Validated composition dependence of the band-gap and band-edge energies of lattice-matched GaInAsSb/GaSb and GaInAsSb/InAs alloys: short-range-order, clustering, and temperature effects

The quaternary alloy $\mathrm{Ga}_{1-y}\mathrm{In}_y\mathrm{As}_x\mathrm{Sb}_{1-x}$, lattice matched to GaSb or InAs, underpins mid-infrared lasers, infrared detectors, and thermophotovoltaic cells, yet its conduction- and valence-band-edge energies as functions of composition are only partially validated. In 2005, Magri, Zunger, and Kroemer computed these band edges atomistically (empirical-pseudopotential plus valence-force-field supercell calculations for a random, T=0 K alloy), predicting upward-concave bowing of both edges, staggered-to-broken-gap lineup transitions at In fraction y≈0.92 (GaSb substrate) and y≈0.62 (InAs substrate), and a minimum gap near (As, In)≈(0.85, 0.87) on InAs, while recording limitations on its validation: sparse experimental coverage caused by the alloy's miscibility gap, short-range-order and clustering effects quantified at only a single composition, temperature comparison reduced to Varshni interpolation, and a dearth of GaInAsSb/InAs experimental data. The subsequent literature shows none of these closed: experimental lineups confirm the GaSb-side transition but place the InAs-side transition near In≈0.25-0.3 versus theoretical 0.62-0.68; an independent tight-binding calculation disagrees with the pseudopotential one on details; short-range-order effects on band edges remain quantified at a single composition; and current device parameter models still rely on interpolation. The open problem is to establish validated band-edge-versus-composition maps for both substrate cases, including short-range-order, clustering, and temperature corrections.

Background

$\mathrm{Ga}_{1-y}\mathrm{In}_y\mathrm{As}_x\mathrm{Sb}_{1-x}$ is a III-V quaternary semiconductor alloy: the cation sublattice mixes Ga and In (In fraction y) and the anion sublattice mixes As and Sb (As fraction x), so the alloy contains four distinct nearest-neighbor bond types (Ga-As, Ga-Sb, In-As, In-Sb). It belongs to the '6.1 Å family' of GaSb- and InAs-based heterostructures used for mid-infrared lasers, infrared photodetectors, and thermophotovoltaics (TPV, the conversion of thermal radiation to electricity in photovoltaic cells). Epitaxial films must be 'lattice matched' to their substrate, i.e. have the same in-plane lattice constant so that misfit strain does not accumulate; lattice matching therefore selects a one-dimensional composition path $x = f(y)$ through the (x, y) plane for each substrate (GaSb or InAs), running from GaSb-like to InAs-like compositions.

Device design along that path requires the conduction-band minimum energy $E_c(x,y)$ and the valence-band maximum energy $E_v(x,y)$, hence the band gap $E_g = E_c - E_v$ and the band offsets (the energies of the alloy band edges relative to the substrate band edges). The offsets determine the band 'lineup' at a junction: type I (straddling; the alloy gap lies inside the substrate gap), type II (staggered; the gaps are offset but neither edge crosses the other), and type III (broken gap; the alloy valence-band maximum lies above the substrate conduction-band minimum, so electrons and holes coexist at the interface). Whether a GaInAsSb/GaSb or GaInAsSb/InAs junction is type II or type III decides whether it acts as a semimetallic tunnel contact or a blocking junction — decisive for Ohmic contacts, injection lasers, and detector dark current.

