Accurate Ab-initio Predictions of III-V Direct-Indirect Band Gap Crossovers
We report the compositional dependence of the electronic band structure for a range of III-V alloys. Density functional theory with the PBE functional is insufficient to mimic the electronic gap energies at different symmetry points of the Brillouin zone. The HSE hybrid functional with screened exchange accurately reproduces the experimental band gaps and, more importantly, the alloy concentration of the direct-indirect gap crossovers for the III-V alloys studied here: AlGaAs, InAlAs, AlInP, InGaP, and GaAsP.
pacs:71.20.Nr, 71.23.-k, 71.55.Eq
Knowing the alloy concentrations where the semiconductor is direct or indirect is essential for optoelectronic device design. Consequently a correct quantitative assessment of the electronic structure across different symmetry points of the Brillouin zone for the alloy is vital. Most studies focused on density functional theory (DFT) calculations using the local density approximation (LDA) or the generalized gradient approximation (GGA) which do poorly on excited states. Only recently have hybrid functionals been used in predicting accurate excited states in individual semiconducting alloys. Lee2006 (); Moses2010 ()
The Heyd-Scuseria-Ernzerhof (HSE) hybrid functional Heyd2003 (); Heyd2006 (); Krukau2006 (); Heyd2004 (); Heyd2005 (), which combines the screened exchange with the Perdew-Burke-Ernzerhof (PBE) GGA functional, Perdew1996 () out performs previous DFT methods in reproducing bulk band gaps. Heyd2004 (); Paier2006-1 (); Paier2006-2 () We report HSE reproduces not only the band gaps across the entire composition range of each alloy studied here but also the direct-indirect band gap crossovers seen experimentally.
Figure 1(a) demonstrates this significant improvement of HSE over PBE in predicting the direct-indirect (-X) crossover (denoted by vertical arrows) for the AlGaAs alloy compared with experiment. Vurgaftman2001 () HSE reproduces the direct-indirect crossover within 5% Al concentration from the most recent experimental value published by Yi et al. Yi2009 () The PBE functional which doesn’t take into account screened exchange overestimates this crossover by 23% Al concentration.
The disordered zinc-blende (cubic) alloys are best modeled by special quasirandom structures (SQS), Wei1990 () ordered structures designed to reproduce the most important pair-correlation functions of a random alloy. The best possible 32-atom SQS’s we produce for concentrations of and match the pair-correlation functions of a random alloy up to 3rd and 7th nearest neighbors, respectively. Nicklas2010 () The SQS with concentration can be used interchangeably with that of .
Table 1 gives the lattice vectors for each SQS used to describe the optical transitions as seen in the zinc-blende primitive cell through folding relations in the Brillouin zone. The SQS with concentration has a base-centered orthorhombic space group symmetry with the following folding relations in reciprocal coordinates,
where the bar denotes superlattice states and the coefficent denotes degeneracies. The SQS with concentration has a triclinic space group symmetry with folding relations given by,
The calculations are performed using the projector augmented-wave (PAW) method. Blochl1994 () The functionals included are the PBE and the HSE06 Heyd2006 () hybrid functional in the vasp code. Kresse1996 () The Ga 3d and In 4d electrons are treated as valence and the wavefunctions are expanded in plane waves up to an energy cutoff of 500 eV. The Brillouin-zone integrations have been carried out on (101010), (644), and (448) -centered meshes for the face-centered cubic primitive cell and SQS supercells for and , respectively.
The lattice constants are linearly interpolated between the experimental parent compound lattice constants taken from Vurgaftman et al.. Vurgaftman2001 () Relaxations are not taken into account after observing only a slight shift of 1% Al concentration in the direct-indirect crossover in the AlInP alloy using the PBE functional. The compositional dependence of the band gap is described by a quadratic fit to the data, whereas a cubic fit is taken for only the direct gap of AlGaAs. Nicklas2010 ()
|0.47||0.65||0.38111From Ref. Vurgaftman2001 (), references therein., 0.42222From Ref. Yi2009 ().|
|0.68||0.76||0.64111From Ref. Vurgaftman2001 (), references therein.|
|0.37||0.44||0.44555From Ref. Onton1970 ()., 0.34666From Ref. Ishitani1996 () for strained AlInP.|
|0.72||0.86||0.67111From Ref. Vurgaftman2001 (), references therein.,0.746333From Ref. Novak2005 ().|
|0.75||0.96||0.77444From Ref. Merle1977 ().|
|0.57||0.84||0.45111From Ref. Vurgaftman2001 (), references therein.|
Figure 1(b) supplements (a) by detailing the individual direct and indirect gap energies for AlGaAs with the calculated values denoted by symbols. It shows accurate HSE gap energies for both the and X symmetry points throughout the entire composition range.
For AlInAs, Figure 2(a) displays the band gap energy for both HSE, PBE, and the recommended experimental bowing parameters. Vurgaftman2001 () HSE not only predicts a real band gap for the concentration range of 0-10% of Al for this alloy, but also reproduces the -X crossover by 4% Al concentration compared with PBE’s overestimation by 12% Al concentration as seen in Table 2.
For GaInP, Figure 2(c), experiments utilizing optical luminescence measurements see only a single -X crossover Hakki1970 (); Williams1970 (); Lettington1971 (); Chevallier1971 (); Alibert1972 (); Macksey1973 (); Lee1974 () whereas high pressure electrical measurements Pitt1974 () and piezoreflectance measurements Merle1977 () observed two-point crossovers for -L and L-X two point crossovers. Both HSE and PBE yield two point crossovers as seen in Fig. 2(c) with only HSE reproducing crossover points that lie nearly on top of experiment as well as reproducing the band gap to within 8% accuracy throughout the whole Ga concentration.
For GaAsP, Figure 2(d), both HSE and PBE predict a two-point crossover in Fig. 2(d); whereas experiments which rely on optical luminescence observe only a single point -X crossover at . HSE not only produces a direct-indirect crossover that agrees better with experiment, but also predicts a two-point crossover that is only separated by 2% P concentration that might be seen as a single point crossover in experiment.
For AlInP, Figure 2(b), HSE predicts a crossover at 37% Al concentration underestimating the experimental value by Onton et al. () Onton1970 () but shows relatively good agreement with the recent results of strained AlInP by Ishitani et al. (). Ishitani1996 () More studies of the AlInP alloy could provide a detailed comparison with theory.
To conclude, HSE reproduces direct-indirect crossovers within 12% atomic composition for the alloys studied here, whereas PBE overestimates crossover points by 39% atomic composition. The HSE functional also substantially improves on band gap energies across the entire composition range. We expect HSE to perform similarly for other semiconducting alloys.
This work was supported by DOE-Basic Energy Sciences, Division of Materials Sciences (DE-FG02-99ER45795). This research used computational resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and the Ohio Supercomputing Center.
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