Residual Oil Saturation Dependence on Initial Water Saturation in
Modeling the Effect of Velocity Saturation in Nanoscale Mosfet
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MODELING THE EFFECT OF VELOCITY SATURATION IN NANOSCALE MOSFET MICHAEL TAN LOONG PENG UNIVERSITI TEKNOLOGI MALAYSIA
Transcript of Modeling the Effect of Velocity Saturation in Nanoscale Mosfet
MODELING THE EFFECT OF VELOCITY SATURATION
IN NANOSCALE MOSFET
MICHAEL TAN LOONG PENG
UNIVERSITI TEKNOLOGI MALAYSIA
PSZ 19:16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
BORANG PENGESAHAN STATUS TESIS♦ JUDUL: MODELING THE EFFECT OF VELOCITY SATURATION
IN NANOSCALE MOSFET
SESI PENGAJIAN: 2006/2007
Saya MICHAEL TAN LOONG PENG
(HURUF BESAR)
mengaku membenarkan tesis (PSM / Sarjana / Doktor Falsafah)* ini disimpan di Perpustakaan Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut:
1. Tesis adalah hakmilik Universiti Teknologi Malaysia. 2. Perpustakaan Universiti Teknologi Malaysia dibenarkan membuat salinan untuk tujuan
pengajian sahaja. 3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara
institusi pengajian tinggi. 4. **Sila tandakan ( )
(Mengandungi maklumat yang berdarjah keselamatan atau kepentingan Malaysia seperti yang termaktub di dalam AKTA RAHSIA RASMI 1972)
(Mengandungi maklumat TERHAD yang telah ditentukan oleh organisasi/badan di mana penyelidikan dijalankan)
(TANDATANGAN PENULIS) (TANDATANGAN PENYELIA)
Alamat Tetap:
293 LORONG MERBAU
TAMAN BERSATU
09000 KEDAH
Tarikh: Tarikh:
CATATAN: * Potong yang tidak berkenaan. **Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak
berkuasa/organisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis ini perlu dikelaskan sebagai SULIT atau TERHAD.
♦ Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara penyelidikan, atau disertai bagi pengajian secara kerja kursus dan penyelidikan atau Laporan Projek Sarjana Muda (PSM)
PROF. MADYA DR. RAZALI ISMAIL
Nama Penyelia
TIDAK TERHAD
SULIT
TERHAD
Disahkan oleh
“I hereby declare that I have read this thesis and in my
opinion this thesis is sufficient in terms of scope and quality for the
award of the degree of Master of Engineering (Electrical)”
Signature : .……………………………………….…
Name of Supervisor : Professor Madya Dr. Razali Bin Ismail .
Date : .
BAHAGIAN A – Pengesahan Kerjasama* Adalah disahkan bahawa projek penyelidikan tesis ini telah dilaksanakan melalui kerjasama antara_______________ dengan _________________ Disahkan oleh: Tandatangan : ………………………………………… Tarikh: …………….. Nama : ……………………………………………… Jawatan : ……………………………………………… (Cop rasmi) * Jika penyediaan tesis/projek melibatkan kerjasama. BAHAGIAN B – Untuk kegunaan Pejabat Fakulti Kejuruteraan Elektrik Tesis ini telah diperiksa dan diakui oleh: Nama dan Alamat Pemeriksa Luar : Nama dan Alamat Pemeriksa Dalam 1 : Pemeriksa Dalam 2 : Nama Penyelia lain : (jika ada) Disahkan oleh Timbalan Dekan (Pengajian Siswazah & Penyelidikan)/Ketua Jabatan
Program Siswazah:
Tandatangan : ………………………………………… Tarikh: ……………..
Nama : …………………………………………
Prof. Dr. Burhanuddin Yeop Majlis Director Institute of Microengineering and Nanoelectronics (IMEN) Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor
Dr. Abdul Manaf bin Hashim Microelectronics & Computer Engineering Department (MiCE) Fakulti Kejuruteraan Elektrik Universiti Teknologi Malaysia 81300 Skudai, Johor
Prof. Madya Dr. Abu Khari bin A’ain Microelectronics & Computer Engineering Department (MiCE) Fakulti Kejuruteraan Elektrik Universiti Teknologi Malaysia 81300 Skudai, Johor
MODELING THE EFFECT OF VELOCITY SATURATION
IN NANOSCALE MOSFET
MICHAEL TAN LOONG PENG
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Engineering (Electrical)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
DECEMBER 2006
ii
I declare that this thesis entitled “Modeling the Effect of Velocity Saturation in
Nanoscale MOSFET” is the result of my own research except as cited in references.
The thesis has not been accepted for any degree and is not concurrently submitted in
candidature of any other degree.
Signature : MICHAEL TAN LOONG PENG .
Name : MICHAEL TAN LOONG PENG .
Date : .
iii
To my wonderful parents and family, for their guidance, support, love and
enthusiasm. I am so thankful for that blessing and for the example you both are to me
over the years. I would not have made it this far without your motivation and
dedication to my success. Thank you, mom and dad, I love you both.
iv
ACKNOWLEDGEMENTS
First of all, I am thankful to my supervisor, Associate Professor Dr Razali
Ismail for his precious insight, guidance, advice and time. I would like to take this
opportunity to record my sincere gratitude for his supports and dedication throughout
the years.
My thankfulness also goes to my manager in Intel Penang Design Center, Mr
Ravisangar Muniandy. His immense support and encouragement was keeping me
going during the times when I was encountering problems at every turn. I wish to
express my most heartfelt thankre abroad and even made themselves available when
I had questions after our meeting.
I thank the many good friends I met here, Kaw Kiam Leong, Ng Choon Peng,
Lee Zhi Feng and many others for the encouragement and unforgettable memories.
They have always given me the chance to discuss my academic issues as well as my
personal issues. Without them, I could not have been completed my study. Very
valuable advice was also given by fellow friend, Dr Kelvin Kwa for his sharing of
experience, sachets of tea and coffee, books and advices. Also, thank to my friends
Sim Tze Yee, Liu Chin Foon, Liew Eng Yew, Gan Hock Lai and Alvin Goh Shing
Cyhe. I cherish the ideas they have given me, their support and warmhearted
friendship.
On a personal note, I would like to thank my family who has always
supported me and the encouragement they have given me.
v
ABSTRACT
MOSFET scaling throughout the years has enabled us to pack million of
MOS transistors on a single chip to keep in pace with Moore’s Law. The introduction
of 65 nm and 90 nm process technology offer low power, high-density and high-
speed generation of processor with latest technological advancement. When gate
length is scaled into nanoscale regime, second order effects are becoming a dominant
issue to be dealt with in transistor design. In short channel devices, velocity
saturation has redefined the current-voltage (I-V) curve. New models have been
modified and studied to provide a better representation of device performance by
understanding the effect of quantum mechanical effect. This thesis studies the effect
of velocity saturation on transistor’s internal characteristic and external factor.
Velocity saturation dependence on temperature, substrate doping concentration and
longitudinal electric field for n-MOSFET are investigated. An existing current-
voltage (I-V) compact model is utilized and modified by appending a simplified
threshold voltage derivation and a more precise carrier mobility model. The compact
model also includes a semi empirical source drain series resistance modeling. The
model can simulate the performance of the device under the influence of velocity
saturation. The results obtained can be used as a guideline for future nanoscale MOS
development.
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ABSTRAK
Penskalaan mendadak MOSFET dari tahun ke tahun selari dengan Hukum
Moore membolehkan berjuta-juta transistor dimuatkan ke dalam serpihan silikon.
Berikutan pengenalan teknologi proses 65 nm and 90 nm, arus pembaharuan yang
dramatik telah membawa kepada penumpuan pemproses yang pantas, berkuasa
rendah dan berdensiti tinggi dengan pendekatan teknologi terbaru. Kesan tertib kedua
menjadi satu isu dominan untuk ditangani dalam rekaan transistor apabila panjang
saluran mencecah nanometer. Salah satu daripandanya ialah halaju tepu yang telah
membawa definasi baru bagi ciri-ciri voltan and arus (I-V) dalam peranti
saluran/kanal pendek. Model-model baru telah diperkenalkan untuk memberi
representasi prestasi yang jelas dengan mengambil kira teori fizik kuantum.
Penyelidikan ini bertujuan untuk meneliti kesan halaju tepu ke atas faktor luaran dan
dalam ke atas transistor. Hubungan hanyut dan halaju tepu dengan suhu, kepekatan
pendopan and medan elektrik diperhatikan. Model voltan-arus (I-V) sedia ada
digabungkan bersama model kebolehgerakan elektron yang lebih terperinci untuk
menganalisis parameter-parameter di atas. Model padat tersebut juga mengandungi
model perintang bersiri sumber-salir semiempirik. Persamaan voltan ambang telah
diterbitkan dan berupaya memberikan ketepatan yang sama dengan silikon sebenar
dan dibuktikan dengan teknik kesesuaian pemadanan. Melalui pendekatan simulasi,
model-model ini dapat memberi prestasi peranti di bawah pengaruh halaju tepu.
Keputusan penyelidikan ini boleh digunakan sebagai panduan untuk perkembangan
MOS pada masa depan.
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TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES xi
LIST OF FIGURES xii
LIST OF ABBREVIATIONS xvi
LIST OF SYMBOLS xviii
1 INTRODUCTION 1
1.1 Background 1
1.2 Problem Statements 4
1.3 Objectives 5
1.4 Research Scope 5
1.5 Contributions 6
1.6 Thesis Organization 7
2 LITERATURE REVIEW 9
2.1 Non Ideal Effects in MOSFET 9
2.2 Velocity Saturation 10
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2.3 Oxide Charges and Traps 13
2.4 Device Modeling 15
2.5 SPICE Model 16
2.6 Threshold Voltage Model 17
2.7 Mobility Model 17
2.8 Drain Current and Voltage Model (I-V) 18
2.9 Velocity Field Model 19
2.10 Source/Drain Series Resistance Model 21
2.11 Hot Carrier Effects 22
2.12 Reducing Hot Carrier Degradation 24
3 RESEARCH METHODOLOGY 29
3.1 Velocity Saturation Electrical Modeling 29
3.2 Intel Proprietary Softwares 31
3.3 Parameters Extraction for Proposed
Model
32
3.4 MOSFET Modeling for Circuit
Simulation
33
3.5 NMOS Transistor 34
4 A PHYSICS BASED THRESHOLD
VOLTAGE MODELING
35
4.1 Introduction 35
4.2 Threshold Voltage Model 36
4.3 Short Channel Threshold Voltage Shift 40
4.4 Narrow Width Effects 41
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5 PHYSICALLY BASED MODEL FOR
EFFECTIVE MOBILITY FOR ELECTRON
IN THE INVERSION LAYER
44
5.1 Introduction 44
5.2 Schwarz and Russek Mobility Model 44
5.3 Proposed Mobility Model 47
5.3.1 Phonon Scattering 52
5.3.2 Coulomb Scattering 52
5.3.3 Surface Roughness Scattering 53
6 A PHYSICALLY BASED COMPACT I-V
MODEL
54
6.1 Introduction 54
6.2 Short Channel I-V Model 55
6.3 Compact I-V Model 59
6.4 Source Drain Resistance 62
7 VELOCITY FIELD MODEL 64
7.1 Introduction 64
7.2 Velocity Saturation Region Length 63
7.3 A Pseudo 2D Model for Velocity
Saturation Region
66
8 RESULT AND DISCUSSION 70
8.1 Introduction 70
8.2 I-V Model Evaluation 70
8.3 Threshold Voltage 74
8.4 Doping Concentration Dependence 76
8.5 Temperature Dependence 79
8.6 Peak Field At The Drain 81
x
9 CONCLUSIONS AND
RECOMMENDATIONS
83
9.1 Summary and Conclusion 83
9.2 Recommendations 84
REFERENCES 86
APPENDICES A-G 92 - 107
xi
LIST OF TABLES
TABLE NO. TITLE PAGE
2.1 Approaches of Compact Modeling 15
2.2 BSIM SPICE Level 16
4.1 Values of α and β for various semiconductors 38
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LIST OF FIGURES
FIGURE NO. TITLE PAGE 1.1 Growth of transistor counts for Intel processors
accordance to Moore's Law 2
1.2 Feature Size Growth 2 1.3 MOSFET Source to Drain Cross Section 4 2.1 Drift velocity versus electric field in Silicon 11 2.2 Drift velocity versus electric field in for three
different semiconductor 11
2.3 Depiction of electrons density in channel under
electric field 12
2.4 Charges and their location in thermally oxidized
silicon 13
2.5 Post oxidation dangling bonds that will become
interfacial traps 14
2.6 Coordinate system used for the model derivation. 20 2.7 Comparison of I-V characteristic for a constant
mobility and for field-dependent mobility and velocity saturation effects.
20
2.8 Simplified Cross Section of Parasitic Resistance of
a NMOS 22
2.9 Hot Carrier Generation 23 2.10 Schematic cross section of “inside” and “outside”
LDD 24
2.11 A schematic diagram of half of a n-channel
MOSFET 25
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2.12 Cross-sectional images showing a strained-Si MOSFET with a 25 nm gate length and its strained-Si layer
26
2.13 TEM micrographs of 45-nm p-type MOSFET 27 2.14 TEM micrographs of 45-nm n-type MOSFET 27 2.15 DI-LDD device cross section 28 2.16 Surface electric field at drain edge for conventional
and DI-LDD devices from 2D device simulation, VSUB = -1V and L=0.6 µm
28
3.1 Electrical Model Development Process 31 3.2 Running Basic Simulation Using Circuit 32 3.3 Parameter Extraction for Proposed Models 32 3.4 Schematic of an NMOS transistor 34 4.1 Metal-semiconductor work function difference
versus doping concentration for aluminum, gold, and n+ and p+ polysilicon gates
37
4.2 Charge sharing in the short channel threshold
voltage model 40
4.3 Cross section of an NMOS showing the depletion
region along the width of the device 42
4.4 Qualitative variation of threshold voltage with
channel length 43
4.5 Qualitative variation of threshold voltage with
channel width 43
5.1 Comparison of calculated and measured μeff versus
Eeff for several channel doping levels without Coulomb scattering and surface roughness scattering
46
5.2 Comparison of calculated and measured μeff versus
Eeff for several channel doping levels with Coulomb scattering and surface roughness scattering
46
5.3 Schematic diagram of Eeff and doping concentration
dependence of mobility in inversion layer by three dominant scattering mechanism
47
xiv
5.4 A diamond structure with (100) lattice plane at [100] direction
49
5.5 A diamond structure with (110) lattice plane at
[110] direction 50
5.6 A diamond structure with (111) lattice plane at
[111] direction 50
5.7 Visualization of surface scattering at Si-SiO2
interface 51
6.1 Charge Distribution in a MOS Capacitor 57 6.2 Source (Rs) and Drain (Rd) Resistance 62 6.3 Current patterns in the source/drain region and its
resistance 63
7.1 Velocity saturation region in MOSFET 65 7.2 Comparison of commonly used velocity field
models 66
7.3 Analysis of velocity saturation region 67 8.1 Comparison of calculated (pattern) versus
measured (solid lines) I-V characteristics for a wide range of gate voltage at resistivity 3.5 x10-5 Ω/cm
71
8.2 Comparison of calculated I-V characteristics for a
wide range of gate voltage at resistivity 3.5e-5 Ω/cm (solid lines) and 4.5 x10-5 Ω/cm (pattern)
72
8.3 Comparison of calculated I-V characteristics for a
wide range of gate voltage at resistivity 4.5e-5 Ω/cm (solid lines) and 5.5 x10-5 Ω/cm (pattern)
72
8.4 Comparison of calculated I-V characteristics for a
wide range of gate voltage at resistivity 3.5 x10-5 Ω/cm with (pattern marking) and without channel length modulation (solid lines)
73
8.5 Threshold Voltage without Short Channel Effect
(SCE) and Narrow Width Effect (NWE) 74
8.6 Short channel and narrow width effect 75 8.7 Threshold voltage modification with doping
concentration 76
xv
8.8 Normalized effective mobility versus transverse electric field at different doping concentration
77
8.9 Normalized drift velocity versus longitudinal
electric field at different doping concentration 78
8.10 Normalized effective mobility versus transverse
electric field at different temperature 80
8.11 Normalized drift velocity versus longitudinal
electric field at different temperature 80
8.12 Normalized longitudinal electric field along the
channel 81
8.13 Normalized saturation drain current versus gate
overdrive 82
8.14 Normalized saturation drain current versus drain
voltage 82
xvi
LIST OF ABBREVIATIONS
ASIC - Application Specific Integrated Circuits BSIM3 v3 - Berkeley Short-Channel IGFET Model Three Version Three CMOS - Complementary Metal Oxide Semiconductor CSV - Comma Separated Values EDA - Electronic Design Automation FET - Field Effect Transistor GCA - Gradual Channel Approximation GUI - Graphical User Interface IC - Integrated Circuit MOS - Metal Oxide Semiconductor MOSFET - Metal Oxide Semiconductor Field Effect Transistor NMOS - n-channel MOSFET PMOS - p-channel MOSFET SOE - Second Order Effects VLSI - Very Large Scale Device VSR - Velocity Saturation Region GHz - Giga Hertz GaAr - Galium Arsenide InP - Indium Phosphide
xvii
k - Boltzmann’s constant L - Channel length S - Spacer thickness Si - Silicon S/D - Source and Drain
xviii
LIST OF SYMBOLS
A - Ampere (unit for current) Cu2S - Copper Sulfide Cox - Gate oxide capacitance E - Electric Field Eeff - Effective transverse electric field Ec - Critical electric field Eg - Energy bandgap variation εSi - Dielectric permittivity of the silicon εox - Dielectric constant of the oxide I-V - Drain current versus drain voltage Ids - MOSFET drain current Idsat - Drain current saturation InP - Indium Phosphide Jn - Drift current density k - Boltzmann’s constant L - Channel length μeff - Effective mobility μm - micrometer μc - Coulombic scattering μph - Phonon scattering
xix
μsr - Surface roughness scattering φf - Fermi potential φfp - Fermi surface potential for p-type semiconductor φfn - Fermi surface potential for n-type semiconductor NA - Acceptor doping concentration ND - Receptor doping concentration NI - Number of electrons per unit area in the inversion layer Nc - Density of states function in conduction band Nv - Density of states function in valence band ni - Intrinsic carrier concentration nm - nanometer n - Free electron density p - Fuchs factoring scattering ρ - Effective resistivity Qtot - Effective net charges per unit area QI - Inversion charge per unit area Qn - Inversion Charge Rsd - Source and drain resistance Rd - Source resistance
Rext - Extrinsic resistance Rint - Intrinsic resistance Rac - Accumulation resistance Rsp - Sreading resistance
Rsh - Sheet resistance Rco - Contact resistance
xx
S - Spacer thickness φms - Work Function tox - Oxide thickness vd - Drift velocity V - Voltage (unit for potential difference) Vds - Drain voltage Vdsat. - Drain voltage saturation Vc - Critical Voltage VFB - Flat band voltage Vgs - Gate Voltage Vt - Threshold Voltage νsat - Carrier saturation velocity W - Channel width xdT - Depletion width xj - Junction depth χ - Oxide electron affinity Z - Averaged inversion layer width Zcl - Classical channel width ZQM - Quantum mechanically broadened width Ω - ohm (unit for resistivity)
xxi
LIST OF APPENDICES
APPENDIX TITLE PAGE A Publication 93 B Roadmap of Semiconductor since 1977 99 C Roadmap of Semiconductor into 32nm Process
Technology 100
D CMOS Layout Design 101 E Mobility in Strained and Unstrained Silicon 102 F Threshold Voltage Model 103 G I-V Model 105
CHAPTER I
INTRODUCTION
1.1 Background Silicon is one of the semiconductor materials which is commonly used in
chip manufacturing to make integrated circuit from miniaturized transistor. Metal
Oxide Semiconductor (MOS) transistor have shrunk from a micrometer into sub
100 nm regime with transistor scaling, which increases the number of transistor per
size by a factor of two every 18 months in accordance to Moore’s Law (Moore,
1965). Many improved lithographic and semiconductor fabrication equipments were
designed to be on track with the curve and one step ahead of the technology. So far,
Moore’s law has been a valuable way of describing the general progress of integrated
circuits and the number of transistors fitted into each generation of Intel processors,
as shown in Figure 1.1.
