Phys. Status Solidi A 206, No. 8, 1892–1897 (2009) / DOI 10.1002/pssa.200881436 p s sapplications and materials science
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Polar AlN/GaN interfaces: Structuresand energeticsJ. Kioseoglou1, E. Kalesaki1, L. Lymperakis2, G. P. Dimitrakopulos1, Ph. Komninou*,1, and Th. Karakostas1
1 Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece2 Max-Planck-Institut fur Eisenforschung, Max-Planck-Strasse 1, Dusseldorf, Germany
Received 17 September 2008, revised 19 December 2008, accepted 27 May 2009Published online 20 July 2009
The structures and energies of {0001} interfaces between GaNand AlN are studied by both ab initio methods and moleculardynamics using the Tersoff empirical inter-atomic potential.Based on experimental observations, structural configurationsdepending on polarity and atomic stacking are considered.It is evidenced by both ab initio and empirical calculationsthat III-polar interfaces are energetically favourable comparedto the N-polar. In addition, the ab initio analysis shows thatthe wurtzite interfacial stacking is energetically preferablecompared to zinc blende. A linear dependence between the
bandgap energy and the strain in AlN/GaN heterostructures isfound. It is shown that the bandgap increases with increasing c/aratio while an inverse proportionality relationship is observedin the case of lattice parameter a. However, biaxial strain isfound to flatten this variation considerably. Empirical potentialcalculations yield the interfacial energies, taking into accountthe relaxation of the lattice mismatch due to arrays of misfitdislocations and in combination with ab initio methods estimatethat the energetically favourable III polarity interface exhibitsat least 18% larger critical thickness than the N polar.
1 Introduction The AlN/GaN heteroepitaxial systemis of particular importance for applications such ashigh electron mobility transistors (HEMTs), deep-UVoptoelectronic devices and IR detectors/emitters based onthe inter-subband principle. Since these device applicationsusually employ polar heterostructures, the polarity of thestacked layers constitutes a key parameter which directlyinfluences the interfacial structures [1, 2]. In addition,although the wurtzite structure is adopted by the epilayers,interfacial structure transformations between wurtzite andzinc blende-type stackings are often observed [3, 4]. Finally,the influence of the strain state and related interfacial defectson the material behaviour should be appreciated.
In a recent contribution we have employed moleculardynamics calculations, using the Tersoff empirical inter-atomic potential [5], in order to study the influence of polarityand interfacial stacking on the structural and energeticcharacteristics of polar InN/GaN heterostructures [4]. Inthe present work we examine these features for the caseof {0001} AlN/GaN using ab initio as well as empiricalpotential calculations. Ab initio calculations using densityfunctional theory (DFT) are employed in order to detect
the properties of pseudomorphic interfacial regions takinginto account various possible strain conditions, polarity andinterfacial structure. Previous efforts have focused only onpseudomorphic heterostructures having either the GaN orAlN lattice parameter, and the influences of polarity andinterfacial stacking were not considered [6, 7]. On the otherhand, the empirical potential method enables one to treatlarge interfacial areas using thousands of atoms, thus takinginto account the structural mismatch. The Tersoff potentialreproduces the lattice and elastic parameters of nitridesemiconductors accurately and describes sufficiently themetallic and inter-metallic interactions [8]. A combinationof first principles and inter-atomic potential calculations isimplemented here for the first time, in order to estimate thecritical thickness of the AlN/GaN heteroepitaxial system.Under the current approach two main issues are considered:(i) the various atomistic configurations along the AlN/GaNinterface and (ii) the feasible strain states that could beobserved along the pseudomorphic AlN/GaN multilayers andare associated with the growth conditions.
The c-plane AlN/GaN heterostructure is characterizedby a −2.4% lattice mismatch of AlN with respect to GaN.
Experimental observations in AlxGa1 − xN/GaN have shownthat misfit relaxation may be accommodated either by curvedmixed-type misfit dislocations with b = a = 1/3
⟨1210
⟩Burgers vectors or by straight edge dislocations withb = a + c = 1/3
⟨1210
⟩ + [0001] [9, 10]. Due to the ratherlow misfit, lattice-matched areas are extensive comparedto the ‘bad fit’ regions corresponding to misfit dislo-cations. Pseudomorphic interfaces are also pertinent tostrained multilayers and epilayers of thicknesses less thancritical.
Section 2 deals with pseudomorphic AlN/GaN het-erostructures under various strain conditions. The empiricalpotential calculations of AlN/GaN heterostructures contain-ing regions of good and bad fit are presented in Section 3.The conclusions are discussed in Section 4.
