Biomedical Engineering Reference
In-Depth Information
carriers causes a field gradient, which limits the current density and the mechanism is then
referred to as space charge limited current. Starting from the basic Gauss's law in one-
dimension, assuming that the insulator contains no free carriers if no current flows the
expression for the space charge limited current can be obtained as shown in Eq. (5)
2
9
εμ
V
J
=
(5)
3
8
d
where
J
is current density,
ε
is relative permittivity,
µ
is charge mobility,
V
is applied voltage
and
d
is electrode spacing. Space charge limited current results from the fact that when the
injected carrier concentration exceeds the thermal carrier concentration, the electric field in
the sample becomes very non-uniform, and the current no longer follows Ohm's law.
2.4 Variable-range hopping mechanism
Charge conduction in semiconducting polymers is thought to take place by hopping of
charge carriers in an energetically disordered landscape of hopping sites (Meisel et al., 2006).
The variable-range hopping (VRH) conduction mechanism originally proposed by Mott for
amorphous semiconductors (Mott & Davis, 1979) assuming a phonon-assisted hopping
process has also been observed in conducting polymers and their composites at low
temperature (Ghosh et al., 2001; Luthra et al., 2003; Singh et al
.
, 2003). Bulk conductivity of
conducting polymers depends upon several factors, such as the structure, number and
nature of charge carriers, and their transport along and between the polymer chains and
across the morphological barriers (Long et al., 2003). When the phonon energy is
insufficient (low temperature), carriers will tend to hop larger distances in order to locate in
sites which are energetically closer than their nearest neighbours. Eq. (6) gives the DC
conductivity based on the VRH conduction model.
1/4
σ
T
T
⎛
⎞
0
σ
=
exp[
−
⎜
⎝⎠
]
(6)
T
where the pre- exponential factor
σ is given by Eq.(7)
1/2
2
qv
N
⎡
⎤
ph
(
EF
)
σ
=
(7)
⎢
⎥
0
1/2
γ
2(8
π
k
)
⎣
⎦
and
q
is the electron charge, k is the Boltzmann's constant,
v
ph
is the typical phonon
frequency obtained from the Debye temperature ( ≈10
13
Hz), γ
is the decay length of the
localized wave function near the Fermi level and
N
(E
F
)
is the density of states at the Fermi
level. The characteristic Mott temperature
T
d
, as shown in Eq.(8) corresponds to the hopping
barrier for charge carriers (also known as the pseudo-activation energy) and measures the
degree of disorder present in the system.
3
γ
(8)
T
=
18.11
d
kN
(
EF
)
Two other Mott parameters, the variable range hopping distance (R
VRH
) and hopping
activation energy (W) are given by Eq. (9) and (10) respectively
