Silicon Devices and Process Integration pot

xiv List of SymbolsCGCh gate to channel capacitance per unit area F/cm2 CGD gate to drain capacitance F CGS gate to source capacitance F CHF high-frequency capacitance per unit area, CV

Silicon Devices and Process Integration Deep Submicron and Nano-Scale Technologies

Badih El-Kareh

Silicon Devices

and Process Integration

Deep Submicron and Nano-Scale Technologies

ABC

° Springer Science+Business Media, LLC 2009

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,

NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,

or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject

to proprietary rights.

Printed on acid-free paper

springer.com

State-of-the-art silicon devices and integrated process technologies are covered inthis book The eight chapters represent a comprehensive discussion of modernsilicon devices, their characteristics, and the relationship between their electricalproperties and processing conditions The material is compiled from industrial andacademic lecture-notes and reflects years of experience in the development of sili-con devices

The book is prepared specifically for engineers and scientists in tor research, development and manufacturing It is also suitable for a one-semestercourse in electrical engineering and materials science at the upper undergraduate orlower graduate level

semiconduc-The chapters are arranged logically, beginning with a review of silicon propertiesthat lays the groundwork for the discussion of device properties, including mobility-enhancement by straining silicon

Junctions and contacts are inherent to practically all semiconductor devices.Chapter 2 covers junctions under forward and reverse characteristics, includinghigh-level injection and high-field effects Understanding the properties of contactshas become increasingly important as the contact size is reduced to deep submicronand nanoscale dimensions The last part of Chap 2 discusses ohmic and rectifyingcontacts

Chapter 3 begins with bipolar fundamentals and moves to an advanced ment of bipolar enhancements with silicon–germanium (SiGe) This chapter is par-ticularly important to analog and mixed-signal applications where complementarymetal-oxide semiconductor (CMOS) and bipolar transistors are integrated in a BiC-MOS process It also benefits engineers in understanding important bipolar effects

treat-in CMOS-only applications, such as subthreshold current and parasitic latch-up.The metal-oxide silicon (MOS) capacitor is a key part of a metal-oxide semicon-ductor field-effect transistor (MOSFET) and a powerful process and device charac-terization tool The physics and characterization of MOS structures are detailed inChap 4, beginning with an ideal stack of a conductor, an insulator and silicon, andgradually moving to real structures and quantum effects

v

vi PrefaceChapter 5 deals with the insulated-gate field-effect transistor It begins with adescription of the modes of transistor operation and the different transistor types.Transistor current–voltage characteristics are detailed, followed by a discussion ofscaling the structure to smaller dimensions, scaling limitations, short-channel, re-verse short-channel, narrow-channel, and reverse narrow-channel effects Mobilityenhancement techniques are described, including strained silicon and optimization

of crystal orientation The discussion extends to ultra-thin gate-oxide, high-K electrics, advanced gate-stacks, and three-dimensional structures

di-Analog devices and passive components are introduced in Chap 6 As an sion of bipolar transistors detailed in Chap 3, the properties of junction field-effecttransistors are described, followed by optimization of MOSFETs for analog appli-cations The design and properties of integrated precision resistors, capacitors, andvaractors are then detailed The chapter concludes with the important topics of com-ponent matching and noise

exten-Chapter 7 covers advanced enabling processes and process integration It gins with integrated CMOS and BiCMOS processes to illustrate typical sequences

be-of processing steps Crystal growth and wafer parameters, including properties

of silicon-on-insulator (SOI), relevant to modern integrated processes are cussed Front-end of the line unit processes include short-duration thermal pro-cesses, atomic-lay deposition (ALD), ionized physical-vapor deposition (IPVD),optical proximity correction (OPC), double exposure and patterning, immersionlithography, and new silicides Back-end of the line processes include copper in-terconnects and low-K dielectrics

dis-The last chapter reviews selected CMOS and BiCMOS digital and memory plications The inverter is used to analyze the important parasitic latch-up effectand methods to suppress it The second part covers memory cells, including dy-namic random-access memory (DRAM), static random-access memory (SRAM),and nonvolatile memory (NVM)

ap-It would not have been possible for me to complete this book in its presentform without the continuous invaluable help with corrections and suggestions forimprovement and encouragement from Dr Wendell Noble, independent consul-tant, retired IBM semiconductor physicist, Professor Carlton Osburn of the NorthCarolina State University, and Dr Albert Puttlitz, IEEE-Components, Packagingand Manufacturing Technology Society, VP of Education I also thank my formercolleagues at IBM, Russell Houghton and Ashwin Ghatalia, for their reviews andinputs My special thanks to the personnel of the University of Texas library fortheir kind support in my research

