8.11 Diffusion
249
Fig. 8.27 Total (circles)
and electronic (squares)
conductivity of CuI
coexisting with copper.
Filled (empty) symbols
refer to polycrystalline
(single crystal) samples.
The different structural
phases (α (cubic), β
(wurtzite), γ (zincblende))
are indicated by shaded
areas as labeled. Dashed
lines are guides to the eye.
Adapted from [808]
10
1
10
0
10
-1
10
-2
1000/T (1000/K)
1.3
1.4
1.5
1.6
1.7
1.8
1.9
405 369350
300 275
CuI
j n = −eμ n n E + eD n ∇n
(8.54a)
j p = eμ p p E − eD p ∇ p .
(8.54b)
This relation can also be deduced more generally from the gradient of the Fermi level as
j n = −eμ n n E − nμ n ∇ E F
(8.55a)
j p = eμ p p E − pμ p ∇ E F .
(8.55b)
Using (7.6) and (7.7) for the concentrations (valid also in the case of degeneracy) and using
dF j (x)/dx = F j−1 (x) we obtain
j n = −eμ n n E − kT μ n
F 1/2 (η)
F −1/2 (η)
∇n
(8.56a)
j p = eμ p p E − kT μ p
F 1/2 (ζ)
F −1/2 (ζ)
∇ p ,
(8.56b)
with η = (E F − E C )/kT and ζ = −(E F − E V )/kT . If the pre-factor of the density gradient is identified
as the diffusion coefficient we find the (generalized) so-called ‘Einstein relations’ (β = e/(kT )) [608,
809]:
D n = −β
−1
μ n
F 1/2 (η)
F −1/2 (η)
(8.57a)
D p = β
−1
μ p
F 1/2 (ζ)
F −1/2 (ζ)
.
(8.57b)
The effect of non-parabolicity has been included in [810].
Useful analytical approximations have been discussed in [811]. We note that, e.g., (8.57a) can also
be written as [812, 813]
D n = −β
−1
μ n n
∂η
∂n
.
(8.58)
249
Fig. 8.27 Total (circles)
and electronic (squares)
conductivity of CuI
coexisting with copper.
Filled (empty) symbols
refer to polycrystalline
(single crystal) samples.
The different structural
phases (α (cubic), β
(wurtzite), γ (zincblende))
are indicated by shaded
areas as labeled. Dashed
lines are guides to the eye.
Adapted from [808]
10
1
10
0
10
-1
10
-2
1000/T (1000/K)
1.3
1.4
1.5
1.6
1.7
1.8
1.9
405 369350
300 275
CuI
j n = −eμ n n E + eD n ∇n
(8.54a)
j p = eμ p p E − eD p ∇ p .
(8.54b)
This relation can also be deduced more generally from the gradient of the Fermi level as
j n = −eμ n n E − nμ n ∇ E F
(8.55a)
j p = eμ p p E − pμ p ∇ E F .
(8.55b)
Using (7.6) and (7.7) for the concentrations (valid also in the case of degeneracy) and using
dF j (x)/dx = F j−1 (x) we obtain
j n = −eμ n n E − kT μ n
F 1/2 (η)
F −1/2 (η)
∇n
(8.56a)
j p = eμ p p E − kT μ p
F 1/2 (ζ)
F −1/2 (ζ)
∇ p ,
(8.56b)
with η = (E F − E C )/kT and ζ = −(E F − E V )/kT . If the pre-factor of the density gradient is identified
as the diffusion coefficient we find the (generalized) so-called ‘Einstein relations’ (β = e/(kT )) [608,
809]:
D n = −β
−1
μ n
F 1/2 (η)
F −1/2 (η)
(8.57a)
D p = β
−1
μ p
F 1/2 (ζ)
F −1/2 (ζ)
.
(8.57b)
The effect of non-parabolicity has been included in [810].
Useful analytical approximations have been discussed in [811]. We note that, e.g., (8.57a) can also
be written as [812, 813]
D n = −β
−1
μ n n
∂η
∂n
.
(8.58)