Ab initio study of InxGa(1-x)N - performance of the alchemical mixing approximation
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Ab initio study of
I n x Ga − x N - performance of thealchemical mixing approximation. P. Scharoch a , M.J. Winiarski b , M.P. Polak a a Institute of Physics, Wroc law University of Technology, Wybrzeze Wyspianskiego 27,50-370 Wroc law, Poland b Institute of Low Temperature and Structure Research, Polish Academy of Sciences,Ok´olna 2, 50-422 Wroc law, Poland
Abstract
The alchemical mixing approximation which is the ab initio pseudopotentialspecific implementation of the virtual crystal approximation (VCA), offeredin the ABINIT package, has been employed to study the wurtzite (WZ)and zinc blende (ZB) In x Ga − x N alloy from first principles. The investi-gations were focused on structural properties (the equilibrium geometries),elastic properties (elastic constants and their pressure derivatives), and on theband-gap. Owing to the ABINIT functionality of calculating the Hellmann-Feynmann stresses, the elastic constants have been evaluated directly fromthe strain-stress relation. Values of all the quantities calculated for par-ent InN and
GaN have been compared with the literature data and thenevaluated as functions of composition x on a dense, 0 .
05 step, grid. Someresults have been obtained which, to authors’ knowledge, have not yet beenreported in the literature, like composition dependent elastic constants in ZBstructures or composition dependent pressure derivatives of elastic constants.The band-gap has been calculated within the MBJLDA approximation. Ad-ditionally, the band-gaps for pure
InN and
GaN have been calculated with
Preprint submitted to Computational Materials Science August 4, 2014 he Wien2k code, for comparison purposes. The evaluated quantities havebeen compared with the available literature reporting supercell-based ab ini-tio calculations and on that basis conclusions concerning the performance ofthe alchemical mixing approach have been drawn. An overall agreement ofthe results with the literature data is satisfactory. A small deviation fromlinearity of the lattice parameters and some elastic constants has been foundto be due to the lack of the local relaxation of the structure in the VCA. Thebig bowing of the band-gap, characteristic of the clustered structure, is alsomainly due to the lack of the local relaxation in the VCA. The method, whenapplied with caution, may serve as supplementary tool to other approachesin ab initio studies of alloy systems.
Keywords: semiconductor alloys, ab initio, virtual crystal approximation,elastic constants, band-gap bowing
1. Introduction
Semiconducting group III metal nitrides have been drawing an interestover the last decades because of their potential applications in optoelectron-ics. The direct band gaps starting from 0.65-0.69eV [1, 2, 3] for
InN , through3.50-3.51eV [4, 5] for
GaN , up to 6.1eV for
AlN [5], together with theirability to form ternary and quaternary alloys, open an interesting perspec-tive for tuning the band gap which is of crucial importance for optoelec-tronic applications. The structural and electronic properties of nitride alloyshave already been intensively studied, both experimentally and theoretically.The idea of tuning the band-gap, although simple in principle, is connectedwith a variety of practical problems like lattice constants mismatch of parent2ompounds, thermodynamically determined phase segregation, the effect ofband-gab bowing, the efficiency of radiative transitions etc. The ab initio in-vestigations are of particular importance in the field owing to their predictivepower. They provide a hint in which direction, technologically and experi-mentally, to proceed. A lot of ab initio works have already been reported,dealing with the structural, elastic, thermodynamical and electronic proper-ties, including bulk systems, thin layers, and interfaces. A popular, supercell(SC) approach in which an alloy is modeled by periodically repeated largecell containing a few primitive cells offers an opportunity to vary composi-tions and ionic configurations. For example in wurtzite (WZ) structure a32-atoms cell (8 primitive cells) contains 16 nitrogen and 16 group III metalatoms which for ternary alloy gives 16 possible compositions and a number ofconfigurations at each [6]. In ZB structure and 8 primitive cells in a supercellthis number is respectively reduced. A great advantage of the supercell ap-proach is the possibility to study the effect of various atomic configurationson physical properties, in particular the extreme cases of the clustered andthe uniform one. However, the configurational space is still significantly lim-ited by the supercell size which, if too big, leads to unrealistic computationtime. For this reason for example, studying