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Abstract
We report determination of parameters in the nearest-neighbor sp3d5s* tight-binding (TB) model for nine binary compound semiconductors which consist of Al, Ga, or In and of P, As, or Sb based on the hybrid quasi-particle self-consistent GW (QSGW) calculations. We have used the determination parameters to calculated band structures and related properties of the compounds in the bulk phase relevant to mid-infrared applications and of the type-II (InAs)/(GaSb) superlattices. For the type-II (InAs)/(GaSb) superlattices with various superlattice periods, good agreement with photoluminescence measurements on the band gaps has been confirmed. Furthermore, two aspects of the band gap properties from other calculations have been reproduced: the band gap energies rising up to some superlattice periods and shrinking beyond them asymptotically. In both the bulk phase and the superlattices, erroneous flat valence bands have appeared within the nearest-neighbor sp3s* TB model. The present TB model has been eliminated these artifacts, potential obstacles to design advanced superlattices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of mid-infrared sensors in wavelength range from 3 to 20µm is actively underway [1–5], because the normal vibrational energies of many molecules overlap mid-infrared. Sensors made of HgCdTe have been conventionally used. However, this approach has disadvantages, containment of hazardous heavy metals, Hg and Cd [6], the requirement of large-scale equipment for cryogenic cooling.
Type-II (InAs)/(GaSb) superlattices are expected to be alternative materials [7]. The type-II (InAs)/(GaSb) superlattice is characterized by larger electron and hole effective masses, which leads to reduction of dark current, lower sensitivity to compositional non-uniformity, and a wide ranging band gap accurately determined by controlling supertlattice period [8–10]. Moreover, toward high temperature operation of infrared sensors, band structure engineering is utilized to suppress Auger recombination [11–13].
There are several methods to predict band structure. Though density functional theory [14,15] (DFT) is accepted as a reliable method, it is also known to underestimate a band gap notoriously, which is especially serious for semiconductors intended for infrared sensors. Theory beyond DFT, such as a GW method [16], can conquer this problem. We have calculated the band structures of type-II (InAs)/(GaSb) superlattices [17] using the latest version [18] of the hybrid quasiparticle self-consistent GW method (QSGW) [19–24] as implemented in ecalj package. A superlattice period is, however, still restricted to a far shorter value than that of real world sensors on account of heavy computational load.
This problem has led us to the previous study [25], where we have determined parameters of empirical tight-binding approach [26, 27] including spin-orbit interaction [28] within a nearest-neighbor sp3s* model [29, 30] based on the hybrid QSGW method by the help of genetic algorithm (GA) [[31–34]]. At that time, however, the nearest-neighbor sp3s* model is known to be accompanied by such a limitation as a poorly described transverse mass at the X point [35,346], which is especially serious to aluminum compounds with conduction band minima at the X valley. The nearest-neighbor sp3s* model is therefore improved either by considering more distant interaction [37] or by adding d orbitals [38]; namely, the second-nearest-neighbor sp3s* and nearest-neighbor sp3d5s* models. Between these two improved models, while computationally more demanding on account of a doubled number of orbitals par atom, the latter is preferable from the viewpoint that the erroneous projection on the atomic symmetries is corrected [39], that the difference between masses of the electron and light hole is correctly reproduced [40], and that distortions, which may be encountered in superlattices, are handled more easily [41]. As mentioned later in Subsection 3.2, however, the known parameters for the nearest-neighbor sp3d5s* model by Jancu et al. [38] are short of accuracy when applied to the type-II (InAs)/(GaSb) superlattices.
In the present study we have extended our previous study by adopting the nearest-neighbor sp3d5s* model. The compound semiconductors taken into account are GaAs, InAs, GaSb, and InSb, whose parameters are necessary to treat the type-II (InAs)/(GaSb) superlattices, AlSb, which can serve as barrier layers in the mid-infrared sensors [7, 42, 43], and AlP, AlAs, GaP, and InP for completeness. Parameters of the aluminum compounds within the nearest-neighbor sp3s* model are also determined for comparison. Test calculations of the compounds in bulk phase and of the type-II (InAs)/(GaSb) superlattices have been also performed. We demonstrate that erroneous flat valence bands in the nearest-neighbor sp3s* model are eliminated. From here, we mean ours, not those in general or some other studies, by TB models, methods, parameters, or the like unless otherwise specified.