List of Symbols xiii

1 Silicon Properties 1

1.1 Introduction 1

1.2 Valence-Bond and Two-Carrier Concept 1

1.2.1 Doping 3

1.3 Energy Bands in Silicon 6

1.3.1 Energy Band Model 6

1.3.2 Metals, Semiconductors and Insulators 10

1.3.3 Band Model for Impurities in Silicon 11

1.3.4 Energy Band Theory 12

1.3.5 Effective Mass 16

1.4 Thermal Equilibrium Statistics 17

1.4.1 The Boltzmann Distribution Function 18

1.4.2 Fermi-Dirac Distribution and Density of States 18

1.4.3 Density of States and Carrier Distribution in Silicon 20

1.4.4 Doped Silicon 23

1.5 Carrier Transport 28

1.5.1 Carrier Transport by Drift: Low Field 29

1.5.2 Matthiesson’s Rule 36

1.5.3 Carrier Transport by Drift: High Field 37

1.5.4 Carrier Transport by Diffusion 42

1.6 Nonequilibrium Conditions 44

1.6.1 Carrier Lifetime 45

1.6.2 Diffusion Length 49

1.7 Problems 50

References 52

2 Junctions and Contacts 55

2.1 Introduction 55

2.2 PN Junction 55

2.2.1 Junction Profiles and Shapes 57

vii

viii Contents

2.2.2 Step-Junction Approximation 57

2.2.3 PN Junction at Thermal Equilibrium 64

2.2.4 PN Junction in Forward Bias 77

2.2.5 PN Junction in Reverse Bias 93

2.3 Contacts 111

2.3.1 Rectifying Contacts, Schottky Barrier Diode 111

2.3.2 Current–Voltage Characteristics 118

2.3.3 Ohmic Contacts 123

2.4 Problems 129

References 131

3 The Bipolar Transistor 135

3.1 Introduction 135

3.2 Transistor Action, a Qualitative Description 136

3.2.1 Nomenclature and Regions of Operation 136

3.2.2 Idealized Structure 138

3.2.3 Ebers-Moll Equations 141

3.2.4 Collector Saturation Voltage, V CEsat 142

3.3 Planar Transistor, Low-Level Injection 143

3.3.1 Low-Level Injection Parameters 144

3.3.2 Collector-Base Reverse Characteristics 151

3.3.3 Emitter-Base Reverse Characteristics 156

3.3.4 Polysilicon Emitter and Interface Oxide 158

3.3.5 Transistor Resistances 165

3.4 High-Level Injection Effects 172

3.4.1 Base Conductivity Modulation 172

3.4.2 Base-Push Effect (Kirk Effect) 173

3.5 Frequency Response of Current Gain 175

3.5.1 Emitter Delay,τE 176

3.5.2 Base Transit Time,τB 177

3.5.3 Collector Delay,τC 179

3.6 The Transistor as a Switch 182

3.6.1 Delay Time, t d 183

3.6.2 Rise Time, t r 185

3.6.3 Storage Time, t s 186

3.6.4 Fall Time, t f 187

3.7 Silicon-Germanium Transistor 188

3.7.1 SiGe Film Deposition and Properties 188

3.7.2 Bandgap Lowering 191

3.7.3 Density of States 192

3.7.4 Mobility 192

3.7.5 Transistor Parameters 196

3.7.6 Transistor Optimization 200

3.8 Problems 204

References 207

Contents ix

4 The MOS Structure 213

4.1 Introduction 213

4.2 Physics of an Ideal MOS Structure 214

4.2.1 Description of Semiconductor Surface Conditions 215

4.2.2 Surface Charge and Electric Field 220

4.2.3 Approximations 222

4.2.4 Excess Surface Carrier Concentrations 224

4.2.5 MOS Capacitance 225

4.3 Calculation of Capacitance 228

4.3.1 Calculation of Low-Frequency Capacitance 228

4.3.2 Description of the Low-Frequency CV-Plot 229

4.3.3 Calculation of High-Frequency Capacitance 237

4.4 Measurement of MOS Capacitance 239

4.4.1 Low-Frequency, or Quasi-Static CV Measurement 239

4.4.2 High-Frequency CV Measurement 240

4.5 Non-Uniform Impurity Profile 241

4.5.1 Profile Approximations 242

4.5.2 Surface Conditions 242

4.6 Non-Ideal MOS Structure 244

4.6.1 Workfunction Difference 244

4.6.2 Dielectric Charge 247

4.7 Characterization and Parameter Extraction 253

4.7.1 Extraction of Equivalent Oxide Thickness, t eq 253

4.7.2 Workfunction Difference 254

4.7.3 Extraction of Dopant Concentration 255

4.7.4 Lifetime Measurements 257

4.7.5 Extraction of Interface-State Distribution 259

4.7.6 Extraction of Mobile Ion Concentration 263

4.8 Carrier Transport Through the Dielectric 264

4.8.1 Tunneling Through the Oxide 265

4.8.2 Avalanche Injection 266

4.9 Problems 268

References 270

5 Insulated-Gate Field-Effect Transistor 273

5.1 Introduction 273

5.2 Qualitative Description of MOSFET Operation 273

5.3 Gate-Controlled PN Junction, or Gated Diode 277

5.3.1 Junction at Equilibrium 277

5.3.2 Reverse Biased Junction: Depleting Gate Voltage 279

5.3.3 Reverse Biased Junction: Accumulating Gate Voltage 281

5.4 MOSFET Characteristics 286

5.4.1 Long and Wide Channel 286

5.4.2 Scaling to Small Dimensions 304

x Contents

5.4.3 Short-Channel Effects, SCE 312

5.4.4 Reverse Short-Channel Effects, RSCE 319

5.4.5 Narrow Channel Effects, NCE 322

5.4.6 Reverse Narrow-Channel Effects, RNCE 326

5.4.7 Small-Size Effects 329

5.5 Mobility Enhancement 330

5.5.1 Mean-Free Time Between Collisions,τ 331

5.5.2 Effective Mass 334

5.6 Ultrathin Oxide and High-K Dielectrics 340

5.6.1 High-K Dielectric Requirements 341

5.6.2 High-K Materials 343

5.7 Gate Stack 345

5.7.1 Polysilicon Workfunction 346

5.7.2 Metal Gates 347

5.8 Three-Dimensional Structures, FinFETS 352

5.9 Problems 353

References 356

6 Analog Devices and Passive Components 369

6.1 Introduction 369

6.2 Analog Devices 370

6.2.1 Junction Field-Effect Transistor, JFET 370

6.2.2 Analog/RF MOSFETs 381

6.2.3 Integrated Passive Components 385

6.3 Matching 409

6.3.1 MOSFET Mismatch 410

6.3.2 Bipolar Transistor Mismatch 416

6.3.3 Resistor Mismatch 417

6.3.4 Capacitor Mismatch 419

6.4 Noise 422

6.4.1 Classification of Noise 423

6.5 Problems 429

References 430

7 Enabling Processes and Integration 439

7.1 Introduction 439

7.2 A Conventional CMOS Logic Process Flow 439

7.3 A BiCMOS Process Flow 446

7.4 Advanced Enabling Processes 451

7.4.1 Crystal Growth and Wafer Preparation 451

7.4.2 Short-Duration Thermal Processes 460

7.4.3 Thin-Film Deposition 466

7.4.4 Integration of Ultra-Shallow Junctions 479

7.4.5 Gate Stack Module 485

Contents xi

7.5 Advanced Interconnects 489

7.5.1 Copper Interconnects 491

7.5.2 Low-K Dielectrics 498

7.6 Problems 501

References 503

8 Applications 523

8.1 Introduction 523

8.2 Logic Units 523

8.2.1 The Inverter 523

8.2.2 The CMOS Inverter 528

8.2.3 The BiCMOS Inverter 533

8.2.4 CMOS NAND and NOR Gates 534

8.2.5 BiCMOS Two-Input NAND 535

8.2.6 The Transmission Gate 536

8.3 Memories 537

8.3.1 Dynamic Random-Access Memories, DRAM 537

8.3.2 Static Random Access Memories, SRAM 546

8.3.3 Nonvolatile Memory, NVM 551

8.3.4 BiCMOS for Analog/RF Applications 566

8.4 Problems 567

References 567

Appendix A: Universal Physical Constants 575

Appendix B: International System of Units, SI 577

Appendix C: The Greek Alphabet 579

Appendix D: Properties of Silicon and Germanium (300 K, Intrinsic Semiconductor Unless Otherwise Stated) 581

Appendix E: Conversion Factors 583

Index 585

AΔR process-related resistor mismatch factor (cm)

AΔVT process-related threshold voltage-mismatch factor (cm)

BVCBO collector-base breakdown voltage, emitter open (V)

BVCBS collector-base breakdown voltage, emitter-base shorted (V)

BVCEO collector-emitter breakdown voltage, base open (V)

BVDGO drain-gate breakdown voltage, source open (V)

BVDGS drain-gate breakdown voltage, source-gate shorted (V)

BVEBO emitter-base breakdown voltage, collector open (V)

BVEBS emitter-base breakdown voltage, collector-base shorted (V)

C capacitance per unit area (F/cm2)

c velocity of light (2.998 × 1010cm/s)

CBL bit-line capacitance (C)

CD diffusion capacitance per unit area (F/cm2)

Cdecap decoupling capacitance (F)

Cdeep deep deletion capacitance per unit area, CV plot (F/cm2)

CFG-CG capacitance between floating and control gate (F)

xiii

xiv List of Symbols

CGCh gate to channel capacitance per unit area (F/cm2)

CGD gate to drain capacitance (F)

CGS gate to source capacitance (F)

CHF high-frequency capacitance per unit area, CV plot (F/cm2)

