UDC 539.893
T.M. Inerbaev1,2*, T. Matsuoka3, Y. Kawazoe3,4,5
1V.S. Sobolev Institute of Geology and Mineralogy SB RAS, Novosibirsk, Russian Federation;
2L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan;
3New Industry Creation Hatchery Center, Tohoku University, Japan;
4Physics and Nanotechnology, SRM Institute of Science and Technology, Tamil Nadu, India;
5School of Physics, Institute of Science, Suranaree University of Technology, Thailand;
*E-mail: inerbaevtm@igm.nsc.ru; inerbaev@yahoo.com
Optical band gap energy values in wurtzite In
xGa
1-xN
The narrow bandgap in InN has been known as a notorious example of local density approximation or gener- alized gradient approximation (LDA or GGA) calculations to give a metallic state. Various density functional methods are applied to optimize the atomic structures of the systems. These numerical results are used as the input values for the subsequent GW calculations, which can be applied to estimate the band gap value without phenomenological parameters. It is found that LDA with GW0 or the hybrid functional with self-consistent GW0 approximation provides sufficient theoretical results for both of the investigated compounds of GaN and InN. Although they are still time-consuming, due to less computational cost the former method is selected as a trial to compute the electronic structure in the entire range in ternary InxGa1-xN alloys without any arbitrary parameters. The present theoretical studies in ternary InxGa1-xN alloy were carried out by LDA with GW0. As a result, a good agreement between theoretical and experimental results is obtained, and it is also shown that zone bending could be well-approximated using a quadratic function with a constant, independent of x, pa- rameter equal to 1.85 eV, which is close to the recent experimental results.
Keywords: InxGa1-xN alloys, density functional theory, GW0 modeling, band structure, bowing parameter, band gap tuning, ternary alloy.
Introduction
The application of optical and electronic properties in nitride semiconductors with wurtzite structure has been focused on optical devices such as blue and white light-emitting-diodes [1–3], and electronic devic- es such as high electron mobility transistors used in the base station of cellular phones. Transistors with high power and high breakdown-voltage are expected in automobile industries for highly efficient inverters in vehicles. In these industrial applications to be realized in short time, the precise theoretical study is required for nitride semiconductors. The fundamental band gap value in InN has been uncertain up to 1980s. In early experiments in 1972 and 1986, the band gap values of about 1.9 eV and 2.1 eV were reported by measuring polycrystalline InN which was epitaxially grown with the sputtering technique and non-leak tight molecular beam epitaxy system (MBE) [4, 5]. Matsuoka predicted that this value is too large from his experiments on InGaN and single crystalline InN showed much smaller band gap energy value [6]. These values are much larger than the value of 0.7 eV reported later, which was measured from single crystalline InN [7–10]. These single crystalline InN samples were prepared by using the Molecular Beam Epitaxy (MBE) and the metalorganic vapor phase epitaxy (MOVPE) methods. A possible explanation of the results is that polycrystal is oxidized and mixed crystals of InN and In2O3 which band gap is 2.7 eV. The purity in single crystalline InN mentioned above is also poor. Its residual carrier concentration of 1017 to 1018 cm-3 is too high to accurately determine the band gap energy value because there is the Burstein-Moss effect due to the pres- ence of residual charge carriers. It is difficult to epitaxially grow InN with high quality because the equilibri- um vapor pressure of nitrogen between solid and gas phases is exceptionally high, compared with AlN and GaN [6]. Therefore, it is essential to theoretically determine the absolute band gap energy value in InN. In the present paper, the values of the band gap energy for the entire range of InxGa1-xN ternary alloys are calcu- lated using ab initio calculations without any empirical fitting parameters.
