Environmental Engineering Reference
In-Depth Information
with
h
v
k
σ
=
∂ε
k
σ
/∂
k
. In the linear regime, the left-hand side of the
Boltzmann equation is
∂g
k
σ
d
k
dt
−
∂f
k
σ
∂ε
k
σ
∂
k
·
e
v
k
σ
·
E
=
eβf
k
σ
(1
− f
k
σ
)
v
k
σ
·
E
.
The collision term on the right-hand side is
coll
=
k
σ
g
k
σ
(1
g
k
σ
)
W
(
k
σ,
k
σ
)
,
dg
k
σ
dt
g
k
σ
)
W
(
k
σ
,
k
σ
)
−
−
g
k
σ
(1
−
where
W
(
k
σ,
k
σ
) is the probability per unit time for an electronic
transition from an occupied state
|
k
σ
>
.
Linearizing the collision term, and using the principle of detailed balance,
so that this term must vanish if
g
k
σ
=
f
k
σ
, we may reduce the Boltzmann
equation to
|
k
σ>
to an unoccupied state
f
k
σ
)
f
k
σ
W
(
k
σ
,
k
σ
)
ψ
k
σ
−
ψ
k
σ
.
eβf
k
σ
(1
−
f
k
σ
)
v
k
σ
·
E
=
−
−
(1
k
σ
It is possible to find an upper bound on the resistivity from this equation,
with the use of a variational principle. Defining
u
to be a unit vector
along one of the principal axes of the resistivity tensor,
k
σ
k
σ
(1
f
k
σ
)
f
k
σ
W
(
k
σ,
k
σ
)
φ
k
σ
−
φ
k
σ
2
V
2
βe
2
−
ρ
uu
≤
,
k
σ
v
k
σ
f
k
σ
)
f
k
σ
φ
k
σ
2
·
u
(1
−
(5
.
7
.
55)
where
φ
k
σ
is an arbitrary trial function, and where the equality applies
if
φ
k
σ
=
ψ
k
σ
. In the case of the free-electron model, the Boltzmann
equation possesses an exact solution,
ψ
k
σ
∝
k
·
u
, if the scattering is
purely elastic. As discussed, for instance, by Hessel Andersen
et al.
(1980), this trial function is still useful for treating possible inelastic
scattering mechanisms, at least as long as the resistivity is dominated
by elastic impurity scattering, so we shall use
φ
k
σ
=
k
·
u
.
In the Born approximation, the transition probability per unit time
is given by the Golden Rule (4.1.1), which we may here write
h
if
W
(
k
σ,
k
σ
)=
2
π
|H
int
|
k
σ
;
f>
2
δ
(
hω
+
E
i
−
P
i
|
<
k
σ
;
i
|
E
f
)
,
where
hω
=
ε
k
σ
−
ε
k
σ
. Instead of basing the derivation of the mag-
netic resistivity on the linearized spin-wave expression (5.7.20) for
H
int
,
we shall be somewhat more general and use
H
sf
from eqn (5.7.6) as the
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