Ci intrinsic capacitance, varactor (F)

CILD inter-level dielectric capacitance (F)

Cinv silicon inversion capacitance per unit area, CV plot (F/cm2)

Cj junction capacitance (F)

CjE emitter-base junction capacitance (F)

CjC collector-base junction capacitance (F)

CL atomic concentration in liquid state (cm−3)

CL load capacitance (F)

CLF low-frequency capacitance per unit area, CV plot (F/cm2)

Cmax maximum capacitance per unit area, CV plot (F/cm2)

Cmin minimum capacitance per unit area, CV plot (F/cm2)

Cox oxide capacitance per unit area, CV plot (F/cm2)

Cpar parasitic capacitance (F)

CPMD pre-metal dielectric capacitance (F)

Cpoly polysilicon, e.g., depletion capacitance per unit area, (F/cm2)

CS atomic concentration in solid state (cm−3)

CS storage node capacitance (F)

CSi silicon capacitance per unit area, CV plot (F/cm2)

CSidep silicon depletion capacitance per unit area, CV plot (F/cm2)

CSiFB silicon capacitance at flatband per unit area, CV plot (F/cm2)

CSimin silicon minimum capacitance per unit area, CV plot (F/cm2)

CSTI shallow-trench capacitance (F)

d distance (cm)

D diffusion constant (cm2/s)

˜

D effective diffusion constant (cm2/s)

D n electron diffusion constant (cm2/s)

D p hole diffusion constant (cm2/s)

E energy (eV)

E electric field (V/cm)

e tensile strain (Pa)

EC critical field (V/cm)

EC bottom of conduction band energy level (eV)

ECNL charge neutrality level (eV)

ED donor energy level (eV)

EF Fermi level (eV)

EFn electron quasi-Fermi level (eV)

EFp hole quasi-Fermi level (eV)

Eg energy gap (eV)

Egrad field induced by grading Ge profile (V/cm)

Ei intrinsic silicon energy level (eV)

List of Symbols xv

Ei ionization energy (eV)

Ei(A) acceptor ionization energy (eV)

Ei(D) donor ionization energy (eV)

En nitride field Qn≈ 0 (V/cm)

Eox oxide field (V/cm)

EOO characteristic tunneling energy (eV)

EP phonon energy (eV)

Epeak peak electric field (V/cm)

Es surface field (V/cm)

ESi field in silicon (V/cm)

ET trap energy level (eV)

EV top of valence band energy level (eV)

E x field in silicon normal to surface (V/cm)

E y surface field parallel to silicon surface (V/cm)

F force (N)

F dimensionless electric field (F-function)

f frequency (Hz)

f (E) Fermi-function

f T gain-bandwidth product, cut-off frequency (Hz)

fmax maximum frequency of operation (Hz)

G constant

G bulk generation rate (cm−3 · s −1)

gD channel (drain) conductance (S)

gD-lin linear channel (drain) conductance (S)

gD-sat saturated channel (drain) conductance (S)

gm transconductance (S)

gm-lin linear transconductance (S)

gm-sat saturated transconductance (S)

GR generation-recombination

G0 lumped JFET parameter

I current (A)

IB base current (A)

IB body current (A)

IBC base-collector current (A)

IBE base-emitter current (A)

IC collector current (A)

ICBO collector-base current, emitter open (A)

ICEO collector-emitter current, base open (A)

ICsat collector saturation current (A)

ID drain current (A)

IDiff diffusion current (A)

IDsat saturated drain current (A)

ID0 drain current per channel-square at threshold (A)

IE emitter current (A)

I emitter-base current, collector open (A)

xvi List of Symbols

IEsat emitter saturation current (A)

IF forward-bias current (A)

IG gate current (A)

Igen generation current (A)

Igen-bulk bulk generation current (A)

Igen-surf surface generation current (A)

IH holding current, latch-up (A)

Ileak total leakage current (A)

In electron current (A)

in noise current (A)

INW current in n-well (A)

Ioff MOSFET off-current (A)

Ip hole current (A)

IPT punch-through current (A)

IPT0 current at onset of punch-through (A)

IPW current in p-well (A)

IR reverse-bias current at VG= VT(A)

Ir surface recombination current (A)

IS source current (A)

IS saturation current (A)

Is surface current (A)

IsB base saturation current (A)

IsC collector saturation current (A)

I0 drain current at VG= VT(A)

j current density (A/cm2)

jdirect direct tunneling current density (A/cm2)

jF forward current density (A/cm2)

jFN Fowler-Nordheim tunneling current density (A/cm2)

jG gate current density (A/cm2)

jn electron current density (A/cm2)

jn(dif) diffusion electron current density (A/cm2)

jp hole current density (A/cm2)

jp(dif) diffusion hole current density (A/cm2)

jR reverse current density (A/cm2)

js saturation current density (A/cm2)

jT total current density (A/cm2)

K dielectric constant =ε/ε0

k Boltzmann constant≈ 8.618 × 10 −5 eV/K

k wave number (cm−1)

k l dimensionless resolution factor

kseg segregation coefficient

kT thermal energy (=0.0258 eV at 300 K)

kT /q thermal voltage (=0.0258 V at 300 K)

L length (cm)

List of Symbols xvii

L inductance (H)

LD Debye length (cm)

LD drawn length (cm)

LE emitter length (cm)

LE electrical length, e.g., resistor (cm)

Le extrinsic Debye length (cm)

Leff effective channel length (cm)

leff effective mean-free path (cm)

LI impact-ionization mean-free path (cm)

Li intrinsic Debye length (cm)

Lmet metallurgical channel length (cm)

Ln electron diffusion length (cm)

LnB electron diffusion length in base (cm)

Lp hole diffusion length (cm)

LpE hole diffusion length in emitter (cm)

Lpoly polysilicon line-width, channel length (cm)

L r optical phonon mean-free path (cm)

LT contact transfer length (cm)

M multiplication factor

mD density of states effective mass (kg)

m0 electron mass (≈ 9.1 × 10 −31kg)

m ∗

n electron effective mass (kg)

mox oxide electron effective mass (kg)

m ∗

p hole effective mass (kg)

N number of electrons per unit area (cm−2)

NA(x) acceptor concentration as function of depth (cm−3)

NB background dopant concentration (cm−3)

NB concentration of lightly-doped region (cm−3)

nb electron concentration in bulk (cm−3)

NC effective density of states at conduction band edge (cm−3)

ND donor concentration (cm−3)

ND+ ionized donor concentration (cm−3)

ND(x) donor concentration as function of depth (cm−3)

ND0 donor concentration at x = 0 (cm −3)

N number of fixed oxide charges per unit area (=Q /q cm −2)

xviii List of Symbols

NI number of mobile ions per unit area (cm−2)

ni intrinsic carrier concentration (cm−3)

ni0 intrinsic carrier concentration without energy-gap lowering (cm−3)

Ninv number of inversion electrons per unit area (cm−2)

Nit number interface traps per unit area (cm−2)

NM noise margin

NMH high noise margin

NML low noise margin

nn majority electron concentration in n-region (cm−3)

¯

nn thermal equilibrium electron concentration in n-region (cm−3)

nn0 electron concentration in n-region at x = 0 (cm −3)

np minority electron concentration in p-region (cm−3)