Band edges of alloys are usually estimated by interpolation: composition-weighted averaging of the binary constituents' energies with possible nonlinear 'bowing' corrections (the bowing parameter quantifies the deviation from a linear composition average). Magri, Zunger, and Kroemer (2005) argued that such schemes are unreliable for a four-bond-type quaternary and instead computed $E_c(x,y)$ and $E_v(x,y)$ atomistically: a large supercell with random atomic occupations, atomic positions relaxed by a valence-force-field (VFF) model (an atomistic bond-stretching/bond-bending elastic model, which also fixes the lattice-matching path $x = f(y)$ more accurately than the linear Vegard rule), and electronic structure from the empirical pseudopotential method (EPM; a plane-wave band-structure calculation with atom-centered pseudopotentials fitted to binary data), including spin-orbit coupling. They predicted upward-concave bowing of both $E_c$ and $E_v$ — contradicting interpolative models, a discrepancy they traced to the highly strained In-Sb and Ga-As bonds that ternary-endpoint interpolations effectively omit — a staggered-to-broken-gap transition at In fraction y≈0.92 (As fraction x≈0.81) on a GaSb substrate and at y≈0.62 (x≈0.59) on an InAs substrate, and a minimum gap at (x, y)≈(0.85, 0.87) on InAs, 72 meV below the InAs gap.

Validation and extension of these predictions are impeded by four factors, each acknowledged in the source work. First, the alloy has a miscibility gap, a thermodynamic two-phase region spanning roughly $0.30 \lesssim y \lesssim 0.70$ on the GaSb-matched path (documented since liquid-phase-epitaxy studies near the gap boundary, e.g. Tournié et al. 1990), which confines equilibrium-grown samples to the composition windows $y \lesssim 0.28$ and $y \gtrsim 0.70$. Second, real samples grown at finite temperature are not perfectly random alloys: short-range order (SRO, the statistical preference for certain nearest-neighbor pairs, quantified in the 2005 work by an excess Ga-Sb bond parameter $\xi$) shifts band edges, and so do clustering (the spatial segregation of like atoms into regions of non-random local composition, characterised by a cluster correlation length $\ell$, in nm) and growth kinetics (specified by the substrate growth temperature $T_{\mathrm{g}}$, in K, which fixes the equilibrium value of $\xi$); the 2005 Monte Carlo plus EPM treatment estimated the SRO shift (15-20 meV gap reduction) at only a single composition (x≈0.13, y≈0.19). Third, the EPM is fitted to T=0 K gaps while experiments are performed at finite temperature, the bridge being only the empirical Varshni equation $E_g(T) = E_g(0) - \alpha T^2/(T+\beta)$ with $\alpha, \beta$ interpolated from the binaries; electron-phonon renormalization (the shift of band edges due to lattice vibrations, including the zero-point motion present even at 0 K), thermal disorder, and configurational fluctuations of the relaxed lattice parameters were left unquantified. Fourth, measured band edges and offsets for the InAs-substrate case were scarce.

The subsequent literature sharpened rather than settled these issues. Experiments by the Ioffe group (Mikhailova et al. 2007) found GaInAsSb/GaSb staggered lineups for In = 0.85-0.92 with broken-gap above 0.92 — consistent with the 2005 prediction — but GaInAsSb/InAs broken-gap only up to In≈0.23 and staggered from In≈0.3, in quantitative tension with the predicted transition at In≈0.62. Shim (2013) recomputed the alignments with a universal tight-binding (a fitted atomic-orbital Hamiltonian method for band structures) plus correlated-function-expansion method, reproducing the GaSb-side transition (In≈0.93) but additionally predicting a type-I window on GaSb (for Ga fractions above ≈0.38, i.e. In below ≈0.62) and placing the InAs-side transition at In≈0.68; the theoretical picture is therefore not converged between methods, and neither calculation treats SRO or temperature. Photoreflectance spectroscopy of InGaAsSb/InAs (Motyka et al. 2014; photoreflectance measures the derivative of reflectance under modulated illumination, resolving interband transition energies) added spin-orbit-splitting data with negative bowing but no band offsets. On the theory side, the inverse-band-structure search of Piquini, Graf, and Zunger (2008) showed that deliberately ordered (non-random) atomic configurations of the same quaternary can reach a target gap at more than one lattice-matched composition — underscoring that configurational order matters, but not quantifying SRO effects on realistic growth. Recent experimental work (Casallas-Moreno et al. 2019; Villa-Martínez et al. 2022; González-Morales et al. 2023) remains concentrated at fixed low-In compositions (In≈0.145), and current device-level parameter models (Olesberg 2024; Parola et al. 2025) refine interpolation formulas or compile experimental data libraries without addressing the atomistic SRO, temperature, or full-map validation questions.

