becomes

more and more important due to the wide application of Si and Ge on contemporary

C'\ OS technology.

Similar to stress and strain, the change of resistivity of a material is a symmetrical

second rank tensor. The tensor connecting the stress and the piezoresistance is of fourth

rank. For Si and Ge, we can simplify the tensor as [7]


11 7r12 712 0 0 0

71l2 rll 712 0 0 0

H 12 712 ri11 0 0 0
n (Bi1)
0 0 0 7T44 0 0

0 0 0 0 7T44 0

0 0 0 0 0 7T44

The most general form of a two-dimensional piezoresistance tensor in the inversion

1-v-r is [8]


iT11 712 7T14

I 721 722 7124 (B-2)

741 7"42 44

For (001), (110), and (111) surface oriented Si (or Ge), F714 741 = F24 i2 = 0

(principle axis (001) for (001) and (110) surface, (110) for (111) surface). We can further

simplify the piezoresistance tensor as [8]

















For (001) surface oriented Si








For (111) surface oriented Si


r11 7r12 0

= 12 722 0

0 0 7144

and Ge, 7rn = 722 and

/ \ll l 0
711 712 0

H 71"12 7i11 0

0 0 7i44

and Ge, 7i44 = T11 712 and


0 0 7in 712
In the piezoresistance tensors, 711 represents the longitudinal piezoresistance coef-

ficient (along ( 100) for (001) and (110) surface). 2,, is the transverse piezoresistnace

coefficient (along ( 010) for (001) and (110) surface). In standard MOSFETs, the channel

direction is along (110) and the uniaxial stress is applied either along (110) or (110). By

rotational transformation of the tensor the new longitudinal and transverse piezoresistance

coefficients become [8]


S= ll +
1-11 t(711 + 712 + 744)
2


1
12 11 + 12 44)
2


(B-6)


(B-7)


(B-3)








(B-4)










For biaxial stress, the piezoresistance coefficient is the sum of the longitudinal and

transverse terms.



7Biazial = 11 + 7112 (B-8)









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BIOGRAPHICAL SKETCH

Guangyu Sun was born in Shandong, China, on June 9th, 1975. In 1992, he was

admitted to University of Science and Technology of C('hlii (USTC) in Hefei, ('C!h i From

1992 to 1996 he studied in USTC and received his B.S. degree in applied physics in 1996.

He subsequently participated in the master's program and obtained the M.S. degree in

1999. In the fall of 1999, he came to the United States and became a Florida Gator. In

the spring of 2004, he entered Prof. Thompson's group and has been studying the strain

effects on Si and Ge MOSFETs, pursuing a Ph.D. degree.





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IamgratefultoallthepeoplewhomadethisdissertationpossibleandbecauseofwhommygraduateexperiencehasbeenonethatIwillcherishforever.FirstandforemostIthankmyadvisor,Dr.ScottE.Thompson,forgivingmeaninvaluableopportunitytoworkonchallengingandextremelyinterestingprojectsoverthepastfouryears.HehasalwaysmadehimselfavailableforhelpandadviceandtherehasneverbeenanoccasionwhenIhaveknockedonhisdoorandhehasnotgivenmetime.Hetaughtmehowtosolveaproblemstartingfromasimplemodel,andhowtodevelopit.Ithasbeenapleasuretoworkwithandlearnfromsuchanextraordinaryindividual.IthankDr.JerryG.Fossum,Dr.HuikaiXie,Dr.ChristopherStanton,andDr.JingGuoforagreeingtoserveonmydissertationcommitteeandforsparingtheirinvaluabletimereviewingthemanuscript.IalsothankDr.ToshiNishidaforalotofhelpfuldiscussionsandkindhelp.MycolleagueshavegivenmealotofassistanceinthecourseofmyPh.D.studies.Dr.YongkeSunhelpedmegreatlytounderstandthephysicsmodel,andwealwayshadfruitfuldiscussions.Dr.ToshiNumataalsogavemegoodadviceandsomeinsightfulideas.IalsothankJisongLim,SagarSuthram,andallothergroupmemberswhomademylifeheremoreinteresting.Iacknowledgehelpandsupportfromsomeofthestamembers,inparticular,ShannonChillingworth,TeresaStevensandMarcyLee,whogavememuchindispensableassistance.Iowemydeepestthankstomyfamily.Ithankmymotherandfather,andmywife,Anita,whohavealwaysstoodbyme.Ithankthemforalltheirloveandsupport.WordscannotexpressthegratitudeIowethem.Itisimpossibletorememberall,andIapologizetothoseIhaveinadvertentlyleftout. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 LISTOFTABLES ..................................... 11 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTIONANDOVERVIEW ....................... 14 1.1HistoryofStraininSemiconductors ...................... 15 1.2ApplyStraintoATransistor .......................... 17 1.3MainContributionsofMyResearch ...................... 18 1.4BriefDescriptionofTheDissertation ..................... 19 2KPMODELANDHOLEMOBILITY ...................... 21 2.1ThekpMethod ................................ 21 2.1.1IntroductiontokpMethod ...................... 21 2.1.2Kane'sModel .............................. 25 2.1.3Luttinger-Kohn'sHamiltonian ..................... 28 2.2HoleMobilityinInversionLayers ....................... 32 2.2.1Self-consistentProcedure ........................ 32 2.2.2HoleMobility .............................. 33 2.3ScatteringMechanisms ............................. 34 2.3.1PhononScattering ............................ 34 2.3.2SurfaceRoughnessScattering ..................... 36 2.4Summary .................................... 38 3STRAINEFFECTSONSILICONP-MOSFETS ................. 39 3.1PiezoresistanceCoecientsandHoleMobility ................ 40 3.1.1PiezoresistanceCoecients ....................... 40 3.1.2HoleMobilityvsSurfaceOrientation ................. 41 3.1.3HoleMobilityandVerticalElectricField ............... 42 3.1.4Strain-enhancedHoleMobility ..................... 42 3.2BulkSiliconValenceBandStructure ..................... 48 3.2.1DispersionRelation ........................... 48 3.2.2HoleEectiveMasses .......................... 49 3.2.3ValenceBandunderSuperLowStrain ................. 56 3.2.4EnergyContours ............................ 56 3.3StrainEectsonSiliconInversionLayers ................... 60 3.3.1QuantumConnementandSubbandSplitting ............ 60 5

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......................... 64 3.3.3Strain-inducedHoleRepopulation ................... 67 3.3.4ScatteringRate ............................. 73 3.3.5MassandScatteringRateContribution ................ 77 3.4Summary .................................... 78 4STRAINEFFECTSONNON-CLASSICALDEVICES .............. 80 4.1SingleGateSOIpMOS ............................. 82 4.1.1HoleMobilityvsSiliconThickness ................... 82 4.1.2Strain-enhancedHoleMobilityofSOISG-pMOS ........... 83 4.2Double-gatep-MOSFETs ........................... 85 4.2.1(001)SDGpMOS ............................ 87 4.2.2StrainEectonFinFETs ........................ 89 4.3Summary .................................... 93 5STRAINEFFECTSONGERMANIUMP-MOSFETS .............. 95 5.1GermaniumHoleMobility ........................... 96 5.1.1BiaxialTensileStress .......................... 96 5.1.2BiaxialCompressiveStress ....................... 97 5.1.3UniaxialCompressiveStress ...................... 100 5.2StrainAlteredBulkGeValenceBandStructure ............... 100 5.2.1E-kDiagrams .............................. 102 5.2.2EectiveMass .............................. 102 5.2.3EnergyContours ............................ 107 5.3DiscussionOfHoleMobilityEnhancement .................. 110 5.3.1Strain-inducedSubbandSplitting ................... 110 5.3.2BiaxialStresson(001)Ge ....................... 111 5.3.3UniaxialCompressionon(001)Ge ................... 112 5.3.4UniaxialCompressionon(110)Ge ................... 114 5.4Summary .................................... 114 6SUMMARYANDSUGGESTIONSTOFUTUREWORK ............ 119 6.1Summary .................................... 119 6.2RecommendationsforFutureWork ...................... 121 APPENDIX ASTRESSANDSTRAIN ............................... 122 BPIEZORESISTANCE ................................ 130 REFERENCES ....................................... 133 BIOGRAPHICALSKETCH ................................ 141 6

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Figure page 1-1SchematicdiagramofbiaxialtensilestressedSi-MOSFETonrelaxedSi1xGexlayer .......................................... 17 1-2UniaxialstressedSi-MOSFETwithSi1xGexSource/Drainorhighlystressedcappinglayer ..................................... 18 3-1Holemobilityvsdevicesurfaceorientationforrelaxedsilicon ........... 41 3-2Holemobilityvsinversionchargedensityforrelaxedsilicon.Bothmeasurementsandsimulationshowlargermobilityon(110)devices. ............... 43 3-3Holemobilityvsstresswithinversionchargedensity11013=cm2. ....... 44 3-4Calculatedstraininducedholemobilityenhancementfactorvs.experimentaldatafor(001){orientedpMOS. ........................... 45 3-5Holemobilityenhancementfactorvsuniaxialstressfordierentchanneldoping. 45 3-6Calculatedstraininducedholemobilityenhancementfactorvs.stressfor(001){orientedpMOSwithdierentinversionchargedensity. .............. 47 3-7E-krelationforsiliconunder(a)nostress;(b)1GPabiaxialtensilestress;and(c)1GPauniaxialcompressivestress. ........................ 50 3-8NormalizedE-kdiagramofthetopbandunderdierentamountofstress.Largerstresswarpsmoreregionoftheband.Theenergyatpointforallcurvesissettozeroonlyforcomparisonpurpose. ...................... 51 3-9Channeldirectioneectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. ......................... 52 3-10Two-dimensionaldensity-of-stateseectivemassesforbulksiliconunder(a)bi-axialtensilestress;and(b)uniaxialcompressivestress. .............. 53 3-11Out-of-planeeectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. ......................... 54 3-12Holeeectivemasschangeunderverysmallstress.Thechangeinthisstressregionexplainsthe\discontinuity"oftheholeeectivemassbetweenthere-laxedandhighlystressedSi. ............................. 57 3-13The25meVenergycontoursforunstressedSi:(a)Heavy-hole;(b)Light-hole. 58 3-14The25meVenergycontoursforbiaxialtensilestressedSi:(a)Topband;(b)Bottomband. ..................................... 59 7

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................................... 59 3-16Quantumwellandsubbandsenergylevelsundertransverseelectriceld. .... 61 3-17Schematicplotofstraineectonsubbandsplitting,theeldeectisadditivetouniaxialcompressionandsubtractivetobiaxialtension. ............ 64 3-18Subbandsplittingbetweenthetoptwosubbandsunderdierentstress. ..... 65 3-19Out-of-planeeectivemassesforh110isurfaceorientedbulksiliconunderuni-axialcompressivestress. ............................... 66 3-20The2Denergycontours(25,50,75,and100meV)forbulk(001)-Si.Uniaxialcompressivestresschangesholeeectivemassmoresignicantlythanbiaxialtensilestress. ..................................... 68 3-21Conned2Denergycontours(25,50,75,and100meV)for(001)-Si.Thecon-toursareidenticaltothebulkcounterparts. .................... 69 3-22The2Denergycontours(25,50,75,and100meV)forbulk(110)-Siunder(a)nostress;(b)uniaxialstressalongh110i;and(c)uniaxialstressalongh111i. .. 70 3-23Conned2Denergycontours(25,50,75,and100meV)for(110)-Si.Thecon-nedcontoursaretotallydierentfromtheirbulkcounterpartswhichsuggestssignicantconnementeect. ............................ 71 3-24Groundstatesubbandholepopulationunderdierentstress. ........... 72 3-25Stresseectonthe2dimensionaldensity-of-statesofthegroundstatesubband. 74 3-26Twodimensionaldensity-of-statesatE=4kT. ................... 75 3-27Straineecton(a)acousticphonon,and(b)opticalphononscatteringrate. .. 76 3-28Straineectonsurfaceroughnessscatteringrate. ................. 77 3-29Holemobilitygaincontributionfrom(a)eectivemassreduction;and(b)phononscatteringratesuppressionforp-MOSFETsunderbiaxialanduniaxialstress. 79 4-1HolemobilityvsSOIthicknessforsinglegateSOIpMOS.Themobilitydecreaseswiththethicknessduetostructuralconnement. ................. 83 4-2HolemobilityforsinglegateSOIpMOSvsuniaxialstressatchargedensityp=11013=cm2. ..................................... 84 4-3HolemobilityenhancementfactorofUTBSOISGdevicesvsuniaxialcompres-sivestressatchargedensityp=11013=cm2. ................... 85 4-4SubbandsplittingUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. .............................. 86 8

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87 4-6HolemobilityofSDGdevicesunderuniaxialcompressivestressatchargeden-sityp=11013=cm2. ................................ 88 4-7HolemobilityenhancementfactorofSDGMOSFETsvsuniaxialcompressivestressatchargedensityp=11013=cm2. ..................... 89 4-8HolemobilityofFinFETsunderuniaxialstresscomparedwithbulk(110)-orienteddevicesatchargedensityp=11013=cm2. .................... 90 4-9HolemobilityenhancementfactorofFinFETsunderuniaxialcompressivestressatchargedensityp=11013=cm2. ......................... 91 4-10Holemobilitygaincontributionfromeectivemassandphononscatteringsup-pressionunderuniaxialcompressionfor(110)/h110iFinFETscomparedwithSG(110)/h110ip-MOSFETsatchargedensityp=11013=cm2. ........ 92 5-1Germaniumholemobilityvseectiveelectriceld. ................ 97 5-2Germaniumandsiliconholemobilityunderbiaxialtensilestresswherethein-versionholeconcentrationis11013=cm2. ..................... 98 5-3Germaniumandsiliconholemobilityunderbiaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. .................. 99 5-4Germaniumandsiliconholemobilityon(001)-orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. ... 100 5-5Germaniumandsiliconholemobilityon(110)-orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. ... 101 5-6E{kdiagramsforGeunder(a)nostress;(b)1GPabiaxialtensilestress;(c)1GPabiaxialcompressivestress;and(d)1GPauniaxialcompressivestress. ... 103 5-7Conductivityeectivemassvsbiaxialtensilestress:(a)Channeldirection(<110>)and(b)out-of-planedirection. ............................ 104 5-8Conductivityeectivemassvsbiaxialcompressivestress:(a)Channeldirection(<110>)and(b)out-of-planedirection. ...................... 105 5-9Conductivityeectivemassvsuniaxialcompressivestress:(a)Channeldirec-tion(<110>)and(b)out-of-planedirection. .................... 106 5-1025meVenergycontoursforunstressedGe:(a)Heavy-hole;(b)Light-hole. .... 108 5-1125meVenergycontoursforbiaxialcompressivestressedGe:(a)Topband;(b)Bottomband. ..................................... 108 9