¯

np thermal equilibrium electron concentration in p-region (cm−3)

np0 electron concentration in p-region at x = 0 (cm −3)

ns surface electron concentration (cm−3)

Ns number of secondary carrier pairs

nsL surface electron concentration at drain (cm−3)

nso surface electron concentration at source (cm−3)

Nt density of generation-recombination centers (cm−3)

Nteff effective density of generation-recombination centers (cm−3)

NV effective density of states at valence band edge (cm−3)

N0 fixed diffusion-source concentration (cm−3)

p thermal equilibrium hole concentration (cm−3)

pb bulk hole concentration (cm−3)

Pj junction perimeter (cm)

pn concentration of minority holes in n-region (cm−3)

¯

pn thermal equilibrium hole concentration in n-region (cm−3)

pn0 hole concentration in n-region at x = 0 (cm −3)

¯

pn0 equilibrium minority hole concentration at x = 0 (cm −3)

pp concentration of majority holes in p-region (cm−3)

¯

pp thermal equilibrium hole concentration in p-region (cm−3)

¯

pp0 equilibrium majority hole concentration at x = 0 (cm −3)

pp0 hole concentration in p-region at x = 0 (cm −3)

ps surface hole concentration (cm−3)

Pstandby standby power (W)

Q charge per unit area (C/cm2)

Q quality factor

List of Symbols xix

q electron charge (≈1.6 × 10 −19C)

QB minority-carrier charge in base (C)

Qb bulk depletion charge per unit area (C/cm2)

Qbmax maximum bulk depletion charge per unit area (C/cm2)

Qb-deep bulk charge in deep depletion per unit area (C/cm2)

Qeff effective dielectric charge per unit area (C/cm2)

Qf oxide fixed charge per unit area (C/cm2)

Qit silicon-oxide interface trap charge per unit area (C/cm2)

Qitm gate-insulator interface trap charge per unit area (C/cm2)

Qm charge induced at gate-oxide interface per unit area (C/cm2)

Qm mobile charge per unit area (C/cm2)

Qmax maximum quality factor

Qn surface electron charge per unit area (C/cm2)

Qot oxide trap charge per unit area (C/cm2)

Qp surface hole charge per unit area (C/cm2)

QS stored charge per unit area (C/cm2)

Qs surface charge per unit area (C/cm2)

RBext extrinsic base resistance (Ω)

RBint intrinsic base resistance (Ω)

RB0 base resistance without applied bias (Ω)

rE emitter dynamic resistance (=kT /qIC,Ω)

Rext extrinsic resistance (Ω)

Rext-S source extrinsic resistance (Ω)

Rext-D drain extrinsic resistance (Ω)

RPBL p-buried layer resistance (Ω)

R intrinsic-base (pinched) resistance (Ω)

RS0 sheet resistance at T = T0(Ω/Square)

Rwire wiring resistance (Ω)

r0 output wiring resistance (Ω)

S subthreshold swing (V/decade)

s, s0 surface recombination velocity (cm/s)

s i current noise power spectral density (A/ √

teq equivalent oxide thickness (cm)

tmetal metal thickness (cm)

TN noise temperature (K)

tn nitride thickness (cm)

tox oxide thickness (cm)

tox-phys physical oxide thickness (cm)

tpoly polysilicon thickness (cm)

tSi path-length in silicon (cm)

tsilicide silicide thickness (cm)

tSTI shallow-trench isolation thickness (cm)

U generation-recombination rate (cm−3 · s −1)

Us surface generation rate (cm−3 · s −1)

u dimensionless Fermi potential (=qφ/kT )

ub dimensionless bulk Fermi potential (=qφb/kT )

us dimensionless surface Fermi potential (=qφs/kT )

u(x) dimensionless Fermi potential versus depth x [=qφ(x)/kT ]

VCBO collector-base voltage, emitter open (V)

VCBS collector-base voltage, emitter shorted to base (V)

VCC power supply voltage, bipolar transistor (V)

VCE collector-emitter voltage (V)

V collector-emitter voltage, base open (V)

List of Symbols xxi

VCEsat collector saturation voltage (V)

VCh channel to source voltage, FET (V)

VD, VDS drain to source voltage

vd drift velocity (cm/s)

VDA measured dielectric absorption (V)

VDD power supply voltage, MOSFET (V)

VDsat saturation drain voltage, MOSFET (V)

vdy drift velocity along surface, in y-direction (cm/s)

VEBS emitter-base voltage, collector shorted to base (V)

VF forward voltage (V)

VFB flatband voltage (V)

VG,VGS gate to source voltage (V)

VH holding voltage, latch-up (V)

VIH high input voltage (V)

VIHmax maximum high input voltage (V)

VIHmin minimum high input voltage (V)

VIL low input voltage (V)

VILmax maximum low input voltage (V)

VILmin minimum low input voltage (V)

Vj junction voltage (V)

VjG junction to gate voltage, gated diode (V)

vn electron velocity (cm/s)

VOH high output voltage

VOHmax maximum high output voltage

VOHmin minimum high input voltage

Vox voltage across oxide (V)

VSS typically ground potential, MOSFET (0 V)

vs dimensionless surface potential (=qψs/kT )

vsat saturation velocity (cm/s)

VT threshold voltage (V)

VT-drain threshold voltage at drain (V)

vth thermal velocity (≈107cm/s at 300 K)

VT-source threshold voltage at source (V)

VT0 threshold voltage for zero floating-gate charge (V)

v(x) dimensionless potential versus depth [=qψ(x)/kT]

v0 initial velocity (cm/s)

W width (cm)

Wb neutral base width (cm)

WContact contact width (cm)

W drawn width (cm)

xxii List of Symbols

WE emitter width (cm)

WE electrical width (cm)

Weff effective width (cm)

WL word-line

WMetal metal width (cm)

Wn width of neutral n-region (cm)

Wp width of neutral p-region (cm)

WTotal sum of all on-chip MOSFET width (cm)

WVia via width (cm)

x depth normal to silicon surface (cm)

xCh channel depth below the surface, thickness (cm)

xd depletion width (cm)

xdD depletion width at MOSFET drain (cm)

xd-deep depletion depth in deep depletion (cm)

xd-field depletion width under field oxide (cm)

xd-inv steady-state depletion depth in inversion (cm)

xd-lat lateral junction depletion width at surface (cm)

xdmax maximum depletion depth below surface (cm)

xdn depletion width in n-side of pn junction (cm)

xdp depletion width in p-side of pn junction (cm)

xdS depletion width at MOSFET source (cm)

xds, xdsurf junction depletion width at surface intercept (cm)

xi depth below surface where n = p = ni(cm)

xJ junction depth (cm)

xjC collector-base junction depth (cm)

xJE emitter-base junction depth (cm)

x jlat , x jl lateral extent of junction at surface (cm)

xm depth of potential peak, image-force barrier lowering (cm)

y direction from source to drain

α grounded base current gain (=IC/IE)

α temperature coefficient of resistance (K−1)

αF forward grounded base current gain (=IC/IE)

αi impact ionization rate (cm−1)

αR reverse grounded base current gain (=IC/IE)

