Hybrid density functional theory studies of AlN and GaN under uniaxial strain

Home Search Collections Journals About Contact us My IOPscienceHybrid density functional theory studies of AlN and GaN under uniaxial strainThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2013 J. Phys.: Condens. Matter 25 045801(/0953-8984/25/4/045801)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 202.206.252.55The article was downloaded on 23/08/2013 at 01:57Please note that terms and conditions apply.IOP P UBLISHING J OURNAL OF P HYSICS:C ONDENSED M ATTER J.Phys.:Condens.Matter25(2013)045801(11pp)doi:10.1088/0953-8984/25/4/045801Hybrid density functional theory studies of AlN and GaN under uniaxial strainLixia Qin1,Yifeng Duan1,Hongliang Shi2,Liwei Shi1and Gang Tang11Department of Physics,China University of Mining and Technology,Xuzhou221116,People’s Republic of China2Institute of High Performance Computing,A*STAR,138632,SingaporeE-mail:yifeng@,shih@.sg and gangtang@Received29August2012,infinal form1November2012Published17December2012Online at /JPhysCM/25/045801AbstractThe structural stability,spontaneous polarization,piezoelectric response,and electronicstructure of AlN and GaN under uniaxial strain along the[0001]direction are systematicallyinvestigated using HSE06range-separated hybrid functionals.Our results exhibit interestingbehavior.(i)AlN and GaN share the same structural transition from wurtzite to a graphite-likephase at very large compressive strains,similarly to other wurtzite semiconductors.Ourcalculations further reveal that this well-known phase transition is driven by thetransverse-acoustic soft phonon mode associated with elastic instabilities.(ii)The appliedtensile strain can either drastically suppress or strongly enhance the polarization andpiezoelectricity,based on the value of the strain.Furthermore,large enhancements ofpolarization and piezoelectricity close to the phase-transition regions at large compressivestrains are predicted,similar to those previously predicted in ferroelectricfields.Ourcalculations indicate that such colossal enhancements are strongly correlated to phasetransitions when large atomic displacements are generated by external strains.(iii)Under thesame strain,AlN and GaN have significantly different electronic properties:both wurtzite andgraphite-like AlN always display direct band structures,while the the bandgap of wurtziteGaN is always direct and that of graphite-like GaN always indirect.Furthermore,the bandgapof graphite-like AlN is greatly enhanced by large compressive strain,but that of wurtzite GaNis not sensitive to compressive strain.Our results are drastically different from those forequibiaxial strain(Duan et al2012Appl.Phys.Lett.100022104).(Somefigures may appear in colour only in the online journal)1.IntroductionAluminum nitride(AlN)and gallium nitride(GaN),which usually crystallize in the wurtzite structure at ambient conditions,have been extensively investigated owing to their abundant physics and practical applications[1–3].The structural properties of wurtzite semiconductors are sensitive to external pressure.Recently,intensive theoretical studies have found that there is an intermediate graphite-like phase between wurtzite and rocksalt at high pressures and the three structures arefivefold,fourfold and sixfold coordinated, respectively[4–8].This is consistent with the well-known fact that external pressure favors more close packed structures, thereby resulting in the increasing coordination number attributed to the bond-length evolution.Since the graphite-like structure is metastable and is difficult to stabilize by applying external pressure,few relevant experimental results have been reported so far.As a valid alternative,accurate first-principles simulations can complement these studies and provide detailed descriptions of the structural and bonding behavior under extreme conditions.The internal lattice parameter u and lattice ratio c/a are good measures for representing the deviation from ideal wurtzite structure but are not sufficient to describe a phase transition to a different symmetry.It is well known that dynamic and elastic instabilities are often responsible for phase transitions. However,due to the discontinuities in the volume,energy and other relevant properties at critical pressures,there are alwaysother driving