9.9 Absorption in the Presence of Free Charge Carriers
295
Fig. 9.43 Principle of
Burstein–Moss shift. Left
panel: Schematic band
structure with completely
filled electron states shown
in grey. The k-vector for
the lowest photon energy
optical absorption process
is indicated as ˆ
k. Right
panel: Electron distribution
function for a degenerate
electron gas with Fermi
level in the conduction
band
E C
k
E
E V
f e
E
E F
E F -4kT
k
Besides the energy shift in the conduction band, the corresponding energy shift in the valence band
ˆ
k
2
/(2m h ) must be considered. Thus, the Burstein–Moss shift of the absorption edge is
= − E g = (E F − 4kT − E C )
1 +
m e
m h
.
(9.82)
The relation between n and the Fermi level is given by (7.6). If E F − E C kT the Fermi integral can
be approximated by
2
√
π
2
3
E F −E C
kT
3/2 . Using (7.8) for N C , the Burstein–Moss shift can be written for
this case as
= n
2/3 h
2
8 m e
3
π
2/3
1 +
m e
m h
≈ 0.97
h
2
8 m r
n
2/3
.
(9.83)
The n
2/3 dependence of the energy shift is found, e.g., for CdO
13 with different carrier concentrations
(due to different deposition temperature, no intentional doping) [921] and depicted in Fig. 9.44a. Similar
behavior is found for ITO (indium-tin-oxide) thin films, deposited at different sputtering conditions,
leading to different carrier concentrations (9.44b).
9.9.3 Inter-Valenceband Transitions
Transitions within the valence band can occur between three bands, i.e. lh→hh, so→hh, and so→lh,
as schematically depicted in Fig. 9.45. Theoretical treatments have been given in [923, 924]. For
GaAs, such intravalence-band absorption occurs at photon energies close to 0 as shown in Fig. 9.46a
for p-type GaAs:Zn [925]. For p-type GaSb, the absorption coefficient below the fundamental band
gap is found almost entirely due to inter-valence band transitions, as shown in Fig. 9.46b for a hole
concentration of p = 3.2 × 10
16 cm
−3 [926].
13 CdO is an indirect semiconductor, the optical band gap is the energy of the direct transition at the typically
obtained from extrapolation in the α 2 versus energy plot. The indirect transitions involve holes from other points in the
Brillouin zone (cmp. Fig. 6.13).
295
Fig. 9.43 Principle of
Burstein–Moss shift. Left
panel: Schematic band
structure with completely
filled electron states shown
in grey. The k-vector for
the lowest photon energy
optical absorption process
is indicated as ˆ
k. Right
panel: Electron distribution
function for a degenerate
electron gas with Fermi
level in the conduction
band
E C
k
E
E V
f e
E
E F
E F -4kT
k
Besides the energy shift in the conduction band, the corresponding energy shift in the valence band
ˆ
k
2
/(2m h ) must be considered. Thus, the Burstein–Moss shift of the absorption edge is
= − E g = (E F − 4kT − E C )
1 +
m e
m h
.
(9.82)
The relation between n and the Fermi level is given by (7.6). If E F − E C kT the Fermi integral can
be approximated by
2
√
π
2
3
E F −E C
kT
3/2 . Using (7.8) for N C , the Burstein–Moss shift can be written for
this case as
= n
2/3 h
2
8 m e
3
π
2/3
1 +
m e
m h
≈ 0.97
h
2
8 m r
n
2/3
.
(9.83)
The n
2/3 dependence of the energy shift is found, e.g., for CdO
13 with different carrier concentrations
(due to different deposition temperature, no intentional doping) [921] and depicted in Fig. 9.44a. Similar
behavior is found for ITO (indium-tin-oxide) thin films, deposited at different sputtering conditions,
leading to different carrier concentrations (9.44b).
9.9.3 Inter-Valenceband Transitions
Transitions within the valence band can occur between three bands, i.e. lh→hh, so→hh, and so→lh,
as schematically depicted in Fig. 9.45. Theoretical treatments have been given in [923, 924]. For
GaAs, such intravalence-band absorption occurs at photon energies close to 0 as shown in Fig. 9.46a
for p-type GaAs:Zn [925]. For p-type GaSb, the absorption coefficient below the fundamental band
gap is found almost entirely due to inter-valence band transitions, as shown in Fig. 9.46b for a hole
concentration of p = 3.2 × 10
16 cm
−3 [926].
13 CdO is an indirect semiconductor, the optical band gap is the energy of the direct transition at the typically
obtained from extrapolation in the α 2 versus energy plot. The indirect transitions involve holes from other points in the
Brillouin zone (cmp. Fig. 6.13).