12.3 Energy Levels in Heterostructures
373
Fig. 12.34 Absorption spectra of a single, double, triple and ten coupled quantum wells. Theoretically predicted transitions with heavy (light) holes are labeled with filled (empty) bars at their respective transition energies. Adapted
from [1192]
Fig. 12.35 Schematic
formation of a triangular
potential well in a
n-(Al,Ga)As/n-GaAs
heterostructure, a before
and b after equilibration of
Fermi levels
(a)
GaAs
AlGaAs
E v
E c
(b)
E v
E c
12.3.4 Heterointerface Between Doped Materials
We consider a single heterointerface between n-doped materials. As an example we take n-(Al,Ga)As/nGaAs (Fig. 12.35). First, we consider the materials without contact, forming a type-I structure. In thermodynamic equilibrium the system must have a constant Fermi level. Thus, charge is transferred from
the region close to the interface from (Al,Ga)As to GaAs. This results in the formation of a triangular
potential well in the GaAs close to the interface. A two-dimensional electron gas (2DEG) forms in this
potential well (Fig. 12.36). The charge transfer in thermodynamic equilibrium adjusts the band bending and the charge density (quantized levels in the well) in such a way that they are self-consistent.
The Poisson equation and the Schrödinger equation are simultaneously fulfilled. Numerically, both
equations are iteratively solved and the solution is altered until it is self-consistent, i.e., it fulfills both
equations.
373
Fig. 12.34 Absorption spectra of a single, double, triple and ten coupled quantum wells. Theoretically predicted transitions with heavy (light) holes are labeled with filled (empty) bars at their respective transition energies. Adapted
from [1192]
Fig. 12.35 Schematic
formation of a triangular
potential well in a
n-(Al,Ga)As/n-GaAs
heterostructure, a before
and b after equilibration of
Fermi levels
(a)
GaAs
AlGaAs
E v
E c
(b)
E v
E c
12.3.4 Heterointerface Between Doped Materials
We consider a single heterointerface between n-doped materials. As an example we take n-(Al,Ga)As/nGaAs (Fig. 12.35). First, we consider the materials without contact, forming a type-I structure. In thermodynamic equilibrium the system must have a constant Fermi level. Thus, charge is transferred from
the region close to the interface from (Al,Ga)As to GaAs. This results in the formation of a triangular
potential well in the GaAs close to the interface. A two-dimensional electron gas (2DEG) forms in this
potential well (Fig. 12.36). The charge transfer in thermodynamic equilibrium adjusts the band bending and the charge density (quantized levels in the well) in such a way that they are self-consistent.
The Poisson equation and the Schrödinger equation are simultaneously fulfilled. Numerically, both
equations are iteratively solved and the solution is altered until it is self-consistent, i.e., it fulfills both
equations.