forces forfirst-order phase transitions[9,10].It is noteworthy that the wurtzite and graphite-like phases,with the space groups P63mc and P63/mmc,respectively,both show hexagonal symmetry.Therefore,study of the effectsof pressure on the dynamic and elastic properties will bemore helpful in understanding the underlying mechanismsfor these phase transitions.Since the graphite-like phaseshows distinct physical properties from the wurtzite androcksalt phases,such pressure-induced structural variationsare attractive technologically,such as in the development ofhigh-quality optoelectronic devices.In this paper,our initialwork is to reveal the structural transformations as well asthe underlying mechanisms for AlN and GaN subjected touniaxial strain.Wurtzite structure possesses the highest symmetrycompatible with the coexistence of spontaneous polarizationand piezoelectric response[11],which are both quite sensitiveto changes of the structural parameters.In the presenceof external pressure,the electric polarization is dividedinto spontaneous and piezoelectric parts.The ground-statepolarization and piezoelectricity of binary III–V nitrideshave been systematically investigated[12–15].Our recentresults show that hydrostatic pressures greatly enhancethe polarization and piezoelectricity of AlN,with themaxima appearing simultaneously at the phase transitionfrom wurtzite to graphite-like[16].The spontaneouspolarization and piezoelectricfields modify the band edges,thereby significantly influencing the optical properties ofthe heterostructures or supercells[12].Therefore,it isvery important tofind a feasible way to control thepolarization and piezoelectricity in electronic and opticaldevices.For piezoelectric devices,ZnO has been predictedas a key enabling material due to its strong spontaneousand piezoelectric polarizations[17].Furthermore,AlNdisplays much better piezoelectric performance than ZnO atambient conditions[18],therefore gaining extensive attentionfor potential applications.As a result,it will be veryuseful tofind a way to further enhance the piezoelectricresponse for technological applications.At present,suchinvestigations are mainly focused on ferroelectric materials,which have a wide range of applications in medical imaging,detectors,actuators,telecommunication,and ultrasonicdevices[19–21].It has been revealed that polarizationrotation near the phase-transition regions results in giantelectromechanical effects for ferroelectric perovskites[22].Our recent calculations point out that the maximumpiezoelectric value of AlN under hydrostatic pressures,appearing at the phase transition to graphite-like,is increasedby a factor of∼97from the equilibrium value and ismuch larger than those of most common ferroelectricperovskites[16].However,the polarization always remainsalong the c axis and no polarization rotation is observed.Therefore,the driving mechanisms are drastically differentfrom those of ferroelectric materials and more work is neededto further explain such phenomena.Such investigations arehelpful in broadening the opportunities for piezoelectricapplications of wurtzite semiconductors and in betterunderstanding the driving mechanisms of giant piezoelectric responses for ferroelectric materials.In this paper,one area we are interested in is how uniaxial strain affects the piezoelectric response of AlN and GaN,especially near the phase transition.AlN and GaN have large direct bandgaps[18];thereby environmentally benign Al(Ga)-based semiconductors al-ways have relatively large bandgaps.Such materials have been widely used in optoelectronic devices such as blue–green light emitting diodes and laser diodes as well as high-power, high-temperature,and high-frequency electronic devices[23]. It is well known that the structural and electronic properties of semiconductors can be altered by either applying external pressure or inducing internal strain,or by both,to obtain the desired physical and electrical properties[5,24].Since the interatomic distances and relative positions of atoms have a strong influence on the band structure,it is possible to adjust the bandgap(E g)by uniaxial strain along the[0001] direction.The effects of external strain on the structural and electronic properties of GaN and Zn X(X=O,S,Se and Te) have been systematically investigated using standard density functional