nMOSFET Schematic
‰ Four structural masks: Field, Gate, Contact, Metal.
‰ Reverse doping polarities for pMOSFET in N-well.
nMOSFET Schematic
polysilicon
gate
ƒ Source terminal:
Ground potential.
ƒ Gate voltage: Vg
ƒ Drain voltage: Vds
ƒ Substrate bias
voltage: −Vbs
gate
oxide
Vg
Vds
z
n+ source
depletion
region
0
y
L
n+ drain
x
inversion
channel
p-type substrate
-Vbs
W
¾ψ(x,y): Band
bending at any point
(x,y).
¾V(y): Quasi-Fermi
potential along the
channel.
¾V(y=0) = 0, V(y=L)
= Vds.
Drain Current Model
Electron concentration:
ni 2 q ( ψ −V ) kT
e
n( x , y) =
Na
Electric field:
2
2kTN a
 dψ 
E ( x, y ) = 
 =
ε si
 dx 
2
 − qψ / kT qψ  ni 2  − qV / kT qψ / kT
qψ
(e
+
− 1 + 2  e
− 1) −
 e
kT
kT
 Na 

Condition for surface inversion:
ψ ( 0 , y ) = V ( y) + 2ψ B
Maximum depletion layer width at inversion:
Wdm ( y ) =
2ε si [V ( y ) + 2ψ B ]
qN a



Gradual Channel Approximation
Assumes that vertical field is stronger than lateral field in the
channel region, thus 2-D Poisson’s eq. can be solved in terms
of 1-D vertical slices.
Current density eq. (both drift and diffusion):
dV ( y)
Jn ( x, y) = −qµ n n( x, y)
dy
Integrate in x- and z-directions,
dV
dV
I ds ( y ) = − µ eff W
Qi ( y ) = − µ eff W
Qi (V )
dy
dy
xi
where Qi ( y ) = − q ∫ n( x , y )dx is the inversion charge/area.
0
Current continuity requires Ids independent of y, integration with
respect to y from 0 to L yields
W Vds
I ds = µeff ∫ ( −Qi (V )) dV
L 0
Pao-Sah’s Double Integral
Change variable from (x,y) to (ψ,V),
ni 2 q (ψ −V )/ kT
n( x, y) = n(ψ , V ) =
e
Na
2
q (ψ −V ) / kT
ψB
ψ s ( n / N )e
dx
i
a
Qi (V ) = − q ∫ n(ψ , V )
dψ = − q ∫
dψ
ψs
ψ
B
E (ψ , V )
dψ
Substituting into the current expression,
2

W Vds  ψ s (ni / N a )e q (ψ −V ) / kT
I ds = qµ eff
dψ  dV
 ∫ψ B
∫
0
L
E (ψ , V )


where ψs(V) is solved by the gate voltage eq. for a
vertical slice of the MOSFET:
Qs
V g = V fb + ψ s −
= V fb + ψ s +
Cox
2ε si kTN a  qψ s
ni 2 q (ψ s −V ) / kT 
+
e


Cox
Na2
 kT

1/ 2
Charge Sheet Approximation
Assumes that all the inversion charges are located at the
silicon surface like a sheet of charge and that there is no
potential drop across the inversion layer.
After the onset of inversion, the surface potential is pinned at
ψs = 2ψB + V(y).
ƒ Depletion charge:
ƒ Total charge:
ƒ Inv. charge:
Qd = −qN a Wdm = − 2ε si qN a (2ψ B + V )
Qs = −Cox (Vg − V fb − ψ s ) = −Cox (Vg − V fb − 2ψ B − V )
Qi = Qs − Qd = −Cox (V g − V fb − 2ψ B − V ) + 2ε si qNa ( 2ψ B + V )
W Vds
Substituting in I ds = µeff ∫0 ( −Qi (V )) dV and integrate:
L

2 2ε si qN a
W 
Vds 
3/ 2
3/ 2 
I ds = µeff Cox Vg − V fb − 2ψ B −
[(2ψ B + Vds ) − (2ψ B ) ]
Vds −
L 
2 
3Cox

