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Chapter 12: Solid state physics
67
12.5.3
The Halleffect
If a magnetic Feld is applied
⊥
to the direction of the current the charge carriers will be pushed aside by the
Lorentz force. This results in a magnetic Feld
⊥
to the ±ow direction of the current. If
±
J
=
J
±
e
x
and
±
B
=
B
±
e
z
than
E
y
/E
x
=
μB
. The Hall coefFcient is deFned by:
R
H
=
E
y
/J
x
B
, and
R
H
=

1
/ne
if
J
x
=
neμE
x
.
The Hall voltage is given by:
V
H
=
Bvb
=
IB/neh
where
b
is the width of the material and
h
de height.
12.5.4
Thermal heat conductivity
With
²
=
v
F
τ
the mean free path of the electrons follows from
κ
=
1
3
C
±
v
²
²
:
κ
electrons
=
π
2
nk
2
T
τ
/
3
m
.
²rom this follows for the
WiedemannFranz ratio
:
κ
/
σ
=
1
3
(
π
k/e
)
2
T
.
12.6
Energy bands
In the
tightbond
approximation it is assumed that
ψ
=e
ikna
φ
(
x

na
)
. ²rom this follows for the energy:
±
E
²
=
±
ψ

H

ψ
²
=
E
at

α

2
β
cos(
ka
)
. So this gives a cosine superimposed on the atomic energy, which
can often be approximated by a harmonic oscillator. If it is assumed that the electron is nearly free one can
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