theory(DFT)[15,25–27].In those works,uniaxial strain along the[0001]direction and equibiaxial strain in the (0001)plane were believed to have an equivalent influence on the band structures.Recently,our results have shown that although AlN and GaN under in-plane strain share the same structural transition from wurtzite to a graphite-like phase,their electronic properties are significantly different. Furthermore,it is more difficult for AlN than for GaN to obtain the graphite-like semi-metallic phase[28].Such effects of strain on the properties of common semiconductors are not only intriguing in physics,but also attractive technologically, since strain-induced E g tuning can be used for photovoltaic applications.In this paper,another area we are interested in is to systematically reveal how uniaxial strain affects the band structures of AlN and GaN,especially near the phase transitions.Since the covalent bonding in semiconductors is highly directional,strain effects depend on the loading direction. In this paper,we study the uniaxial-strain effects on the structural transitions,polarizations,piezoelectric responses and electronic structures of AlN and GaN using total energy as well as linear response calculations.Our work differs from the previous studies in three main aspects.(1)Previous investigations have just obtained the structural transition from wurtzite to the graphite-like phase from the evolution of lattice constants with strain.However,the underlying mechanism still remains unknown.In our work,one key aim is to reveal the driving forces of such phase transitions based on dynamic and elastic properties.(2)We focus on the effects of uniaxial strain on the polarization and piezoelectric response,especially near the phase-transition regions,in order to point out a feasible way to enhance the electromechanical response.However,such systematic studies have seldom been made in previous work.(3)To overcome the bottleneck of severe underestimation of E g in standard DFT,Heyd–Scuseria–Ernzerhof(HSE06)hybrid functionals[29,30]are adopted in this work,which have been proved to be highly reliable[28,31,32].However,the previous studies have mainly been performed using standardttice constants(a,c/a and u),E g and elastic constants(c ij)for ground-state wurtzite AlN and GaN.a,E g and c ij are in˚A,eV and GPa,respectively.The experimental values are from[18]and other theoretical data are from[42].Figure1.The calculated E g for equilibrium AlN and GaN as a function of the exchange mixing ratio AEXX.The open symbols refer to the calculated results and the solid ones refer to the ground-state experimental data.DFT,which is unable to accurately describe the variation of E g with strain.The structure of this paper is organized as follows.In the next section we describe the computational methods used in this work.In section3,we report the main results of this work.Finally,a brief summary is given in section4.putational methodsThe total energy and band structure calculations are performed using the projector augmented-wave method[33] as implemented in the V ASP code[34].The electronic wavefunctions are described using a plane wave basis set with an energy cutoff of650eV.A Monkhorst–Pack k-point mesh of6×6×6is used throughout the calculations to obtain well-converged results.The standard exchange mixing,which contains25%Hartree–Fock and 75%Perdew–Burke–Ernzerhof-GGA(PBE-GGA)[35],is employed in the HSE06hybrid functional.To check the setting of the mixing ratio AEXX(denoted in the V ASP code),figure1shows the calculated E g as a function of AEXX for equilibrium AlN and GaN;the ground-state experimental E g values are listed for comparison.Good agreement is achieved at AEXX=0.25,especially for AlN,supporting the standard exchange mixing adopted in our work.The incorporation of an appropriate portion of the nonlocal exact exchange into the local or semilocal exchange(HSE06calculations)can yield reasonable bandgaps compared to standard DFT(which willbe systematically discussed in the following).Hartree–Fockcannot describe the van der Waals forces since it only accountsfor exchange.We have also checked the influence of van derWaals forces on the results for the wurtzite and graphite-likephases,similarly to previous work[36],and found that themain results do not change after such forces have beenincorporated