Abstract
Structural and electronic properties of hexagonal (h) and cubic (c) phase AlGaInN quaternary alloys are investigated using a unified and accurate localdensity approximation1/2 approach under the densityfunctional theory framework. Lattice bowing parameters of h (and c) phase AlGaN, AlInN, InGaN, and AlGaInN alloys are extracted as 0.006 (−0.007), 0.040 (−0.015), 0.014 (−0.011), and −0.082 (0.184) Å, respectively. Bandgap bowing parameters of h (and c) phase AlGaN, AlInN, InGaN, and AlGaInN alloys are extracted as 1.775 (0.391), 3.678 (1.464), 1.348 (1.164), and 1.236 (2.406) eV, respectively. Directtoindirect bandgap crossover Al mole fractions for cphase AlGaN and AlInN alloys are determined to be 0.700 and 0.922, respectively. Under virtual crystal approximation, electron effective masses of h and cphase AlGaInN alloys are extracted and those of cphase alloys are observed to be smaller than those of the hphase alloys. Overall, cphase AlGaInN alloys are shown to have fundamental material advantages over the hphase alloys such as smaller bandgaps and smaller effective masses, which motivate their applications in light emitting and laser diodes.
Introduction
Ever since the inventions of highefficiency and highbrightness blue and white light emitting diodes (LEDs), hexagonal (h) phase (wurtzite) gallium nitride (GaN) and its ternary alloys–AlGaN, AlInN, and InGaN–have successfully revolutionized the way we generate efficient light source, which spans all across visible light and extends in ultraviolet spectra^{1}. To date, the feasible way to generate a variety of visible light, including natural white light, is to superimpose the primary colors of light: red (1.91 eV), green (2.24 eV), and blue (2.75 eV), socalled fullcolor tuning. Highlyefficient red and blue LEDs are fabricated by AlGaInP and InGaN materials, respectively; while, green LEDs are manufactured by phosphorcoated blue LED, phosphidebased materials, or nitridebased materials^{2}. However, the efficiency of green LEDs is still more than twice less efficient than that of the red and blue LEDs due to the energy loss of phosphorcoated blue LED during wavelength conversion and the indirect bandgap of phosphidebased materials. The factors that plagued nitridebased green LEDs are more complicated; to be simplified, the strong spontaneous polarization and piezoelectric polarization of hphase nitrides resulting from the lack of inversion symmetry and the lattice mismatch under highIn alloying deteriorate the radiative recombination efficiency, the phenomenon is known as quantumconfined Stark effect^{3}. As a result of lacking suitable materials for green emitters, the low efficiency of green LEDs is described as a “green gap”. An urgent engineering challenge is to discover a material that has a compatible and direct bandgap to emit green wavelength, meanwhile, depress the quantumconfined Stark effect.
IIInitrides can also crystallize in the cubic (c) phase (zincblende) structure, which has potential advantages in optical and electrical applications. Specifically, cphase GaN exhibits high electron mobility^{4}, smaller acceptor activation energy^{5}, high hole mobility^{6}, and small Auger losses^{7}. Additionally, the cphase IIInitrides are centrosymmetric, which makes them polarizationfree in the growth <001> direction. This leads to a larger electronhole wavefunction overlap, which increases radiative recombination efficiency and optical gain^{8}. Thus, cphase GaN and its ternary and quaternary alloys are ideal photonic materials for highefficiency vertical transport devices, which includes light emitting and laser diodes. However, due to the metastability of cphase IIInitrides, the crystal structure suffers from a high level of defectivity and can easily relax to the hphase structure. Recently, Liu et al. have synthesized cphase GaN with high crystalquality and high phasepurity using the Ugroove hexagonaltocubic phase transition approach. A bandtoband emission in the ultraviolet (UV) from the cphase GaN has shown a record high internal quantum efficiency of 29%, which is higher than that of bulk hphase GaN at 12%^{9}. Further increase of the radiative efficiency using carrier confinement necessitates the use of quantum well structures. Yet, there are limited computational and experimental studies on the structural and electronic properties of these cphase IIInitrides; there are no numerical expressions for the bandgap, latticeconstant, and effective mass of cphase IIInitrides as well. In literature, bandgaps, directtoindirect bandgap crossover points, and effective masses of h and cphase AlN, GaN, and InN are studied using various simulation approaches such as localdensity approximation (LDA)^{10,11}, G_{0}W_{0} approximation^{12}, and generalized gradient approximation (GGA) approaches^{13}, which led to dissimilar bandgap values. For instance, the calculated bandgap of hphase InN ranges from 0.69^{12} and 1.02^{13} to 2.00^{11} eV; the broad range of bandgap is also reported in experiments, while the widely accepted value is 0.78 eV^{14}. The inconsistent bandgap can virtually affect the accuracy in determining the bandgap of Inrich ternary and quaternary IIInitrides, which alters the bowing parameters and the directtoindirect bandgap crossover points^{15}. In addition, the evolution of electron effective mass in cphase ternary and quaternary IIInitrides with respect to Al, Ga, and In mole fractions is still unclear.
