246
8 Transport
Fig. 8.24 Temperature
dependence of the planar
resistance for Si films
deposited at room
temperature. Solid line is
linear fit with
T 0 = 8 × 10 7 K according
to (8.45) (s = 1/4).
Adapted from [798]
10
7
10
8
10
9
10
10
10
11
10
12
10
6
T (K )
-1/4
-1/4
0.25
0.27
0.29
0.31
0.33
0.35
a-Si
300
200
100
T
A commonly observed phenomenon is the variable range hopping with a conductivity given by
σ = σ 0 exp
−(T 0 /T )
s
(8.45)
with s = 1/4. Such law is fulfilled for amorphous silicon (Fig. 8.24). Mott has derived [799] the
exponent s = 1/4 using the following argument: The probability p to hop from one localized site to
another is proportional to
p ∝ exp(−2αR − W/kT ) .
(8.46)
The first term stems from the probability to find the electron within radius R from its initial site, α
being the decay constant of its wave function, ) ∝ exp(−α r ). The second term is the Boltzmann
factor for bridging the energy mismatch W between localized states with a phonon-assisted process,
assuming a low temperature limit (kT W ). There is a trade-off between hopping to levels closer in
energy but spatially further away (on average), preferred at low temperature and the hopping to energy
levels with larger W but spatially closer at higher temperatures. Thus the hopping range changes with
temperature, giving the mechanism its name.
D(E F ) shall be the (constant) density of localized states around the Fermi level. Within a radius
R, there is on average one state of energy between 0 and W (R) when (for three-dimensional bulk
material)
W (R) =
1
D(E F ) (4π/3) R 3 .
(8.47)
Substituting (8.47) in (8.45) and searching for the maximum yields the most probable hopping distance
R ≈ (α kT D(E F ))
−1/4
,
(8.48)
showing again, the varying range of hopping with temperature. Thus we find for T 0 in (8.45),
T 0 ≈
α
3
k D(E F )
.
(8.49)
Other types of hopping mechanisms are the Efros–Shklovskii variable range hopping (s = 1/2),
emerging for an energy dependent density of states D(E) ∝ (E − E F )
2 due to Coulomb interaction
between hopping sites [800], or the next neighbor hopping (s = 1).
8 Transport
Fig. 8.24 Temperature
dependence of the planar
resistance for Si films
deposited at room
temperature. Solid line is
linear fit with
T 0 = 8 × 10 7 K according
to (8.45) (s = 1/4).
Adapted from [798]
10
7
10
8
10
9
10
10
10
11
10
12
10
6
T (K )
-1/4
-1/4
0.25
0.27
0.29
0.31
0.33
0.35
a-Si
300
200
100
T
A commonly observed phenomenon is the variable range hopping with a conductivity given by
σ = σ 0 exp
−(T 0 /T )
s
(8.45)
with s = 1/4. Such law is fulfilled for amorphous silicon (Fig. 8.24). Mott has derived [799] the
exponent s = 1/4 using the following argument: The probability p to hop from one localized site to
another is proportional to
p ∝ exp(−2αR − W/kT ) .
(8.46)
The first term stems from the probability to find the electron within radius R from its initial site, α
being the decay constant of its wave function, ) ∝ exp(−α r ). The second term is the Boltzmann
factor for bridging the energy mismatch W between localized states with a phonon-assisted process,
assuming a low temperature limit (kT W ). There is a trade-off between hopping to levels closer in
energy but spatially further away (on average), preferred at low temperature and the hopping to energy
levels with larger W but spatially closer at higher temperatures. Thus the hopping range changes with
temperature, giving the mechanism its name.
D(E F ) shall be the (constant) density of localized states around the Fermi level. Within a radius
R, there is on average one state of energy between 0 and W (R) when (for three-dimensional bulk
material)
W (R) =
1
D(E F ) (4π/3) R 3 .
(8.47)
Substituting (8.47) in (8.45) and searching for the maximum yields the most probable hopping distance
R ≈ (α kT D(E F ))
−1/4
,
(8.48)
showing again, the varying range of hopping with temperature. Thus we find for T 0 in (8.45),
T 0 ≈
α
3
k D(E F )
.
(8.49)
Other types of hopping mechanisms are the Efros–Shklovskii variable range hopping (s = 1/2),
emerging for an energy dependent density of states D(E) ∝ (E − E F )
2 due to Coulomb interaction
between hopping sites [800], or the next neighbor hopping (s = 1).