Linear Region I-V Characteristics
For Vds << Vg ,
4ε si qN aψ B
W
I ds = µ eff Cox Vg − V fb − 2ψ B −
L 
Cox
4ε si qNaψ B
Cox
is the MOSFET threshold voltage.
1E+0
0.8
1E-2
0.6
1E-4
0.4
1E-6
0.2
1E-8
1E-10
0
0.5
1
1.5
2
Gate Voltage, Vg (V)
V on≈V t
2.5
3
0
Linear Ids (arbitrary scale)
Log(Ids ) (arbitrary scale)
where
Vt = V fb + 2ψ B +

W
Vds = µ eff Cox (Vg − Vt )Vds
L

Saturation Region I-V Characteristics
Keeping the 2nd order terms in Vds: I ds = µ eff Cox
where m = 1 +
ε si qN a / 4ψ B
Cox
= 1+
Cdm
3t
= 1 + ox
Cox
Wdm
W
L
m 2

(
V
−
V
)
V
−
Vds 
t
ds
 g
2

is the body-effect coefficient.
(Vdsat, Idsat)
I ds = I dsat
2
W (Vg − Vt )
= µ eff Cox
2m
L
when
Vds = Vdsat = (Vg − Vt)/m.
Drain Current
Vg4
Vg3
Vg2
Vg1
Typically, m ≈ 1.2.
Drain Voltage
Pinch-off Condition
From inversion charge density point of view,
Q i (V ) = −C ox (V g − Vt − mV )
W
while I ds = µeff
L
V ds
∫ (−Q (V ))dV
i
0
−Qi (V )
At Vds = Vdsat = (Vg − Vt)/m,
Qi = 0 and Ids = max.
Cox (Vg − Vt )
∝ Ids
0
Source
Vdsat
Vds
Drain
=
Vg − Vt
m
V
Pinch-off from Potential Point of View
V ( y) =
Vg − Vt
m
2
 V − Vt 
y  Vg − Vt
−  g
−
2

L  m
 m 
At the pinch-off point,
dV/dy → ∞
V (y )
V g −V
t
m
⇒ Gradual channel
approximation breaks
down.
L′
Vds
|Qi|/mCox
V(y)
0

y 2
+
V
 ds L Vds

0
Source
Current is injected into
the bulk depletion
region.
Vds
L
Drain
y
Beyond Pinch-off
Subthreshold Region
Vds
Saturation
region
Subthreshold
region
Linear
region
Vt
1E+0
Vg − Vt
m
Drain Current (arbitrary units)
Vds =
Vg
Low Drain
Bias
1E-1
1E-2
Diffusion
Component
1E-3
1E-4
1E-5
1E-6
Drift
Component
1E-7
1E-8
0
0.5
1
Gate Voltage (V)
1.5
2
Subthreshold Currents
 qψ s ni 2 q (ψ s −V ) / kT 
− Qs = ε si E s = 2ε si kTN a 
+ 2e

kT
N
a


1/ 2
Power series expansion: 1st term Qd, 2nd term Qi,
2
ε si qN a  kT  ni  q (ψ s −V )/ kT
− Qi =
e




2ψ s  q  N a 
⇒
W
I ds = µ eff
L
ε si qN a  kT 
2ψ s  q 
2
2
 ni  qψ s / kT
− qV ds / kT
e
e
1
−
N 
 a
(
2
)
 kT  q (Vg −Vt )/ mkT
W
( 1 − e − qVds / kT )
or, I ds = µ eff Cox (m − 1)  e
L
 q 
Inverse subthreshold slope:
−1
 d (log I ds ) 
mkT
kT  Cdm 
= 2.3 1 +
S =
 = 2.3

 dV
q
q
C
g
ox




Body Effect: Dependence of Threshold
Voltage on Substrate Bias
If Vbs ≠ 0,
I ds = µeff Cox
W
L

2 2ε si qN a
Vds 
−
V
−
V
−
2
ψ
−
V
( 2ψ B + Vbs + Vds )3/ 2 − ( 2ψ B + Vbs )3/ 2
 g
 ds
fb
B
2 
3Cox