In this work, lattice constants and bandgaps of h and cphase IIInitrides are investigated under the densityfunctional theory framework. For the crystal relaxation, LDA is applied to approximate the exchangecorrelation energy of the manyelectron system due to its reliability and low computational cost of determining the groundstate electronic properties. However, it is wellknown that LDA underestimates the bandgap since the exchangecorrelation potential is not discontinuous between the conduction and valence band. LDA1/2 approach, on the other hand, is applied to correct the bandgap by halfionizing electron to conduction band with the same reliability and computational cost as the LDA method. It can retrieve the accurate bandgap of IIIV alloys because the relation between the singleparticle energy obtained from KohnSham equation and the electron occupation can be linearly approximated^{16}. Figure 1(a) illustrates a unit cell of hphase binary IIInitrides. The red dash lines highlight the primitive cell that contains 2 group III atoms and 2 N atoms and defines the hphase crystal structure. Similarly, Fig. 1(b) shows a unit cell of cphase binary IIInitrides, where the primitive cell contains 1 group III atom and 1 N atom. Figure 1(c,d) demonstrate the electronic structure of h and cphase GaN along the highsymmetry lines, where the energy states shift with respect to the valence band maximum (E_{VBM}). The electronic structure of ternary (AlGaN, AlInN, and InGaN) and quaternary (AlGaInN) alloys with different mole fractions are calculated by substituting the group III atoms in a unit cell, which allows an accurate interpolation of directtoindirect bandgap crossover points, an extraction of bowing parameters, and an numerical expression of lattice constants and bandgaps. Transverse and longitudinal electron effective masses are also extracted for h and cphase IIInitrides.
Methods
Firstprinciples calculations are carried out under the densityfunctional theory framework implemented in the Vienna ab initio Simulation Package (VASP)^{17} and are carried out on binary, ternary and quaternary IIInitrides. Al 3s^{2}3p^{1}, Ga 4s^{2}4p^{1}, In 4d^{10}5s^{2}5p^{1}, and N 2s^{2}2p^{3} valence electrons are characterized using projector augmented wave pseudopotentials (PAW)^{18}; while the cutoff kinetic energy of 500 eV is secured for the planewave expansion. All atoms are fully relaxed so that the interatomic forces and energy difference are smaller than 0.01 eVÅ^{−1} per ion and 10^{−6} eV/atom, respectively. Figure 2(a) illustrates 2 × 2 × 1 primitive cell of hphase binary IIInitrides used to construct and simulate a 16atom supercell of hphase quaternary alloys. Similarly, Fig. 2(b) illustrates 2 × 2 × 2 primitive cell of cphase binary IIInitrides used to construct and simulate a 16atom supercell of cphase quaternary alloys. Each supercell contains 8 group III atoms and 8 N atoms. The mole fractions of Al, Ga, and In can be adjusted by substituting the 8 group III atoms in the supercell, which gives the finest mole fraction tunability of 0.125. For instance, the unit cell of Al_{0.375}Ga_{0.625}N contains 3 Al atoms, 5 Ga atoms, and 8 N atoms; while, the unit cell of Al_{0.5}Ga_{0.5}N has 4 Al atoms, 4 Ga atoms, and 8 N atoms. Overall, 9 and 45 cases are sampled individually for each phase of ternary (AlGaN, AlInN, and InGaN) and quaternary (AlGaInN) alloys. Although the initial size of supercells used to build ternary and quaternary alloys is identical, the size of supercells changes after the atoms and the cells are relaxed to its equilibrium positions and volumes. For