Extracting $E$ versus $\vec{k}$ effective band structure from supercell calculations on alloys and impurities

The supercell approach to defects and alloys has circumvented the limitations of those methods that insist on using artificially high symmetry, yet this step usually comes at the cost of abandoning the language of $E$ versus $\vec{k}$ band dispersion. Here we describe a computational method that maps the energy eigenvalues obtained from large supercell calculations into an effective band structure (EBS) and recovers an approximate $E\left(\vec{k}\right)$ for alloys. Making use of supercells allows one to model a random alloy A${}_{1-x}$B${}_x$C by occupying the sites A and B via a coin-toss procedure, affording many different local environments (polymorphic description) to occur. We present the formalism and implementation details of the method and apply it to study the evolution of the impurity band appearing in the dilute GaN:P alloy. We go beyond the perfectly random case, realizing that many alloys may have nonrandom microstructures, and investigate how their formation is reflected in the EBS. It turns out that the EBS is extremely sensitive in determining the critical disorder level for which delocalized states start to appear in the intermediate band. In addition, the EBS allows us to identify the role played by atomic relaxation in the positioning of the impurity levels.

Figure 1

Two-dimensional representation of (a) a simple unfolding of bands for a periodic fictitious system ABC and (b) the analogy with periodic systems when treating an alloy A${}_{1-x}$B${}_x$C by effective medium and polymorphic model approaches. The connection between the two representations of a primitive cell (PC) and a supercell (SC) in (c) direct space and (d) reciprocal space. The PC and SC are related by a transformation similar to Eq. (1). The corresponding primitive (pbz) and supercell (SBZ) Brillouin zones, their associated wave vectors $\vec{k}$ ($\vec{K}$) and translation vectors $\vec{g}$ ($\vec{G}$) as well as folding (unfolding) relations are also illustrated. In panel (d), ${\vec{k}}_C$ and ${\vec{K}}_C$ are wave vectors equivalent to $\vec{k}$ and $\vec{K}$ by a symmetry operation $C$ of both pbz and SBZ.

Figure 2

An example, for the In${}_{0.1}$Ga${}_{0.9}$N alloy, of how the effective band structure (EBS) is obtained. (a) Supercell eigenstates $\{E_m\left({\vec{K}}_J\right),|{\vec{K}}_{j}m\rangle \}$ calculated at various ${\vec{K}}_J$ are projected and then unfolded on the fcc Brillouin zone vectors ${\vec{k}}_i={\vec{K}}_J+{\vec{G}}_i$ [Eq. (5)], providing the spectral functions $\overline{A}({\vec{k}}_i,E)$ [Eq. (16), red (dark gray)] and the cumulative sums $S_{{\vec{k}}_i}\left(E\right)$ [Eq. (17), blue (black)]; the latter quantity is used to determine the alloy “bands” positions and widths [cyan (light gray) shaded areas]. (b) Results for all ${\vec{k}}_i$ vectors (thin vertical lines) are put together into an $E$ versus $\vec{k}$ plot, having along the abscissa also the spectral function amplitude.

Figure 3

Effect of clustering on EBS for GaN${}_{0.98}$P${}_{0.02}$ derived from a 512-atom SC. This contains, at the P composition of $0.02$, five P atoms in the SC. Leftmost panel: The perfectly random alloy shows the Anderson-like impurity band. Rightmost: All of the five P atoms are grouped in a cluster, leading to discrete levels. In between these extrema we have systems consisting of an $n_c$-atom P cluster kept fixed while the rest of the P atoms, $5-n_c$, were placed randomly in the SC (taking 12 different realizations). We note the formation of the impurity band with the decreasing size, from right to left, of the P cluster. The dashed horizontal lines mark the position of the GaN VBM and CBM.

Figure 4

Same as Fig. 3 but for GaN${}_{0.95}$P${}_{0.05}$. The SC contains at the P composition of $0.05$ 13 P atoms. Leftmost panel: The perfectly random alloy shows the Anderson-like impurity band. Rightmost: All of the 13 P atoms are grouped in a cluster, leading to discrete levels.

Figure 5

Detailed representation of the spectral function $\overline{A}(\Gamma ,E)$ at the $\Gamma$ point in the energy range of the impurity band for the systems of Fig. 3.

Figure 6

Effect of VFF relaxation on the EBS of GaN:P for different P compositions ($x=0.004$, $0.02$, $0.05$) (from left to right) and a fixed random realization. When the relaxation is neglected (the panels labeled “unrelaxed”) the impurity band disappears from the gap.