[
⇓
ε si qN a / 2(2ψ B + Vbs )
dVt
=
dVbs
Cox
tox=200 Å
Threshold Voltage, Vt (V)
2ε si qNa ( 2ψ B + Vbs )
Cox
]
1.8
⇓
Vt = V fb + 2ψ B +



1.6
Na=1016 cm−3
Vfb=0
1.4
1.2
Na=3×1015 cm−3
1
0.8
0.6
0
2
4
6
Substrate Bias Voltage, Vbs (V)
8
10
Dependence of Threshold Voltage on
Temperature
For n+ poly gated nMOSFET, Vfb = − (Eg /2q) − ψB
⇒
4ε si qNaψ B
Vt = −
+ ψB +
Cox
2q
Eg
ε si qNa / ψ B
1 dEg 
dVt
=−
+ 1 +
2q dT 
dT
Cox
⇒
 dψ
1 dE g
dψ
B
=−
+ ( 2 m − 1) B

2q dT
dT
 dT
dVt
k   N c N v  3  m − 1 dE g
= − (2m − 1)  ln
+ +
dT
q   N a  2 
q dT
From Table 2.1, dEg/dT ≈ −2.7×10−4 eV/K and (NcNv)1/2 ≈
2.4×1019 cm−3.
For Na ∼ 1016 cm−3 and m ∼ 1.1,
dVt/dT is typically −1 mV/K.
MOSFET Channel Mobility
µ eff
∫
=
xi
µ n n( x )dx
0
∫
xi
0
n( x )dx
It was empirically found that when µeff is plotted against an
effective normal field Eeff, there exists a “universal
relationship” independent of the substrate bias, doping
concentration, and gate oxide thickness (Sabnis and
Clemens, 1979).
1 
1

Here
Eeff =  Qd + Qi 
ε si 
2 
Since Qd = 4ε si qN a ψ B = Cox (Vt − V fb − 2ψ B ) and |Qi| ≈ Cox(Vg − Vt),
⇒
For
n+
Eeff =
Vt − V fb − 2ψ B
3tox
poly gated nMOSFET,
+
Vg − Vt
6tox
Eeff =
Vt + 0.2 Vg − Vt
+
3tox
6tox
N-channel MOSFET Mobility
ƒ Low field region (low
electron density): Limited by
impurity or Coulomb
scattering (screened at high
electron densities).
ƒ Intermediate field region:
Limited by phonon
scattering,
µ eff ≈ 32500 × E −1/ 3
ƒ High field region (> 1
MV/cm): Limited by surface
roughness scattering (less
temp. dependence).
Temperature Dependence
of MOSFET Current
P-channel MOSFET Mobility
Eeff =
1 
1 
Q
Qi 
+
 d
3 
ε si 
In general, pMOSFET mobility does not exhibit as
“universal” behavior as nMOSFET.
Electron and Hole Mobilities vs. Field
1000
-0.3
tox=35 Å
tox=70 Å
Eeff
100
2
(cm /V-s)
300
µeff
500
200
Electron
-0.3
Eeff
-2
Eeff
Hole
-1
50
30
0.1
1 µm CMOS
0.2
0.3
0.1 µm CMOS
0.5
1
Eeff (MV/cm)
Eeff
2
3
Intrinsic MOSFET Capacitance
−1
 1
1 
C
=
WL
+
‰ Subthreshold region:
g
 ≈ WLCd

 Cox Cd 
‰ Linear region: Cg = WLCox
‰ Saturation region:
y
Qi ( y ) = −Cox (Vg − Vt ) 1 −
L
⇒
2
Cg = WLCox
3
V (y )
V g −V
t
m
L′
|Qi|/mCox
V(y)
0
Vds
0
Source
Vds
L
Drain
y
Inversion Layer Capacitance
Gate
In the charge-sheet model, Ci = ∞
and Qi = Cox(Vg − Vt).
Vg
Cox
−Qs
(inversion)
C
d
Q
d
Q
i
(low freq.)
p-type
substrate
n+
channel
C
i
In reality, inversion layer has a
finite thickness and finite
capacitance.