2. Method
Since an outline of a procedure to extract the TB parameters is almost the same as in our previous study, we describe it only briefly. The QSGW method in its infancy is known to overestimate the band gap [20, 21]. To remedy this problem, the exchange-correlation potential term is diluted [20] slightly with that of local density approximation [15]:
where α is an adjustable parameter. This hybrid QSGW method at α = 0.8 is shown to describe energy band properties universally well for a wide variety of semiconductors and insulators [18]. In the present study, we have shifted α against each compound to reproduce accurately a band gap at the Γ point except that for AlP a band gap at the X point is considered, because with the Γ point α is not well adjusted. Specifically, after a lattice constant and band gap of a compound of interest are taken from Vurgaftman et al. [43] as input parameters for the hybrid QSGW method, α is so fixed that the specified and resultant band gaps agree at the specified lattice constant. Table 1 shows the specified lattice constants and fixed α’s. At those input values, the energy band structures are calculated by the hybrid QSGW method along a pathway in the Brillouin zone adopted by Jancu et al. [38] and effective masses of the split-off hole, light hole, heavy hole, and electron at the Γ point are derived. For the light and heavy holes three orientations [100], [110], and [111] are considered. The target values subsequently fitted by the nearest-neighbor sp3d5s* model with all the TB parameters set to be adjusted by the GA include the energy levels along the pathway within an open interval (1eV= 1.60218 × 10−19J) where the VBM and CBM denote the valence band maximum and conduction band minimum, respectively, and the effective masses. Note that whereas in the nearest-neighbor sp3s* model setting some of the TB parameters to be adjusted by the GA suffices to determine them completely by virtue of exact analytic expressions which associate the energy band properties with the TB parameters [35], to our knowledge that is not the case in the nearest-neighbor sp3d5s* model because of lack of such expressions for compound semiconductors.
Table 1. Input parameters employed in the hybrid QSGW calculations. The lattice constant a in Å(1Å= 1 × 10−10m); the adjustable factor α in arbitrary unit. The parameters for gallium and indium compounds in our previous study are given again for reader’s convenience.
3. Results
3.1. Parameters and bulk properties
The determined TB parameters are listed in Tables 2 and 3 for the aluminum and remaining compounds, respectively.
Table 2. Tight-binding parameters for aluminum compounds in units of eV (1eV= 1:60218 × 10−19J).
Table 3. Same as Table 2 except that only the sp3d5s* model is dealt with for the gallium and indium compounds.
The corresponding main energies and effective masses of the compunds relevant to the mid-infrared sensors are summarized in Tables 4 and 8. The values from the hybrid QSGW and sp3s* TB calculations besides AlSb in our previous study are given again for reader’s convenience. The properties are described slightly better by the sp3d5s* model except for the Γ7v levels, probably because the exact analytic expressions which associate the properties with the TB parameters are unavailable as already mentioned. Since just comparing those limited properties may be insufficient to illustrate the quality of the sp3d5s* model, we show the band structures in Figs. 1 and 5. In AlSb as an example in Fig. 1, whereas the lowest conduction band is fitted moderately well by the sp3s* TB method along the left two segments (L → Γ → X) in the pathway, that is not the case further to the right than the first X point. In particular, the lowest valence band is almost flat between the X, W, and U, K points within the sp3s* TB method, reflecting the poorly described transverse mass. This deficiency is particularly serious to AlSb, because as evident in Fig. 1 an excited electron which should stay at the X valley may be incorrectly predicted to float along the wrong flat band. The sp3d5s* TB method resolves this problem because it fits the lowest conduction band well along the whole pathway, including the case of GaAs, GaSb, InAs, and InSb.
Table 4. Bulk material properties of AlSb obtained by the hybrid QSGW and TB calculations. Energies are in units of eV; masses in terms of the free electron mass.
Table 5. Same as Table 4 except for GaAs.
Table 6. Same as Table 4 except for GaSb.
Table 7. Same as Table 4 except for InAs.
Table 8. Same as Table 4 except for InSb.
Fig. 1 Band structures of AlSb obtained by the hybrid QSGW (black) and TB calculations (blue for the sp3s* model and red for the sp3d5s* one). Energies are in units of eV.
3.2. Superlattice properties
Using the TB parameters for GaAs, InAs, GaSb, and InSb we have calculated the band gaps of a series of the (InAs)/(GaSb) and (InAs)/(InSb)1/(GaSb) superlattices, assumed to be grown on GaSb substrates in a pseudomorphic way. The deformation of the InAs and interface layers is treated with classical elasticity [44]:
where ’s are the uniform strain in a material i made of two adjacent atomic layers, asub and ai lattice constants of the substrate (GaSb) and material i, respectively, and and elastic constants of the material i. The lattice and elastic constants are taken from Refs. [43] and [45], respectively. No further atomic relaxation is considered. It should be noted that fractional coordinates of constituent atoms from such a classical elasticity and relaxed by generalized gradient approximation as implemented in Vienna Ab-initio Simulation Package [44–48] are in surprisingly good agreement [17]. The modifications of the TB parameters have been accounted for by including generalized Harrison’s d−2 law [49]. Furthermore, the on-site energies of the d orbitals Exy, Exz, and Eyz are assumed to dependent linearly on the strain in the same way as proposed in Ref. [36]. The TB parameters related to the strain will be shown elsewhere.Figure 6 shows a whole comparison of the calculated band gap with the photoluminescence (PL) data for the (InAs)/(GaSb) and (InAs)/(InSb)1/(GaSb) superlattices with various periods. The PL data in Fig. 6 are extrapolated into T = 0K after Ref. [50] for the (InAs)/(GaSb) and Ref. [51] for the (InAs)/(InSb)1/(GaSb). The TB band gaps agree well with those of the PL in discrepancy of 16meV and 25meV for the (InAs)/(GaSb) and (InAs)/(InSb)1/(GaSb) superlattices, respectively. The larger error for the latter may be attributed to large strain of 6% in bond length suffered by an InSb layer, which appears inevitably at the two interfaces.
