156
Strain-Engineered MOSFETs
voltage fluctuations is calculated by summing the contributions from all
traps in the gate oxide by transforming Equation (6.17) [24]:
S
S C
q
W L
N f E
f E
f
dx dy dz dE
4
( )(1 ( )) 1 (2 )
Q
V
ox
t
t
L
W
E
E
2
2
2 2
2
0
0
0
ox
fb
ox
V
C
∫
∫
∫
∫
=
=
−
τ
+ π τ
(6.21)
where f(E) = 1/[1 + e (E–Efn,p)/kT ] is the Fermi function, and N t is the density of
traps in the gate dielectrics at the quasi-Fermi level (in cm –3 eV –1 ). Only these
traps contribute to the 1/f noise, with the other traps being permanently
filled or empty. In the McWorther model it is assumed that trapping and
de-trapping occur through tunneling processes; the trapping time constant
is given as
E e
with
h
m
( )
,
,
4 2 *
z
B
0
/
1
τ = τ
λ =
π
Φ
λ
−
(6.22)
for tunneling from the interface to the trap located at position z in the gate
oxide. The tunneling attenuation length λ is predicted by the Wentzel–
Kramers–Brillouin (WKB) theory, Φ B is the tunneling barrier height seen by
the carriers at the interface, and m* is the effective mass of channel carriers
in the gate oxide. The time constant τ 0 is often assumed as 10 –10 s, and λ ≈ 1 Å
for the Si/SiO 2 system. This yields z = 2.6 and 0.7 nm for frequencies of 0.01
Hz and 1 MHz, respectively. Thus, oxide traps located too close to the channel interface are too fast to give 1/f noise, and those located more than ~3 nm
from the interface are too slow to contribute. By inserting Equation (6.22), the
integral in Equation (6.21) can be evaluated as
S
q kT N
f WLC
V
t
ox
2
2
fb =
λ
γ
(6.23)
The frequency exponent γ deviates from 1 if the trap density is not uniform
in depth. γ is less than 1 when the trap density near the oxide/semiconductor
interface is higher than inside the trap density inside the gate oxide, and γ is
greater than 1 for the opposite case.
The simulated bias dependence of the normalised drain current noise
PSD, S ID /I D
2 , in the number fluctuation model with drain currents ranging from subthreshold to strong inversion regimes using Equation (6.20),
α = 0, and a constant arbitrary N t , is shown in Figure 6.5. S ID /I D
2 varies
approximately as 1/(V GS – V T ) 2 ∝ 1/Q i
2 in strong inversion. S ID /I D
2 decreases
more rapidly with drain current as g m is reduced at high-gate-voltage overdrives. In the subthreshold region, on the other hand, S ID /I D
2 is almost
constant since g m = I D q/mkT. The physical explanation is that change in
Strain-Engineered MOSFETs
voltage fluctuations is calculated by summing the contributions from all
traps in the gate oxide by transforming Equation (6.17) [24]:
S
S C
q
W L
N f E
f E
f
dx dy dz dE
4
( )(1 ( )) 1 (2 )
Q
V
ox
t
t
L
W
E
E
2
2
2 2
2
0
0
0
ox
fb
ox
V
C
∫
∫
∫
∫
=
=
−
τ
+ π τ
(6.21)
where f(E) = 1/[1 + e (E–Efn,p)/kT ] is the Fermi function, and N t is the density of
traps in the gate dielectrics at the quasi-Fermi level (in cm –3 eV –1 ). Only these
traps contribute to the 1/f noise, with the other traps being permanently
filled or empty. In the McWorther model it is assumed that trapping and
de-trapping occur through tunneling processes; the trapping time constant
is given as
E e
with
h
m
( )
,
,
4 2 *
z
B
0
/
1
τ = τ
λ =
π
Φ
λ
−
(6.22)
for tunneling from the interface to the trap located at position z in the gate
oxide. The tunneling attenuation length λ is predicted by the Wentzel–
Kramers–Brillouin (WKB) theory, Φ B is the tunneling barrier height seen by
the carriers at the interface, and m* is the effective mass of channel carriers
in the gate oxide. The time constant τ 0 is often assumed as 10 –10 s, and λ ≈ 1 Å
for the Si/SiO 2 system. This yields z = 2.6 and 0.7 nm for frequencies of 0.01
Hz and 1 MHz, respectively. Thus, oxide traps located too close to the channel interface are too fast to give 1/f noise, and those located more than ~3 nm
from the interface are too slow to contribute. By inserting Equation (6.22), the
integral in Equation (6.21) can be evaluated as
S
q kT N
f WLC
V
t
ox
2
2
fb =
λ
γ
(6.23)
The frequency exponent γ deviates from 1 if the trap density is not uniform
in depth. γ is less than 1 when the trap density near the oxide/semiconductor
interface is higher than inside the trap density inside the gate oxide, and γ is
greater than 1 for the opposite case.
The simulated bias dependence of the normalised drain current noise
PSD, S ID /I D
2 , in the number fluctuation model with drain currents ranging from subthreshold to strong inversion regimes using Equation (6.20),
α = 0, and a constant arbitrary N t , is shown in Figure 6.5. S ID /I D
2 varies
approximately as 1/(V GS – V T ) 2 ∝ 1/Q i
2 in strong inversion. S ID /I D
2 decreases
more rapidly with drain current as g m is reduced at high-gate-voltage overdrives. In the subthreshold region, on the other hand, S ID /I D
2 is almost
constant since g m = I D q/mkT. The physical explanation is that change in