2 DFT calculations The (0001) and (0001) pseudo-morphic interfaces between AlN and GaN were studied byfirst-principles DFT calculations using the multiscale librarySP/hi/NGX [11]. The Perdew–Burke–Ernzehof generalizedgradient approximation was used for the exchange andcorrelation with a plane wave basis set (energy cut-off equalto 50 Ry), and soft Troullier–Martins pseudopotentials. TheBrillouin zone was sampled by a 4 × 4 × 2 Monkhorst-Packk-point sampling. This approach is sufficiently accurate forwurtzite nitride structures [12]. The equilibrium values oflattice constants and internal parameter u of the wurtzitestructure were obtained following the methodology ofRef. [17]. For AlN we found a = 3.0869 A, c/a = 1.6055 andu = 0.3815, while for GaN a = 3.1858 A, c/a = 1.6258 andu = 0.3770.
In our DFT study initial structural conditions weredefined as follows:
(i) Concerning the in-plane lattice parameter, in allconfigurations the two lattices have the same a latticeparameter. Three different cases were examined. In thefirst case, the average a lattice parameter of AlN andGaN was used. This case can be considered to describea pseudomorphic AlN/GaN multilayer where both crystalsare strained. In the other two cases, the AlN parameter wasadjusted to conform to that of GaN and vice versa.
(ii) Regarding the initial value of the c lattice parameter,two different conditions were investigated. In the first, theinitial [0001] parameter of each crystal (c′) is defined usingthe biaxial strain formulation:
c′ = c
[1 − 2
C13
C33
(a′ − a
a
)]
where a and c are the unstrained lattice parameters, a′ isthe strained in-plane lattice parameter and C13, C33 are theelastic constants. The latter were obtained using the samepseudopotentials and DFT code, and are reported in Ref. [13].However, experimental results have demonstrated that thestrain state of epitaxially grown III-N films may deviate from
biaxial, and may be influenced by a hydrostatic component[14, 15]. While the biaxial strain is caused by the latticeand thermal mismatch, the hydrostatic component may beattributed to the presence of point defects [14], or to theinfluence of a thick matrix to a thin layer or quantumdot [15]. In both cases, a considerable strain in the basalplane is combined with only a slight variation of the clattice parameter. Therefore, we have also considered thisplane strain case, i.e. the initial c lattice parameter was setto the strain-free value of each lattice. Subsequent atomicgeometry optimization relaxed the intrinsic strain and newc lattice parameters were obtained. The heterostructuresstudied under the first approach for [0001] will be hereafterreferred to as ‘biaxially strained’ (BS), while for the secondapproach, they will be referred to as ‘plane strain’ (PS)heterostructures.
2.1 Role of strain and interfacial stackingCombining (i) and (ii), six sets of initial conditions wereobtained. The AlN and GaN cells were created, eachcontaining eight monolayers (MLs) along [0001], and thesix aforementioned structural cases were constructed, havingeither wurtzite or zinc blende interfacial stacking, therebyincreasing the total number of studied configurations to 12.Periodic boundary conditions (PBCs) were applied, and allatomic positions were allowed to fully relax. However, byusing PBCs, total energy calculations can only provide anaverage interfacial energy of both polarities since if we startwith a III-polar GaN/AlN bicrystal, the PBCs introduce anadditional AlN/GaN bicrystal where the interface is inverted.Thus we use these conditions only to compare interfacialstackings and for band gap calculations. The influence ofpolarity was studied by a slab geometry methodology and ispresented in Section 3.2.
The energy differences between wurtzite and zincblende interfacial stackings (�Eint
ZB−W) are presented inTable 1. In all cases, the wurtzite stacking was found to beenergetically favourable compared to zinc blende, and thisenergy difference increases proportionally to the in-planelattice parameter and inverse-proportionally to the c/a ratio.It is noted that this energy difference is not resolvable withinthe accuracy of empirical potential calculations as discussedin Section 3.
Table 1 The differences �EZB−W of interfacial energy p.u.a.between wurtzite and zinc blende stacking sequences obtained byusing PBCs. Values are given for the PS condition, while the energydifferences of the corresponding BS cases with respect to the PSones
(�EBS
ZB−W − �EPSZB−W
)are given in parentheses. In the last
two columns the values of c/a ratio for relaxed GaN and AlN in thePS configurations are presented.
1894 J. Kioseoglou et al.: Polar AlN/GaN interfaces
Table 2 Band gap energy values (eV) at Γ point for wurtzite GaN and AlN compared to band gap values for PS AlN/GaN relaxedsupercells with wurtzite and zinc blende interfacial stackings. The differences in band gap values of the corresponding BS cases are givenin parentheses.