the alloy thermodynamics fromfirst principles becomes a challenging task requiring various approximations[7, 8, 9]. Moreover, a simulated alloy is never a random alloy, i.e. the mi-croscopic configuration of atoms in a supercell is periodically repeated whichhas an effect on the electronic structure [10, 11].In this paper we employ an approach which is called the alchemical mix-ing approximation, following the nomenclature introduced by the authors of3he ABINIT package [12, 13]. This is the modern, ab initio pseudopotentialbased, implementation of the old idea of the virtual crystal approximation(VCA) whose main advantage is that the alloy can be studied within a primi-tive cell, which significantly reduces the computational costs. In the cell, at ametal site, the norm-conserving pseudopotential which is constructed of twopseudopotentials and which represents the scattering properties of two metalatoms entering the alloy is placed, at a given proportion. Thus, the com-position becomes a continuous (not a discrete, like in supercells) parameter.One of important shortcomings of the approximation is that the ”alchemical”atom is always on the ideal position, which means that the lattice distortioncaused by different sizes of atoms, very characteristic of alloy systems, is notrepresented here, and which is (as we discuss later) the main reason of thedeviation of the results from those obtained within the SC approach. Also,studying the thermodynamics is not possible within the approximation sincethe lattice dynamics would be very poorly represented (the ”alchemical”atom would have to have an unphysical intermediate mass). The aim of thiswork was to study the performance of the approximation in various applica-tions, to find its strong or weak points and possible reasons of deficiencies,believing that when applied with caution can provide a useful reference forexperiment and for the other ab initio studies. We have concentrated on thestructural, elastic and electronic properties. The ground state calculationsgave us the opportunity to evaluate the LDA band-gap within the MBJLDAapproximation [14]. An overall agreement of the results with the literaturedata has appeared very satisfactory. A small deviation from linearity of thelattice parameters and some elastic constants, showing an intermediate be-4avior between the clustered and the uniform structure of the alloy, has beenfound to be due to the lack of the local relaxation of the structure in theVCA. The apparent big bowing of the band-gap, characteristic of the clus-tered structure, points at certain inconsistency in the behavior of the VCA,which is supposed to simulate rather a perfectly uniform medium. An argu-mentation is given according to which this effect is also mainly due to thelack of the local relaxation in the VCA.
2. Computational methods
The alchemical mixing of pseudopotentials implemented in the ABINITpackage has been employed to emulate the In x Ga − x N alloy. The proto-type of the idea is the virtual-crystal approximation (VCA), used to describemixed crystals within empirical potential approach. In the approximation themain idea is to introduce an object (an ion, scattering center) whose prop-erties would reflect the properties of two atoms simultaneously, at a givenproportion. The VCA is simply a linear combination of two one-electronpotentials describing pure crystals. The alchemical mixing of pseudopoten-tials implemented in the ABINIT package uses the following construction[15]: the local potentials are mixed in the proportion given by mixing coeffi-cients, the form factors of the non-local projectors are all preserved, and allconsidered to generate the alchemical potential, the scalar coefficients of thenon-local projectors are multiplied by the proportion of the correspondingtype of atom, the characteristic radius of the core charge is a linear combina-tion of the characteristic radii of the core charges, the core charge function f ( r/rc ) is a linear combination of the core charge functions. In all the lin-5ar combinations the mixing coefficients reflecting the proportion at whichparticular atoms enter the alloy are used. Norm conserving pseudopotentialswith the same valence electronic configuration must be used, like e.g. In andGa. It would be impossible then to emulate e.g. the In x Al − x N with In d -electrons included.The norm conserving pseudopotentials have been generated with theOPIUM package [16]. The Perdew-Zunger form [17] of the local densityapproximation (LDA) for the exchange-correlation functional was employedin the scalar relativistic mode. The cut-off radii: 2 .
0, 1 .
8, and 1 . Bohr wereselected respectively for In (4 d : 5 s : 5 p ), Ga (3 d : 4 s : 4 p ), and N (2 s : 2 p )pseudo-orbitals. The non-linear core-valence correction (NLCV) radii [18]were: 1 .
