Ab initio study of structural parameters and gap ...

Ab initio study of structural parameters and gap bowing in zinc-blende Al x Ga 1 − x N and Al x In 1 − x N alloys M. B. Kanoun, S. Goumri-Said, A. E. Merad, and H. Mariette Citation: Journal of Applied Physics 98, 063710 (2005); doi: 10.1063/1.2060931 View online: http://dx.doi.org/10.1063/1.2060931 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/98/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spontaneous polarization and band gap bowing in Y x Al y Ga1- x - y N alloys lattice-matched to GaN J. Appl. Phys. 110, 074114 (2011); 10.1063/1.3651154 Ab initio study of lattice vibration and polaron properties in zinc-blende Al x Ga 1 − x N alloys J. Appl. Phys. 108, 113710 (2010); 10.1063/1.3517065 Ab initio calculations of structural properties of Sc x Ga 1 − x N J. Appl. Phys. 103, 063510 (2008); 10.1063/1.2884580 Lattice parameter and energy band gap of cubic Al x Ga y In 1−x−y N quaternary alloys Appl. Phys. Lett. 83, 890 (2003); 10.1063/1.1597986 Phase diagram, chemical bonds, and gap bowing of cubic In x Al 1−x N alloys: Ab initio calculations J. Appl. Phys. 92, 7109 (2002); 10.1063/1.1518136

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JOURNAL OF APPLIED PHYSICS 98, 063710 共2005兲

Ab initio study of structural parameters and gap bowing in zinc-blende AlxGa1−xN and AlxIn1−xN alloys M. B. Kanouna兲 Unité de Recherche Matériaux et Energies Renouvelables, Département de Physique, Faculté des Sciences, Université A. Belkaid, Tlemcen, B.P. 119, 13000 Tlemcen, Algeria

S. Goumri-Saidb兲 Condensed Matter Theory Group, Department of Physics, Kaiserslautern University of Technology, P.O. Box 3049, D-6765 Kaiserslautern, Germany

A. E. Merad Unité de Recherche Matériaux et Energies Renouvelables, Département de Physique, Faculté des Sciences, Université A. Belkaid, Tlemcen, B.P. 119, 13000 Tlemcen, Algeria

H. Mariette Commissariat à l’Energie Atomique-Centre National de la Recherche Scientifique (CEA-CNRS) Research Group Nanophysique et Semiconducteurs, Laboratoire de Spectrométrie Physique, Unite Mixté de Recherche (UMR) 5588 CNRS, Université J. Fourier, CEA Département de Recherche Fondamentale sur la Matière Condensée, Service de Phisique des Matériaux Microstructures (DRFMC/SP2M), 17 Avenue des Martyrs, 38054 Grenoble, France

共Received 9 February 2005; accepted 11 August 2005; published online 28 September 2005兲 We present first-principles calculations of the structural and electronic properties of zinc-blende AlxGa1−xN and AlxIn1−xN alloys by application of the all-electron full-potential linearized augmented plane-wave method within density-functional theory and the local-density approximation. When the parameter x varies, both the lattice constant a and the bulk modulus B are found to vary linearly for AlxGa1−xN, while for AlxIn1−xN the lattice parameters show an upward bowing. The calculated band-gap variation for the two alloys varies nonlinearly as a function of composition x, with a strong downward bowing for AlxIn1−xN. © 2005 American Institute of Physics. 关DOI: 10.1063/1.2060931兴 I. INTRODUCTION

Recently the group-III nitrides and their alloys have attracted both scientific and technological interests. Indeed, the nitrides based on AlN, GaN, and InN represent promising wide-band-gap semiconductors both for optoelectronic applications in a short-wavelength range as well as for hightemperature, high-power, and high-frequency electronic devices.1 Bright and highly efficient blue and green lightemitting diodes 共LEDs兲 are already commercially available and diode lasers have been reported, which emit in the blueviolet range initially under pulsed conditions and subsequently under continuous operation.2,3 An interesting feature of the device applications is generally the use of the ternary AlxGa1−xN and AlxIn1−xN.3 Little attention has been paid to AlGaN and AlInN alloys which are very important for the UV emitters, in comparison to InGaN. The InxAl1−xN alloy exhibits a very large band-gap variation and it is expected that the alloy lattice matched to GaN would have a sufficiently large energy gap to serve as an insulating barrier in GaN-based electronic devices. Besides that it could be used as a cladding layer with no strain on the laser diode structure, which may lead to a reduction of defects.4,5 The natural tendency for AlN, GaN, and InN compounds is to crystallize in the wurtzite structure. A stabilization in the a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0021-8979/2005/98共6兲/063710/5/$22.50

