Nanoscale MOS Transistors: Semi-Classical Transport and Applications

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgements......Page 16
Notation......Page 17
Symbols......Page 18
Physical constants......Page 19
1.1 The historical CMOS scaling scenario......Page 21
1.2 The generalized CMOS scaling scenario......Page 25
1.3 Support of modeling to nano-scale MOSFET design......Page 27
1.4 An overview of subsequent chapters......Page 29
References......Page 32
2.1.1 Bravaix lattice......Page 39
2.1.2 Reciprocal lattice......Page 41
2.1.3 Bloch functions......Page 44
2.1.4 Density of states......Page 49
2.2.1 The pseudo-potential method......Page 50
2.2.2 The k·p method......Page 54
2.3.1 Conduction band......Page 57
2.3.2 Valence band......Page 59
2.4.1 The equivalent Hamiltonian......Page 61
2.4.2 The Effective Mass Approximation......Page 63
2.5.1 Wave-packets and group velocity......Page 65
2.5.2 Carrier motion in a slowly varying potential......Page 70
2.5.3 Carrier scattering by a rapidly fluctuating potential......Page 74
2.5.4 The Fermi golden rule......Page 75
2.5.5 Semi-classical electron transport......Page 78
2.6 Summary......Page 80
References......Page 81
3 Quantum confined inversion layers......Page 83
3.1 Electrons in a square well......Page 84
3.2 Electron inversion layers......Page 85
3.2.1 Equivalent Hamiltonian for electron inversion layers......Page 86
3.2.2 Parabolic effective mass approximation......Page 87
3.2.3 Implementation and computational complexity......Page 89
3.2.4 Non-parabolic effective mass approximation......Page 90
3.3.1 k·p method in inversion layers......Page 92
3.3.2 Implementation and computational complexity......Page 94
3.3.3 A semi-analytical model for hole inversion layers......Page 97
3.4 Full-band energy relation and the LCBB method......Page 101
3.4.1 Implementation and computational complexity......Page 104
3.4.2 Calculation results for the LCBB method......Page 105
3.5 Sums and integrals in the k space......Page 106
3.5.1 Density of states......Page 107
3.5.2 Electron inversion layers in the effective mass approximation......Page 108
3.5.3 Hole inversion layers with an analytical energy model......Page 111
3.5.4 Sums and integrals for a numerical energy model......Page 112
3.6 Carrier densities at the equilibrium......Page 114
3.6.1 Electron inversion layers......Page 115
3.6.2 Hole inversion layers......Page 117
3.6.3 Average values for energy and wave-vector at the equilibrium......Page 118
3.7 Self-consistent calculation of the electrostatic potential......Page 120
3.7.1 Stability issues......Page 121
3.7.2 Electron inversion layers and boundary conditions......Page 123
3.8 Summary......Page 128
References......Page 129
4 Carrier scattering in silicon MOS transistors......Page 132
4.1.1 The Fermi golden rule in inversion layers......Page 133
4.1.3 Physical interpretation and validity limits of Fermi's rule......Page 134
4.1.4 Inter-valley transitions in electron inversion layers......Page 135
4.1.5 Hole matrix elements for a k·p Hamiltonian......Page 143
4.1.6 A more general formulation of the Fermi golden rule......Page 144
4.1.8 Elastic and isotropic scattering rates......Page 147
4.2 Static screening produced by the free carriers......Page 148
4.2.1 Basic concepts of screening......Page 149
4.2.2 Static dielectric function for a 2D carrier gas......Page 150
4.2.3 The scalar dielectric function......Page 155
4.2.4 Calculation of the polarization factor......Page 159
4.3.1 Potential produced by a point charge......Page 163
4.3.2 Scattering matrix elements......Page 168
4.3.3 Effect of the screening......Page 171
4.3.4 Small areas and correlation of the Coulomb centers position......Page 173
4.4.1 Bulk n-MOSFETs......Page 176
4.4.2 SOI n-MOSFETs......Page 182
4.4.3 Effect of the screening in n-MOSFETs......Page 185
4.4.4 Surface roughness in p-MOSFETs......Page 186
4.5.1 Classical model for the lattice vibrations......Page 189
4.5.2 Quantization of the lattice vibrations......Page 193
4.6.1 Deformation potentials and scattering potentials......Page 196
4.6.2 General formulation of the phonon matrix elements......Page 198
4.6.3 Electron intra-valley scattering by acoustic phonons......Page 200
4.6.4 Electron intra-valley scattering by optical phonons......Page 207
4.6.5 Electron inter-valley phonon scattering......Page 209
4.6.6 Hole phonon scattering......Page 213
4.6.7 Selection rules for phonon scattering......Page 215
4.7 Screening of a time-dependent perturbation potential......Page 216
4.7.1 Dynamic dielectric function for a 2D carrier gas......Page 217
4.7.2 Screening for phonon scattering......Page 220
4.8 Summary......Page 221
References......Page 222
5.1 The BTE for the free-carrier gas......Page 227
5.1.1 The BTE for electrons......Page 228
5.1.2 The BTE for holes......Page 231
5.2.1 Real and wave-vector space in a 2D carrier gas......Page 234
5.2.2 The BTE without collisions......Page 235
5.2.3 Driving force......Page 236
5.2.4 Scattering......Page 239
5.2.6 Detailed balance at equilibrium......Page 240
5.4 Momentum relaxation time approximation......Page 243
5.4.1 Calculation of the momentum relaxation time......Page 244
5.4.2 Momentum relaxation time for an electron inversion layer......Page 249
5.4.3 Momentum relaxation time for a hole inversion layer......Page 253
