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, 2000.- multiple research articles (reference list at the end of this lecture)

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Courtesy of S. Salahuddin (UC Berkeley)

Dispersion Relationship: E vs. k

~ eikx called the plane wave solutionk is called the wave vector

Ec

Ev

K.E. of electrons

K.E. of holes

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• E vs. distance in semiconductors

• Schrödinger equation

If V(r) = 0 (free electron): m → m0If V(r) = U( ) (electron in crystal): m → m*

Semiconductor Band Structure

L L

43210-1-2-3-4

GaAs Si

Ener

gy (e

V)

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Zinc-blende structure

• E vs. k

• Reciprocal space Fourier Transform as

Silicon Band Structure• Equi-energy contours

Electrons HolesHeavy Hole

Light Hole

Split-off

• E-k relationship is direction-dependent for electrons in solids.• In Si, electrons have a more parabolic E-k relationship than holes.

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Valley degeneracy: 2Spin degeneracy: 2

Valley degeneracy: 1Spin degeneracy: 2

T. Guillaume, SSE (2006)

Quantum Confinement (QC) Effects• Sub-bands

• Reduced density-of-states (DOS)

In a MOS Inversion Layer:

Position

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Sub-band top/valley energies due to QC:

F. Stern, PRB (1972)

3D Bulk:

2D Inversion Layer:

Gate

S D EFM

EC

QC Effect on Si Band Structure: Electrons

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0 5 10 15 20 25

0.00

0.05

0.10

0.15

0.20

"Bulk" EC

4, 1st

2, 2nd

2, 1st

Di t fDepth from oxide/Si interface (nm)

Electron Sub-band Energies

(100) Si Ns=2×1012 cm-2

∆

∆Valley degeneracy:

Δ2: 2 Δ4: 4Spin degeneracy: 2Δk

E+ΔEE

QC Effect on Si Band Structure: Holes

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kz=0

N. Xu, Ph.D. thesis (2012)

QC

0 5 10 15 20 25-0.20

-0.15

-0.10

-0.05

0.00

0.05

LH-SO, 1st

"Bulk" EV

HH, 2nd

HH, 1st

Depth from oxide/Si interface (nm)

Hole Sub-band Energies

(100) Si Ns=2×1012 cm-2

Valley degeneracy: HH: 1 LH-SO: 2

Spin degeneracy: 2

CTinv

Impacts of QC on MOS Electrostatics

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• Inversion Thickness (Tinv)

• Quantum Capacitance

Measured Tinv in Si Bulk CMOSFETs

K. Yang, VLSI-T (1999)Cox

Cdep

Cox

Cdep

CTinv

Cox

Cdep

Cox

Cinv Cdep Cinv

Classic QM

OFF

ON

Physical Oxide Thickness (tox) Effective Oxide Thickness (EOT) Electrical Effective Oxide

Thickness (EOTelec)

Hence there are 3 metrics to characterize Cgate …

0 5 10 150

1

2

3

4

5

6

7

0 5 10 15Distance from Top SiO2/Si Interface (nm)

Inve

rsio

n H

ole

Con

cent

ratio

n (a

.u.)

Increasing gate voltage

Carrier Mobility: A Quantum Mechanical View

2*

, ' '( ) ( )n n n nz

F dz z z

, ,2 2

1 ( ) (1 ( ))(2 )i j

n nn n ni j n n

B n i jk

e g dk E E f E f Ek T n k k

, ,

1i j i j

tot nn

ntot

nn

' '

' ' '',

2 ( , , , ) ( ( ) ( ) )n nnk n kS M n k n k E k E k

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• Fermi Golden Rule

• mobility

for the nth sub-band

for the overall inversion layer

E

k(n,k)

(n’,k’)

' '

'' '

2 ,'

1 ( , )(2 )( ) nk n k

nn k

d k S nk n kk

scattering rates for sub-band n at k

(n,k)

sub-b 1 sub-b n’… …wavefuntion overlap along confinement direction

: transition rates between two quantum states

: scattering-induced quantum state transitions

Transition Types of Carriers

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∆

∆Valley degeneracy:

Δ2: 2 Δ4: 4

Electrons Holes

1. intra-valley scattering2. g-type inter-valley scattering3. f-type inter-valley scattering

1. intra-sub-band scattering2. Inter-sub-band scattering

……

' '

'' '

2 ,'

1 ( , )(2 )( ) nk n k

nn k

d k S nk n kk

Momentum Relaxation Factor

Acoustic and Optical Phonon Scatterings

' '

2' '

. 2 , , ,( , , , ) B

elast n k n kl

k TM n k n k Fu

' '

22' '

. , , ,

1 1( , , , ) ( )2 ( ) 2 2

tinelast opn k n k

D kM n k n k F n

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Acoustic Phonon Scattering Optical Phonon Scattering

Sca

tterin

g R

ates

(S-1

)

Sca

tterin

g R

ates

(S-1

)

Carrier Kinetic Energy (eV) Carrier Kinetic Energy (eV)

elastic scattering inelastic scattering

AP: coherent movements of atoms, i.e. adjacent atoms move together.