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....................................... 109 5-1325meVenergycontoursforuniaxiallycompressivestressedGe:(a)Topband;(b)Bottomband. ................................... 109 5-14Gesubbandsplittingunderdierentstress. ..................... 110 5-15NormalizedgroundstatesubbandE-kdiagramvsbiaxialcompressivestress. .. 112 5-16Twodimensionaldensity-of-statesofthegroundstatesubbandforSiandGeat(a)E=5meV;(b)E=2kT=52meVunderuniaxialcompressivestress. ....... 113 5-17Phononscatteringratevsuniaxialcompressivestress:(a)Acousticphonon,and(b)opticalphonon. ............................... 115 5-18SurfaceroughnessscatteringratevsuniaxialcompressivestressforGeandSi. 116 5-19Mobilityenhancementcontributionfromeectivemasschange(solidlines)andphononscatteringratechange(dashedlines)forSiandGeunderuniaxialcom-pressivestress. .................................... 117 5-20Conned2Denergycontoursfor(001){orientedGepMOSwithuniaxialcom-pressivestress. .................................... 117 5-21Conned2Denergycontoursfor(110){orientedGepMOSwithuniaxialcom-pressivestress. .................................... 118 A-1Stressdistributiononcrystals. ............................ 123 10

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Table page 2-1Luttinger-Kohnparameters,deformationpotentialsandsplit-oenergyforsili-conandgermanium. ................................. 31 3-1CalculatedandmeasuredpiezoresistancecoecientsforSipMOSFETswith(001)or(110)surfaceorientation.Therstvalueofeachpairisfrommeasure-mentsandthesecondisfromcalculation. ...................... 40 A-1ElasticstinessesCijinunitsof1011N=m2andcompliancesSijinunitsof1011m2=N 11

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Myresearchexploresthestrainenhancedholemobilityinsilicon(Si)andgermanium(Ge)p-typemetal-oxide-semiconductoreld-eect-transistors(p-MOSFETs).Thepiezore-sistancecoecientsarecalculatedandmeasuredviawaferbendingexperiments.Withgoodagreementinthemeasuredandcalculatedsmallstresspiezoresistancecoecients,kpcalculationsareusedtogivephysicalinsightsintoholemobilityenhancementatlargestress(3GPaforSiand6GPaforGe)forstressesoftechnologicalimportance:in-planebiaxialandchannel-directionuniaxialstresson(001)and(110)-surfaceorientedp-MOSFETswithh110iandh111ichannels. Themathematicaldenitionofstrainandstressisintroducedandthetransformationbetweenthestrainandstresstensorisdemonstrated.Self-consistentcalculationofSchr~odingerEquationandPoissonEquationisappliedtostudythepotentialandsubbandenergylevelsintheinversionlayers.Subbandstructures,two-dimensional(2D)density-of-states(DOS),holeeectivemass,phononandsurfaceroughnessscatteringrateareevaluatednumericallyandtheholemobilityisobtainedfromalinearizationofBoltzmannEquation. Theresultsshowthatholemobilitysaturatesatlargestress.Underbiaxialtensilestress,theholemobilityisdegradedatsmallstressduetothesubtractivenatureofthestrainandquantumconnementeects.Atlargestress,holemobilityisimprovedviathesuppressionofthephononscattering.Biaxialcompressivestressimprovesholemobility 12

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Straineectsonnon-classicaldevices(single-gate(SG)silicon-on-insulator(SOI)anddouble-gate(DG)p-MOSFETs)arealsoinvestigated.ThecalculationshowsthatthemobilityenhancementforSGSOIandDG(001)p-MOSFETsissimilartotraditionalSip-MOSFETs.HolemobilityenhancementinFinFETsismorethantraditional(110)p-MOSFETsduetothesubbandmodulation. 13

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Metal-oxide-semiconductoreld-eecttransistors(MOSFETs)havebeenscaleddownaggressivelytoachievedensity,speedandpowerimprovementsince1960s[ 1 ].Asthechannellengthisscaledtosubmicronevennanoscalelevel,thesimplescalingofcomplementarymetal-oxide-semiconductor(CMOS)devicesbringssevereshort-channeleects(SCEs)suchasthresholdvoltageroll-o,degradedsubthresholdslope,anddraininducedbarrierlowering(DIBL).Oxidethicknesshastobereducedtosub-10nm(about1nminthestate-of-the-arttechnology)andchanneldopinghastobeincreasedupto1019=cm3inordertomaintaingoodcontrolofthechannel[ 1 ].Thethinoxideandthehighchanneldopingresultinhighverticalelectriceldinthechannelthatseverelyreducesthecarriermobility.Furtherscalingofthedevicesdoesnotbringperformanceimprovementduetocarriermobilitydegradation. WithnothingtoreplacesiliconCMOSdevicesinthenearfutureandtheneedtomaintainperformanceimprovementsandMoore'slaw,featureenhancedSiCMOStechnologyhasbeenrecognizedasthedriverforthemicroelectronicsindustry.StrainisonekeyfeaturetoenhancetheperformanceofSiMOSFETs.BiaxialtensilestrainhasbeeninvestigatedbothexperimentallyandtheoreticallyinCMOStechnology[ 2 3 4 ].Itimprovestheelectronmobility[ 5 ],butdegradestheholemobilityatlowstressrange(<500MPa)[ 3 ].Recently,uniaxialstresshasbeenappliedtoIntel's90,65,and45{nmtechnologiestoimprovethedrivecurrentwithoutsignicantlyincreasedmanufacturingcomplexity[ 5 6 ]. ThegoalofthisdissertationistoprovidephysicalinsightsintothestrainenhancedholemobilityinSiandGep-MOSFETs.Beforeweinvestigatetheholemobility,thehis-toryofstraintechnologyandthemethodstoapplystraintoatransistorareisdiscussedinthischapter.Theorganizationofthedissertationisalsointroduced. 14

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7 8 ],andthetheoreticalapproachtothestraineect,i.e.,deformationpotentialtheory,canbetracedbacktothe1950s.Butnotuntilintheearly1980'sdidscientistsandengineersstarttorealizethatstraincouldbeapowerfultooltomodifythebandstructureofsemiconductorsinabenecialandpredictableway[ 9 10 ]. Deformationpotentialtheory,whichdenestheconceptofstraininducedenergyshiftofthesemiconductor,wasrstdevelopedtoaccountforthecouplingbetweentheacousticwavesandelectronsinsolidsbyBardeenandShockley[ 11 ],whostatedthatthelocalshiftofenergybandsbytheacousticphononwouldbeproducedbyanequivalentextrinsicstrain,hencetheenergyshiftsbybothintrinsicandextrinsicstraincanbedescribedinthesamedeformationpotentialframework.ThedeformationpotentialtheorywasappliedbyHerringandVogt[ 12 ]in1955intheirtransportstudiesofsemiconductorconductionbands.Asetofsymbols,,wasusedtolabelthedeformationpotentials.HerringandVogt[ 12 ]alsosummarizedtheindependentdeformationpotentialsconstrainedbysymmetryatdierentconductionbandvalleys.Atthepoint,anothersetofsymbolsarecommonlyused:ac;av;b;andd,whereav;b;anddarethreeindependentvalencebanddeformationpotentialswhichhaveacorrespondencetotheLuttingerparameters[ 13 ]employedinbandcalculations.Thekpmethodweuseinthisworkreliesonthesethreedeformationpotentialstoaccountforthestraineects. Smithmeasuredthepiezoresistancecoecientsforn{andp{typestrainedbulksiliconandgermaniumin1954[ 7 ].Thiswastherstexperimentalworkthatstudiedstraineectsonsemiconductortransport.HerringandVogtusedShockley'sbandmodelandascribedtheelectronmobilitychangetotwostraineects,\electrontransfereect"andinter-valleyscatteringratechangecausedbyvalleyenergyshift[ 12 ].This 15

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Piezoresistancecoecientsarewidelyusedintheindustryduetoitssimplicityinrepresentingthesemiconductortransportproperties(mobility,resistance,andetal:)understrain.Itisdenedastherelativeresistancechangewiththestressappliedonthesemiconductor.Piezoresistancecoecient()canbeexpressedas whereistheappliedstressandistheresistivityofthesemiconductor. In1968,Colman[ 8 ]measuredthepiezoresistancecoecientsinp{typeinversionlayers.Thiswasthersttimethatstraineectonholetransportwasinvestigatedintheinversionlayers.Thesimilarityanddierenceofthepiezoresistancecoecientscomparedwiththebulksiliconwasexplainedqualitativelyinthatwork. Therstsiliconn{channelMOSFETwhichusedbiaxialstresstoimprovetheelectronmobilitywasdemonstratedbyWelseretal:[ 14 ]in1992.Theworkshowedthattheelectronmobilitywasimprovedby2.2times.Abiaxialstressedsiliconp{channelMOSFETwasrstreportedbyNayaketal:[ 15 ]in1993wheretheholemobilitywasenhancedby1.5times.In1995,Rim[ 16 ]showedtheholemobilityenhancementinsiliconp-MOSFETsontopofSi1xGexsubstratewithdierentgermaniumcomponents.TheideaofusinglongitudinaluniaxialstresstoimprovetheperformanceofMOSFETswasactivatedbyItoetal:[ 17 ]andShimizuetal:[ 18 ]inthelate1990'sthroughtheinvestigationsofintroducinghighstresscappinglayersdepositedonMOSFETstoinducechannelstress.Gannavarametal:[ 19 ]proposedSi1xGexinthesourceanddrainregionforhigherboronactivationandreducedexternalresistancewhichalsofurnishedatechnicallyconvenientmeanstoemployuniaxialchannelstress.ThesestudiesopenedthegatetousestrainasactivefactorinVLSIdevicedesignandresultedinextensiveindustrialapplications. 16

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For(001)wafer,biaxialtensilestressinSiMOSFETsisappliedtothechannelbyusingtheSi1xGexsubstrate.Thelatticemismatchstretchessiliconatomsinbothh100iandh010idirectionswhichisillustratedinFigure 1-1 .Thepercentageofgermaniumcontentinthesubstratedeterminesthemagnitudeofthestrain.Thisin-planetensilestraincanalsobeachievedbyapplyinguniaxialcompressivestressfromtheout-of-planedirection[ 20 ]withcappinglayer.Theout-of-planeuniaxialcompressionisequivalenttothein-planebiaxialtensionindeterminingthetransportpropertiesofSi.Thedetailsareshownintheappendix.ForGeMOSFETs,biaxialtensilestressisnotapplicableduetoitslargelatticeconstant.BiaxialcompressivestressisusuallyintroducedbyapplyingSiorSi1xGexsubstrate. Figure1-1. SchematicdiagramofbiaxialtensilestressedSi-MOSFETonrelaxedSi1xGexlayer Uniaxialstresscanbeappliedfromout-of-plane,in-planelongitudinal(paralleltothechannel),orin-planetransverse(perpendiculartothechannel)direction.Thein-planelongitudinalstressisappliedtothechannelbyeitherdopinggermaniumtosourceand 17

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1-2 [ 20 ]. Figure1-2. UniaxialstressedSi-MOSFETwithSi1xGexSource/Drainorhighlystressedcappinglayer Withoutfurtherclarication,uniaxialstressinthisworkrepresentsin-planeuniaxiallongitudinalstress.Itisnormallyalongh110isinceitistheclassicalchanneldirection.Biaxialstressmeansin-planebiaxialstress.For(110){orientedwafer,biaxialstressisemployedinbothparallelandperpendiculardirectiontothechannel(<110>{and<100>{directions).Thestraininthosetwodirectionsarenotassameas(001){orientedwafer(<100>{and<010>{directions)duetothedierentYoung'sModulusin<110>{and<100>{directions. 18

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Straininducedholemobilityenhancementisstudiedtheoreticallyforthersttimeinultra-thin-body(UTB)non-classicalp-MOSFETs,includingsingle-gate(SG)silicon-on-insulator(SOI),(001)symmetricaldouble-gate(SDG)p-MOSFETs,and(110)p-typeFinFETs.ForSGSOIp-MOSFETs,thestraineectsareassameastraditionalSip-MOSFETs.For(001)SDGp-MOSFETsand(110)FinFETs,subbandmodulationisfoundwhenthechannelthicknessissmallerthan20nm.Asthestressincreases,themobilityenhancementin(001)SDGp-MOSFETsiscomparabletotraditionalSGp-MOSFETs.ForFinFETs,theformfactorsaremuchsmallerthanSG(110)p-MOSFETsandthechangewithstressislargerwhichsuggestsmorereductionofthephononscatteringrate.Therefore,thestrain-inducedholemobilityenhancement(200%)islargerthansinglegate(110)p-MOSFETs(100%). 19

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Strainenhancedholemobilityinsingle-gateSip-MOSFETsisthendiscussed.TheunstrainedSiholemobilityversusdevicesurfaceorientationandverticalelectriceldiscalculated.Holemobilityunderbiaxialanduniaxialstressin(001)and(110)p-MOSFETsisshowed.Thebandstructureofbulksiliconunderstrainisdiscussed.IntheSiinversionlayers,theconnedenergycontours,subbandsplitting,holepopulationingroundstatesubband,two-dimensional(2D)density-of-states(DOS),phononandsurfaceroughnessscatteringrateareevaluated.Thedierenceofstraininducedholemobilityenhancementin(001)and(110)p-MOSFETsunderbiaxialanduniaxialstressisexplained. Uniaxialstrain{inducedholemobilityenhancementiscalculatedforUTBnon-classicalp-MOSFETs,includingsingle-gateSOI,(001)SDGp-MOSFETs,and(110)p-typeFinFETs.ThesimilarityanddierencefromthetraditionalSip-MOSFETsarediscussedandphysicalinsightsaregiven. StraininducedholemobilityenhancementinGep-MOSFETsisdiscussed.Un-strainedholemobilityin(001)and(110)Gep-MOSFETsiscalculated.StraineectonholemobilityinSi1xGexwitharbitraryGecomponentsisevaluated.Tounderstandthephysics,thebulkvalencebandstructureandholeeectivemasswithstraineectsarecalculated.Intheinversionlayers,thesubbandstructure,2DDOSandscatteringratearecalculatedandtheirrelationtoholemobilityisanalyzed. WeconcludewiththeresultsthatweobtaininthisdissertationandsuggestpossiblefutureresearchonstrainedSiandGe. 20