αT base transport factor

αT pre-tunneling factor

β grounded emitter current gain (= IC/IB)

β =μeff·Cox·W/L (MOSFET)

βF forward grounded emitter current gain (=IC/IB)

βR reverse grounded emitter current gain (=IC/IB)

βR ratio of NMOSβto PMOSβ

γ injection efficiency

γn electron injection efficiency, NPNγn= In/(In+ Ip)

γp hole injection efficiency, PNPγp= Ip/(In+ Ip)

δ thickness of interface barrier gap (cm)

List of Symbols xxiii

δL length of pinch-off region (cm)

Δ width of gap layer (cm)

Δ voltage drop across interface oxide (V)

ΔEC change (offset) in minimum conduction band edge (eV)

ΔEg change in energy bandgap (eV)

ΔEV change (offset) in maximum valence band edge (eV)

ΔIB increment in base current (A)

ΔIC increment in collector current (A)

ΔIE increment in emitter current (A)

ΔL change in channel length (cm)

Δn change in electron concentration (cm−3)

Δnp0 change in minority-electron concentration at x = 0 (cm −3)

Δp change in hole concentration (cm−3)

Δpn0 change in minority-hole concentration at x = 0 (cm −3)

ΔVBE increment in emitter-base forward voltage (V)

ΔVG increment in gate voltage (V)

ΔVR increment in reverse voltage (V)

ΔVT change in threshold voltage (V)

ΔW change in channel width (cm)

Δφ barrier lowering (eV)

ε0 permittivity of free space (≈8.86 × 10 −14 F/cm)

εgap dielectric constant of interface gap

εox oxide dielectric constant (≈3.9)

εn nitride dielectric constant (≈7.0)

εSi silicon dielectric constant (≈11.7)

η multiplier of inversion-layer concentration to calculate field

μeff effective mobility (cm2/V · s)

μh high mobility, normal to crystallographic axis (cm2/V · s)

μI ionized-impurity scattering limited mobility (cm2/V · s)

μl lattice-scattering limited mobility (cm2/V · s)

μl low mobility, along crystallographic axis (cm2/V · s)

μln electron lattice mobility (cm2/V · s)

μlp hole lattice mobility (cm2/V · s)

μn electron mobility (cm2/V · s)

˜

μn effective electron mobility (cm2/V · s)

xxiv List of Symbols

ρ volume charge concentration (C/cm3)

ρC specific contact resistance (Ω-cm2)

τB base transit time (s)

τC collector transit time (s)

τE emitter transit time (s)

τn electron transit time, lifetime (s)

τnB electron transit time, lifetime in base (s)

τp hole transit time, lifetime (s)

τpE hole transit time, lifetime in emitter (s)

τSRH Shockley-Read-Hall recombination lifetime (s)

τ0 assumed same lifetime for electrons and holes (s)

φ potential (V)

φ dose (cm−2)

φI pulsed-shaped implant dose (cm−2)

φB barrier height (V)

φb bulk Fermi potential (V)

φbn bulk electron Fermi potential (V)

φbp bulk hole Fermi potential (V)

φFn electron quasi Fermi potential (V)

φFp hole quasi Fermi potential (V)

φm metal (gate) workfunction (V)

φms workfunction difference between metal (gate) and Si (V)

φm-app apparent metal (gate) workfunction (V)

φn electron Fermi potential (V)

φp hole Fermi potential (V)

φs surface Fermi potential (V)

Chapter 1

Silicon Properties

1.1 Introduction

A review of silicon properties is important to understanding silicon components,

in particular modern components such as strained-silicon MOSFETs and

hetero-junction bipolar transistors Several books cover this subject in detail The objective

of this chapter is to highlight those features that are most important to silicon deviceoperation and characteristics

1.2 Valence-Bond and Two-Carrier Concept

The valence-bond model is frequently used to qualitatively describe the properties ofsemiconductors [1] In this model, the covalent bond between two adjacent atomsformed by two valence electrons, one from each atom contributing to the bond,

is visualized as localized bars along which electrons shuttle back and forth withopposite spins (Fig 1.1)

When a pure silicon crystal is near 0 K, all valence electrons remain locally bound

to their covalent bonds since they do not have sufficient energy to break loose Inthis case, no quasi-free electrons are generated and the crystal behaves like a perfectinsulator As the temperature is increased, the amplitude of vibration of lattice atomsincreases around their equilibrium positions A fraction of the vibrational energy

is transferred to valence electrons Some electrons can acquire sufficient energy tobreak loose from their bonds and move quasi freely in the crystal Hence, the number

of quasi free electrons and holes (missing bond electrons) in the crystal increases asthe temperature is increased

The energy required to break a silicon bond is an ionization energy (∼1.1eV)

which differs from the ionization energy of an isolated silicon atom (∼8eV) because

B El-Kareh, Silicon Devices and Process Integration: Deep Submicron 1

and Nano-Scale Technologies,

c

Springer Science+Business Media, LLC 2009

2 1 Silicon Properties Fig 1.1 Three dimensional

representation of the silicon

crystal Dark atoms define

the unit cell Lattice constant

a = 0.54307 nm

a

a /2

it is influenced by other forces in the crystal.1When ionization occurs, the crystal

as a whole remains neutral, although locally the ion becomes positively charged Avacancy is left where an electron breaks loose from a bond It behaves as a positivefree carrier that is referred to as a hole (or “defect electron”) with a mass comparable

to that of the electron The hole can move while, under normal operating conditions,the positive ion remains fixed

To visualize the motion of holes, imagine the lattice sites to be occupied by onlybound electrons and disregard the ions [2] Suppose that one lattice point is void of

an electron, that is, a hole is created If now a field is applied to such an “electroncrystal,” the electron that is on the negative side of the hole will move into the hole,thus creating another hole The hole moves as if it were a positively charged particle,although it is the bound electron that has moved in the opposite direction Imaginenow that in this “thought electron crystal” an electron is set free from its latticesite by some external force, for example, thermal agitation, and wanders aroundindependently along interstitial sites This quasi-free electron will be repulsed by allother electrons but not by the hole The hole will appear to the interstitial electron

as a positive charge During its random motion, the interstitial electron may fillthe hole, thus annihilating a positive and a negative charge simultaneously In puresilicon, the concentration of electrons and holes are equal since they are generatedand annihilated in pairs In this case, silicon is said to be intrinsic and

n and p are the electron and hole concentrations, respectively, and ni the intrinsiccarrier concentration (∼1.4 × 1010cm−3at 300 K)

1 One electron-Volt (eV) is the energy dissipated or acquired by one electron that goes through a

potential difference of one Volt Since the charge of one electron is 1.6 × 10 −19Coulomb, 1 eV =

1.6 ×10 −19Joule In this book, eV and cm are frequently used in place of J and m, as a convenient

departure from SI units.