In silicon chip manufacturing, feature size and wafer size is the two most
important parameter as they determined the cost of a plant and production line
equipments. Presently, 300 mm wafers is the largest silicon wafers which produce
more than double as many chips as the older 200 mm wafers. Since the end of 2005,
Intel is the first manufacturer offering single and core 2 duo processors based on 65
nm production technologies. 65 nm generation transistors come with gates that are
able to turn a transistor on and off measuring only 35 nm which is roughly 30 percent
smaller than 90 nm technology gate lengths. Intel claims that 65 nm transistors cut
current leakage by four times compared to previous process technology. According
to Figure 1.2, new technology generation is introduced every 24 months and this
2
successfully extends their 15-year record of mass production in Intel. As
performance technology improves, the gate length start to get smaller than predicted
by the ideal feature size trend of each process technology. This allows higher
transistor density, squeezing more of them on a single chip. The roadmap of
semiconductor from 1977 to 2018 can be seen in Appendix B and Appendix C.
Figure 1.1 Growth of transistor counts for Intel processors accordance to
Moore's Law
Figure 1.2 Feature Size Growth
Gate Length Scales Faster
Feature Size
Gate Length
3
In nanoscale dimension, new problems began to occur. The magnitude of the
electric field is comparably higher in short channel devices than long channel devices
where the channel length is comparable to the depletion region width of the drain and
source. Here, Secondary Order Effect (SOE) exists and must be considered to model
the generation of a more precise short channel Metal Oxide Semiconductor Field
Effect Transistor (MOSFET). Velocity saturation is a vital parameter of the SOE
which occur in high electric field.
At low electric field, the drift velocity of electron, vd is proportional to the
electric field, E as shown in Eq. (1.1).
d effv Eμ= (1.1) where µeff is the effective mobility. When the electric field applied is increase,
nonlinearities appear in the mobility and carriers in the channel will have an
increased velocity. In high field, charge carriers gain and lose their energy rapidly
particularly through phonon emission until the drift velocity reaches a maximum
value called velocity saturation. Velocity saturation in MOSFET will yield a smaller
lower drain current and voltage. A cross section of a MOSFET is illustrated in
Figure 1.3. This research focuses on the role of velocity saturation has on the
characteristic of MOSFET in term of the carrier velocity field, carrier doping
concentration, drain current versus drain voltage curve. Intel proprietary software is
used to generate the experimental drain current versus drain voltage characteristic for
90nm generation of MOSFET. After parameter extraction is carried out, several
compact models are employed to study the effect of velocity saturation and the
impact of high electric field. The modified device models will be able the predict
behavior of electrical devices based on fundamental physics. Characteristics and
gives us the mobility and the drift velocity of the electrons versus transverse and
longitudinal electric field respectively.
4
Figure 1.3 MOSFET Source to Drain Cross Section
1.2 Problem Statements
This research utilizes a newly developed short channel models to study
velocity saturation in 90 nm process technology and reports the results it has on the
transistor basic characteristic. Questions that are bound to be answered through the
high field analysis are:
(i) What is velocity saturation?
(ii) What is the different between short and long channel devices? What is
the limitation of nanometer MOSFET?
(iii) How does velocity saturation affect the transistor?
(iv) What models can be represented to predict the characteristics and
behaviors of Complementary Metal Oxide Semiconductor (CMOS)
transistor in nanometer dimension?
(v) What are the limitations of conventional long channel models?
P
Source Gate Drain Metal Lines
Gate Oxide
Poly Si Gate
Interlayer Dielectric (ILD) Contact/via
Isolation
Substrate/ Well
Gate MOS Capacitor
Drain Regions
N+
Drain: PN Junction
(Diode)
Source: PN Junction
(Diode)
N+ N+
5
1.3 Objectives The following are the objectives of this study.
(i) To understand high field effects in nanoscale transistor in 90nm
process technology.
(ii) To formulate simple analytical and semi-empirical equations for
device model applicable to nanoscale devices by taking into account
velocity saturation.
(iii) To analyze velocity saturation effects on temperature, doping
concentration, longitudinal and transverse electric field.
1.4 Research Scope
The goal of this research is to investigate the role and characteristic of
velocity saturation on the following parameters.
(i) Longitudinal electric field
(ii) Transverse electric field
(iii) Doping concentration
(iv) Mobility
(v) Drain current and voltage (I-V) curve
(vi) Temperature
The research is divided into three major phases. In the beginning, literature
review and previous researches in this field is carried out. Strengths and weaknesses
of available model and equations are compared. The second phase begins with the
modeling based on the literature review. A semi-empirical model for velocity
saturation due to high field mobility degradation is presented. Best fit model
parameters are extracted from the experimental results. The results compared with a
6
set of published experimental data points and validated. The final phase is preceded
with analysis incorporating all the derived and modified models.
Future improvements and suggestion for the model are presented at the end of
the thesis. N-channel MOSFET with polysilicon gate is used in this research. The
Ids compact model is derived by studying and analyzing Berkeley Short-Channel
IGFET Model 3 Version 3 (BSIM3v3) standard (Berkeley, 2005). The one region
equation from linear to saturation is based on the modification of the conventional
long channel model with the addition of second order effects, each parameter with its
physical meanings. Other semi empirical models include threshold voltage, mobility
and source drain series resistance.
1.5 Contributions
The semiconductor industry particularly microchip industry strives to develop
high performance processor which is capable to cater for the demanding market.
MOSFET-based integrated circuits have become the dominant driving force in the
industry. It is important for the research and development’s designer team to
investigate how transistor behaves differently in nanometer dimension. These
characteristic includes the electric field, carrier velocity field, carrier mobility, carrier
concentration, carrier in saturation velocity region and drain current versus drain
voltage in short channel devices.
The modified short channel models are able to overcome the limitations of
previously long channel analytical and semi empirical models without velocity
saturation. It is also the interest of this study to find out the critical voltage and
electric field when velocity saturation appears. Based on this evaluation, it can
provide a guideline for designers in term of graphical representation on figures and
as well as parameter dependency relationship on the mobility behaviour, including
threshold voltage, doping concentration and temperature. Designer can examine the
behavior of the device when it is saturated at high electric field under specific
7
temperature and doping concentration. They can improve their design after a
thorough testing to prevent device breakdown.
Long channel I-V (current voltage) model is based on one dimensional
theory. The modified short channel model is more accurate for nanoscale MOSFET
than long channel model which cover Poisson's equation using gradual channel
approximation and coherent Quasi 2 Dimensional (2D) analysis in the velocity
saturation region. We have included the drain source resistance as well as the high
longitudinal electric field effects into the I-V model. Furthermore, the threshold
voltage model also includes the aspect of non ideal effects such as short channel
threshold voltage shift and narrow width effects. In addition, through these calculated
results, a bigger picture is given on how velocity saturation can be sustained. On top
of that, several enhancement and insight is discussed to overcome the challenges in
MOSFET design particularly the reduced drive strength. By investigating the effects
of velocity saturation, the author attempts to give general guidelines of how
parameters should be chosen.
1.6 Thesis Organization
This thesis consists of 8 chapters. Chapter 1 introduces the background of this
research. The problem statements, research objectives, research scope, research
contributions and thesis organization are also provided. Chapter 2 provides an
overview of the literatures reviewed throughout the research. A detailed description
on velocity saturation effects is presented. Previous long channel model
characteristics and related research are summarized.
Chapter 3 deals with the work flow of this research. It also introduce the
modeling process as well as the Intel Proprietary Schematic Editor and Circuit
Simulator. Chapter 4 marks the beginning of the proposed models formulation with
the introduction of threshold voltage modeling. Chapter 5 discusses about the
electron mobility model in MOS inversion layer. A comprehensive semi empirical
8
drain current and voltage model is explained in Chapter 6 by studying the long
channel characteristics and short channel effects. In addition, this chapter also
includes the derivation of source and drain resistance equation which normally
omitted in long channel devices. In Chapter 7, we study the velocity field model
which describes the effects of high vertical and lateral field in inversion layer pinch
off and velocity saturation.
In Chapter 8, the calculated data is observed and validated against the
experimental data generated from simulations to investigate effects of velocity
saturation. Analysis was carried out on the simulated results and the findings are
discussed. Finally, Chapter 9 concludes the thesis with summary of contributions and
suggestions for possible future development.
CHAPTER II
LITERATURE REVIEW 2.1 Non Ideal Effects in MOSFET
Integrated circuit (IC) is an electronic circuit built on a thin semiconductor
substrate. It can be categorized into two groups based on the type of transistors they
contain which are the bipolar integrated circuits and MOS integrated circuits. There
are three types of ICs. Memory, processor and application specific integrated
circuits (ASIC). In the new millennium, IC has indeed transformed our modern life,
making it easy to perform complicated tasks. IC can be found in everything from
satellite to personal computers to cell phones.
Advances in IC manufacturing process have led to tremendous growth in the
semiconductor industry. As CMOS is scaled, the density and speed of the IC is
increased. A schematic of a nanoscale CMOS can be seen in Appendix D. With the
latest technological advancement, 65 nm and 90 nm generation of processor has a
major improvement over performance, thermal design power and the system
functionality. In this context, thermal design power refers to the maximum amount
of power the thermal solution in a computer system is required to dissipate. The
introduction of multiple core architecture in single processor has a positive impact on
power efficiency. Nevertheless, these same advances come at a price as they affect
the stability and reliability of the devices. As process technology improved to a point
where devices could be fabricated into nano dimension, MOSFET began to exhibit a
phenomena not predicted by long channel models. Such phenomenon is called
10
second order effects (SOE) and occurs in short channel device is where the effective
channel length is approximately equal to the source and drain junction depth.
Among the SOE which need to be examined carefully in short channel device
design are mobility degradation, velocity saturation, hot carrier effects, short channel
effects and narrow width effects (Taur et al., 1997). Without doubt, one of the most
widely discussed effects is velocity saturation that has a significant implication upon
the current voltage characteristics of the short channel MOSFET.
2.2 Velocity Saturation
It is up most important to predict the drain current of scaled devices, both in
linear and saturation regions particularly short channel MOS transistor. Velocity
saturation reduces the saturation current below the value predicted by the
conventional long channel. As a result, this issue reduces the speed of digital IC. At
the performance level, whereby charging and discharging of parasitic capacitances
takes a longer time. Velocity saturation also lift the pinch off condition in long
channel model as the carrier density does not appear to be vanished at the saturation
point. (Arora, 1989). This research addresses the effects of velocity saturation on
temperature, doping concentration, longitudinal and transverse electric field in short
channel MOSFET. Old models of long channel MOSFET fails to give an accurate
numeral equation for MOSFET in sub-100 nm regime (Arora, 2000).
In this dimensions, the functionality of the device is now governed by
quantum mechanic as described by Yu et al. (1997). When devices are reduced in
size the electric field, E increases and the carriers in the channel have an increased
velocity. However, at high electric fields in the gate and channel of the MOSFET,
there is no longer a linear relation between the electric field and the drift velocity as
the velocity gradually saturates reaching the saturation velocity as shown in Figure
2.1.
11
Figure 2.1 Drift velocity versus electric field in Silicon
Velocity saturation is caused by the increased scattering rate of highly
energetic electrons, primarily due to optical phonon emission and degrading effect of
the carrier mobility, resulting in the breakdown of Ohm's law behaviour (Bringuier,
2002). The electric field at which saturation occurs differ with different
semiconductors; Silicon, Gallium Arsenide and Indium Phosphide. This is illustrated
in Figure 2.2.
Figure 2.2 Drift velocity versus electric field in for three different semiconductor
1x107
2x107
3x107
Dri
ft V
eloc
ity, V
d (cm
/s)
Si
GaAs
InP
0 5 10 15 20
Electric Field E (104 V/cm)
E (V/cm)
v d (c
m/s
)
12
When E is applied, the random electrons drifts in a direction that is opposite
to the direction of applied electric, which in this situation is the direction from right
to left. In Figure 2.3, the random carrier is accelerated by the electric field in a
streamlined motion. The electrons will collide with each other and at a critical
electric field, the drift velocity becomes saturated and thus making Ohm’s law
invalid (Arora, 2000). The density of arrows in an indicator of electrons and left-right
motion in an electric field when a concentration gradient is present.
Figure 2.3 Depiction of electrons density in channel under electric field
(Arora, 2000)
Previous studies namely by Frank et al. (2000), Agnello (2002) and
Lochtefeld et al. (2002) provide proper guidelines for further scaling of CMOS and
its limitation in regards to carrier velocity. Takeuchi and Fukuma (1994) has
incorporated velocity saturation model into the I-V model and the extracted
parameters was validated by numerical data from the device simulator. In this work,
the impact of velocity saturation on device performance is broadened to include
variation of drift velocity and effective mobility versus external and internal factor
such as temperature and doping concentration
EV
n (x-l) n (x+l)n (x)
l
l
EF
EC
E
13
2.3 Oxide Charges and Traps
In silicon technology, one of the critical components in device fabrication lies
in the silicon dioxide which acts as an insulator between the silicon and the
polycrystalline silicon. In an ideal insulator, the silicon oxide and the silicon oxide
interface is electrically neutral and there is no charge exchange between them.
However, unwanted charges always present in the practical devices. In general, there
are four general categories of oxide charge as shown in Figure 2.4. They are the
mobile ionic charge, oxide trapped charge, fixed oxide charge and interface trap
charge.
Figure 2.4 Charges and their location in thermally oxidized silicon
Mobile ionic charge, Qm is introduced during device fabrication due to
sodium or potassium contaminants. The major sources of sodium in a clean room
environment is human being through sweat and skin peeling (Contant et al., 2000)
while potassium contamination is due to potassium hydroxide (KOH) or water deep
etching environment (Aslam et al., 1993). When electric field is applied, these
mobile ionic charges can move from one location to the other end of the oxide with
increasing temperature. Oxide trapped charges, Qot is also distributed around the
SiO2 layer. They are created during ionizing radiation during electron hole pairs
Oxide trapped charge (Qot) SiO2
Si
Fixed oxide charge (Qf)
Interface trap charge (Qit)
K+ Na+ Mobile ionic charge (Qm)
Metal
SiOX
14
generation, by hot electron injection or high current passing into the SiO2. This has
become a more prominent problem as oxide thickness is scaled down.