Problem Statement

Establish the composition dependence of the band gap and band-edge energies (band offsets and band-lineup types) of $\mathrm{Ga}_{1-y}\mathrm{In}_y\mathrm{As}_x\mathrm{Sb}_{1-x}$ lattice matched to GaSb and to InAs, validated against experiment — the problem left open by Magri, Zunger, and Kroemer (2005), whose predictions (upward-concave bowing of $E_c$ and $E_v$; staggered-to-broken-gap transitions at $(x, y) \approx (0.81, 0.92)$ on GaSb and $\approx (0.59, 0.62)$ on InAs; minimum gap at $\approx (0.85, 0.87)$ on InAs) rested on a random-alloy, T=0 K model with sparse experimental validation. The unresolved question has four parts, all of them required, with 'validated against experiment' meaning agreement, within the stated experimental and theoretical uncertainties, between the predicted band edges and transition compositions and independent measurements in each composition window where data exist. (i) Extend measured band gaps and band offsets across the lattice-matched composition range, including the miscibility-gap window roughly $0.30 \lesssim y \lesssim 0.70$ where equilibrium single-phase samples cannot be grown; in that window, band-edge energies and their uncertainties are required, either from non-equilibrium samples or as an explicit quantitative bound. (ii) Quantify how short-range order (the excess Ga-Sb bond parameter $\xi$ defined in Background), the clustering scale (the cluster correlation length $\ell$, in nm), and growth kinetics (the substrate growth temperature $T_{\mathrm{g}}$, in K) shift $E_c(x,y)$ and $E_v(x,y)$ (in meV) and the type-II-to-type-III transition compositions (in $y$); the only existing estimate is a 15-20 meV gap reduction at a single composition (x≈0.13, y≈0.19). (iii) Quantify the temperature dependence of the band edges beyond the Varshni interpolation used in 2005, whose parameters $\alpha, \beta$ were linearly interpolated from the binaries. (iv) For the InAs-substrate case — distinct from (i), because it targets the predicted minimum gap and the crossover composition rather than general composition coverage — determine band edges and offsets sufficient to test the predicted minimum gap at $(x, y) \approx (0.85, 0.87)$ and to resolve the transition composition, where existing experimental indications (broken-gap up to In≈0.23, staggered from In≈0.3) disagree quantitatively with the theoretical values (In≈0.62-0.68).

