6.9 Electron Dispersion
161
Fig. 6.35 Energy
isosurfaces in k-space in
the vicinity of the
conduction-band minima
for a GaAs with isotropic
(spehrical) minimum at
-point, b ZnO with
anisotropic (ellipsoidal)
minimum at -point
(anisotropy exaggerated), c
silicon with six equivalent
anisotropic minima
(m l /m t = 5 not to scale)
along 100 and d
germanium with eight
equivalent anisotropic
minima along 111. The
cube indicates the 100
directions for the cubic
materials. For the wurtzite
material (part b) the
vertical direction is along
[00.1]
(a)
(b)
(c)
(d)
The electron mass is given by
4
m 0
m ∗
e
= 1 +
E P
3
2
E g
+
1
E g + 0
(6.43)
= 1 + E P
E g + 2 0 /3
E g
E g + 0
≈ 1 +
E P
E g + 0 /3
≈
E P
E g
.
Comparison with the fit from Fig. 6.34 yields that E P is similar for all semiconductors [511] and of the
order of 20 eV (InAs: 22.2 eV, GaAs: 25.7 eV, InP: 20.4 eV, ZnSe: 23 eV, CdS: 21 eV).
In silicon there are six equivalent conduction-band minima. The surfaces of equal energy are
schematically shown in Fig. 6.35c. The ellipsoids are extended along the 100 direction because
the longitudinal mass (along the path) is larger than the transverse mass in the two perpendicular
directions (Table 6.5). For example, the dispersion relation in the vicinity of one of the minima is given
as (k
0
x denotes the position of one of the conduction-band minima close to a X-point)
E(k) =
2
(k x − k
0
x )
2
2 m l
+
k
2
y + k
2
z
2 m t
.
(6.44)
For germanium surfaces of constant energy around the eight conduction-band minima in the 111
directions are depicted in Fig. 6.35d. The longitudinal and the transverse masses are again different.
For GaAs, the conduction-band dispersion around the point is isotropic, thus the surface of constant
energy is simply a sphere (Fig. 6.35a). In wurtzite semiconductors the conduction-band minimum is at
the -point. The mass along the c-axis is typically smaller than the mass within the (00.1) plane [512]
4 0 is the spin-orbit splitting discussed in Sect. 6.10.2.
161
Fig. 6.35 Energy
isosurfaces in k-space in
the vicinity of the
conduction-band minima
for a GaAs with isotropic
(spehrical) minimum at
-point, b ZnO with
anisotropic (ellipsoidal)
minimum at -point
(anisotropy exaggerated), c
silicon with six equivalent
anisotropic minima
(m l /m t = 5 not to scale)
along 100 and d
germanium with eight
equivalent anisotropic
minima along 111. The
cube indicates the 100
directions for the cubic
materials. For the wurtzite
material (part b) the
vertical direction is along
[00.1]
(a)
(b)
(c)
(d)
The electron mass is given by
4
m 0
m ∗
e
= 1 +
E P
3
2
E g
+
1
E g + 0
(6.43)
= 1 + E P
E g + 2 0 /3
E g
E g + 0
≈ 1 +
E P
E g + 0 /3
≈
E P
E g
.
Comparison with the fit from Fig. 6.34 yields that E P is similar for all semiconductors [511] and of the
order of 20 eV (InAs: 22.2 eV, GaAs: 25.7 eV, InP: 20.4 eV, ZnSe: 23 eV, CdS: 21 eV).
In silicon there are six equivalent conduction-band minima. The surfaces of equal energy are
schematically shown in Fig. 6.35c. The ellipsoids are extended along the 100 direction because
the longitudinal mass (along the path) is larger than the transverse mass in the two perpendicular
directions (Table 6.5). For example, the dispersion relation in the vicinity of one of the minima is given
as (k
0
x denotes the position of one of the conduction-band minima close to a X-point)
E(k) =
2
(k x − k
0
x )
2
2 m l
+
k
2
y + k
2
z
2 m t
.
(6.44)
For germanium surfaces of constant energy around the eight conduction-band minima in the 111
directions are depicted in Fig. 6.35d. The longitudinal and the transverse masses are again different.
For GaAs, the conduction-band dispersion around the point is isotropic, thus the surface of constant
energy is simply a sphere (Fig. 6.35a). In wurtzite semiconductors the conduction-band minimum is at
the -point. The mass along the c-axis is typically smaller than the mass within the (00.1) plane [512]
4 0 is the spin-orbit splitting discussed in Sect. 6.10.2.