into the HSE06calculations.Therefore,ourresults demonstrate that the long-range van der Waals forcesoriginating from nonlocal electron–electron correlation do nothave a significant influence on the properties of the GaN andAlN systems.The phonon-dispersion calculations are performed withthe direct method implemented in the PHONOPY pack-age[37,38].This method uses the Hellmann–Feynman forcescalculated for an optimized supercell through V ASP[34].In theory,the larger the supercell,the more accurate thedispersion curves that are obtained.However,since largesupercell HSE06calculations are currently not computation-ally feasible,we adopt the2×2×3supercell for allphonon calculations and an accuracy of0.05THz for thehighest optical zone-center phonon frequency is achieved.The piezoelectric constants e iνand elastic constants cµν(hereRoman indices go from1to3,and Greek ones from1to6)are calculated using the density functional perturbationtheory(DFPT)[39]of the linear response of strain typeperturbations[40].The electric polarization(P)is calculatedusing the well-known Berry-phase approach[41].Tofind thestrain-free lattice constants,the lattice vectors and atomiccoordinates are fully relaxed until the Hellmann–Feynman force acting on each atom is reduced to less than0.02eV˚A−1.In the presence of uniaxial strain( ),relaxation is performedwith the lattice constant cfixed until the following conditionsare satisfied within a small tolerance:σ11=σ22<0.02GPa andσij=0for i=j,whereσ11andσ22are the stresses inthe(0001)plane.The tensorσ33is the externally applieduniaxial stress along the[0001]direction and is adjusted bychanging step by step.We have examined the accuracyof our calculations by comparing the ground-state resultswith experimental and other theoretical data[18,42].Assummarized in table1,good agreement between theoreticaland experimental methods is achieved.3.Results and discussionIn the presence of out-of-plane stress,a new lattice parameterc corresponds to a specified uniaxial strain given by =Figure2.(a)Out-of-plane stressσ33,and(b)elastic constants c13and c33as a function of .33=(c−c0)/c0and a commensurate equibiaxial in-plane stress-free strain given by 11= 22=(a−a0)/a0,where a0 and c0are the strain-free lattice constants.The top panels in figure2summarize the calculated dependence ofσ33,which displays the same trends for AlN and GaN.The tensileσ33 allfirst increase and then decrease with ,with a maximum of∼45GPa(34GPa)appearing at =0.16(0.14)for AlN(GaN).This indicates that the calculated ideal tensile strengths along the polar c axis are∼45GPa for AlN and ∼34GPa for GaN,which are the maximum stresses required to break the perfect crystals.The structural instabilities at large are emphasized by the dependence of the elastic constants c13and c33,which become negative when >0.16 for AlN and >0.14for GaN,as shown in the bottom panels offigure2.On the other hand,for compressive ,the σ33magnitudes drastically increase when−0.08< <0for AlN and−0.12< <0for GaN,which is well known and anticipated,consistent with previous theoreticalfindings[16]. The most unexpectedfinding is that the magnitude ofσ33first decreases and then increases with further increase of ,with the minimum appearing at =−0.14(−0.18)for AlN(GaN).The abnormal behavior ofσ33when−0.14< <−0.08for AlN and−0.18< <−0.12for GaN is attributed to the evolution of the elastic constant c33,which always decreases with compressive and becomes negative in these ranges,as shown infigure2(b).In the abnormal ranges,although the elastic stability criteria c44>0and c11>|c12|are true,(c11+c12)c33>2c213is false,indicating that the wurtzite structures become elastically unstable.More interestingly,after the discontinuous changes from negative to positive at =−0.14(−0.18)for AlN(GaN),the c13 and c33increase linearly with further increase of compressive .The new elastic trends lead to great enhancement of the σ33magnitudes and the reappearance of elastic stabilities for high-pressure structures in these ranges.As a conclusion, phase transitions are usually predicted at the critical .Since dynamic instabilities are often responsible for structural transformation,five phonon-dispersion curves at different values are plotted infigure3for AlN and GaN.The previous related works were mainly performed using standard DFT,where the DFT-PBE(LDA)always overestimates (underestimates)the optimized volume of the unit cell atfixed .This leads to very weak(strong)nearest-neighbor force constants and a significant underestimation(overestimation) of phonon frequencies,especially for the optical modes[43]. The HSE06adopted in this work has been proven to be more accurate than the DFT-PBE and LDA for the phonon frequency and dispersion[43].The reciprocal lattice point (0,0,0.06)near the zone center is denoted hereafter as Y,owing to its interesting behavior,as shown in panel(b). There are twelve phonon branches for the wurtzite unit cell: one longitudinal-acoustic(LA),two transverse-acoustic(TA), three longitudinal-optical(LO),and six transverse-optical (TO)ones,as shown in panel(d).Since the Al(Ga)atom is heavier than the N atom,the internal vibrations of Al(Ga) atoms are mainly correlated to the low-frequency modes.On the other hand,due to the difference between Al and Ga atoms,the modes of AlN shift to higher frequencies than those of GaN at different .The six TO modes are separated from the other branches by the well-known phonon bandgap for the wurtzite phase,which is most noticeable in the equilibrium structure,as shown infigure3(d).At ambient conditions, the gap of GaN is larger than that of AlN.Our calculated results further reveal that the tensile always reduces the gaps for AlN and GaN,whereas the gaps arefirst reduced when −0.14< <0for AlN and−0.18< <0for GaN andFigure3.Phonon-dispersion curves for AlN and GaN.AlN:(a) =−0.18,(b) =−0.14,(c) =−0.11,(d) =0.0and(e) =0.17; GaN:(a) =−0.20,(b) =−0.18,(c) =−0.17,(d) =0.0and(e) =0.16.Figure4.Atomic displacements along the[110]direction,perpendicular to the Al–N(Ga–N)plane,at the critical TA(Y)mode softening strains:(a) =−0.14for AlN and(b) =−0.18for GaN.then enlarged unexpectedly as the compressive increases further.This abnormal phonon behavior is consistent with the novel elastic trends and emphasizes the occurrence of phase transition at the critical .It is shown that the phonon modes become unstable when >0.16for AlN and >0.14for GaN(seefigure3(e)). In other words,the wurtzite phase is stabilized by tensile when <0.16(0.14)for AlN(GaN),consistent with the results from the elastic properties.Our investigations further reveal that the wurtzite structure remains stable for a broad compressive range until the TA frequencies at Y become imaginary near =−0.14for AlN and =−0.18for GaN, as shown infigure3(b).This is drastically different from the results for the elastic properties,where elastic instabilities are observed when−0.14< <−0.08for AlN and−0.18< <−0.12for GaN;phonon instabilities are still not obtained at =−0.11(−0.17)for AlN(GaN)(seefigure3(c)).The most interestingfinding is that when <−0.14for AlN and <−0.18for GaN,the modes become stable once again,as shown infigure3(a).The anomalous behavior of the phonon modes confirms the occurrence of a phase transition at the critical .Schematic representations of the eigenvectors for the TA(Y)soft phonon modes at the critical are shown infigure4.The eigenvectors of the Al(Ga)cation and the N anion are parallel and along the[110]direction.Since the particular atomic displacements are responsible for the soft modes,which result in the structural instabilities,the atomic movements along this particular direction are closely correlated with the new-found phase transitions.To reveal the predicted structure,we systematically study the evolution of the in-plane strain 11,lattice ratio c/a,internal lattice parameter u,and Born effective charge Z∗33with respect to ,as shown infigure5.When >−0.14for AlN and >−0.18for GaN,the calculated 11,u,and Z∗33display a nonlinear response with respect to ;a linear trend is observed for c/a.When <−0.14(−0.18)for AlN(GaN),c/a<1.20and u=0.50. This implies the structural transition from wurtzite to a graphite-like phase,similar to those previously predicted for other forms of pressures[15,16,26–28].Therefore,the newFigure5.(a)In-plane strain 11,(b)lattice ratio c/a,(c)internal lattice parameter u(in units of the lattice c),and(d)Born effective chargeZ∗33as a function of .The transition from wurtzite to a graphite-like phase occurs at ∼=−0.14(−0.18)for AlN(GaN)(dashed line)andthe charge satisfies Z∗33(Al(Ga))=−Z∗33(N).Figure6.