160
6 Band Structure
Fig. 6.33 Schematic
diagram of the electron
dispersion E(k) in a
typical semiconductor
(blue) and corresponding
carrier velocity (∝ ∂ E/∂k)
(red) and effective mass
(∝ 1/(∂ 2 E/∂k 2 )) (green)
Fig. 6.34 Effective
electron mass (in units of
the free-electron mass m 0 )
as a function of the
(low-temperature) band
gap for several (direct band
gap) semiconductors. The
dashed line fulfills
m ∗ /m 0 = E g /20 eV
|c = u c,k c (r) exp (i k c r)
(6.40a)
|v = u v,k v (r) exp (i k v r) .
(6.40b)
Typically, the k-dependence of the matrix element is small and neglected. The momentum matrix element will also be important for optical transitions between the valence and conduction bands (Sect. 9.6).
Other related quantities that are often used are the energy parameter E P
E P =
2 |p cv |
2
m 0
,
(6.41)
and the bulk momentum matrix element M
2
b that is given by
M
2
b =
1
3
|p cv |
2
=
m 0
6
E P .
(6.42)
6 Band Structure
Fig. 6.33 Schematic
diagram of the electron
dispersion E(k) in a
typical semiconductor
(blue) and corresponding
carrier velocity (∝ ∂ E/∂k)
(red) and effective mass
(∝ 1/(∂ 2 E/∂k 2 )) (green)
Fig. 6.34 Effective
electron mass (in units of
the free-electron mass m 0 )
as a function of the
(low-temperature) band
gap for several (direct band
gap) semiconductors. The
dashed line fulfills
m ∗ /m 0 = E g /20 eV
|c = u c,k c (r) exp (i k c r)
(6.40a)
|v = u v,k v (r) exp (i k v r) .
(6.40b)
Typically, the k-dependence of the matrix element is small and neglected. The momentum matrix element will also be important for optical transitions between the valence and conduction bands (Sect. 9.6).
Other related quantities that are often used are the energy parameter E P
E P =
2 |p cv |
2
m 0
,
(6.41)
and the bulk momentum matrix element M
2
b that is given by
M
2
b =
1
3
|p cv |
2
=
m 0
6
E P .
(6.42)