Anticrossing and coupling of light-hole and heavy-hole states in (001)GaAs/AlxGa1−xAsheterostructures

Heterostructures sharing a common atom such as AlAs/GaAs/AlAs have a $D_{2d}$ point-group symmetry which allows the bulk-forbidden coupling between odd-parity light-hole states (e.g., lh1) and even-parity heavy-hole states (e.g., hh2). Continuum models, such as the commonly implemented (“standard model”) $\mathbf{k}\cdot \mathbf{p}$ theory miss the correct $D_{2d}$ symmetry and thus produce zero coupling at the zone center. We have used the atomistic empirical pseudopotential theory to study the lh1-hh2 coupling in (001) superlattices and quantum wells of $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}.$ By varying the Al concentration x of the barrier we scan a range of valence-band barrier heights $\Delta E_v\mathrm{}\left(x\right).$ We find the following: (i) The lh1 and hh2 states anticross at rather large quantum wells width or superlattice periods $60<\mathrm{}{n}_c<70\mathrm{}$ monolayers. (ii) The coupling matrix elements $V_{\mathrm{lh}1,hh2\mathrm{}}^{{\mathbf{k}}_{\Vert }=0\mathrm{}}$ are small (0.02–0.07 meV) and reach a maximum value at a valence-band barrier height $\Delta E_v\approx 100\hspace{0.5em}\mathrm{meV},$ which corresponds to an Al composition $x_{\mathrm{Al}}=0.2\mathrm{}$ in the barrier. (iii) The coupling matrix elements obtained from our atomistic theory are at least an order of magnitude smaller than those calculated by the phenomenological model of Ivchenko et al. [Phys. Rev. B 54, 5852 (1996)]. (iv) The dependence of $V_{\mathrm{lh}1,hh2\mathrm{}}$ on the barrier height $\Delta E_v\mathrm{}\left(x\right)\mathrm{}$ is more complicated than that suggested by the recent model of Cortez et al., [J. Vac. Sci. Technol. B 18, 2232 (2000)], in which $V_{\mathrm{lh}1,hh2\mathrm{}}$ is proportional to the product of $\Delta E_v\mathrm{}\left(x\right)\mathrm{}$ times the amplitudes of the lh1 and hh2 envelopes at the interfaces. Thus, atomistic information is needed to establish the actual scaling.