0, 0 . . Bohr , for In, Ga, and N, respectively. Psedopotentialswere optimized with the Rappe-Rabe-Kaxiras-Joannopoulos method [19].All the calculations have been performed with the ABINIT package [12,13]. The total energy values were converged with the accuracy ≈ meV on the 8 × × ǫ, , , , ,
0) and a shear strain (0 , , , ǫ, ,
0) (in the Voight,vector notation). For WZ structure one more tensile strain (0 , , ǫ, , , ǫ , in the range ( ± . , ± . ǫ the ions have been re-laxed to their equilibrium positions. The values of the stress tensor from theground state calculations (Hellmann-Feynman stresses) were used to calcu-late the C ′ ij ( ǫ ) constants from the stress-strain relation. Obtained in thatway ǫ dependent C ′ ij values have been extrapolated to ǫ = 0 giving the elas-tic constants at equilibrium state C ij , corresponding to infinitesimal strains.The pressure derivatives of elastic constants have been calculated as direc-tional coefficients of straight lines fitted to 3-points. The values of elasticconstants, necessary for that purpose, have been evaluated at 3 hydrostaticpressures (not exceeding 5 GPa) in the same way as described above, ex-cept the preliminary ground state calculations were performed to find thereference states of a crystal at given pressure targets.The related quantities, like the bulk modulus and Poisson’s ratio havebeen calculated within the Voight-Reus-Hill approximation [21] (accordingto [22]). First, the Reuss (lower) [23] and Voight (upper) [24] bounds, for thebulk ( B ) and for the shear ( G ) modulus have been evaluated, correspondingto policrystalline values at uniform stress and uniform strain respectively.Thus, for the cubic phase we have: B V = B R = ( C + 2 C ) / G V = ( C − C + 3 C ) / G R = 5( C − C ) C / [4 C + 3( C − C )] (1)and for the hexagonal phase: 7 V = (1 / C + C ) + 4 C + C ] G V = (1 / M + 12 C + 12 C ) B R = C /MG R = (5 / C C C ] / [3 B V C C + C ( C + C )] M = C + C + 2 C − C C = ( C + C ) C − C (2)Then the Young modulus ( E ) and the Poisson’s ratio ( ν ) have been calcu-lated from the average values of B and G, M H = (1 / M R + M V ) , M = B, G (Voight-Reus-Hill approximation): E = 9 BG/ (3 B + G ) , ν = (1 / B − G ) / (3 B + G ) (3)Additionally, the ratio of shear modulus to bulk modulus B/G has beencalculated to estimate the brittle or ductile behavior of the material. A highB/G ratio is associated with ductility, whereas a low value corresponds to thebrittle nature. The critical value which separates ductile and brittle materialis 1 .
75. If
B/G > .
75, the material tends to be ductile, otherwise, it behavesin a brittle manner [25].The biaxial relaxation coefficients have been calculated from the formulae: R W Zc = 2 C /C for WZ and R ZBc = 2 C /C for ZB structure.The LDA band-gap as a function of composition has been calculatedwithin the MBJLDA approximation [14]. The C m parameter for the par-ent compounds has been fitted so that the values of band-gap it producedmatched the experimental ones from []. It was then interpolated linearly tobecome a function of composition x . 8 . Results and discussion The ab inito values of equilibrium lattice parameters a and c/a ratio andinternal parameter u for parent GaN and
InN compounds are presented inTab.1. The quality of used pseudopotentials is confirmed by a good agree-ment with former results both experimental and theoretical. Fig.1 shows thecomposition dependence of the lattice constants of WZ and ZB structureswhich agree very well with independent supercell-based ab initio calcula-tions [6]. In Fig.1 it can be seen that the alloy lattice parameters for ZB andWZ structures change nearly linearly with the indium content x , although asmall deviation from linearity can be observed, especially for the c parame-ter. The results reported in [6] show that the linear composition dependenceof the lattice constants (obeying Vegard’s law) is characteristic of the uni-form configuration of indium whereas a small deviation from linearity of c parameter appears in clustered configuration (Fig.1 in [6]). The effect canbe explained by the fact that when InN component is gathered in clustersthen it tends to keep its original lattice constant which is higher than thatof
GaN . Similarly, in the alchemical mixing approximation the atoms stayat their ideal lattice positions (do not relax), and the ”rigid“ contribution ofindium pseudopotential results in the bowing characteristic of the clusteredcase. Similar effect has been observed in
