12.3 Energy Levels in Heterostructures
371
(a)
TE
TE
TM
(b)
1/2
hh
m j
1/2
1/2
3/2
3/2
1/2
electrons
lh
hh
lh
holes
m j
Fig. 12.30 a Directions of electric-field vector relative to the quantum-well plane for TE and TM polarization. b Optical
selection rules for band–band transitions in a quantum well. If the (in-plane averaged) relative strength of the e–hh
transitions (solid lines) is 1, the relative strength of the TE-polarized e–lh transitions (dashed lines) is 1/3 and that of the
TM-polarized e–lh transitions (dash-dotted lines) is 4/3
Table 12.3 Squared momentum matrix elements ||c|ˆ e · p|v 2 in a quantum well for various propagation directions in
units of M 2
b . The quantum-well normal is along z
Propagation
ˆ
e x (TE)
ˆ
e y (TE)
ˆ
e z (TM)
e–hh
x
–
1/2
0
y
1/2
–
0
z
1/2
1/2
–
e–lh
x
–
1/6
2/3
y
1/6
–
2/3
z
1/6
1/6
–
e–so
x
–
1/3
1/3
y
1/3
–
1/3
z
1/3
1/3
–
5
0
5
3
0
3
5
2
5
1
0
2
0
1
40
35
30
25
5
GaAs/Al Ga As
Fig. 12.31 Experimental values for the acceptor binding energy in GaAs/Al 0.3 Ga 0.7 As quantum wells (solid circles)
from [1186] as a function of well width. Solid line is theory (variational calculation) for the well-center acceptor
including top four valence bands and finite barriers, dashed line is hydrogen-like model with infinite barrier height.
Adapted from [1187]