EE-612: Lecture 11: Effective Mobility Mark Lundstrom Electrical - - PowerPoint PPT Presentation

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EE-612: Lecture 11: Effective Mobility

Mark Lundstrom Electrical and Computer Engineering Purdue University West Lafayette, IN USA Fall 2008

www.nanohub.org

NCN

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• utline

1) Review of mobility 2) “Effective” mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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mobility

μ = qτ m*

(1/τ ) Δt = probability per second of scattering [(1/τ ) is ‘scattering rate’]

μ1 = qτ1 m* μ2 = qτ 2 m* μ12 = ? μ12 ≠ μ1 + μ2 μ12 = 1 μ1 + 1 μ2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−1

τ = average time between scattering events

Mathiessen’s Rule

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diffusion coefficient

D = υTλ 2 = υRλ D μ = kBT q

D ↔ μ

unidirectional thermal velocity Richardson velocity mean-free-path for scattering

υT υR λ

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mobility vs. doping density (silicon)

μ(NI ) NI = ND NA

( )

NCR ≈ 1017 cm-3

1360(480 holes) 100(80)

μII : T 3/2 NI μ = 1 μL + 1 μII ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−1

μ NI << NCR

( )≈ μL

μL : T −3/2 μ NI >> NCR

( )≈ μI

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μII temperature dependence (silicon)

μ(T ) T 1 2 m*υ 2 = 3 2 kBT μII : T 3/2 +

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lattice scattering (silicon)

μ(T ) T μII : T 3/2 μL : T −3/2 μTOT (T ) NP ~ kBT hω (acoustic phonons) ~ (optical phonons)

B

k T P

N e

ω −

μTOT = 1 μL + 1 μII ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−1

Mathiessen’s Rule

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comment (added after lecture)

E(k) k ω(k) β

electrons phonons

EC π a π a − π a − π a

ω0

υS = ω k

AP OP

Electrons in a periodic structure are characterized by “dispersion curve” E(k). Phonons in a periodic structure are characterized by “dispersion curve” ω (β). Two types of phonons, acoustic (AP) and optical (OP).

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high-field transport (silicon)

−υy Ey υy = −μoEy 1+ Ey Ecr

( )

2

104 υy = −μoEy υsat 107 Ey >> Ecr → υy → μoEcr ≡ υsat Ecr = υsat μ

electric field along the direction of current flow

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field-dependent mobility

μ(Ey) Ey 104 μo μ(Ey) ≡ −υy Ey = μo 1+ Ey Ecr

( )

2

υy = μ Ey

( )Ey

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• utline

1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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mobility in an inversion layer

VD VS VB VG

source drain

y x μn NA,T,Ey,Ex ?

( )

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effective mobility

ID = −WQi y

( )υy = WCG VG − VT − mV(y)

[ ]μn(x,y) dV

dy ID dy

L

∫

= WCG μeff VG − VT − mV(y)

[ ]dV

VDS

∫

μeff = n(x)μn(x)dx

∞

∫

n(x)dx

∞

∫

(If we only consider the x- dependence, i.e. normal to the surface)

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effective normal field

μeff (Eeff ) Eeff Eeff ≡ n(x)Ex(x)dx

∞

∫

n(x)dx

∞

∫

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effective normal field

ES = qNAW εSi E W / 2

( )= qNAW

2εSi W x NA

−

below threshold

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effective normal field (ii)

ES = QS εSi WDM x NA

−

above threshold electron inversion layer

QI

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effective normal field (iii)

ES = QS εSi Ex x WDM tinv ES = E 0

( )

E tinv

( )

ES = 1 εSi QDM + QI

( )

E(tinv) = QDM εSi Eeff = Ex tinv / 2

( )

Eeff = 1 εSi QDM + QI 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

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effective normal field (iv)

Eeff = 1 εSi QDM + QI 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ VT = VFB + 2ψ B + QDM Cox QI = CG VGS − VT

( )

Eeff = Cox εSi VT − VFB − 2ψ B + VG 2 − VT 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

(1)

QDM = Cox VT − VFB − 2ψ B

( )

QI ≈ Cox VGS − VT

( )

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effective normal field (v)

Eeff = COX εSi VT − VFB − 2ψ B + VG 2 − VT 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ VFB = − EG 2q −ψ B Eeff = εOX εSi tOX VG + VT 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + EG 2q −ψ B ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

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• utline

1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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transport in bulk Si

z x y

ml

* = 0.9m0

mt

* = 0.19m0

Si conduction band mC

* =

2 6ml

* +

4 6mt

*

⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−1

= 0.26m0

dominant scattering processes:

(low-field, room temperature)

• acoustic phonons (ADP)
• ionized impurities (II)
• intervalley phonons (IV)

under low (and modest) fields:

• 6 equivalent ellipsoids
• n/6 electrons in each one

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transport in Si inversion layers

is different from transport in bulk Si….. z x y

ml

* = 0.9m0

mt

* = 0.19m0

Si conduction band

x

energy -->

W ∞ ∞ ε1 ε3 ε2 ′ ε1 ′ ε2

EF

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transport in Si inversion layers

z x y

ml

* = 0.9m0

mt

* = 0.19m0

Si conduction band

expectations: most carriers in unprimed subbands

mC

* ≈ 0.19m0

• lighter conductivity m*
• suppressed intersubband scattering

X

• enhanced intra subband phonon

scattering

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surface roughness

VD VS VB VG

source drain

μeff = 1 μL + 1 μII + 1 μSR ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

−1

Si : SiO2 interface is rough! (possibly present too)

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effective mobility vs. effective field

μeff Ex or VG μII μSR μL

1) screening 2) electron confinement 3) proximity to the surface

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surface roughness scattering

δtOX SiO2 Si Δ r r

( )

VG = VFB +ψ S − QS Cox δtOX ⇒ δψ S

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from Jing Wang, et al., Appl. Phys. Lett., Aug. 2005

surface roughness scattering (ii)

δtOX ⇒ δψ S

EC EV EF VG

εi

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‘universal’ mobility

μeff Eeff

increasing NA

NA1 NA4 Eeff = 1 εSi QDM + QI 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

electrons:

Eeff = 1 εSi QDM + QI 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

holes: universal behavior

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• utline

1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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measuring μeff

ID = W L μeffQiVDS

ID VDS

VGS

Qi(VG) ≈ CG VGS −VT

( )

RCH = VDS ID = L WμeffQ

i

μeff (VG) = L W RCH VG

( )

Qi(VG) RCH >> RSD

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measuring μeff (ii)

VG ID VT

VDS = 10mV

QI VG

( )=

Cgs

VG

∫

VG

( )dVG

VG QI

QI VG

( ) = Cox VGS − VT ( )

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universal mobility for electrons

• S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev.,

41, pp. 2357-2362, 1994

Effective field (MV/cm) Effective mobility (cm2/V -s)

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universal mobility for holes

• S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev.,

41, pp. 2357-2362, 1994

Effective field (MV/cm) Effective mobility (cm2/V -s)

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• utline

1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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“field-effect” mobility

ID = W L μeffQI VGS

( )

VDS ≈ W L μeffCG VGS −VT

( )

VDS gm = ∂ID ∂VGS VDS = W L μ CGVDS μFE ≡ L WCGVDS gm

(assumes μ is independent of VG)

For a discussion of mobility measurement techniques, see: Narain Arora, MOSFET Models for VLSI Circuit Simulation, Theory and Practice, Springer-Verlag, New York, 1993.

“field-effect mobility”

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mobility and on-current

ID = W L μeffCG VGS −VT

( )

VDS

Linear region current is proportional to the effective mobility: What about the on-current?

ID = WCGυsat VGS −VT

( )

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mobility and on-current (ii)

ID = WCG T 2 −T % υT V

GS −V T

( )

Dn = υTλ 2 = kBT q

( )μn

μn = qτ m*

smaller m* implies larger ballistic velocity:

υT = 2kBT πm*

• ε1 x

( )

x E

T = λ λ + l

( )

also:

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effective mobility for the 45 nm technology node

Eeff = εOX εSi tOX (VG + VT ) 2 + EG 2q −ψ B

[ ]

tOX = 1.1 nm V

T ≈ 0.25 V

V

G = VDD = 1.0 V

NA ≈ 2.7 ×1018 cm-3 →ψ B = 0.48 Eeff VG = 1V

( )≈ 2 MV/cm

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universal mobility for electrons

• S. Takagi, A. Toriumi, M. Iwase, and H. Tango, IEEE Trans. Electron Dev.,

41, pp. 2357-2362, 1994

Effective field (MV/cm) Effective mobility (cm2/V -s)

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the tyranny of the effective mobility*

Eeff = εOX εSi tOX (VG + VT ) 2 + EG 2q −ψ B

[ ]

each technology generation, device scaling increases NA, decreases tox --> Eeff increases and mobility decreases mobility decreases each technology generation, unless we can find ‘new’ materials with higher mobilities (e.g. strained silicon).

(* Dimitri Antoniadis at MIT describes the problem this way.)

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strain engineering

C.-H. Jan, et al., “A 65nm Ultra Low Power Logic Platform Technology using Uni-axial Strained Silicon Transistors,” Intel Corporation, IEDM 2005,

PMOS

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• utline

1) Review of mobility 2) Effective mobility 3) Physics of the effective mobility 4) Measuring effective mobility 5) Discussion 6) Summary

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summary

1) Effective mobility is an important device parameter for both the linear and saturated region currents 2) Effective mobility is strongly reduced by surface roughness scattering 3) Mobility enhancement through strain engineering has been an important performance booster in recent years.