1.2 Valence-Bond and Two-Carrier Concept 3

1.2.1 Doping

Intrinsic silicon has very limited use in device applications since the conductivity

is very low and conduction of electrons and holes essentially occurs in pairs Onecan, however, modify the type and magnitude of conductivity by adding small andcontrolled amounts of certain elements to the otherwise pure silicon The crystalcan be doped to have more conduction electrons than holes and vice versa To beactive, the dopants must occupy substitutional sites, that is, occupy a lattice sitenormally occupied by silicon The doping process is described in more detail in [3]

Of particular importance to silicon devices is doping with elements from the thirdand fifth columns of the periodic table (Fig 1.2)

1.2.1.1 Dopants from the Fifth Column: Donors

Phosphorus, arsenic, antimony and bismuth are elements of the fifth column Theydiffer mainly in their diffusivity and solid solubility in silicon [3] These elementshave five electrons in their outer shell, that is, five valence electrons When theyoccupy a substitutional site in silicon, only four of the valence electrons are needed

to complete the covalent bonding in the crystal The fifth electron does not contribute

7 14.007 N Nitrogen

83 208.980 Bi Bismuth

15 30.974 P Phosphorus

33 74.922 As Arsenic

51 121.757 Sb Antimony

5 10.811 B Boron

81 204.383 Ti Titanium

13 26.982 Al Aluminum

31 69.723 Ga Gallium

49 114.820 In Indium

6 12.011

C Carbon

82 207.200 Pb Lead

14 28.086 Si

Silicon

32 72.610 Ge

Germanium

50 118.710 Sn Tin

4 1 Silicon Properties

+

a

− Free electron

b

Fig 1.3 Simplified model for dopants in silicon a Quasi-free electron liberated from donor.

b Nearby bound electron transferred to acceptor to complete bond, hole created

to the bonding and is set free, or “donated” (Fig 1.3) The ionization energy required

to set the fifth electron free is considerably smaller than the energy required to break

a silicon bond

Most of donor atoms remain ionized even at temperatures as low as 100 K Eachpositive ion left behind has four bound valence electrons, the same as the originalsilicon atom Note that by ionizing the donor, a free electron is generated without

creating a hole Thus, for a donor concentration N D, the free electron concentration

is n ∼ = N D+, where N D+∼ = N Dis the ionized donor concentration Ionized donors are

fixed positive charges that do not contribute to conduction The crystal becomes rich

in electrons, n-type, but as a whole remains neutral At temperatures below∼100K,

the probability for electrons to break loose from donors begins to decrease and anincreasing fraction of electrons remains “frozen” to donors

1.2 Valence-Bond and Two-Carrier Concept 5One can estimate the energy required to ionize a donor by treating the fifthelectron as the electron in a hydrogen atom [4] The electron moves around the

central field of the donor core with a net charge of q The central force is

where q is the electronic charge (1.60218 × 10 −19C), r the radius of the

impu-rity atom,ε0the permittivity of free space (8.85418 × 10 −14 F/cm), εSi the

rela-tive dielectric constant of silicon (11.7), and m 0 the free electron mass (9.1095 ×

in the crystal It is a measure of the ease with which an external field can accelerateelectrons and holes along an axis in the crystal and allows the use of a relationbetween force and momentum that is similar to the classical Newton’s law Theeffective mass is further discussed in Sect 1.3.5 Assuming an electron effective

mass of 0.26m0and substituting the values for E i(H) , ε0, andεSi into (1.2) gives

E i (D) = 0.025 eV [5] The actual measured donor ionization energy ranges from

0.044 to 0.067 eV for different group V elements in silicon [1]

1.2.1.2 Dopants from the Third Column: Acceptors

Boron and indium are acceptor elements from group III in the periodic table Whensubstituted for silicon, the three available valence electrons in the outer shell takepart in the bond structure, leaving one vacancy since the fourth bond is not filled.The vacancy is “attractive” to an adjacent bound electron that easily moves to fill

it Boron or indium “accepts” the fourth electron, creating a hole where the fillingelectron came from without producing a free electron (Fig 1.3b) When silicon isdoped with boron, holes become the majority carriers and silicon is said to be p-type.For an acceptor concentration NA, the hole concentration at room temperature is

p ∼ = N A − , where N A − ∼ = N Ais the ionized acceptor concentration Acceptors are fixed,

negatively charged ions that do not contribute to conduction As for donors, theionization energy for acceptors can be estimated in terms of the ionization energy for

6 1 Silicon Propertieshydrogen by using the conductivity effective mass of holes in silicon and assumingthat the hole is “liberated” from the acceptor The acceptor ionization energy isestimated as Ei(A)≈ 0.05eV, compared to the actual measured ionization energy

of 0.045 eV for boron [4]

1.3 Energy Bands in Silicon

The properties of semiconductors are more accurately described with the mechanical energy-band model than with the over-localized valence-bond modeldiscussed in the previous section Energy bands in semiconductors are discussedextensively in reference books on solid-state physics [4–6] The sole objective ofthis section is to highlight, in simple terms, the energy-band model concepts that aremost pertinent to understanding the properties of silicon devices

quantum-1.3.1 Energy Band Model

A free electron is allowed to occupy a continuum of energy levels, similar to theclassical case of molecules in an atmospheric column that can occupy any energylevel without restrictions From quantum mechanics it is known that when an elec-tron is bound to, for example, an isolated hydrogen atom, it is allowed only discreteenergy levels separated by energy gaps When two hydrogen atoms are far fromeach other, they behave as two isolated entities with independent, identical sets ofdiscrete energy levels As the atoms are brought close to each other, their wavefunctions begin to overlap so that the electrons of the two atoms begin to interact.Electrons in the first atom can also occupy energy levels in the second and viceversa A study of energy levels shows that, in the limit, when a hydrogen molecule

is formed, each energy level of the isolated hydrogen atom splits into a pair of levelswhen the atoms are bound to form the molecule The total number of energy levels

in the molecule is the same as in the system of two isolated atoms A mechanicalanalogy may help illustrate this situation [2] Consider, for example, the coupling

of two identical pendulums that are connected by an elastic band and can oscillatewith negligible friction in planes normal to the paper (Fig 1.4a) When not coupled

by the band, each pendulum can be treated as a harmonic oscillator of constant plitude Coupled with the band, when one pendulum is made to oscillate while theother starts at rest, the oscillation energy of the first pendulum is gradually trans-ferred to the second pendulum The second pendulum in turn gradually transfers theoscillation energy to the first, and so on This results in an oscillation pattern similar

am-to that of beats (Fig 1.4b) When one pendulum loses all its energy am-to the other, itcomes to rest while the other reaches its maximum amplitude Beats are thus pro-duced by the coupling of two oscillating systems that have exactly the same naturalfrequency when they are isolated from each other

1.3 Energy Bands in Silicon 7

a

b

Fig 1.4 Mechanical analogy of energy splitting a Model of coupled pendulums to visualize the splitting of energy levels in coupled atomic systems b Schematic representation of resulting beats [2]

The result is similar to beats caused by tuning two separate forks of equal tude but slightly different natural frequencies Thus, when two oscillating systems

ampli-of equal natural frequencies are brought to coupling, the result is similar to the nance of two oscillating systems of equal amplitudes but frequencies that are slightlydifferent [2] This suggests that coupling two oscillating systems of the same natu-ral frequency results in the splitting of the initially undisturbed frequency into twoslightly different frequencies, one higher and one lower than the natural frequency.The difference between the two new frequencies increases as the coupling-strengthincreases, that is, as the frequency of amplitude exchange between the two pendu-lums increases This is not surprising since the oscillation frequency of the coupledsystem is lower when the pendulums are in phase and higher when they are out ofphase The pendulums move toward each other when they oscillate in phase andaway from each other when they are out of phase

reso-One can project the above observations to atomic systems When two atomicsystems are brought in proximity of each other, coupling of the Ψ-waves inSchr¨odinger’s wave function results in an amplitude-exchange ofΨ-oscillationsbetween two atomic systems, with an exchange frequencyΔν Since the square of

Ψ represents the probability of finding an electron (electron density),Δνrepresents

8 1 Silicon Propertiesthe electron–exchange frequency between the two systems This exchange results

in a split of energy levelsΔE = hΔν, where h is Planck’s constant.

