Courtesy of S. Salahuddin (UC Berkeley) Lecture 4

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Courtesy of S. Salahuddin (UC Berkeley) Lecture 4 MOSFET Transport Issues semiconductor band structure quantum confinement effects low-field mobility and high-field saturation Reading: - M.

1 Courtesy of S. Salahuddin (UC Berkeley) Lecture 4 MOSFET Transport Issues semiconductor band structure quantum confinement effects low-field mobility and high-field saturation Reading: - M. Lundstrom, Fundamentals of Carrier Transport, 2nd edition, Cambridge University Press, multiple research articles (reference list at the end of this lecture) 9/23/2013 Nuo Xu EE 290D, Fall

2 Dispersion Relationship: E vs. k E vs. distance in semiconductors E c K.E. of electrons E v K.E. of holes Schrödinger equation ~ eikx called the plane wave solution k is called the wave vector If V(r) = 0 (free electron): m m 0 If V(r) = U() (electron in crystal): m m* 2

3 Semiconductor Band Structure Reciprocal space Zinc-blende structure Fourier Transform as E vs. k GaAs Si Energy (ev) L L 3

4 Silicon Band Structure Equi-energy contours Electrons Valley degeneracy: 2 Spin degeneracy: 2 Holes Valley degeneracy: 1 Spin degeneracy: 2 Heavy Hole Light Hole Split-off T. Guillaume, SSE (2006) E-k relationship is direction-dependent for electrons in solids. In Si, electrons have a more parabolic E-k relationship than holes. 4

5 Quantum Confinement (QC) Effects Sub-bands In a MOS Inversion Layer: Gate E C S D E FM Position Sub-band top/valley energies due to QC: Reduced density-of-states (DOS) 3D Bulk: F. Stern, PRB (1972) 2D Inversion Layer: 5

QC Effect on Si Band Structure: Electrons 0.20 Electron Sub-band Energies 0.15 "Bulk" E C 0.10 0.

6 QC Effect on Si Band Structure: Electrons 0.20 Electron Sub-band Energies 0.15 "Bulk" E C , 2 nd 4, 1 st 2, 1 st E Δk E+ΔE Valley degeneracy: Δ 2 : 2 Δ 4 : 4 Spin degeneracy: (100) Si N s = cm Di t f Depth from oxide/si interface (nm) 6

k z =0 QC Effect on Si Band Structure: Holes QC

2 0.05 0.00 Hole Sub-band Energies "Bulk" E V -0.

7 k z =0 QC Effect on Si Band Structure: Holes QC Valley degeneracy: HH: 1 LH-SO: 2 Spin degeneracy: Hole Sub-band Energies "Bulk" E V HH, 1 st HH, 2 nd LH-SO, 1 st (100) Si N s = cm Depth from oxide/si interface (nm) N. Xu, Ph.D. thesis (2012) 7

8 Impacts of QC on MOS Electrostatics Inversion Thickness (T inv ) 7 Measured T inv in Si Bulk CMOSFETs Quantum Capacitance Classic OFF ON Inversion Hole Concentration (a.u.) Distance from Top SiO 2 /Si Interface (nm) C dep C ox C dep C ox Increasing gate voltage C ox C dep C ox QM C T inv C T inv C inv C dep C inv K. Yang, VLSI-T (1999) Hence there are 3 metrics to characterize C gate Physical Oxide Thickness (t ox ) Effective Oxide Thickness (EOT) Electrical Effective Oxide Thickness (EOT elec ) 8

9 n Carrier Mobility: A Quantum Mechanical View Fermi Golden Rule mobility 2 ' ' ' S M( n, k, n, k ) ( E ( ) ' ', n k En' ( k ) ) nk n k 2 F dz () z () z wavefuntion overlap along confinement direction i, j 2 2, * nn, ' n n' z ' 1 dk ' ' S ( nk, n k ) 2 ' ', ' (2 ) nk n k n( k) n k e 1 gn dk n En En i j f ( E n) (1 f ( E n)) kt n (2 ) k k scattering rates for sub-band n at k B n k i j tot 1 n : transition rates between two quantum states : scattering-induced quantum state transitions i, j i, j tot n n n n for the overall inversion layer for the n th sub-band 9 (n,k) (n,k) E (n,k ) sub-b 1 sub-b n k