zinc-blende structure of the nitride compounds by epitaxial growth of thin films on 共001兲 crystal planes of cubic substrates such as 3C-SiC and GaAs 共Refs. 6–8兲 has been achieved. These progresses in the synthesis of such cubic layers are related to the use of plasma-assisted molecularbeam epitaxy on 3C-SiC共001兲 substrates.7 Many experimental and theoretical studies have been performed on these ternary alloys crystallized in the wurtzite phase.3,9 In this way, there are only few available information about the structural and electronic properties of cubic AlGaN and AlInN alloys.10,11 Moreover, there are still considerable controversies in the literature concerning fundamental parameters such the gap of AlN 共Ref. 12兲 and InN 共Ref. 13兲 binaries and consequently the bowing of alloys gap. In fact, it is not clear how the band gap varies as a function of the alloy composition. Although early optical studies of AlGaN suggested a small energy-gap bowing parameter.10,14 For InAlN, the deviation between different measurements of the band gap is significant. For example, a large band-gap bowing was observed by Yamaguchi et al.,15 a similarly strong bowing was found by Peng et al.,16 while Lukitsch et al. measured a smaller bowing parameter.17 This discrepancy between the different measurement, especially for InAlN, may also be related to the fluctuations in the local aluminum molar fraction and to the possible gap miscibility enduring a spinodal decomposition as well as to the importance of strain effects. The measured values of the band-gap bowing also depend on the band gap of measurement of the

98, 063710-1

© 2005 American Institute of Physics

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J. Appl. Phys. 98, 063710 共2005兲

Kanoun et al. TABLE I. Structural parameters and band gaps of zinc-blende AlN, GaN, and InN. a共Å兲

B共GPa兲

B⬘

E⌫−⌫共eV兲

E⌫−x共eV兲

AlN

This work Calc. Expt.

4.35 4.32–4.42a 4.37b

211.78 202–214a ¯

3.90 3.2–3.3a ¯

4.36 4.3–4.75a ¯

3.23 3.1–3.2a 5.34d

GaN

This work Calc. Expt.

4.461 4.45–4.537a 4.50b

202 200–207a 190c

4.32 3.9–4.6a

1.93 1.28–1.9a 3.25d

3.23 3.1–4.9a

InN

This work Calc. Expt.

4.947 4.92–4.98a 4.98b

144 137–161a 137c

4.56 3.9–4.4a ¯

0.013 0.20–0.16a 0.9e

2.81 2.85–2.87a ¯

a

References 25–32. Reference 33. c Reference 34. d Reference 35. e Reference 36. b

technique. Photoluminescence and absorption techniques give different band-gap values in these ternary alloys, due to carrier localization effects. In order to help understand and control the material and behavior of bowing and related properties, we report in this paper a numerical investigation based on a first-principles study of the structural and electronic properties of zincblende AlGaN and AlInN bulk nitride alloys. As recently reported by three of us for AlN and GaN,18 the study is performed by using a full-potential linearized augmented plane-wave method within the framework of densityfunctional theory and local-density approximation. The aim of this study is to analyze the behavior of the lattice parameters, bulk modulus, and band-gaps as a function of aluminum molar fractions of the AlGaN and AlInN alloys in the zinc-blende structure. II. COMPUTATIONAL DETAILS

