368
12 Heterostructures
Fig. 12.26 Schematic
energy levels and
wavefunctions in a
potential well with a
infinite barriers, and b
finite barrier height
(a)
L z
n=1
n=2
n=3
(b)
L z
n=1
n=2
12.3.2 Quantum Wells
The energy in a single quantum well of thickness L z (along the growth direction z) can be calculated with
the quantum-mechanical particle-in-a-box model. In the envelope function approximation (Appendix I)
the wavefunction is written as a product of the Bloch function and the envelope function χ(z).
A,B
(r) = exp (i k ⊥ r) u nk (r) χ n (z) ,
(12.4)
where ‘A’ and ‘B’ denote the two different materials. The envelope function χ fulfills, approximately,
the one-dimensional Schrödinger-type equation,
−
2
2m ∗
∂
2
∂z 2 + V c (z)
χ n (z) = E n χ n (z) ,
(12.5)
where m
∗ denotes the effective mass. V c is the confinement potential determined by the band discontinuities. Typically, V c = 0 in the well and V 0 > 0 outside in the barrier. E n are the resulting energy
values of the quantized levels. In the case of infinite barriers (V 0 → ∞, Fig.12.26a) the boundary
conditions χ(0) = χ(L z ) = 0 yield
E n =
2
2m ∗
n π
L z
2
(12.6)
χ n (z) = A n sin
n π
L z
z
,
(12.7)
where E n is called the confinement energy.
For finite barrier height V 0 (Fig. 12.26b) the calculation leads to a transcendental equation. The
wavefunction tunnels into the barrier. While for infinite barrier height the lowest level diverges for
L z → 0, for finite barrier height E 1 → V 0 . A complication arises from the different effective mass
in the well and barrier material. This is taken into account by forcing the continuity of χ and χ
/m
∗
across the interfaces
4 (‘BenDaniel-Duke’ boundary conditions [1180]). The Schrödinger equation and
4 The kinetic energy term in (12.5) is written as
2
2
∂
∂z
1
m ∗ (z)
∂χ
∂z for varying mass across the structure [1179].
12 Heterostructures
Fig. 12.26 Schematic
energy levels and
wavefunctions in a
potential well with a
infinite barriers, and b
finite barrier height
(a)
L z
n=1
n=2
n=3
(b)
L z
n=1
n=2
12.3.2 Quantum Wells
The energy in a single quantum well of thickness L z (along the growth direction z) can be calculated with
the quantum-mechanical particle-in-a-box model. In the envelope function approximation (Appendix I)
the wavefunction is written as a product of the Bloch function and the envelope function χ(z).
A,B
(r) = exp (i k ⊥ r) u nk (r) χ n (z) ,
(12.4)
where ‘A’ and ‘B’ denote the two different materials. The envelope function χ fulfills, approximately,
the one-dimensional Schrödinger-type equation,
−
2
2m ∗
∂
2
∂z 2 + V c (z)
χ n (z) = E n χ n (z) ,
(12.5)
where m
∗ denotes the effective mass. V c is the confinement potential determined by the band discontinuities. Typically, V c = 0 in the well and V 0 > 0 outside in the barrier. E n are the resulting energy
values of the quantized levels. In the case of infinite barriers (V 0 → ∞, Fig.12.26a) the boundary
conditions χ(0) = χ(L z ) = 0 yield
E n =
2
2m ∗
n π
L z
2
(12.6)
χ n (z) = A n sin
n π
L z
z
,
(12.7)
where E n is called the confinement energy.
For finite barrier height V 0 (Fig. 12.26b) the calculation leads to a transcendental equation. The
wavefunction tunnels into the barrier. While for infinite barrier height the lowest level diverges for
L z → 0, for finite barrier height E 1 → V 0 . A complication arises from the different effective mass
in the well and barrier material. This is taken into account by forcing the continuity of χ and χ
/m
∗
across the interfaces
4 (‘BenDaniel-Duke’ boundary conditions [1180]). The Schrödinger equation and
4 The kinetic energy term in (12.5) is written as
2
2
∂
∂z
1
m ∗ (z)
∂χ
∂z for varying mass across the structure [1179].