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Mark LundstromNetwork for Computational Nanotechnology
Purdue UniversityWest Lafayette, Indiana USA
NCN@Purdue - Intel Summer School: July 14-25, 2008
Physics of Nanoscale Transistors: Lecture 3A:
Theory of theBallistic MOSFET
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outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V characteristic5) Discussion6) Summary
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review: the ballistic MOSFET
IDS =WCox VGS −VT( )υT1− e−qVDS /kBT
1+ e−qVDS /kBT
⎛⎝⎜
⎞⎠⎟
IDS
VDS
VGS
VDSAT ≈ kBT q
“on-current”
finite channel resistance
VGS = VDD
(Boltzmann statistics)
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review: the ballistic MOSFET
IDS =WCox VGS −VT( )υT1− e−qVDS /kBT
1+ e−qVDS /kBT
⎛⎝⎜
⎞⎠⎟
Questions:
1) How do we generalize eqn. (1) for Fermi-Dirac statistics (essential above threshold)? Part 1
2) How do we properly treat MOS electrostatics?(subthreshold as well as above threshold and 2D in addition to 1D) Part 2
(1)
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generic model of a nano-device
Gate
EF1 EF1-qVDD(E −U )
Supriyo Datta, “Concepts in Quantum Transport” www.nanoHUB.org
τ1
γ 1 = h τ1
τ 2
γ 2 = h τ 2
I =2qh
T E( ) f1 − f2( )dE−∞
∞
∫ T E( )= πD E −U( ) γ 1γ 2
γ 1 + γ 2
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thermal equilibrium reservoirs, reflectionless
the ballistic MOSFET and the generic model
E TOP
EC
E C (x)
x
EF1
EF2
‘device’
‘contacts’
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expression for current
ID =2qh
T (E)M (E) f1 − f2( )dE−∞
∞
∫
T E( )= T (E)M (E)
1) semiclassical treatment of T(E)
2) parabolic energy bands for M(E)
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transmission in the ballistic MOSFET
E TOP
EC
E C (x)
x
E TOP
T (E)
E
1
0
EF1
EF2
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energy subbands in a MOSFET
N+ N+
xy
z
TSi
εn =h2n2π 2
2m*TSi2
n = 1,2, 3...
E
x
E C (x)ε1(x)ε2 (x)
ε3(x)
EC (x)→ εn (x)
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outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V Characteristic5) Discussion6) Summary
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current in a ballistic MOSFET
ε1(0)
EC
ε 1(x)
x
ε1(0)
T (E)
E
1
0
EF1
EF2
ID =2qh
T (E)M (E) f1 − f2( )dE−∞
∞
∫
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transverse conducting channels (modes)
ID =2qh
T (E)M (E) f1 − f2( )dE−∞
∞
∫ ID =2qh
M (E) f1 − f2( )dEε1 0( )
∞
∫
M (E) = ?
x
yz WTSi
Assume that there is one subband associated with confinement in the z-direction. Manysubbands associated with confinement in the y-direction
M = # of electron half wavelengths that fit into W.
λ 2lowest mode
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transverse conducting channels below energy, E
E − ε1(0) =
h2k2
2m*
x
yz WTSi
λ 2
M = # of electron half wavelengths that fit into W.
lowest mode
M (E) = intWλ 2( )
k =2πλ M (E) =W
2m* E − ε1(0)[ ]πh
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transverse modes below E (another approach)
M (E) =W g1D E( )
ε1 (0)
E
∫ dE =W2m* E − ε1(0)[ ]
πh
x
yz WTSi
g1D (E) =
1πh
m*
2 E − ε1(0)[ ] #/eV-cm
M(E) = number of transverse modes in the y-direction with energy cut-off less than E.
(Divided by 2 to account for the fact the spin has already been included in the current formula.)
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the drain current
M (E) =W 2m* E − ε1(0)[ ] πhID =2qh
M (E) f1 − f2( )dEε1 0( )
∞
∫
I + =W
2qh
1πh
2m* E − ε1(0)[ ]1+ e E−EF( ) kBT dE
ε1 0( )
∞
∫
I + =2qh
M (E) f1(E) dEε1 0( )
∞
∫
ID = I + − I −
I − =2qh
M (E) f2 (E) dEε1 0( )
∞
∫
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the drain current: cont.