Experimental
The density-functional theory (DFT) within the local density approximation (LDA) or the generalized gradient approximation (GGA) has been extensively employed to investigate various electronic properties in post-transition metal (TM) compounds including both oxides and nitrides. DFT is a complete many-body the-
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ory, but it is basically applicable only to the non- degenerated ground state. When DFT is used to estimate the band gap, the main drawback of DFT is that the band gap value is seriously underestimated in the ranges over 50 %-70 % for most of the post-TM compounds and reaches the value of 80 % reduction in the case of ZnO [11]. In the case of InN, DFT gives even the negative band gap, i.e., predicts InN to be metallic [12, 13, 14]. For gallium nitride, the DFT calculation yields a strongly underestimated band gap value [13, 15]. Basi- cally this extensional usage of DFT for the band gap estimation is theoretically incorrect since it is a ground state theory, and higher level of theoretical methods such as GW approximation is requested.
The band gap underestimation in post-TM compounds could be addressed in part by considering the on- site electron correlation energy U. Such an empirical treatment tries to include the effect of correlations of d- electrons which increases in the band gap value. However, we note that the DFT+U method violates the neces- sary condition of the virial theorem (2T+V=0), and tends to give the effect on other physical parameters to be non-physical. For example, the lattice-constants are reduced and also contribute widening the band gap.
Alternatively, the hybrid functional scheme incorporating the exact exchange term from Hartree-Fock ap- proximation is used to improve the band gap. However, the fraction of the exact exchange energy again should be varied among post-TM compounds to fit the band gap and it is only a phenomenological treatment [11, 12].
Modern methods of the theoretical modeling allow predicting the fundamental band gap value with high accuracy without any parameter by combining DFT and the quasiparticle (QP) theory with the exchange- correlation self-energy in GW approximation [16]. Taking into account QP corrections, this band gap prob- lem can be solved with an accuracy of about 0.1 eV [17]. The first attempt to calculate the electronic proper- ties of InxGa1-xN ternary alloys with parameter-free theoretical technique was made in Ref. [18] using the LDA-1/2 method [19]. This method approximately includes the self-energy of excitations in semiconductors and gives results close to calculations by the GW method.
The many-body perturbation theory in the GW approach presents a QP theory that overcomes the defi- ciencies of LDA and GGA and provides a suitable description of the band structure for weakly correlated sol- ids such as GaN and InN. The GW approximation is formally the first term in an expansion of the nonlocal and energy-dependent self-energy in the screened Coulomb interaction [20]. The self-energy operator is de- scribed as follows:
(1) where G is Green’s function, W is screened Coulomb interaction, and δ is infinitesimal. The evaluation of self-energy requires the wave functions and corresponding eigenvalues. If these quantities are fixed to DFT results, it is usually called as single-shot G0W0 calculations. GW0 and GW approximations correspond to the case of iterative updates of the eigenvalues in the computation of G and W, respectively. A full update of the orbitals can be performed by specifying self-consistent GW calculations. If the orbitals and eigenvalues are updated in G and W, it is scGW approximation whereas scGW0 corresponds to orbitals and eigenvalues up- date only in G calculations. In addition to the level of self-consistency of calculations, the results also depend on the initial conditions, i.e., from which method the eigenvalues and orbitals are derived in GW calculations.
Numerous studies using GW method have shown that the numerical results strongly depend on the initial geometry of the structure as well as on the choice of the starting wave functions [12, 13, 21-30]. Thus, the use of source structures optimized with the help of various functionals leads to a variation of the resulting numerical values of the band gap, for example from 3.366 eV (GGA-PBE) to 3.847 eV (LDA) optimized GaN wurtzite structure [30].
The choice of starting wave function for GW calculations is also crucial. Kang et al. [11] tested various levels of self-consistency and starting conditions to establish the most proper GW calculation scheme for the best exact description of the band gap value for post-TM oxides. It was found that the GW0 scheme with GGA+U as the DFT functional turned out to give the best results in every aspect of the band structures. In one specific case of ZnO, it was proposed a modified scheme where on-site term U on Zn-d orbital was used within GW0 scheme. Higher level of self-consistent scGW and employment of hybrid Heyd-Scuseria-Ernzerhof (HSE06) functional [31] to calculate wave functions and eigenvalues as starting conditions for G0W0 calcula- tions were found to overestimate the band gap value. Pure DFT calculations with LDA or GGA functionals pre- dict metallic state for InN [12, 21]. Kumar et al. [27] used the DFT+U to calculate the starting wave function, which made it possible to open the optical gap in the electron structure of indium nitride.
