372
12 Heterostructures
(a)
15
10
5
5
10
15
z
0
Al Ga As/GaAs
0.6
0.4
m m
b
w
m m
m =m
m =m
b
w
b
w
b
w
, b w
,
,
, =
=
b
w
b
w
b
w
(b)
0
5
1 0
W
10
9
8
7
6
InGaAs/
GaAs
Fig. 12.32 a Theoretical (variational) calculation (solid line) of the heavy-hole exciton binding energy versus QW
thickness in a GaAs/Al 0.4 Ga 0.6 As quantum well (using also different approximations, other lines). Adapted from [1188].
b Experimental exciton binding energy in In x Ga 1−x As/GaAs quantum wells of different thickness. Circles: data and
error bars from [1189], x unspecified, squares: data from [1190], x = 0.18
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
E 1
2
3
4
5
6
7
8
9 10
1
0
E 2
E 4
E 3
Fig. 12.33 Band structure of a superlattice with a potential depth of 0.4 eV and well and barrier width w (L QW = L barr ).
Adapted from [1191]
In a realistic calculation the effect of different dielectric constants in the well and barrier (image charge
effect) need to be considered.
12.3.3 Superlattices
In a superlattice, the barrier thickness is so small that carriers can tunnel in neighboring wells or, in
other terms, that there exists a significant wavefunction overlap between adjacent wells. This leads to
a band structure (Fig. 12.33), similar to the Kronig-Penney model (Appendix F). For the superlattice
the bands are called minibands, the gaps are called minigaps. The density of states does not make a
step at the subband edge but follows an arccos function. The modification of the density of states, as
seen in the absorption spectrum, are shown in Fig. 12.34 for 1, 2, 3 and 10 coupled wells.
12 Heterostructures
(a)
15
10
5
5
10
15
z
0
Al Ga As/GaAs
0.6
0.4
m m
b
w
m m
m =m
m =m
b
w
b
w
b
w
, b w
,
,
, =
=
b
w
b
w
b
w
(b)
0
5
1 0
W
10
9
8
7
6
InGaAs/
GaAs
Fig. 12.32 a Theoretical (variational) calculation (solid line) of the heavy-hole exciton binding energy versus QW
thickness in a GaAs/Al 0.4 Ga 0.6 As quantum well (using also different approximations, other lines). Adapted from [1188].
b Experimental exciton binding energy in In x Ga 1−x As/GaAs quantum wells of different thickness. Circles: data and
error bars from [1189], x unspecified, squares: data from [1190], x = 0.18
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
E 1
2
3
4
5
6
7
8
9 10
1
0
E 2
E 4
E 3
Fig. 12.33 Band structure of a superlattice with a potential depth of 0.4 eV and well and barrier width w (L QW = L barr ).
Adapted from [1191]
In a realistic calculation the effect of different dielectric constants in the well and barrier (image charge
effect) need to be considered.
12.3.3 Superlattices
In a superlattice, the barrier thickness is so small that carriers can tunnel in neighboring wells or, in
other terms, that there exists a significant wavefunction overlap between adjacent wells. This leads to
a band structure (Fig. 12.33), similar to the Kronig-Penney model (Appendix F). For the superlattice
the bands are called minibands, the gaps are called minigaps. The density of states does not make a
step at the subband edge but follows an arccos function. The modification of the density of states, as
seen in the absorption spectrum, are shown in Fig. 12.34 for 1, 2, 3 and 10 coupled wells.