References

1. Rita Magri, Alex Zunger, H. Kroemer, "Evolution of the band-gap and band-edge energies of the lattice-matched GaInAsSb/GaSb and GaInAsSb/InAs alloys as a function of composition", Journal of Applied Physics 98, 043701 (2005). DOI: 10.1063/1.2010621, https://doi.org/10.1063/1.2010621
2. Kyurhee Shim, "Composition dependence of band alignments in GaxIn1-xAsySb1-y heterojunctions lattice matched to GaSb and InAs", Journal of Applied Physics 114, 203703 (2013). DOI: 10.1063/1.4834536, https://doi.org/10.1063/1.4834536
3. M. P. Mikhailova, K. D. Moiseev, T. I. Voronina, T. S. Lagunova, Yu. P. Yakovlev, "Transition from the type-II broken-gap heterojunction to the staggered one in the GaInAsSb/InAs(GaSb) system", Semiconductors 41, 161-166 (2007). DOI: 10.1134/S1063782607020091, https://doi.org/10.1134/S1063782607020091
4. Paulo Piquini, Peter A. Graf, Alex Zunger, "Band-Gap Design of Quaternary (In,Ga)(As,Sb) Semiconductors via the Inverse-Band-Structure Approach", Physical Review Letters 100, 186403 (2008). DOI: 10.1103/PhysRevLett.100.186403, https://doi.org/10.1103/PhysRevLett.100.186403
5. Paulo Piquini, Alex Zunger, Rita Magri, "Pseudopotential calculations of band gaps and band edges of short-period (InAs)n/(GaSb)m superlattices with different substrates, layer orientations, and interfacial bonds", Physical Review B 77, 115314 (2008). DOI: 10.1103/PhysRevB.77.115314, https://doi.org/10.1103/PhysRevB.77.115314
6. M. Motyka, M. Dyksik, F. Janiak, K. D. Moiseev, J. Misiewicz, "The spin-orbit splitting band in InGaAsSb alloys lattice-matched to InAs", Journal of Physics D: Applied Physics 47, 285102 (2014). DOI: 10.1088/0022-3727/47/28/285102, https://doi.org/10.1088/0022-3727/47/28/285102
7. E. Tournié, F. Pitard, A. Joullié, R. Fourcade, "High temperature liquid phase epitaxy of (100) oriented GaInAsSb near the miscibility gap boundary", Journal of Crystal Growth 104, 683-694 (1990). DOI: 10.1016/0022-0248(90)90012-A, https://doi.org/10.1016/0022-0248(90)90012-A
8. Y. L. Casallas-Moreno, G. Villa-Martínez, M. Ramírez-López, P. Rodríguez-Fragoso, M. L. Gómez-Herrera, M. Pérez-González, A. Escobosa-Echavarría, S. A. Tomás, J. L. Herrera-Pérez, J. G. Mendoza-Álvarez, "Growth mechanism and physical properties of the type-I In0.145Ga0.855AsySb1-y/GaSb alloys with low As content for near infrared applications", Journal of Alloys and Compounds 808, 151690 (2019). DOI: 10.1016/j.jallcom.2019.151690, https://doi.org/10.1016/j.jallcom.2019.151690
9. G. Villa-Martínez, D. M. Hurtado-Castañeda, Y. L. Casallas-Moreno, M. Ramírez-López, M. A. González-Morales, M. L. Gómez-Herrera, J. S. Arias-Cerón, V. M. Sánchez Reséndiz, P. Rodríguez-Fragoso, J. L. Herrera-Pérez, J. G. Mendoza-Álvarez, "Si-doped In0.145Ga0.855As0.123Sb0.877: A novel p-type quaternary alloy with high crystalline quality", Solid State Sciences 123, 106797 (2022). DOI: 10.1016/j.solidstatesciences.2021.106797, https://doi.org/10.1016/j.solidstatesciences.2021.106797
10. M. A. González-Morales, G. Villa-Martínez, J. J. Cruz-Bueno, M. Ramírez-López, M. Camacho-Reynoso, P. Rodríguez-Fragoso, J. G. Mendoza-Álvarez, Y. L. Casallas-Moreno, J. L. Herrera-Pérez, "Determination of lateral strain in InGaAsSb alloys and its effect on structural and optical properties", Journal of Materials Science 58, 16172-16183 (2023). DOI: 10.1007/s10853-023-08991-6, https://doi.org/10.1007/s10853-023-08991-6
11. Jonathon T. Olesberg, "Interpolation of compound semiconductor alloy parameters from those of their constituents", Journal of Applied Physics 136, 215105 (2024). DOI: 10.1063/5.0217016, https://doi.org/10.1063/5.0217016
12. Stéphanie Parola, Lucas Gavotto, Julie Tournet, Alexandre Vauthelin, Frédéric Martinez, Joanna Kret, Yves Rouillard, Eric Tournié, Yvan Cuminal, "Material parameter models and simulations of III-Sb solar cells based on AlGaAsSb, AlInAsSb and GaInAsSb quaternary alloys", Solar Energy Materials and Solar Cells 288, 113613 (2025). DOI: 10.1016/j.solmat.2025.113613, https://doi.org/10.1016/j.solmat.2025.113613