AlN − x P x [26] and BN − x P x alloys[27]. According to our experience the effect of bowing in the VCA can beartificially suppressed by setting small orbital (hard) but big NLCC (soft)cut-off radii in the construction of pseudopotentials.In Tab.2 the values of elastic constants calculated in this work for parent GaN and
InN are compared with the literature data, both theoretical and9xperimental. One can see that the present results fit well into the rangesof values reported earlier. One exception are the ZB C constants whosevalues (210 GP a for
GaN and 141
GP a for
InN ) are significantly larger thanother reported values (by more than 30%). The problem has already ap-peared and has been discussed in the literature, namely, similar large values(respectively 206 and 177
GP a ) have been reported in [28] and then correctedin [29] to 142 and 79
GP a . According to discussion in [29] the discrepancy isdue to simultaneous effect of semicore Ga
3d and In
4d states and high-lyingconduction-band like Ga
4d and In
5d states on the valence band, whichwhen poorly represented lead to high values of ZB C constants. In thepseudopotential approximation used in this work, although the semicore Ga
3d and In
4d states are included, the high-lying conduction-band states arenot represented sufficiently well. In Figs.2 and 3 the composition dependentelastic constants, bulk modulus, shear modulus and Young modulus are pre-sented for ZB and WZ structures respectively. The results for WZ structurecan be directly compared with the supercell based calculations reported in[30] and an excellent agreement can be observed. In the work [30] a dis-tinction has been made between the case of the uniform and the clusteredconfiguration of In in In x Ga − x N . The particular elastic constants showeither linear (Vegard’s law) or sublinear behavior with composition depend-ing on the indium configuration. We find our results to correspond neitherto clustered nor to uniform case, although they are close to both, i.e. theyrepresent an intermediate (or mixed) state. To authors knowledge, thereare no data for the composition dependent elastic constants in ZB structure(Fig.2) reported in the literature. In that case a small bowing (deviation10rom Vegard’s law) is characteristic of all the dependencies.There are rather few works reporting pressure derivatives of elastic con-stants. Some reference data for parent GaN and
InN are gathered in Tab.3,to compare them with the results of this work. The agreement of most ofpresent results with the literature data is satisfactory, although some values( dC /dP , dC /dP ) are higher (by 10 − dC /dP which might be due to the reasons discussed in theprevious paragraph. The original result of this work seem to be the com-position dependent pressure derivatives of elastic constants, bulk modulus,Young modulus and shear modulus for ZB and WZ structures which areshown in Figs.4 and 5. In Fig.4 points represent the ab initio data, whereasin Fig.5, to make the graph clearer, the second order polynomials have beenfitted to ab initio results with the standard deviation not exceeding 0 .
3. Thebowing (deviations from Vegard’s) of majority of the quantities is rathersmall, except for dC /dP in WZ structure exhibiting an anomalously largebowing (a maximum of the pressure derivatives appears at x = 3 . GP a ) its value isbetween 0 . R c andPoisson’s ratio ν are presented for ZB and WZ structures. The results canbe compared with those reported in [33], where the parameters have been11alculated for WZ structure ab initio within the supercell approach and theeffect of In distribution investigated. An excellent agreement of our resultswith those corresponding to the uniform distribution of In, for both param-eters can be seen. Some reference values for parent GaN and
InN in WZstructure can be found in [34] and [35] where the reported (calculated) valuesof the R c are respectively for GaN: 0 . , .
509 and for InN: 0 . , . B/G ratio is shown. Except for small range of x ∈ (0 . , .
2) inZB structure the material shows ductile character, according to the criteriumpresented in the previous chapter.Finally, in Tab.4 data for the band-gaps of parent
GaN and
InN ob-tained in this work and reported in the literature are collected. The valuesin this work has been obtained within MBJLDA approximation [14], withthe use of ABINIT and Wien2k codes. The values obtained with ABINITcoincide with experimental ones owing to appropriate fitting of the C m pa-rameter mentioned earlier. Its values are: InN (ZB) 1 .
505 ( Eg = 0 . eV ), InN (WZ) 1 .
36 ( Eg = 0 . eV ), GaN (ZB) 1 .
67 ( Eg = 3 . eV ), GaN (WZ)1 .