Further down into the oxide layer at close range to the SiO2/Si interface, there
exist fixed oxide charges, Qf due to the abruptly incomplete oxidized silicon. The
presence of these charges affects the flat band voltage. As a charged scattering
center, it is responsible for the mobility degradation. Interface trapped charge, Qit is
represented by X at the SiO2/Si interface in Figure 2.16 and is caused by excess Si,
oxygen and impurities. Here, Qit have energy states in the forbidden energy gap of
the Si which is called the surface states. This arises from the dangling/incomplete
bonds as a result of a disruption of the crystal lattice as pictured in Figure 2.5.
Surface states act like a localized generation and recombination centers. They
captured electron from the conduction band or a hole from the valence band. The
electrons or holes trapped in this state are called interface trap charge.
Figure 2.5 Post oxidation dangling bonds that will become interfacial traps
O Si O O Si
O O
O O
Si Si Si Si Si
Si Si Si Si Si
SiO2
Si
Interfacial trap
15
2.4 Device Modeling
Semiconductor device modeling creates models to characterize the behavior
of electrical devices based on fundamental physics. A meticulous method to describe
the operation of the transistor is to write semiconductor equations in three
dimensions and solve it numerically by using program. This approach is not
recommended for general circuit simulation. The efficient way is to use compact
model or Computer Aided Design (CAD) model. There are a various types of
compact models. A physical model is based purely on device physics. An empirical
model on the other relies on curve fitting, coefficient and has no physical
significance. Semi empirical model however is the combination of both models
mention beforehand. The last compact model is a table model which places the input
and output data is in the form of a table. The value is read by a program instead of
calculating them and this saves a great deal of processing time. There are a couple of
approaches of selecting the types of compact modeling, which are based on the
charged based model or the surface potential based models as shown in Table 2.1.
These two categories of advanced MOSFET model are being model today and are
available commercially as Electronic and Electrical Computer Aided Design (ECAD)
tools.
Table 2.1 Approaches of Compact Modeling
Charge-Based Models
Surface-Potential Based Models
ACM - Advanced Compact Model
HiSIM - Hiroshima-university STARC
IGFET Model
EKV - Enz-Krummenacher-Vittoz
Model
MM11 - MOS Model 11
BSIM - Berkeley Short-channel
IGFET Model
SP - An Advanced Surface-Potential
Based Compact MOSFET Model
16
2.5 SPICE Model
SPICE (Simulation Program with Integrated Circuits Emphasis) is a general
purpose analog circuit simulator (Sheu et al., 1987). It is a great program that is
widely used in IC and board level design to check circuit designs and to predict
circuit behavior. It can simulate the performance analog and mixed analog/digital
systems. By solving sets of non-linear differential equations in the frequency domain,
steady state and time domain, SPICE can simulate the behavior of transistor and gate
designs.
BSIM is selected to be the core model formulation since it well considered
being the standard model from deep submicron into nanoscale CMOS circuit design.
It is widely adopted by IC companies such as Intel, IBM, AMD, National
Semiconductor, Texas Instrument, TSMC, Samsung, Siemen and NEC for modeling
devices with a good accuracy (Xuemei and Mohan, 2006). It is developed by the
BSIM Research Group in the Department of Electrical Engineering and Computer
Sciences (EECS) at the University of California, Berkeley. Table 2.2 shows the
SPICE level of each BSIM since it is first released in 1984.
Table 2.2 BSIM SPICE Level
MOSFET Model Description SPICE Level
BSIM1 13
BSIM2 29
BSIM3 39
BSIM3v2 47
BSIM3v3 49
BSIM4 54
17
2.6 Threshold Voltage Model
There are varieties of definition models that can be used to measure and
predict the threshold voltage of a MOSFET. Old long channel threshold voltage
model need to be modified to give a more accurate value for nanoscale devices as
new physic is being discovered and new materials being used (Wang et al., 1971).
Presently, analytical models can adequately describe the threshold behaviour very
well both in definition and extraction method (Zhou et al., 1999). In this work, a
short-channel threshold voltage model for short channel MOSFET is derived for n-
channel MOSFET with n+ polysilicon gate and p-type silicon. For precision
formulation, short channel effect and narrow width effect is taken into consideration
(Yu et al., 1997).
The analytical definition of the long channel threshold voltage is given as
( )2 2 2T f FB Si A f oxV V qN Cφ ε φ= + + (2.1)
where φf is the Fermi potential, NA is the substrate doping concentration, k is the
Boltzmann’s constant, is the Cox gate oxide capacitance, VFB is the flat band
voltage, εSi is the dielectric permittivity of the silicon.
2.7 Mobility Model
The accuracy of inversion layer mobility modeling is crucial for simulating
today’s Very Large Scale Device (VLSI) circuits. An effective carrier mobility as
function of transverse electric field is utilized in the work. The mobility is based on
the original model developed by Scwarz and Russek (1983) with some improvement
on the scattering mechanism as previous phonon scattering models does not include
surface roughness and Coulomb scattering. Operation in high transverse field and
channel doping concentration requires all of these scatterings mechanism to be
accounted as devices are scaled down.
18
The electron effective mobility due to the transverse field is modeled semi-empirically
and calculated using Matthiessen's rule by incorporating the individual scatterings as
shown below.
11 1 1
effph sr c
μμ μ μ
−⎡ ⎤
= + +⎢ ⎥⎢ ⎥⎣ ⎦
(2.2)
where μc is Coulombic scattering due to doping concentration, μph is phonon scattering and
μsr is surface roughness scattering.
2.8 Drain Current and Voltage Model (I-V)
This velocity saturation has a very significant implication upon the current
voltage characteristics of the short channel MOSFET. It reduces the saturation mode
current below the current value predicted by the conventional long channel equation
Simpler approximations for long channel MOS transistor are no longer valid as the
accuracy of a drain current model is directly affected by occurrence of velocity
saturation effect. The long channel numerator will be divided by velocity saturation
to get newly improved short channel drain current equation. Below are equation for
short and long channel current in the linear region for NMOS. The current in the
saturation region is a linear function of Vgt instead of Vgt2 as in the long channel.
222
eff oxds gt ds ds
C WI ( long channel ) V V VL
μ⎡ ⎤= ⋅ ⋅ −⎣ ⎦ (2.3)
22
2 1
gt ds dseff oxds
ds
c
V V VC WI ( short channel ) VLV
μ ⎡ ⎤−⎣ ⎦= ⋅ ⋅+
(2.4)
19
The saturation drain current in for long and short channel can be expressed as
( )2
2eff ox
ds gs T
C WI ( long channel ) V VL
μ= ⋅ ⋅ − (2.5)
( )2
eff oxds gs T c
C WI ( short channel ) V V VL
μ= ⋅ ⋅ − (2.6)
where Vc is the critical voltage defined in Eq. (6.12), Vds is the drain voltage, Vgs is
gate voltage, W is the width and L is the length of the channel. In this research, a
newly developed unified one-region current-voltage (I-V) compact model for linear
and saturation region is adopted (Zhou and Lim, 1997). The model is formulated on
the analysis of second order effects on the long channel model.
2.9 Velocity Field Model
The accuracy of a drain current model is directly affected by how the velocity
saturation effect is implemented. The most commonly used carrier velocity model is
in Eq. (2.7) and Eq. (2.8) as shown below (Jeng, Ko and Hu, 1998).
(2.7)
(2.8)
where νsat is the carrier saturation velocity and Ec is critical electric field for velocity
saturation defined by
satc
eff
vEμ
= (2.9)
v =( ) ( )
1eff
cc
E, E E
E Eμ
+<
( )sat cv , E E≥
20
The channel of the MOS transistor can be generally divided into two regions, a
gradual channel region and a velocity saturated region, as shown in Figure 2.6. In the
gradual channel region ν<νsat and E < Ec, while in the velocity saturation region, ν
=νsat and E ≥Ec (Takeuchi and Fukuma, 1994).
Figure 2.6 Coordinate system used for the model derivation.
Velocity saturation will yield an Idsat value smaller than that predicted by the
ideal relation, and it will yield a smaller Vdsat value than predicted. Figure 2.7 shows
a comparison of drain current versus drain to source voltage characteristic for
constant mobility and for field dependent mobility.
Figure 2.7 Comparison of I-V characteristic for a constant mobility and for field-
dependent mobility and velocity saturation effects
0 LC LEFF
VS Region E=EC V=VC v=vsat
Gradual Channel Region v<vsat
Source Drain I d
s (m
A)
Vds (V)
Constant Mobility Velocity Saturation
21
2.10 Source and Drain Series Resistance Model
When current flow into the channel through the terminal contact, there is a
tiny voltage drop in the source and drain region. This is associated with the resistance
that appears in the device. This resistance has several components:
(i) The resistance of the doped source/drain region.
(ii) The resistance of the metal-to-silicon contact.
(iii) The “spreading resistance” due to current traveling from the doped
source/drain region to the inversion charge in the channel.
In long channel device, source drain resistance is negligible compared to the
channel resistance. Nevertheless, in short channel devices the source drain parasitic
is becoming comparable with the channel resistance. In reality, the assumption of
source and drain being a perfect conductor is not applicable and the resistivity should
be taken into consideration.
In general, series resistance is undesirable because it reduces the drive
strength and contributes to the RC delay. Therefore, the inclusion of a compact
series-resistance model for nanoscale MOSFET would give a better accuracy on the
modeling. The physical model has a bias-dependent intrinsic and a bias-independent
extrinsic component (Zhou and Lim, 2000). The final S/D series resistance is given
by
2ext ints / d
j TGS
SR R Rx W V Vρ υ= + = +
− (2.10)
where Rext is the extrinsic resistance, Rint is intrinsic resistance, xj is the junction
depth, S is the spacer thickness and ρ is taken as effective resistivity of the S/D
regions (including contacts). The component of the drain source resistance
expression will be discussed in Section 6.4.
22
The parasitic source and drain resistance of a NMOS is shown in Figure 2.8.
Figure 2.8 Simplified Cross Section of Parasitic Resistance of a NMOS
2.11 Hot Carrier Effects
It was shown that the peak electric field, Emax in Figure 8.12 can attain high
value at low drain voltage for short channel device. When the carrier in velocity
saturation region gain adequate kinetic energy to generate electron-hole pairs,
avalanche process occurs. Electrons from the conduction band gains energy as they
move down the channel. They possess high kinetic energy and travel at saturation
velocity. On impact with the lattice, they generate a secondary electron and holes by
breaking the bond, ionizing the valence electron from the valence band into the
conduction band. As of this moment, there are now three carriers; the original
electron and the electron-hole pair. Subsequently, the newly generated pair that gains
enough energy will collide with the lattice to generate more electron hole pairs. This
phenomenon is referred to as impact ionization. The effect is more pronounced at
drain end where fields are highest. The generated electrons which are attracted to the
drain will cause a rise in drain current while the generated holes which flow through
Source Gate Drain
N+
Metal Lines
Gate Oxide
Source Resistance
Interlayer Dielectric (ILD)
Substrate/ Well
Drain Resistance
N+
Poly Si Gate Contact/via
N+ Inversion Layer
RS RD
23
the substrate to the body terminal will form substrate hole current. The potential
difference between substrate and source will create a forward bias near the source
terminal. All these effects are shown in Figure 2.9 below.
Figure 2.9 Hot Carrier Generation
Under this circumstances, electrons from the source inject themselves into the
substrate and a fraction of them will diffuse into the drain space charge region. This
triggers a positive feedback and increases the avalanche process. Another issue with
hot carrier effect that concern designer is when electron that has enough energy to
overcome the Si-SiO2 interface barrier enter the oxide layer. The trapped electrons in
the oxide produce a net negative charge and contribute to the total oxide charge in
Eq. (4.13). The trapped oxide charge results in a positive shift in threshold voltage,
which in turn increases surface scattering, reduces mobility as well as drain current
and transconductance. Eventually, this will lead to the reduction of speed and cause
the circuit to fail the speed test specification. Since these processes are continuous,
device will degrade over a period of time and stability is at stake. Ultimately, the
lifespan of the device will decrease and breakdown may occur sooner than expected.
1
--SiO2
n+
Source Drain
p-type Si substrate
|Vg|
Vs = 0 V Vds
n+
Vb
Depletion region
--- -
-- -
- - - -
- - - -
2
3
4
Avalanche
Forward injection
Substrate Current
24
2.12 Reducing Hot Carrier Degradation
Hot carrier effect in nanoscale devices is a serious problem. We identify that
as device is scaled down, electric field increases dramatically. This enhances the
breakdown effect of near avalanche breakdown and near punch through breakdown.
There are various methods that can be employed to suppress and reduce the hot
carrier degradation. This involves introducing a new structure and strengthening the
Si-SiO2 interface barrier to improve device performance. One of a way to diminish
hot carrier effects is to reduce the magnitude of the maximum electric in the channel
which is contributing to the breakdown mechanism. The most reasonable approach
is to limit the power supply voltage. However, this option might not best as it leads
to a performance trade-off. Next, we can modify the structure of a conventional
MOSFET so it is less sensitive to hot carrier degradation. Lightly Doped Drain-
Source (LDD) has been reported to effectively improve NMOSFET with reduced
lateral electric fields, higher operating voltage and a channel length reduction
(Ogura, 1980).
Narrow self aligned n- region are placed between the channel and n+ source
drain diffusion as shown in Figure 2.10. Optimized n- doses in LDD spreads the high
field at pinchoff along the n- region and reduces the peak value of the longitudinal
electric field along the channel length. It also increases the breakdown voltage and
limit the impact ionization.
Figure 2.10 Schematic cross section of “inside” and “outside” LDD
p-
n+ n+ n- n-
p-
n+ n+ n- n-
25
A close up view is shown in Figure 2.11. The gate is now extend from the
LDD which makes it the effective channel shorter than conventional MOSFET where
Loverlap is the length of the LDD-gate overlap region. Leff is the length of silicon
surface region in which its conductivity is effectively controlled by the gate bias and
Lmet is the physical separation between the source channel junction and the drain-
channel junction near silicon surface.
Figure 2.11 A schematic diagram of half of a n-channel MOSFET
Another innovation that is well received in the industry is the introduction of
strained-silicon layers into a MOSFET, demonstrated in modern CMOS integration
by IBM and Intel in 2004. It is known that high-k gate dielectrics causes significant
carrier degradation in the channel. The improvement of High-K transistor over
conventional gate oxides will not be possible without strained silicon technology.
Here, channel made of a thin layer of strained silicon is used to improve the current
drive in the transistor. The epitaxial strained Si layers is grown on relaxed SiGe as
depicted in Figure 2.13 so that the amount of current flows more smoothly from
source to drain. Higher drive current causes a transistor to switch between its on-off
states faster, ultimately creating a higher frequency and speed device.
gate electrode spacer
Loverlap
p- substrate
LDD
n+ source x
y
Lgate /2 Lmet /2
26
Figure 2.12 Cross-sectional images showing a strained-Si MOSFET with a 25 nm
gate length and its strained-Si layer (Goo et al., 2003).
There are several different approaches for introducing strain into the Si
channel of nanoscale MOSFETs. Of these, two of the well known strained silicon
options are process-induced (uniaxial) strain and bulk wafer (biaxial) strain.
Recently, biaxial strained-Si has received substantial attention. In biaxial strained
silicon, the electronic band structure is modified by pulling the individual silicon
atoms moderately apart. As a result, higher currents in the transistor can be achieved
as electrons and holes can move faster in the layer. This can be in Appendix E where
mobility is higher for strained device compared to unstrained silicon and the
universal mobility curve by Takagi et al. (1994).
More attention has been paid to uniaxial strained-Si as it enables the amount
of strain for the n-type and p-type MOSFET can be controlled independently on the
same wafer. Furthermore, uniaxial stressed Si for PMOS demonstrates tremendous
hole mobility improvement at a given strained and mobility enhancement is present
at large vertical fields. The process flow consists of selective epitaxial SiGe in the
source/drain regions to create longitudinal uniaxial compressive strain in the PMOS
as illustrated in Figure 2.14. On the other hand, specially engineered high tensile Si
nitride-capping layer is used to introduce tensile uniaxial strain as shown in Figure
2.15 (Thompson et al., 2004). The technology is proven to be remarkable and has
increases saturated n-type and p-type MOSFET drive currents by 10 and 25%,
respectively. Since then, it has been implemented and ramped into high volume
27
manufacturing to fabricate the next generation Pentium® and Intel® CentrinoTM
processor families.
Figure 2.13 TEM micrographs of 45-nm p-type MOSFET (Thompson et al., 2004)
Figure 2.14 TEM micrographs of 45-nm n-type MOSFET (Thompson et al., 2004)
For short channel VLSI transistors, LDD MOSFETs are commonly
implemented to reduce high fields inside devices. The drain to source punchthrough
is a serious problem when device scaling goes on continuously. Punchthrough
suppression becomes less effective at gate length approaches nanometer dimension.
Therefore, double implanted LDDs using halo implant is introduced. A p-type
dopants is locally implanted under the lightly doped region of the LDD as shown in
Figure 2.16. The halo implant uses the same implant type as the original well
dopant. For instance, halo device uses p-type dopant as punchthrough stoppers for p
well of the NMOS device adjacent to the n+ source and drain. They improve the
28
short channel effect such as threshold voltage roll-off characteristics, DIBL and
punch-through voltage and also electric field peak at drain end as illustrated in Figure
2.17.