AlN, GaN (a = 3.136 A) 2.37 (+0.00) 2.41 (−0.01)GaN (a = 3.186 A) 2.22 (+0.06) 2.27(+0.01)
The band gap energy values at Γ point were calculatedfor the relaxed supercells. As expected, the band gap ofAlN is screened by that of GaN in all cases. In addition,our results (Table 2) show that the compressive strain ofGaN leads to a widening of the band gap in agreementwith previous studies for the case of bulk GaN [16, 17].This widening exhibits an almost linear dependence to thestrain. Figure 1 shows the dependence of the band gapenergy on the a lattice parameter and the c/a ratio, for thewurtzite interfacial stacking models. The band gap valuespresented have been normalized to the experimental valueof the GaN band gap [18]. The band gap, increases withincreasing c/a ratio for both the PS and BS cases, whilean inverse proportionality relationship is observed in thecase of lattice parameter a. In the case of zinc blendeinterfacial stacking, linearity is retained for the PS case buta strong divergence is observed for the BS case (Table 2).It is also noticed that the zinc blende stacking increasesthe AlN/GaN band gap with respect to the wurtzite one by37 meV, and 13 meV on average for the PS and the BS casesrespectively.
2.2 Role of polarity As already mentioned, PBCscannot account for the role of polarity on interfacial energy. Inorder to distinguish between the III- and N-polarities, a slabgeometry methodology was implemented. The interfaceswere treated using periodically repeated slabs having surfaceunit cell periodicity (1 × 1) with five MLs of GaN and fiveMLs of AlN as shown in Fig. 2(a). Pseudohydrogens withfractional charges were used in order to saturate the danglingbonds on the top and bottom slab surfaces.
For the Al or Ga (N) terminated surfaces the bonds weresaturated with pseudohydrogens with a fractional charge of5/4 (3/4). A vacuum region of 12 Awas used in all cases andall atoms were allowed to relax, except for the terminatingatoms on the top and bottom slab surfaces as well as thecorresponding pseudohydrogens, which were kept fixed afterhaving their positions already optimized in a first relaxationstep.
We formed slab configurations of the aforementionedpseudomorphically strained structural cases only for thewurtzite interfacial stacking. PBC-relaxed supercells wereused as initial input for the slab atomic configurations in
Figure 1 (online colour at: www.pss-a.com) Plots of the band gapenergy of AlN/GaN heterostructures, with a wurtzite interfacialstacking versus (a) the imposed lattice parameter a and (b) the c/aratio of relaxed GaN and AlN supercells for the PS and the BScases. The band gap values presented have been normalized to theexperimental value of GaN’s band gap.
order to ensure a well-relaxed interface before the additionof the pseudohydrogens. Convergence with respect to k-pointsampling, supercell size, slab and vacuum thickness wasexplicitly checked. The total energy of the slab E
Figure 2 (online colour at: www.pss-a.com) Atomistic models ofthe three different configurations used to calculate the interfacialenergy Eint for the two polarities. (a) Slab system consisting of theinterface and two free surfaces. (b) Slab systems consisting solelyof GaN or AlN (Esurf denotes the energy of the free surfaces).
given by the following equations for the III- and N- polarityinterfaces respectively (Fig. 2(a)):
EGaN/AlNslab−III = Ebulk + Eelastic + AEsurf
III + AEintIII (1)
and
EGaN/AlNslab−N = Ebulk + Eelastic + AEsurf
N + AEintN , (2)
where Eint is the interfacial energy per unit area (p.u.a.), Ais the interfacial area and Esurf corresponds to the energyp.u.a. of two free surfaces that depends on the N- or III-polarcharacter of the interface (i.e. Esurf
III = EN−H0.75 + EAl−H1.25 forIII- and Esurf
N = EN−H0.75 + EGa−H1.25 for N- polarity). Herewe assume that the energy difference between AlN andGaN of the N-terminated c-plane surfaces passivated bypseudohydrogens is negligible [19]. Ebulk = EGaN
bulk + EAlNbulk is
the total energy of the bulk GaN and AlN in the system, andEelastic represents the excess elastic energy (EGaN
elastic + EAlNelastic).