I + =W
2qh
1πh
2m* E − ε1(0)[ ]1+ e E−EF( ) kBT dE
ε1 0( )
∞
∫
I + =W
2qh
2m* kBT( )3/2
πhη1/2
1+ eη−ηF1dη
0
∞
∫
I + =Wq
m*kBT2πh2
⎛⎝⎜
⎞⎠⎟
2kBTπm* F 1/2 ηF1( )=Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟F 1/2 ηF1( )
η ≡ E − ε1(0)[ ] kB T
ηF1 ≡ EF1 − ε1(0)[ ] kB T
F 1/2 ηF( )≡ 2
πη1/2dη
1+ eη−ηF0
+∞
∫
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Fermi-Dirac integrals
F 1/2 ηF( )≡ 2
πη1/2dη
1+ eη−ηF0
+∞
∫
Fermi-Dirac integral of order 1/2
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the drain current: cont.
ηF1 ≡ EF1 − ε1(0)[ ] kB T
ID = I + − I −
I + =Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟F 1/2 ηF1( )
ηF2 ≡ EF2 − ε1(0)[ ] kB T
= EF1 − qVDS − ε1(0)[ ] kB T= ηF1 − qVDS kBT
I − =Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟F 1/2 ηF2( )
ID =Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟
F 1/2 ηF1( )−F 1/2 ηF2( )⎡⎣ ⎤⎦
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the drain current: cont.
EF1 = 0
EF2 = EF1 − qVDS
ε1 x( )
kx
E
x
ηF1 ≡ EF1 − ε1(0)[ ] kB T
ηF2 = ηF1 − qVDS kB T ID =Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟
F 1/2 ηF1( )−F 1/2 ηF 2( )⎡⎣ ⎤⎦
ε1 0( )
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the Landauer picture of a nano-MOSFET
EC
x
EF1 = 0
EF2qVDS
ε 1(0)
top of the barrier
ballistic channel
The gate voltage controls the top of the barrier, and, therefore, ε1(0).
reservoirreservoir
ε 1(x) If we know ε1(0), then we know ηF1 and ηF2.
If we know ηF1 and ηF2, then we know ID.
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outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V Characteristic5) Discussion6) Summary
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the generic model
Gate
EF1 EF1-qVDD(E −U )
τ1 = γ 1 h
N = f1(E)D1(E) + f2 (E)D2 (E)[ ]dE−∞
∞
∫
τ 2 = γ 2 h
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E(k)
k
filling states in a ballistic MOSFET (k-space approach)
ε1(0)
E
x
ε1(x)
f + =1
1+ e E−EF1( ) kBT
f − =1
1+ e E−EF 2( ) kBT
EF1EF1 − qVDS
N = f1(E)D1(E) + f2 (E)D2 (E)[ ]dE−∞
∞
∫
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E(k)
k
filling states in a ballistic MOSFET (k-space approach)
ε1(0)
E
x
ε1(x)
f + =1
1+ e E−EF1( ) kBT
f − =1
1+ e E−EF 2( ) kBT
EF1EF1 − qVDS
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filling states: summary
1) In a ballistic device with contacts in equilibrium and identical connections to the contacts, each state inside the device is in equilibrium - with one of the two contacts.
2) Things are especially simple at the top of the barrier: + velocity states are filled by the source, and - velocity states are filled by the drain.
N = f1(E)D1(E) + f2 (E)D2 (E)[ ]dE−∞
∞
∫
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filling states in a planar MOSFET
Increasing VDS
X (nm) --->-10 -5 0 5 10
ε1 vs. x for VGS = 0.5V
ε 1(e
V)
--->
f (υx, υy)
1) 2)
3) 4)
(Numerical simulations of an L = 10 nm double gate Si MOSFET from J.-H. Rhew and M.S.Lundstrom, Solid-State Electron., 46, 1899, 2002)
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mathematics of filling states (energy space)
ε1(0)
E
xε1(x) L
EF1
EF2 = EF1 − qVDS
nS+ 0( )= D2D E − ε1(0)[ ]
2f1 E( )dE
ε1 (0)
ε1 (top)
∫
y
nS− 0( )= D2D E − ε1(0)[ ]
2f2 E( )dE
ε1 (0)
ε1 (top)
∫
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integration in energy space
nS+ 0( )= D2D E − ε1(0)[ ]
2f1 E( )dE
ε1 (0)
ε1 (top)
∫ D2D E( )= m*
πh2
nS+ 0( )= m*
2πh2
dE1+ e E−EF1( ) kBT
ε1 (0)
∞
∫
nS+ 0( )= m*kBT
2πh2
dη1+ eη−ηF1
=N2 D
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∞
∫ ln 1+ eηF1( )= N2D
2F 0 ηF1( )
η ≡ E − ε1(0)[ ] kB T
ηF1 ≡ EF1 − ε1(0)[ ] kB T
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filling states in a planar MOSFET
f (υx, υy)
1) 2)
3) 4)
nS+ 0( )= N2D
2F 0 ηF1( )
nS− 0( )= N2D
2F 0 ηF2( )
ηF1 ≡ EF1 − ε1(0)( ) kB T
N2D =
m*kBTπh2 #/cm2
ηF2 ≡ ηF1 − qVDS kB T
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filling states in a planar MOSFET
f (υx, υy)
1) 2)
3) 4)
Note that the positive half in case 4) (VDS >> kBT/q) is larger than the positive half in case 1) (VDS = 0). Why?