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Globaldescriptionsofthedispersionrelationsofbulkmaterialscanbeobtainedviapseudo-potentialortight-bindingmethods[ 21 ].However,suchglobalsolutionoverthewholeBrillouinzoneisunnecessaryformanyaspectsofsemiconductorelectronicproperties.Whatisneededistheknowledgeofthedispersionrelationsoverasmallkaroundthebandextrema[ 21 ].kpmethodiswidelyusedinnowadaysquantumwellandquantumdotscalculationduetoitssimplicityandaccuracyregardingthepropertiesinthevicinityofconductionbandandvalencebandedgeswhichgovernmostopticalandelectronicphenomena. Tostudytheuniaxialorbiaxialstraineectonholemobilityintheinversionlayers,a6-bandkpmodel,Luttinger{Kohn'sHamiltonian[ 13 ],isutilizedinthiswork.Inthischapter,kpmethodandthederivationoftheluttinger{Kohn'sHamiltonianisintroducedrst.Thentheprocedurecalculatingtheholemobilityisexplained.Finally,theevaluationofscatteringmechanisms,mainlythephononandsurfaceroughnessscattering,isdiscussed. Inthecalculationoftheholemobilitywithstraineectintheinversionlayers,SchrodingerEquationandPoissonEquationaresolvedself-consistentlytosimulatethepotentialenergyinthechannel.Thesubbandstructureandthetwo-dimensionaldensity-of-states(2DDOS)ofeachsubbandarecalculatedandthescatteringrelaxationtimeisevaluatedinkspace.Finally,holemobilityisobtainedfromalinearizationoftheBoltzmannequation. 2.1.1IntroductiontokpMethod 21 22 23 ]isessentiallybasedontheperturbationtheoryandwasrstintroducedbyBardeen[ 24 ]andSeitz[ 25 ].Itisalsoreferredtoaseectivemasstheoryinliteratures.Thekpmethodismostusefulforanalyzingthebandstructure 21

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Foranelectroninaperiodicpotential whereR=n1a1+n2a2+n3a3,anda1,a2,a3arethelatticevectors,andn1,n2,andn3areintegers,theelectronwavefunctioncanbedescribedbytheSchrodingerequation (2{2) wherep=hr=iisthemomentumoperator,m0isthefreeelectronmass,andV(r)representsthepotentialincludingtheeectivelatticeperiodicpotentialcausedbythenuclei,ionsandcoreelectronsorthepotentialduetotheexchangecorrelation,impurities,etc. ThesolutionoftheSchrodingerequation (2{3) satisesthecondition (2{4) (2{5) where andkisthewavevector.Equations 2{4 2{5 and 2{6 istheBlochtheorem,whichgivesthepropertiesofthewavefunctionofanelectroninaperiodicpotentialV(r). TheeigenvaluesforEquation 2{3 canbecategorizedintoaseriesofbandsEn;n=1;2;:::[ 26 ]duetotheperturbationoftheperiodicpotentialattheBrillouinzoneedge. 22

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InsertingtheBlochfunctionEquation 2{5 intoEquation 2{7 ,wehave m0kp+V(r)#unk(r)=En(k)unk(r): Includingthespin-orbitinteractionterm h intheHamiltonianandsimplifyingtheequation,Equation 2{8 becomes =En(k)unk(r): wherecisthespeedoflightandisthePaulispinmatrix.hasthecomponents[ 22 ] RewritingtheHamiltonianinEquation 2.1.1 ,wehave [H0+W(k)]unk=Enkunk; where (2{13) and SinceonlyW(k)dependsonwavevectork,Equation 2{13 canbeusedtoevaluatethebandpropertyatk0.IftheHamiltonianH0hasacompletesetoforthonormal 23

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theoreticallyanyfunctionwithlatticeperiodicitycanbeexpandedusingeigenfunctionsun0.Substitutingtheexpression intoEquation 2.1.1 ,multiplyingfromtheleftbyun0,integratingandusingtheorthonor-malityofthebasisfunctions,wehave Solvingthismatrixequationgivesusboththeexacteigenstatesandeigenenergies.Aswementionedearlier,onlythedispersionrelationsoverasmallkrangearoundthebandextremaareimportantdescribingtheelectronicpropertiesofthesemiconductor.Onlyenergeticallyadjacentbandsarenormallyconsideredwhenstudyingthekexpansionofonespecicbandforsimpleness.Topursueacceptablesolutionswhenkincreases,onehastoincreasethenumberofthebasisstates,orconsiderhigherorderperturbations,orevenboth. Neglectingthenon-diagonaltermsinEquation 2{17 forsmallk,theeigenfunctionisunk=un0,andthecorrespondingeigenvalueisgivenbyEnk=En0+h2k2 where 24

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(2{20) thesecondordereigenenergiescanbewrittenas Equation 2{21 canalsobeexpressedas where 1 2{23 istheeectivemasstensor,and;=x;y;z.Theeectivemassgenerallyisanisotropicandk{dependent.Inthevicinityofthepoint,sometimesmcanbetreatedask-independent,sinceatthislevelofapproximation,theeigenenergiesclosetothepointonlydependquadraticallyonk[ 22 23 ]. 2{17 givesexactsolutionsofbotheigenenergiesandeigenfunctions.Inreality,itisalmostimpossibletoincludeacompletesetofbasisstates,thereforeonlystronglycoupledbandsareincludedinusualkpformalism,andtheinuenceoftheenergeticallydistantbandsistreatedasperturbation. InKane'smodelforSi,Ge,orIII-Vsemiconductors,fourbandsareconsideredasstronglycouplesbands{theconduction,heavy-hole(HH),light-hole(LH),andtespin-orbitsplit-o(SO)bandsareconsidered,whichhavedoubledegeneracywiththeirspincounterparts.Therestbandsaretreatedasperturbationandcanbeanalyzedwiththesecondorderperturbationtheory. 25

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2.1.1 witheigenfunction (2{24) Thebandedgefunctionsun0(r)are Conductionband:jS"i,jS#iforeigenenergyEs(s-type), Valenceband:jX"i,jY"i,jZ"i,jX#i,jY#i,jZ#iforeigenenergyEp(p-type). Normallythefollowingeightbasisfunctionsarechosen 2;1 2i=jS"i=jS"i;u2=j3 2;3 2i=jHH"i=1 2;1 2i=jLH"i=1 3jZ"i;u4=j1 2;1 2i=jSO"i=1 2;1 2i=jS#i=jS#i;u6=j3 2;3 2i=jHH#i=1 2;1 2i=jLH#i=1 3Z#i;u8=j1 2;1 2i=jSO#i=1 Thissetofbasisstatesisaunitarytransformationofthebasisfunctionsandtheeigen-functionsoftheHamiltonian 2{13 .TheeigenenergiesforjSi,jHHi,jLHiandjSOiatk=0areEg,0,0,,respectively,whereEgisthebandgap,andtheenergyofthetop 26

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Atthislevelofapproximation,thebandsarestillatbecausetheHamiltonian 2{13 isk-independent.IncludingW(k)inEquation 2{14 intotheHamiltonian,anddeningKane'sparameteras m0hSjzjZi; weobtainamatrixexpressionfortheHamiltonianH=H0+W(k),i.e., 3Pk00q 3Pkz1 3Pkz000q 3Pk+00+h2k2 3Pkz1 3Pk+0000Pk+h2k2 3Pkz0001 3Pk00+h2k2 wherek+=kx+iky,k=kxiky,andkx,ky,kzarethecartesiancomponentsofk.TheHamiltonian 2{27 iseasytodiagonalizetondtheeigenenergiesandeigenstatesasfunctionsofk.Wehaveeighteigenenergies,butduetospindegeneracy,thereareonlyfourdierenteigenenergieslistedbelow.Fortheconductionband, Forthelightholeandsplit-obands, (2{29) 27

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Theseresultsarenotcompletesincetheeectsofhigherbandshavenotbeenincluded.TheywillbetakenintoaccountnextwhendiscussingtheLuttinger-Kohnmodel. 27 ]wherethesixvalencebandsaretreatedinclassAandtherestbandsareputinclassB. WelabelclassAwithsubscriptnandclassBwithsubscript.Wavefunctionuk(r)canbeexpandedas ChoosetheeigenstatesforclassA,wehave 2;3 2i=jHH"i=1 2;1 2i=jLH"i=1 3jZ"i;u3=j3 2;1 2i=jLH#i=1 3Z#i;u4=j3 2;3 2i=jHH#i=1 2;1 2i=jSO"i=1 2;1 2i=jSO#i=1 WithLowdin'smethodweonlyneedtosolvetheeigenequation 28

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(2{34) where (2{36) m0kju0i=Xhk m0pj(j2A;62A) (2{37) LetUAjn=Djn,Djncanbeexpressedas whereDjnisdenedas (2{39) ToexpressDjnexplicitly,wedine ThendenetheLuttingerparameters1,2,and3as 29

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3(a0+2B0)h2 6(a0B0)h2 (2{41) FinallyweobtaintheLuttinger-KohnHamiltonian 2SR+0PQSq 2S+p 2Sp 2S+p 2;3 2ij3 2;1 2ij1 2;1 2ij3 2;1 2ij3 2;3 2ij1 2;1 2ij1 2;1 2i where, Whenstrainispresentinthesemiconductor,P,Q,R,andSinEquation 2{43 canberesolvedtotwoparts:kpterms(Pk,Qk,Rk,andSk)andstrainterms(P,Q,R,andS).Theycanbeexpressedas[ 13 ] 30

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2b(xxyy)idxy;S=d(zxiyz); 28 ]. Table2-1. Luttinger-Kohnparameters,deformationpotentialsandsplit-oenergyforsiliconandgermanium. Si4.220.391.442.46-2.35-5.30.044 Ge13.354.255.692.09-2.55-5.30.296 31

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2.2.1Self-consistentProcedure @zinEquation 2{45 .Coordinatesystemtransformationisneededtocalculatecaseswithothersurfaceorientation. SubbandenergycanbeevaluatedbysolvingSchrodingerEquation, [H(k;z)+V(z)]k(z)=E(k)k(z) (2{45) whereV(z)denesthepotentialenergyinthequantumwell.Triangularpotentialapproximationiswidelyusedinsimulationsforsimplicity.Stern[ 29 ]statedthatitshouldnotbeusedwhenmobilechargesarepresent.Inordertoaccuratelysimulatethepotentialinthequantumwell,SchrodingerEquation 2{45 issolvedself-consistentlywithPoissonEquation (2{46) wherep(z)andn(z)aremobileholeandelectrondensity,N+D(z)andNA(z)arespacechargedensity. TonumericallyevaluateSchrodingerEquationandPoissonEquation,Finite-DierenceMethodisutilized.TheequationsareevaluatedonazmeshofNzpointsintheinterval(0;zmax)[ 3 30 31 ],wherezmaxhereisthesumofthethicknessofsiliconlayerandoxidelayer.Thisyieldsa6Nz6Nzeigenvalueproblemofthetridiagonalblockform[ 3 ].SchrodingerEquationbecomes 32

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whereeachi=(zi)isasix-componentcolumn-vectorj(zi),theindexjrunningoverthekpbasis,and^H,^Hi,^H+=^H+are66block-diagonaldierenceoperators,functionsofthein-planewave-vectork. Inprinciple,thepotentialV(z)resultsfromthreeterms:animage-term,vimg(z);anexchangeandcorrelationpotential,Vxc(z);andtheHartreeterm,VH(z)[ 3 30 ].Fischetti[ 3 ]suggeststhattheimagepotentialcancelsthemany-bodycorrectionsgivenbytheexchangeandcorrelationtermandtheHartreetermisfocussedasthesolutionoftheself-consistentcalculationofSchrodingerEquation 2{45 andPoissonEquation 2{46 3 ] (2{48) whereps=Ppisthetotalholeconcentrationintheinversionlayer,pistheholedensityofsubband,()x(K;)isthex-componentofthemomentumrelaxationtimeinsubband,and 1+expEEF 33

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Theevaluationofdensity-of-states(DOS)and@E @ktermneedfurtherconsideration.InenergyspaceamaximumkineticenergyEmaxforeachsubbandisselectedinordertoaccountcorrectlyforthethermaloccupationofthetop-mostsubband.Inourcalculation,weassumedEmax=120meVanddividedtheenergyspaceto1200uniformparts,thenevaluatedDOSand@E @kineachpart. 3 ],hencetheyareneglectedinourcalculation.Onlyphononscatteringandsurfaceroughnessscatteringareinvestigated. 32 ].Vibrationoftheionsabouttheirequilibriumpositionsintroducesinteractionbetweenelectronsandtheions.Thisinteractioninducestransitionsbetweendierentstates.Andthisprocessiscalledphononscattering. Phononscatteringcanbecategorizedtoacousticphononscatteringandopticalphononscatteringbasedonthephaseofthevibrationofthe2dierentatomsinoneprimitivecell.Bothcontributetothemomentumrelaxationtime.Acousticphononenergyisnegligiblecomparedwithcarrierenergy,whileopticalphononenergyisabout61.3meVforsiliconand37meVforgermaniumatlongwavelengthlimit.Whenstrainisappliedtothecrystal,theHHandLHdegeneracyisliftedatpoint,aswementionedpreviously.Therefore,theinter-bandopticalphononscatteringwillbelimitedduetobandsplittingandmobilityisenhanced.Infact,thisisonlysignicantwhenstrainishighandthebandsplittingisbeyondtheopticalphononenergy.Thereasoningwillbeshowninthefollowingsection.Oneshouldalsonoticethattheanisotropicnatureof 34

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32 ]isusedwherewereplacetheanisotropichole-phononmatrixelementwithappropriateangle-averagedquantities. Firstforacousticphonon,relaxationtimecanbeexpressedas[ 3 ] 1 (2{50) whereeff=7:18eVistheeectiveacousticdeformationpotentialofthevalenceband,isthe2-dimensionaldensity-of-statesofsubbandwhichisdenedas (2)2Z20dK(E;) Thetwo-dimensionalcarrierscatteringrateforthephonon-assistedtransitionsofacarrierfromaninitialstateinthe-thsubbandandanalstateinthe-thsubbandisproportionaltotheformfactor 2W=1 2Z+1jI(qz)j2dqz; where TheformfactorFillustratestheinteractionbetweeninitialstateandnalstateduetothewavefunctionoverlapping.where()k(z)or()k(z)istheenvolopefunctionatkforsubbandor,respectively.zisthecoordinateperpendiculartotheSi=SiO2interface,andqzisthechangeinthecomponentperpenticulartotheinterfacesofthecarriermomentuminatransitionfromthe-thsubbandtothe-thsubband. FollowingPrice'spioneeringwork,Wcanbeexpressedas 35

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Ifthenalstateisalso-thsubband,Wrepresentstheeectivequantumwellwidthforthe-thsubband. SincetheacousticphononenergyissmallcomparedwithsubbandsplittingoreventhethermalenergykT,acousticphononscatteringisanequal-energyscatteringprocess[ 32 ].Thescatteringratesolelydependsonthedensity-of-statesofthenalstates.Straineectonacousticphononscatteringissmallerthanthatonopticalphononscatteringwhichisshowninoursimulation. Second,theopticalphononscatteringrelaxationtimeisexpressedas[ 3 ] 1 1f0[E(K)]nop+1 21 2 Forabsorptionandemission,respectively,whereDop=13:24108eV=cmistheopticaldeformationpotentialconstantofthevalenceband,h!op=61:3eVisthesiliconopticalphononenergy.Opticalphononscatteringisnotsignicantlyreducedforstress<1GPasincethesubbandsplittingislessthantheopticalphononenergy. 36

