First-principles survey of the structure, formation energies, and transition levels of As-interstitial defects in InGaAs

While point defects in elemental (Si) and compound (GaAs, GaN, AlN) semiconductors have been extensively studied both experimentally and theoretically, only limited theoretical studies of these defects exist for technologically important binary $\left(\mathrm{S}{\mathrm{i}}_x\mathrm{G}{\mathrm{e}}_{1\text{−}x}\right)$ and pseudobinary $(\mathrm{I}{\mathrm{n}}_x\mathrm{G}{\mathrm{a}}_{1\text{−}x}\mathrm{As}$, $\mathrm{I}{\mathrm{n}}_{\mathrm{x}}\mathrm{G}{\mathrm{a}}_{1\text{−}x}\mathrm{N}$, $\mathrm{A}{\mathrm{l}}_x\mathrm{G}{\mathrm{a}}_{1\text{−}x}\mathrm{N})$ semiconductor alloys. Here, we use density-functional theory and a recently developed bounds-analysis approach to survey the atomic structures, formation energies, and charge-state transition levels of the stable and metastable states of As interstitials in the pseudobinary alloy $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$. Our studies consider seven different candidate defect structures for the As interstitial, with calculations performed for selected defect charge states in the range $q=-2$ to $+3$. In each case, the mean and standard deviations of the defect-formation energy are determined using statistical sampling methods that place the defect into a wide variety of differing local-alloy environments. When examined from the point of view of the mean formation energy of the defect, the stable configurations of the As interstitial in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ are found to resemble previous findings for GaAs, with a ${\mathrm{C}}_{1\mathrm{h}}\text{−}\mathrm{p}{001}_{\mathrm{III}}$ interstitial structure in a $q=+1$ charge state favored near midgap and below, and a ${\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ split-interstitial structure in a $q=-1$ charge state favored above midgap (the named point-group symmetries refer to the underlying symmetry that the alloy defect would possess if within GaAs). The statistical sampling reveals a strong dependence of the defect-formation energy on the local-alloy environment, with the standard deviation $\sigma$ of the formation energy approaching 0.21 eV for the most stable As-interstitial structures. Because the range of ground-state energies encountered by an As-interstitial defect when moving through the alloy is found to be quite large, approaching $\sim 1.2\mathrm{eV}(\pm 3\sigma )$, defect-diffusion pathways in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ will have spatial and temporal complexities not found in GaAs.

Figure 1

Ball-and-stick models of candidate As-interstitial defect structures in InGaAs alloys. The black balls indicate interstitial As atoms; the dark gray balls (brown online) indicate As atoms located on the group-V sublattice; and the light gray balls (light green and light blue online) indicate Ga and In atoms located on the group-III sublattice. DFT calculations were performed for each defect structure for each of the indicated charge states $q$.

Figure 2

Upper and lower bounds on admissible localized defect levels in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ versus the size of the finite periodic supercell (upper axis) used for DFT calculations of defect levels. The added or removed charge densities (lower axis) result when one electron is either added to or removed from each of the corresponding finite-sized bulk-alloy supercells. The bounds are compared to the Kohn-Sham band gap (blue long-dashed line) and the measured band gap (green dotted line) in undoped $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$.

Figure 3

Mean defect-formation energies vs Fermi level for the ${\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ split As interstitial in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ with charge states $q=-2$, $-1$, 0, $+1$, and $+2$. Calculated lower and upper bounds on defect charge-transition levels in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ are indicated by the solid vertical red lines located at ${\varepsilon }_F=-0.06$ and $0.86\mathrm{eV}$. Black dots indicate the charge-state transition levels of the defect given by intersection of the mean formation energies; corresponding solid blue Gaussian curves show relative probability distributions of the defect levels as obtained from a separate statistical analysis of the population of levels found for each set of sampled alloy configurations; $\sigma$ indicates the standard deviation of each distribution.