d ( − Qi )
C ox Ci
1
=
≈ C ox 1 −

dV g
C ox + Ci + C d
 1 + Ci / C ox 
Inversion Layer Capacitance
1st order approximation, Ci ≈ |Qi|/(2kT/q) and
|Qi| ≈ Cox(Vg − Vt), therefore, Ci/Cox = (Vg − Vt)/(2kT/q).
Inversion Charge Density, Qi (µC/cm2)

2 kT  q (Vg − Vt ) 
− Qi = Cox  (Vg − Vt ) −
ln1 +

2
q
kT



0.8
Na=5×1016 cm−3
Note:
Linearly extrapolated
threshold voltage is
typically (2-4)kT/q
higher than the
threshold voltage Vt
at ψs(inv.) = 2ψB.
tox=100 Å
0.6
Vfb=0
Cox(Vg−Vt)
0.4
0.2
0
Vt
0
0.5
1
1.5
2
2.5
Gate Voltage, Vg (V)
3
3.5
Short-Channel Effect
If L ↓, I ds = I dsat
2
W (Vg − Vt )
= µ eff Cox
↑
2m
L
source
drain
Threshold voltage becomes
sensitive to channel length
and drain bias.
1E-3
1
Log
scale
1E-4
0.8
1E-5
0.6
Slope
~q/kT
1E-6
1E-7
1E-8
Vt
0
0.2
0.4 0.6 0.8
Gate Voltage (V)
0.4
Linear
scale
1
0.2
0
1.2
Source-Drain Current (mA/µm)
gate-controlled
barrier
Source-Drain Current (A/µm)
2
And Cg = WLCox ↓. But ……
3
Short-Channel Vt Roll-off
Drain-Induced Barrier Lowering
1E+1
Drain current (A/cm)
1E+0
Vds=
]3.0 V
L=0.2 µm
]50 mV
1E-1
1E-2
1E-3
1E-4
L=2.0 µm
1E-5
1E-6
1E-7
1E-8
-0.5
L=0.35 µm
0
0.5
Gate voltage (V)
tox=100 Å
Na=3×1016 cm−3
1
1.5
Lateral Field Penetration
2-D Poisson’s Eq.:
qN
∂E x ∂E y ρ
+
=
=− a
∂ x ∂ y ε si
ε si
ƒ εsi∂Ex/∂x: gate controlled
depletion charge.
ƒ εsi∂Ey/∂y: S/D controlled
depletion charge.
Note that the characteristic
length of exponential decay
is independent of channel
length.
MOSFET
and
Scale Geometry
Length
2-D
AnalysisGeometry
in a Simplified
MOSFET
L
Gate
n+ poly
Source
n+
t ox
Na
Drain
Wd
n+
Substrate
A 2-D boundary-value problem
with Poisson’s equation
Gate
n+ poly
-t ox
Source
0H
A
n+
B
Wd
C
Na
D
Drain
y
GL
F
E
n+
x
Substrate
To eliminate the boundary condition
at the Si/oxide interface, the oxide
region is replaced by an equivalent
Si region (εsi/εox)tox ≈ 3tox thick.
Subthreshold region:
In AFGH
(oxide),
∂ 2ψ ∂ 2ψ
+
=0
∂ x 2 ∂ y2
In ABEF
(silicon),
∂ 2ψ ∂ 2ψ
qN a
+
=
∂ x 2 ∂ y2
ε si
Boundary conditions:
ψ ( −3tox , y ) = Vg − V fb
ψ ( x ,0 ) = ψ bi
ψ ( x , L ) = ψ bi + Vds
ψ (Wd , y ) = 0
along GH,
along AB,
along EF,
along CD.
General approach to a 2-D boundary value problem
Gate
Source
n+ poly
-t ox
0H
A
n+
Wd
B
C
Na
D
Drain
y
GL
F
E
n+
x
Substrate
Let:
ψ ( x , y ) = v ( x , y ) + uL ( x , y ) + uR ( x , y ) + uB ( x , y )
v(x,y) is a solution to the inhomogeneous equation and satisfies
the top boundary condition.
uL, uR, uB are solutions to the homogeneous equation such that
ψ(x,y) satisfies the other B.C.’s.
Gate
Source
For satisfying the
Boundary conditions:
 nπ ( L − y ) 
sinh
 W + 3t  
∞
nπ ( x + 3tox ) 
*
ox 
 d
uL ( x , y ) = ∑ bn
sin
Wd + 3tox 
 nπ L 
n =1