instance, h and cphase AlN have the smallest supercells of 163.36 and 163.75 Å^{3}; while h and cphase InN have the largest supercells of 240.58 and 240.93 Å^{3}, respectively. The Brillouin zone of the supercells is sampled with an 8 × 8 × 8 gammacentered MonkhorstPack set of kpoints. Notably, the bandgap of binary alloys (AlN, GaN, and InN) calculated by the supercell approach is consistent with the bandgap calculated using the primitive cells, which indicates the numerical accuracy of the supercell calculations. LDA is performed to calculate the exchangecorrelation energy of manyelectron system for structural relaxation. To fix the bandgap underestimation of LDA, LDA1/2 method is utilized to calculate the electronic structure and bandgap of IIInitrides. LDA1/2 method inherits from Slater halfoccupation scheme^{16,19}, which assumes that singleparticle KohnSham energy linearly depends on the occupation expressed as:
where E(N) and E(N−1) are the total system energies under the occupation number of N and N−1, respectively; ε_{α} is the singleparticle KohnSham energy at state α. Using the linear dependence of the singleparticle KohnSham energy on the occupation number, the ionization energy is approximated by the singleparticle KohnSham energy with a half ionization, as shown in the last term. Starting from this postulation, Ferreira et al. further demonstrated that the singleparticle KohnSham energy with a half ionization is equivalent to the singleparticle KohnSham energy at groundstate, ε_{α}(N), minus a hole selfenergy (S_{α}) at state α expressed as^{20}:
S_{α} can be formulated by quantummechanical average:
where n_{α}, V_{s}, and Θ are the electron density at state α, the selfenergy potential, and the trim function, respectively. The trim function is used to cut the Coulomb tail of V_{s} in an infinite crystal, which has a negligible contribution to the selfenergy due to the localization of wavefunction, defined by:
where CUT and n are the cutoff radius and cutoff sharpness; while, A is the amplitude of trim function, which can be exploited to adjust the amplitude of selfenergy and bandgap linearly and semiempirically. The bandgap (E_{g}), defined by the energy difference between the system energy that has one electron excited from the valence to the conduction band, E(N−1, 1), and the groundstate system energy, E(N,0), can be formulated as the bandgap calculated by LDA (\({E}_{g}^{LDA}\)) with corrections from electron (S_{c}) and hole (S_{v}) selfenergies:
where ɛ_{c} and ɛ_{v} are singleparticle KohnSham energies for electron and hole, respectively.
According to the suggestions by Ferreira et al., A = 1 and n = 8 are fixed in the first place^{20}. The CUT values of 4.40, 1.53, and 3.0Å are benchmarked for Al, Ga, and N ions to make the bandgap of hphase AlN (6.2 eV) and hphase GaN (3.5 eV) experimentallyverified^{14}. However, several combinations of CUT and n have been benchmarked for In ion, none of them gives a satisfactory result. The optimal bandgap is 1.24 eV, which is similar to other reports^{13,15,21}. But, it significantly deviates from the experiment of 0.78 eV. The bandgap overestimation may originate from the assumption that the ionization energy is equal to the singleparticle KohnSham energy with a half ionization. In other words, the singleparticle KohnSham energy of In ion may not linearly dependent upon the occupation number. To determine the correct selfenergy, the trim function amplitude A is benchmarked semiempirically to adjust the selfenergy amplitude and the bandgap linearly. Finally, A = 2.3 and CUT = 2.926Å are benchmarked for In ion.