In a crystal of N identical lattice atoms, there exists the possibility of exchange

of every valence electron with valence electrons of the remaining N − 1 atoms One

expects then that each energy level of the isolated atom would split in the crystal into

N energy levels that can be distinguished by a quantum number k, and each of which can be occupied by two electrons of opposite spins Since N is a very large number

(∼1023cm−3), the levels are too close to each other to be distinguished They arethus described by a band of energy levels, bounded by a maximum and minimumlevel The difference between maximum and minimum energy levels (the width ofthe band) depends on the degree of coupling of wave-functions between atoms, that

is, on the distance between atoms and probability of electron exchange which is

a function of temperature, pressure and stress, and independent of N One expects

therefore that the energy levels of innermost electrons remain sharp because theprobability for them to interact is very small, and that the band-width increases asthe principal quantum number increases, as shown schematically in Fig 1.5 Thenumber of quantum states in the energy band is the same as the number of statesfrom which the band was formed

Of primary interest for conduction are the uppermost two bands, the conductionband of quasi-free electron energy levels, and the band just below the conductionband, referred to as the valence band of bound electrons It will be shown that forsilicon (and germanium) at crystal temperatures near absolute zero, the valence band

Edge: holes at rest

Hole energy increases

Edge: electrons at rest Electron energy increases

1.3 Energy Bands in Silicon 9

is full (all bound electrons in place) and the conduction band is empty; the crystalbehaves like a perfect insulator As the temperature increases, some electrons ac-

quire sufficient energy from crystal vibrations to overcome the energy gap Egandare elevated from the valence band to the conduction band where they are quasi-free

to move Holes are created in the valance band where electrons are missing

A simplified one-dimensional representation of the energy bands relative to theperiodic potential in the crystal is shown in Fig 1.6 It will be shown later that most

of the transitions occur between the upper edge of the valence band and the loweredge of the conduction band In such situations only the edges of the conduction andvalence bands are drawn, as indicated with dashed lines in Fig 1.5

A theoretical analysis of energy levels as a function of atomic space in the amond structure, to which C, Si and Ge belong, was made by varying the atomicspacing, in a thought experiment, from infinity to below the actual spacing in the

di-crystal, as illustrated for carbon in Fig 1.7 [7, 8] At large spaces, the s(l = 0) and

Valence band

2N states, 2N electrons 4N states, 4N electrons

6N states, 2N electrons

8N states, 4N electrons

4N states,

0 electrons

Assumed atomic spacing (nm)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 a

Fig 1.7 Schematic representation of theoretical energy levels for carbon versus assumed atomic spacing a: actual spacing (Adapted from [1, 6])

inter-10 1 Silicon Properties

p(l = 1) levels of the valence shell (n = 2) in carbon are sharp For N atoms, the 2s level contains 2N allowed states and is fully occupied by 2N electrons while the 2p level has 6N states and is partially occupied by the remaining 2N valence elec-

trons As the atomic space is theoretically reduced below∼1.2nm, the atoms begin

to interact with each other and the energy levels split in bands that contain the samenumber of energy levels as for the isolated atoms

As the space is further reduced, the bands merge The energy gap disappears andthere is no distinction between the two levels This merger is not predicted fromthe discussion in the preceding section and cannot be explained in simple terms.The total number of available states remains as the sum of states in both levels

(8N), and the total number of occupied states remains 4N At the actual space of about 0.37 nm, however, the bands split again, exhibiting an energy gap Egand a

repartitioning of energy levels into 4N in the lower band (the valence band) and 4N

in the upper band (the conduction band) The lower band is now filled with the 4N

electrons and the upper band completely empty

1.3.2 Metals, Semiconductors and Insulators

The simplified energy-band model is now used to distinguish between metals, conductors and insulators A solid conducts electricity only if carriers are free tomove under the influence of an electric field, that is, if the carriers can acquire ki-netic energy from the field and be accelerated in the solid When an electric field isapplied to the crystal, electrons can gain electric energy only if they can be placed

semi-at a higher energy level in the band If the band is completely filled with electrons,the carriers cannot gain energy from the field since there is no “place” for higher-energy electrons to be placed Therefore, if we assume that electrons do not getenough thermal or optical energy to make the transition from the completely filledband to a high-level empty or partially-filled band, the solid behaves like an insula-tor A crystal can therefore conduct electricity only if its highest energy band is notcompletely filled

It was shown in the preceding section that in a metal crystal of N atoms, the bands consist of N levels that can each be occupied by two electrons of opposite spins The band, therefore, has 2N available states For monovalent metals only half the band is

filled since the metal can only provide one electron per atom (Fig 1.8a) The metal,therefore, exhibits good conductivity One would expect divalent metals to be in-

sulators since the 2N available states in the upper band would be completely filled with 2N electrons The strong coupling between valence electrons, however, results

in an overlap of the upper bands, so that the net is a band that is not completely fulland the metal behaves as a good conductor (Fig 1.8b) A solid in which the upperband is not completely filled exhibits metallic character

It can be shown that in silicon, germanium and carbon the valence band contains

4N states that are completely filled by 4N electrons when the temperature is near

1.3 Energy Bands in Silicon 11

Empty

Eg > ∼ 3 eV Filled

germa-of germanium, silicon and carbon is, respectively,∼0.74, ∼1.17eV, and ∼5.48eV.

When the bandgap is small, of the order of 1 eV, as for silicon and germanium, thecrystal exhibits semiconductor properties (Fig 1.8c) In this case, as the tempera-ture is raised above 0 K, an increasing number of electrons can gain sufficient en-ergy from crystal vibrations to be excited from the valence band into the conductionband, increasing the crystal conductivity If, however, the energy gap is larger than

∼3eV, as for carbon, the probability of raising one electron from the valence band

to the conduction band becomes negligible and the crystal behaves like an insulator

at normal temperatures (Fig 1.8d)

1.3.3 Band Model for Impurities in Silicon

When shallow donors such as arsenic (As), phosphorus (P), antimony (Sb) or muth (Bi) are incorporated into substitutional sites in the silicon crystal, their “fifthelectron” is not bound sufficiently tight to be in the valence band It is almost free to

bis-move in the conduction band At low to moderated concentrations (< ∼1017cm−3),

donor levels are represented by short bars, indicating localized energy states ED

that do not interfere with each other, just below the conduction band (Fig 1.9).The numbers next to the bars are ionization energies measured from the conductionband edge It will be shown by statistical analysis in Sect 1.4.4 that for low to mod-erate concentrations at a temperature not too far below∼100K, practically all donor

atoms are positively ionized Thus, the probability that their energy levels are notoccupied by electrons is almost 100%

