Chemistry Reference
In-Depth Information
for infinitesimal changes in
µ
and
σ
can be written as,
δσ
σ
=
δµ
µ
+
δQ
n
Q
n
(1)
For suciently large oxide thickness,
|δQ
n
|
=
|δQ
T
|
,where
δQ
T
is the
change in trapped charge.
To calculate the contribution from number fluctuations, let's take a
volume element ∆
V
and energy element ∆
E
inside the SiO
2
such that
the trapping events inside ∆
V
∆
E
can be characterized by a single time
constant
τ
T
. We assume that the fluctuations come from the traps situated
afew
above or below the quasi-Fermi level of graphene. The power
spectral density of the fluctuation arising from ∆
k
B
T
V
∆
E
is found by taking
the Fourier transform of the autocorrelation of
δn
∆
V
∆
E
,givenby
τ
T
1+
S
n
∆
V
∆
E
=
ω
2
τ
T
n
tr
(
x, E
)
f
T
(1
− f
T
)∆
V
∆
E
(2)
where
n
tr
is the trap density of states per unit volume times energy,
ω
is
the frequency expressed in radians,
f
T
is the probability of a trap being
filled.
In our model, we assume that the Coulomb potential due to a trapped
charge will also give rise to the fluctuation in mobility. Using Mathiessen's
rule
51
we can write
1
µ
1
µ
avg
1
µ
T
=
+
(3)
where
µ
avg
is the time averaged mobil-
ity, from all static scattering mechanism. The term
µ
is the instantaneous mobility,
µ
−
1
T
arises from the
Coulomb scattering off the trapped charge that fluctuates between the ox-
ide and graphene. Consider a channel element ∆
, where we concentrate
on the fluctuation in number and mobility of the graphene channel due to
the fluctuation of trap charges in ∆
V
∆
E
inside SiO
2
.So
1
µ
T
y
∆
z
)(
δQ
n
∆
y
∆
z
=Σ(
x
)
(4)
WL
x
where Σ(
) is related to the scattering rate entirely due to the Coulomb
potential of the trapped charge located inside the substrate at a distance x
from graphene.
Simple mathematical calculations show that power spectral density in
the source-drain current consists of three terms
S
V
=
S
V
1
+
S
V
2
+
S
V
3
(5)