5.4.4 Calculation of mobility......Page 255
5.4.5 Mobility for an electron inversion layer......Page 256
5.4.7 Multiple scattering mechanisms and Matthiessen's rule......Page 259
5.5.1 Drift–Diffusion model......Page 261
5.5.2 Analytical models for the MOSFET drain current......Page 264
5.6 The ballistic transport regime......Page 266
5.6.1 Carrier distribution in a ballistic MOSFET......Page 267
5.6.2 Ballistic current in a MOSFET......Page 270
5.6.3 Compact formulas for the ballistic current......Page 272
5.6.4 Injection velocity and subband engineering......Page 274
5.7.1 Compact formulas for the quasi-ballistic current......Page 276
5.7.2 Back-scattering coefficient......Page 279
5.7.3 Critical analysis of the quasi-ballistic model......Page 281
5.8 Summary......Page 283
References......Page 284
6 The Monte Carlo method for the Boltzmann transport equation......Page 288
6.1 Basics of the MC method for a free-electron-gas......Page 289
6.1.1 Particle dynamics......Page 290
6.1.2 Carrier scattering and state after scattering......Page 293
6.1.3 Boundary conditions......Page 299
6.1.4 Ohmic contacts......Page 302
6.1.5 Gathering of the statistics......Page 303
6.1.6 Enhancement of the statistics......Page 305
6.1.7 Estimation of the current at the terminals......Page 307
6.1.8 Full band Monte Carlo......Page 308
6.1.9 Quantum corrections to free carrier gas MC models......Page 310
6.2 Coupling with the Poisson equation......Page 311
6.2.1 Poisson equation: linear and non-linear solution schemes......Page 312
6.2.3 Charge and force assignment......Page 313
6.2.5 Stability......Page 316
6.3.1 Flowchart of the self-consistent MSMC method......Page 321
6.3.2 Free-flight, state after scattering and boundary conditions......Page 323
6.3.4 Multi-subband Monte Carlo transport for holes......Page 324
6.4 Summary......Page 326
References......Page 327
7.1.1 Measurement and representation of mobility data......Page 334
7.1.2 Low field mobility in bulk devices......Page 339
7.1.3 Low field mobility in SOI devices......Page 344
7.2 Far from equilibrium transport......Page 348
7.2.1 High field transport in uniform samples......Page 349
7.2.2 High field transport in bulk and SOI devices......Page 350
7.3.1 Ballistic and quasi-ballistic transport......Page 352
7.3.2 Voltage dependence and gate length scaling......Page 358
7.4 Summary......Page 361
References......Page 362
8.1.1 Definitions......Page 368
8.1.2 Subband energy and in-plane dispersion relationship......Page 370
8.1.3 Carrier dynamics......Page 372
8.1.4 Change of the coordinates system......Page 373
8.1.5 Scattering rates......Page 377
8.2 Hole inversion layers......Page 378
8.3 Simulation results......Page 379
8.3.1 Mobility in electron and hole inversion layers......Page 380
8.3.2 Drain current in n- and p-MOSFETs......Page 382
8.4 Summary......Page 384
References......Page 385
9.1 Fabrication techniques for strain engineering......Page 386
9.1.1 Global strain techniques......Page 387
9.1.2 Local strain techniques......Page 388
9.2.1 Stress: definitions and notation......Page 389
9.2.2 Strain: definitions and notation......Page 390
9.2.3 Strain and stress relation: the elastic constants......Page 392
9.2.4 Change of coordinate systems for strain and stress......Page 394
9.2.5 Biaxial strain......Page 396
9.2.6 Uniaxial strain......Page 399
9.3 Band structure in strained n-MOS transistors......Page 402
9.3.1 Strain effects in the bulk silicon conduction band......Page 403
9.3.2 Biaxial and uniaxial strain in n-MOS transistors......Page 407
9.4.1 The k·p model for holes in the presence of strain......Page 412
9.4.2 Biaxial and uniaxial strain in p-MOS transistors......Page 413
9.5 Simulation results for low field mobility......Page 414
9.6 Simulation results for drain current in MOSFETs......Page 418
9.7 Summary......Page 419
References......Page 421
10.1 Alternative gate materials......Page 426
10.2 Remote phonon scattering due to high-κ dielectrics......Page 427
10.2.1 Field propagation in the stack......Page 429
10.2.2 Device structure with an infinite dielectric......Page 431
10.2.3 Device structure with ITL/high-κ/metal-gate stack......Page 436
10.2.4 Calculation of the scattering rates......Page 440
10.3.1 Scattering matrix elements......Page 443
10.4 Simulation results for MOSFETs with high-κ dielectrics......Page 445
10.5 Alternative channel materials......Page 450
10.5.1 Ballistic transport modeling of alternative channel devices......Page 451
10.5.2 Energy reference in alternative channel materials......Page 454
10.6.1 Conduction band and phonon parameters......Page 455
10.6.2 Electrons: velocity and low field mobility......Page 457
10.6.3 Holes: band structure and low field mobility......Page 459
10.7.1 Conduction band parameters......Page 460
10.7.2 Phonon scattering......Page 461
10.7.3 Simulation results......Page 463
10.8 Summary......Page 464
References......Page 465
A.1 Fourier transform......Page 471
A.3 Fermi integrals......Page 473
References......Page 474
B Integrals and transformations over a finite area A......Page 475
C.1 Three dimensional hole gas......Page 477
C.2 Two dimensional hole gas......Page 478
D Matrix elements beyond the envelope function approximation......Page 481
E Charge density produced by a perturbation potential......Page 484
Index......Page 488