Answer: Because the total number of carriers is fixed at:
nS (0) =COX
qVGS −VT( )
How does this happen?
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low VDS
ε1
ε 1(x)x
EF1 EF2
E
k
nS (0) = nS+ (0) + nS
− (0)nS+ (0) ≈ nS
− (0)
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high VDS
ε1
ε 1(x)x
EF1
EF2
E
k
nS (0) ≈ nS+ (0)
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outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V Characteristic5) Discussion6) Summary
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re-cap
nS 0( )= N2D
2F 0 ηF1( )+F 0 ηF2( )⎡⎣ ⎤⎦
(1)
(2)
Solve (2) for N2D, then insert in (1):
ID =Wq
N2 D
2υT
⎛⎝⎜
⎞⎠⎟
F 1/2 ηF1( )−F 1/2 ηF2( )⎡⎣ ⎤⎦
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I-V characteristic
ID =WqnS 0( )υT
F 1/2 ηF1( )−F 1/2 ηF 2( )F 0 ηF1( )+F 0 ηF2( )
⎡
⎣⎢
⎤
⎦⎥
qnS 0( )≈ Cox VGS −VT( ) (simple, 1D MOS electrostaticsVGS > VT)
ID =WqnS 0( ) υT
F 1/2 ηF1( )F 0 ηF1( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
1−F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
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final result
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 10 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( ) = υT
F 1/2 ηF1( )F 0 ηF1( )
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using the final result
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 10 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( ) = υT
F 1/2 ηF1( )F 0 ηF1( )
(1)
(2)
Cox VGS −VT( )= q
N2D
2F 0 ηF1( )+F 0 ηF2( )⎡⎣ ⎤⎦ (3)
ηF2 = ηF1 − qVDS kBT (4)
Given, VGS and VDS, solve (3) for ηF1then solve (2) for the ‘ballistic injection velocity.’Finally, solve (1) for ID.
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outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V Characteristic5) Discussion6) Summary
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key results
Boltzmann limit: EF − ε1(0)[ ] kBT << 0 η << 0 F j η( )→ eη
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( ) = υT
F 1/2 ηF1( )F 0 ηF1( )
See: “Notes on Fermi-Dirac Integrals, 2nd Edition”
by Raseong Kim and Mark Lundstrom
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Boltzmann limit
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( ) = υT
F 1/2 ηF1( )F 0 ηF1( )
υT →υT =
2kBTπm*
ID →WCox VGS −VT( )υT1- e−qVDS kBT
1+ e−qVDS kBT
⎡
⎣⎢
⎤
⎦⎥
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examine result
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
%υT = υT
F 1/2 ηF1( )F 0 ηF1( )
1) Ballistic injection velocity (now gate voltage dependent)
2) High drain bias (on-current)
3) Low drain bias (ballistic channel conductance)
4) Ballistic mobility
5) Drain saturation voltage (was ~ kBT/q)
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injection velocity
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( ) = υT
F 1/2 ηF1( )F 0 ηF1( )
most convenient to plot vs. nS (0).
Assume:
1) Si (100)
2) 1 subband occupied
3) parabolic energy bands
<υ(0)> for high drain bias.