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In the present paper, we tested various GW approaches on GaN and InN to compare their validity and accuracy. One important point is to minimize the computational time required for electronic structure calcula- tions, since we should use an extra-ordinally high demand computing resources for the ternary systems of InxGa1-xNas. With all of these careful theoretical considerations, the optimal computational method was em- ployed to calculate the accurate value of the band gap energy for the InxGa1-xN ternary alloy. To investigate the electronic structure in InGaN alloys, we used 2×2×2 supercells containing 32 atoms. Supercells with 25 %, 50 %, and 75 % concentration of In atoms were considered and should provide a good approximation for the random alloy [12].
To obtain the initial calculations, all numerical calculations were performed using HSE06, Perdew- Burke-Ernzerhof (PBE) [32] and local density approximation (LDA) functionals by employing the projector augmented wave (PAW) method [33] as implemented in the Vienna Ab initio Simulation Package, VASP [34, 35]. The atomic structures of both InN and GaN were modeled by a fully relaxed wurtzite structure of 4 atoms (C46v-P63mc: space group number 186) [36, 37]. The In-4d and Ga-3d electrons were treated as valence elec- trons. The electronic wave functions are described using a plane wave basis set with an energy cut-off of 600 eV. Г-point centered k-point meshes of 6×6×6 are used throughout the calculations to obtain well-converged results for GaN and InN. Since the DFT calculations predict a metallic state for InN, to open the optical gap, the additional on-site correlation DFT+U in the Dudarev parameterization [37] was taken into account with the value of U = 3.5 eV.
In Ref. [21], the dependencies of theoretical band gap values were calculated depending on the choice of cutoff energies and k-space partitions. It was shown that the convergence of the calculated band gap occurs at the cut-off energy of 600 eV. By selecting a 4×4×4 k-point mesh, the calculated fundamental band gap values differ from those obtained on a 6×6×6 by 1.5 %. For this reason, the present numerical calculations are performed using the 2×2×2 k-point mesh.
From the theoretical point of view, state-of-the-art calculations within many-body perturbation theory allow to rigorously obtain band gap energy values. The GW approximation method used to compute self- energy corrections is a quite well-established and standard technique, giving energy levels generally in good agreement with experiments, even for complicated systems like reconstructed surfaces and clusters [39]. Due to high complexity and large computational require ments of ab initio calculations of the self-energy, this approach has rare been used to study systems with a large number of atoms in a unit cell. In the present study, were performed the investigation of the InxGa1-xN alloy to estimate the band gap value in the entire composi- tion range with GW approximation, predicting its properties without any experimental/fitting parameters.
Previous studies of InxGa1-xN compounds showed poor agreement of the DFT band gap values with ex- perimental data [12, 22, 40–44]. To improve consistency between theory and experiment, hybrid functionals in DFT have been used as an empirical simulation method. In the case of zinc blende InxGa1-xN alloys, the overestimation of the band gap in the In-rich region was found [45]. For AlN, GaN, and InN with wurtzite structure, the resulting band gap values depend on the mixing ratio of the exact exchange energy functional, Mixing ratio is required to match the experimental band gaps increases with experimental band gaps [12].
Result and Discussion
To solve the problem of describing the band gap in the ternary alloy system, it is necessary to choose a suitable theoretical method that would equally accurately describe both compounds of GaN and InN. The available results in published literatures on InN and GaN GW band gap calculations are summarized in Table 1. It is astonishing to recognize that to date, only one study has been published in the literature which shows that the optimized effective potential (OEP) calculations predict well both the lattice geometry and the band gap for both nitrides.
[13] Unfortunately, this method requires enormous computational effort and cannot be applied to the study of large model cells as required by the present research on ternary alloys. Base on this reason, we tested different approaches as initial stages to GW calculations in the framework of GGA, LDA, and hybrid functionals.
Besides choosing the initial wave function for calculating GW, which means selecting the appropriate density functional, the geometric structure of the model cells also largely determines the theoretical values of the band gap. The lattice constants of InxGa1-xN ternary alloys can be obtained both as a result of theoretical calculations for lattice optimization and using interpolation formulae if we assume that the lattice parameters
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of the alloys are governed by Vegard’s laws. Another potential source of calculation errors is the inability to perform structural optimization in a real calculation process using the GW method.