(a)The equilibrium wurtzite unit cell and(b)the graphite-like phase unit cell under large compressive uniaxial strain.The c axis is along the[0001]direction in each case.Adapted with permission from[27].Copyright2010American Institute of Physics.structure predicted by elastic and phonon instabilities at large compressive is the graphite-like phase,whose structure is shown infigure6,together with the wurtzite structure for comparison.Furthermore,the predicted transition values are in good agreement with those from the evolution of c/a and u.As a result,the TA soft phonon modes associated with the elastic instabilities are the driving forces for such well-known phase transitions.In the graphite-like phase, each cation(anion)has bonds with two adjacent anions (cations)with the same bonding length along the c axis,inaddition to bonds with three anions(cations)in the basal plane.The electric polarization(P)originates from two contributions:the lack of centrosymmetry and the deviation from the ideal wurtzite structure for which c/a=√8/3and u=0.375.The former is characterized by u,which describes the shortest bonding length between two adjacent atoms alongthe c axis;the latter has a strong influence on Z∗33.The u andFigure7.(a)Electric polarization P(C m−2)and(b)piezoelectric coefficient e33(C m−2)as a function of .The open symbols refer to the wurtzite phase and the solid ones refer to the graphite-like phase.Z∗33,whose dependence is shown infigures5(c)and(d),are responsible for rge u corresponds to high symmetry along the polar direction,which suppresses P,whereas largeZ∗33enhances P.As increases from compressive to tensile,u and Z∗33bothfirst decrease and then increase for wurtziteAlN and GaN.On the other hand,the graphite-like phase shows a higher symmetry than the wurtzite phase,with the coordination number increasing from four tofive.It is well known that the wurtzite phase displays the highest symmetry compatible with the coexistence of spontaneous P and piezoelectric response.The graphite-like structure does not possess polar and piezoelectric properties.In other words, the foregoing structural transformation is a polar–nonpolar phase transition,which is similar to that from the ferroelectric to the paraelectric phase in a ferroelectricfield[19–21].Figure7(a)shows P as a function of .For equilibrium wurtzite AlN and GaN,the spontaneous P values are 0.098C m−2and0.027C m−2,respectively,which are consistent with the other corresponding theoretical values of0.081and0.027C m−2[13,3].For the ideal wurtzite structure,the spontaneous P is nonvanishing due to the nonsuperposition of positive and negative electric charge centers along the polar c axis.When <−0.14(−0.18)for AlN(GaN),P=0,emphasizing the structural transition from the wurtzite to the graphite-like phase.Our calculated results further reveal that P displays the same trend as u and Z∗33 for the wurtzite phase,whose valuesfirst decrease and thenincrease with .This implies that Z∗33plays the dominant rolein the behavior of P.The minimum P values appear at ∼= 0.06for AlN and ∼=0.02for GaN,whereas the maximum values appear at the phase transition.Figure7(b)shows the piezoelectric coefficient e33 as a function of .For equilibrium wurtzite AlN and GaN,the e33values are1.65C m−2and0.76C m−2,respectively,in good agreement with the other correspondingresults of 1.46and0.73C m−2[3,13].When <−0.14(−0.18)for AlN(GaN),e33=0,emphasizing the predicted phase transition.However,as increases,the e33value of the wurtzite phasefirst strongly decreases untilit reaches zero at ∼=0.06(0.02)for AlN(GaN),andthen its absolute value gradually increases.The maximumvalue of∼122C m−2(91C m−2)appears at the phase transition,which is increased by a factor of∼74(118)from the ground-state value for AlN(GaN),and is muchlarger than those of most common ferroelectric perovskites.Therefore,two important conclusions are obtained.(1)Theexternally applied tensile can either suppress or enhancethe piezoelectric response,based on the value of .