Figure 2.15 DI-LDD device cross section (Codella and Ogura, 1985)
Figure 2.16 Surface electric field at drain edge for conventional and DI-LDD
devices from 2D device simulation, VSUB = -1V and L=0.6 µm (Ogura et al., 1982)
CHAPTER III
RESEARCH METHODOLOGY
3.1 Velocity Saturation Electrical Modeling
The objective of this research is to develop electrical models to investigate
the velocity saturation effects in nanoscale NMOS and it dependence on temperature,
doping concentration and longitudinal electric field. The electrical model mentioned
here is a circuit representation that has the electrical properties and characteristics of
the component. The electronic design automation (EDA) tools run on UNIX
workstation that can accessed via Microsoft Windows XP operating system.
In short, this research will develop an analytical short-channel MOSFET for
90nm process technology to predict the drain current of scaled devices, both in the
linear and saturation regions in the perspective of Ohm’s law. The investigation is
carried out on temperature, substrate doping concentration and longitudinal electric
field dependence. The approximation is based of several assumptions, which are
considerably acceptable for a wide range of analysis.
To begin with, hands-on review of UNIX environment is essential which
include basic UNIX commands, directory structuring, file editor and interactive
communications. At the same time, theoretical backgrounds of the available
analytical and semi empirical models derivations are attained through literature
review. Comparison between the models is made and proposed semi empirical
models are implemented.
30
Physical parameters or elements are extracted from the I-V curve in order to
analyze the velocity saturation response. In this stage, schematic design of the
NMOS is carried out by using Intel Proprietary Schematic Editor. Circuit netlist is
then generated and the I-V curve is displayed by running Intel Proprietary Circuit
Simulator. The numerical values of the extracted parameters are feed into the semi
empirical models.
Verification is performed on the model to ensure it is able to reflect the actual
characteristics of the electrons behaviour during velocity saturation. After the models
are verified, it is validated and analyze to obtained a better insights of the velocity
response to the variation of input parameters. Data analysis of the figures are
generated in Microsoft Excel. Figure 3.1 shows the generic steps used in the
research to develop the electrical model.
Figure 3.1 Electrical Model Development Process
Create and Edit Schematics
Check Schematics
Netlist Schematics
Circuit Simulator
Invoke Schematics Editor
Invoke Circuit Simulator
Proposed Models
Parameters Extraction
31
3.2 Intel Proprietary Softwares
Intel Proprietary Schematic Editor and Intel Proprietary Circuit Simulator is
used to provide the experimental data for this research. Intel Proprietary Circuit
Simulator provides a full Spice simulation and support industry standard. It has
customizable graphical user interface (GUI) makes it easier to perform most tasks.
The GUI can be customized to include your own buttons and menus. Once schematic
symbols are created in the schematic editor, the circuit description is written into a
netlist file that served as the input file to circuit simulator program. Then, a design is
build. The type of simulation process is defined and the input sources as well as
output probing point are chosen. By defining the probes, the simulator will provide
the type of data to be shown at a particular circuit object. The range of the display
output waves need to be set so that the simulation results can presented satisfactorily.
Finally, measurement can be done. The data output can be exported to Microsoft
Excel in Comma Separated Values (CSV) format.
Figure 3.2 Running Basic Simulation Using Circuit Simulator
Build a Design From Netlist
Define Analysis Type Define Input and Output Probes
Set Parameter
Invoke Circuit Simulator
Measure Results
Run Simulation
Export to CSV
32
3.3 Parameters Extraction for Proposed Models Threshold voltage, effective mobility, doping concentration, source/drain series
resistance, diffused junction depth, gate oxide thickness, gate oxide capacitance and
others related empirical parameter would be among parameters extracted from the
experimental data. Subsequently, curves based on the newly developed threshold
voltage model, effective mobility model and I-V model are plotted. The velocity
longitudinal field equation is then incorporated to model the velocity saturation
region (VSR). Finally, based on the graphs depicted, the velocity saturation
relationship between the models is analyzed. Figure 3.3 shows the flowchart of each
extraction process. The threshold voltage is an analytical model while the rest;
velocity field, I-V with Rds series resistance and mobility model are based on
semi empirical formulation. The I-V model is a unified SPICE model which uses
single equation for both linear and saturation region. Several assumption and
approximation is used and this is discussed in detailed in the following Chapter 4 to
Chapter 6. Appropriate values are chose so that the models are able to predict
behaviour as close as possible to measurements.
Figure 3.3 Parameter extraction
Mobility Model
Parameters Extraction
Simulated Experimental Data
Threshold Voltage Model
I-V Model Velocity Field Model
Analysis on the Velocity Saturation
33
3.4 MOSFET Modeling for Circuit Simulation Two main types of compact model are utilized in this research.
a) Physical model
This model is based on device physics formulation and each parameter in the
model has a physical significance such as flat band voltage, doping
concentration and Fermi potential. The threshold voltage model discussed
here is a physical model.
b) Physical based semi-empirical model
This model is based on device physics formulation and partly on empirical
measurements as a curve fitting expression. The parameter includes
additional coefficients that can be used by the equation to fits data on the
curve. The mobility, drain current-voltage, velocity field and source and drain
series resistance models discussed here are physical based semi-empirical
models.
In developing the models, several effects that occur in a given region of operation is
combined into one physical model. Several phenomena which simpler model ignore
is taken into account to derive a good model in term of accurate parameter extraction.
3.5 NMOS Transistor
The NMOS transistors consist of three terminals: gate, drain, and source. The
source is biased at a lower potential (often 0V) than the drain. In this case, the source
and body are both grounded. The drain current is induced based on voltages applied
at the gate and drain of the transistor. Every NMOS transistor contains a threshold
voltage. The voltage applied to the gate terminal determines whether current can
flow between the source and drain terminals. In order for the transistor to operate, Vgs
must be greater than Vt. Once this condition has been met, the resulting drain current
34
can be controlled by the voltages supplied at the gate and the drain. The relationship
between Vgs, Vds, and Ids is described by three regions of operation; the cut off region,
the triode region and the saturation region. Figure 3.4 illustrates a schematic of an
NMOS transistor.
Figure 3.4 Schematic of an NMOS transistor
CHAPTER IV
A PHYSICS BASED THRESHOLD VOLTAGE MODELING
4.1 Introduction
A new effective threshold voltage model is derived in this section based on
the long channel device by considering the effects of short and narrow channel
effects. The following discussion applies for NMOS. The threshold voltage equation
for n-channel MOSFET is defined as
( )2 2
2si A f
T f FBox
qNV V
C
ε φφ= + + (4.1)
where φf is the Fermi potential, NA is the substrate doping concentration, k is the
Boltzmann’s constant, Cox is the gate oxide capacitance, VFB is the flat band voltage,
εSi is the dielectric permittivity of the silicon. The full derivation can be seen is
Appendix F.
36
4.2 Threshold Voltage Model
The MOSFET is fabricated with polysilicon gates over a p-type silicon
substrate or well. The gate is heavily degenerately doped as the source and drain
regions.
The metal semiconductor work function difference is determined by
' ( ' )2
' ' )2
2
gms fp
gfp
gfp
Ee
Ee
Ee
φ χ χ φ
χ χ φ
φ
= − + +
= − − −
⎛ ⎞= − +⎜ ⎟
⎝ ⎠
where χ is the oxide electron affinity. φfp and φfn is the Fermi surface potential for p-
type and n-type semiconductor substrate respectively. They are written as
( ) ln Afp
i
NkTTq n
φ⎛ ⎞
= ⎜ ⎟⎝ ⎠
(4.3)
( ) ln Dfn
i
NkTTq n
φ⎛ ⎞
= ⎜ ⎟⎝ ⎠
(4.4)
where NA is the acceptor doping concentration, ND is the donor doping concentration.
The intrinsic carrier concentration, ni is given by
1
2( ) exp2
gi c v
En N N
kT⎛ ⎞
= −⎜ ⎟⎝ ⎠
(4.5)
Figure 4.1 shows experimentally determined values of the gate-to-substrate
work function difference φms for various type of gate materials. The Fermi level in
the n+ polysilicon is essentially at the edge of conduction band while for p+ substrate
is below the intrinsic Fermi level. Polysilicon gate has a smaller work function
(4.2)
37
compared to the substrate, which make it negative. The magnitude of φms for
n+ polysilicon with p+ substrate decreases with doping as the Fermi level for the
substrate proceed to travel down further from the mid-gap.
Polysilicon is preferable to a metal gate as it is able to give a lower threshold
voltage due to its small work function difference. When metal was used for long
channel devices, gate voltage is still relatively large to overcome the work function
difference between metal and silicon substrate. A transistor with a high threshold
voltage would not be practical in nanoscale devices.
Figure 4.1 Metal-semiconductor work function difference versus doping
concentration for aluminum, gold, and n+ and p+ polysilicon gates (Sze, 1981)
38
The energy bandgap variation, Eg with temperature in Eq. (4.5) is defined as
2
( ) (0)( )g g
TE T ETα
β= −
+ (4.6)
where Eg (0) is the initial value of the band gap at 0 K while α and β are constant.
(Zeghbroeck, 2004). The band gap of a semiconductor is the energy difference
between the top of its valence band and the bottom of its conduction. The bandgap is
a forbidden energy range between the valence and conduction bands. In order to be
in the conduction band and taking part in conduction, electron in the valence band
must absorb sufficient energy to leap across the band gap. The fitting parameters for
germanium, silicon and gallium arsenide in listed in Table 4.1.
Table 4.1 Values of α and β for various semiconductors (Zeghbroeck, 2004) Germanium Silicon GaAs
Eg(0) [eV] 0.7437 1.166 1.519
α [eV/K] 4.77 x 10-4 4.73 x 10-4 5.41 x 10-4
β [K] 235 636 204
The bandgap of semiconductor decreases as temperature increases. A raise in
temperature enable atoms to have a higher thermal energy. This resulted in larger
atomic vibrations. The greater atomic vibrations have the tendency to broaden the
distances between the atoms which causes lattice expansion. This increase in inter-
atomic separations lower the potential. Less energy is needed to break the bonds and
release electrons from the valence to the conduction band, thus reducing the
bandgap. The temperature dependence for silicon energy bandgap can be
approximated as
-4 24.73 x 10( ) 1.17
( 636)gTE T
T= −
+ (eV) (4.7)
39
The effective hole and electron masses used in the calculations are given by
01.08nm m= (4.8)
and
00.81pm m= (4.9)
The parameter Nc in Eq. (4.5) is the effective density of states function in the
conduction band and can be defined as
( )315 325.42 x10cN T cm−≅ (4.10)
Nv, the effective density of states function in valence band is defined as
( )315 323.52 x10vN T cm−≅ (4.11)
The product of equation Eq. (4.10) and Eq (4.11) yields
( )31 3 61.908x10c vN N T cm−= (4.12)
The flat band voltage in Eq. (4.1) can be calculated by
totFB ms
ox
QVC
φ= − (4.13)
where Qtot is effective net charges per unit area at the Si-SiO2 (C/cm2) and is a sum
of fixed oxide charge, Qf and interface trap charge, Qit which is discussed is Section
2.4. Using Eq. (4.2) and Eq. (4.13), Eq. (4.1) can be written as
( )2 2
2si A f g tot
T fpox ox
qN E QVC e C
ε φφ
⎛ ⎞= + − +⎜ ⎟
⎝ ⎠ (4.14)
40
4.3 Short Channel Threshold Voltage Shift
Additional effects on VT occur as the device shrink in size. The short channel
threshold voltage shift reduce VT predicted by the long channel. The source and
drain distance becomes comparable to the MOS depletion width. The net charge in
the depletion region has to be considered since a portion of the charge is shared
among the gate, source and drain charge. The fraction of the charged induced by the
source and drain becomes significant as the channel length reduces. Therefore, the
charge-sharing model (Yau, 1974) is utilized. This model assumes the charge-
induced by gate voltage is contained in a region that can be best described by a
trapezoid, as shown in the Figure 4.2. The total charge contributing to the threshold
under the gate contact is represented by the rectangle with a width of xdT and a length
of L.
Figure 4.2 Charge sharing in the short channel threshold voltage model
The bulk charge inside the trapezoid is controlled by the gate is given by
'2
'B A dTL LqN xQ +⎛ ⎞
⎜ ⎟⎝ ⎠
= (4.15)
xdm n+ rj n+
p substrate
VS VG
VD
VB
L
L’
41
The depletion width, xdT extending into the p substrate is written as
1/ 24 si fp
dTA
xqNε φ⎛ ⎞
=⎜ ⎟⎝ ⎠
(4.16)
The threshold voltage shift due to short channel effects (Neaman, 2003) can be
expressed as
21 1jA dT dTT
ox j
rqN x xVC L r
⎡ ⎤Δ = − + −⎢ ⎥
⎢ ⎥⎣ ⎦ (4.17)
By taking Eq. (4.17) into account, the threshold voltage now can be approximated by
22 1 1 1jA dT dTT fp FB
ox j
rqN x xV VC L r
φ⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥= + + − + −⎜ ⎟⎨ ⎬⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
(4.18)
where
( ) ( )T Short Channel T Long Channel TV V V= + Δ (4.19)
4.4 Narrow Width Effects
The effect of narrow-width, which causes an increase in the threshold voltage
as the channel width is reduced, is also included in this modeling. As the depletion
region edge approaches the edge of the device, it makes a transition from the deep
depletion under the gate to the shallow depletion region under the thick oxide. This
transition region is shown in the Figure 4.3.
For wide widths, the size of this augmented charge at each end of the channel
width relative to the bulk charge is small and can be neglected. However, as the
width is reduced, this ratio increases and becomes significant. This additional charge
increases the bulk charge and causes VT to increase. As the width becomes smaller,
the shift in threshold voltage becomes larger.
42
The additional space charge due to narrow width effect is given by Neamen (2003)
and it is shown to be
dT dTT
ox
qNx xVC W
ξ⎡ ⎤Δ = ⎢ ⎥⎣ ⎦ (4.20)
where W is the channel width while ξ is a fitting parameter for lateral space charge
width and normally given to be π/2 for a semicircle width. By adding this extra
charge into Eq. (4.18), the final threshold voltage expression can be approximated as
22 1 1 1jA dT dT dTT fp FB
ox j
rqN x x xV VC L r W
ξφ⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥= + + − + − +⎜ ⎟⎨ ⎬⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
(4.21)
Figure 4.3 Cross section of an NMOS showing the depletion region along the
width of the device
Tox SiO2
WTH Si
Ideal Depletion Boundary
Actual Depletion Boundary
W
TTH
WD
43
The final model is able to provide a better representation of the threshold
voltage instead of Eq. (4.14) as it takes into account the influences of other effects as
mentioned above. The effect of short channel and narrow width is shown in Figure
4.4 and Figure 4.5 respectively. A larger threshold voltage is noted for narrow
channel device. On the hand, the short channel device has a decreasing threshold
voltage with channel length. Note that adding the doping concentration value does
not necessarily increase the threshold voltage. This device parameter should be
considered thoroughly to give the optimized results.
Figure 4.4 Qualitative variation of threshold voltage with channel length
Figure 4.5 Qualitative variation of threshold voltage with channel width
L
VT
W
VT
CHAPTER V
PHYSICALLY BASED MODEL FOR EFFECTIVE MOBILITY FOR
ELECTRON IN THE INVERSION LAYER
5.1 Introduction
The effective mobility of inversion layer carriers is a substantial element in
the performance of a MOSFET. A physically based semi-empirical equation is
employed to model surface roughness, phonon and coulomb scattering. In long
channel device, mobility is assumed to be constant along the channel. However, this
is not the case for short channel device. The term effective is used to describe the
average mobility in the channel. On the whole, mobility depends on many process
parameters and bias conditions. For example, mobility can be a dependence on the
gate oxide thickness, substrate doping concentration, threshold voltage, gate and
substrate voltages. The proposed effective mobility model is transverse electric field
dependent based on the concept introduced by Sabnis and Clemens (1979) which
lumps many process parameters and bias conditions together. The modeling begin
with a brief study of previous mobility model.
5.2 Schwarz and Russek Mobility Model
A portion of Schawrz and Russek (1983) model is adopted in this research.
Developed in 1983, it is consist of bulk phonon scattering, μbph and a surface
mobility term, μs which represent the scattering by surface phonons and fixed
interface charge.
45
The equations are as follows
1
1 1
bph seff μ μ
μ−
⎛ ⎞+⎜ ⎟⎜ ⎟
⎝ ⎠= (5.1)
2.51150bph nTμ −= (5.2)
* 9 1 2
23.2 x10th n
sqZ Z
pm pTυμ −= = (5.3)
1.5 8 11.5x100.09 In n f
NT T NZ
p − −⎛ ⎞+ ⎜ ⎟⎝ ⎠
= (5.4)
cl QMZ Z Z+= (5.5)
13 2 0.0388cl n effeff
kTZ Tq
EE
−= = (5.6)
( )1
2 315 3
*
9 21.24 x10
4QM effeff
hZ
m qE
Eπ −−
⎡ ⎤=⎢ ⎥
⎢ ⎥⎣ ⎦= (5.7)
where p is the Fuchs factoring scattering which describe the probability of diffuse
scattering, Z is the averaged inversion layer width, Nf is the interface charge density,
Zcl is the classical channel width and ZQM is the quantum mechanically broadened
width due to the two dimensional quantization of the energy levels in the inversion
layer, h is the Planck's constant, m* is the electron effective mass and υth is thermal
velocity and q is electronic charge.
The above-mentioned equations lack of a surface roughness and columbic
scattering. Therefore, these scatterings are accounted in this semi empirical equation
as the nanoscale MOSFET is operating at high transverse electric field and high
doping concentration level. This is supported by the finding from Takagi et al.