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33 3 ].TheearlyformulationbyPrangeandNee,Saitoh,andAndoisstillthebestmodelavailable[ 3 ].Dierentroughnessparametersareusedindierentreferences.Here,we'lluseGamiz'modelandcorrespondingparameters[ 34 ]. Asweknow,thesurfaceroughnessscatteringiscausedbytheroughnessofthesurfaceandhencetheabruptpotentialchangeatSi=SiO2interface.2assumptionsareneededinthesimplicationoftheproblem[ 34 ].Therstassumptionistoconsidertheinterfacebetweensiliconandoxideisanabruptboundarywhichrandomlyvariesaccordingtoafunctionoftheparallelcoordinate,r,(r).AnotherassumptionisthatthepotentialV(z)closetotheinterfacecanbeexpressedby Thescatteringratecanbeexpressedas[ 34 ], 1 m(z)dz2mL2Z20d InthisequationVm(z) misapproximatelyequaltotheeectiveelectriceld,whichmeansthescatteringrateisproportionaltothesquareoftheelectriceld.Therefore,surfaceroughnessscatteringbecomesmoresignicantwhenelectriceldreacheshigherlevel. DierentvaluesforLandaretakenbydierentresearcherstoexplaintheexper-imentaldata.Here,weuseL=20:4nmand=4nm[ 3 ]forsiliconassuggestedbyFischetti.n=0:5[ 34 ]ischoseninthiswork. 37

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Inthiswork,MATLABandCcodesarewrittentocalculatetheholeeectivemass,bandandsubbandstructures,andholemobilityinSiandGep-MOSFETs.Tocalculateholemobilitydependenceondevicesurfaceorientation,coordinatetransformationisperformedtocalculateholemobilityin(110),(111),and(112)orientedSiandGetoaccountforthedierentquantumconnementconditions.Fordierentsurfaces,dierentsurfaceroughnessparametersareutilizedtottheinterfaceroughnesscondition.DOSandformfactorsarecalculatedinthewholekspace. 38

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Holetransportintheinversionlayerofsiliconp-MOSFETsunderarbitrarystressanddevicesurfaceorientationisdiscussedinthischapter.Piezoresistancecoecientsarecal-culatedandmeasuredatstressupto300MPaviawafer-bendingexperimentsforstressesoftechnologicalimportance:uniaxialcompressiveandbiaxialtensilestresson(001)-and(110)-surfaceorienteddevices.Withgoodagreementinthemeasuredvscalculatedlowstresspiezoresistancecoecients,kpcalculationareusedtogiveinsightathighstress(1{3GPa).Theresultsshowthatbiaxialtensilestressdegradestheholemobilityatlowstressduetothequantumconnementosettingthestraineect.Uniaxialstresson(001)/<110>,(110)/<110>,and(110)/<111>devicesimprovestheholemobilitymono-tonically.Unstressed(110)-orienteddeviceshavesuperiormobilityover(001)-orienteddevicesduetothestrongquantumconnementcausingsmallerconductivityeectivemassoftheholes.Whenthestressispresent,theconnementof(110)-orienteddevicesunder-minesthestresseect,hencetheenhancementfactorfor(110)-orienteddevicesislessthan(001)-orienteddevices.Holemobilityenhancementsaturatesasthestressincreases.Athighstress,themaximumholemobilityfor(001)/<110>,(110)/<110>,and(110)/<111>devicesiscomparable. Physicalinsightsaregiventoexplainthedierencebetweenbiaxialanduniaxialstress,andthedierenceof(110)and(001)p-MOSFETs.Thebulksiliconvalencebandstructureunderuniaxialcompressiveorbiaxialtensilestrainisshownandthedierenceineectivemasschangeiscalculated.Thedierenceoftheverticalelectriceld(quantumconenment)eecton(001)-and(110)-orientedp-MOSFETsisexplained.Subbandsplitting,groundstatesubbandholepopulation,andtwodimensional(2D)density-of-states(DOS)ofsubbandsarecalculatedunderstress.Scatteringratechangewithstressisalsodiscussed. 39

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7 8 ].Table 3-1 comparedmeasuredandcalculatedpiezoresistancecoecients.Inthemeasurements,thestressisappliedusing4-pointorconcentric-ringbendingofthewafers.Thepiezoresistancecoecientsareobtainedthroughthelinearregressionofthemeasuredresistanceversusstress.Theactualstraininthedevicesismeasuredthroughtheresistancechangeofastraingaugemountedonthesample,andviathelaser-detectedcurvaturechangeofbentwafer.InTable 3-1 ,L,T,andBiaxialrepresentlongitudinal,transverse,andbiaxialpiezoresistancecoecients,respectively. Table3-1. CalculatedandmeasuredpiezoresistancecoecientsforSipMOSFETswith(001)or(110)surfaceorientation.Therstvalueofeachpairisfrommeasurementsandthesecondisfromcalculation. Substrate(001)(110) Channel<110><110><110> L71.7[ Bothmeasuredandcalculatedresultsintable 3-1 showthatunderuniaxiallongitu-dinalstress,(110)/<111>deviceshavethelargetpiezoresistancecoecient,followedbythe(001)/<110>devices.Thepiezoresistancecoecientof(110)/<110>devicesisthelowest.Underuniaxialtransversestress,thepiezoresistancecoecientsaresmallerthanlongitudialstressforallp-MOSFETs.Thetablealsoshowsthatthebiaxialtensilestrainincreasesthechannelresistanceandhencedegradestheholemobilityatlowstress. 40

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36 ]reportedthatforp-typedeviceswith<110>channel,themobilityisthehighestin(110)-orientedandlowestin(001)-orientedp-MOSFETs.TheholemobilityonafewsurfaceorientationsissimulatedandcomparedwithSato'sexperimentalresults[ 36 37 ]inFigure 3-1 .Goodagreementisfoundbetweenthecalculationandtheexperimentaldata.Twodierentsurfaceroughnessmodels[ 3 34 ]areusedinthecalculation.Bothmodelsarequiteaccurateandinthefollowingresults,Gamiz'surfaceroughnessmodelisutilized. Figure3-1. Holemobilityvsdevicesurfaceorientationforrelaxedsiliconwith<110>channel.Theholemobilityishigheston(110)andloweston(001)devices.Dierentsurfaceroughnessscatteringmodelsareusedinthesimulation(solid:Gamiz1999;dotted:Fischetti2003). 41

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38 39 40 ]inFigure 3-2 .Theagreementbetweenthecalculationandtheexperimentalresultssuggeststhisworkusereasonablescatteringmechanisms.Normally(110){Sihassmootherinterfacewiththegatedielectricmaterials[ 41 42 ],hencethesurfaceroughnessscatteringrateislowerthan(001){orienteddevices.Lee[ 43 ]evensuggestedthattheeectiveeldin(110){orienteddevicesissmallerthan(001){orienteddevices,whichalsoindicatessmallersurfaceroughnessscatteringrateconsideringthatthescatteringrateisinverselypropor-tionaltotheeectiveelectriceld[ 3 34 ].Thesmallersurfaceroughnessscatteringrateispartlyresponsibleforthehigherholemobilityonunstressed(110){orienteddeicesthanthatofthe(001){orienteddevices.Tottheappropriatesurfaceroughnesscondition,theroughnessparametersusedareL=2:6nm;=0:4nmfor(001){orientedp-MOSFETsandL=1:03nm;=0:27nmfor(110){orientedp-MOSFETsinthiswork.Thesamesur-faceroughnessscatteringmodelisutilizedinthemobilitycalculationevenwhenthestrainispresent,assumingthattheprocess-inducedstrain(uniaxialstrain)doesnotchangetheSi=SiO2interfaceproperties[ 3 44 ]. 3-3 showstheholemobilityversus(upto3GPa)stressatinversionchargedensitypinv=11013=cm2andchanneldopingdensityND=11017=cm3)for(001)/<110>,(110)/<110>,and(110)/<111>p-MOSFETs.Uniaxialcompressivestressimprovestheholemobilitymonotonicallyasthestressincreases.Theholemobilityenhancementsaturatesatlargestress(3GPa).Underuniaxiallongitudinalcompres-sivestress,themaximumholemobilityenhancementfactoris350%for(001)/<110>p-MOSFETs,150%for(110)/<111>p-MOSFETs,and100%for(110)/<110>p-MOSFETs.At3GPauniaxialstress,(001)and(110)p-MOSFETshavecomparablehole 42

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Holemobilityvsinversionchargedensityforrelaxedsilicon.Bothmeasurementsandsimulationshowlargermobilityon(110)devices. 43

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Figure3-3. Holemobilityvsstresswithinversionchargedensity11013=cm2.Theenhancementfactoristhehighestfor(001)/<110>devicesandlowestfor(110)/<110>devices.Athighstress(3GPa),threeuniaxialstresscaseshavesimilarholemobility. Calculatedstrain-inducedholemobilityenhancementfactorof(001){orientedpMOSFETsisshowninFigure 3-4 comparingwithexperimentaldata[ 45 46 47 48 49 5 50 51 ].Goodagreementisfoundbetweenthecalculatedandmeasureddata. InFigure 3-3 and 3-4 ,thechanneldopingdensityissettobe11017=cm3inthecalculation.Theinversionchargedensityis11013=cm2.Incontemporarytechnology,theactualchanneldopingisupto11019=cm3.Themobilityenhancementfactoriscalculatedwithdierentchanneldopingdensityatinversionchargedensityof11013=cm2inFigure 3-5 .Theenhancementfactorsaresimilarforallthreedopinglevels.Forsimplicity,therestoftheworkwillusechanneldoping11017=cm3. 44

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Calculatedstraininducedholemobilityenhancementfactorvs.experimentaldatafor(001){orientedpMOS. Figure3-5. Holemobilityenhancementfactorvsuniaxialstressfordierentchanneldoping. 45

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3-6 comparestheholemobilityenhancementfactorfordierentinversionchargedensity.Thegureshowsthattheenhancementfactordecreasesastheinversionchargedensityincreases.Thisisbecausewithmoreinversioncharge,holesarepopulatedtothehigherenergylevelsinthevalenceband,whilethestressonlyaectsthevicinitiesofpoint.Thiscausestheaveragechangeoftheholeeectivemassdecrease.Moreinversionchargesincreasestheelectriceldinthechannelwhichunderminesthestraineect.Thedetailwillbeaddressedlaterinthischapter. Withthestrain-inducedholemobilitychangeasweshowedhere,physicalinsightsofthedierenceofbiaxialanduniaxialstress,andthedierencebetween(001){and(110){orientedpMOSFETsisgiveninthenextsections.Strain-inducedsiliconvalencebandstructurechange,subbandstructurecausedbythetransverseelectriceld,andtheholeeectivemassandscatteringratechangewiththestrainareanalyzed. 46

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(b) Calculatedstraininducedholemobilityenhancementfactorvs.stressfor(001){orientedpMOSwithdierentinversionchargedensity. 47

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m whereisthecarriermomentumrelaxationtimethatisinverselyproportionaltoscatteringrateandmisthecarrierconductivityeectivemass.Insiliconinversionlayers,carriersareconnedinapotentialsuchthattheirmotioninonedirection(perpendiculartothesilicon|oxideinterface)isrestrictedandtheelectronicbehaviorofthesecarriersistypicallytwo-dimensional(2D).Themobilityofthe2Dholegasisdierentfromthe3Dholesinbulksilicon.Butthesimplicityofthebulkbandstructurecalculationcangiveusinsightstohowtheeectivemassesoftheholeschangewiththestressandhelpunderstandhowthequantumconnementmodiesthesubbandpositionandsplittingwhichisimportantto2Dholemobility.Therefore,bulkvalencebandstructureisdiscussedinthissectionbeforewemovetotheSipMOSFETs. 3-7 .Fortheunstressedsilicon,theHeavy-hole(HH)andLight-hole(LH)bandsaredegenerateatpoint.Thisis4-folddegeneracytakingintoaccountthespin.TheSpin-orbitalSplit-o(SO)bandis44meVbelowHHandLHbands.Whenstressisapplied,thedegeneracyofHHandLHbandsisliftedasshowningure 3-7 (b)and(c).Thesetwobandsarealsoreferredtoasthetopandthesecondbandindicatingthesplitenergylevels.Thebandsplittingresultsinthebandwarpingwhichchangestheeectivemassoftheholes.Inthemeantime,thesplittingcausestherepopulationoftheholesinthesystem.Whenthestressislargeandthesplittingishigh,mostholeswilllocateinthetopbandbasedonFermi-Diracdistributionfunctionaslongasthedensity-of-states(DOS)ofthetopmostbandisnot 48

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Figure 3-7 showsthatthestressonlyaectthebandpropertyclosetopoint.Theguresshowthatawayfromthezonecenter,thebandstructureisalmostidenticaltotheunstressedsilicon.Figure 3-8 illustratesthatmoreregionaroundthezonecenterandmorecarriersareaectedbythestresswhenthestressincreases.Therefore,thestraineectcannotbeexplainedonlybythepropertiesatthepoint.Instead,thestatisticsofthewholesystemshouldbeconsidered.Figure 3-8 suggeststhatasthestressincreasesfrom500MPato1.5GPa,thebandwarpingandeectivemassatpointchangeverylittle.Thenextsubsectionwillalsoshowthis.Intheprocess,moreholesareaectedbythestress,thereforetheaverageholebehaviorswillstillchange.WeshowedinFigure 3-6 thatthemobilityenhancementfactordecreasesastheamountofinversionchargesincreases.Thiscanbeunderstoodasfollows.Fordeviceswithmoreinversioncharges,moreholesoccupythehigherenergystateswhentheinversionchargedensityincreases.Atthesamestress,theaveragechangeinducedbystressissmallerthanthecaseswithfewerinversioncharges. 3-9 3-10 and 3-11 .Figure 3-9 showsthe<110>{directioneectivemasses,gure 3-10 showsthe2-dimensionaldensity-of-stateeectivemasses,andtheout-of-plane<001>{directioneectivemassesareillustratedingure 3-11 Figure 3-9 3-10 and 3-11 alsosuggestthatwithstrain,theHHandLHbandsarenolonger\pure"HHorLHanymoreduetostrongcouplingofthewavefunctions.Thepropertyofeachbanddependsheavilyonthecrystalorientation.AsinglebandcanbeHH-likealongonedirection,butLH-likealonganother.Ingeneral,ifthecrystalshowscompressivestrainalongonedirection,thetopbandisLH-likealongthisspecic 49

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(b) (c) Figure3-7. E-krelationforsiliconunder(a)nostress;(b)1GPabiaxialtensilestress;and(c)1GPauniaxialcompressivestress. 50

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(b) NormalizedE-kdiagramofthetopbandunderdierentamountofstress.Largerstresswarpsmoreregionoftheband.Theenergyatpointforallcurvesissettozeroonlyforcomparisonpurpose. 51

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(b) Channeldirectioneectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 52

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(b) Two-dimensionaldensity-of-stateseectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 53

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(b) Out-of-planeeectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 54