Figure 4

Mean defect-formation energies vs Fermi level for the ${\mathrm{C}}_{1\mathrm{h}}\text{−}\mathrm{p}{001}_{\mathrm{III}}$ As interstitial in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ with charge states $q=0,+1,\mathrm{and}+2$. Calculated lower and upper bounds on defect charge-transition levels in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ are indicated by the solid vertical red lines located at ${\varepsilon }_F=-0.06$ and $0.86\mathrm{eV}$. Black dots indicate the charge-state transition levels of the defect given by intersection of the mean formation energies; corresponding solid blue Gaussian curves show relative probability distributions of the defect levels as obtained from a separate statistical analysis of the population of levels found for each set of sampled alloy configurations; $\sigma$ indicates the standard deviation of each distribution. The $q=-1{\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ ground state is replotted (inclined short-dashed green line) from Fig. 3 for comparison.

Figure 5

Mean defect-formation energies vs Fermi level for the ${\mathrm{T}}_{\mathrm{d}}$-III As interstitial $(q=+3)$ and the ${\mathrm{T}}_{\mathrm{d}}$-As interstitial $(q=+2\mathrm{and}+3)$ in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$. Calculated lower and upper bounds on defect charge-transition levels in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ are indicated by the solid vertical red lines located at ${\varepsilon }_F=-0.06$ and $0.86\mathrm{eV}$. Black dots indicate the charge-state transition levels of the defect given by intersection of the mean formation energies; corresponding solid blue curves show relative probability distributions of the defect levels as obtained from a separate statistical analysis of the population of levels found for each set of sampled alloy configurations; $\sigma$ indicates the standard deviation of each distribution. Ground states of the $q=+1{\mathrm{C}}_{1\mathrm{h}}\text{−}\mathrm{p}{001}_{\mathrm{III}}$ and $q=-1{\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ defects are replotted (lower inclined solid green lines) from Figs. 3 and 4 for comparison.

Figure 6

Mean defect-formation energies vs Fermi level for the ${\mathrm{C}}_{3\mathrm{v}}$-hex As interstitial in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ with charge states $q=0,+1,+2,\mathrm{and}+3$. Calculated lower and upper bounds on defect charge-transition levels in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$ are indicated by the solid vertical red lines located at ${\varepsilon }_F=-0.06$ and $0.86\mathrm{eV}$. Black dots indicate the charge-state transition levels of the defect given by intersection of the mean formation energies; corresponding solid blue Gaussian curves show relative probability distributions of the defect levels as obtained from a separate statistical analysis of the population of levels found for each set of sampled alloy configurations; $\sigma$ indicates the standard deviation of each distribution. Ground states of the $q=+1{\mathrm{C}}_{1\mathrm{h}}\text{−}\mathrm{p}{001}_{\mathrm{III}}$ and $q=-1{\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ defects are replotted (lowermost inclined solid green lines) from Figs. 3 and 4 for comparison.

Figure 7

Relative probability distribution of the defect formation energy for (a) ${\mathrm{C}}_{2\mathrm{v}}\text{−}110_{\mathrm{a}}$ and (b) ${\mathrm{C}}_{1\mathrm{h}}\text{−}\mathrm{p}{001}_{\mathrm{III}}$ As-interstitial defect structures in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$. The formation-energy distributions are computed for a Fermi level located at the valence-band maximum $({\varepsilon }_F=0)$.

Figure 8

Relative probability distribution of the defect formation energy for (a) ${\mathrm{T}}_{\mathrm{d}}$-III and ${\mathrm{C}}_{3\mathrm{v}}$-hex, as well as (b) ${\mathrm{T}}_{\mathrm{d}}$-As-interstitial defect structures in $\mathrm{I}{\mathrm{n}}_{0.5}\mathrm{G}{\mathrm{a}}_{0.5}\mathrm{As}$. The formation-energy distributions are computed for a Fermi level located at the valence-band maximum $({\varepsilon }_F=0)$.