Biomedical Engineering Reference
In-Depth Information
qfð
r
Þ
½Cl
ð
Þ¼½Cl
0
exp
r
;
(7.3)
kT
where
T
is temperature and
k
is Boltzmann constant. These virtual solid-state
parameters for the solution permit us to formulate a whole semiconductor model
for the charge and electric potential.
7.3.2.2 Semiconductor Charge Model
In the solid-state material regions, the electron and hole concentrations are given by
the basic semiconductor physics theory [
28
]. In the Si-layers we assume the donor
doping concentration
N
d
¼
10
20
cm
3
, so the carriers are degenerate, and their
distribution follows the Fermi-Dirac distribution. The electron concentration
n
(r)
and the hole concentration
p
(r) are given by [
28
]
2
2
nð
r
Þ¼N
c
p F
1
=
2
ð
c
ð
r
ÞÞ;
(7.4)
2
pð
r
Þ¼N
v
p F
1
=
2
ð
v
ð
r
ÞÞ;
(7.5)
where
N
c
and
N
v
are the density of states in the conduction and valence bands of the
solid-state material;
F
l/2
is the 1/2 order Fermi-Dirac function, the parameters
c
(r)
and
v
(r) are related to the local potential
f
(r)by
E
f
E
c
ð
r
Þ
c
ð
r
Þ¼
;
with
E
c
ð
r
Þ¼qfð
r
ÞE
g
;
(7.6)
kT
E
v
ð
r
ÞE
f
kT
v
ð
r
Þ¼
;
with
E
v
ð
r
Þ¼qfð
r
Þ;
(7.7)
where
E
c
(r) is the solid-state material conduction band edge profile and
E
v
(r) is the
corresponding valence band edge profile;
E
g
is the band gap of the material;
E
f
is the
Fermi level, which, because of the nature of the system, is assumed to be constant
and is taken as our reference energy level.
To model the electrolyte/semiconductor interface we also use the conduction
band offset between materials with respect to the Si:
DE
SiO
2
c
2
eV; DE
solution
c
¼
3
:
¼
0
:
3
eV:
(7.8)
This essential feature of the solid-state materials accounts for Fermi level
pinning and band bending in the presence of the positive space charge inside the
semiconductor and close to the interface.
DE
SiO
2
c
is a measured value, whereas
DE
solutio
c
is a model parameter. The choice of its value does not affect the induced
voltage dramatically.
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