(1994) which reveal that the deviation at the low and higher field is caused by
Coulomb scattering from dopant in the channel and surface roughness scattering
respectively. Figure 5.1 illustrates the failure of the equation to fit nicely into the
experimental data in both low and high field region roll off as there is no Coulomb
scattering and surface roughness scattering taken into account. On the other hand,
46
Figure 5.2 includes both scattering and show an excellent agreement with measured
data.
Figure 5.1 Comparison of calculated and measured μeff versus Eeff for several
channel doping levels without Coulomb scattering and surface roughness scattering
(Shin et al., 1991).
Figure 5.2 Comparison of calculated and measured μeff versus Eeff for several
channel doping levels with Coulomb scattering and surface roughness scattering
(Shin et al., 1991).
47
5.3 Proposed Mobility Model
In order to develop the model, Matthiessen’s rule in Eq. (2.2) is applied. It
gives the approach to extract the individual contribution to the mobility as well as
data on the charged interface scattering density, surface roughness scattering
coefficient and effective fixed oxide charge density. Figure 5.3 shows a set of data
on mobility versus Eeff and its contributing element.
Figure 5.3 Schematic diagram of Eeff and doping concentration dependence of
mobility in inversion layer by three dominant scattering mechanism
The universal curve can be divided into rising curve influenced by phonon
scattering term, the mid section governed by columbic scattering term and the roll off
contributed by the surface roughness term. The equations to calculate effective
mobility of electron in MOS (Shin et al., 1991) based on the three dominant
scattering are given below.
1
1 1 1
ph sr ceff μ μ μ
μ−
⎛ ⎞+ +⎜ ⎟⎜ ⎟
⎝ ⎠= (5.8)
( )
( )
111
*
1112.5
9 1 2
2
14003.2 x10
TKph B n
th
cl QMn
n
qZpm
Z ZT
pT
K Tμυ
−−−
−−−−
−
⎡ ⎤⎛ ⎞⎢ ⎥+ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤+⎛ ⎞⎢ ⎥= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
=
(5.9)
MO
BIL
ITY
(cm
2 /V-s
) Coulomb Scattering
Surface Roughness Scattering
Phonon Scattering
High
Low
Total Mobility
EFFECTIVE FIELD (V/cm)
48
0.25
1.75 8 14.53x100.09 In n f
NT T NZ
p−
− −⎛ ⎞+ ⎜ ⎟⎝ ⎠
= (5.10)
13 2 0.0388cl n effeff
kTZ Tq
EE
−= = (5.11)
1 153 31.73x10QM QM eff effZ K E E− −−= = (5.12)
2 14 26 x10sr sr eff effK E Eμ − −= = (5.13)
1.5
22
2
21 1.5
22
2
1
ln (1 )1
1.1x10 1
ln (1 )1
c nc
BH ABH
BH
n
BH ABH
BH
K TN
TN
μγγ
γ
γγγ
−
=+ −
+
=+ −
+
19
2 2 22x10BH n n
I I
KT T
N z N zγγ = = (5.15)
where Tn is the normalized temperature at 300 K given as
( )300n
T KT = (5.16)
γ2BH is the Brooks Herring constant while KB, KQM, Ksr, Kc and Kγ are numerical
coefficients that provide best agreements with the experimental data. The total
number of electrons per unit area in the inversion layer, NI in Eq. (5.10) can be
calculated by using
( )2 siI eff 0N
qE Eε
= − (5.17)
(5.14)
49
The effective transverse field can be expressed as
1 ( )eff inv Bsi
E Q Qηε
= + (5.18)
where η is a fitting parameter. The best values are η = 1/2 for (100) electrons
(Sabnis and Clemens, 1979) and η = 1/3 for holes and (111) or (110) electrons
(Takagi et al., 1994). A set of three vector integers in parentheses describe the
crystal planes in a lattice while the vector component in brackets are used to
designate direction. The lattice point of the diamond structure is represented by a dot.
The three types of lattice planes along the a , b and c axis is shown in Figure 5.4,
Figure 5.5 and Figure 5.6.
Figure 5.4 A diamond structure with (100) lattice plane at [100] direction
a
b
c
[100]
50
Figure 5.5 A diamond structure with (110) lattice plane at [110] direction
Figure 5.6 A diamond structure with (111) lattice plane at [111] direction
The effective electric field can also be defined as
2
s 0eff
E EE += (5.19)
where Es is the transverse electric field at the gate dielectric silicon interface.
a
b
c
[110]
a
b
c
[111]
51
The transverse electric field at the edge of the inversion layer, E0 is given as
1/ 2
1/ 2
4
4
A dTB B0
si si si
si fpA
si A
A fp
si
qN xqN Q
qNqN
qN
Eε ε ε
ε φε
φε
= = =
⎛ ⎞= ⎜ ⎟
⎝ ⎠
⎛ ⎞=⎜ ⎟
⎝ ⎠
Carrier transport primarily occurs in the surface on the inversion layer. The
induced transverse and longitudinal electric field influence the velocity of the carrier
toward and parallel to the silicon-silicon oxide interface respectively. The carrier
near the surface suffers collision with the silicon interface which resemble a zig zag
motion which is shown in Figure 5.7.
Figure 5.7 Visualization of surface scattering at Si-SiO2 interface
The importance of three scattering mechanisms mentioned above is determined by
the temperature and the strength of electric field which interact with the charges in
the silicon-silicon oxide interface.
(5.20)
n+
S D Gate
n+
Depletion region
Silicon Oxide
52
5.3.1 Phonon Scattering
Mobility that is due to the various modes of lattice vibration including surface
acoustic phonons and optical phonons is called phonon scattering. Phonon scattering
is dominant in strong inversion where interaction between electrons and vibrating
lattice atom is frequent. It is dominant at high temperature and weak at low
temperature. Phonon scattering is also known as lattice scattering. KB and KT in
Eq. (5.9) is fitting parameter for bulk phonon scattering. KB=1400 and KT=2.5 and
found to have good agreement with measured data as described by Shin (1991).
5.3.2 Coulomb Scattering
Coulomb scattering is due to collision between charged centers near the
silicon-silicon oxide interface. The proposed model not only includes fixed oxide
charge, and interface-state charge but also localized charge due to ionized doping
impurities. The effects of Coulomb scattering are significant for lightly inverted
surfaces. A large percentage of the total carriers are able to interact with these
ionized impurity and interface traps.
However, it is less effective for a heavily inverted surface because of carrier
screening from the ionized impurities. Carrier screening is essentially the weakening
of electric charges due to large numbers of carriers in the surrounding of the charge.
The Brooks-Herring equation for screened Coulomb scattering is adopted in Eq.
(5.14) and Eq. (5.15). High surface-charge densities or substrate doping
concentrations increases Coulomb scattering as the probability of a carrier
encountering a charge is higher. When Coulomb scattering rises, mobility of the
remaining free carriers is reduced.
53
5.3.4 Surface Roughness Scattering
Surface roughness scattering is due to the microscopic roughness of the
silicon-silicon oxide interface. This type of scattering is important under high field
and strong inversion conditions because the strength of the interaction is governed by
the distance of the carriers from the surface. In nanoscale device, transverse electric
field can go up to 106 V/cm because of a thinner gate oxide and short channel
length. As carriers get closer to the interface layer, the stronger will be the scattering
due to surface roughness. The value of Ksr= 6 x 1014 in Eq. (5.13) provide the best
agreement with the I-V curve generated by Intel Proprietary Software.
CHAPTER VI
A PHYSICALLY BASED COMPACT I-V MODEL
6.1 Introduction
Long channel MOSFET drain current derivation can be described by solving
Poisson equation and drift-diffusion current density equations in one-dimensional
form. The drain current can be modeled based on gradual channel approximation
(GCA). GCA assumes that the variation of the electric field in the y direction (Ey)
along the channel is much less than the corresponding variation in the x-direction
(Ex) which is perpendicular to the channel as in Eq. (6.1).
y xE Ey x
∂ ∂∂ ∂
(6.1)
In the GCA, a two dimensional problem can be separated into two
independent one dimensional problems and solve them in sequence. The first piece
would be the vertical electrostatics problem relating the gate voltage to the channel
charge and the depletion region. The second piece is the longitudinal problem
involving the voltage drop along the channel and drift current. By solving the latter
equation for the inversion charge per unit area, QI, the drain current can be
calculated. Nevertheless, GCA is valid for most channel region except beyond the
pinch off point as the gradient of the longitudinal field becomes comparable to the
gradient of the transverse electric field. Pao and Sah (1966) model was used
alternatively to calculate QI numerically. The charge sheet MOSFET introduced by
55
Baccarani and Brews (1978) has make calculation simpler by avoiding numerical
analysis needed in Pao-Sah model and became one of the most widely adopted
model. Charged sheet approximation assumes that all the inversion charges are
located at the silicon surface like a sheet of charge and there is no potential drop and
no band bending across the inversion layer.
6.2 Short channel I-V Model
In this section, the charge sheet model is employed to derive the expression
for drain current in linear and saturation region. To gain insight into short channel
effects, velocity saturation coefficient is incorporated. The current flow in silicon is
driven by two mechanism; the drift of carriers and the diffusion of carrier. In this
section, the carrier concentration within the silicon is assumed to be uniform.
Therefore, diffusion current which is caused by an electron or hole concentration
gradient in silicon is omitted from the calculation.
Under thermal equilibrium, carrier move in a random direction through the
silicon crystal with an average thermal velocity, vth of the order of 107 cm/s and
kinetic energy proportional to kT at room temperature. When an electric field is
applied to an inversion layer containing free carriers, the electrons in this case are
accelerated in the direction opposite the field and acquire a drift velocity over a mean
free path between collision. The drift velocity of the electron with mass m* gain
during a mean free time τ between collision is
*y
d
qEv
mτ−
= (6.2)
Drift velocity returns to zero after a collision event where velocity is randomized and
the whole process repeats all over again.
56
The I-V derivation is shown below while a more detailed explanation can be seen
Appendix G. The mobility of the electron can be defined as
* *th
q qlm m vτμ = = (6.3)
where the mean free path, l is
thl v τ= (6.4)
For n-type silicon with a free electron density n, drift current density Jn, under an
electric is given by
n dJ qnv= (6.5)
where vd under the influence of velocity saturation is a function of
1
eff yd
y c
Ev
E Eμ
=+
(6.6)
where E is the longitudinal electric field while Ec is the critical electric field before
velocity saturation occurs. The total charge per unit area at a point y along the
inversion layer can be given as
( ) ( )[ ]( )
s ox GS T
ox GT
Q y C V V V y
C V V y
⎡ ⎤= − −⎣ ⎦= −
where Cox is the capacitance per unit area and given as
oxox
ox
Ctε
= (6.8)
εox is dielectric constant of the oxide and tox is oxide thickness as shown in
Figure 6.1.
(6.7)
57
Figure 6.1 Charge Distribution in a MOS Capacitor
The total current at a point y along the inversion layer can be represented as
( )1
eff yds v d v i
y c
EI n qv A n q zx
E Eμ⎛ ⎞
= = ⋅⎜ ⎟⎜ ⎟+⎝ ⎠
where A is the area of the inversion width, xi as shown in Figure 6.1. Ids can also be
written as
( )( ) ( ) ( )1 1ds c ox GT effI V y E C V V y z V yμ⎡ ⎤ ⎡ ⎤+ ∂ ∂ = − ⋅ ⋅ ∂ ∂⎣ ⎦ ⎣ ⎦ (6.10)
By integrating from 0 to L the drain current for short channel device is given by
( )
[ ]
2. 2ox eff GT ds dsds
ds c
C z V V VI
L L V LE
μ −=
+ (6.11)
where the critical voltage, Vc is given as
satC c
eff
vV LE Lμ
= = (6.12)
Therefore, the equation given for short channel current in the linear region is
22
2 1
GT ds dseff oxds
ds
C
V V VC WI VLV
μ ⎡ ⎤−⎣ ⎦= ⋅ ⋅+
(6.13)
(6.9)
GATE
+ + + + + + + + + + + +
- - - - - - - - - - - - -L
Zxi
58
A better approximation of the drain current in second order term in the power series
expansion would include a body-effect coefficient, m (Tsividis, 1999 and Zhou et al.,
1999) as shown below.
22
2 1
GT ds dseff oxds
ds
C
V V mVC WI VLV
μ ⎡ ⎤−⎣ ⎦= ⋅ ⋅+
(6.14)
Next is the derivation of short channel saturation voltage and current. Note that the
short channel current without body-effect coefficient in the linear region is used here
for an easier calculation. Idsat can written as
( )dsat s sat
ox GT dsat sat
I n qv WC V V v W
=
= −
Drain saturation current can also be approximated by letting V= Vdsat in Eq. (6.14)
22
2 1
GT dsat dsateff oxdsat
dsat
C
V V mVC WI VLV
μ ⎡ ⎤−⎣ ⎦= ⋅ ⋅+
(6.16)
Substituting Ids from Eq (6.15) into Eq. (6.16), one can obtain Vdsat in quadratic form
2 2 2 0dsat GT C dsat CV V V V V− + = (6.17)
Vdsat can be found by solving the quadratic equation
22 4 82
1 2 1
C C GT Cdsat
GTC
C
V V V VV
VVV
− ± +=
⎛ ⎞= + −⎜ ⎟⎜ ⎟
⎝ ⎠
(6.15)
(6.18)
59
From Eq. (6.15), when VGT >> Vc and Vc is small, Vdsat can be approximated by
2dsat GT CV V V≈ (6.19)
Rearranging Eq (6.17), one obtain
( )2 2dsat C GT dsatV V V V= − (6.20)
Therefore, if one uses the Eq (6.18) for Vdsat and Eq (6.20) for V2dsat, the integration
in Eq. (6.15) for short channel saturation drain current can be carried out to yield
dsat ox sat GTI C v W V= (6.21) 6.3 Compact I-V Model
The electrical characteristics of bulk MOS transistors are also influenced by
the bias applied to the substrate. The body effect coefficient or body factor represents
the dependence of the source voltage on the gate bias. For example, the more
positive the source-substrate bias the larger number of acceptor atoms that are
depleted. As such, Idsat in Eq (6.21) can be written as Idsat0 to include the body effect
coefficient (Zhou and Lim, 2001).
( )0dsat ox sat G T dsatI C v W V V mV= − − (6.22)
When drain source resistance, Rds is considered, Vgs and Vd in Eq (6.22) are replaced
as shown below
( ) ( ) dsat ox sat G dsat s T dsat dsat S DI C v W V I R V m V I R R= − − − − +⎡ ⎤⎣ ⎦ (6.23)
60
Rearranging Eq. (6.23), one obtains
( )
0
1 1dsat
dsatox sat S D
IIC v W m R mR
=− − +⎡ ⎤⎣ ⎦
(6.24)
Substituting Eq. (6.24) into Eq. (6.11), Vdsat becomes
2 0dsat dsataV bV c+ + = (6.25)
The extracted Vdsat can be obtained by solving the quadratics equation
2 4
2dsatb b acV
a− − −
= (6.26)
where
sat ox Sa v WC mR= (6.27)
( )( )2g t sat ox g t S SD satb V V v WC V V R mR mE L⎡ ⎤= − − + − + +⎣ ⎦ (6.28)
( ) ( )22g t sat sat ox SD g tc V V E L v WC R V V= − + − (6.29)
In order to have one drain current equation which cover both linear and saturation
region, a smoothing transition function is utilized.
( )21 42deff dsat dsat ds s dsat ds s s dsatV V V V V V Vδ δ δ⎡ ⎤= − − − + − − +⎢ ⎥⎣ ⎦
(6.30)
Vdeff is used to replace the Vds. δ is a fixed parameter.
61
By incorporating Eq. (6.30), into Eq. (6.16), the unified one region equation is given
as
2
0
22 1
GT deff deffeff oxds
deff
C
V V mVC WI VLV
μ ⎡ ⎤−⎣ ⎦= ⋅ ⋅+
(6.31)
Ids0 denoted the condition for Rsd = 0, therefore when Rsd is calculated, the expression
becomes
00
01ds
dsds sd deff
III R V
=+
(6.32)
where Rch is the resistance along the channel. Channel length modulation (CLM) is a
non ideal characteristic that present in short channel MOSFET. Therefore, the
effective early voltage, VAeff is included into the formulation. It is considered to be
positive and defined as
( )sat eff sat eff ds
Aeffdeff
E L E L VV
Vξ+
= (6.33)
where ξ is a fitting parameter. With CLM effect, Eq. (6.31) can be rewritten as
01 ds deffdeff ds
Aeff
V VI I
V⎛ ⎞−
= +⎜ ⎟⎜ ⎟⎝ ⎠
(6.34)
With the inclusion of Eq (6.33), the drain current is given by
( )1deff
dssd deff deff
II
R I V=
+ (6.35)
62
6.4 Source Drain Resistance
In reality, there are parasitic resistances associated with the source and drain
regions other than the channel resistance (Rch) as illustrated in Figure 6.2. These
source drain resistance should be taken into consideration in short channel modeling
as it causes a substantial drain current degradation besides contributing to the RC
delay.