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3-9 .Alongz{direction(out-of-plane),thesampleshowstensilestrainasweshoedinChapter1.ThetopbandisLH-likealongthisdirectionasshowninFigure 3-11 .Thisisaveryimportantissueforbiaxialtensilestress.Aswewillshowinthefollowingsection,thetransverseelectriceldeectosetsthebiaxialstresseectandcausestheholemobilitydegradationatlowstress.Similaranalysiscanbeappliedtouniaxialcompressivestress.Underuniaxialcompression,the<110>channeldirectionexperiencescompressivestrain,thereforethetopbandisLH-likealongthechannel.Atthesametime,theout-of-planedirectionexperiencestensilestrain,thetopbandisHH-likeout-of-plane. Thespin-orbitalsplit-o(SO)bandisalsocoupledwithHHandLHbandwhenstrainispresent.ThisbandisnotasimportantduetothelargeenergyseparationfromHHandLHbandsandhenceveryfewholeslocateinthisband. AsstatedpreviouslythatnormalMOSFETshave<110>directionasthechanneldirection,conductivityeectivemassalongthisdirectionaectstheholemobilitydirectlyaccordingtoDrude'smodel,a.k.aequation 3{1 .Figure 3-9 tellsusthatcomparedwiththebiaxialtensilestress,theuniaxialcompressivestressinducesmuchsmallertopbandeectivemasswhichsuggestsgreaterholemobilityimprovementisexpectedforuniaxialcompressivestress. Two-dimensionaldensity-of-stateseectivemassesasshowninFigure 3-10 givesaqualitativeestimationofthe2Ddensity-of-statesoftheholesineachband.The2DDOSisnotdirectlyrelatedtothebulkelectronicpropertiesofsemiconductors.Intheinversionlayers,large2DDOSofthegroundstatesubbandsuggestsmostholeslocatinginthissubband.Thisreducesinter-subbandphononscatteringpossibility.Inthemeantime,ifthegroundstatesubbandhasverylowconductivityeectivemass,thelargeDOS 55

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3-9 3-10 and 3-11 ,asignicantdiscontinuitycanbefoundatlowstrain(stress<1MPa).Aswementionedbefore,HHbandbecomesLH-likealong<110>directionunderuniaxialcompressionandalongout-of-planedirectionunderbiaxialtension.Intheholemobilitycalculation,thediscontinuityoftheholeeectivemassisalsoaconfusingquestion,althoughitisnotimportantinindustriessinceanysingletransistorwouldhavemuchlargerstraininthechannelintheprocess.Tounderstandthe\discontinuity",holeeectivemassatpointiscalculatedforsuperlowstress[ 52 ]asshownin 3-12 Theguresshowthatunderuniaxialcompressivestress,theHHbandisalwaysHH-likeandtheLHbandisalwaysLH-likeout-of-plane.Alongthe<110>direction,theeectivemasscurvescrossoveratabout3kPawhereHHbandbecomesLH-likeandLHbandbecomesHH-like.Biaxialtensilestressactsdierently.Thein-planeHHandLHbandsarestillHH-likeandLH-like,respectively.Intheout-of-planedirection,theHHbandbecomesLH-likeandLHbandbecomesHH-likeasthestressisgreaterthan1kPa. Asthestressincreasesbeyond100kPa,theconductivityeectivemassdoesnotchangeatpoint.Theaverageeectivemasschangeofthesystemcomesfromthefactthatmoreregionofthebandsisaectedbythestress. 3-13 forunstressedbulksilicon.TheanisotropicnatureoftheSivalencebandisclearlyshown.UsingthesimpleparabolicapproximationE=h2k2 56

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(b) Holeeectivemasschangeunderverysmallstress.Thechangeinthisstressregionexplainsthe\discontinuity"oftheholeeectivemassbetweentherelaxedandhighlystressedSi. 57

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3-14 for1GPabiaxialtensilestressandgure 3-15 for1GPauniaxialcompressivestress.Thecontours,aswellasE-krelationcurves,showstraininduceslowerconductivityeectivemassalong<110>directionforthetopband.Theeectivemassesforunstressedbulksiliconare0:59m0forHHand0:15m0forLHbandwherem0isthefreeelectronmass.Thosetwonumbersbecome0:28m0=0:22m0for1GPabiaxialtensilestressand0:11m0=0:2m0for1GPauniaxialcompressivestress.ThebottombandeectivemassesdonotshowenhancementcomparedwiththeLHbandmassoftheunstressedsilicon.Again,forbulkelectronictransport,uniaxialcompressivestressshouldenhancetheholemobilityasstressincreases,sincethetopbandisLH-likealong<110>direction.BiaxialtensilestressdoesnothavethemassadvantagesincethetopbandisHH-likein-plane.Thepossiblemobilityenhancementcomesonlyfrombandsplittingcausingphononscatteringratereduction.Forholesintheinversionlayers,thestatementisstilltrueaswewillshownext. (b) The25meVenergycontoursforunstressedSi:(a)Heavy-hole;(b)Light-hole. 58

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(b) The25meVenergycontoursforbiaxialtensilestressedSi:(a)Topband;(b)Bottomband. (b) The25meVenergycontoursforuniaxiallycompressivestressedSi:(a)Topband;(b)Bottomband. 59

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29 ].Thequantumconnementleavesasetoftwodimensionalsubbandsink-space(kx;ky).Thesubbandstructuresareaectedbyboththestressandthetransverseelectriceld.InpMOSFETs,thetopmosttwosubbands(4countingthespin),thegroundstateandtherstexcitedstatesubbands,containmostoftheholesandanalyzingthosetwosubbandsgivesusqualitativeunderstandingoftheholetransportproperties.Therefore,thosetwosubbandswillbefocusedinthefollowingdiscussionstoexplainthestraineects,althoughupto12subbandsareactuallytakenintoaccountintheholemobilitycalculation. Inthissection,weshallexplainwhythebiaxialtensilestressanduniaxialcom-pressivestressaectthesubbandstructureandtheholemobilitydierentlyunderthetransverseelectriceld.Thedierenceof(001)and(110){orienteddevicesunderuniaxialstresswillalsobestudied. 1 ].Takingholes(pMOS)asanexample,theconductionandvalencebandsbendup(benddownfornMOS)towardsthesurfaceduetotheappliednegativegatebiasatstronginversionregion.Thismeansholemotioninz-directionthatisperpendiculartothesiliconsurfaceisrestrictedandthusisquantized,leavingonlya2-dimensionalmomentumork-vectorwhichcharacterizesmotioninaplanenormaltotheconningpotential.Therefore,theinversionlayerholes(orelectrons)mustbetreatedquantummechanicallyas2-dimensional(2D).Figure 3-16 illustratesthequantumwellandquantizedsubbands[ 51 ], 60

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Figure3-16. Quantumwellandsubbandsenergylevelsundertransverseelectriceld. Thecomplexcalculationproceduresomehowpreventspeopleunderstandingthephysicsbehindstressandelectriceldeect.Togivethephysicalinsightsintotherelationbetweenthosetwoeects,triangularpotentialapproximationisutilizedtoestimatethesubbandenergylevels.Thetriangularpotentialapproximationstatesthatthebandbendingsolelydependsondepletionchargesundersubthresholdconditionwhenthemobilechargedensityisnegligible.ThepotentialV(z)isreplacedbyeEeffz,whereEeffistheeectiveelectriceldinthedepletionlayer.Triangularpotentialapproximation 61

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SolvingSchrodinger'sequation, [H(k;z)+V(z)]k(z)=E(k)k(z) (3{2) onewillgetthesubbandenergies.Theenergyofsubbandicanbeexpressedas[ 1 ], 4#2=3i=0;1;2;::: wherehisplankconstant,eistheelectroncharge,andmzistheout-of-planeholeeectivemass,alsoknownasconnementeectivemass.ThiseectiveeldisdenedastheaverageelectriceldperpendiculartotheSiSiO2interfaceexperiencedbythecarriersinthechannel.Itcanbeexpressedintermsofthedepletionandinversionchargedensities: (3{4) where=1 2forelectronsand1 3forholes[ 1 53 ].WefocusontheinversionregionofMOSFETswheretheeectiveeldisover0.5MV/cmthroughoutthiswork.Thisequationfortheeectiveelectriceldisanempiricalequation.Itmaynotbeaccuratetomodelthecarriertransportfordeviceswithsurfaceorientationotherthan(001)orotherdevicestructuressuchassilicon-on-insulator(SOI)devicesordouble{gated(DG)devices. Equation 3{3 showsthatthesubbandenergyofholesisinverselyproportionaltotheout-of-planeeectivemassoftheholes.Withthetransverseelectriceld,thesubbandthatisHH-likeout-of-planeisshiftedup(lowerenergyforholes)andthesubbandthatisLH-likeout-of-planeisshifteddown(higherenergy).Figure 3-11 and 3-9 showthatin(001){orienteddevices,biaxialtensilestrainshiftstheout-of-planeLH-likebandupwhichisthein-planeHH-likeband.Theelectriceldeectosetsthebiaxialtensile 62

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3-17 schematically.Ifthestraincontinuesincreasing,thestrainbecomesdominantdeterminingthesubbandenergiesandstructures.Duringtheprocess,theaverageholeeectivemassincreasessinceholestransferfromthein-planeLH-likesubbandtotheHH-likesubband.Thisincreasingeectivemassisresponsibletotheinitialmobilitydegradationunderbiaxialtensilestrainwhichisobservedbothinexperimentsandourcalculation.ThemobilityenhancementshowninFigure 3-3 comesfromthesuppressedinter-subbandphononscatteringrateduetothehighsubbandsplittingaswillbeshownlater.Underuniaxialcompressivestrain,thetopbandisHH-likeout-of-planeandLH-likealongthechannel,whichsuggeststhestrainandtheelectriceldeectsareadditive.Basedonthesimilaranalysis,boththeuniaxialcompressivestrainandthequantumconnementeectsshiftuptheout-of-planeHH-likebandwhichisLH-likealongthechannel.ThereforethegroundstatesubbandisalwaysLH-likealongthechannelandtheaverageeectivemassdecreasesmonotonicallyasthestressincreases. ThecalculatedsubbandsplittingbetweenthegroundstateandtherstexcitedstateisshowedinFigure 3-18 fordierentstressandsurfaceorientation.Forbiaxialstress,thesplittingiszeroat500MPawhichsuggeststhecrossing-overoftheHH-likeandLH-likesubbands.Foralluniaxialstresscases,thesubbandsplittingincreaseswiththestress.Like(001)/<110>devices,thegroundstatesubbandofboth(110)/<110>and(110)/<111>devicesisHH-likeout-of-planeandLH-likealongthechannelunderuniaxialcompressivestress.Thedierenceistattheout-of-planeeectivemassofthegroundstatesubbandin(110){orienteddevicesismuchlarger 3-19 thanthatofthe 63

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Schematicplotofstraineectonsubbandsplitting,theeldeectisadditivetouniaxialcompressionandsubtractivetobiaxialtension. (001){orienteddevices,whichresultsinmuchlargersubbandsplittingatlowstress.Thesplittingfor(110){orienteddevicesdoesnotchangeasmuchas(001){orienteddevices,andthesplittingsaturatesmuchfasterwiththestresscomparedwith(001){devices.Thisisduetothestrongquantumconnementundermingingthestraineect,whichisnotobservedin(001){orienteddevices. Ingeneral,in-planecompressivestressisdesirableforpMOS,sinceitcausesthesilicontopvalencebandtobeHH-likeout-of-planeandLH-likein-plane,whichisadditivetotheelectriceldeect.<110>uniaxialcompressivestressisthebestchoicebecauseitgivesverysmallconductivityeectivemass. 3-18 showsthedierenceofthesubbandsplittingbetween(001)-and(110){orienteddevices.Figure 3-3 showsthatthemaximumenhancementfactorat3GPastressfor(001){orienteddevicesunderuniaxialstressismuchlargerthan(110){orienteddevices. 64

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Subbandsplittingbetweenthetoptwosubbandsunderdierentstress. Toexplainthephysics,thebulkandconned2Denergycontoursofthegroundstatesubbandfor(001)and(110){orientedSiareshowninFigure 3-20 3-21 3-22 ,and 3-23 .Theguresshowthatfor(001)/<110>devices,thegroundstateholeeectivemassdecreaseswithuniaxialcompressivestress(LH-like)alongthechannel),butthereductionisnotasnotableunderbiaxialstress.ComparedwiththebulkSienergycontours,theelectricelddoesnotmodifythesubbandstructureinkxkyplanefor(001){orienteddevices(itdoesaectthesubbandsplittingthough).Theconductivityeectivemassesalongthechanneldirectionarealmostidenticaltothoseofbulkcounterparts.Theconnementeectismuchmoresignicanton(110){orienteddevices.Theconnedeectivemassofthegroundstatesubbandisverylowalong<110>and<111>directionevenforunstressedSi,whichexplainswhyunstressed(110){orienteddeviceshavesuperiorholemobilityover(001){orienteddevices(theconnementeectisalsosignicantin(111)and(112)p-MOSFETs( 3-1 ),thoughtheholeeectivemassislargerthanthatin(110) 65

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Out-of-planeeectivemassesforh110isurfaceorientedbulksiliconunderuniaxialcompressivestress. 66

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3-24 .For(001)devicesunderuniaxialstress,theinitialdecreaseoftheholepopulationisduetothedecreasedDOSnearpoint.Asstressincreases,theincreasingsubbandsplittingcausestheholepopulationincreasingandtheaverageconductivityeectivemasskeepsdecreasingsincethegroundstatesubbandisLH-likealongthechannelunderuniaxialcompressviestress.Forbiaxialstress,thedecreaseoftheholepopulationatlowstressagainreectstheinitialconnementeectliftingthein-planeLH-likesubbandandreducingthesubbandsplitting( 3-18 ).Thisin-planeLH-likesubbandisshifteddownasthestressincreasesandthein-planeHH-likesubbandisshiftedup.Afterthecrossing-overofthetwosubbands,thegroundstatesubbandpopulationstartsincreasingwiththestress.For(110){orienteddevicesunderuniaxialcompressivestress,thegroundstateholepopulationincreaseswiththestress,butitsaturatesatmuchlowerstresscomparedwith(001){orienteddeviceswhichisconsistentwiththesubbandsplittingchange.Theholepopulationof(110){orienteddevicesisalwayslowerthan(001){orienteddevicesunderuniaxialcompressivestress,althoughthesubbandsplittingismuchlarger.Thesubbandsplittingandholepopulationdierenceof(001)-and(110){orienteddevicescanbeexplainedbythegroundstatesubband2DDOSasshowninFigure 3-25 .DOSdierencealsosuggeststhedierentstrain-inducedmobilitychange.Bothguresshowthat(001){orienteddeviceshavelargerDOSthan(110){orienteddevices.For(001)/<110>devicesunderuniaxialcompressivestress,althoughtherstexcitedstatesubbandisHH-like 67

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(b) (c) The2Denergycontours(25,50,75,and100meV)forbulk(001)-Si.Uniaxialcompressivestresschangesholeeectivemassmoresignicantlythanbiaxialtensilestress. 68

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(b) (c) Conned2Denergycontours(25,50,75,and100meV)for(001)-Si.Thecontoursareidenticaltothebulkcounterparts. 69