63 ( Eg = 3 . eV ). For the alloy, its values are obtained form the linearinterpolation. In Fig.8 the In x Ga − x N band-gap vs. composition is plotted,calculated within VCA. For comparison, the values of Eg from SC calcula-tions, for x = 0 .
25 and x = 0 .
75 and two In configurations (clustered anduniform), are given. The SC values of the band-gaps are in good agree-ment with the values obtained in [6] and with experiment. However, theband-gap bowing obtained within VCA is bigger than even that in the SCclustered configuration, which somehow disagrees with expectation that the12CA should rather simulate a perfectly uniform alloy. For comparison, thecalculations reported in [36], based on LMTO-CPA-MBJ for WZ In x Ga − x N show smaller bowing of the band-gap (corresponding rather to the uniformconfiguration of In [6]) than in the present work. Bellow, an argumentationaccording to which the anomalous bowing in VCA can be attributed to thefact that the ”alchemical” atoms are always in ideal (unrelaxed) positions andthus the distance between metal and nitrogen atoms is averaged, is given.
4. An anomalous In x Ga − x N band-gap bowing in alchemical mix-ing approximation The admixture of
InN in GaN leads to a lowering of the band gap fortwo reasons: first,
InN has in nature much lower band-gap than
GaN , andsecond, the involved expansion of the lattice constants leads to a loweringof the band gap in
GaN . The latter effect appears also in
InN and isresponsible for the difference in the band-gap between the uniform and theclustered configuration of In [6], i.e. the bigger bowing for the clusteredcase is due to the locally expanded bonds between In and N atoms in the InN cluster region. Thus, the band-gap appears to be very sensitive tobond lengths. As mentioned above, in the VCA the bond lengths betweenthe metal and the nitrogen atom are averaged, i.e. they are the same forpartially contributing In and Ga. For example, at x = 0 .
25, in SC uniformcase the distances are: Ga - N . A , In - N . A , whereas in VCA, metal- N .
01. The differences do not exceed 5% but as will be shown below they arecrucial for the band-gap behavior.In Fig.9 the effect of In doping in GaN is presented in terms of the total13ensity of states. The large CB negative offset (the electron trapping case)appears to be the same in the VCA and the SC approaches, however, theVB offsets are very different. Thus, the direct reason for the large band-gapbowing in VCA is the wrong behavior of the VB with doping and can beunderstood by analyzing the DOSes projected on the angular momentumeigenstates of chosen atoms (partial DOSes). From Fig.10, which shows anear the band-gap fragment of the projected DOS for
GaN , it is clear thatthe main contribution to the bottom of CB comes from the metal s -stateand the N s -state, whereas the top of the VB is formed mainly by the
N p -states and the metal p , d -states. The situation is the same in alloy simulatedeither within the VCA or SC. Since the reason for the deep bowing in VCAis the VB behavior we will concentrate on that region now. In Fig.11 theprojected DOSes are plotted in the region of the VB for three cases: 1. VCA,relaxed lattice, 2. VCA, lattice compressed by 5%, 3. SC, with the uniform In distribution. A large shift of the VB towards the correct SC positioncan be seen for the compressed VCA lattice, which confirms the fact of highsensitivity of the bond lengths on the band-gap and the presented abovehypothesis of the bond length effect on the VCA band-gap.It should be mentioned that the VCA band-gap behavior in alloy systemscan be different in different systems. For example, in AlN x P − x the tendencyis opposite, i.e. the VCA shows smaller bowing than in SC based calculations[26]. An analysis similar to that presented in this work should be done toexplain this fact. It’s worth to add that some purely technical procedure,based on averaging over the transition energies near the transition point, canbe applied within VCA approach to obtain the correct composition dependent14and-gap of InGaN alloy, as we have shown in [37]. Thus, in spite of thediscussed difficulties, the alchemical pseudopotential method can be used forband-gap calculations.
