Epitaxial Stabilisation of G e 1−x S n x Alloys
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Epitaxial Stabilization of Ge − x Sn x Alloys
Alfonso Sanchez-Soares,
Conor O’Donnell, and James C. Greer
3, 4 EOLAS Designs, Grenagh, Co. Cork, T23 AK70, Ireland Tyndall National Institute, University College Cork, Dyke Parade, Cork, T12 R5CP, Ireland Nottingham Ningbo New Materials Institute, The University of Nottingham Ningbo China, Ningbo, 315100, China Department of Electrical and Electronic Engineering,The University of Nottingham Ningbo China, Ningbo, 315100, China ∗ The thermodynamic stability of Ge − x Sn x alloys is investigated across the full composition rangeby employing density functional theory (DFT) in conjunction with the cluster expansion formal-ism (CE). Configurational, vibrational, and electronic entropy contributions are estimated to allowcomputation of alloy free energy at finite temperatures. Germanium and tin are found to be im-miscible in the bulk up to temperatures approaching germanium’s melting point, and much higherthan that of tin. Since the main contribution to alloy destabilization is found to be related to thelarge difference in atomic radii between atomic constituents, the possibility of stabilizing the alloyby reducing segregation through epitaxial constraints in thin films is explored. For germanium-tinalloys, the (001) substrate orientation is preferred for epitaxial growth as it allows for the largestdegree of out-of-plane relaxation. Epitaxial films have been simulated by biaxially straining bulkalloy cells as to constrain their lattice spacing to that of substrates lattice-matched to x = 0, andapproximately x = 0 .
5, and x = 1. We conclude that due to the large difference in elastic constantsbetween the components, epitaxial films with high tin content grown on lattice-matched substratesexhibit the greatest stability.
I. INTRODUCTION
Germanium-tin alloys have received significant at-tention over the past decade due to their potentialapplications in optoelectronic devices. It represents theonly group IV binary alloy predicted to exhibit a directelectronic band gap, making it particularly suitable forphotonics devices which can be readily integrated intoan industry and infrastructure dominated by silicon.Furthermore, germanium and tin are already present inelectronics manufacturing processes as technology boost-ers to improve the performance of silicon-based devices;a material with such properties and level of integra-tion into semiconductor processing presents numerousopportunities for novel electronic and optoelectronicdevice designs. Although silicon-germanium alloys weresuccessfully introduced into electronics manufacturingdecades ago, progress towards technological applicationsinvolving germanium-tin alloys has been much slowermainly due to two of its key characteristics: i) its con-stituents exhibit a large difference in ionic radii whichresults in a low solid solubility of tin in germanium( ≈ and ii) tin’s stable phase understandard conditions is β -Sn (metallic; tetragonal crystalstructure) instead of the preferred for electronics cubic α -Sn (semimetallic; cubic crystal structure), stable attemperatures below 13 . ◦ C at ambient pressure. Suchcharacteristics result in significant difficulty in incorpo-rating tin into germanium crystals without the formationof metallic β -Sn clusters within the alloy. Great progresshas been made in past decades towards fabricatinggermanium-tin alloys by employing non-equilibriumepitaxial growth techniques such as molecular beamepitaxy (MBE) and chemical vapor deposition (CVD), allowing realization of thin films with thicknesses upto hundreds of nanometers. Moreover, recent studieshave reported lasers comprised of Ge − x Sn x thin filmswith low tin content (0 . < x < . Although recent efforts have been focused on fabricat-ing films with low tin content alloys for photonics appli-cations, the electronic structure character of germanium-tin alloys has been predicted to vary significantly withcomposition: the addition of a few atomic percent oftin into germanium induces a transition to a directband gap, while alloys with larger tin contents exhibit-ing semimetallic behavior. While integration of directband gap Ge − x Sn x into electronic devices already car-ries great potential such as enhanced carrier mobilities,optical interconnects, and efficient designs for devicesbased on tunneling such as tunneling field-effect tran-sistors (TFETs) or tunnel diodes, the possibility ofincreasing tin composition to induce semimetallic regionsor to counteract confinement-induced band gap wideningin nanoelectronic device designs further increases thesealloys’ technological value. With features in modern de-vices already below 10 nm and designs capable of exploit-ing semimetals by confinement- and surface-assisted bandgap engineering already proposed, the ability to furthertune band gaps by controlling alloy composition intro-duces another dimension of customization exploitable innanoelectronic designs. Fabrication of crystalline Ge − x Sn x alloys in the com-position range predicted to exhibit semimetallic behaviorwhen in bulk form has also been experimentally achieved.Recently, atomically flat epitaxial films grown on Ge(100)with tin content as high as x = 0 .
46 were reported in theliterature. In their study, the high levels of compres-sive strain resulting from incorporation of such high tincontent into films epitaxially constrained to germaniumsubstrates resulted in maximum crystalline film thick-nesses of 3 nm. In contrast, the use of lattice-matchedsubstrates has been reported to allow growth of alloyswith compositions 0 . < x < .