Similarly, when an electron occupies an acceptor level to complete its bond ture, it is only a little less tightly held from being free than in a normal bond Accep-tor levels are thus represented by short bars, indicating discrete energy states EAjustabove the valence band The numbers below the bars are ionization energies mea-sured from the valence band edge Statistical analysis shows that for lightly dopedsilicon with shallow acceptors, such as boron and indium, the probability for their

Fig 1.10 Schematic representation of partially compensated impurities a N D> NA, n ≈ ND−

N A ; b N A> ND, p = NA− ND

energy levels to be occupied by electrons from the valence band is almost 100% attemperatures above∼100K Thus, practically all acceptors are negatively ionized,

each creating a hole in the valence band

In most cases, both donors and acceptors are present in the same region of thecrystal Since acceptor levels are below both the conduction band and the donorlevels, any donor or conduction band electron will tend to fill an acceptor level.Both donors and acceptors will then be ionized, but only the difference between theirconcentrations will be available for conduction (Fig 1.10) The crystal is said to becompensated In case donors and acceptors are equal in concentration, all donorswould be positively ionized without contributing free electrons, and all acceptorsnegatively ionized without contributing holes for conduction

As the donor or acceptor concentration increases above∼1018cm−3, the rity atoms come closer to each other, their wave functions begin to overlap and theybegin to share each other’s electrons Because of this coupling, the levels begin tosplit and form bands The donor ionization energy decreases because the field thatbinds the fifth electron to a particular donor atom is reduced by the charged donorneighbors [1] Similarly, the acceptor ionization energy decreases at high concen-tration Properties of heavily doped silicon are further discussed in Sect 1.4.4

impu-1.3.4 Energy Band Theory

The characteristics of most silicon devices can be adequately described with the plified energy band model presented in the preceding section There are, however,

sim-1.3 Energy Bands in Silicon 13several situations where a more in-depth discussion of the band theory would bebeneficial For example, the dependence of carrier mobility on crystallographic di-rections and the modulation of mobility by mechanical stress in silicon can be bestunderstood with a more detailed energy band diagram than shown in Figs 1.6–1.8.The potential energy of a free electron is arbitrary within a constant which is set

to zero for convenience Thus, the time-independent one-dimensional Schr¨odingerwave equation simplifies to

d2ψ

dx2 +8π2m

where E is the fixed total energy Solutions to the above differential equation are

periodic traveling plane waves of the form

tional to the electron momentum and inversely proportional to the electron

wave-length The relation between E and k is then

E = ¯h22m k

The E–k diagram should therefore be a parabola, as shown in Fig 1.11 The free

electron can acquire a continuum of energy levels without restrictions

In a crystal, electrons are not quite free but move in the periodic potential of

lattice ions and the E–k plot of Fig 1.11 does not quite apply One can, however,

make the approximation of a free electron in which the electron wave functions are

Fig 1.11 E–k diagram for a

free electron

E

k

14 1 Silicon Propertiestraveling plane waves, and treat their interaction with the crystal as a diffraction

of waves The crystal behaves then as a three-dimensional diffraction grating Thisapproximation is only valid for metals It can be seen, for example, that when theelectron De Broglie wavelengthλ approaches the critical value 2a/n and k = n π/a, where a is the lattice constant (Fig 1.1) and n = 1, 2, 3, , the waves are reflected

in phase by all lattice points, with 180◦ phase-shift relative to the incident wave.Because of this reflection, referred to as Bragg reflection, standing waves are pro-duced for critical values ofλ and hence k, where electrons will be Bragg-reflected

and cannot propagate in the crystal [2, 4, 5, 7, 9] The standing waves can be resented as a combination of two waves traveling in opposite directions, in one di-mension as

rep-e ikx = cos(kx) + i sin(kx) and e −ikx = cos(kx) − i sin(kx) (1.7)

The sum of the above equations is 2 cos(kx) and the difference 2i sin(kx), both representing standing waves Either cos(kx) or sin(kx) can represent the electron at the critical De Broglie wavelength of 2a/n The probability of finding an electron is

energy has therefore a large negative value q2/x in the first case, since x is small and

a smaller value in the second case, that is, a higher potential energy, since x is larger.

The difference between the two potential energies is the energy gap (Fig 1.12)

The E–k diagram in Fig 1.12 is plotted for the region between k = −π/a and

+π/a The plots are shown flat at the critical values k = π/a and −π/a because the

electron velocity, and hence the slope dE/dk, is zero at those points.

E

k π/a

1.3 Energy Bands in Silicon 15

k

Top of

valence bands

Indirect transition

Ev Ec

E Bottom of

conduction band

Fig 1.13 E–k diagram for silicon along the 100 direction [6]

The actual E–k diagram for a silicon crystal is, however, more complicated than

in Fig 1.12 For example, the E–k plot obtained for silicon along the 100 axis

exhibits a multi-valley band diagram as shown in Fig 1.13 From this figure, severalimportant observations can be made:

(a) There exist three valence bands, V 1 , V 2 , and V 3whose band maxima occur at

k = 0 V 1 and V 2 meet at k = 0 while the maximum of V 3is about 0.04 eV belowthat of the other two bands Their relative population depends on temperature

(b) There are two equivalent minima along the k x, equally spaced on opposite sides

of k = 0 Similarly, there are equivalent minima along the k y and k zaxes

(c) The actual energy gap Egis measured from the maximum of the valence band to

the minimum of the conduction band Egdecreases with increasing temperaturebecause the crystal expands and the average distance between atoms increases

The temperature dependence of Egis approximated by [11]

the minimum of the conduction band does not occur at k = 0, but at a distance Δk from k = 0, the generation of an electron–hole pair by an indirect transition from the

valence-band maximum to the conduction-band minimum must be accompanied by

16 1 Silicon Properties

kyE

sili-the gain of momentumΔk Similarly, the electron–hole recombination process must

involve the loss of momentumΔk Since a phonon (crystal vibration modes) has a

large momentum, the transitions can occur by absorption or emission of phonons

At very low temperature only few phonons are present and indirect transitions areimprobable As the temperature increases, transitions become possible through anincrease in the number of phonons The constant energy surfaces near the bandminima and the valence band maxima are shown in Fig 1.14

In Si there are six conduction band valleys, two along each of the100

crys-tallographic directions In the vicinity of the minima, the surfaces of the six valleyscan be approximated by ellipsoids of revolution The constant energy surfaces of va-lence bands are more complicated because of the existence of multiple bands Theycan be approximated as spheres although they appear somehow “warped” [13]

1.3.5 Effective Mass

When an electric field is applied to a crystal, carriers do not only feel the externalelectric force but also forces that originate at the atomic cores and other carriers inthe periodic crystal array The effect of all these internal forces can be represented

by a quantum-mechanical parameter that has the unit mass and is known as the

effective mass, m ∗ The effective mass is inversely proportional to the curvature of

the E–k curve and defined as [6]:

m ∗= ¯h2

∂2E/∂k2. (1.11)