10 Transition Types of Carriers Electrons Holes Valley degeneracy: Δ 2 : 2 Δ 4 : 4 1. intra-valley scattering 2. g-type inter-valley scattering 3. f-type inter-valley scattering 1. intra-sub-band scattering 2. Inter-sub-band scattering 1 n ' dk (2 ) S 2 ' ' nk, n k ( k) n ' k ' ' ( nk, n k ) Momentum Relaxation Factor 10

11 Acoustic and Optical Phonon Scatterings Acoustic Phonon Scattering Optical Phonon Scattering AP: coherent movements of atoms, i.e. adjacent atoms move together. elastic scattering OP: out-of-phase movements of atoms, i.e. adjacent atoms move in opposite directions. inelastic scattering M ( n, k, n, k ) B F elast 2 ' ' kt 2 ' ' Dk t inelast. u nkn k l. 2 ' ',,, 1 1 M ( n, k, n, k ) F n ( ) ' ' 2 ( ) nkn,,, k op Scattering Rates (S -1 ) Scattering Rates (S -1 ) Carrier Kinetic Energy (ev) Carrier Kinetic Energy (ev) 11

12 Source Surface Roughness Scattering L. Donetti, JAP (2009) Type: elastic scattering, mostly intra-valley/band Mechanism: Real KE PE Ideal causing very fast carrier Drain momentum change! E c 2 d ( z) M ( nkn,,, k) ( z) F ( zdz ) E E ( zdz ) Sq ( ) ' ' 0 0 n ' surf. rough. n n ' n n ' n dz z z Power Spectrum of Surface Roughness E Considering screening effect (important at high gate voltage): M ' ' screen( n, k, n, k ) 2 ' ' 2 M unscr. ( nkn,,, k) 2 ( q) Dielectric function 12

Depletion Charge MRT (1/Scattering Rates) N it Q dep F.

13 Coulomb Scatterings Type: elastic scattering, mostly intra-band/valley Mechanism: SiO 2 /Si Interface Charge Depletion Charge MRT (1/Scattering Rates) N it Q dep F. Driussi, TED (2009) D. Esseni, TED (2003) F. Driussi, TED (2009) 13

14 Effective Transverse Field (E eff ) In a MOS Inversion Layer: S Gate E C Depth Surface (oxide/si interface) field: Bottom (of inversion layer) field: Average field: More generally: D E X dep Reality Depletion Approximation 2D Sheet Charge Density: For (100) bulk Si MOSFET: Electrons: α = 0.5 Holes: α =

15 Universal Mobility Curve: vs. E eff Unify the electric field value by including the oxide thickness (compared to vs. V G ) and depletion charge effect (compared to vs. N inv ). Measured Si Universal Curves - enhanced screening effect - Increased SR scattering - Increased DOS effect S. Takagi, TED (1994) 15

16 Universal Mobility Curve: Dependent Factors Temperature Band Structure Y. Zhao, IEDM (2008) O. Weber, VLSI-T (2007) Si universal mobility curves are often used to show a new technology s enhancement. 16

Lu, EDL (2006) -(t SiO2 +t HK ) -t SiO2 0 Remote (Surface Optical) Phonon

17 High--induced Scatterings Remote Coulomb Scattering N it Metal high-κ SiO 2 Gate Si C.-Y. Lu, EDL (2006) -(t SiO2 +t HK ) -t SiO2 0 Remote (Surface Optical) Phonon Scatterings R. Chau, EDL (2004) Hκ Modes Coupled Modes S Gate High-κ SiO 2 IL Si Modes (AP, OP) SiO 2 Modes D Courtesy of D. Vesileska (ASU) 17

18 Dependence of High- Thickness Mobility doesn t follow universal curve. K. Choi, VLSI-T (2009) By using metal-gate technology, RCS becomes the limiting factor in state-of-the-art MOSFET s mobility. 18

19 Si Carrier Velocity Saturation Under high lateral electric field, carrier velocity saturates to a constant value, due to dramatically enhanced optical phonon scatterings. B. Ho, TED (2011) M. Saitoh, IEDM (2009) State-of-the-art technologies actually pushes away MOSFETs from velocity saturation, due to: Increased doping/junction defects High--induced scatterings Ballistic transport (will be discussed in Lec. 5) Reduced V DD 19