The energies of the free surfaces entering Eqs. (1)and (2), of thick GaN and AlN slabs exposing the (0001)and (0001) surfaces (Fig. 2(b)) were calculated separately.After this the total energies of the corresponding GaN andAlN slabs are EGaN
slab = EGaNbulk + A (EN−H0.75 + EGa−H1.25 ) and
EAlNslab = EAlN
bulk + A (EN−H0.75 + EAl−H1.25 ).Finally, the interfacial energies for the III- and the N-
polarity cases will be respectively:
EintIII = E
GaN/AlNslab−III − Eelastic − EAlN
slab − EGaNbulk
A(3)
and
EintN = E
GaN/AlNslab−N − Eelastic − EGaN
slab − EAlNbulk
A. (4)
The elastic energy terms in Eqs. (3) and (4) are calculatedas total energy differences between each slab and thecorresponding unstrained crystal.
In Table 3 the interfacial energies for the PS wurtziteinterfaces and the interfacial energy differences between N-and III-polarity are given. It is observed that the interfacialenergies depend on the pseudomorphic strain state only inthe millielectron volt order of magnitude. This is because themain contribution to the interfacial energies arises from theinterfacial chemistry, i.e. the type of bonds formed acrossthe interface, which is the same for all pseudomorphic statesconsidered. III-polar interfaces are by about 0.12 eV · A−2
lower in energy than the N-polar ones. The latter can beexplained in terms of the difference in enthalpies of the bondsformed across the interface. In III-polar interfaces, the AlNbicrystal exhibits the N polar c-plane surface, and GaN thecorresponding Ga-polar surface. Thus, each Al interfacialatom forms three bonds and each Ga interfacial atom formsone bond to N interface atoms respectively. On the other hand,in the N-polar interfaces the situation is reversed and each Alinterfacial atom forms one bond while interfacial Ga atomsform three bonds to N interfacial atoms respectively. AlNhas larger cohesive energy than GaN, i.e. the Al–N bondsare stronger than Ga–N ones. A rough estimate, referring tothe bulk materials and considering only the enthalpies of thebonds across the interfaces, confirms the III polarity as thepreferable one.
Table 3 The interfacial energy p.u.a. for the PS wurtzite interfaceshaving either N- or III-polarity and the interfacial energy differencebetween N- and III-polarity.
1896 J. Kioseoglou et al.: Polar AlN/GaN interfaces
3 Empirical interatomic potential calculationsThe empirical potential calculations allow us to studythe AlN/GaN interfaces taking into account the structuralmismatch that leads to the coexistence of ‘good fit’ and‘bad fit’ regions. For this purpose the bond-order Tersoffpotential was employed. The parameterizations of differentinteractions were taken from Refs. [8, 20]. In these, the inter-metallic interactions (i.e. Al–Ga) were explicitly considered.This is a prerequisite for the potential to be appropriate for thedescription of the lattice mismatched AlN/GaN interfaces,since the introduction of extended defects is expectedto introduce inter-metallic interactions. Using the Tersoffpotential, the lattice parameters and internal parameter uwere calculated and are aAlN = 3.112 A, cAlN/aAlN = 1.630 andu = 0.375 for AlN, and aGaN = 3.189 A, cGaN/aGaN = 1.635 andu = 0.375 for GaN. From these values the calculated misfitis δ = (αAlN − αGaN)/αGaN = −2.4%.
Supercells were constructed taking into account thepolarity and interfacial stacking. Furthermore, as describedin Ref. [4] the interfacial plane can be considered as adividing plane, cutting either single or double bonds. Thisdistinction is of importance only when the supercell is notpseudomorphic since the structural mismatch defines thespacing of the lattice planes (AlN has extra half-planes)and hence the strain of the bonds at the interfacial plane.We designate interfaces cutting single bonds as type 1 andinterfaces cutting double bonds as type 2. The interfacesof the unrelaxed supercells were taken to be abrupt andwithout any inter-diffusion of atomic species. The AlN andGaN bicrystal components were constructed in the form ofrectangular parallelepiped volumes, each containing 11 MLsalong [0001]. The exact dimensions of each cell volumewere for GaN 40 × a GaN along [1210] by 40 × a GaN31/2 along [1010]; for AlN they were 41 × a AlN along[1210] by 41 × a AlN 31/2 along [1010]. These supercellscomprised a total of about 144,000 atoms. PBCs wereapplied in the interfacial plane, while fixed boundaries wereimposed along the normal to the interface direction, inthe manner as described in Ref. [21]. Initially the separationdistance between the AlN and GaN bicrystal components wasoptimized by the conjugate gradient relaxation procedure. Inorder to drive the system closer to the global minimum, theatomic geometry of the interface at the optimum separationdistance was further optimized by performing simulatedannealing, whereby the system was slowly quenched from600 to 0 K at a rate of 10 K · fs−1 with a time stepof 0.1 fs.