OP: out-of-phase movements of atoms, i.e. adjacent atoms move in opposite directions.

Surface Roughness Scattering

2' '

2 .' '2

( , , , )( , , , )

( )unscr

screen

M n k n kM n k n k

q

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Considering screening effect (important at high gate voltage): Dielectric function

Power Spectrum of Surface Roughness

Source

DrainEc

KEPE Ideal

Real

2

' ' 0 0 '. . ' '

( )( , , , ) ( ) ( ) ( ) ( )nsurf rough n n n n n

z z

d zM n k n k z F z dz E E z dz S qdz

E

causing very fast carrier momentum change!

L. Donetti, JAP (2009)• Type: elastic scattering, mostly intra-valley/band• Mechanism:

Coulomb Scatterings

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• Type: elastic scattering, mostly intra-band/valley• Mechanism:

Nit

F. Driussi, TED (2009)

SiO2/Si Interface Charge

D. Esseni, TED (2003)

Depletion Charge

Qdep

MRT (1/Scattering Rates)

F. Driussi, TED (2009)

Effective Transverse Field (Eeff)

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≡

Gate

S D

ECDepth

In a MOS Inversion Layer:

Surface (oxide/Si interface) field:

Bottom (of inversion layer) field:

Average field:

Xdep

E

∙More generally: For (100) bulk Si MOSFET:

Electrons: α = 0.5Holes: α = 0.33

2D Sheet Charge Density:

Depletion Approximation

Reality

Universal Mobility Curve: vs. Eeff

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S. Takagi, TED (1994)

- enhanced screening effect

- Increased DOS effect

- Increased SR scattering

• Unify the electric field value by including the oxide thickness (compared to vs. VG) and depletion charge effect (compared to vs. Ninv).

Measured Si Universal Curves

Universal Mobility Curve: Dependent Factors

Y. Zhao, IEDM (2008)

O. Weber, VLSI-T (2007)

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Temperature Band Structure

• Si universal mobility curves are often used to show a new technology’s enhancement.

Gate

S D

High--induced Scatterings

R. Chau, EDL (2004)

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• Remote Coulomb Scattering

• Remote (Surface Optical) Phonon Scatterings

-(tSiO2+tHK) -tSiO2 0

Metal high-κ SiO2 SiGate

Nit

Coupled ModesHκ Modes

SiO2 Modes

Si Modes(AP, OP)

SiO2 ILHigh-κ

Courtesy of D. Vesileska (ASU)

C.-Y. Lu, EDL (2006)

Dependence of High- Thickness

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K. Choi, VLSI-T (2009)

• Mobility doesn’t follow universal curve.

• By using metal-gate technology, RCS becomes the limiting factor in state-of-the-art MOSFET’s mobility.

Si Carrier Velocity Saturation

M. Saitoh, IEDM (2009)

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B. Ho, TED (2011)

• Under high lateral electric field, carrier velocity saturates to a constant value, due to dramatically enhanced optical phonon scatterings.

• 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 VDD

ReferencesBand Structure1. 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. 701-708, 2006.2. N. Xu, “Effectiveness of Strain Solutions for Next-Generation MOSFETs,” Ph.D. Thesis, University of

California at Berkeley, 2012.3. F. Stern, “Self-Consistent Results for n-Type Si Inversion Layers,” Physical Review B, Vol. 5. No. 12,

pp. 4891-4899, 1972.4. K. Yang, Y.-C. King, C. Hu, "Quantum Effect in Oxide Thickness Determination From Capacitance

Measurement," Symposium on VLSI Technology Digest, pp. 77-78, 1999.

Mobility5. (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. 2445-2455, 2003.

6. (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, 023705, 2009.

7. (CS) F. Diussi, D. Esseni, “Simulation Study of Coulomb Mobility in Strained Silicon,” IEEE Transactions on Electron Devices, Vol.56, no.9, pp. 2052-2059, 2009.

8. (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. 1665-1674, 2003.

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References9. (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. 2357-2362, 1994.

10. 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. 577-580, 2008.

11. O. Weber, S. Takagi, “New Findings on Coulomb Scattering Mobility in Strained-Si nFETs and its Physical Understanding,” Symposium on VLSI Technology Digest, pp. 130-131, 2007.

12. (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. 859-861, 2006.

13. (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. 408-410, 2004.

14. 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. 138-139, 2009.

15. 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. 2895-2902, 2011.

16. 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. 469-472, 2009.

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