Phase diagram and polarization of stable phases of (Ga 1−x In x ) 2 O 3
PPhase diagram and polarization of stable phases of (Ga − x In x ) O Maria Barbara Maccioni and Vincenzo Fiorentini
Department of Physics, University of Cagliari and CNR-IOM,UOS Cagliari, Cittadella Universitaria, 09042 Monserrato (CA), Italy
Using density-functional ab initio calculations, we provide a revised phase diagram of(Ga − x In x ) O . Three phases –monoclinic, hexagonal, cubic bixbyite– compete for the groundstate. In particular, in the x ∼ β , and bixbyite (thelatter separating into binary components). Over the whole x range, mixing occurs in three discon-nected regions, and non-mixing in two additional distinct regions. We then explore the permanentpolarization of the various phases, finding that none of them is polar at any concentration, despitethe possible symmetry reductions induced by alloying. On the other hand, we find that the ε phaseof Ga O stabilized in recent growth experiments is pyroelectric –i.e. locked in a non-switchablepolarized structure– with ferroelectric-grade polarization and respectable piezoelectric coupling. Wesuggest that this phase could be used profitably to produce high-density electron gases in transistorstructures. PACS numbers: 61.66.Dk,77.22.Ej,81.05.Zx
I. INTRODUCTION
The Ga and In sesquioxides have recently been un-der intense scrutiny as, among others, UV absorbersand trasparent conductors. With a view at exploitingmaterials engineering concepts from other semiconduc-tor systems, a growing body of work is being devotedto the (Ga − x In x ) O alloy. Theoretical studies on itsphase stability and optical properties have been pub-lished recently by at least two groups , but the pic-ture is apparently still far from complete. Recent growthexperiments on the (Ga − x In x ) O alloy in the vicinityof x =0 and x =0.5 have suggested that phases other thanthose so far assumed as ground state may in fact be sta-ble or stabilized by constrained (e.g. epitaxial) growth.In one paper , three competing phases are reported toappear near x ∼ x =0.5; a monoclinic close relative of the β -Ga O structure; and that derived from the bixbyitestructure of In O , mostly in the form of phase-separatedIn and Ga oxides. Another paper reported that the ε phase of Ga O can be obtained at 820 K via epitax-ial growth on GaN, although a bulk phase transition tothat phase from the ground state β phase is not expectedbelow 1500 K .In this work, we report i ) a phase diagram of(Ga − x In x ) O accounting for new findings around x (cid:39) ii ) the polarization of the competing phases,plus the ε -Ga O phase. The results are in a nutshell that a ) the hex and β structures do indeed compete energeti-cally with the bixbyite phase expected based on previousresults, and this competition occurs predominantly in thevicinity of x ∼ b ) none of the alloy phases is polar, butthe ε phase of Ga O is. As dictated by its symmetry,this phase (only slightly energy-disfavored over the sta-ble β phase) has a large spontaneous polarization andsizable piezoelectric coupling. Importantly, it cannot be transformed into (is not symmetry-related to) the stable β phase. These results open up some interesting per-spectives, such as growing the hex phase epitaxially, orexploiting the polarization properties of ε -Ga O . II. METHODS
Geometry and volume optimizations as well as elec-tronic structure calculations are performed using density-functional theory (DFT) in the generalized gradient ap-proximation (GGA), and the Projector Augmented-Wave(PAW) method as implemented in the VASP code . Inall calculations the cut-off is 471 eV and the force thresh-old is 0.01 eV/˚A. For all phases, 80-atom cells are used.The k-point summation grids are a Γ-centered 2 × × × × ε phase (4 × × β phase and 4 × × the k-grid is a 4 × x , F mix = F alloy − F bulk = [ E alloy − T S alloy ] − F bulk , (1)where E alloy is the internal energy calculated from firstprinciples as just described, and S alloy = − x log x − (1 − x ) log (1 − x ) + (2)+3 [(1 + n ) log (1 + n ) − n log n ] = S mix + S vib with n ( T, x ) = 1 / ( e Θ m ( x ) /T − , Θ m ( x ) = (1 − x ) Θ Ga O + x Θ In O (3)the Planck distribution and the mixture’s Debye tem-perature Θ m ( x ) interpolated between the parent com-pounds. (The approximation of the vibrational entropy a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec with that of a single-Debye-frequency oscillator is admis-sible, as the growth temperatures are comparable to orhigher than the Debye temperatures Θ In O =420 K andΘ Ga O =870 K.) The bulk free energy F bulk ( x ) = x F In O + (1 − x ) F Ga O (4)interpolates between the binary-compound values, cal-culated as for the alloy. We finally recall that a mix-ture separates into phases if the specific free energy isa negative-curvature function of x . The x values wherethe curvature becomes negative and, respectively, goesback to positive (i.e. the inflection points of the mixingfree energy) delimit the phase separation region; thesebounds, which may depend on temperature T, define arange known as miscibility gap. III. RESULTSA. Hexagonal phase near x (cid:39) We first consider a hexagonal phase of (Ga − x In x ) O .The motivation comes from recent growth experiments of (Ga − x In x ) O near x ∼ β and bixbyite crystal portions, of sig-nificant hexagonal microcrystallites. A candidate phasehad been identified earlier on for InGaO , and classi-fied in the non-polar space group P / mmc . This phaseis depicted at x =0.5 in Fig.1. c ba FIG. 1. The hex structure at 50-50 concentration.
