(p.458) Appendix K Semiconductor quantum wells
(p.458) Appendix K Semiconductor quantum wells
Semiconductor quantum wells are an important class of nanostructures, with many technological applications in optoelectronics and elsewhere. Our particular interest in quantum wells is not so much because of all these wonderful device applications, but because they represent very simple, quasionedimensional model systems which are nice for benchmark calculations in TDDFT. In this appendix, we will review the basic concepts of the electronic structure and excitations of quantum wells, focusing on a particular type of excitation called intersubband plasmon. Readers who wish for a more indepth treatment of semiconductor nanostructures can choose among a large number of texts; a good place to start is the books by Davies (1998) and Harrison (2005).
The basic geometry of a quantum well is illustrated in Fig. K.1: a layer of semiconductor material A, typically only a few tens of nanometers thick, is sandwiched between layers of another material B. The combination GaAs/Al_{x}Ga_{1–x}As, with 0 < x < 0.3, is very popular since the two materials are almost perfectly latticematched. Here and in the following, we take the zaxis to be perpendicular to the layers, i.e., along the direction in which the heterostructure has been grown. The system is assumed to be infinitely extended in the x–y plane. In practice, the system B–A–B of Fig. K.1 will be grown on a substrate and surrounded by additional spacer layers, which, however, are not important for the following considerations.
We will consider quantum well systems that are ndoped; the doping centers are sufficiently far away (this is called “remote doping” or “modulation doping”) that the ionized impurities have little influence on the electron dynamics in the well. The electrons from these remote doping centers accumulate in the quantum well, and the electronic sheet density N _{s} (which measures the number of electrons per unit area) is assumed to be given. In the following, we will describe how to calculate the electronic ground state and dynamics using (TD)DFT in the effectivemass approximation.
K.1 Effectivemass approximation and subband levels
The electronic structure in a periodic solid reflects the basic symmetries of the underlying crystal lattice. According to Bloch's theorem, the electronic states in the periodic potential have the form of modulated plane waves, ψ _{nq}(r) = u _{nq}(r)e ^{iqr}, where u _{nq} is a latticeperiodic function, with band index n and wave vector q (disregarding spin for now). The associated singleparticle energies ε _{nq} can be computed using standard band structure techniques, which rely heavily on the symmetry of the perfect crystal.
At first sight it appears that the situation becomes extremely complicated if this symmetry is broken, for example if a charged impurity is introduced, or if a heterostructure is formed by combining different materials (see Fig. K.1). This would of course be true if we insisted on calculating a fully ab initio electronic structure for these situations. However, we can often get away with an enormously simplified approach called the effectivemass approximation, which allows us to describe charge carriers in nanostructures with relatively minor sacrifices in accuracy.
Figure K.2 shows a small portion of the electronic band structure of a socalled direct semiconductor (for example, GaAs) close to the zone center. In this region, the valence and conduction bands are parabolic, which means that they have energy dispersion relations similar to that of a free electron (ħ ^{2} q ^{2}/2m), except that the curvatures suggest that their masses are different from the freeelectron mass m. The effective electron mass (m*) and hole masses $\left({m}_{hh}^{*},{m}_{lh}^{*},{m}_{so}^{*}\right)$ in a semiconductor are usually much smaller than the freeelectron mass m. In the following, we focus on the conduction electrons.
(p.460) Let us now apply the effectivemass approximation to describe electrons in a quantum well. The singleparticle states are freeelectronlike in the x–y plane, but experience finitesize quantization along the z direction. This suggests the following form:
The Kohn–Sham potential is given by υ _{s}(z) = υ _{conf}(z) + υ _{H}(z) + υ _{xc}(z). Here, υ _{conf} is the bare confining potential of the quantum well, typically a squarewell potential, but other shapes are also possible by gradually varying the material composition.^{2} The Hartree potential follows from integrating Poisson's equation:
For the xc potential, we use the LDA in its threedimensional form.
The groundstate density n _{0}(z) is obtained as
Equations (K.3)–(K.7) constitute the most elementary DFT approach to describing the selfconsistent subband structure of quantum wells, and are in fact quite accurate for the widely studied GaAs/AlGaAs systems. There are, however, situations where one needs to do better than this, for example in the case of materials with a smaller band gap such as InAs/AlSb. A powerful method to calculate the electronic structure of semiconductors close to the band extrema is the socalled k·p approach, where one expands in terms of a basis of valence and conduction band states at the zone center, including some semiempirical parameters to get the correct band gap. We will not discuss any technical details here (see Davies, 1998; Harrison, 2005), but will merely point out two interesting consequences emerging from a more rigorous treatment:

1. The subband energy dispersions deviate from parabolicity, owing to an energydependence of the effective mass. The bands are also warped to some extent, although this effect is much stronger for the valence bands.

2. In the presence of spinorbit coupling, spin is no longer a good quantum number.For systems without inversion symmetry, the quantum well subbands acquire aq_{ǁ}dependent spin splitting.
(p.462) K.2 Intersubband dynamics
Let us now consider intersubband excitations in a quantum well. Since our quantum wells are translationally invariant in the x–y plane, we write the TDDFT linear spindensity response equation (7.84) in a mixed representation:
The Kohn–Sham response function (7.86) becomes
The envelope functions φ _{j}(z) follow from eqn (K.3), and with ω _{lj} = ε _{l} – ε _{j} we have
The q_{ǁ}integral in eqn (K.10) can be evaluated analytically for all k_{ǁ} (Eguiluz, 1985). The linearized effective potential is given by υ _{s1σ} = υ _{1σ} + υ _{Hxc1σ}, where
We choose the following external perturbation to excite intersubband excitations with a finite inplane wave vector:
The intersubband plasmons correspond to sharp peaks in σ(k_{ǁ}, ω), whereas the singleparticle continuum causes a broad, diffuse bump in the absorption spectrum.
Of particular interest is the case of zero momentum transfer (see Fig. K.3), corresponding to vertical intersubband transitions. We have
In this case, the external perturbation reduces to the usual dipole approximation for linearly polarized electromagnetic waves, ${v}_{1\sigma}\left(z,w\right)={S}_{\sigma}^{\pm}{E}_{0}z$ , and the photoabsorption cross section becomes
The plasmon excitations are separated from the Landaudamping region. We can use the smallmatrix approximation of Section 7.6 to solve the response equation analytically. Keeping only those terms that contain the first and second subband in the Kohn–Sham response function, we have, for k _{ǁ} = 0,
Defining
The Hartree contribution in S _{σσ}′ is known as the depolarization shift, and the xc contribution is sometimes (somewhat misleadingly) called the excitonic shift. The Hartree part always induces an upshift in the plasmon frequency with respect to ω _{21}, and the (p.464)
Figure K.5 shows how the intersubband plasmon excitations look like in real time. The charge plasmon is a simple collective sloshing motion of the density perpendicular to the quantum well plane, along the direction of confinement. The spinup and spindown densities move together in the charge plasmon, but they move in opposite directions in the spin plasmon. This is indicated by snapshots of the densities during the first halfcycle of the oscillations.
Notes:
(^{1}) The peculiar form of the kineticenergy operator used here ensures conservation of current at the interfaces (Davies, 1998).