In our calculations we distinguish, respectively, Al 共1s2 2s2 2p6兲, Ga 共1s2 2s2 2p6 3s2 3p6兲, In 共1s2 2s2 2p6 3s2 3p63d10 4s2 4p6兲, and N 共1s2兲 as inner-shell electrons from the valence electrons of Al 共3s2 3p1兲, Ga 共3d10 4s2 4p1兲, In 共4d10 5s2 5p1兲, and N 共2s2 2p3兲 shells. Our calculations are based on the recently developed Vienna package WIEN2k 共Ref. 19兲 for the scalar relativistic hybrid full-potential 共linearized兲 augmented plane-wave plus local-orbitals method within density-functional theory 共DFT兲.20 The electrons exchange-correlation energy is described with local-density approximation 共LDA兲 for which we adopt the Ceperley-Alder21 forms and Perdew and Wang22 parametrization. Moreover, we employ the semirelativistic approximation 共i.e., spin-orbit effects are not included兲 and the core levels are treated fully relativistically.23 In particular, the semicore d electrons of the Ga and In are included in the valence bands. The remaining core states are selfconsistently relaxed in the spherical approximation, where inside the nonoverlapping spheres of muffin-tin radius, RMT, around each atom, spherical harmonics expansion is used.

We choose the plane-wave basis set for the remaining space of the unit cell. The maximum value l for the wave-function expansion inside the atomic spheres is taken to lmax = 10. In order to achieve energy eigenvalues convergence, the wave functions in the interstitial region are expanded in plane waves with a cutoff of 共Kmax兲共RMT兲 = 8共Kmax is the maximum modulus for the reciprocal lattice vector兲. In the binary compounds, the RMT is considered to be equal to 1.82, 1.8, 2.05, and 1.62 bohrs for Al, Ga, In, and N, respectively. For ternary alloys, we have chosen muffin-tin radii of 1.78 bohrs for Ga, Al, and In and 1.65 bohrs for N. In order to model AlxGa1−xN and AlxIn1−xN alloys with the different compositions x = 0.0, 0.25, 0.5, 0.75, and 1.0, we consider a supercell with eight atoms and a cubic symmetry. For each configuration, the fundamental physical properties 共total energy and band gap兲 are calculated. The configurationally averaged quantity is computed using the ConollyWilliams approach24 for each given x. The k integration over the Brillouin zone is performed using that used by Monkhorst and Pack.25 A mesh, yielding to 10 k points for the binary compounds and 11 k points for the ternary alloys, is taken in the irreducible wedge of Brillouin zone. The iteration process is repeated until the calculated total energy of the crystal converges to less than 0.1 mRy. The ground-state properties in the pressure-free case are obtained considering the minimum of the total energy with respect to the cell volume. The theoretical equilibrium lattice constant, the bulk modulus at zero pressure, and its derivative are computed from the fit of the total energy to volume curve, using the Murnaghan’s equation of state.26 III. RESULTS AND DISCUSSION

In Table I the structural optimization for zinc-blende AlN, GaN, and InN is reported. They are compared with the results of different theoretical works27–34 as well as the available experimental data.35,36 We observe that all lattice parameters are underestimated compared to the experimental values.35,36 These results agree well with the theoretical in-

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Kanoun et al.

FIG. 1. Composition dependence of the calculated lattice constants of 共a兲 AlGaN and 共b兲 AlInN alloys 共solid squares兲 compared to VCA 共dashed line兲.

vestigations reported in Refs. 27–34. One should notice here that the experimental values of the bulk modulus are somehow uncertain due to the difficulty of growing high-quality single crystals of III-V nitrides. Using these lattice parameters we obtain an indirect band gap of 3.231 eV for AlN, and direct band gaps of 1.93 eV for GaN and −0.013 eV for InN. Compared to the experimental data,37,38 we obtained, as usual, an underestimation of the lattice constants and energy band gaps which is a consequence of the overbinding tendency of the application of DFT and LDA. After that we optimize the lattice parameters for each composition x using the calculated total energy which has to be relaxed in the unit-cell volume. The calculated equilibrium lattice constants are compared 关see Figs. 1共a兲 and 1共b兲兴 with those of the average values, aVCA 关using virtual crystal approximation 共VCA兲 aVCA = xaAC + 共1 − x兲aBC兴. In order to investigate the ternary alloys properties, we need to reproduce the lattice mismatch. Our calculated lattice mismatch between GaN and AlN is about 2.5% and it is of 12% between InN and AlN. These results agree strongly with the theoretical values reported in Ref. 39. In Figs. 1共a兲 and 1共b兲 we display the variation of the calculated equilibrium lattice constants versus Al concentration curves for AlGaN and AlInN, respectively. We observe that the AlxGa1−xN lattice parameter is what the Vegard’s law follow 共i.e., a linear interpolation of the binaries兲. Contrary to AlInN, a slight deviation from Vegard’s law is clearly visible, i.e., an upward bowing. The physical origin of this deviation in the AlInN