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(b) (c) The2Denergycontours(25,50,75,and100meV)forbulk(110)-Siunder(a)nostress;(b)uniaxialstressalongh110i;and(c)uniaxialstressalongh111i. 70

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(b) (c) Conned2Denergycontours(25,50,75,and100meV)for(110)-Si.Theconnedcontoursaretotallydierentfromtheirbulkcounterpartswhichsuggestssignicantconnementeect. 71

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3-24 .ThedecreasingDOSinFigure 3-25 (b)forbothbiaxialanduniaxialstressof(001){orienteddevicesalsosuggeststhatthephononscatteringratedecreaseswithtestress.TheDOSof(110){orienteddevicesdoesnotchangewiththestressespeciallyathighstressregion(1{3GPa)whichsuggeststhephononscatteringrateshouldnotchangemuch. Figure3-24. Groundstatesubbandholepopulationunderdierentstress. Aswementionedintheprevioussection,thestressdoesnotwarpthebandstructureevenlyinthewholek-space.ThiscanalsobeseenfromtheDOSchangeinFigure 3-25 (b)wheretheDOSatEnergyE=52meV(2kTwhereT=300k)isshown.Takinguniaxialstresson(001)devicesasanexample,whenthestressislow,onlyasmallregionclosetopointisaectedandbecomesLH-likealong<110>direction(stillHH-likealongtransverseandout-of-planedirection),whiletherestofthebandwithhigherenergy(includingtheenergylevelshowedhere)doesnotrespondtothestressyet.Asthestress 72

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3-25 (b)suggestswhenthestressislowerthanabout500MPa,thestressistoosmalltowarpthebandatthisenergylevel.Whenthestressincreases,DOSstartsdecreasingbecausethestressstartswarpingthebandatthisenergyandthe<110>directionbecomesLH-like.TheDOScurvebecomesatagainwhenthestresseectsaturatesforthisenergylevel.For(001)p-MOSFETsunderbiaxialstress,Figure 3-25 (b)doesnotshowaDOSpeaklikeFigure 3-25 (a)whichmeansthepositioncrossing-overofthetoptwosubbandsonlyhappensclosetopoint,andtheHH-likebandisalwaysontopoutofthatregion. For(110)p-MOSFETs,theDOSisconstantwiththestress,whichisduetothestrongquantumconnementeect.Todiscoverthestraineect,2DDOSat4kT(102meVatT=300K)isshowninFigure 3-26 .For(001)p-MOSFETs,thecurveshavethesimilartrendcomparedwiththeDOScurvesat2kT.TheonlydierenceisthattheDOSstartstodecreaseathigherstress.For(110)p-MOSFETs,DOSdecreasesatlowstressandthechangeisnotassignicantlyas(001)p-MOSFETs.Figure 3-25 and 3-26 suggestthatthestrainin(110)p-MOSFETsonlywarpsthesubbandathighenergyregionduetothestrongquantumconnement.Thestraininducedmobilitychangeshouldbelessthan(001)p-MOSFETs,sincesmallerportionofholeslocateathighenergycomparedwithpoint. 3 1 ]. Figure 3-27 showsthatfor(001){orienteddevices,thephononscatteringratedoesnotchangemuchwhenthestressislowerthan500MPa.Thisindicatesthatatlowstress,theholemobilityenhancement(ordegradation)isalmostpurelycausedbytheeectivemass 73

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(b) Stresseectonthe2dimensionaldensity-of-statesofthegroundstatesubbandat(a)thetopofthesubband(E=0);(b)E=2kT.(110){deviceshavemuchsmaller2DDOSwhichlimitsthegroundstateholepopulation(largerinter-subbandphononscattering).AnotherobservationisthatDOSof(110){devicesdoesnotchangewithstress. 74

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Twodimensionaldensity-of-statesatE=4kT. change.Whenthestressincreasesfrom500MPato3GPa,thephononscatteringratedecreasesby50%forbothacousticphononandopticalphononscattering.thephononscatteringratereductionoverweighstheeectivemasschangetobecomethemaindrivingforcetoimprovetheholemobilityinthisstressrange,especiallyforbiaxialstress. Unlike(001){orienteddevices,phononscatteringratechangesmoreatlowstressregionratherthanhighstressregionfor(110){orienteddevicesunderuniaxialcompressivestress.ThisisconsistentwithFigure 3-18 and 3-24 thatthesubbandsplittingandthegroundstatesubbandholepopulationonlyincreaseatlowstress.Theconstantphononscatteringrateathighstressexplainswhytheholemobilityof(110)/<111>devicesat3GPaisnotsignicantlylargerthan(001)/<110>or(110)/<110>devices,regardlessofthelargestpiezoresistancecoecientatlowstress. Figure 3-28 showsthatthesurfaceroughnessscatteringrateincreaseswithstressfor(001){orienteddevices.Thisisduetotheincreasingholepopulationintheground 75

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(b) Straineecton(a)acousticphonon,and(b)opticalphononscatteringrate.Opticalphononscatteringisthedominantscatteringmechanismimprovingthemobility.Phononscatteringratechangesmainlyinhighstressregionfor(001)-devicesandlowstressregionfor(110)-devices. 76

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Straineectonsurfaceroughnessscatteringrateofholesintheinversionlayer.Asstressincreases,thescatteringrateincreasesfor(001)-devicesduetotheincreasingoccupationinthegroundstatesubbandwhichbringsthecentroidsoftheholesclosertotheSi=SiO2interface. statesubbandwhichbringsthecentroidsoftheholesclosertotheSi=SiO2interface.Themagnitudeofthesurfaceroughnessscatteringrateismuchsmallerthanthephononscatteringrateandthereforetheincreasingsurfaceroughnessscatteringdoesnotaecttheholemobilityasmuch.Thesurfaceroughnessscatteringratefor(110){orienteddevicesdoesnotchangemuchwiththestress,whichisconsistentwiththefactthatthegroundstatesubbandholepopulationisrelativelyconstantwiththestress. 3-29 illustratesthestress{inducedholemobilityenhancementcontributionfromholeeectivemassandphononscatteringratereduction,respectively.Underuniaxialcompression,(001)/h110ip-MOSFETshavethelargestmobilityimprovementfrombothaspects.Comparedwith(110)/h111ip-MOSFETs,(110)/h110ip-MOSFETs 77

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Uniaxialstresson(110)devicesimprovestheholemobilitytoo.Buttheimprovementisnotasmuchas(001)-orienteddevices.Thisisduetothestrongconnementeecton(110)-orienteddevicesunderminingthestresseect.Whennostressispresent,theconnementeectswapsthesubbandstructureandreducestheholeeectivemassaroundthe{point.Thiseectivemassadvantageoverthe(001)-orientedunstressedpMOScausesthattheholemobilityismuchlarger.Whenthestressisapplied,theeectivemasschangeisnotassignicant,neitherdoesthesubbandsplitting.Therefore,themobilityenhancementwiththestressisnotsupposedtobeasmuchasthe(001)-orientedpMOS. Itisalsonoticedthatthesubbandsplittingsaturateswhenthestressreaches2or3GPa,sodoestheeectivemass.Thisleadstothesaturationofthestressenhancedholemobility. 78

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(b) Holemobilitygaincontributionfrom(a)eectivemassreduction;and(b)phononscatteringratesuppressionforp-MOSFETsunderbiaxialanduniaxialstress. 79

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AsthesiliconCMOStechnologyisscaledtosub{100nm,evensub{50nmscale,furthersimplescalingoftheclassicalbulkdevicesislimitedbytheshortchanneleects(SCEs)anddoesnotbringperformanceimprovement.Theultra-thinbody(UTB)silicon-on-insulator(SOI)transistorarchitecture[ 54 55 56 57 58 ]hasbeenconsideredpossiblereplacementforthebulkMOSFETs.ThebasicideaofSOICMOSfabrication[ 54 56 ]istobuildtraditionaltransistorstructureonaverythinlayerofcrystallineSiwhichisseparatedfromthesubstratebyathickburiedoxidelayer(BOX).ComparedwiththebulkCMOS,UTBSOItechnologybringsbenetssuchasreducedjunctioncapacitancewhichincreasesswitchingspeed,nobodyeectsincethebodypotentialisnottiedtothegroundorVddbutcanrisetothesamepotentialasthesource,lowsubsurfaceleakagecurrent,andetal.. SOIMOSFETsareoftendistinguishedaspartiallydepleted(PD)transistorsthattheSithicknessislargerthanthemaximumdepletionwidthandfully-depleted(FD)SOItransistorsthattheSiisthinnerthanthemaximumdepletionwidth.FDSOItechnology[ 1 ]addadditionalperformanceenhancementsoverPDSOIincludinglowverticalelectriceldinthechannel(highermobility)duetothefactthatmostFD-SOItransistorshaveundopedchannel,furtherreductionofthejunctioncapacitance,andbetterscalability.AlthoughFDSOItechnologyhasbetterscalabilitythanclassicaldevicestructures,itisstilldiculttoscalethedevicetosub{20nmscale.Inshort-channelFDSOIMOSFETs,thethickBOXactslikeawidegatedepletionregionandisvulnerabletosource-draineldpenetrationandresultsinsevereshort-channeleects[ 1 59 60 ].Tobettercontrolthechannel,double-gate(DG)transistors,especiallyFinFETs,havebeeninvestigatedtheoreticallyandexperimentally[ 61 62 63 64 ].DG-MOSFETshavebetterscalabilitythansingle-gate(SG)SOItransistorsandareconsideredpromisingcandidatesforsub-20nmtechnologies[ 62 ].Overall,SOISGdevicesandDGdeviceshavebeenshown 80

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Withtheresearchofstraineectsonbulksilicondevices,strainedsiliconUTBFETsdrawtheattentionofresearchersassuchdevicesmaycombinethestraininducedtransportpropertyenhancementswiththeirscalingadvantages.StressenhancedholemobilityinSOI{deviceshasbeeninvestigatedexperimentallyinrecentyears[ 65 66 67 68 69 70 ].In2003,Rim[ 45 ]reportedthebiaxialtensilestressedSOI{pMOSholemobilitywithdependenceofstrainandinversionchargedensity.Zhang[ 71 ]showedtheholemobilityenhancementunderlowuniaxiallongitudinalandtransversestress.(110)-surfaceSOIdeviceswithstraineectsarealsoinvestigated[ 72 ].ThoseresultsareconsistentwiththemeasuredandcalculatedresultsforbulkSidevicesthatareshowedinthelastchapter. StrainresearchondoublegatedeviceslagsthatonbulkdevicesandevensinglegateSOIdevicespartlyduetothedicultyemployingstresstothechannelwithoutdamagingthepropertiesofthechannelandSi=SiO2interfaces.Duetothebetterscalabilityandhigherholemobility,moreattentionhasbeendrawnto(110){orientedFinFETsoverplanarDGFETs.Collaert[ 73 ]investigatedstraineectonelectronandholemobilityenhancementonFinFETs.Shin[ 74 ]andhiscolleaguesinvestigatedmultiplestresseectsonp-typeFinFETsusingwaferbendingmethod.Verheyen[ 75 ]reported25%drivecurrentimprovementofp-typemultiplegateFETdeviceswithgermaniumdopedsourceanddrain.Althoughholemobilityenhancementisobservedinthoseexperiments,theactualstressinthenisunknown.Theoretically,straineectsonFinFETsaremuchlessunderstood.Withtheknowledgeofstressenhancingholemobilityinbulkdevices,it'simportanttounderstandhowthatstressalterstheholemobilityinFinFETs.Uniaxialcompressivestresswillbefocusedinthisworksinceitprovidesthegreatestholemobilityimprovementthanotherstressonbulkdevices.Anotherreasonisthatfor(110){orientedFinFETs,thestressinthechannelisnormallyuniaxiallongitudinalstressevenifSiGe 81

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4-1 [ 67 76 ]showsthattheholemobilityisalmostindependentofthesiliconthicknesswhenSOIthicknessisover10nm.Ifthesiliconthicknessissmallerthan10nm,holemobilitydecreasesastheSOIthicknessdecreases.Themainreasonisthattheincreaseintheformfactor(/1=ph)causestheincreasingphononscatteringrate[ 77 ]duetothestructuralconnement.Anotherreasonistheincreasingsurfaceroughnessscatteringcausingsignicantloweringofthesurfaceroughnesslimitedmobility,sinceholesaremucheasierinvolvedinthesurfaceroughnessscatteringasthesiliconthicknessdecreases. SubbandsplittingiscalculatedforSOIpMOSandcomparedwiththebulkpMOS.Withthesameinversionchargeanddopingdensity,thesplittingisverysimilarforbothcases.IftheSOIthicknessdecreasesfrom20nmto5nm,thechangeofthesubbandsplittingislessthan5%.Thestructureofeachsubbandisalsoidenticaltothebulkdevices.IftheSOIthicknessissmallerthan5nm,subbandsplittingincreasesasthe 82

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HolemobilityvsSOIthicknessforsinglegateSOIpMOS.Themobilitydecreaseswiththethicknessduetostructuralconnement. SOIthicknessdecreases.Thisdoesnotbringsmallerinter-subbandscatteringrate.Therapidlyincreasingformfactoractuallykeepsthescatteringrateincreasing. Anotherissuerelatedtothesiliconthicknessissubbandmodulation.Bothmeasure-mentsandMonte{Carlosimulationshowthatthephonon-limitedmobilityincreasesatverythinSOIthickness[ 67 69 77 ].ThisissueonlyhappenstonMOS.Uchida'smea-surementsshowthereisnosuchmobilitypeakinp-typeUTBSOIFETs[ 67 ],whichisconsistentwithourcalculation. 45 ]reportedthatbiaxialtensilestrainimproves(ordegrades)theholemobilityassameasitdoestothebulkdevices,whichissupportedbyourcalculation.Uniaxialcompressivestrainisfocusedinthischapterduetoitsmuchlargermobilityenhancementfactorthanbiaxialtensilestrain. Figure 4-2 showsthesingle-gateSOIpMOSholemobilityvsuniaxialcompressivestresscomparingwithbulkSidevices.CalculatedcurvesforSOIthicknessof3nmand5 83

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Figure4-2. HolemobilityforsinglegateSOIpMOSvsuniaxialstressatchargedensityp=11013=cm2. TheholemobilityenhancementfactorforSOIpMOSwithSOIthicknessof3nmisshowninFigure 4-3 .TheenhancementfactorforSOIdevicesissimilartothecaseofbulkdevicesatlowstress,butlargerthanbulkFETsathighstress.AswementionedinChapter3thatfor(001){orientedSipMOS,themobilityisenhancedmainlyduetothedecreasedholeeectivemassatlowstress.Athighstress,phononscatteringratereductionduetotheincreasingsubbandsplittingisthemaindrivingforcetoimprovethemobility.Theoverlappingcurvesatlowstresssuggesttheeectivemassgainshouldbesimilarforbothcases.CalculationshowsthatthestructureofeachsubbandinSOIpMOSisassameasthebulkcounterpartwhichalsosuggeststheeectivemasschangeforbothcasesshouldbethesame.Figure 4-4 showsthesubbandsplittingofthegroundstateandtherstexcitedstatesubbandsforSOIandbulkFETs.ThelargersplittingforSOI 84