sinh

 Wd + 3tox 
 nπy 
sinh
 W + 3t  
∞
nπ ( x + 3tox ) 
*
ox 
 d
uR ( x, y ) = ∑ cn
sin

 nπL   Wd + 3tox 
n =1
sinh

 Wd + 3tox 
 nπ ( x + 3tox ) 
sinh


∞
L
*
 sin nπy 

uB ( x, y ) = ∑ d n


 nπ (Wd + 3tox )   L 
n =1
sinh

L


n+ poly
-t ox
0H
A
n+
Wd
B
C
Na
D
Drain
y
GL
F
E
n+
x
Substrate
Note that for u=sin(kx),
d2u/dx2=-k2u;
And that for u=sinh(ky),
d2u/dy2=k2u.
MOSFET Scale Length
Short-channel threshold roll-off (V)
24t ox
∆Vt =
Wdm
ψ bi (ψ bi + Vds ) e
1
1.2
1.4
To keep short-channel
effect under control,
Lmin should be kept
larger than about 2λ.
1.5
0.4
0.2
0
Wdm + 3t ox
λ ≡ Wdm + (εsi/εox)tox
1.3
0.6
πL / 2
Define scale length,
m=
0.8
−
0
1
2
Lmin / (Wdm + 3tox )
3
4
m = ∆Vg /∆ψs = 1 + 3tox /Wdm
Depletion Width Scaling
0
Wdm =
4ε si kT ln( N a / ni )
q2 N a
Maximum Depletion Width (µm)
10
1
0.1
0.01
1.0E+14
1.0E+15
1.0E+16
1.0E+17
1.0E+18
Substrate Doping Concentration (cm-3)
1.0E+19
Generalized Scale Length
εi
Source
Gate
ψ1
εsi ψ2
In the one-region model,
the eigenvalues are:
ti
Wd
kn =
Drain
For two regions, assume
eigenvalues:
Body
L
kn =
π
λn
D. Frank et al., EDL 10/98
∞
 π ( L − y)   π ( x + t i ) 
 sin 

u L1 ( x, y ) = ∑ bn1 sinh 
λ
λ
n =1
n
n

 

 π ( L − y )   π ( x − Wd ) 
 sin 

u L 2 ( x, y ) = ∑ bn 2 sinh 
λ
λ
n =1
n
n

 