J. Appl. Phys. 98, 063710 共2005兲

FIG. 2. Bulk modulus vs concentrations for 共a兲 AlGaN and 共b兲 AlInN alloys.

should be mainly due to the relaxation of the In–N and Al–N bond lengths of RIn−N = 2.14 and RAl−N = 1.88, respectively. Such relaxation is also related to the large mismatch of the lattice constant between InN and AlN. Now we report the bulk modulus evolution as a function of Al concentration in AlGaN and AlInN alloys. This is illustrated in Figs. 2共a兲 and 2共b兲, where the calculated bulk modulus values for AlGaN and AlInN are plotted as a function of the alloy composition x. Our results show that an increase of the aluminum concentration in both ternary AlGaN and AlInN leads to a larger bulk modulus. In fact, we find that the bulk modulus of AlN is 5% and 47% higher than that of GaN and InN. These results suggest the existence of a significant upwards and downwards bowing of the bulk modulus B共x兲 curve for the Al1−xGaxN and Al1−xInxN systems, respectively. The corresponding configurationally averaged values of the isothermal bulk modulus are found to be not linear as a function of the alloy composition and hence a bowing occurs. It nearly holds B共x兲 = 共1 − x兲B关共Ga or In兲N兴 + xB共AlN兲 − bx共1 − x兲 with the bowing b equal to −5 and 19.1 GPa for AlGaN and AlInN, respectively. Consequently the weak variation of the bulk modulus with the alloy composition is an indication that the applied approach to the mixed crystals is reliable. In Figs. 3共a兲 and 3共b兲 we report the variation of the calculated energy band gap as a function of the x concentration of AlxGa1−xN and AlxIn1−xN alloys, respectively. These calculations predict that the band gap varies almost linearly with the composition in zinc-blende AlxGa1−xN. Adopting

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FIG. 3. Composition dependence of the calculated averaged energy band gaps for zinc-blende 共a兲 AlGaN and 共b兲 AlInN alloys 共solid squares兲 compared to VCA 共dashed line兲. The band gaps at both the ⌫ and x points are shown.

the conduction-band energy at x point as plotted in these figures, one can observe the direct-indirect crossover. The bowing parameters calculated for the direct and the indirect transitions of the AlGaN and AlInN alloys are given in Table II compared to other calculations. We observe also a small bowing in AlGaN than in AlInN. Therefore, the bowing for the ⌫-point transition is larger and downward compared to the x-point transition which has a small upward bowing for AlInN. The direct to indirect band-gap transition is predicted TABLE II. Band-gap bowing parameters b 共in eV兲 of AlGaN and AlInN alloys in the zinc-blende structure. This work b

E⌫−⌫

E⌫−x

AlGaN

0.514

–0.189

AlInN

3.851

0.104

a

Reference 43. Reference 42. c Reference 38. d Reference 44. e Reference 35. f Reference 41. b

Other calculations

Expt.