Figure 6.2 Source (Rs) and Drain (Rd) Resistance
The model is based on the concept introduced by Zhou and Lim (2000). Rsd is the
sum of Rs and Rd when both resistance are equal in magnitude. It is also compose of
extrinsic (Rext) and intrinsic resistance (Rint)
ext intsdR R R= + (6.36)
The bias dependent or Rint is given by
int sp acR R R= + (6.37)
where Rac and Rsp is accumulation and spreading resistance respectively. On the other
hand, the bias independent or extrinsic resistance (Rext) is given as
ext coshR R R= + (6.38)
where Rsh and Rco is sheet and contact resistances respectively.
Rch
Vds
Rs Rd
63
The extrinsic resistance Rext and intrinsic resistance Rint can be rewritten as
2ext
j
SRx Wρ= (6.39)
intTGS
RV V
υ=−
(6.40)
where ρ is average resistivity in the spacer region, S is the spacer thickness, xj is the
junction depth and υ is treated as a fixed fitting parameter in Ω - V unit. All of these
resistance component is illustrated in Figure 6.3.
Figure 6.3 Current patterns in the source/drain region and its resistance (Ng and Lynch, 1986)
Rco/2 Rsh/2 Rsp/2
Rac/2
Lwin S
Metallurgical Junction
Xj
GATE
CHAPTER VII
VELOCITY FIELD MODEL
7.1 Introduction
Drift velocity saturation in the channel of a MOSFET is an important factor
in the modeling of short channel devices. It does not only limit the current at the point of
current saturation in a transistor but also reduces the quadratic relationship of the saturation
current on gate voltage into a virtually linear relationship. The velocity model for electron is
presented as well as analysis on the velocity saturation region.
7.2 Velocity Saturation Region Length
Carrier velocity saturation occurs at a section called the velocity saturation
region (VSR) length. It is also known as the width of the high-field region where
impact ionization takes place as shown in Figure 7.1. The patterned box depicted in
Figure 7.1 as ΔL in length is the portion affected with velocity saturation.
65
Figure 7.1 Velocity saturation region in MOSFET The two commonly used empirical velocity longitudinal field relationship models in
the inversion layer described by Trofinen (1965) and Caughey and Thomas (1967)
respectively as
( )1eff y
y cd
EE E
vμ
+= (7.1)
and
( )
1 221
yeffd
y c
Ev
E E
μ
⎡ ⎤⎢ ⎥⎣ ⎦
=+
(7.2)
Ec is the critical electric field before the occurrence of velocity saturation defined by
satc
eff
vEμ
= (7.3)
Ey is the longitudinal electric field given by
d sdy
V IREL
−= (7.4)
ΔL
L′
SiO2
n+
Source Drain
p-type Si substrate
Vg
Vs Vd Gate
n+
Depletion layer
L-ΔL L
66
The accuracy of Eq. (2.7), Eq. (2.8), Eq. (7.1) and Eq (7.2) is depicted in
Figure 7.2. Electric field should be large enough than Ec in order for Eq. (7.1) to
approach νsat. Eq. (2.7) is able to fit the experimental data nicely in the linear region
than the complicated Eq. (7.1) while Eq. (7.2) fits the data quite accurately. The
simpler Eq (7.2) reaches saturation at a much later point than Eq (7.1).
Figure 7.2 Comparison of commonly used velocity field models
7.3 A Pseudo 2D Model for Velocity Saturation Region
A pseudo 2 dimensional model for the MOSFET operating in the velocity
saturation region illustrated in Figure 7.3 can be adopted to yield E(y). ABCD
represent the Gaussian box, with boundary AB marks the starting of the saturation
point extending to CD which is the drain junction depth. The value of Esat and Vdsat is
located at the edge of VSR point A. In the following analysis, the mobile carrier are
assumed to spread evenly in the VSR. The drain junction is heavily doped with a
square corner and conducting perfectly.
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06
Electric Field (V/cm)
Vel
ocity
(cm
/sec)
( ) ( )( ) ( )
2 40 c
7 4sat c
710 cm /Vsec E = 2 8x10 V/cm for 2 7 2 8
1x10 cm/sec E = 1 1x10 V/cm for 7 1 7 2
. Eq. . and Eq. .
. Eq. . and Eq. .
μ
ν
=
=
Eq. (2.7) & Eq. (2.8) Eq. (7.1)
Eq. (7.2)
1x102 1x103 1x104 1x105 1x106 1x104
1x105
1x106
1x107
1x108
Eq (2.7) & Eq (2.8)
67
Figure 7.3 Analysis of velocity saturation region
Pseudo 2 dimensional model includes the effect of gate field into the analysis and
comprises both depletion charge density, qNa and mobile charge density qNm. The
Poisson’s equation with field gradient in the x direction is now given by
( ) ( ) ( ) ( )
( ) ( )
2 2
2 2
yx
x y x y x y x yx y
yxx y
a m
si si
a m
si si
V , V , qN , qN ,
EE qN qN
ε ε
ε ε
∂ ∂+ = +
∂ ∂
∂∂+ = +
∂ ∂
It is assumed that the value of x xE∂ ∂ at each point can be represented by the
average value x xE∂ ∂ from x=0 to x=xj. The value Ex (xj,y) is assumed to be close to
zero while Ex at the Si-SiO2 interface is denoted by Ex (0,y). The average value
x xE∂ ∂ can be approximated by
( ) ( ) ( )
( )
0
0
x x jx
j
x
j
E ,y E x ,yE xx x
E ,yx
∂ − ∂∂=
∂
∂=
(7.5)
(7.6)
(0,0)
xj
Ex (0,y)
Esat Emax
A D
B C
tox
xj
y
x ΔL
GATE
VSR
68
Ex (0,y) can be rewritten as Esi (0,y). The field at the surface of the silicon is related
to the field in the oxide by the equation
( ) ( )0 y 0 yoxsi ox
si
E , E ,εε
= (7.7)
( )0oxE ,y can be expressed as
( ) ( )2 y0 GS FB f
oxox
V V VE ,y
tφ− − −
= (7.8)
Substituting Eq. (7.8) into Eq. (7.5) yields
( ) ( )y y 2 y1y
GS FB fox a m
si ox j si si
E V V V qN qNt x
φεε ε ε
⎛ ⎞∂ − − −⎛ ⎞⎛ ⎞+ = +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠⎝ ⎠
(7.9)
Since V(y=0) = Vdsat, Eq. (7.8) can be given as
( )2
y 0 GS FB f dsatox
ox
V V VE
tφ− − −
= = (7.10)
and
( )y 0 cox
ox
a mj
ox
QE
qN qNx
ε
ε
= =
⎛ ⎞+= ⎜ ⎟
⎝ ⎠
Hence
21 GS FB f dsatox a m
si ox j si si
V V V qN qNt x
φεε ε ε
⎛ ⎞− − −⎛ ⎞⎛ ⎞ ⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
(7.12)
(7.11)
69
Substituting the right hand side of Eq. (7.9) with Eq (7.12) yields
( ) ( )
( )
y
2
y 1 1 yy
y
oxdsat
si ox j
dsat
EV V
t x
V Vl
εε
⎛ ⎞∂ ⎛ ⎞⎛ ⎞= −⎡ ⎤⎜ ⎟⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟∂ ⎝ ⎠⎝ ⎠⎝ ⎠
−⎡ ⎤⎣ ⎦=
where l is the characteristic effective length of the VSR. Eq. (7.13) is a linear first
order differential equation which can be solved with boundary condition V(0) = Vdsat
and E(0) = Esat, The general solution of Eq. (7.13) is
( ) y y
2
y dsat l lV V
Ae Bel
−−⎡ ⎤⎣ ⎦ = + (7.14)
The coefficients in Eq. (7.14) are found to be A= lEsat/2 and B=-lEsat/2. Thus, the
expression for V (y) and Ey (y) in the VSR is
( ) ( )y ydsat satV V lE sinh l= + (7.15)
and
( ) ( ) ( )yy yy dsat sat
VE V lE cosh l
x∂
= = +∂
(7.16)
By solving V (y=ΔL) = Vds and Ey (y=ΔL)= Emax, they yield
2
ln 1d dsat d dsat
sat sat
V V V VL llE lE
⎡ ⎤⎛ ⎞− −⎢ ⎥Δ = + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(7.17)
( )2
2max y y = d dsat
satsat
V VE E L ElE
⎛ ⎞−= Δ = +⎜ ⎟
⎝ ⎠ (7.18)
(7.13)
CHAPTER VIII
RESULT AND DISCUSSION 8.1 Introduction
In this chapter, the simulated device measurements from the mathematical
models calibrated to silicon are compared with the experimental/measured data
generated from Intel Proprietary Softwares. This is the case for I-V model which is
gate biased at 0.7 V, 0.8 V, 0.9 V, 1.0 V and 1.2 V. Two average resistivity values
are being employed for evaluation. The short channel NMOS L is 80 nm with the
widths of 1.3 micron. The mobility model has parallel field dependence in the I-V
model to describe the effect of high longitudinal field for velocity saturation effects.
The investigation of velocity saturation is based on the electron drift velocity,
effective mobility and electric field in the VSR. The I-V curve, saturation electron
velocity, effective mobility, effective electric field corresponding to the gate voltage
and drain voltage are presented in normalized value due to a Confidentiality/ Non-
Disclosure Agreement. All of the simulated results are generated using Microsoft
Excel while the experimental data is taken from Intel Proprietary Software for 90 nm
technology process.
8.2 I-V Model Evaluation The measured I-V curve generated by Intel Proprietary Schematic Editor and
Intel Proprietary Circuit Simulator is shown as solid lines in Figure 8.1, Figure 8.2
and Figure 8.3 for gate voltage 0.7 V, 0.8 V, 0.9 V, 1.0 V, 1.1 V and 1.2 V. In order
71
to validate the short channel MOSFET developed for this work, the I-V model
calculated in Section 6.2 is compared to those calculated by a compact I-V model in
Section 6.3. Initially, the analytical short channel model from Eq. (6.10) and Eq.
(6.15) are used to calculate the drain current. However, the model is not able to
predict the drain voltage precisely beyond saturation drain current due to channel
length modulation. Furthermore, drain series resistance is not included in this model
which affects the degree of accuracy. Therefore, a unified MOSFET compact I-V
model from cutoff to saturation has been utilized to fit the graph. In addition to that,
a physically based semi empirical series resistance model is also incorporated with
parasitic resistance, ρ of 3.5x10-5 Ω/cm, 4.5x10-5 Ω/cm and 5.5x10-5 Ω/cm. The
calculated curve shows an excellent agreement with the I-V experimental data across
a wide range of gate voltage in Figure 8.1. The calculated results show that the
lower ρ in Figure 8.1 gives a fairly accurate prediction than Figure 8.2 and 8.3. The
reason for this is that drain current is inversely proportional to the source drain series
resistance. Lower Rsd will give a higher drain current and vice versa. Further analysis
show that there is a slight Ids inconsistency for Vg above 1.0 V from Figure 8.2 and
Figure 8.3 with Vds ranging from 0.8 V-1.5 V.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Drain Voltage (V)
Nor
mal
ized
Dra
in C
urre
nt
VGS = 1.2V
VGS = 1.1V
VGS = 1.0V
VGS = 0.9V
VGS = 0.8V
VGS = 0.7V
Figure 8.1 Comparison of calculated (pattern) versus measured (solid lines) I-V
characteristics for a wide range of gate voltage at resistivity 3.5x10-5 Ω/cm
Measured Data Calculated Data
72
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Drain Voltage (V)
Nor
mal
ized
Dra
in C
urre
nt VGS = 1.2V
VGS = 1.1V
VGS = 1.0V
VGS = 0.9V
VGS = 0.8V
VGS = 0.7V
Figure 8.2 Comparison of calculated I-V characteristics for a wide range of gate
voltage at resistivity 3.5x10-5 Ω/cm (solid lines) and 4.5x10-5 Ω/cm (pattern)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Drain Voltage (V)
Nor
mal
ized
Dra
in C
urre
nt VGS = 1.2V
VGS = 1.1V
VGS = 1.0V
VGS = 0.9V
VGS = 0.8V
VGS = 0.7V
Figure 8.3 Comparison of calculated I-V characteristics for a wide range of gate
voltage at resistivity 4.5x10-5 Ω/cm (solid lines) and 5.5x10-5 Ω/cm (pattern)
4.5e-5 Ω/cm 5.5e-5 Ω/cm
3.5e-5 Ω/cm 4.5e-5 Ω/cm
73
The author also analyze the effect of Channel Length Modulation (CLM) has on the
I-V curve. Notable differences can be found in Figure 8.4 where the saturation drain
current is predicted higher in the saturation region compared to the shaded lines
(without CLM). Since the channel length is modulated by the drain current when it is
in saturation, Idsat increases with increasing Vds. In other word, the deviation of the
drain current from its ideal I-V curve is due to increased average electric field at
decreasing effective channel length. The velocity saturation extracted from these
figure is found to be satisfactory with the default velocity suggested by the process
technology file.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Drain Voltage (V)
Nor
mal
ized
Dra
in C
urre
nt VGS = 1.2V
VGS = 1.1V
VGS = 1.0V
VGS = 0.9V
VGS = 0.8V
VGS = 0.7V
Figure 8.4 Comparison of calculated I-V characteristics for a wide range of gate
voltage at resistivity 3.5e-5 Ω/cm with (pattern marking) and without channel length
modulation (solid lines)
Without CLM With CLM
74
8.3 Threshold Voltage
Short channel effect also alter the threshold voltage characteristic of long
channel MOSFET. The combined effects of reduced gate length and gate width
produce a change in VT. Figure 8.5 show the long channel like threshold voltage
behaviour generated using Eq. (4.14). In the beginning, threshold voltage is negative
as device is lightly doped in which the positive oxide charge and work function
difference is overwhelming.
-0.3
0.0
0.3
0.6
0.9
1.E+16 1.E+17 1.E+18 1.E+19Doping Concentration (cm-3)
Vol
tage
(V)
Threshold Voltage
Figure 8.5 Threshold Voltage without Short Channel Effect (SCE) and Narrow
Width Effect (NWE)
To get a good calculation, two short channel effects on the threshold voltage
of MOSFET namely the short channel voltage shift and narrow gate width effect.
Short channel effect decreases the threshold voltage due to the charge sharing under
gate oxide and can be plotted using Eq. (4.17). The threshold voltage shift decreases
dramatically at higher channel doping. When the channel length approaches the
junction depletion region width of source and drain, a greater portion of the channel
Thr
esho
ld V
olta
ge (V
th)
1x10171x1016 1x10191x1018
75
depletion region begins to overlap the space charge in the junction depletion regions.
As a result, less gate charge is needed to cause inversion in short channel MOSFET.
Hence, a smaller threshold voltage is needed to turn on a short channel device and
changes the ideal threshold voltage as shown in Figure 8.6. An additional effect that
also modifies the expected VT is the narrow width effect and is represented using
Eq. (4.20). Although, this effect is less critical than short channel effect as illustrated
in Figure 8.6, it causes a positive shift for threshold voltage. In Figure 8.7, the effect
of short channel and narrow width effects from becomes more pronounced as the
doping concentration goes beyond 1x1016 cm-3. The final VT in Figure 8.7 is given
by Eq. (4.21).
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19Doping Concentration (cm-3)
Vol
tage
(V)
Short Channel Effect
Narrow Width Effect
Figure 8.6 Short channel and narrow width effect
1x10161x1014 1x10181x10171x1015 1x1019
76
-0.9
-0.6
-0.3
0.0
0.3
0.6
1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19Doping Concentration (cm-3)
Vol
tage
(V)
Threshold Voltage
Figure 8.7 Threshold voltage modification with doping concentration
8.4 Doping Concentration Dependence
The transport of free charge carriers in semiconductor is described by
studying semiconductor’s conductivity via mobility. The mobility is higher for
electrons than holes due to lower effective mass. Many microelectronics applications
favor semiconductor with higher mobility of charge carrier that provides moderate
and high device performance. Scattering is the motion-impending collisions
interaction within the crystal. These collisions can be an electron bumping into
another electron, a hole, with the thermal vibrations of the lattice and with the
ionized impurities atom in a direction dictated by the electric field. The three major
scattering mechanisms that affect the mobility of the carrier are Coulomb, surface
roughness and phonon scatterings. Scattering may increase or decrease with
increasing electric component perpendicular to the direction of current flow and
temperature. These scattering limits the mobility at certain portion along the
transverse electric field. For example, the mobility drop or roll-off of the effective
mobility in the low field region as shown in Figure 8.8 is attributed by the Coulomb
Thr
esho
ld V
olta
ge (V
th)
1x1015 1x10171x10161x1014 1x1018 1x10191x1013
77
scattering while the mobility degradation is contributed by the surface scattering.
Eq. (5.8) is employed to plot the mobility curve.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00E+04 1.00E+05 1.00E+06 1.00E+07Effective Transverse Electric Field (V/cm)
Nor
mal
ized
Eff
ectiv
e M
obili
ty
Na = 1.0 E+18cm-3
Na = 1.0 E+16 cm-3
Na = 1.0 E+17 cm-3
Na = 1.0 E+15 cm-3T = 300K
Figure 8.8 Normalized effective mobility versus transverse electric field at
different doping concentration
Although each of the individual mobility curves for the four doping
concentration in Figure 8.8 has different initial mobility, they formed a unique
universal mobility curve versus transverse electric field. Coulomb scattering
considerably degrades the mobility on higher impurity concentration as the deviation
from the universal curve becomes larger. It is caused by the Coulomb interaction
between the carrier in the inversion layer and the interface charges localized in SiO2.