In the present study, in contrast to Ref. [13], where experimental values of lattice constants were used, we explore various functionals concerning their accuracy in predicting the geometry of the unit cells. The lattice geometry found in this way is then used for GW calculations with the initial wave functions obtained with various functionals. The functional that provides the best agreement between the calculated bandgap values and the experimental ones were used for calculations using a model supercell of InxGa1-xN (x=0.5) compound with lattice constants calculated according to Vegard’s law for comparison.
The results of the present numerical results on the lattice parameters in GaN and InN in comparison with the experimental data are given in Table 2. The data obtained are typical for calculations carried out by DFT. In the case of InN, the hybrid functional predicts almost exact lattice-constant values, whereas the PBE functional overestimates, while the LDA functional overestimates the value of the parameter a and underes- timates c. In the case of GaN, the best values of the lattice-constants are given by the calculation using the PBE functional, while the HSE06 and LDA functionals underestimate these parameters. Thus, none of the selected methods for the optimization of the crystal lattice is preferable in comparison with experimental data for both considered compounds.
T a b l e 1 Summary of the literature available high-accuracy calculations of
fundamental band gap value for InN and GaN of wurzite structure
Compound Band gap, eV Method (based on the WF
obtained with this method) Reference
InN
0.71 GW0+RPA (HSE3) [24]
0.99 scGW (LDA) [25]
0.74 Hybrid approacha)
0.66 HSE06 [12]
0.58 LDA with SICb) [22, 23]
0.74 Simplified Gcc)
1.50 GW SIC (LDA)
0.82 GW SIC (LDA) [26]
0.8 scGW0 [27]
0.7 (0.8 correction factor was applied) GW+RPA (LDA) [28]
0.638, 0.765; 0.494d) G0W0+SOCe) (HSE) [30]
0.711 HSE06 [21]
0.694 G0W0 (HSE06)
0.805 scGW0
0.7 OEPx6+G0W0(LDA) [13]
0.0 PBE [14]
0.5 HSE06 [14]
0.95 LDA-1/2 [19]
GaN
3.5 GW [29]
3.23 HSE06 [12]
3.81 3.42
scGW (LDA)
Hybrid approacha) [25]
3.6 (0.8 correction factor was applied) GW+RPA (LDA) [28]
3.659; 3.847; 3.366d) G0W0+SOCe) (HSE) [30]
3.24 OEPxf)+G0W0(LDA) [13]
3.52 LDA-1/2 [19]
a) Combines 80 % of the GW self-energy with 20 % of the LDA self-energy as it is described in
Ref. [53]; b) Self-interaction corrections; c)Quasi-particle corrections to DFT within a simplified GW approximation; d) Data ob- tained for geometries optimized using different methods (AM05, LDA, PBE); e) Spin-orbit coupling; f)exact-exchange optimized ef- fective potential
Further, all obtained equilibrium lattice geometries were used to calculate the band gap values. For each structure, three possible wave functions were used as the starting values for the numerical calculations within
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GW approximation. The results are shown in Table 3. From the presented data it can be seen that only two approaches provide sufficiently good values was an error less than 0.1 eV for both indium and gallium ni- trides; namely, (case 1) scGW0 approach with optimized geometry by HSE06 and starting wave functions by LDA, and (case 2) GW0 calculations with LDA for both geometry optimization and starting wave functions.