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Figure4-3. HolemobilityenhancementfactorofUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. UchidareportedthatastheSOIthicknessreducesdownto2{3nm,theuctuationoftheSi=SiO2interfaceisthemainfactortolimitthecarriermobility[ 67 69 ].Therefore,thelargeholemobilityenhancementasshowninFigure 4-3 cannotbeobtainedinrealdevices.Anewsurfaceroughnessmodelisneededtosolvethisproblem.Inourdiscussionofthedouble-gatedevicesincludingFinFETslaterinthischapter,thesmallestSithicknessweconsiderwouldbe5nm. 85

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SubbandsplittingUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. 86

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4-5 .ThesubbandsplittingforSDGMOSFETsandFinFETsisverysmallwhentheSithicknessisover5nm(5meVwhentSi=5nm,3meVwhentSi=15nm).IftheSithicknessisbelow5nm,thestronginteractionofthetwosurfacechannelcausesthesubbandsplittingincreasingdrastically(i.e.18meVfortSi=3nm): ComparisonofthesubbandsplittingofdoublegateandsinglegateMOSFETs. 4-6 and 4-7 ,respectively.Doublegatedeviceshavehighermobilitythantraditionalbulktransistorsmainlyduetotheundopedbody,muchsmallerchanneleectiveelectriceldandbulkinversion[ 1 ].Figure 4-6 showsthattheholemobilitydecreasesasthesiliconthicknessdecreases.ThereasonisassameassinglegateSOIdevicesandhasbeenexplainedinlastsection. ThemobilityenhancementfactorofSDGpMOSinFigure 4-7 isverysimilartothebulkcase,butthemechanismsarealittledierent.Therstexcitedsubband(veryclosetothegroundstate)providessmalleraverageeectivemasstohelpthemobility 87

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HolemobilityofSDGdevicesunderuniaxialcompressivestressatchargedensityp=11013=cm2. 88

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Figure4-7. HolemobilityenhancementfactorofSDGMOSFETsvsuniaxialcompressivestressatchargedensityp=11013=cm2. 4-8 ,comparingwiththesingle-gate(110)-and(001)-orientedp-typedevicesattheinversionchargedensityof11013=cm22.Inthecalculationofthesingle-gatedevices,thedopingdensityistakentobe11017=cm3.ThisisalowdopingdensitycomparedwiththecontemporaryCMOStechnology.Evenso,theFinFETshowssignicantlygreatermobilitythanthebulkdevices.Iflargerdopingdensityisapplied,themobilityadvantageoftheFinFETwouldbeevenlarger.When3GPauniaxialcompressivestressisapplied 89

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4-9 .Eventhoughthe(001)-orientedpMOSshowsgreaterrelativeenhancement(over400%),theabsolutemobilityisstilllowerthanthatoftheFinFETduetoitslowmobilitywithnostress. Figure4-8. HolemobilityofFinFETsunderuniaxialstresscomparedwithbulk(110)-orienteddevicesatchargedensityp=11013=cm2. Wementionedinthelastchapterthat2DDOSofthetopmostsubbandin(110)-orienteddevicesisverysmallnearpointnomatterifthestressispresentandthestressdoesnotwarpthesubbandsmuch.Thereforetheaverageeectivemassdoesnotchangeasmuchasstandard(001){orienteddeviceswhenuniaxialstressispresent.RegardingFinFETs,strongsubbandmodulationisobservedwherethetopmost2subbandsareclosetoeachother(like(001)SDGp-MOSFETs)asweillustratedinFigure 4-5 .ThisextrasubbandissoclosetothegroundstatesubbandanditactslikeincreasingtheDOSofthegroundstatesubband.Moreimportantly,thebandbendingattheSi=SiO2interfaceis 90

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HolemobilityenhancementfactorofFinFETsunderuniaxialcompressivestressatchargedensityp=11013=cm2. 91

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Figure4-10. Holemobilitygaincontributionfromeectivemassandphononscatteringsuppressionunderuniaxialcompressionfor(110)/h110iFinFETscomparedwithSG(110)/h110ip-MOSFETsatchargedensityp=11013=cm2. TounderstandtheholemobilitydierencebetweenFinFETsandtraditionalsinglegate(110)/h110i,theholemobilitygaincontributionfromeectivemasschangeandphononscatteringratechangeisshowninFigure 4-10 .ItshowsthatphononscatteringratechangeisthemainfactortoimprovetheholemobilityforbothFinFETsandbulk 92

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Thecalculationalsoshowstheenhancementisnotastrongfunctionofthesiliconthicknessofthenasthenthicknessisabove5nm.Ifthenisthinnerthanthat,moresubbandsplittingisobserved(about18meVfor3nmofthenthickness).Sincethesplittingisstillnottoolarge,ouranalysisabouttheeectivemassstaystrue.Surfaceroughnessscatteringrateismuchlargerandtheholemobilityenhancementwouldnotbeaslargeasthatforthickern.Anaccuratesurfaceroughnessmodelforsuchdeviceswouldbenecessarytoevaluatethemobilitychangenumerically. Fordoublegatedevices,subbandsplittingisdrasticallysmallerthanthebulkdevicesduetotheinteractionofthequantumstatesofthetwosurfacechannels.For(001){orientedplanarsymmetricalDGpMOS,thestructureofeachsubbandisstillidenticaltothecounterpartinthebulkdevices.Theextraeectivemassgainiscanceledbytheinter-subbandphononscatteringandthetotalholemobilityenhancementissimilartothebulkFETsatlowstress.Butwhenthestressisover2GPa,theeectivemassgainissaturate.ThemobilitygainislessthanthatofbulkFETsduetothelargerinter-subbandopticalphononscattering.Thiseectisnotthatsignicantduetothesmallerformfactors. 93

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94

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Asshort-channel-eects(SCEs)preventthesimplescalingoftraditionalSiMOSFETsachievinghistoricalperformanceimprovement,newmaterial,aswellasfeatureenhancedtechnology(straintechnology),attractattentionoftheresearchers.Germaniumisoneofthosenewmaterialsduetoitslargeelectronandholemobility.Withthestrainedsilicontechnologyintheindustry,it'sainterestingtopictodiscoverhowthestrainaectstheelectronandholemobilityingermaniumMOSFETs. GermaniumhasbeenofspecialinterestinhighspeedCMOStechnologyforyears[ 78 79 ].Thebulkgermaniumholemobilityislargerthanthatofothersemiconductormaterials,anditselectronandholemobilityaremuchlessdisparatethanothermaterials.In1989,germaniumholemobilityof770cm2=VsecinapMOSFETwasexhibitedbyMartin[ 80 ]andhisco{workersusingSiO2asthegateinsulator.Sincethen,moreandmorework[ 81 82 ]hasbeendoneongermaniumorSiGechannelpMOS[ 83 84 85 ].Inordertoreducethesurfaceroughnessandlimittheband{to{bandtunnelingissue,silicon{germaniumorSi{SiGedualchannelisalsousedinsomeapplications.Dierentgatedielectricmaterials[ 86 87 88 ]havebeenutilizedtondthebestmaterialtolimitthesurfaceroughnessattheinterfacebetweengateinsulatorandgermaniumchannel.Duetotheuncertaintyinthesurfaceroughnessandthesurfacestates,dierentholemobilityvalueshavebeenreportedinthosepublications.Inrecentyears,withthestraintechnologyappliedtosiliconCMOS,straineectisalsoinvestigatedongermaniumMOSFETs[ 87 89 90 91 92 ].ThestrainisnormallyachievedbyapplyingSiGesubstrateunderneaththegermaniumorSiGechannel.Butmostoftheworkstaysonlyinexperiments,thephysicalinsightsofthestraineectongermaniumMOSFETshavenotbeendiscussedcarefully.TheonlyavailabletheoreticalworksaresomeMonte-Carlosimulations[ 93 94 95 ].Thegoalofthischapteristogivephysicalinsightsofstraineectsongermaniumutilizingkpcalculation. 95

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86 96 ]vsverticalelectriceldanddevicesurfaceorientationisshowningure 5-1 .Experimentalworksgivealotofdierentmobilityvaluesrangingfrom70cm2=Vsectoover1000cm2=Vsec,dependingonwhatthegatedielectricmaterialsareused[ 86 87 88 ]andifSibuerisapplied[ 97 98 ]betweentheGe(orSiGe)andthegateoxide.WithSibuer,thedeviceactsasaburied-Gechanneltransistorandnormallyshowslargeholemobilityduetothelackofconnementandsurfaceroughnessscattering.Duetothebadscalabilityofburied-channeldevices,onlysurfacechannelGe-pMOSisdiscussedhere.CalculatedGeholemobilitymatchesthemeasureddataandthemobilityismuchlargermobilitythansilicon.(110)-orienteddeviceshowshighermobilitythan(001)-orienteddevice,whichisconsistentwiththeresultsofSi.Weshallshowthatthelargerholemobilityofgermaniummainlycomesfromthesmallereectivemassoftheholes.Therelativesmallerinter-subbandphononscatteringrateduetothelargersubbandsplitting(andsmalleropticalphononenergy)alsoimprovesthegermaniummobility. 96

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Germaniumholemobilityvseectiveelectriceld. thebiaxialtensilestraineectongermaniumholemobilityiscalculatedandshowedinFigure 5-2 Likesilicon,thedegradationoftheholemobilityatlowbiaxialtensilestressisduetothesubtractivenatureofstraineectandtransverseelectriceldeectresultingintheincreaseoftheaverageeectivemass,togetherwithalittleincreasedinter-subbandphononscattering.Athighstress,themobilityenhancementisobtainedduetoreducedinter-subbandopticalphononscattering. 5-3 97

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Germaniumandsiliconholemobilityunderbiaxialtensilestresswheretheinversionholeconcentrationis11013=cm2. 98

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Germaniumandsiliconholemobilityunderbiaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 99

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5-4 for(001){orientedGeandFigure 5-5 for(110){orientedGe.For(001){orienteddevices,bothSiandGeshowlargeenhancement.OnedierencebetweenthetwocurvesisthatthemobilityenhancementforSisaturatesatabout3GPa,butitdoesnotsaturateuntil6GPaofstressisappliedtoGe. Figure5-4. Germaniumandsiliconholemobilityon(001)-orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 100

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Germaniumandsiliconholemobilityon(110)-orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 101

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5-6 showsthedispersionrelationdiagramsfor(001)-Geunderdierentstress.Likesilicon,theheavyholeandlightholebandsofrelaxedGearedegenerateatpointasshowninFigure 5-6 (a).Thedegeneracyisliftedwhenstrainisapplied.Thebandsplittingleadstobandwarpingandthechangeofholeeectivemassandphononscatteringrate.TheSObandenergyis296meVlowerthantheHHandLHbandsforrelaxedgermaniumwhichimplieslesscouplingwithHHandLHbandscomparedwithsilicon.Underbiaxialtensilestrain,thetopbandisLH-likeout-of-planeandHH-likealongh110i.ForbothcompressivestraininFigure 5-6 (c)and(d),thetopbandisHH-likeout-of-planeandLH-likealongh110i.Uniaxialcompressivestrainbringsthemostwarpingonthetopvalenceband.Thewarpingisthesmallestunderbiaxialcompressivestrain,whichsuggeststheleastmobilityenhancementasshowninFigure 5-3 5-7 forbiaxialtension, 5-8 forbiaxialcompression,and 5-9 foruniaxialcompres-sion.Comparedwithsilicon,theeectivemassforgermaniumisobviouslymuchsmalleralongbothdirections.ThissuggestslargerholemobilityforgermaniumthansiliconaccordingtoDrude'smodel.OnesignicantdierencefromSieectivemassisthattheholeeectivemasschangeofGesaturateswithstressatmuchhigherstressthansilicon.Forsomeofthecurves,i.e.\top"bandofFigure 5-7 (a)and 5-9 (b),or\bottomband"ofFigure 5-8 ,theeectivemasschangedoesnotsaturateuntilthestressgoesupto7GPa.Butforsilicon,normallytheeectivemasschangesaturatesat2or3GPa.Thissuggestshigherstressforthemobilitysaturation. Thetrendoftheeectivemasschangewithstressissimilarforbothsiliconandgermanium.Ifwelookatthechanneldirection(h110i)eectivemass,thetopband 102

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(b) (c) (d) E{kdiagramsforGeunder(a)nostress;(b)1GPabiaxialtensilestress;(c)1GPabiaxialcompressivestress;and(d)1GPauniaxialcompressivestress. 103

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(b) Conductivityeectivemassvsbiaxialtensilestress:(a)Channeldirection(<110>)and(b)out-of-planedirection. 104

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(b) Conductivityeectivemassvsbiaxialcompressivestress:(a)Channeldirection(<110>)and(b)out-of-planedirection. 105

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(b) Conductivityeectivemassvsuniaxialcompressivestress:(a)Channeldirection(<110>)and(b)out-of-planedirection. 106

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5-4 becauseofthemuchsmaller2DDOSandinitiallargesubbandsplittingintheinversionlayers,whichwillbeaddressedinthenextsection.Underbiaxialtensilestress,thetopbandhashigherchanneldirectioneectivemass(increasingwithstress)andlowerout-of-planeeectivemasswhichissimilartosilicon.ThismeansthestresseectandthetransverseelectriceldeectintheinversionlayershouldbesubtractiveandtheholemobilityshouldbedegradedatlowstressasshowninFigure 5-2 .Underbiaxialcompressivestress,thetopbandhasverylowconductivityeectivemassatpointalongh110i.Aswementionedbefore,thebandwarpingisnotsignicantandonlyhappensveryclosetothepoint,whichsuggeststheaverageeectivemassofthesystemmaynotdecreasemuchwiththestress. 5-10 .Contours(25meV)underbiaxialcompressiveandtensilestressareshowningure 5-11 and 5-12 .Figure 5-13 showsthecontoursunderuniaxialcompressivestress.Theenergycontoursaresimilartothoseofsilicon,buttheshapeofthecontourschangesmorethanSicontourswhenthesameamountofstrainispresent.Anotherdierenceisthatunderuniaxialcompressivestress,the2DDOSofGelooksmuchsmallerthanSi.FromtheanalysisofSi,lowerpointDOSleadstosmallerstraininducedmobilityimprovementduetofewerholesareaectedbystrain.ThismayexplainwhythemobilityenhancementfactorforGeisnotlargerthanSi,althoughtheeectivemasschangeismuchlargeratpoint. 107

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(b) 25meVenergycontoursforunstressedGe:(a)Heavy-hole;(b)Light-hole. (b) 25meVenergycontoursforbiaxialcompressivestressedGe:(a)Topband;(b)Bottomband. 108

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(b) 25meVenergycontoursforbiaxialtensilestressedGe:(a)Topband;(b)Bottomband. (b) 25meVenergycontoursforuniaxiallycompressivestressedGe:(a)Topband;(b)Bottomband. 109