Both out of interest for its possible energetic sta-bility, and for the possibility that the structure mightbecome polar, we investigated this phase in the range x ∈ [0.45,0.55]. In this region the hex phase is lower in en-ergy than, and therefore favored over, the bixbyite andvirtually degenerate at x =0.5 with the β phase (discussedbelow). We quantify this calculating the Helmholtz mix-ing free energy by the model described above and inRef.3. (As in previous work, large error bars are ex-pected from the limited configurational sampling, but aserrors should largely cancel out when comparing the var-ious phases, we deem the relative energetics to be rather reliable.) As shown in Fig.2, bottom panel, we find thatthe lowest energy structures of the sample of configura-tions for the hex symmetry in the vicinity of x =0.5 arelower in mixing free energy by about 0.1 eV than the free-standing bixbyite configurational sample, and thereforemore stable than the bixbyite alloy. B. β phase near x ∼ In previous work we found that the alloy adoptingthe β structure of Ga O is disfavored over bixbyite for x above 0.1 or so. The internal energy of that phase in-creases drastically and monotonically in that region of x ,so we refrained from pursuing it further. However, thesame paper reporting the occurrence of hex phase crys-tallites also signaled β -phase inclusions near x =0.5, so werevisited our previous assessment and studied the β phasein that region of concentration. It turns out that at ex-actly x =0.5 the β phase is more stable than bixbyite andas stable as the hex phase discussed above (see Fig.2, bot-tom; a similar occurrence was reported in Ref.5). At thisconcentration, In atoms occupy all the octahedral sites,and Ga atoms occupy all the tetrahedral sites. However,consistently with our previous conclusions, as soon as wemove away from exact 50-50 concentration the energyshoots up immediately on both sides of the x =0.5 mini-mum, accompanied by a volume collapse (mainly of thetetrahedra) by over 10% at x =0.47 and x =0.53. There-fore, the β phase itself should only occur at the “magic”50-50 concentration, or in the vicinity of that concentra-tion if one assumes that some other phase will take upthe local cation excess. C. Revised phase diagram
Based on our calculations discussed above we pro-vide an improved phase diagram accounting for the newphases. The diagram is reported in Fig.2, bottom panel,as mixing free energy vs x . The temperature is 800 K, atypical growth temperature. As shown previously, thephase boundaries are insensitive to temperature withinour model, and hence apply to all practical growth tem-peratures. Put differently, the miscibility gaps and mis-cibility regions are persistent with temperature.The stability of the β phase only at low x is confirmed,and so is the phase separation into components of thebixbyite phase in most of its own range (signaled by thefree energy being everywhere upward-convex except for x ≥ x ∼ x ∼ β phase has a very narrow stability slot at x =0.5.As dictated by the curvature of the mixing free energy,there is full miscibility of the two binary oxides at all G a p ( e V ) M i x i ng fr ee e n e r gy ( e V / ca ti on ) Bix β hex FIG. 2. Revised phase diagram at T=800 K (bottom) andenergy gap (top) at T=0 for (Ga − x In x ) O . Mixing regionsare shaded, and the gap is only drawn for those regions. Leg-end applies to both panels; empty diamonds: dipole-forbiddentransition in bixbyite. The lines are quadratic fits for hex and x ∼ x β . temperatures in the ranges x ∈ (0,0.18), x ∈ (0.4,0.6), and x ∈ (0.9,1), where, respectively, the β , the hex, and thebixbyite structures are adopted. (These x values corre-spond to the inflection points of the mixing free energy.)In the rest of the x range, separation into binaries is ex-pected from the convex mixing free energy.In the central region of the x range, there are severalcompeting possibilities. The hex and β mixed phases areobviously favored over the bixbyite alloy. But the lat-ter should phase-separate into binary components, withIn O certainly adopting the bixbyite structure. Ga O may go either bixbyite or β : in the first case, the energy(from an interpolation between the end values) is about0.02 eV, i.e. falls between the β alloy and the hex; inthe second case, the free energy is zero by construction,making phase separation at 50% slightly favored. Theseconsiderations, however, neglect