T a b l e 2 The equilibrium values of the lattice-constants for InN and GaN at atmospheric pressure calculated by various
methods in comparison with experimental data. Parentheses indicate the relative error of calculations
Compound a, Å
(error, %)
c, Å (error, %)
Unit cell
volume, Å3 c/a u Method
InN
3.533 5.693 61.50 1.611 0.375 Exp. [36]
3.530 (-0.08) 5.700 (0.1) 61.52 1.614 0.379 HSE06
3.578 (1.3) 5.777 (1.5) 64.06 1.614 0.379 PBE
3.503 (0.8) 5.655 (-0.7) 60.09 1.614 0.379 LDA
3.62 5.83 1.61 PBE/HSE06
[14]
GaN
3.19 5.28 46.53 1.655 0.375 Exp. [54]
3.176 (-0.4) 5.173 (-2.0) 45.20 1.628 0.377 HSE06
3.214 (0.8) 5.235 (0.9) 46.84 1.628 0.377 PBE
3.156 (-1.1) 5.140 (-2.7) 44.34 1.628 0.377 LDA
Thus, two theoretical approaches provide approximately the same values of the perception of both the lattice geometry and the band gap in both nitrides under study. Moreover, for the calculated values of the cell parameters, the use of the hybrid functional is somewhat closer to experimental values. On the other hand, the inclusion of the Hartree-Fock exchange in the hybrid functional raises the computation cost very signifi- cantly, especially when one uses the plane-wave basis sets [46, 47]. Besides, for the exact prediction of the width of the band gap, in this case, the scGW0 method is used, which includes an additional self-consistency procedure in comparison with GW0, which also significantly increases computational costs. Thus, based on consideration of the balance of the ability of prediction and the required computational costs, in the present study, we employed the LDA for structure optimization and wave functions calculation for GW0. In the latter case, the LDA+U method was used on indium atoms, as indicated in “Computational method” section.
T a b l e 3 The values of the band gap for nitrides, calculated using various methods. The values that most closely
coincide with the experimental values for both considered compounds are highlighted in bold
Compound Input geometry Input wavefunction
Bang gap calculation method
DFT G0W0 GW0 scGW0
Band gap, eV
1 2 3 4
InN
HSE06
HSE06 0.76 0.71 0.71 0.89
PBE+U 0.09 0.21 0.27 0.50
LDA+U 0.11 0.43 0.51 0.73
PBE
HSE06 0.60 0.49 0.48 0.66
PBE+U 0.04 0.07 0.30 0.30
LDA+U 0.03 0.14 0.14 0.40
LDA
HSE06 0.85 2.65 2.59 2.63
PBE+U 0.16 0.29 0.36 0.25
LDA+U 0.26 0.67 0.76 0.94
GaN HSE06 HSE06 3.30 3.77 3.85 3.97
PBE 1.97 3.13 3.33 3.50
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1 2 3 4
LDA 1.99 3.16 3.38 3.44
PBE
HSE06 3.02 3.47 3.67 3.67
PBE 1.73 4.66 5.00 5.26
LDA 3.03 3.47 3.57 3.63
LDA
HSE06 3.46 3.93 4.01 4.14
PBE 2.10 3.29 3.51 3.74
LDA 2.12 3.33 3.55 3.69
The estimated lattice-constant values for the InxGa1-xN ternary alloy as a function of x are shown in Figure 1. The present numerical calculations were carried out for a 2×2×2 supercell and the data presented was reduced to a single cell. When creating model structures, the atoms in the cases x = 0.25 and = x = 0.5, the substitution of gallium atoms was carried out in such a way as to distribute the indium atoms uniformly throughout the cell, while avoiding both the clustering of indium atoms and their ordering. The latter could lead to the appearance of periodic superstructures. Earlier results show that the clustering of indium InGaN alloy leads to a greater curvature of the curve of the band gap dependence on x value [51]. The case x = 0.75 is identical to x = 0.25 up to the mutual permutation of indium and gallium atoms. Test calculations on three different supercells with x = 0.5 showed a weak (< 2 %) dependence of the band gap on the choice of the atomic configuration. The geometries of all supercells have been optimized. The optimization concerned both the positions of the ions inside the supercells and the volume of the supercells. In this case, the lattice con- stants changed so that the volume of the optimized cell corresponded to the system at zero external pressure.
The lattice symmetry was maintained constant.
It can be seen that the cell parameters are linearly dependent on the In content according to Vegard’s law.
The resulting atomic structures were used to calculate the band gap using the start wave functions obtained using the LDA functional. The obtained data are presented in Figure 2 in comparison with the available exper- imental data and earlier calculations using hybrid HSE06 functional [12]. The present theoretical results, without any parameters are in excellent agreement with the experimental results.