5. Conclusions
The main objective of this work was to test the performance of the al-chemical mixing of pseudopotentials approximation in theoretical ab initio studies of structural, elastic and electronic properties of semiconductor al-loys, on the example of In x Ga − x N . We find the results for various calculatedparameters to be in an overall good or very good agreement with other abinitio calculations, performed within the supercell approach, and with theexperimental values. The behavior of the lattice parameters and the elasticconstants together with the related quantities as a function of compositionappears to be intermediate between the uniform and clustered structure ofthe alloy, whereas the band-gap would rather behave like the alloy with clus-tered structure, which is, as discussed above, an artefact connected mainlywith the VCA inherent feature of the lack of the local relaxation of atomicpositions. This seems to be also the main reason for the discrepancy (al-though rather small) between other VCA and SC results. As an additionalresult of this work, some composition dependent quantities, such as compo-sition dependent elastic constants and related quantities in ZB structures, orcomposition dependent pressure derivatives of elastic constants, have beencalculated. Their values, to authors’ knowledge, have not been reported pre-viously. The obtained results lead to a conclusion that the ab initio alchemi-cal mixing approximation, if used with caution, can serve as a supplementary15 l a tt i c e c on s t an t ( Å ) x ZB "a"WZ "a"WZ "c"
Figure 1: Equilibrium lattice constants for In x Ga − x N alloy; calculated values are markedwith points tool for semiconductor alloy studies.
6. Acknowledgments
Calculations have been carried out in Wroclaw Centre for Networking andSupercomputing.
ReferencesReferences [1] V. Y.Davydov, Phys. Stat. Sol. B , R1 (2002).16 able 1: Lattice parameters of GaN and InN; a in WZ and ZB structures; the c/a ratioand internal parameter u in WZ structure. a (˚A) c/a u GaN-wz this work 3.17 1.632 0.375other calc. 3.16 d ,3.17 c d ,1.628 c cd exp. 3.19 a a a InN-wz this work 3.53 1.615 0.377other calc. 3.50 d ,3.52 gc c ,1.614 g ,1.619 d d ,0.380 gc exp. 3.53 f f -GaN-zb this work 4.49 - -other calc. 4.461 j ,4.46 k ,4.46 m - -exp. 4.5 l - -InN-zb this work 4.97 -other calc. 4.932 d ,4.95 m - -exp. 4.98 l - - a Ref. [38]. b Ref. [39]. c Ref. [40]. d Ref. [41]. e Ref. [42]. f Ref. [43]. g Ref. [44]. h Ref. [45]. i Ref. [46]. j Ref. [47]. k Ref. [34]. l Ref. [48]. m Ref. [49]. able 2: Elastic constants and bulk modulus of GaN and InN in WZ and ZB structures(in GPa).System Data from C C C C C C B this work 290 169 - - 210 - 209other calc. 293 a , 282 b a , 159 b - - 155 a , 142 b - 184 n ,197.88 o GaN-zb 305 f ,264 g f ,153 g - - 147 f ,68 g -exp. - - - - - - 237 n ,245 n ,195 n this work 188 134 - - 141 - 152other calc. 187 a , 182 b a , 125 b - - 86 a , 79 b - 137 n InN-zb 217 f ,172 g f ,119 g - - 104 f ,37 g -exp. - - - - - - 125.5 n this work 364 150 111 412 90 107 210other calc. 367 a ,346 b a ,148 b a ,105 b a ,389 b a ,76 b f a ,210 m GaN-wz 357 f ,337 h f ,113 h f ,97 h f ,353 h f ,95 h exp. 390 c ,374 d c ,106 d c ,70 d c ,379 d c ,101 d c ,180 d i ,390 j i ,145 j i ,106 j i ,398 j , 81 i ,105 j this work 231 124 106 242 46 54 154other calc. 223 a ,220 b a ,120 b a ,91 b a ,249 b a ,36 b f a ,152 l InN-wz 257 f ,211 h f ,95 h f ,86 h f ,220 h f ,48 h exp. 190 e ,223 j e ,115 j e ,92 j e ,224 j e ,48 j e ,126 ka Ref. [35]. b