When the doping concentration is low, the mobility is higher as Coulomb scattering
is lower. The mobility of the carriers begins to decrease gradually when
concentration of the dopants increases. Consequently, there are more carriers to
bump into and scattering becomes more frequent. Therefore, the Coulomb scattering
equation described in the mobility model includes both scattering caused by fixed
interface charge and doping impurities for better modeling. As the channel inversion
1x106 1x104 1x107 1x105
NA = 1x1015 cm-3
NA = 1x1016 cm-3
NA = 1x1018 cm-3
NA = 1x1017 cm-3
T = 300 K
78
increases, the number of impurities and charge which are able to interact with the
carriers decreases. This can be ascribed to the carrier screening effect which is less
effective for a heavily inverted surface. The velocity saturation dependence on
doping concentration is depicted in Figure 8.9 and determined by using Eq. (7.2).
0.0
0.2
0.4
0.6
0.8
1.0
1.0E+03 1.0E+04 1.0E+05 1.0E+06Effective Longitudinal Electric Field (V/cm)
Nor
mal
ized
Dri
ft V
eloc
ity
1.0e15cm-3
1.0e16cm-3
1.0e17cm-3
1.0e18cm-3
1.0 E+15 cm-3
1.0 E+16 cm-3
1.0 E+17 cm-3
1.0 E+18 cm-3
T=300K
Figure 8.9 Normalized drift velocity versus longitudinal electric field at different
doping concentration
The drift velocity as a function of the longitudinal field for different doping
concentrations are essentially similar due to the domination of surface roughness
scattering at higher field. Surface-roughness scattering within the inversion layer is a
function of the effective field independent of the doping concentration. When drift
velocity is plotted as a function of the longitudinal field for different doping
concentrations of 1015 cm-3, 1016 cm-3, 1017 cm-3 and 1018 cm-3 as shown in Figure
8.9, they are essentially similar due to the limitation of surface roughness scattering.
At first glance, doping concentration does not have a significant influence on drift
velocity at high field as they overlap each other. This is due to the fact that the
mobility for different doping concentrations is taken from the mobility curve in
1x105 1x103 1x106 1x104
1x1015 cm-3
1x1016 cm-3
1x1018 cm-3
1x 1017 cm-3
T = 300 K
79
Figure 8.8 at a fixed transverse field. In reality, the transverse field at a given gate
bias for different doping concentrations will be different due to the different VT at
different substrate doping concentration. For the same gate bias, the higher doped
substrate will experience a higher transverse field, and lower mobility at the high
fields experienced when the device is on. However, this also means that the critical
field will be higher for higher doped substrates and higher doped substrate will reach
saturated velocity later. The saturation field, Esat is chosen to be twice of the critical
field, 2Ec as it has been the range of vertical field in MOSFET device reported in the
literature.
8.5 Temperature Dependence
Normalized effective mobility versus longitudinal electric field at 77 K,
300 K, 400 K and 450 K is illustrated in Figure 8.10 by utilizing Eq. (5.8). At low
temperature of 77 K, saturation velocity is substantially influence by surface
roughness scattering. Devices at low temperature have a higher drift velocity as a
result of a high effective mobility as illustrated in Figure 8.10. However, as
temperature is increase to 300K and above, the calculated mobility shows that
scattering mechanism transit to phonon scattering for moderately doped devices
biased at moderate vertical field. The drift velocity curve for 77 K, 300 K, 400 K and
430 K is depicted in Figure 8.11 by employing Eq. (7.2).
At 77 K, drift velocity reaches saturation earlier compared to 300 K. This is
then followed by 400 K and 430 K. Velocity saturation is achieved for all four
temperatures at 1x106 V/cm onwards. In short, temperature plays a vital role in
device operation. On one hand, mobility increases inversely proportional temperature
due increased phonon scattering at higher temperature. On the other hand, drift
velocity increases proportionally with mobility only to be limited by velocity
saturation. Critical longitudinal electric field also increases with temperature. The
finding shows that the operation of the MOSFET revolves at high field beyond
1x106 V/cm for both Ex and Ey as a result of a very short L at 80nm.
80
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0E+04 1.0E+05 1.0E+06 1.0E+07Effective Transverse Electric Field (V/cm)
Nor
mal
ized
Eff
ectiv
e M
obili
ty70K
300K
400K
450K
Figure 8.10 Normalized effective mobility versus transverse electric field at
different temperature
Figure 8.11 Normalized drift velocity versus longitudinal electric field at different
temperature
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Effective Longitudinal Electric Field (V/cm)
Nor
mal
ized
Dri
ft V
eloc
ity
77K
300K
400K
430K
Gate Voltage = 1.1V
70 K
300 K
400 K 450 K
1x106 1x104 1x107 1x105
T = 70 K
T = 300 K
T = 400 K
T = 450 K
T = 300 K
T = 400 K
T = 450 K
T = 70 K
1.2 V
1x105 1x103 1x107 1x104 1x106 1x108
81
8.6 Peak Field At The Drain
Figure 8.12 illustrates the normalized longitudinal electric field along the channel
position for drain voltage of 1.2 V. The strength of the longitudinal electric field is
determined by the device VT, source and drain series resistance and L as expressed in
Eq. (7.4). Shorter channel length yield larger Ey and correspondingly higher velocity.
By using Eq. (7.16), the longitudinal electric field versus L is plotted. It has a linear
relationship with the channel up to Esat where Vdsat occurred. From this point
onward, the electric field grows exponentially to Emax near the drain end in the
velocity saturation region. Emax is calculated using Eq. (7.18). Under the influence of
saturation velocity, electron drift velocity becomes constant and independent of the
effective longitudinal field, Ey. In short channel devices, the saturation of drain
currents occur at a much lower voltage due to velocity saturation. The Idsat is
approximately linear with gate overdrive, instead of having a square law dependence
predicted in long channel device in Figure 8.13. The saturation drain current is
equally spaced voltage against drain voltage at different gate voltages as shown in
Figure 8.14. The value of Idsat is determined by using Eq. (6.26) and Eq. (6.33).
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05Channel Position y (m)
Nor
mal
ized
Lon
gitu
dina
l Ele
ctri
c Fi
eld
Esat
EmaxVds = 1.2V
Figure 8.12 Normalized longitudinal electric field along the channel
0x100 2x10-6 4x10-6 6x10-6 8x10-6 1x10-5
Emax 1.2 V
82
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Gate Overdrive VG-VT (V)
Nor
mal
ized
Sat
urat
ion
Dra
in C
urre
nt
Doping Concentration: 1 E+16 cm-3
Figure 8.13 Normalized saturation drain current versus gate overdrive
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6Drain Voltage Vd (V)
Nor
mal
ized
Sat
urat
ion
Dra
in C
urre
nt
Doping Concentration: 1 E+16 cm-3
Vgs = 0.7 V
Vgs = 1.1 V
Vgs = 0.8 V
Vgs = 1.0 V
Vgs = 0.6 V
Vgs = 0.9 V
Vgs = 0.5 V
Vgs = 1.2 V
Figure 8.14 Normalized saturation drain current versus drain voltage
1x1016 cm-3
1x1016 cm-3
CHAPTER IX
CONCLUSIONS AND RECOMMENDATIONS 9.1 Summary and Conclusions
Velocity saturation is caused by the increased scattering rate of energetic
electrons, particularly optical phonon emission resulting in the degrading effect of
the carrier mobility. Under the influence of saturation velocity, electron drift velocity
becomes constant and independent of the effective longitudinal field. Velocity
saturation becomes more pronounced in short channel devices as they operate at high
electric field. It yields smaller drain saturation current and voltage value than that
predicted by the ideal long channel relation. The capability of long channel models is
limited as they are unable to provide an accurate representation of drain current
model for short channel devices due to a lack of physicals component consideration.
Initially, the work done by Shin (1991) is limited to effective mobility. He
studied a variation of effective mobility versus temperature, doping concentration
and longitudinal electric field. On the other hand, the investigation of the effect of
the velocity saturation by Takeuchi and Fukuma (1994) is carried out only for drain
voltage dependence and scaling. Hence, this research extends the work into drift
velocity versus longitudinal electric field for a variation of temperature, substrate
doping concentration and longitudinal electric field in nanoscale MOSFET.
84
Several newly developed models are modified and employed to analyze the
characteristics and behaviors of transistor in sub-100 nm. They are the threshold
voltage model, physically based model for effective mobility and compact I-V,
velocity field model and source drain resistance model. It is vital to investigate the
physical insight into the device’s operating principles at high field to diminish hot
carrier effect and avoid catastrophic breakdown of the device via punchthrough. The
abovementioned models are improvement upon existing long channel models and
include various effects discussed in previous chapters.
Intel Proprietary Schematic Editor and Circuit Simulator are used to generate
experimental data. The relevant parameters are then extracted from the I-V curve.
Finally, the modified models are utilized to characterize the effects of velocity
saturation and effective mobility on drain voltage, gate voltage, temperature and
doping concentration. Simulated results show that the temperature effect on velocity
saturation depends on the predominant scattering mode. It is shown that at high fields
where surface roughness is predominant, the mobility has a negative temperature
coefficient and the velocity saturates at lower temperatures. Substrate doping
concentration changes the transverse field and scattering modes for a given gate bias,
and hence the saturation velocity is achieved at different longitudinal fields
9.2 Recommendations for Future Work
(i) The VT model can be modified to account for nonuniform channel
doping profile. When the MOSFET channel is nonuniformly doped,
designer has the flexibility to tailor the desired threshold voltage and
the off current requirement. Among the advanced MOSFET variation
is threshold voltage adjustment implantation, punchthrough
implantation and halo implantation.
(ii) The electron mobility model can be extended into strained and doped
silicon transport. The presence of a Ge alloy in strained silicon has
additional implications for the effective mobility over the universal
85
mobility curve. Alloy scattering can be added describe the carrier
collision with the SiGe alloy substrate.
(iii) The implication of velocity saturation effects on PMOS can also be
examined. With the simulated results, research can be carried out in
order to observe the hot carrier effects focusing on CMOS in
processor particularly at high temperature operation.
(iv) By using PSpice Model Editor, the derived model can be exported to
the library and configured to simulate the design output for DC
analysis. Instead of using Microsoft Excel, the analytical and semi
empirical models can be redefined into SPICE models and added into
a circuit as a schematic symbol to perform basic simulation.
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APPENDICES
93
APPENDIX A
PUBLICATION A1 Michael Tan Loong Peng, Razali Ismail, Ravisangar Muniandy and Wong Vee Kin. (2005). Velocity Saturation Dependence on Temperature, Substrate Doping Concentration and Longitudinal Electric Field in Nanoscale MOSFET. Proceedings IEEE National Symposium on Microelectronics. pp. 210-214
94
Velocity Saturation Dependence on Temperature, Substrate Doping Concentration and Longitudinal Electric Field in
Nanoscale MOSFET
Michael Tan Loong Peng1, Student Member, IEEE, Razali Ismail1, Ravisangar Muniandy2 and Wong Vee Kin2
1Department of Microelectronics and Computer Engineering,
Universiti Teknologi Malaysia, 81300 Skudai, Johor, MALAYSIA Email: [email protected], [email protected]
2Penang Design Center (PG12), Intel Microelectronics (M) Sdn. Bhd.,
Bayan Lepas Free Industrial Zone, 11900 Penang, MALAYSIA Email: [email protected], [email protected]
Abstract – This paper reports the drift and velocity saturation dependence on temperature, substrate doping concentration and longitudinal electric field for n-MOSFET. This study observes the velocity in response to the variation of above input parameters. An existing current-voltage (I-V) compact model is utilized and modified by appending a simplified threshold voltage derivation and a more precise carrier mobility model. The threshold voltage derivation provides similar accuracy when compared to actual devices and comprises major short channel effect as well as narrow width effect. The compact model also includes a semi empirical source drain series resistance modeling. The models show good agreement with the experimental data over a wide range of gate and drain bias.
I. INTRODUCTION VELOCITY saturation occurs in high electric fields when the drift velocity of electron in the inversion channel reaches a limiting value, at 107 cm/s due to mobility degradation. Linear velocity field relationship is applicable as long as the electric field is low. Here, Ohm Law’s is still valid. Electrons in nanoelectronic devices are being driven by a tremendously high electric field [1] and can go up to 106 V/cm. At high critical electric field, the average carrier energy and scattering rate of highly energetic electrons increases. Under these circumstances, the carrier loses their energy by optical-phonon emission nearly as fast as they acquire from the field. This resulted in mobility degradation. Second order effects (SOE) in 50-100nm generation of ultra
submicron transistor designs such as velocity saturation, short channel effect (SCE), channel length modulation (CLM) and narrow width effect (NWE) is increasingly becoming a great concern for designers [2]. Velocity saturation deteriorates the drive strength in the saturation velocity region. SOE could be ignored in previous long channel CMOS generation as the effects are not that significant. Nevertheless, these undesirable effects are inevitable when devices are gradually scaled down to gain higher speed, performance and chip density [3]. Many research have been carried out [4-5] for the development of semi empirical model for nanoscale MOSFET under the influence of velocity saturation. More investigation has to be done to examine the velocity saturation dependence on transverse electric field, substrate doping concentration and temperature. This analysis is crucial when taking into account MOSFET scaling on device performance. In this paper, the impact of velocity saturation, on the parameters mentioned above is presented. The paper is organized as follows. We began with a survey of studies on limitation of scaling of MOSFET and effect of the velocity saturation [6]. Section II provides further description of the employed models. Firstly, a simple semi empirical threshold voltage is derived for nanoscale MOSFET [7]. It is then incorporated with a modified semi empirical I-V model by considering a more accurate description of mobility model [8]. Finally, a velocity field model is used to analyze the carrier drift velocity in the channel. Result are presented and discussed in Section III. Section IV concludes the study.
95
II. MODEL AND METHODS
A. The Threshold Voltage Model The threshold voltage, VT is one of the key parameters in MOSFET design and modeling to predict the device performance. There are various kinds of definitions and extraction methods proposed [9], each with a focus on different aspects. This model formulation starts with the modeling of threshold voltage for n-channel MOSFET with n+ polysilicon gate and p-type silicon substrate. The model assumes that there is no substrate bias effect. The analytical definition of the threshold voltage is
( )2 2 2T f FB Si A f oxV V qN Cφ ε φ= + + (1)
where φf is the Fermi potential, NA is the substrate doping concentration, k is the Boltzmann’s constant, is the Cox gate oxide capacitance, VFB is the flat band voltage, εSi is the dielectric permittivity of the silicon. Next, SCE which decreases the MOSFET threshold voltage as the channel length is reduced is included as follow:
21 1jdT dTT for sc
ox j
rqNx xVC L r
⎡ ⎤Δ =− + −⎢ ⎥
⎢ ⎥⎣ ⎦ (2)
where xdT is the lateral space charge width and rj is the diffused junction. In addition, there is also a shift in threshold voltage in the positive direction for the devices due to a narrow-width effect.
dT dTT for nw
ox
qNx xVC W
ξ⎡ ⎤Δ = ⎢ ⎥⎣ ⎦ (3)
where ξ is the empirical parameter that account for the shape of the fringe depletion region and q is the electronic charge. Hence, the final threshold voltage expression is as shown
( )2 22
fT fp FB
ox
qNV V
C
ε φφ= + ±
21 1jdT dT dT
ox j
rqNx x xC L r W
ξ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟− + − − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦ (4)
B. The Mobility Model The outcome of this model will depend primarily on the original mobility model developed by Schwarz and Russek described by [10] with some improvement on the scattering mechanism. The electron effective mobility due to the vertical field [11] is modeled semi-empirically incorporating three basic scatterings mechanisms in the inversion layer is namely, Coulombic scattering due to doping concentration, phonon scattering and surface roughness scattering as shown below
1
1 1 1eff
ph sr c
μμ μ μ
−⎡ ⎤
= + +⎢ ⎥⎢ ⎥⎣ ⎦
(5)
Matthiessen’s rule is used to calculate each different scattering contribution to the effective mobility. The effective mobility is a function of the effective transverse electric field, E⊥eff [12] and is defined by the following equations
1 1( ) ( )6eff inv B gs t
si oxE Q Q V V
tη
ε⊥ = + = + (6)
where η is ½ for <100> electrons. From (6), it can be seen that the effective electric field is proportional to the channel inversion charge, Qinv and depletion charge, QB. It can be written as a function of gate overdrive and gate oxide thickness, tox for as long as Vgs > Vt. The model developed by [13] has been used. The mobility due to phonon and fixed interface charge scattering, μph is
111
2.5 19 23.2x10B
phn n
zT pT
μμ
−−−
− −
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟= +⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
(7)
while the surface roughness scattering, μsr is defined as 2
sr sr effK Eμ −⊥= (8)
and coulomb scattering, μc can be expressed as
21 1.5
22
2
1.1x10
ln (1 )1
nc
BHBH A
BH
T
Nμ
γγγ
−
=⎡ ⎤
+ −⎢ ⎥+⎣ ⎦
(9)
96
where p is the Fuchs factor, z is the average inversion layer width including quantum channel broadening effects. In the above Tn is the normalized temperature at 300K. The mobile inversion layer charge density is given as 02 ( )inv I si effQ qN E Eε ⊥= = + (10) and depletion region charge density is 2 2B A d A si FQ q N x q N ε φ= ⋅ ⋅ = ⋅ ⋅ ⋅ (11) C. The I-V model The I-V model is much simpler than the BSIM3v3 expression and taken from the compact model given by Xing Zhou et al. [14]. The final complete Ids model includes the effect of CLM and Rs/d and is given by
( )1
deffds
s / d deff deff
II
R I V=
+ (12)
where Vdeff is the smoothing function to replace Vds for a smooth transition from linear to saturation region while Ideff is the smoothing function for Ids0iwith effective early voltage. A physically-based source/drain (S/D) series resistance, Rs/d model for deep submicron MOSFET is included comprising a bias-dependent (intrinsic) component and a bias-independent (extrinsic) to give a precise reading [15]. The final S/D series resistance is given by
2s / d ext int
j GS T
SR R Rx W V Vρ υ
= + = +−
(13)
D. The Velocity-Field Model The electron velocity-longitudinal field relationship in the inversion layer given in [16] is described as
( )
1
1
eff yd nn
y c
Ev
E E
μ=
⎡ ⎤+⎢ ⎥⎣ ⎦
(14)
with n=2 for electrons, Ec is the critical electric field for velocity saturation defined by
satc
eff
vEμ
= (15)
and Ey is the longitudinal electric field. Ey is given by
d s / dy
V IREL
−= (16)
When E >> 2Ec, vd becomes μeffEc. Quasi two dimensional approach by Ko [18] is applied here to model the velocity saturation region (VSR).
