Alloy scattering in Quantum Cascade Lasers in a magnetic field : a numerical study
Résumé
It is well known that alloy scattering is a powerful scattering mechanism in ${\rm Ga(In)As}$ nanostructures; in fact it dominates the dc mobility of modulation doped heterojunctions. Recent experiments performed on Quantum Cascade Lasers (QCL) subjected to a strong magnetic field have evidenced strong oscillations of the output power of these lasers as a function of the field. The minima are associated with efficient non-radiative paths when the $n=0$ Landau level of the upper subband is lined up by the field with the $n$-th Landau level (LL) of the lower subbands (elastic scattering) or when these two levels have energies that differ by one LO phonon energy. Here we focuse our attention on the theoretical evaluation of the alloy scattering limited resonant relaxation in ${\rm Ga(In)As}$-based QCLs. The 2D Landau levels have highly singular density of states (discrete peaks with a macroscopic degeneracy). This prevents the use of conventional Born approximation to compute the transition rates. So far in QCLs, inhomogeneous kind of broadenings have been used to smooth out the LL density of states, based on the fact that it is very likely that an actual QCL device is actually the sum of very many micro-samples that differ from one another because of a randomly varying well thickness [1]. In this work we instead investigate homogeneous broadening and solve numerically the one electron Schrödinger equation for alloy broadened Landau levels. The computation is done (in the Landau gauge) by diagonalizing the Hamiltonian in a large box ($100{\rm nm} \times 100 {\rm nm} \times L$, where $L$ is the QW thickness). For convenience the $z$ QW has infinite barriers. We study the coupling between the states with the states in the strong field limit (no other LL retained so far in the basis). To model the ${\rm Ga}_{1-x}{\rm In}_x{\rm As}$ alloy, we partition the large box into tiny cubes ($5Å$ side). In each cube the scattering potential is a random variable equal to $x\Delta V$ with probability $1-x$ and to $(1-x)\Delta V$ with probability $x$ where $\Delta V = 0.6{\rm eV}$. There is (so far) no correlation between the x values of different cubes. A Lanczos algorithm is used to extract the eigenvalues and eigenfunctions. The calculations are done with $m^* = 0.05 m_0$, $x = 0.53$, $E_2-E_1 = 143 {\rm meV}$ For inter LL effects we find that the LL width is roughly independent of the LL index, a result that holds in the self consistent Born approximation. The width of the $E_2-E_1$ absorption line varies roughly like $\sqrt{B}$, at least for $B > 15 {\rm T}$. When the $E_1$ LL crosses the $n=0$ LL of $E_2$, there is no detectable anti-crossing although the disorder has sizeable non diagonal elements. This is because the intra-LL broadening dominates over the inter-LL intersubband matrix elements. Based on this, we shall compare the lifetime of $n=0$ level of $E_2$ numerically evaluated with the result of a Born approximation calculation that uses the density of states broadened by the intra-LL contributions. An extension of our calculations to the LO phonon emission between broadened Landau levels will be the next step.