Environmental Engineering Reference
In-Depth Information
Since the other modification, the appearance of
k
Fσ
instead of the para-
magnetic value
k
F
in (5
.
7
.
57
b
), generally only causes a minor correction
to the value of the integral in this equation, the magnetic contribu-
tion to
ρ
total
(
T
) is approximately independent of the spin-polarization,
in this model. However, the spin-polarization in the real metals may
be suciently great to alter the topology of the Fermi surface, as dis-
cussed in Section 1.4, so that the resistivity may change abruptly with
temperature or magnetic field. Under these circumstances, the resistiv-
ity must be calculated from first principles, using a realistic model of
the spin-polarized energy bands. The
zz
-contribution should be treated
separately, as the
q
-integral for this case should go from 0 to 2
k
F
,even
when the electron spins are polarized, since no spin-flip is involved in
the scattering process. This modification is, however, unimportant as
the dominating contributions, in the ordered phase, arise from the per-
pendicular spin-wave components of the susceptibility.
The above results also apply, to a good approximation, when the
moments are ordered antiferromagnetically, if the value of
ζ
(
T
)iscalcu-
lated for a spatial modulation of the moments. The spin-polarization of
the band electrons is determined by the MF Hamiltonian, and assuming
J
iz
=
J
z
cos (
Q
·
R
i
), we may replace (5.7.7) by
1)
I
(
n
k
,n
k
)
c
n
k
↑
c
n
k
↓
c
n
k
↓
H
sf
(MF) =
−
(
g
−
c
n
k
↑
−
nn
kk
δ
k
,
k
+
Q
+
τ
+
δ
k
,
k
−
Q
+
τ
×
2
J
z
,
(5
.
7
.
62)
showing that the modulated moments induce a coupling between the
band electrons at the wave-vectors
k
and
k
±
Q
+
τ
.Inthesameway
as the periodic lattice potential lifts the degeneracy of the band states
at the Brillouin-zone boundaries (passing through
k
=
/
2), the above
MF Hamiltonian gives rise to energy gaps at the
superzone boundaries
,
the planes perpendicular to, and passing through, the vectors
k
s
τ
=
(
±
Q
+
τ
)
/
2. If
k
s
is along the
c
-axis, the value of the energy gap
δ
is (
g
in the
n
th band. The importance of the
superzone gaps for the resistivity was first pointed out by Mackintosh
(1962), and detailed theories were developed by Elliott and Wedgwood
(1963) and Miwa (1963). These theories utilized the free-electron model
and the
relaxation time
approximation,
dg
k
σ
/dt
−
1)
|
I
(
n
k
,n
−
k
)
|
J
z
|
coll
=
−
(
g
k
σ
−
f
k
σ
)
/τ
k
σ
,
giving a conductivity
e
2
β
V
τ
k
σ
v
k
σ
·
u
2
f
k
σ
1
f
k
σ
σ
uu
=
−
k
σ
or, if the relaxation time
τ
k
σ
is assumed to be constant over the Fermi
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