To model the electronic properties of multinary alloys, the unit cell of multinary alloys is constructed by the supercell method. However, the corresponding Brillouin zone is folded with the increasing size of the supercell. As the consequence, it is challenging to analyze the electronic properties, such as extracting effective mass and determining indirect bandgap, from the Ek dispersion because the energy states outside of the first Brillouin zone are folded into the first Brillouin zone socalled band folding. fold2Bloch utility^{22} is employed to unfold the band structure of supercell back to its primitive basis representation by calculating and filtering the Bloch spectral density, the procedure is known as virtual crystal approximation. After unfolding the electronic structure, the curvatures of the lowest conduction band for electron effective mass calculations are extracted using the finitedifference method and parabolic approximation. The lattice constants, bandgaps, transverse (\({m}_{t}^{\ast }\)) and longitudinal (\({m}_{l}^{\ast }\)) electron effective masses for h and cphase IIInitrides with the corresponding corroboration from experiments are tabulated (Table 1). For hphase alloys, \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) are calculated along [100] and [001] directions. For cphase alloys, \({m}_{l}^{\ast }\) is computed along [101] direction and \({m}_{t}^{\ast }\) is calculated along [111] direction, at Γvalley (direct bandgap alloys), or [211] direction, at Xvalley (for indirect bandgap alloys).
Results and Discussion
Figure 3(a–f) summarizes the lattice constants along [100] direction, bandgaps, and fits for both lattice constant and bandgap using Vegard’s law of h [Fig. 3(a–c)] and cphase [Fig. 3(d–f)] AlGaN, AlInN, and InGaN. All six ternary alloys have a linear relationship between lattice constants and Al or In mole fractions. The lattice bowing parameters of h (and c) phase AlGaN, AlInN, and InGaN alloys are extracted as 0.006 (−0.007), 0.040 (−0.015), and 0.014 (−0.011) Å fitted by lattice Vegard’s law tabulated in Table 2. Since the defectivity and piezoelectric polarization of IIInitrides strongly depend on the lattice mismatch between two substrates, it is essential to estimate the lattice mismatch between the potential materials of quantum well and GaN. The lattice mismatch can be estimated by \(\frac{a{a}_{GaN}}{{a}_{GaN}}\times 100 \% ,\) where a and a_{GaN} are the lattice constants of the potential material and GaN, respectively. As the results, the strains of hphase AlN, cphase AlN, hphase InN, and cphase InN interfaced with GaN are −3.00%, −3.49%, 10.00%, and 9.77%, respectively. The strains between the ternary alloys and GaN can be linearly interpolated due to the linear relationship between lattice constants and Al and In mole fractions.
The accuracy of LDA1/2 method on the bandgap calculations of ternary alloys is demonstrated by comparing the simulation results with the experimental measurements^{23,24,25,26,27,28,29,30,31,32}, except for cphase AlInN and Inrich cphase InGaN due to the lack of experimental reference data. The nonlinear dependence of bandgap on the Al or In mole fraction is summarized using bandgap Vegard’s law tabulated in Table 2. The bandgap bowing parameters of hphase AlGaN, AlInN, and InGaN are extracted as 1.775, 3.678, and 1.348 eV, respectively, which are larger than their cphase counterparts of 0.391 (AlGaN), 1.464 (AlInN), and 1.164 eV (InGaN). The simulated bandgaps agree well with the available experiments. It is worth mentioning that most of the exchangecorrelation functionals, including highlyaccurate hybrid functionals and GW quasiparticle approximation, fail to recover the bandgap of InN^{21}, which also leads to the bandgap overestimation of Inrich hphase InGaN by more than 20%^{15}. The similar issue also occurs in the LDA1/2 method. To address this issue, the selfenergy energy of In ion is linearly and semiempirically increased by 2.3 times so that the bandgap of hphase InN matches the experiment of 0.78 eV. The numerical adjustment corrects the underestimation of selfenergy and, eventually, improves the bandgap overestimation. Therefore, the correction of selfenergy using hphase InN bandgap fixes the bandgap overestimation of Inrich hphase InGaN. The calculated bandgap of cphase InN also matches the experiment, which further indicates the consistency and reliability of the semiempirical correction.
