12.3 Energy Levels in Heterostructures
369
Fig. 12.27 Schematic representation of the development of hole levels in a quantum well: a degenerate bulk levels at
b splitting at the subband edge (due to different quantized values of k z ), c in-plane dispersion (mass reversal), d
anticrossing behavior. Based on [1123]
(semi-)analytical solutions for special and numerical methods for arbitrary potential distributions are
discussed in [1181]. The application of k · p theory (Appendix H) to heterostructures is discussed
in [1182].
The motion of carriers in the plane is still free and has a two-dimensional dispersion. Thus, each
quantized level contributes m
∗
/(π
2
) to the density of states at each subband edge E n .
For holes, the situation is a little more complicated than for electrons (Fig. 12.27). First, the degeneracy of heavy and light holes is lifted since their mass enters the confinement energy. The effective
hole masses along the z direction, i.e., those that enter (12.5), are
1
m
z
hh
= γ 1 − 2 γ 2
(12.8a)
1
m
z
lh
= γ 1 + 2 γ 2 .
(12.8b)
The light holes have the higher quantization energy. The angular momentum is quantized along the z
direction. The transverse masses for the dispersion in the interface plane are
1
m
xy
hh
= γ 1 + γ 2
(12.9a)
1
m
xy
lh
= γ 1 − γ 2 .
(12.9b)
Now the heavy hole, i.e., the J z = ±
3
2
state, has the smaller mass and the light hole (J z = ±
1
2
) the larger
(Fig. 12.27c). However, this consideration is only an approximation since the lifting of degeneracy and
the dispersion have to be treated on the same level. Higher terms of the perturbation calculation lead
to band mixing and remove the band crossing that seems to originate from the situation at the point.
In reality, the bands show anticrossing behavior and are strongly deformed. The hole dispersion in a
superlattice and the anticrossing behavior is shown in Fig. 12.28.
Experimentally observed transition energies in quantum wells of varying thickness are shown in
Fig. 12.29 and are in good agreement with the theoretical calculation. We note that for infinite barriers
optical transitions are only allowed between confined electron and hole states with the same quantum
number n. For finite barriers this selection rule becomes relaxed, and other transitions become partially
allowed, e.g., e 1 –hh 3 . The optical matrix element from the Bloch part of the wavefunction, which was
isotropic for (cubic) bulk material (9.37), is anisotropic for quantum wells. TE (TM) polarization is
defined with the electromagnetic field in (perpendicular to) the plane of the quantum well (Fig. 12.30a).
At the subband edge, i.e., for in-plane wavevector k || = 0 the matrix elements for the various polariza-
369
Fig. 12.27 Schematic representation of the development of hole levels in a quantum well: a degenerate bulk levels at
b splitting at the subband edge (due to different quantized values of k z ), c in-plane dispersion (mass reversal), d
anticrossing behavior. Based on [1123]
(semi-)analytical solutions for special and numerical methods for arbitrary potential distributions are
discussed in [1181]. The application of k · p theory (Appendix H) to heterostructures is discussed
in [1182].
The motion of carriers in the plane is still free and has a two-dimensional dispersion. Thus, each
quantized level contributes m
∗
/(π
2
) to the density of states at each subband edge E n .
For holes, the situation is a little more complicated than for electrons (Fig. 12.27). First, the degeneracy of heavy and light holes is lifted since their mass enters the confinement energy. The effective
hole masses along the z direction, i.e., those that enter (12.5), are
1
m
z
hh
= γ 1 − 2 γ 2
(12.8a)
1
m
z
lh
= γ 1 + 2 γ 2 .
(12.8b)
The light holes have the higher quantization energy. The angular momentum is quantized along the z
direction. The transverse masses for the dispersion in the interface plane are
1
m
xy
hh
= γ 1 + γ 2
(12.9a)
1
m
xy
lh
= γ 1 − γ 2 .
(12.9b)
Now the heavy hole, i.e., the J z = ±
3
2
state, has the smaller mass and the light hole (J z = ±
1
2
) the larger
(Fig. 12.27c). However, this consideration is only an approximation since the lifting of degeneracy and
the dispersion have to be treated on the same level. Higher terms of the perturbation calculation lead
to band mixing and remove the band crossing that seems to originate from the situation at the point.
In reality, the bands show anticrossing behavior and are strongly deformed. The hole dispersion in a
superlattice and the anticrossing behavior is shown in Fig. 12.28.
Experimentally observed transition energies in quantum wells of varying thickness are shown in
Fig. 12.29 and are in good agreement with the theoretical calculation. We note that for infinite barriers
optical transitions are only allowed between confined electron and hole states with the same quantum
number n. For finite barriers this selection rule becomes relaxed, and other transitions become partially
allowed, e.g., e 1 –hh 3 . The optical matrix element from the Bloch part of the wavefunction, which was
isotropic for (cubic) bulk material (9.37), is anisotropic for quantum wells. TE (TM) polarization is
defined with the electromagnetic field in (perpendicular to) the plane of the quantum well (Fig. 12.30a).
At the subband edge, i.e., for in-plane wavevector k || = 0 the matrix elements for the various polariza-