6.9 Electron Dispersion
163
2 k
2
2m
∗
0
= E
1 +
E
E
∗
0
,
(6.46)
where E
∗
0 > 0 parameterizes the amount of nonparabolicity (a parabolic band corresponds to E
∗
0 = ∞).
The nonparabolic dispersion for GaAs is shown in Fig. 6.37a. The curvature is reduced for larger k
and thus the effective mass is energy dependent and increases with the energy. Equation (6.46) leads
to the energy-dependent effective mass
m
∗
(E) = m
∗
0
1 +
2E
E
∗
0
,
(6.47)
where m
∗
0 denotes here the effective mass at k = 0. Theory and experimental data for the effective
electron mass of GaAs are shown in Fig. 6.37b.
6.10 Holes
6.10.1 Hole Concept
Holes are missing electrons in an otherwise filled band. A Schrödinger-type wave-equation for holes
(unoccupied electron states) was derived by Heisenberg [70] to interpret Hall effect data. The hole
concept is useful to describe the properties of charge carriers at the top of the valence band. The hole
is a new quasi-particle whose dispersion relation is schematically shown in Fig. 6.38 in relation to the
dispersion of electrons in the valence band.
The wavevector of the hole (filled circle in Fig. 6.38) is related to that of the ‘missing’ electron
(empty circle in Fig. 6.38) by k h = −k e . The energy is E h (k h ) = −E e (k e ), assuming that E V = 0,
otherwise E h (k h ) = −E e (k e ) + 2E V . The hole energy is larger for holes that are further away from
the top of the valence band, i.e. the lower the energy state of the missing electron. The velocity of the
hole, v h =
−1 dE h /dk h , is the same, v h = v e , and the charge is positive, +e. The effective mass of the
(a)
k (10 cm )
2
1 3
- 2
18
-3
0
0.1
0.2
0.3
GaAs
[001]
[111]
(b)
10
16
-3
)
10
18
10
19
200
E (meV)
F
10
17
0.14
0.08
0.12
0.10
0.06
m*/m
0
GaAs
Fig. 6.37 a Dispersion relations for the conduction band of GaAs. The solid line is parabolic dispersion (constant
effective mass). The dashed (dash-dotted) line denotes the dispersion for k along [001] ([111]) from a five-level k · p
model (5LM). b Cyclotron resonance effective mass of electrons in GaAs as a function of the Fermi level (upper abscissa)
and the corresponding electron concentration (lower abscissa). The dashed line is from a 2LM according to (6.47) with
E ∗
0 = 1.52 eV. The solid lines are for a 5LM for the three principal directions of the magnetic field. The symbols represent
experimental data from different sources. Data from [519]
163
2 k
2
2m
∗
0
= E
1 +
E
E
∗
0
,
(6.46)
where E
∗
0 > 0 parameterizes the amount of nonparabolicity (a parabolic band corresponds to E
∗
0 = ∞).
The nonparabolic dispersion for GaAs is shown in Fig. 6.37a. The curvature is reduced for larger k
and thus the effective mass is energy dependent and increases with the energy. Equation (6.46) leads
to the energy-dependent effective mass
m
∗
(E) = m
∗
0
1 +
2E
E
∗
0
,
(6.47)
where m
∗
0 denotes here the effective mass at k = 0. Theory and experimental data for the effective
electron mass of GaAs are shown in Fig. 6.37b.
6.10 Holes
6.10.1 Hole Concept
Holes are missing electrons in an otherwise filled band. A Schrödinger-type wave-equation for holes
(unoccupied electron states) was derived by Heisenberg [70] to interpret Hall effect data. The hole
concept is useful to describe the properties of charge carriers at the top of the valence band. The hole
is a new quasi-particle whose dispersion relation is schematically shown in Fig. 6.38 in relation to the
dispersion of electrons in the valence band.
The wavevector of the hole (filled circle in Fig. 6.38) is related to that of the ‘missing’ electron
(empty circle in Fig. 6.38) by k h = −k e . The energy is E h (k h ) = −E e (k e ), assuming that E V = 0,
otherwise E h (k h ) = −E e (k e ) + 2E V . The hole energy is larger for holes that are further away from
the top of the valence band, i.e. the lower the energy state of the missing electron. The velocity of the
hole, v h =
−1 dE h /dk h , is the same, v h = v e , and the charge is positive, +e. The effective mass of the
(a)
k (10 cm )
2
1 3
- 2
18
-3
0
0.1
0.2
0.3
GaAs
[001]
[111]
(b)
10
16
-3
)
10
18
10
19
200
E (meV)
F
10
17
0.14
0.08
0.12
0.10
0.06
m*/m
0
GaAs
Fig. 6.37 a Dispersion relations for the conduction band of GaAs. The solid line is parabolic dispersion (constant
effective mass). The dashed (dash-dotted) line denotes the dispersion for k along [001] ([111]) from a five-level k · p
model (5LM). b Cyclotron resonance effective mass of electrons in GaAs as a function of the Fermi level (upper abscissa)
and the corresponding electron concentration (lower abscissa). The dashed line is from a 2LM according to (6.47) with
E ∗
0 = 1.52 eV. The solid lines are for a 5LM for the three principal directions of the magnetic field. The symbols represent
experimental data from different sources. Data from [519]