∞
nπ
Wd + 3t ox
B.C. at x = 0: uL1 = uL2
(duL1/dy = duL2/dy)
and εiduL1/dx = εsiduL2/dx
⇒ εsi tan(π ti /λn) + εi tan(π Wd /λn) = 0
Generalized Scale Length
Lowest eigenvalue: εsi tan(π ti /λ1) + εi tan(π Wd /λ1) = 0
Normalized Gate Insulator Thickness,
ti /λ1 1
∆ψSCE∝exp(−πL/2λ1)
Lmin ~ 2λ1
ε i /ε si=
0.8
3
1
1/3 (SiO2)
Note that:
ƒ λ1>Wd, and λ1>ti
0.6 10
0.4
ƒ λ1=2Wd =2ti is always
a point of symmetry
regardless of εi, εsi.
0.2
0
0
0.2
0.4
0.6
0.8
Normalized Si Depletion Depth, Wd /λ1
1
ƒ If εi=εsi, λ1=Wd + ti
MOSFET Body Effect
Depletion
boundary
Gate
Oxide
Silicon
Potential
∆ψ change
s
∆Vg
3tox
Wdm
(εsi /εox =3)
m = ∆Vg /∆ψs = 1 + 3tox /Wdm
MOSFET Design Space with SiO2
Gate Oxide Thickness (nm)
10
λ=20 nm
8
SiO2
To obtain a good
subthreshold slope,
the body-effect
coefficient,
15 nm
10 nm
6
5 nm
m = ∆Vg /∆ψs = 1 + 3tox /Wdm
4
Wd =10tox
2
0
m=1.3
0
5
10
15
Depletion Depth in Si (nm)
In the intercept region,
λ = Wd + 3tox
is a good approximation.
should be kept close
to unity.
20
Lmin ~1.5λ ~ 20t ox
High-k Gate Insulator
Normalized Gate Insulator Thickness,
ti /λ 1
High-k gate insulator
is an active area of
Si research because
it may replace SiO2
thereby
circumventing the
tunneling problem.
ε i /ε si=
0.8
3
1
1/3 (SiO2)
0.6 10
0.4
0.2
0
0
0.2
0.4
0.6
Normalized Si Depletion Depth,
0.8
Wd /λ
1
But
λ ~ Wd + (εsi/εi)ti
is valid only when
ti << λ.
In general, requires ti < λ/2, regardless of εi.
Velocity Saturation
ƒ Because of velocity
saturation, the saturation of
drain current in a shortchannel device occurs at a
much lower voltage than Vdsat
= (Vg − Vt)/m for long channel
devices.
ƒ This causes the saturation
current, Idsat, to deviate from
the ∝ (Vg − Vt)2 behavior and
from the 1/L dependence.
Velocity-Field Relationship
v=
µ eff E
  E n 
1 +   
  Ec  
1/ n
ƒ At low fields, v = µeffE : Ohm’s law.
ƒ As E → ∞, v = vsat = µeffEc.
Critical Field: Ec =
vsat
µ eff
It is commonly believed that:
ƒ n = 2 for electrons, n = 1 for holes.
ƒ vsat is independent of µeff (vertical field), but Ec
depends on µeff.
Only the n = 1 case can be solved analytically.
Analytical Solution for n=1
I ds = −WQi v = −WQi (V )
µ eff ( dV / dy )
1 + ( µ eff / v sat )( dV / dy )
Current continuity requires that Ids be a constant,
independent of y.
µeff I ds  dV