0.0 0.2 0.4 0.6 0.8 1.0
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6
Lattice parameter, Å
In fraction, x a, Å c, Å
Experiment
Figure 1. Calculated lattice parameters of InxGa1-xN within LDA in comparison with experimental values [35, 53]
We also investigated the dependence of the theoretical band gap on the starting geometry of the model cell, implying that the lattice constants depend on the concentration of In and Ga according to Vegard's law.
The obtained band gap for the ternary alloy of the composition In0.5Ga0.5N, obtained using GW calculations based on the LDA functional, is 1.36 eV. This value is 0.29 eV less than the corresponding value calculated for a structure whose lattice constants are theoretically obtained.
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Figure 2. Theoretical and experimental values of the fundamental band gap values in Ga1-xInxN ternary alloy as a function of the indium molar fraction. Theoretical results are obtained using GW0 method (this work), HSE06 hybrid functional [12] and LDA-1/2 method [18]. The band gap GW(VL) value was obtained for a model cell with lattice constants calculated using the Vegard’s law. Experimental values are adopted from Refs. [6, 8–10, 55, 56]
The band gap in InxGa1-xN as a function of the In content is of key importance for the analysis and de- sign of efficient electronic devices. Qualitatively, there is an agreement that the band gap in InxGa1-xN is a nonlinear function of the alloy composition. Conventionally, alloy band gap values are expressed as
– , (2) where b is the so-called bowing parameter. Since the most important parameter in the semiconductor alloy system is the band gap and its deviation from linear dependence characterized by the band gap bowing pa- rameter b, which generally is a function of concentration. The various studies [22, 40, 43, 48–51] have disa- greed on the magnitude and In concentration dependence of the bowing parameter. Some investigations show that the bowing cannot accurately be described by a composition-independent bowing parameter [12, 22, 40, 48, 52]. Wu et al. by optical absorption and photoluminescence measurements showed that band gap versus composition is well-described by a constant bowing parameter of 1.4 eV [10]. However, McCluskey et al., obtained the bowing parameter value of 2.6 eV by carrying out optical absorption spectroscopy measure- ments [51]. Kazazis et al. measured that the intrinsic band gap value dependency on the In content is ade- quately expressed, for the entire range, by a bowing parameter value of 1.66±0.08 eV [8]. Fitting of our nu- merical results gives b=1.85 eV with a maximal absolute error less than 0.04 eV that is a good agreement with the most recent experiment of Kazazis et al..
The above analysis corresponds to the use of a single parameter bowing calculating would be fitting (2).
Moses et al. demonstrated that the bowing parameter varies with indium concentration. Namely, with an in- crease in the indium content, its value decreases [12]. Our results confirm this trend, although to a much lesser extent, and give b equal to 1.92, 1.85, and 1.79 eV for 25, 50, and 75 % of the indium content.
Conclusions
To solve the problem of band gap values in ternary InxGa1-xN alloys theoretically without phenomeno- logical parameters, various functionals have been preliminarily tested both concerning using them to opti- mize the geometry of model cells and to calculate the initial wave functions for GW calculations applied to binary nitrides. For both InN and GaN, the acceptable accuracy is achieved either by GW0 approximation with LDA or by scGW0 method with hybrid HSE06 functional, especially the narrow band gap of InN was successfully reproduced without using the phenomenological parameters. With LDA the optical band gap in InN is estimated to be negative (metallic). To open it, it is necessary to use the LDA+U method. Since the former approach requires less computational costs, the present theoretical studies in ternary InxGa1-xN alloy were carried out by LDA with GW0. As a result, a good agreement between theoretical and experimental re- sults was obtained, and it is also shown that zone bending could be well-approximated using a quadratic
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function with a constant, independent of x, parameter equal to 1.85 eV, which is close to the recent experi- mental results.
Acknowledgements
The authors are grateful to JHPCN project as “jh190019-NAJ” on the supercomputer at Research Insti- tute for Information Technology Kyushu University and the Center for Computational Materials Science, Institute for Materials Research, Tohoku University for their continuous support of the supercomputing sys- tem to be used for our simulation works.