The electronic structures of cphase AlGaN and cphase AlInN are unfolded to examine the crossover point at which the directtoindirect bandgap transition occurs. The evolutions of conduction band minimum and valence band maximum are traced with respect to Al mole fraction. It is observed that the conduction band minimum at Xvalley decreases as the Al mole fraction increases; while, the conduction band minimum at Γvalley shifts oppositely. The crossover points are interpolated using the energy difference of conduction band minimum at the Xvalley and the Γvalley. When the Al mole fraction reaches the crossover point of 0.700 and 0.922 for cphase AlGaN and cphase AlInN, respectively, the Xvalley takes the possession of conduction band minimum, which indicates indirect bandgap. The crossover point of cphase AlGaN agrees well with the reported value of 0.692^{33}. However, the crossover point of cphase AlInN is slightly greater than the reported values of 0.81‒0.85^{34}, which are calculated based on the overestimated bandgap of cphase InN. However, because the LDA1/2 approach fully recovers the deeplybend Γvalley contributed from cphase InN, a higher content of cphase AlN is required to lift up the Γvalley and drag down the Xvalley. Ultimately, the cphase AlGaN and AlInN turn into indirectgap alloys as the Al mole fraction goes beyond 0.700 and 0.922, which correspond to the bandgap energies of 4.585 and 4.792 eV, respectively.
Figure 4(a,b) plot the lattice constants (white dashed lines) and bandgaps (color contour and black solid lines) of h and cphase Al_{x}Ga_{y}In_{1−x−y}N, respectively. Lattice Vegard’s law for quaternary alloys is used to have a numerical expression for the lattice constants of quaternary alloys. The lattice bowing parameters of h (and c) phase AlGaInN are extracted to be −0.082 (0.184) Å with R^{2} values of 0.998 (0.998). The lattice constants of h and cphase Al_{x}Ga_{y}In_{1−x−y}N are summarized in the unit of Å by the numerical expressions tabulated in Table 3. It worth mentioning that the h (and c) phase quaternary alloys obeying y = −1.325x + 1 (y = −1.350x + 1) are lattice matched to GaN.
The bandgaps of hphase quaternary alloys are 19.77% (AlN) and 22.58% (InN) larger than their cphase counterparts. Similarly, bandgap Vegard’s law is applied for the numerical expression of quaternary alloy bandgap. The bandgap bowing parameters of h and cphase AlGaInN are extracted to be 1.236 and 2.406 eV with R^{2} values of 0.968 and 0.989, respectively, where the numerical expressions of the bandgaps in the unit of eV are tabulated in Table 3 for h and cphase AlGaInN. Notably, the bandgap boundary (black dashed line) between the direct and indirect cphase Al_{x}Ga_{y}In_{1−x−y}N alloys obey y = −1.351x + 1.246, where x and y are the Al and Ga mole fractions defined within [0, 1]. Black solid lines highlight the bandgaps of common emitters: red (1.91 eV), green (2.24 eV), blue (2.75 eV), and UV (4.43 eV). For engineering applications, the h (and c) phase quaternary alloys obeying y = −1.829x + 0.678 (y = −1.752x + 0.726) have the bandgap of 2.24 eV that is promising for green emitters; while, y = −1.587x + 0.838 (y = −1.555x + 0.867) guarantees the bandgap of 2.75 eV for blue emitters. For UV emitters, h (and c) phase quaternary alloys obeying y = −1.550x + 1.297 (y = −2.407x^{2} + 1.947x + 0.095) satisfies the bandgap of 4.43 eV, where x is defined within [0, 1] ([0.624, 0.855]). The mole fractions for green, blue, and UV emitters can be further optimized by minimizing the lattice mismatch. As the results, h (and c) phase In_{0.322}Ga_{0.678}N (In_{0.274}Ga_{0.726}N), In_{0.162}Ga_{0.838}N (In_{0.133}Ga_{0.867}N), and Al_{0.837}In_{0.163}N (Al_{0.855}In_{0.145}N) are the best candidates for green, blue, and UV emitters in terms of the matched bandgap and have the least latticemismatch with GaN, respectively.
