Microscopic theory of impurity states in coupled quantum wells and superlattices

We present a theory of impurity states in quantum wells and superlattices which treats the confining heterostructure potential and the random impurity potential on the same footing. The relevant three-dimensional Hamiltonian is diagonalized in the low-doping regime. The results are used to calculate infrared absorption spectra which contain contributions of impurity and intersubband transitions. We mainly discuss the excited impurity states, which are pinned to higher subbands and are resonant states in the continuum. After a detailed analysis of a coupled quantum well system, we study the transition to a superlattice. In particular, we are able to explain existing experimental data on a quadruple quantum well.

Figure 1

Sketch of the subband levels and most important impurity levels (grey) for a single (a) and double (b) quantum well. In (c) the minibands (hatched) and impurity bands (grey shaded) for a superlattice are shown.

Figure 2

Calculated absorption of a double quantum well for different temperatures.

Figure 3

Wave functions of the initial (a) and final states (b)–(d) that contribute the spectrum of the double QW at $T=4\mathrm{K}$.

Figure 4

Final states of the most important transitions contributing to the absorption spectrum of the double QW at $T=300\mathrm{K}$.

Figure 5

Absorption spectra for the double QW with different barrier thicknesses and of a single QW for comparison at $T=4\mathrm{K}$. The spectra a vertically shifted for clarity.

Figure 6

Calculated normalized absorption of a double quantum well for different electron densities and temperatures.

Figure 7

Calculated absorption of a double quantum well for different lateral distances of the donors in each well at an electron temperature of $4\mathrm{K}$. The inset shows the energy level of the ground state for different distances.

Figure 8

Experimental (a) and calculated (b) absorption spectra of a quadruple quantum well shown for different temperatures.

Figure 9

Calculated absorption spectra at $T=300\mathrm{K}$ for different numbers of quantum wells.

Figure 10

Calculated absorption spectra at $T=4\mathrm{K}$ for different numbers of quantum wells.

Figure 11

Wave functions of the final states contributing to a transition at $135\mathrm{meV}$ at $4\mathrm{K}$ (a) and to a transition at $132\mathrm{meV}$ at $T=300\mathrm{K}$ (b).

Figure 12

Absorption spectra of a superlattice for magnetic fields $B=0$, 4, and $7\mathrm{T}$, exhibiting no change with magnetic field. $\Delta T$ is the ratio of $p$- and $s$-polarized transmission, the curves are vertically shifted for clarity. The fine structure around $150\mathrm{meV}$ is probably due to some traces of organic solvents on the sample surface and not relevant here.