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Т. Инербаев, Т. Матсиока, И. Кавазое
In
xGa
1-xN вуртциттегі оптикалық жолақ саңылауының энергия мәндері
InN тар тыйым салынған аймақ жергілікті тығыздыққа жақындағанда немесе жалпыланған градиент жақындағанда (LDA немесе GGA) есептеулер жартылай өткізгіштің орнына заттың металдық күйін болжайтын белгілі мысал болып табылады. Бұл жұмыста зерттелетін жүйелердің геометриясын оңтайландыру үшін тығыздық функционалының әртүрлі әдістері қолданылды және бұл сандық нәтижелер феноменологиялық параметрлерсіз тыйым салынған аймақ енінің мәнін бағалау кезінде болуы мүмкін GW әдісімен кейінгі есептеулер үшін кіріс мәндері ретінде пайдаланылды. LDA комбинациясын GW0 есептеулерімен немесе гибридті функционалды GW0 өзіне сәйкес келетін жуықтауымен пайдалану зерттелген GaN және InN қосылыстары үшін жеткілікті дәл теориялық нәтижелер беретіні анықталды. Мұндай есептеулер әлі де компьютердің көп уақытын қажет ететініне қарамастан, есептеу құнының төмен болуына байланысты, бірінші әдіс InxGa1-xN үштік қорытпаларындағы барлық диапазондағы электрондық құрылымды кез келген реттеу параметрлерінсіз есептеудің сынақ әдісі ретінде таңдалды. InxGa1-xN үштік қорытпасының нақты теориялық зерттеулері LDA GW0 жуықтау комбинациясында орындалды. Нәтижесінде теориялық және эксперименттік нәтижелер арасында өте жақсы келісім алынды, сонымен қатар тыйым салынған аймақтың енін x функциясы ретінде иілу квадраттық функцияны қолдана отырып, x-ге тәуелді емес, 1,85 эВ параметрімен жақсы жақындастыруға болатындығы көрсетілген, бұл соңғы эксперименттік нәтижелерге жақын.
Кілт сөздер: InxGa1-xN қорытпалары, тығыздықтың функционалдық теориясы, GW0 модельдеу, жолақ құрылымы, иілу параметрі, жолақ саңылауын орнату, үштік қорытпа.
Т. Инербаев, Т. Матсиока, И. Кавазое
Значения энергии оптической запрещенной зоны в вюрците In
xGa
1-xN
Узкая запрещенная зона InN представляет собой известный пример, когда расчеты в приближении ло- кальной плотности или приближении обобщенного градиента (LDA или GGA) предсказывают метал- лическое состояние вещества вместо полупроводникового. В настоящей работе для оптимизации гео- метрии исследуемых систем применены различные методы функционала плотности, и эти численные результаты использованы в качестве входных значений для последующих вычислений методом GW, которые могут быть при оценке значения ширины запрещенной зоны без феноменологических пара- метров. Установлено, что применения комбинации LDA с GW0 расчетами или гибридного функциона- ла с самосогласованным GW0 приближением дают достаточные точные теоретические результаты для обоих исследованных соединений GaN и InN. Хотя такие расчеты по-прежнему занимают очень много компьютерного времени, из-за меньшей вычислительной стоимости первый метод выбран в качестве пробного для расчета электронной структуры во всем диапазоне в тройных сплавах InxGa1-xN без ка- ких-либо подгоночных параметров. Настоящие теоретические исследования тройного сплава InxGa1-
xN были выполнены в комбинации LDA с GW0 приближением. В результате получено очень хорошее согласие между теоретическими и экспериментальными результатами, а также показано, что изгиб ширины запрещенной зоны как функции x можно хорошо аппроксимировать с помощью квадратич- ной функции с постоянным, не зависящим от x, параметром, равным 1,85 эВ, что близко к последним экспериментальным результатам.
Ключевые слова: сплавы InxGa1-xN, теория функционала плотности, моделирование GW0, зонная структура, параметр изгиба, настройка запрещенной зоны, тройной сплав.
