Effective Band Structure of Random Alloys

Random substitutional $A_x{B}_{1-x}$ alloys lack formal translational symmetry and thus cannot be described by the language of band-structure dispersion $E(\overrightarrow{k})$. Yet, many alloy experiments are interpreted phenomenologically precisely by constructs derived from wave vector $\overrightarrow{k}$, e.g., effective masses or van Hove singularities. Here we use large supercells with randomly distributed $A$ and $B$ atoms, whereby many different local environments are allowed to coexist, and transform the eigenstates into an effective band structure (EBS) in the primitive cell using a spectral decomposition. The resulting EBS reveals the extent to which band characteristics are preserved or lost at different compositions, band indices, and $\overrightarrow{k}$ points, showing in (In,Ga)N the rapid disintegration of the valence band Bloch character and in Ga(N,P) the appearance of a pinned impurity band.

Figure 1

Two-dimensional illustration of the relation between a supercell (SC) and a primitive cell (PC) direct (a) and reciprocal spaces (b) of an ordered $AB$ lattice. In panel (b), squares (circles) denote SC (PC) reciprocal lattice vectors. $\overrightarrow{K}=\overrightarrow{k}+{\overrightarrow{G}}_0$ illustrates a folding relation $\mathrm{PC}\mapsto \mathrm{SC}$. Brillouin zone vectors $\overrightarrow{k}$ and ${\overrightarrow{k}}_C$ ($\overrightarrow{K}$ and ${\overrightarrow{K}}_C$) are symmetry related if $C$ is a symmetry operation of the crystal.

Figure 2

EBS for (a) pure GaN, (e) pure InN, and for the ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloy (b)–(d). The different $\overrightarrow{k}$ vectors in the primitive cell are represented by thin vertical lines. Red (gray) lines represent the SFs $A(\overrightarrow{k},E)$ at each such $\overrightarrow{k}$ point. For the random alloys, the SFs exhibit a finite, $\overrightarrow{k}$ dependent band width [light blue (gray) shaded areas]. (f): Effective effective mass $m^{**}$ of electrons and holes in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ at the $\Gamma$ point (filled symbols). Open symbols refer to effective masses $m^{*}$ for individual bands in the pure binaries.

Figure 3

Same as Fig. 2 but for (a) P-dilute GaN and (b) N-dilute GaP. $E_1$ and $E_2$ label two localized states corresponding to P-P pairs. Note the two different energy scales in (a) and (b).

Figure 4

Isosurfaces [cyan (gray)] of the two localized wave functions labeled $E_1$ and $E_2$ in Fig. 3a for ${\mathrm{GaN}}_{0.98}{\mathrm{P}}_{0.02}$, corresponding to ${\mathrm{PP}}_1$ ($E_1$) and ${\mathrm{PP}}_4$ ($E_2$) pairs. P atoms are depicted in yellow (light gray), all the other atoms in the SC are dark grey.