Theory of impurity states in coupled quantum wells and superlattices and their infrared absorption spectra

The problem of shallow impurities in confined semiconductor systems has been extensively investigated over the past two decades. It is well known that in two dimensions the binding energy is increased and the degeneracy of the states is lifted due to the symmetry breaking. Yet virtually all calculations were based on variational approaches for the impurity levels; some more sophisticated calculations, on the other hand, were mostly done solely for the ground state and other low-lying states.

Here we present an essentially exact calculation, treating the quantum well (QW) and impurity potential on the same footing in a unified framework [1], by diagonalizing the fully three-dimensional Schrödinger equation exactly on a 100 x 100 nm grid (however neglecting electron-electron interaction). This results in some thousand states. To facilitate comparison with existing experiments we calculate the infrared (IR) absorption spectra by properly evaluating the matrix elements between all relevant states, taking into account their proper thermal occupation.

These calculations have lead to some remarkable insights:

In a superlattice (SL; we use 20 QWs for the calculation) we show that there is not only a an excited 2pz state slightly below the second miniband [2], but rather an excited, resonant impurity band which partly overlaps the second miniband [1]. Consequently the low-temperature IR absorption spectrum in not too highly doped SLs exhibits a transition from the 1s ground state to this resonant impurity band, showing two maxima related to the band edges. This implies that there are impurity states not only attached to the lower miniband edge, but also to the higher miniband edge. The localized nature of the high-energy peak can only be identified by analyzing the final-state wavefunction in the xy plane, since it occurs at nearly the same spectral position as the kz=0 interminiband peak. This observation requires re-interpretation of previous interminiband absorption experiments and is important for the experimental diagnostics of the magnetic-field induced metal-insulator transition. Note that the excited impurity band does not result from impurity levels overlapping in the xy plane, but from the state coupling in z-direction, just like the miniband.

By studying the energy levels and IR spectra for systems with a varying number of quantum wells, we can map out the transition from coupled QWs to a SL, i.e. from sharp intersubband and/or impurity lines to band-like spectra including their van Hove singularities. In a coupled double QW, where the 1-3 intersubband transition is symmetry forbidden, the related impurity transition can nevertheless be observed.

A particularly spectacular example is provided by the analysis of previously not understood experimental IR absorption spectra of a quadruple QW system. The complex temperature dependent absorption pattern is reproduced by our theory with a remarkable degree of accuracy.

[1] D. Stehr et al., Phys. Rev. Lett. 95, 257401 (2005)
[2] M. Helm et al., Phys. Rev. B 48, 1601 (1993)