Other calculations

E⌫−⌫

E⌫−x

E⌫−x

1.4e

–0.10,b–0.92,c 0.61d

¯

–0.51f

0.32,a0.53,b –0.40,c0.05d 3.6,a2.53,b1.7f

to occur at x = 0.6 for AlGaN and at x = 0.85 for AlInN, which agree with theoretical values.40–43 The lattice mismatches in AlN/ InN system are large, so are the bowing factors for AlInN. Notice that the bowing parameters increase with chemical and size differences of the constituents. The AlGaN bowing factor value of the energy gap for direct transitions is found to be 0.514 eV, which is in good agreement with the results obtained by Wright and Nelson44 共i.e., 0.53 eV兲, results obtained by applying the ab initio plane-wave pseudopotential calculations. This bowing factor is larger than the one obtained by Van Schilfgaarde et al.45 关i.e., 0.32 eV calculated by linear muffin-tin orbital 共LMTO兲 method with a large supercell兴. With first-principles LMTO calculations Albanesi et al.40 obtained a negative bowing factor of −0.40 eV. This value disagrees with our result and can probably be related to their upward bowing of the energy gap. In the same way Fan et al.46 have found a small bandgap bowing 共about 0.05 eV兲 using the empirical pseudopotential method. Due to the difficulty of AlGaN growth, only few studies about its optical properties are available, which suggests a direct-type absorption edge. The optical results of Martinez-Guerrero et al.37 suggest a strong band-gap bowing for the direct band minimum ⌫ 共with a bowing parameter equal to 1.4 eV兲, a value much larger than the ones reported by Okumura et al.6 and our results. Moreover it was not possible to show 共or not兲 the evidence of the indirect character of the AlN band gap.37 For the cubic AlInN, we find 3.85 eV the bowing parameter of the 共average兲 energy gap of the direct transition. This value is in good agreement with the recent calculations 共i.e., 3.6 eV兲 of Van Schilfgaarde et al.,45 and larger than the value found by Wright and Nelson44 共i.e., 2.53 eV兲 and Teles et al.,43 共1.7 eV兲, using the cluster expansion method within the pseudopotential-plane-wave DFT-LDA approach. Using the pseudopotential-plane-wave LDA calculation of a large 64atom cubic supercell 共including atomic relaxation兲, Ferhat and Bechstedt39 obtained a bowing factor ranging from 4.67 eV 共for x = 0.25兲 to 2.20 eV 共for x = 0.75兲. One can notice here that the experimental values for the wurtzite structure are few and scattered 共bowing varying from 0 to 6.9 eV兲.16,47,48 An explanation for these huge discrepancies found for the bowing may be related to an inaccurate determination of the average composition x by disregarding the strain influence on the lattice constants. In other words the microstructures of the samples are not the same17 as well as the immiscibility of AlN in InN or GaN 共or vice versa兲.15 However, the latter effect has to be excluded outside the miscibility gap. Finally, for cubic AlInN, a composition equal to x = 0.85 is predicted to give an alloy lattice matched to GaN. Such an alloy will have a direct gap of 3.2 eV, large enough that the alloy can be used as an insulating barrier in GaN-based electronic devices. Besides that, we can apply a linear shift in the band gap of the alloy with the aim to obtain the gap energy values of the binary compounds AlN and InN as is done with the experimental data. If we adopt the gap values corresponding to the zinc-blende phases AlN and InN, 5.34 and 0.9 eV,35,38 respectively, and do not consider the gap composition, we obtain a value for the band-gap energy of Al0.85In0.15N about 4.74 eV which is larger than the experi-

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mental value of 3.25 eV for cubic GaN.37 Therefore we conclude that the InAlN alloy lattice matched to GaN is very suitable to be used in the insulating barrier. IV. CONCLUSION

In this paper, we have studied the structural and electronic properties of III-nitride ternary alloys, AlGaN and InAlN in zinc-blende structure using a hybrid all-electron full-potential linearized augmented plane-wave method within LDA and DFT approaches. We found that the AlGaN lattice constants follow closely the Vegard’s law, whereas for Al-InN, a deviation in the lattice constant is found due to the large mismatch. The calculated band-gap bowing for AlGaN agree well with other theoretical works using different approaches. For AlInN, we found that gap variations are very important due to the large internal lattice mismatch. Consequently a definition of an averaged energy gap and a minimum gap with completely different composition dependences can be given. Moreover these different gaps are able to explain the seemingly contradicting experimental findings. It is worth pointing out that although the results presented here for AlGaN and AlInN have been obtained for bulk materials, and surface effects were not taken into consideration, they may certainly be used to guide heterostructure design for future experiments. Moreover, as demonstrated in previously reported works, the theoretical approach adopted here has been able to not only interpret recent experimental data on these nitride alloy layers, but also to predict several results which can be proven experimentally. 1

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