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injection velocity vs. nS(0)
%υT ≡
2kBTπm*
F 1/2 ηF1( )F 0 ηF1( )
υT = 2kBT πm*
υT → 4 3π( )υF
υF =hkF
m*
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high drain bias
ηF 2 = ηF1 − qVDS kBT = EF1 − qVDS − ε1(0)[ ] kBT << 0
ID =WCox %υT VGS −VT( ) 1-F 1/2 ηF2( ) F 1/2 ηF1( )
1+F 0 ηF2( ) F 10 ηF1( )⎡
⎣⎢
⎤
⎦⎥
F 1/2 ηF 2( ),F 0 ηF 2( )→ eηF 2 → 0IDS
VDSVDSAT ≈ kBT q
ID →WCox %υT VGS −VT( )
VGS = VDD
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high gate and drain bias
IDS
VDSVDSAT
ID =WCox %υT VGS −VT( )
υT →υT
43 π
ηF11/2 ~ VGS −VT( )1/2
ID ∝ VGS −VT( )3/2
Recall, in the non-degenerate case,
ID ∝ VGS −VT( )1
ID ∝ VGS −VT( )α
1 ≤ α ≤ 1.5
VGS = VDD
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low drain bias
IDS
VDS
VGS
VDSAT
finite channel resistance
VGS = VDD
GCH = WCox VGS −VT( ) υT
2kBT q( )⎛
⎝⎜⎞
⎠⎟F −1/2 ηF1( )F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
Fermi-Dirac correction
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typical numbers
GCH = WqnS 0( ) υT
2kBT q( )⎛
⎝⎜⎞
⎠⎟F −1/2 ηF1( )F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
assume Si (100), single subband,T = 300K, nS(0) ~ 1013 cm-2
GCH WBoltz
= qnS (0)υT
2kBT q( )⎛
⎝⎜⎞
⎠⎟≈ 0.038 mhos/μm
F −1/2 ηF1( ) F 0 ηF1( )⎡⎣ ⎤⎦ ; 0.5
RCHW = GCH W( )−1≈ 52 Ω− μm
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ITRS: 2022
ID
VDS
VGS
VDSAT
ION / W = 2786 μA / μm
1 RCH
http://public.itrs.net (ITRS Report, 2007 Ed.)
RCHW <VDD ION W( )≈ 230 Ω− μm
RCHWballistic
≈ 52 Ω− μm
RSDW < 135 Ω−
channel resistance:
series resistance:
μm
VDD
ballistic resistance:
= 0.65 V
VGS = VDD
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ballistic mobility
ID = WCox VGS −VT( ) υT
2kBT q( )⎛
⎝⎜⎞
⎠⎟F −1/2 ηF1( )F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥VDS
ID =WLμeff Cox VGS −VT( )VDS
ballistic:
diffusive:
μB ≡
υT L2kBT q( )
F −1/2 ηF1( )F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
ballistic mobility:
at nS ≈ 1013 cm-2
F −1/2 ηF1( ) F 0 ηF1( )≈ 0.5
μB ≈ 12 × L nm( ) cm2 /V-s
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drain saturation voltage
IDS
VDS
VGS
VDSAT
VGS = VDD
VDSAT ≈ kBT q( )ln exp2COX VGS −VT( )
qN2D
⎛
⎝⎜⎞
⎠⎟−1
⎡
⎣⎢⎢
⎤
⎦⎥⎥≈ β VGS −VT( )
β =2Cox kBT q( )
qN2D
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drain saturation voltage
VDSAT = VGS −VT( )
traditional:
VDSAT ≈ kBT q( )
ballistic (Boltzmann statistics):
VDSAT ≈2Cox kBT q( )
qN2D
⎡
⎣⎢
⎤
⎦⎥ VGS −VT( )
ballistic (Fermi-Dirac statistics):
0.026 V
0.97 V
0.12 V
Ex.: 100nm NMOS
VGS = VDD = 1.2 V
VT = 0.23 V
N2D = 4.1×1012 cm-2
Cox = 1.5 ×10−6 F / cm-2
VDSAT ≈ 0.35 V
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52
outline
1) Introduction / Review2) Drain current3) Filling states at the top of the barrier4) I-V Characteristics5) Discussion6) Summary
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summary
1) We have generalized
to include Fermi-Dirac statistics.
ID =WCox VGS −VT( )%υT
1-F 1/2 ηF2( ) F 1/2 ηF1( )1+F 0 ηF2( ) F 0 ηF1( )
⎡
⎣⎢
⎤
⎦⎥
IDS =WCox VGS −VT( )υT1− eqVDS /kBT
1+ eqVDS /kBT
⎛⎝⎜
⎞⎠⎟
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summary
2) We discussed key device parameters for ballistic MOSFETs:
-ballistic injection velocity-ballistic on-current-ballistic channel resistance-ballistic mobility-drain saturation voltage
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questions
1) How do we treat more realistic band structures (e.g. conduction band of Si?)
2) How to we treat subthreshold conduction and 2D electrostatics?
These questions are addressed in Part 2