Magnitude and Size Scaling of Intervalley Coupling in Semiconductor Alloys and Superlattices

Coupling between different \ensuremath{\Gamma}, $X,$ and $L$ band-structure valleys is responsible for (a) level anticrossing in superlattices as a function of period, pressure, and electric field and for (b) ``optical bowing'' of band gaps in random alloys. We investigate the symmetry, magnitude, and size scaling of intervalley coupling in semiconductor superlattices and alloys by direct supercell calculations, performed with screened pseudopotentials and a plane-wave basis, considering up to ${10}^{6}$ atoms/supercell. Projecting the calculated electronic wave functions ${\ensuremath{\psi}}_{i}$ of alloys or superlattices onto the bulk states of the constituent zinc-blende materials shows that ${\ensuremath{\psi}}_{i}$ contain a ``majority representation'' from one or more zinc-blende states \ensuremath{\gamma}. The intervalley coupling $E(i,j)$ between the alloy states ${\ensuremath{\psi}}_{i}$ and ${\ensuremath{\psi}}_{j}$ then includes a term $2F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ due to the ``majority representations'' \ensuremath{\gamma} and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ of ${\ensuremath{\psi}}_{i}$ and ${\ensuremath{\psi}}_{j},$ respectively, plus residual terms due to the minority representations. We find the following: (i) In alloys, the orbital overlap function $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ is large, since the wave functions are extended. The intervalley coupling element $V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ exhibits simple selection rules: being zero for $({\ensuremath{\Gamma}}_{1c}{,X}_{1c}),$ $({\ensuremath{\Gamma}}_{1c}{,L}_{3c}),$ ${(X}_{1c}^{x}{,X}_{1c}^{y}),$ etc. (``weak coupling''), and nonzero for $({\ensuremath{\Gamma}}_{1c}{,X}_{3c}),$ $({\ensuremath{\Gamma}}_{1c}{,L}_{1c}),$ ${(L}_{3c}{,X}_{1c}),$ etc. (``strong coupling''). This explains why the $\overline{\ensuremath{\Gamma}}$-like conduction band of mixed-cation alloys contains zinc-blende ${\ensuremath{\Gamma}}_{1c}$ and ${L}_{1c}$ character, but not ${X}_{1c}.$ In the case of strong coupling, $E(i,j)$ scales as $1/\sqrt{\ensuremath{\Omega}},$ where \ensuremath{\Omega} is the volume, while in the weak-coupling case the entire coupling originates from the ``minority representation,'' and is 20--100 times smaller. The minority representation, however, contributes to the bowing of the band gap vs composition. (ii) In superlattices, although the above selection rule for $V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ still exists, the magnitude of the intervalley coupling is governed by the overlap function $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}}).$ For simple superlattices, $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ is small, since the wave functions are localized in particular segments (``weak coupling''). Consequently, the ``majority representation'' contributes 5--100 times less than in the analogous case of alloys. Furthermore, $E(i,j)$ scales as ${1/n}^{3},$ where $n$ is the superlattice period.