(2)The compressive greatly enhances the piezoelectricity,especially near the phase-transition region.The strongest piezoelectric responses near the phase-transition region were originally predicted in ferroelectricmaterials[19,21,22],which as well as wurtzite semi-conductors possess spontaneous P and piezoelectricity.Thedriving mechanism has been revealed to be polarizationrotation[19,22].However,since the P of the wurtzitephase only remains along the c axis(the[0001]direction),it is impossible to observe polarization rotation,i.e.,onlythe magnitude of P changes with .This is because thereis only one polar direction for the wurtzite phase.It iswell known that the e33describes the derivative of P withrespect to .The maximum e33,appearing at the phasetransition,corresponds to the maximum slope of the P versus curve,as shown infigure7(a).On the other hand,the zero slope,related to the minimum P,leads to e33=0 at ∼=0.06(0.02)for AlN(GaN).At large tensile ,the e33becomes negative,which just reflects the change in the sign of the slope;what we care about most is its absoluteFigure8.Band structures for AlN and GaN.AlN:(a) =−0.18,(b) =−0.08,(c) =0.00and(d) =0.17;GaN:(a) =−0.20,(b) =−0.12,(c) =0.00and(d) =0.16.The crystal structures are graphite-like in(a),and wurtzite in(b),(c)and(d).The Fermi level is set to zero in eachfigure.value.Very similar trends are predicted for the strain-inducedcolossal enhancement in piezoelectricity when approachingthe phase-transition region;therefore,our investigationsuggests that polarization rotation is not a necessary conditionfor such great enhancement in piezoelectricity;instead,it isthe change in the P magnitude with that leads to such anamazing phenomenon.The predicted novel piezoelectric response can alsobe summarized from the elastic behavior.As shown infigure2(b),the c33of the wurtzite phasefirst increases untilit reaches the maximum at ∼=0.06(0.02)for AlN(GaN),and then it monotonically decreases with further increase of .This suggests that these wurtzite materialsfirst become hardand then soft along the polar c axis in these ranges.Basedon the conclusion for ferroelectric materials,the softer thecrystal lattice,the larger the electromechanical response[44],it is easy tofind that as increases,the piezoelectricresponse isfirst suppressed and then strongly enhanced forthe wurtzite phase,with the weakest behavior appearing at ∼=0.06(0.02)for AlN(GaN).The predicted trends are in good agreement with the directly calculated results,asshown infigure7(b),thereby indicating that this conclusionis applicable to wurtzite semiconductors.Such investigationsare technologically very helpful,in view of the conversionbetween electric energy and mechanical energy,in enhancingthe performance of piezoelectric devices.The electronic properties can also be modified by theuniaxial .Four band structures at different values areplotted infigure8for AlN and GaN.The reciprocal latticepoint(0,1/2,1/3)is denoted hereafter as X,due to itsinteresting behavior in the conduction bands,as shown in panel(c).For wurtzite AlN and GaN(seefigures8(b)–(d)),the E g is always direct,with the valence-band maximum(VBM)and conduction-band minimum(CBM)at .This isdrastically different from wurtzite AlN under equibiaxial ,where the E g is either direct or indirect based on the valueof [28].As a conclusion,uniaxial and equibiaxial showdifferent influences on the band structure for AlN;this isattributed to the different elastic responses to uniaxial andbiaxial .However,in other theoretical work[26,45],uniformuniaxial along the[0001]direction was believed to beequivalent to equibiaxial in the(0001)plane.Our resultsfurther reveal that the band structures of graphite-like AlNand GaN are significantly different(seefigure8(a)).The E gof AlN is direct,with the VBM and CBM at ,whereas thatof GaN is indirect,with the VBM at H and the CBM at .This is also different from graphite-like AlN under equibiaxial ,where an indirect E g is observed at large tensile [28]. Therefore,one key difference between AlN and GaN underuniaxial is that both wurtzite and graphite-like AlN alwaysdisplay direct band structures,while the E g of wurtzite GaN isalways direct and that of graphite-like GaN is always indirect.To better reveal the underlying mechanisms for theabnormal band structures,figure9shows the minima of theconduction states and the maxima ofthe valence states atdifferent points,where the CBM or VBM most probablyappear,as a function of .As shown infigure9(a),the CBMof AlN(GaN),whose states are mainly attributed to the N sand Al(Ga)s orbitals,is always located at .This suggests adominant role of N s and Al(Ga)s orbital hybridization in the state.This is consistent with previously reported work[46], where the electronic structure of ground-state GaN was。