4.2 Free Electrons in a Magnetic Field 91
As can be seen from Fig. 4.7, M
para
z
depends on the density of states (or
theelectronmass)andontheg factor at the Fermi energy. In solids, we
expect material specific deviations of the experimental values for the Pauli
susceptibility
χ
spin
=
dM
para
z
dB
=
2
3
D(E
F
)μ
2
B
, (4.54)
from the free electron value indicating the already mentioned modifications
of t hese parameters due to the periodic p otential and the electron–electron
interaction. In fact such deviations are particularly strong, e.g., for heavy
fermion systems, as was the case for the specific heat.
• Landau–Peierls
5
diamagnetism: The second term is negative, acting against
the external magnetic field according to the Lenz rule applied to the
cyclotron motion of the electrons. The magnetic moment connected with
this motion has a direction opposite to the external magnetic field and
results in a diamagnetic contribution. For free electrons in a periodic poten-
tial, which can be described as particles with an effective mass m
∗
,the
diamagnetic contribution is to be multiplied by a factor (m
∗
/m)
2
:
M
dia
z
= −
1
2
N
V
μ
2
B
B
E
F
m
∗
m
!
2
(4.55)
For mass ratios m
∗
/m ≃ 1 (see Table 4.2), we expect comparable values of
the para- and dia-magnetic contributions. But there are also systems with
m
∗
/m ≫ 1 due to orbital contributions of d and f electrons, for which the
diamagnetic term dominates.
• de Haas–van Alphen
6
effect: The third term describes the already men-
tioned oscillating contribution, periodic in 1/B with a period determined
by the Fermi energy, which is characteristic for Landau quantization. In
fact, this behavior of the magnetic susceptibility is found in metals and
known as de Haas–van Alphen effe ct. It can be used to determine the Fermi
energy or, more precisely, the parameters of the Fermi surface,which
for solids can deviate from the spherical for m (see Sect. 5.7). The sum
in the oscillating contribution converges rapidly due to the denominator
and usually it suffices to con sider only the first term with n =1:
χ
osc
(B) ≃−μ
0
3
2
N
V
π
2
k
B
T
B
2
m
∗
m
x
!
1/2
cos(π
m
∗
m
)
cos(
π
4
−
m
∗
π
m
x)
sinh(
m
∗
π
m
y)
. (4.56)
Here we have considered again possible deviations of the effective mass
from the free electron mass. An example of de Haas–van Alphen oscillations
measured for Cu is shown in Fig.
4.8.
5
Sir Rudolf Ernst Peierls 1907–1995
6
Wander Johannes de Haas 1878–1960, P.M. van Alphen 1906–1967