6.2 From Bulk to Zero-Dimensional Structures
193
for instance in y direction, with n y = 0, ±1, ±2, . . .. The solution of the wavefunction which fulfils these requirements are of the form
ψ ∝ exp(ik y y).
(6.4)
This makes clear that the possible energies of a massive-particle system (e.g.
electrons) range between fully free in 3D in the form of
E(k) =
2 k
2
2m
, (free massive particle dispersion),
(6.5)
where m is the particle mass and k = k x + k y + k z the overall momentum in bulk,
to partially free such as in wells according to
E n z (k || ) =
2
π
2
2m L 2
z
n
2
z +
2 k
2
||
2m
(1D confined 2D system),
(6.6)
and to the case of full quantisation in 0D represented by
E n x ,n y ,n z =
2
π
2
2m L 2
x
n
2
x +
2
π
2
2m L 2
y
n
2
y +
2
π
2
2m L 2
z
n
2
z (3D confined 0D system). (6.7)
Here, the terms with quantum numbers n x/y/z denote the confinement energies for
the respective directions x, y, z, whereas for the particle Hamiltonian (with E tot =
E kin (k) + V (r)) in this steady-state consideration (i.e. without time-dependence) the
potential V (r, t) = 0 relative to the bulk ground-state in the well region.
For a schematic overview on the quantum-well confinement of electronic particles
in semiconductor heterostructures see Fig. 6.2.
While size reduction itself enables the modification of energy states in systems,
e.g. for charge carriers, quasi-particles or light, one can further play with energy
levels by means of periodic arrangement.
Amazingly, artificial atom-like structures or quantum wells are not limited to
a single entity. Indeed, coupled quantum wells can provide molecule-like hybrid
states (besides increasing the number of confined particles in the total potentialwells structure with regard to a single well).
Going one step further, superlattices of wells or dots (e.g. deterministically
achieved by epitaxy or naturally formed by moiré patterns) lift the degeneracy of
energy states contributed from each quantum box completely by giving rise to miniband formation (due to the periodic arrangement of these entities). This is analogous
to band structure formation in crystals, in which the elements of a crystal are periodically arranged in space, as repetitions of a unit cell. Subbands in the conduction or
valence band are for instance exploited in quantum-cascade lasers, unipolar devices
that totally rely on intersubband transitions and cascaded (stimulated) relaxation
processes for (coherent) light extraction.
193
for instance in y direction, with n y = 0, ±1, ±2, . . .. The solution of the wavefunction which fulfils these requirements are of the form
ψ ∝ exp(ik y y).
(6.4)
This makes clear that the possible energies of a massive-particle system (e.g.
electrons) range between fully free in 3D in the form of
E(k) =
2 k
2
2m
, (free massive particle dispersion),
(6.5)
where m is the particle mass and k = k x + k y + k z the overall momentum in bulk,
to partially free such as in wells according to
E n z (k || ) =
2
π
2
2m L 2
z
n
2
z +
2 k
2
||
2m
(1D confined 2D system),
(6.6)
and to the case of full quantisation in 0D represented by
E n x ,n y ,n z =
2
π
2
2m L 2
x
n
2
x +
2
π
2
2m L 2
y
n
2
y +
2
π
2
2m L 2
z
n
2
z (3D confined 0D system). (6.7)
Here, the terms with quantum numbers n x/y/z denote the confinement energies for
the respective directions x, y, z, whereas for the particle Hamiltonian (with E tot =
E kin (k) + V (r)) in this steady-state consideration (i.e. without time-dependence) the
potential V (r, t) = 0 relative to the bulk ground-state in the well region.
For a schematic overview on the quantum-well confinement of electronic particles
in semiconductor heterostructures see Fig. 6.2.
While size reduction itself enables the modification of energy states in systems,
e.g. for charge carriers, quasi-particles or light, one can further play with energy
levels by means of periodic arrangement.
Amazingly, artificial atom-like structures or quantum wells are not limited to
a single entity. Indeed, coupled quantum wells can provide molecule-like hybrid
states (besides increasing the number of confined particles in the total potentialwells structure with regard to a single well).
Going one step further, superlattices of wells or dots (e.g. deterministically
achieved by epitaxy or naturally formed by moiré patterns) lift the degeneracy of
energy states contributed from each quantum box completely by giving rise to miniband formation (due to the periodic arrangement of these entities). This is analogous
to band structure formation in crystals, in which the elements of a crystal are periodically arranged in space, as repetitions of a unit cell. Subbands in the conduction or
valence band are for instance exploited in quantum-cascade lasers, unipolar devices
that totally rely on intersubband transitions and cascaded (stimulated) relaxation
processes for (coherent) light extraction.