Figure 5(a–d) plots the anisotropic electron effective masses of h and cphase AlGaInN. \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) are in the unit of free electron mass (m_{0}). For the hphase AlGaInN, \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\,\,\)increase uniformly with the increasing mole fractions of Al and Ga. By decomposing the density of states of hphase quaternary alloys with respect to electron orbitals, dominant contributions from sorbitals are observed at the conduction band minimum. Since the sorbitals have spherical symmetry, the difference between \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) is insignificant even though the atomic configurations of hphase AlGaInN lack centrosymmetry. Tabulated in Table 3, a similar form of Vegard’s law is exploited to express the electron effective mass numerically in the unit of m_{0} with a bowing parameter of −0.4684 m_{0} and an R^{2} value of 0.990. The \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) of cphase AlGaInN, except for the Alrich quaternary alloys, are identical due to the centrosymmetry of zincblende structure. However, both the \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) lift sharply at the crossover points of the directtoindirect bandgap transition because the conduction minimum shifts from the Γvalley to the Xvalley at high Al mole fraction, where Xvalley has a smaller curvature. At Xvalley, the \({m}_{l}^{\ast }\) is significantly greater than the \({m}_{t}^{\ast }\). The drastic difference between \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) is ascribed to the anisotropic nature of porbitals that dominantly contribute to Xvalley, as confirmed by the partial density of states. The electron effective masses of c and hphase alloys at Γvalley are similar. However, with the increasing Ga mole fraction, the electron effective masses of the cphase alloys become increasingly smaller than those of the hphase alloys. As a result, although the electron effective mass of cphase InN is 68% larger than that of hphase InN, the electron effective mass of cphase GaN is 9.3% smaller than that of hphase GaN, which evidences an additional advantage of cphase GaN in electronic transport.
Conclusion
In conclusion, lattice constants, bandgaps, and electron effective masses of binary, ternary, and quaternary IIInitrides have been investigated using the accurate and unified LDA1/2 approach. Cphase IIInitride alloys have smaller bandgaps and electron effective masses than hphase nitrides. The lattice constants and bandgaps of h (and c) phase AlGaInN are shown to follow Vegard’s law with the bowing parameters of −0.082 (0.184) Å and 1.236 (2.406) eV, where the corresponding R^{2} values are 0.998 (0.998) and 0.968 (0.989), respectively. Directindirect bandgap crossing points in cphase AlGaN and AlInN are identified at Al mole fractions of 0.700 and 0.922, respectively. Both the \({m}_{t}^{\ast }\) and \({m}_{l}^{\ast }\) of hphase AlGaInN are expressed numerically with the bowing parameter of −0.4684 m_{0}, where the R^{2} value is 0.990. The electron effective masses of cphase AlGaInN is found to resemble its hphase counterparts, except for the Alrich region because the conduction band minimum is shifted from Γvalley to Xvalley, where Xvalley has a heavier electron effective mass. Cphase IIInitrides benefit polarizationfree nature making it promising materials for green, blue, and UV emitters. Specifically, h (and c) phase In_{0.322}Ga_{0.678}N (In_{0.274}Ga_{0.726}N), In_{0.162}Ga_{0.838}N (In_{0.133}Ga_{0.867}N), and Al_{0.837}In_{0.163}N (Al_{0.855}In_{0.145}N) alloys having the least latticemismatch with GaN substrates are shown to be promising active layer materials for green, blue, and UV emitters, respectively.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work is supported by the National Science Foundation Faculty Early Career Development (CAREER) Program under award number NSFECCS1652871. The authors acknowledge the computational resources allocated by Extreme Science and Engineering Discovery Environment (XSEDE) with No. TGDMR180050 and TGDMR180075.
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C. Bayram conceived the idea and supervised the research. Y.C. Tsai carried out the calculations, performed the analysis, and wrote the manuscript. Both authors participated in the discussions of the results and commented on the manuscript.
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Tsai, YC., Bayram, C. Structural and Electronic Properties of Hexagonal and Cubic Phase AlGaInN Alloys Investigated Using First Principles Calculations. Sci Rep 9, 6583 (2019). https://doi.org/10.1038/s4159801943113w
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