440
15 External Fields
(a)
k x
E F
k z
B
n=n m
n=0
n=...
n=1
k y
(b)
E
F
h c
1/2
E
0
3/2
5/2
7/2
3D
Fig. 15.11 3D electron gas in an external magnetic field. a Allowed states in k-space for magnetic field along the z
direction. b Density of states (DOS) ρ versus energy (in units of c ). Dashed line is three-dimensional DOS without
magnetic field. Based on [1123]
1
B
=
e
m ∗ E F
.
(15.33)
This periodicity is used to determine experimentally, e.g., the properties of the Fermi surface in metals
using the Shubnikov-de Haas oscillations (of the magnetoresistance) or the de Haas-van Alphén effect
(oscillation of the magnetic susceptibility).
Equation (15.31) needs to be extended to account for the splitting (15.7) of the Landau level due to
the electron spin. According to [1452], the electron Landau level energy can be written as
E n =
n +
1
2
e B
m ∗ (E)
± g
∗
e (E) μ B B ,
(15.34)
with energy dependent effective mass and g-factor
1
m ∗ (E)
=
1
m ∗ (0)
E g (E g + 0 )
3E g + 2 0
2
E + E g
+
1
E + E g + 0
(15.35a)
g
∗
e (E) = g
∗
e (0)
E g (E g + 0 )
0
1
E + E g
−
1
E + E g + 0
(15.35b)
The band edge value m
∗
(0) of the effective mass is given by (6.43) and that of the g-factor by
g
∗
e (0) = 2
1 −
2 0
3 E g (E g + 0 )
E P
.
(15.36)
For large spin-orbit splitting, the value of the g-factor deviates strongly from 2 and becomes negative.
15.2.4 Magnetic Field Effect on Impurities
The Zeeman effect is the magnetic field dependence (and the lifting of degeneracies) of spectral
lines. For semiconductor physics this is a tool to investigate properties of defect related states with
sufficiently sharp spectral features. As example we refer to the investigation of the Si:P donor system
15 External Fields
(a)
k x
E F
k z
B
n=n m
n=0
n=...
n=1
k y
(b)
E
F
h c
1/2
E
0
3/2
5/2
7/2
3D
Fig. 15.11 3D electron gas in an external magnetic field. a Allowed states in k-space for magnetic field along the z
direction. b Density of states (DOS) ρ versus energy (in units of c ). Dashed line is three-dimensional DOS without
magnetic field. Based on [1123]
1
B
=
e
m ∗ E F
.
(15.33)
This periodicity is used to determine experimentally, e.g., the properties of the Fermi surface in metals
using the Shubnikov-de Haas oscillations (of the magnetoresistance) or the de Haas-van Alphén effect
(oscillation of the magnetic susceptibility).
Equation (15.31) needs to be extended to account for the splitting (15.7) of the Landau level due to
the electron spin. According to [1452], the electron Landau level energy can be written as
E n =
n +
1
2
e B
m ∗ (E)
± g
∗
e (E) μ B B ,
(15.34)
with energy dependent effective mass and g-factor
1
m ∗ (E)
=
1
m ∗ (0)
E g (E g + 0 )
3E g + 2 0
2
E + E g
+
1
E + E g + 0
(15.35a)
g
∗
e (E) = g
∗
e (0)
E g (E g + 0 )
0
1
E + E g
−
1
E + E g + 0
(15.35b)
The band edge value m
∗
(0) of the effective mass is given by (6.43) and that of the g-factor by
g
∗
e (0) = 2
1 −
2 0
3 E g (E g + 0 )
E P
.
(15.36)
For large spin-orbit splitting, the value of the g-factor deviates strongly from 2 and becomes negative.
15.2.4 Magnetic Field Effect on Impurities
The Zeeman effect is the magnetic field dependence (and the lifting of degeneracies) of spectral
lines. For semiconductor physics this is a tool to investigate properties of defect related states with
sufficiently sharp spectral features. As example we refer to the investigation of the Si:P donor system