internal interfaces, grainboundaries, strain effects in the binaries, and growth ki-netics (all of which are exceedingly complicated and wellbeyond our present scope), which will tend to disfavorphase separation. Thus, at this level of accuracy, it seemsvery plausible that –as experiments suggest– the hex, β ,and phase-separated binaries will coexist in this region, depending on the growth conditions.In Fig.2, top panel, we also report the calculated fun-damental gap in the stable phases in the regions wheremixing occurs. The gap is calculated as difference ofKohn-Sham GGA eigenvalues plus the empirical correc-tion used in Ref.3 to adjust the gap of the binaries tothe experimental values. At low x and at high x oneexpects optical absorption typical of Ga O and In O ,respectively. Around x ∼ β alloy absorp-tions should be present; since the bixbyite alloy phaseseparates, absorption may also be observed at energiestypical of Ga O (4.5-4.7 eV) and In O (2.9 eV forbid-den, 3.5 eV allowed)); so in the central x region, distincttransitions may be expected at roughly 3.5 eV, 4 eV and4.5 eV. D. Polarization: ε -Ga O One reason of interest in the hex phase is checkingwhether it distorts into a non-centrosymmetric symme-try group as a consequence of alloying. We investigatea range of alloying of between 43% and 57%, enablingall symmetry lowerings starting from P / mmc . Wefind that the polarization is always numerically zero re-ferred to the non-polar high-symmetry phase, and arethus forced to conclude that this structure, somewhatanticlimatically, is robustly non polar. In fact, our con-clusion agrees with the symmetry determination of Ref.12at x =0.5, and shows that this applies at generic concen-trations in that vicinity. We also sampled the polariza-tion in a few bixbyite and β alloy samples, unsurprisinglyfinding them to be always zero (referred to the cubic andmonoclinic- β phases).We therefore turned to another potentially polar phaseof this system, ε -Ga O , recently grown epitaxially onGaN by Oshima et al. . The ε phase is structurallyakin to the same phase of Fe O and its space groupis P na , which does not contain inversion. We calcu-late its structural parameters, finding them in essentialagreement with a previous study, and the energy differ-ence with the β phase at zero temperature, which is just90 meV per formula unit.At low enough temperature, the epitaxial stabiliza-tion of the ε phase is not endangered by a possible de-cay in the β ground state, for the simple reason thatthere is no possible ε -to- β symmetry path, since the twospace groups are P na and C /m , respectively. Thisis quite analogous to the situation of wurtzite III-V ni-trides (group P mc ) which cannot transform, again forsymmetry reasons, into the close relative structure ofzincblende (group F m ), despite the volume being al-most the same and the energy difference being about only10 meV/atom (the two ε and β phases also have the samevolume and an energy difference of about 15 meV/atom).Just as ε -Ga O , zincblende nitrides can be grown underappropriate epitaxial constraints.Since the group P na does not contain inversion, ob- P o l a r i za ti on ( C / m ) FIG. 3. Polarization calculated along a path connectingthe centrosymmetric parent phase to non-centrosymmetric ε -Ga O. servable polar vector quantities are allowed in ε -Ga O .The polar axis is the c axis, so we calculate the sponta-neous polarization P =(0,0,P) as difference of the polar-izations calculated in this phase and in a symmetry-connected centrosymmetric parent phase (we verifiedthat the other components are indeed zero). The latteris chosen to have symmetry group P nma (a supergroupof
P na ). The evolution of P with a path connectingthe two structures is in Fig.3.The final result is that ε -Ga O has a remarkableP=0.23 C/m , a value similar to that of BaTiO , a fac-tor of 3 larger than of AlN, and nearly a factor 10 largerthan of GaN. The structure of the ε phase is not struc-turally switchable (in the same sense that wurtzite is not,though of course the polar axis can be inverted, again asin nitrides, by inverting the growth direction using e.g. abuffer layer); therefore P is expected to mantain its ori-entation along the polar axis within any given crystallinedomain. ε -Ga O can thus be classified as a pyroelectricmaterial, one more time like III-V nitrides. The sym-metry of the ε -Ga O