Fig. 6 Band gaps of (InAs)/(GaSb) (closed circle) and (InAs)/(InSb)1/(GaSb) (open circle) with various superlattice periods calculated by the TB method compared with photoluminescence (PL) data extrapolated into T = 0K after Ref. [48] for (InAs)/(GaSb) and Ref. [51] for (InAs)/(InSb)1/(GaSb). A diagonal line is drawn to guide the eye.
More specifically, Fig. 7 compares the band gap energies of the (InAs)n/(GaSb)n superlattices in the TB model together with the results of the TB model parametrized by Jancu et al., of empirical pseudopotential (EP) calculations [52], of the PL, and of the hybrid QSGW [17] method at α = 0.8. First, all the empirical calculations show shrinking band gap energies in an asymptotic way with the superlattce period n sufficiently large, which is in line with the results of the PL and hybrid QSGW methods. Second, when we begin with n = 1, however, the band gap energies obtained by the hybrid QSGW and two TB methods exhibit bell lines, in other words, rise and then fall beyond some n’s, with the EP method an exception. This findings will be discussed in detail elsewhere [53]. Third, as expected from Fig. 6, the present TB band gap energies reproduce the PL data well within one-third error compared to the other empirical methods. Difference between the hybrid QSGW and TB methods in the band gap energy itself may be attributed to impossibility to vary α in Eq. (1) spatially into a more appropriate value for each constituent material at least within the present implementation.
Fig. 7 Band gaps of superlattice (InAs)n/(GaSb)n calculated by the TB method (solid line) compared with those calculated by the TB method in Ref. [40] (dashed line), with those calculated by the empirical pseudopotential (EP) method in Ref. [52] (dash-dotted line), and with photoluminescence (PL) data extrapolated into T = 0K after Ref. [51] (open circle), and with those calculated by the hybrid QSGW method [17] (closed circle).
So far, we have mentioned an advantage of the sp3d5s* TB model over the other known one. Now we superpose the band structures of the (InAs)4/(GaSb)4 superlattice in Fig. 8 calculated by the sp3s* and sp3d5s* TB methods over that obtained by the hybrid QSGW method [17]. The modifications of the TB parameters have been accounted for by including Harrison’s d−2 law in common for a fair comparison between the two TB models. The sp3s* TB model exhibits valence bands with almost flat dispersion between 1.0 and 1.5eV. Since these flat valence bands do not appear in the hybrid QSGW method, they are artifacts reflecting the similar flat valence bands in the bulk band structures as explained in the previous subsection. On the other hand, the sp3d5s* model eliminates these artifacts. Although at the present stage these artifacts may seem harmless, that is not the case when designing the advanced superlattices including the barrier layers which consist of AlSb, whose erroneous energy levels around the X point in the bulk phase will be folded down into the Γ point and its neighborhood in the superlattices. Although one might hit upon applying the sp3d5s* model to the barrier layers and the sp3s* to the remaining layers, the interface between the different TB models would become hard to deal with. One should apply the sp3d5s* model to the superlattices including the barrier layers altogether.
Fig. 8 The band structure of (InAs)4/(GaSb)4 superlattice obtained by the hybrid QSGW (black) and TB methods (blue for the sp3s* model and red for the sp3d5s* one). See Ref [17] for the complete results of the hybrid QSGW method.
4. Conclusion
We report determination of parameters in the sp3d5s* TB model for the nine binary compound semiconductors which consist of Al, Ga, or In and of P, As, or Sb based on the hybrid QSGW calculations. For the compounds in the bulk phase actually or potentially relevant to the type-II (InAs)/(GaSb) superlattices, we have confirmed that the erroneous flat valence bands, which appear within the sp3s* TB model, are eliminated. We have confirmed that the band gap energies of the type-II (InAs)/(GaSb) superlattices of various superlattice periods using the present sp3d5s* TB model agree with those of the corresponding PL data within discrepancy of 25meV. We have further compared the band gap energies of the type-II (InAs)/(GaSb) superlattices of common superlattice periods for each layer calculated by the present TB model with those calculated using the other known TB model, the EP method, and the hybrid QSGW method and with the PL data. The TB model reproduces an asymptotic decrease in the band gap energies for the large superlattice period obtained by the other TB and the EP methods and a bell line for the small superlattice period predicted by the other TB and the hybrid QSGW methods. The band gap energies calculated by the present TB model agree best with the PL data. Moreover, last but not least, the erroneous flat valence bands, which are potentially harmful to design the barrier layers, within the sp3s* TB model are eliminated using the present sp3d5s* TB model again along with the bulk phase. The present results indicate that the TB model is a reliable method to guide the superlattice design.
Acknowledgments
The authors are grateful to M. Shiozaki and Dr. A. Yamaguchi for helpful suggestions and encouragement and acknowledge the computing time provided by the Computing System for Research in Kyushu University.
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