Taking �D to be the total excess energy of a relaxedsupercell at zero temperature, i.e. the difference betweenthe total energy found by inter-atomic potential calculationsED and the energies of the bicrystal components, �D isindependent of the Al, Ga, N chemical potentials, andthus independent of the III/V ratio, for stoichiometricinterfaces as has been previously shown [4]. This isimportant for isolating the structural contribution to theinterfacial energy. The interfacial energy p.u.a. is givenby Eint = �D
A.
Table 4 Reduced interfacial energies p.u.a. for the type 1 andtype 2 interfaces having as reference the most favourable AlN/GaNinterface (type 1, III-polarity, Eint = 0.28 eV · A−2).
In Table 4, the reduced interfacial energies are giventaking as reference, the energy of the lowest energy interfacei.e. the III-polar/type 1/interface (Eint = 0.28 eV · A−2).Type 1/N-polar interfaces are second in preference, whiletype 2 interfaces generally exhibit higher energies. It is notedthat, within the error interval of the method, the wurtzite andzinc blende atomic stackings are indistinguishable by theseempirical calculations.
In all cases the topological property of the misfitdislocations, the Burgers vector, was the same, i.e.1/3
⟨1210
⟩. The contour plot of the excess energy per atom at
the Ga interfacial layer is illustrated in Fig. 3, projected along[0001]. This layer exhibits the highest excess energy amongthe interfacial layers. It is known that the dislocation energyperpendicular to the dislocation line decreases radially, whilethe core along the dislocation line stores a high amount ofplastic energy. It is evident that in our simulated supercell,the high energy path follows the three
⟨1210
⟩directions and
consequently we conclude that the misfit dislocation arraysadopt the
⟨1210
⟩line directions and they are of mixed-type.
The interface between a relaxed layer and the substratecontains a network of misfit dislocations and has higherenergy than the strained pseudomorphic interface, whichhas an elastic energy proportional to the thickness ofthe strained layer. The strained layer cannot relax until,
Figure 3 (online colour at: www.pss-a.com) Contour plot,projected along [0001], of the excess energy per atom of the Gainterfacial layer. The dislocation lines along
⟨1210
⟩are denoted by
dashed lines. The size of the contour plot along [1210] and [1010]corresponds to the supercell of the wurtzite type 1 interface.
at the critical thickness, there is enough stored elasticenergy to create the higher energy dislocated interface. Bycombining the plastically relaxed interfaces with the variouscases of pseudomorphic interfaces we have calculated thecorresponding elastic energy thresholds. When the strainenergy stored in the epitaxial layers exceeds these values,the AlN/GaN strain tends to give way to misfit dislocations.These thresholds yield critical thickness values (Table 5) ifwe employ first principle calculations assuming the biaxialformulation for the various strain states. Table 5 implies thatthe energetically favourable III polarity interface exhibits atleast 18% larger critical thickness than the N polar. The resultconcerning the average lattice constant is pertinent to strainpartitioning between GaN and AlN, and demonstrates themaximum achievable critical thickness that may be obtainedthrough strain engineering.
4 Conclusions AlN/GaN interfaces were studiedusing DFT and empirical potential calculations in anintegrated manner. The DFT calculations were employed totreat pseudomorphic bicrystals, taking into account possibleconditions of strain corresponding to experimental systemsof interest. The empirical potential calculations on theother hand could provide the interfacial energy of bicrystalscomprising both ‘good fit’ and ‘bad fit’ areas.
The calculations indicate the single-bond III-polarinterfaces to be the most favourable. The a-type misfitdislocations were found to comprise
⟨1210
⟩line directions.
The III-polarity interface exhibits at least 18% larger criticalthickness in comparison to the N polar.
DFT served to distinguish between wurtzite andzinc blende interfacial stackings, and it was shownthat the wurtzite one has lower energy. The interfacialenergy difference between wurtzite and zinc blendeinterfacial stackings increases proportionally to the basallattice parameter of pseudomorphic layers and inversely-proportional to the c/a ratio.
Band gap calculations performed on strained AlN/GaNmultilayers, with basal lattice parameter ranging from aGaN
to aAlN, led to the conclusion that the band gap of theheterostructure increases and an almost linear dependence
of EAlN/GaNg on the lattice parameter was found. It was also
found that the band gap, increases linearly with increasingc/a ratio for both the PS and BS cases. However, for biaxialstrain this linear variation is flattened considerably.
Acknowledgements This work was supported by EC underthe contract MRTN-CT-2004-005583 (PARSEM) and the FP7STREP Project DOTSENSE, Grant No. FP7–IST-224212.
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