I
=
−
µ
WQ
(
V
)
+
⇒
ds
i
 eff

vsat  dy

Multiplying dy on both sides and integrating from y = 0 to
L and from V = 0 to Vds, one solves for Ids:
V ds
I ds =
− µeff (W / L )∫ Qi (V )dV
0
1 + ( µeff Vds / vsat L )
Charge-sheet model:
Therefore,
I ds =
Qi (V ) = −Cox (Vg − Vt − mV )
[
µ eff Cox (W / L ) (Vg − Vt )Vds − ( m / 2 )Vds 2
1 + ( µ eff Vds / vsat L )
]
Saturation Drain Voltage and Current
The saturation voltage, Vdsat, can
be found by solving dIds/dVds = 0:
And the saturation current is:
Drain saturation current (mA/µm)
0.6
Vdsat =
2(Vg − Vt ) / m
1 + 1 + 2 µeff (Vg − Vt ) / ( mvsat L )
I dsat = CoxWvsat (Vg − Vt )
1 + 2µeff (Vg − Vt ) / ( mvsat L ) − 1
1 + 2µeff (Vg − Vt ) / ( mvsat L ) + 1
tox=100Å
0.5
0.4
L=2.5 µm
L=0
0.3
L=0.5 µm
(Dashed: long-ch.
model,
solid: velocity sat.
model)
0.2
0.1
L=10 µm
0
0
1
2
3
Gate Overdrive, Vg−Vt , (V)
4
5
Velocity-Saturation-Limited Current
At the drain end of the channel when Vds = Vdsat,
Qi ( y = L ) = −Cox (Vg − Vt − mVdsat )
and Idsat = −WvsatQi(y = L),
i.e., carriers move at the saturation velocity.
This implies that dV/dy → ∞ at the drain.
Therefore, the gradual channel approximation breaks down and
the carriers are no longer confined to the surface channel.
I dsat = CoxWvsat (Vg − Vt )
When (Vg − Vt) << mvsatL/2µeff,
2
W (Vg − Vt )
I dsat = µ effCox
L
2m
Long channel limit.
1 + 2µeff (Vg − Vt ) / ( mvsat L ) − 1
1 + 2µeff (Vg − Vt ) / ( mvsat L ) + 1
In the limit of L → 0,
I dsat = CoxWvsat (Vg − Vt )
Velocity saturation limited current.
Velocity Overshoot: Monte Carlo Simulation
Velocity saturation is
derived from the drift
and diffusion model
which assumes that
carriers are always in
thermal equilibrium
with the silicon
lattice.
But if the MOSFET is only a few mean free path (~10 nm) long,
carriers do not travel enough distance to establish equilibrium
⇒ velocity overshoot, i.e., carrier velocity at the high field
region near the drain can exceed the saturation velocity.
Velocity Overshoot
Monte-Carlo simulation:
¾ Even in a 30 nm device,
nFET/pFET velocity and
therefore current ratio is
still ≈ 2 because of the
difference in effective
masses.
¾ The velocity at the source does not greatly exceed 107 cm/s;
therefore, current does not greatly exceed I dsat = CoxWvsat (Vg − Vt )
Distribution Function
Fermi-Dirac distribution under equilibrium:
f ( E) =
1
1+ e
( E − E f )/ kT
The standard semi-classical transport theory is based on the
Boltzmann transport equation (BTE):
eE
∂f
+ v ⋅ ∇r f +
⋅ ∇ k f = ∑ {S ( k ′, k ) f ( r , k ′, t )[1 − f ( r , k , t ) ] − S ( k , k ′) f ( r , k , t )[1 − f ( r , k ′, t ) ]}
h
∂t
k′
where r is the position, k is the momentum, f(r,k,t) is the distribution
function, v is the group velocity, E is the electric field, S(k,k’) is the
transition probability between the momentum states k and k’.
The summation on the right hand side is the collison term, which
accounts for all the scattering events. The terms on the left hand
side indicate, respectively, the dependence of the distribution
function on time, space (explicitly related to velocity), and
momentum (explicitly related to electric field).
Velocities at the Source and at the Drain
Ids = WQi v
Ids = WCox (Vg - Vt )vs
Source
E0
Ids = WCox (Vg - Vt - Vdsat)vd
1
Drain
r
‰ Inversion charge density at the source is given by Cox(Vg-Vt).
‰ Inversion charge density at the drain is much lower because
of the drain bias.
‰ Current continuity is maintained consistent with band bending.
Scattering theory
At high drain bias, T’=0,
Source
E0
I ds = TI +
Let r=ns-/ns+, the backscattering
coefficient.
1
Drain
r
Ref. Lundstrom, EDL, p.361, 1997
Then
1− r 
I ds / W = Cox (Vg − Vt )v T 

r
+
1


Note that r depends on the low-field mobility near the source.
In the ballistic limit, no collisions in the channel, i.e., r = 0, and
I ds / W = Cox (Vg − Vt )vT
Carrier Thermal Injection Velocity
For 2-D nondegenerate carriers,
∞
EN ( E ) f ( E ) dE
∫
⟨E⟩ =
= kT
∫ N ( E ) f ( E )dE
0
∞
0
so
vrms =
2 kT
m
For uni-directional injection,
vT =
∫
∞
0
∫
2
v x exp(−mv x / 2kT )dv x
∞
0
2
exp(−mv x / 2kT )dv x
=
2kT
πm
Injection Velocity in the Degenerate Case
At 0 K, all states below the
Fermi energy are filled. In
2-D, define a Fermi circle
with velocity vF.
∫∫ v dv dv
x
⟨ vx ⟩ =
x
v x >0
∫∫ dv dv
x
y
y
4
=
vF
3π
v x >0
Since
Cox (Vg − Vt )
m 1
1
2
N ( E ) EF = 2 mvF = ns =
πh 2
q
2
vT =
4h 2Cox (Vg − Vt )
3m
qπ
I-V Curves of a Ballistic MOSFET
Natori, JAP, p. 4879, 1994. I ds / W =
4h 2Cox
Cox (Vg − Vt )3 / 2 (T=0 K)
3m qπ
Independent of L!