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5.3.1Strain-inducedSubbandSplitting 5-8 and 5-9 showthatbothbiaxialcompressivestressanduniaxialcompressivestressshiftuptheout-of-planeHH-likeband.Thiseectisclearlyadditivetotheelectriceldeect.Forbiaxialtensilestress,theelectriceldeectissubtractivetothestraineectandthereforeiftheelectriceldisxed,thesubbandsplittingshoulddecreaseatlowstresslevelandatsomestressvalue,thetoptwosubbandswouldcrossovereachotherjustlikeSi.Thesubbandsplittingbetweenthegroundandtherstexcitedstateof(001)GeisillustratedinFigure 5-14 Figure5-14. Gesubbandsplittingunderdierentstress. Figure 5-14 showsthatthesubbandsplittingforrelaxedGep-MOSFETsismuchlargerthanthatoftheSip-MOSFETs.Thesplittingislargerthantheopticalphonon 110

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5-15 showsthenormalizedin-planeE-kdiagramunderbiaxialcompressivestress.Ontheonehand,thegureshowsasthestressincreases,moreregionisnearthezonecenteriswarpedandhaslowerDOS.Ontheotherhand,outofthewarpedarea,thebandcurvesupalittleasthestressincreases,whichsuggeststheincreaseoftheDOS.Theoveralleectisthattheeectivemassgainclosetopointduetostressiscompromisedbytheheaviermassoftheholesawayfromthepoint.Atlowstress,themasschange,togetherwiththeincreasingsubbandsplitting,enhancestheholemobilityslightly.Underhigherstress,theenhancementisminimal.ForSipMOSunderbiaxialcompressivestress,theE-kdiagramissimilartoGe.ThedierenceisthatSihasmuchlargerDOSnearpoint,thereforethereisalwayseectivemassgain.ThedierentDOSresultsinthestrainenhancedholemobilitydierenceasinFigure 5-3 BiaxialtensilestressaectstheGeholemobilitysimilartoSidevices:thesubtractivenatureofthestrainandtransverseelectriceldeectsdegradestheholemobilityatlowstress,andthedecreaseofthephononscatteringrateenhancesthemobilityathighstress.Uniaxialcompressivestresson(001){orientedGeisfocusednextbecauseitprovidesthe 111

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NormalizedgroundstatesubbandE-kdiagramvsbiaxialcompressivestress. mostmobilityenhancementandthemobilityenhancementmechanismisalittledierentfromSi. 5-16 atdierentenergies.DOSofGeismuchlowerthanSi.ThetrendoftheDOSchangewithstressissimilarforSiandGe.Figure 5-16 (a)showsthattheDOSofGesaturateswithstressathigherstressthanSi,sincetheeectivemasschangeswithstressathigherstress.ThisisconsistentwiththemobilitysaturationcurvesinFigure 5-4 PhononandsurfaceroughnessscatteringratechangevsuniaxialstressisshowninFigure 5-17 and 5-18 .ForbothSiandGe,phononscatteringratedoesnotchangemuchatlowstress,andathighstressthephononscatteringratedecreasesastheuniaxialstressincreases.GehaslowerscatteringratethanSiduetothesmallerDOSofGe.ForSi,bothacousticphononandopticalphononscatteringratedecreasesby50%whenthestress 112

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(b) Twodimensionaldensity-of-statesofthegroundstatesubbandforSiandGeat(a)E=5meV;(b)E=2kT=52meVunderuniaxialcompressivestress. 113

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Figure 5-19 showsthemobilityenhancementcontributionfromeectivemass(solidlines)andphononscatteringrate(dashedlines)forSiandGe.ForSi,eectivemassgainisthemaindrivingforceofthemobilityenhancementatlowstress,andthescatteringratechangeisdominantathighstressrange(1GPa{3GPa).Fromunstressedcaseto3GPaofstress,eectivemassgainandphononscatteringratedecreasehavecomparableenhancementtotheholemobility.ForGe,thephononscatteringonlycontribute1.5timesoftheenhancement.Theeectivemasschangeisdominantinthewholestressrange.Aswementionedbefore,thisisbecausetheeectivemasschangeratioislargeunderstress(0:38m0tosmallerthan0:04m0).Anotherobservationoftheeectivemassisthatasthestressisover1GPa,increasingthestressdoesnotchangetheholeeectivemassforSi,buttheeectivemassofGecontinuetodecreaseasthestressincreases.ThisextraeectivemassgaincontributetotheholemobilityenhancementforGeatveryhighstress. 5-20 for(001){orientedMOSFETsandFigure 5-21 for(110){orientedp-MOSFETs.For(110)Gep-MOSFETsunderuniaxialstress,thestraineectissimilarto(110)Sip-MOSFETs.Thestrongquantumconnementwarpsthesubbandstructureandresultsinsmallholeeectivemass,whichexplainsthehigherunstrainedholemobilitythan(001)Gep-MOSFETs.Astheuniaxialcompressivestressisapplied,thestraineectisunderminedbythestrongquantumconnementandonlywarpsthehighenergyregionofeachsubband.Asaresult,theholemobilityisnotenhancedassignicantlyas(001){orientedp-MOSFETs. 114

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(b) Phononscatteringratevsuniaxialcompressivestress:(a)Acousticphonon,and(b)opticalphonon. 115

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SurfaceroughnessscatteringratevsuniaxialcompressivestressforGeandSi. 116

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Mobilityenhancementcontributionfromeectivemasschange(solidlines)andphononscatteringratechange(dashedlines)forSiandGeunderuniaxialcompressivestress. (b) Conned2Denergycontoursfor(001){orientedGepMOSwithuniaxialcompressivestress. 117

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(b) Conned2Denergycontoursfor(110){orientedGepMOSwithuniaxialcompressivestress. eachstresstypeforbothgermaniumandsiliconissimilar|uniaxialcompressivestresson(001)-orientedtransistorshasthemostholemobilityimprovementmainlyfromthereducedholeconductivityeectivemass.Uniaxialcompressivestresson(110)-orienteddevicesdoesnotprovideasmuchimprovementduetothestrongquantumconnementunderminingthestraineect.Holemobilityisdegradedunderlowbiaxialtensilestressduetothesubtractivenatureofthestrainandverticalelectriceldeectsandhencetheincreaseoftheaverageeectivemass.Themobilityisenhancedathighstressbecauseofthereductionoftheinter-subbandscatteringrate.BiaxialcompressivestressdoesnotimprovetheholemobilitymuchduetothesmallDOSafterband/subbandwarpingandnotmucheectivemassgain. 118

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119

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Straineectonholemobilityimprovementin(001)and(110)GeandSi1xGexp-MOSFETsiscalculatedforthersttime.ThemobilityenhancementatlowstressissimilartoSi.Athighstress,themaximummobilityenhancementfactorfor(001)GeislargerthanSiduetothegreatereectivemasschange,especiallyathighstress.ThephononscatteringratechangeforGep-MOSFETsisalittlesmallerthanSi.For(110)Gep-MOSFETs,strongquantumconnementisfoundandthestraininduced 120

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GermaniumisonenewmaterialthathasbeenconsideredtoreplacesiliconinCMOStechnology.UniaxialstrainevenhashigherenhancementongermaniumpMOS.Buttheexperimentalworkisstilllackforgermanium.Peoplearestilltryingtondoutthebestlayout,properdielectricandgatematerials.Itwillbealongwaybutdenitelyworthworkingon. Howaboutafterallofthis?Therewillbeanultimatelimitforthescalingthatballistictransportwilltakeplaceandthemobilityconceptwillnotbevalid.Willstrainstillbeusefulatthatstage?Theanswerisprobablyyes,sincethestraincanreducetheeectivemassofthecarriersandthiswillstillhelpthetransport.Thatbeingsaid,seriouscalculationwillbenecessarytofurtherexplainthis. 121

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StressisdenedastheforceFappliedonunitareaA. A Anystressonanisotropicsolidbodyinacartesiancoordinatesystemcanbeexpressedasastressmatrix[ 13 99 ], whereii=limAi!0Fi 100 ]asshowninFigure A-1 .Thisstressmatrixcompletelycharacterizesthestateofstressatcrystals. ForstressSalong<100>-direction,thematrixcanbewrittenas ForstressSalongboth<100>and<010>{direction(biaxialstress), 122

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Stressdistributiononcrystals. StressSalong<110>-directionisalittlecomplicated.Thestressisappliedonboth(100)and(010)planes.Ifweresolveeachcomponentalongxandyaxestogetbothnormalandshearterms,eachtermhasthesamemagnitudeofS=2.Thestresstensorcanbeexpressedas, ForstressSalong<111>-direction,basedonthesimilaranalysis,thestressisactuallyactedon(100),(010),and(001)planes.Eachcomponentcanberesolvedalongx,y,andzaxesandthestressalongeachdirectionisS=3.Thereforethestresstensoris, 123

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Strainisdenedasthedistortionofastructurecausedbystress.Normalstrainisdenedastherelativelatticeconstantchange[ 13 99 ], wherea0andaarelatticeconstantbeforeandafterthestrain. However,thedeformationofthecrystalcannotbefullyrepresentedwiththenormalstrain.Italsohassheartermsthataredenedaschangeintheinterioranglesoftheunitelement.Likestress,straincanalsobeexpressedwithasymmetric33tensoror61vector[ 100 ]. 124

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99 ]. or, Fordiamondorzinc-blende-typecrystal,stinessmatrixandcompliancematrixcanbesimpliedas[ 99 ] 125

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ElasticstinessesCijinunitsof1011N=m2andcompliancesSijinunitsof1011m2=N C11C12C44S11S12S44 Ge1.2920.4790.6700.964-0.2601.49 Thestinessandcompliancecoecientsforsiliconandgermaniumarelistedinthefollowingtable. Let'sgobacktothestraintensor.Eachstraincanbedecomposedtotwocompo-nents:hydrostatictermandshearterm.Thesheartermcanbefurtherdecomposedtoshear{100termwhichonlyhasdiagonalelementsandshear{111termwhichonlycontainsnon-diagonalelements.

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30BBBBBB@xx+yy+zz000xx+yy+zz000xx+yy+zz1CCCCCCAhydrosatic+1 30BBBBBB@2xx(yy+zz)0002yy(xx+zz)0002zz(xx+yy)1CCCCCCAshear100+0BBBBBB@0xyxzyx0yzzxzy01CCCCCCAshear111 Thehydrostaticterminthestraintensorshiftstheenergyofallthebandsinsemiconductorsbythesameamountsimultaneouslybutdoesnotcausebandsplitting,sinceitisactuallyaconstantandinthecalculationofthebandenergyitonlyactslikeaddinganadditionalpotentialtermtothehamiltonian.Thesemiconductortransportpropertyisindependentonthehydrostaticstrainterm.Fortwodierentstress,aslongasthesheartermsoftheirstraintensorsareequal,theirimpacttothecarriermobilityshouldbeidentical. Stresscanbeappliedtosemiconductorsfromanydirection.ForasiliconMOS-FET,onlyin-planebiaxialstressorchanneldirectionuniaxialstresshastechnologicalimportance.Thecommonsiliconwafersthatareusedinindustryare(001){oriented,andnormallythechanneloftheMOSFETisalong<110>{direction.Biaxialstressheremeansthatthestressisappliedinboth<100>{and<010>{directionsofthewaferwiththesamemagnitude.Uniaxialstressrepresentsthestressalongthe<110>channeldirec-tion.Thisstressisalsocalleduniaxiallongitudinalstress.Inthesamemanner,uniaxialtransversestressnormallymeanstheuniaxialstressappliedperpendiculartothechanneldirection.Bothofthosestressesareappliedintheplaneofthewafer,thereforetheyarealso\in-plane"stresses.Anotherkindofuniaxialstressiscalled\out-of-plane"uniaxial 127

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Fortheout-of-planeuniaxialstressandthein-planebiaxialstresson(001)wafer,thestrainmatricesonlyhavediagonaltermsandallnon-diagonaltermsarezero.Thequestionis,howdothesetwostressesdierfromeachother?Let'sassumewehaveout-of-planeuniaxialstressononesampleandin-planebiaxialstressonanothersample.Forcase1,basedon(1.4)and(1.15),thestraintensorcanbeexpressedas,intheformof(1.16), Forin-planebiaxialstress,

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(1.17)and(1.18)showthatthehydrostatictermsofthosetwostraintensorsaredierent,butthesheartermsareidentical.Thistellsusthatthebiaxialtensile(com-pressive)stressshouldhavethesameeectastheout-of-planeuniaxialcompressive(ortensile)stressindeterminingthetransportpropertyoftheholes. 129

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Thepiezoresistance,orpiezoresistiveeect,describestheelectricalresistancechangeofmaterialscausedbyappliedmechanicalstress.TherstmeasurementofpiezoresistancewasperformedbyBridgmanin1925andextensivestudyonthistopicwasdoneeversince.In1954,SmithmeasuredthepiezoresistanceeectonSiandGe[ 7 ].ThiseectbecomesmoreandmoreimportantduetothewideapplicationofSiandGeoncontemporaryCMOStechnology. Similartostressandstrain,thechangeofresistivityofamaterialisasymmetricalsecondranktensor.Thetensorconnectingthestressandthepiezoresistanceisoffourthrank.ForSiandGe,wecansimplifythetensoras[ 7 ] Themostgeneralformofatwo-dimensionalpiezoresistancetensorintheinversionlayeris[ 8 ] For(001),(110),and(111)surfaceorientedSi(orGe),14=41=24=42=0(principleaxish001ifor(001)and(110)surface,h110ifor(111)surface).Wecanfurthersimplifythepiezoresistancetensoras[ 8 ] 130

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For(001)surfaceorientedSiandGe,11=22and For(111)surfaceorientedSiandGe,44=1112and Inthepiezoresistancetensors,11representsthelongitudinalpiezoresistancecoef-cient(alongh100ifor(001)and(110)surface).12isthetransversepiezoresistnacecoecient(alongh010ifor(001)and(110)surface).InstandardMOSFETs,thechanneldirectionisalongh110iandtheuniaxialstressisappliedeitheralongh110iorh110i.Byrotationaltransformationofthetensorthenewlongitudinalandtransversepiezoresistancecoecientsbecome[ 8 ] 2(11+12+44) (B{6) and 2(11+1244) (B{7) 131

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. 132

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GuangyuSunwasborninShandong,China,onJune9th,1975.In1992,hewasadmittedtoUniversityofScienceandTechnologyofChina(USTC)inHefei,China.From1992to1996hestudiedinUSTCandreceivedhisB.S.degreeinappliedphysicsin1996.Hesubsequentlyparticipatedinthemaster'sprogramandobtainedtheM.S.degreein1999.Inthefallof1999,hecametotheUnitedStatesandbecameaFloridaGator.Inthespringof2004,heenteredProf.Thompson'sgroupandhasbeenstudyingthestraineectsonSiandGeMOSFETs,pursuingaPh.D.degree. 141