Ref. [29]. c Ref. [50]. d Ref. [51]. e Ref. [52]. f Ref. [53]. g Ref. [54]. h Ref. [55]. i Ref. [56]. j recommended values Ref. [57]. k Ref. [44]. l Ref. [40]. m Ref. [42]. n according to Ref. [49]. o Ref. [58]. able 3: Pressure derivatives of elastic constants and bulk moduli of GaN and InN in WZand ZB structures.System Data from dC /dP dC /dP dC /dP dC /dP dC /dP dC /dP dB/dP GaN-zb this work 4.3 4.8 0 0 3.4 0 4.6other calc. 3.88 c ,3.64 d c ,4.87 d c ,-0.55 d c ,4.32 d InN-zb this work 4.2 5.2 0 0 3.4 0 4.9other calc. 3.81 c c c c GaN-wz this work 4.5 4.4 4.4 5.4 0.024 0.22 4.5other calc. 3.74 a ,4.54 b b b a ,5.4 b a ,0.49 b a c c c c c c InN-wz this work 3.6 5.2 5.2 3.9 0.36 0.72 4.7other calc. 3.86 a ,3.88 b b b a ,3.69 b a ,0.1 b -0.08 a c c c c c ca Ref. [31]. b Ref. [32]. c Ref. [59]. d Ref. [47]. able 4: The calculated band gaps in comparison with experimental values and othertheoretical results for GaN and InN in WZ and ZB structures (all values in eV). In thiswork the calculations have been done within MBJLDA [14], with the use of Abinit andWien2k codes. In Abinit the C M parameter has been fitted to give experimental values.System Exp. Abinit Wien2k other calc.GaN-zb 3.30 j l , 3.06 l m ,3.03 n InN-zb 0.78 j l , 0.63 l GaN-wz 3.50 f ,3.51 j a ,3.47 h ,3.50 i k ,3.26 l l , 3.21 n InN-wz 0.65 c ,0.63 d b ,0.69 a ,0.65 g e k ,0.74 l l , 0.71 na Reference [6] b Reference [44] c Reference [1] d Reference [2] e Reference [3] f Reference [4] g Reference [60] h Reference [61] i Reference [62] j Reference [5] k Reference [63] l Reference [36] m Reference [14]
50 100 150 200 250 300 350 0 0.2 0.4 0.6 0.8 1 C ij ( G P a ) x C11C12C44BGE
Figure 2:
Ab initio elastic constants ( C ij ), bulk modulus ( B ), shear modulus ( G ) andYoung modulus ( E ) of In x Ga − x N alloy (ZB structure) C ij ( G P a ) x C11C12C13C33C44C66BGE
Figure 3: Elastic constants ( C ij ), bulk modulus ( B ), shear modulus ( G ) and Young mod-ulus ( E ) of In x Ga − x N alloy (WZ structure) d C ij / d P x dC11/dPdC12/dPdC44/dPdB/dPdG/dPdE/dP Figure 4: Pressure derivatives of elastic constants ( C ij ), bulk modulus ( B ), shear modulus( G ) and Young modulus ( E ) of In x Ga − x N alloy (ZB structure) d C ij / d P x dC11/dPdC12/dPdC13/dPdC33/dPdC44/dPdC66/dPdB/dPdG/dPdE/dP Figure 5: Pressure derivatives of elastic constants ( C ij ), bulk modulus ( B ), shear modulus( G ) and Young modulus ( E ) of In x Ga − x N alloy (WZ structure); curves fitted to ab initio data with standard deviation not exceeding 0.3. x ZB "poisson’s ratio"WZ "poisson’s ratio"ZB "Rc"WZ "Rc"
Figure 6: Poisson’s ratio and biaxial relaxation coefficient of In x Ga − x N alloy (ZB andWZ structures) ZB "B/G"WZ "B/G"
Figure 7: The
B/G of In x Ga − x N alloy (ZB and WZ structures). band gap ( e V ) x ZB VCAWZ VCAWZ SC clusteredWZ SC uniform Figure 8: The band gap of In x Ga − x N alloy (ZB and WZ structures) calculated withABINIT (MBJLDA); the results of supercell calculations (WZ) are shown for comparison. t o t a l D O S E (eV) GaNIn0.25Ga0.75N VCAIn0.25Ga0.75N SC clusteredIn0.25Ga0.75N SC uniform
Figure 9: The effect of In doping ( x = 0 .
25) in
GaN on the total DOS (the 32-atomssupercell DOS has been normalized to an elementary cell). pa r t i a l D O S E (eV)Ga sGa pGa dN sN p
Figure 10: Near the band-gap fragment of the partial DOSes for
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Figure 11: Partial DOSes near the VB for In x Ga − x N at x = 0 .
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