III. RESULTS AND DISCUSSION
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08Effective Longitudinal Electric Field (V/cm)
Nor
mal
ized
Dri
ft V
eloc
ity
77K
300K
400K
430K
Gate Voltage = 1.1V
Fig. 1 Normalized υd versus Ey for different temperature as indicated. The saturation electron velocity, effective mobility, effective electric field corresponding to the gate voltage and drain voltage are summarized here. Under the influence of saturation velocity, electron drift velocity becomes constant and independent of the effective longitudinal field, Ey. The saturation field, Esat is chosen to be twice of the critical field, 2Ec as it has been the range of vertical field in MOSFET device reported in literature [17]. In short channel devices, the saturation of drain currents occur at a much lower voltage due to velocity saturation. It can be concluded that for a higher vertical field, the effective mobility is lower but the critical field becomes higher as shown in (15).
At low temperature at 77K, saturation velocity is substantially influence by surface roughness scattering. Devices at low temperature have a higher drift velocity as a result of a high effective mobility as illustrated in Figure 2. However, as temperature is increase to 300K and above, the extracted mobility show that scattering mechanism transit to phonon scattering for moderately doped devices biased at moderate vertical field. The drift velocity curve for 300K, 400K and 430K appear lower compared to 77K as depicted in Figure 1 but
97
velocity saturation is achieved for all four temperature at 1 x106 V/cm onwards.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00E+04 1.00E+05 1.00E+06 1.00E+07Effective Transverse Electric Field (V/cm)
Nor
mal
ized
Eff
ectiv
e M
obili
ty
Na = 1.0 E+18cm-3
Na = 1.0 E+16 cm-3
Na = 1.0 E+17 cm-3
Na = 1.0 E+15 cm-3
Fig. 2 Normalized μeff versus E⊥eff for different doping concentration. (NA=1015cm-3, 1016cm-
3, 1017cm-3 and 1018cm-3)
0.00
0.20
0.40
0.60
0.80
1.00
1.0E+03 1.0E+04 1.0E+05 1.0E+06Effective Longitudinal Electric Field (V/cm)
Nor
mal
ized
Dri
ft V
eloc
ity
1.0e15cm-3
1.0e16cm-3
1.0e17cm-3
1.0e18cm-3
1.0 E+15 cm-3
1.0 E+16 cm-3
1.0 E+17 cm-3
1.0 E+18 cm-3
Fig. 3 Plot of the normalized υd dependence of Ey for four doping level At a high enough transverse field, the mobility for different doping concentrations are taken from the mobility curve in Figure 2. If we plot the drift velocity as a function of the longitudinal field for different doping concentrations as shown in Figure 3, they are essentially similar due to surface roughness scattering as discussed in [18]. In reality, the transverse field at a given gate bias for different doping concentrations will be different due to the different VT at different substrate doping concentration; for the same gate bias, the higher doped substrate will experience a higher transverse field, and lower mobility at the high fields experienced when the device is ON. However, this also means that the critical field will be higher for higher doped substrates and higher doped substrate will reach saturated velocity later. Conventional deep submicron devices use halo implantation to control short channel effects and this has the undesired effect of raising the substrate concentration in the
channel, thus, making the velocity saturation effects less pronounced. Nevertheless, the drive current of the transistor still suffers from the reduced transconductance.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05Channel Position y (m)
Nor
mal
ized
Lon
gitu
dina
l Ele
ctri
c Fi
eld
Esat
EmaxVds = 1.2V
Fig. 4 Normalized Ey along the channel and illustration of velocity saturation region.
Figure 4 illustrates the normalized Ey along the channel position for Vds=1.2V. The strength of the longitudinal electric field is determined by the device VT, source/drain series resistance and L as expressed in (16). Shorter L will yield larger Ey and correspondingly higher velocity. Ey shown here has a linear relationship with the channel up to Esat or where Vdsat occurred. From here onward, the velocity saturation region begins and rises exponentially to Emax near the drain end.
IV. CONCLUSION
In this paper, the VT model, μeff model, I-V
model and the Rd/s model are employed to investigate the velocity saturation in nanoscale MOSFET. Major physical mechanisms contributing to velocity saturation have been considered, discussed and analyzed. The temperature effect on velocity saturation depends on the predominant scattering mode. We have shown that at high fields where surface roughness is predominant, the mobility has a negative temperature coefficient and the velocity saturates at lower temperatures. Substrate doping changes the transverse field and scattering modes for a given gate bias, and hence the saturation velocity is achieved at different longitudinal fields. For digital circuit designers, these effects of saturation velocity moves the regions of operation for the MOSFET, and can be compensated by increasing the drive current using novel process technologies such as using strained silicon
98
ACKNOWLEDGEMENT
The authors would like to thank Intel Penang Design Center (PDC) for providing the experimental data and simulation tools for this work. The authors are also indebted to Prof Vijay K. Arora who gave them continuous encouragement during the study of saturation velocity. Fruitful discussions on the modeling with Dr. Kelvin Kwa Sian Kiat, Lock Choon Hou, Kaw Kiam Leong, Alvin Goh Shing Chye and Lim Kiang Leng are greatly appreciated.
REFERENCE
[1] V. K. Arora, “High-Field Effects in Sub-
Micron Devices,” IEEE Proceedings Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD), pp. 33-40 (2000).
[2] Y. Taur, D. A. Buchanan, W. Chen, D. J. Frank, K. E. Ismail, S.-H. Lo, G. A. Sai-Halasz, R. G. Viswanathan, H.-J. C. Wann, S. J. Wind, and H.-S. Wong, “CMOS Scaling into the Nanometer Regime,” Proceedings of the IEEE, vol. 85, pp. 486-504 (1997).
[3] Y. Taur, “CMOS Design Near the Limit of Scaling,” IBM J. Res. & Dev., vol. 46, no. 2/3, pp. 213-222 (2002).
[4] J. B. Roldan, F. Gamiz, J.A. Lopez-Villanueva, and J.E Carceller, “Modeling Effects of Electron-Velocity Overshoot in a MOSFET,” IEEE Transactions on Electron Devices, vol. 44, no. 5, pp 841-846 (1997).
[5] M. Pattanaik and S. Banerjee, “A new approach to model I-V characteristics for nanoscale MOSFETs,” VLSI Technology, Systems, and Applications 2003 International Symposium, pp. 92-95 (2003).
[6] A. Lochtefeld, I. J. Djomehri, G. Samudra and D. A. Antoniadis, “New insights into carrier transport in n-MOSFETs,” IBM J. Res. & Dev., vol. 46, no. 2/3, pp. 347-358 (2002).
[7] R. Wang, J. Dunkley, T. A. DeMassa and L.F. Jelsma, “Threshold Voltage Variations with Temperature in MOS Transistors,” IEEE Trans. Electron Dev, vol. 18, no. 6, pp. 386-388 (1971).
[8] J. R. Hauser, “Extraction of experimental mobility data for MOS devices,” IEEE Transactions Electron Devices, vol. 43, no. 11, pp. 1981-1988 (1996).
[9] X. Zhou, K. Y. Lim, D. Lim, “A simple and unambiguous definition of threshold voltage and its implications in deep-submicron MOS device modeling,” IEEE Trans. Electron Devices, vol. 46, no. 4, pp. 807-809 (1999).
[10] S. A. Schwarz and S. E. Russek, “Semi-empirical equations for electron velocity in silicon: Part II—MOS inversion layer,” IEEE Trans. Electron Devices, vol. 30, no. 12, pp. 1634-1639 (1983).
[11] K. Y. Lim and X. Zhou, “A physically-based semi-empirical effective mobility model for MOSFET compact I–V modeling,” Solid-State Electron, vol. 45, no. 1, pp. 193-197 (2001).
[12] A. G. Sabnis and J. T. Clemens, “Characterization of the electron mobility in the inverted <100> Si surface,” IEDM Tech. Dig, pp. 18-21 (1979).
[13] H. Shin, G. M. Yeric, A. F. Tasch, and C. M. Maziar, “Physically-based Models for Effective Mobility and Local Field Mobility of Electrons in MOS Inversion Layers,” Solid-State Electron, vol. 34, no 6, pp. 545-552 (1991)
[14] X. Zhou and K. Y. Lim, “Unified MOSFET compact I-V model formulation through physics-based effective transformation,” IEEE Transactions on Electron Devices, vol. 48, no 5, pp. 887-896 (2001).
[15] X. Zhou and K. Y. Lim, “Physically-based semi-empirical series resistance model for deep-submicron MOSFET I-V modeling,” IEEE Transactions on Electron Devices, vol. 47, no. 6, pp.1300-1302 (2000).
[16] F. Assaderaghi, P.K. Ko, C. Hu, “Observation of velocity overshoot in silicon inversion layers,” IEEE Electron Device Letters, vol. 14, no 10, pp. 484-486 (1993).
[17] P. K. Ko, “Approaches to scaling,” Advanced MOS Device Physics, vol. 18, no. 1, pp. 1-37 (1989).
[18] F. Gamiz, J. A. Lopez-Villanueva, J. Banqueri, J. E. Carceller, P. Cartujo, “Universality of electron mobility curves in MOSFETs: A Monte Carlo study”, IEEE Transactions on Electron Devices, vol. 42, no. 2, pp. 258-265 (1995).
99
APPENDIX B
ROADMAP OF SEMICONDUCTOR SINCE 1977
Past Trend of MOSFET Scaling (Hu, 1995)
MOSFET Technology Projection (Hu, 1995)
100
APPENDIX C
ROADMAP OF SEMICONDUCTOR INTO 32NM PROCESS
TECHNOLOGY
High Performance Logic Technology Requirement (ITRS, 2003)
101
APPENDIX D
CMOS LAYOUT DESIGN
Schematic cross section of a planar PMOS and NMOS (Zeitzoff and Chung, 2005)
102
APPENDIX E
MOBILITY IN STRAINED AND UNSTRAINED SILICON
Electron mobility enhancement in strained Si MOSFET. Electron enhancement of
∼1.8 x persist up to high Eeff (∼ 1 MV/cm). Strained Si allows “moving off” of the
universal mobility curve. (Zeitzoff and Chung, 2005)
103
APPENDIX F
THRESHOLD VOLTAGE MODEL
F.1 Modified Threshold Voltage Model
cN , density of states in conduction band
( )
32 2
32
-31 -23 32
2-34
315 32
2 2 /
2 x 1.08 x 9.11 x 10 x1.38 x 10 /26.625x 10
5.42 x10
pm kT h
kg J K TJ S
T cm
π
π
−
⎡ ⎤⎣ ⎦
⎡ ⎤⎢ ⎥≅⎢ ⎥⎡ ⎤⋅⎣ ⎦⎣ ⎦
≅
vN , density of states in valence band
( )
32 2
32
-31 -23 32
2-34
315 32
2 2 /
2 x 0.81 x 9.11 x 10 x1.38 x 10 /26.625x 10
3.52 x10
pm kT h
kg J K TJ S
T cm
π
π
−
⎡ ⎤⎣ ⎦
⎡ ⎤⎢ ⎥≅⎢ ⎥⎡ ⎤⋅⎣ ⎦⎣ ⎦
≅
104
Eq. (4.1) can expanded by using Eq. (4.2) and Eq. (4.13). It yield
( )
( )
( )
( )
2 22
2 22
2 22
2
2 2
2
si A fT fp FB
ox
si A ftotfp ms
ox ox
si A fg totfp fp
ox ox
si A f g totfp
ox ox
qNV V
C
qNQC C
qNE Qe C C
qN E QC e C
ε φφ
ε φφ φ
ε φφ φ
ε φφ
= + +
⎛ ⎞= + − +⎜ ⎟
⎝ ⎠
⎛ ⎞= − + − +⎜ ⎟
⎝ ⎠
⎛ ⎞= + − +⎜ ⎟
⎝ ⎠
The final threshold voltage expression can be approximated as
( ) ( )
( ) ( ) ( )
T Short Channel T Long Channel T
T Long Channel T Short Channel Effects T Narrow Width Effects
V V V
V V V
= + Δ
= + +
( )
( )
2 2 22 1 1
2 2 22 1 14
2
si A fp jA dT dT A dT dTfp FB
ox ox j ox
si A fp jdT dTAfp FB dT
ox si fp ox j
A dT dT
ox
A dTfp FB
o
qN rqN x x qN x xVC C L r C W
qN rqNx xqNV xC C L r
qN x xC W
qN xVC
ε φ ξφ
ε φφ
ε φ
ξ
φ
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= + + − + − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞⎢ ⎥= + + ⋅ ⋅ − + −⎜ ⎟
⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞+ ⎜ ⎟⎝ ⎠
= + +21 1
22 1 1 1
jA dT dT A dT dT
x ox j ox
jA dT dT dTfp FB
ox j
rqN x x qN x xC L r C W
rqN x x xVC L r W
ξ
ξφ
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− + − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎛ ⎞⎢ ⎥= + + − + − +⎜ ⎟⎨ ⎬⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
105
APPENDIX G
I-V MODEL G.1 Modified I-V Model
The total current at a point y along the inversion layer can be represented as
( )
( ) ( )( )
( )( )
( ) ( )( )
1
1 1
1 1
1 1
eff yds v d v i
y c
v i effc
s effc
ox GT effc
EI n qv A n q zx
E E
V yn x q zV y E
V yn q zV y E
V yC V V y zV y E
μ
μ
μ
μ
⎛ ⎞= = ⋅⎜ ⎟⎜ ⎟+⎝ ⎠
⎛ ⎞∂ ∂= ⋅ ⋅ ⎜ ⎟⎜ ⎟+ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂= ⋅ ⋅⎜ ⎟⎜ ⎟+ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂⎡ ⎤= − ⋅ ⋅⎜ ⎟⎣ ⎦ ⎜ ⎟+ ∂ ∂⎝ ⎠
where .s v in n x= The integration of Eq. (6.10) from 0 to L yield
( )( ) ( ) ( )
( )
[ ] ( )( )
[ ]
0 0 0
2
2
1 1
1
. 2
. 2
ds D
ds c ox eff GT
L V V
ds c ox eff GT
ds ds c ox eff GT ds ds
ox eff GT ds dsds
ds c
I V y E y C z V V y V y y
I y E V C z V V y V
I L L V LE C z V V V
C z V V VI
L L V LE
μ
μ
μ
μ
⎡ ⎤ ⎡ ⎤+ ∂ ∂ ∂ = − ⋅ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
⎡ ⎤∂ + ∂ = − ∂⎣ ⎦
+ = −
−=
+
∫ ∫∫ ∫ ∫
106
Vdsat in quadratic form is obtained by substituting Ids from Eq (6.15) into Eq. (6.16).
( ) ( )
( )
2
2
22
2 1 2
eff oxGT dsat dsat ox GT dsat sat
sat dsatGT dsat GT dsat
eff c
C W V V V C V V v WL
v L VV V V VV
μ
μ
⋅ ⋅ − = −
⎛ ⎞− + = −⎜ ⎟
⎝ ⎠
2
2
2 2 2 2 0
2 2 0GT C dsat C GT C dsat
dsat GT C dsat C
V V V V V V V
V V V V V
− + − =
− + =
Vdsat in Eq. (6.18) can be approximated when VGT >> Vc and Vc is small to be
1 2 1
2
2
GTdsat C
C
GTC
C
GT C
VV VV
VVV
V V
⎛ ⎞= + −⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞
≈ ⎜ ⎟⎜ ⎟⎝ ⎠
≈
2dsatV is obtained by rearranging Eq (6.17)
( )
2
2
2
2 2 0
2 2
2
dsat GT C dsat C
dsat GT C dsat C
dsat C GT dsat
V V V V V
V V V V V
V V V V
− + =
= −
= −
The short channel saturation drain current is given as
( )
( )
212
22
dsat s sat
effox GT dsat c
effox dsat
effox GT C
ox sat GT
I n qv W
C W V V VL
C W VL
C W V VL
C v W V
μ
μ
μ
=
= −
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
=
107
When Rsd is calculated, the expression Ids0 in Eq. (6.31) becomes
( )
0
0
0
0
0
1
1
1
deffds
ch sd
deff ch
sd ch
ds
sd deff ds
ds
ds sd deff
VI
R RV R
R RI
R V I
II R V
=+
=+
=+
=+