20 Band Structure References 1. T. Guillaume, M. Mouis, Calculations of Hole Mass in [110]-Uniaxially Strained Silicon for the Stress Engineering of p-mos Transistors, Solid-State Electronics, Vol. 50, pp , N. Xu, Effectiveness of Strain Solutions for Next-Generation MOSFETs, Ph.D. Thesis, University of California at Berkeley, F. Stern, Self-Consistent Results for n-type Si Inversion Layers, Physical Review B, Vol. 5. No. 12, pp , K. Yang, Y.-C. King, C. Hu, "Quantum Effect in Oxide Thickness Determination From Capacitance Measurement," Symposium on VLSI Technology Digest, pp , Mobility 5. (PS) D. Esseni, A. Abramo, L. Selmi, E. Sangiorgi, Physically Based Modeling of Low Field Electron Mobility in Ultrathin Single- and Double-Gate SOI n-mosfets, IEEE Transactions on Electron Devices, Vol.50, no.12, pp , (SRS) L. Donetti, F. Gamiz, N. Rodriguez, A. Godoy, C. Sampedro, The Effect of Surface Roughness Scattering on Hole Mobility in Double Gate Silicon-on-Insulator Devices, Journal of Applied Physics, Vol.106, , (CS) F. Diussi, D. Esseni, Simulation Study of Coulomb Mobility in Strained Silicon, IEEE Transactions on Electron Devices, Vol.56, no.9, pp , (CS) D. Esseni, A. Abramo, Modeling of Electron Mobility Degradation by Remote Coulomb Scattering in Ultrathin Oxide MOSFETs, IEEE Transactions on Electron Devices, Vol.50, no.7, pp ,

21 References 9. (universal curve) S. Takagi, A. Toriumi, M. Iwase, H. Tango, On the Universality of Inversion Layer Mobility in Si MOSFET s: Part I Effects of Substrate Impurity Concentration, IEEE Transactions on Electron Devices, Vol.41, no.12, pp , Y. Zhao, M. Takenaka, S. Takagi, Comprehensive Understanding of Surface Roughness and Coulomb Scattering Mobility in Biaxially-Strained MOSFETs, IEEE International Electron Device Meeting Technical Digest, pp , O. Weber, S. Takagi, New Findings on Coulomb Scattering Mobility in Strained-Si nfets and its Physical Understanding, Symposium on VLSI Technology Digest, pp , (high- trap) C.-Y. Lu, K.-S. C.-Liao, P.-H. Tsai, T.-K. Wang, Depth Profiling of Border Traps in MOSFET With High-κ Gate Dielectric by Charge-Pumping Technique, IEEE Electron Device Letters, Vol. 27, no.10, pp , (high-) R. Chau, S. Datta, M. Doczy, B. Doyle, J. Kavalieros, M. Metz, High-/Metal Gate Stack and Its MOSFET Characteristics, IEEE Electron Device Letters, Vol.25, no.6, pp , K. Choi, H. Jagannathan, C. Choi, L. Edge, T. Ando et al., Extremely Scaled Gate First High-/Metal Gate Stack with EOT of 0.55nm Using Novel Interfacial Layer Scavenging Techniques for 22nm Technology Node and Beyond, Symposium on VLSI Technology Digest, pp , B. Ho, N. Xu, T.-J. King Liu, Study of High-Performance Ge pmosfet Scaling Accounting for Direct Source-to-Drain Tunneling, IEEE Transactions on Electron Devices, Vol.58, no.9, pp , M. Saitoh, N. Yasutake, Y. Nakabayashi, K. Uchida, T. Numata, Understanding of Strain Effects on High-Field Carrier Velocity in (100) and (110) CMOSFETs under Quasi-Ballistic Transport, IEEE International Electron Device Meeting Technical Digest, pp ,

Lecture 9. Strained-Si Technology I: Device Physics

Lecture 9. Strained-Si Technology I: Device Physics Strain Analysis in Daily Life Lecture 9 Strained-Si Technology I: Device Physics Background Planar MOSFETs FinFETs Reading: Y. Sun, S. Thompson, T. Nishida, Strain Effects in Semiconductors, Springer,