structure allows for five distinctpiezoelectric coefficients; here we calculate the diagonalcoefficient e as the finite-differences derivative of thepolarization with respect to the axial strain (cid:15) =( c – c )/ c .The result is e =0.77 C/m , which is in line with typicalcoefficients of strongly polar semiconductors (oxides andnitrides), although over an order of magnitude smallerthan those of strong ferroelectrics (see e.g. Ref.13).These results open up interesting perspectives. Thepolarization of ε -Ga O can be exploited growing the ox-ide epitaxially on GaN (or, equivalently, growing GaN onthe oxide) to build a high-mobility transistor. The polaraxis of ε -Ga O is found to be parallel to that of GaNin growth experiments, so their polarizations are eitherparallel or antiparallel. Since the polarization differenceis very large in both cases, a correspondingly large po-larization charge will appear at the GaN/ ε -Ga O inter- face and thus attract free carriers (provided by dopants,e.g.) to form an interface-localized two-dimensional gasat potentially huge concentrations. The polarization dif-ference across the interface, i.e. the polarization chargeto be screened by free carriers, and hence the potentially-reachable local electron-gas concentration, is 0.2 to 0.26C/m , i.e. 1.2 to 1.6 × cm − (for P vectors in thetwo materials being parallel or antiparallel, and neglect-ing possible interface traps, native charges, etc.). In addi-tion, since the gap of ε -Ga O is much larger than that ofGaN, the interface confinement should be quite efficient.Finally, the sign of the polarization charge and hence ofthe accumulation layer will depend on the chosen polarityof the substrate. The above scenario is a “writ-large” ver-sion of the GaN/AlGaN HEMTs currently in use, whosehigh-frequency, high-power operation is enabled primar-ily by the high polarization-induced interface charge. Ofcourse, all else assumed to be equal, Ga O /GaN transis-tors could be much superior to AlGaN/GaN ones, whichenjoy a much lower areal density of order 10 cm − . IV. SUMMARY
Using density-functional ab initio theoretical tech-niques, we have revised the phase diagram of(Ga − x In x ) O , showing that the β phase is stable (with-out phase separation into binary components) at low x and exactly at 50-50 concentration; a new hexagonalphase is stable (again without phase separation into bi-nary components) for x from about 0.4 to 0.6, where itis robustly non-polar; and bixbyite will be favored for x between 0.2 and 0.4 and upward of 0.6, but shouldphase-separate into binary components. Around x ∼ β and phase-separated binary bixbyites shouldbe closely competing. Optical signatures are expected ataround 4.6 eV at low x ( β phase), around 3.5 eV at large x (bixbyite), and at 3.5, 4, 4.5 eV from the competingphases at x ∼ (cid:15) -phase of Ga O , andconfirmed it as the second most stable structure beside β -Ga O . We find it to have a large spontaneous po-larization (0.23 C/m ) and a sizable diagonal piezoelec-tric coefficient ( e =0.77 C/m ). Symmetry dictates thatthis phase, once epitaxially stabilized, will not transformback into the ground-state β , despite having the samevolume and a small energy difference; in this sense, the ε - β relation is similar to that between zincblende andwurtzite III-V nitrides. ACKNOWLEDGMENTS
Work supported in part by MIUR-PRIN 2010 project
Oxide , CAR and PRID of University of Cagliari, Fon-dazione Banco di Sardegna grants, CINECA computinggrants. MBM acknowledges the financial support of herPhD scholarship by Sardinia Regional Government underP.O.R. Sardegna F.S.E. Operational Programme of theAutonomous Region of Sardinia, European Social Fund 2007-2013 - Axis IV Human Resources, Objective l.3,Line of Activity l.3.1. M. B. Maccioni, F. Ricci, and V. Fiorentini, Appl. Phys.Express , 021102 (2015) M. B. Maccioni, F. Ricci, and V. Fiorentini, J. Phys. Conf.Ser. , 012016 (2014). M. B. Maccioni, F. Ricci, and V. Fiorentini, J. Phys.: Con-dens. Matter F. Ricci, F. Boschi, A. Baraldi, A. Filippetti, M. Higashi-waki, A. Kuramata, V. Fiorentini, and R. Fornari, J. Phys.:Condens. Matter H. Peelaers, D. Steiauf, J. B. Varley, A. Janotti, and C. G.Van de Walle, Phys. Rev. B , 085206 (2015). R. Schewski, T. Markurt, T. Schulz, T. Remmele, G.Wagner, M. Baldini, H. von Wenckstern, M. Grundmann,and M. Albrecht,
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