99 and even stabiliza-tion of pure α -Sn films with thicknesses in excess of 100nm; the reduction of strain associated with growthon lattice-matched substrates thus allows fabrication ofsemimetallic Ge − x Sn x with thicknesses on the order ofhundreds of nanometers, which is well above the lengthscales required for modern nanoelectronics.In this work we investigate the structural proper-ties and thermodynamic stability of bulk and epitaxialGe − x Sn x alloys. We employ density functional theory-based simulations in conjunction with the cluster expan-sion formalism in order to explore the relative stability ofalloys across the full alloy composition range and devisestrategies for maximizing the miscibility of germaniumand tin. II. METHODS
In order to assess the thermodynamic stability ofGe − x Sn x alloys across the concentration range we em-ploy the Helmholtz free energy F (x) = E (x) − T S (x) , (1)where E is the internal energy, S the entropy, and bothterms include vibrational, electronic, and configurationalcontributions. We describe the alloy in terms of an Isingmodel and compute the internal energy’s configurationaldependency by employing two levels of approximation:the Bragg-Williams model, and the cluster expan-sion formalism. The Bragg-Williams (BW) model is a mean-field modelin which the configurational energy of a substitutionalalloy on an Ising lattice is described by nearest-neighborinteractions as E config = X i,j = { Ge , Sn } V ij N ij , (2)where V ij and N ij are the interaction energies (bondstrengths) and number of bonds between species i and j , respectively. By assuming the alloy to be random –anassumption supported by experimental results– wemay write the concentration-dependent energy per atomas E config (x) = (1 − x) V GeGe + x V SnSn + 2x(1 − x) V, (3)such that computing the alloy’s configurational energyonly requires knowledge of the interaction energies V ij ,which can be obtained from total energy calculations per-formed on crystalline structures ( e.g. diamond or zinc-blende crystal structures). The sign and magnitude of the relative binding energy V = V GeSn −
12 ( V GeGe + V SnSn ) (4)provide a quick and intuitive measure of the system’spreference to either segregate or form a solid solution bycomparing the energy of heteronuclear bonds with theaverage energy of homonuclear bonds. An improved de-scription of the alloy’s energetics can be achieved by em-ploying a generalization of the model to include inter-actions beyond nearest neighbors: the cluster expansionformalism. By assigning spin-like occupation variables σ i to sites i in the Ising lattice, we may expand the internalenergy of an alloy with configuration σ = { σ i } as E config ( σ ) = X α J α ˆ S α ( σ ) , (5)where lattice sites are grouped into clusters α , ˆ S α is theirassociated pseudospin, and J α their effective cluster in-teractions (ECI) ( i.e. interaction energies). Since themagnitude of the ECI is found to decrease with clusterrange, the internal energy of any configuration σ can beapproximated by truncating the sum in eq. (5) to onlyinclude relatively compact clusters. The ECI in the trun-cated sum can then be determined from explicit total en-ergy calculations performed on a relatively small set ofconfigurations in an approach known as the structure in-version method or the Connolly-Williams method. Inorder to select the optimal set of crystalline structuresand interactions to include when building the cluster ex-pansion, we have employed the algorithm described inref. .The interaction energies required by both methodsoutlined above have been obtained employing density-functional theory (DFT) simulations within the usualKohn-Sham framework in an implementation usingnorm-conserving pseudopotentials and linear combina-tion of numerical atomic orbitals (NAO) basis sets. Basis sets used to expand wavefunctions include s4p4d3f2NAO for germanium and s2p3d3f2 NAO for tin, wherethe notation indicates the number of s -type, p -type, d -type, and f -type orbitals centered about atoms of eachspecies. Brillouin zone integrations are performed overa grid generated according to the Monkhorst-Pack scheme maintaining a density of at least 7 k-points / ˚A − ,whilst real-space quantities are discretized on a grid witha corresponding energy cut-off of at least 100 Ha. Thelocal density approximation (LDA) was employed for theexchange-correlation potential. Including spin-orbit in-teractions in our simulations was found to have a negli-gible effect on the structural properties and energetics ofa selected set of alloy structures and has thus been ne-glected, similarly to another recent study on this alloy. Structural relaxations are performed on all simulatedstructures until forces acting on atoms are below 5 × − eV / ˚A and all stress tensor elements are below 0 . TABLE I. Structural parameters of germanium and tin’s α phase as computed with DFT-LDA in this work, and previ-ously reported experimental values. a (˚A) B (GPa)This work Exp. This work Exp.Ge 5 .
64 5 .
74 75 . Sn 6 .
47 6 . .
16 42 . − . The structural properties of tin and germanium’s dia-mond structures computed with this method are shownin table I. Within this approximation computed equilib-rium lattice parameters show agreement with experimen-tally reported values to within 1%. While the bulk mod-ulus computed for germanium exhibits a deviation of lessthan 3% with respect to experimentally reported valuesobtained via ultrasound, our computed bulk modulus fortin’s alpha phase lie within previously reported values ob-tained by fitting of neutron scattering data. Structuralparameters for both alloy components are thus accuratelydescribed in our approximation.Finite temperature effects are incorporated into ourstudy by estimating contributions arising from vibra-tional and electronic degrees of freedom, as well as config-urational contributions to entropy. Electronic contribu-tions to free energy are computed within the one-electronand temperature-independent bands approximations byemploying the electronic density of states computedwithin DFT. Vibrational contributions are computedusing the bond stiffness versus bond length approachwhereby phonon frequencies in the alloy are estimatedvia a nearest-neighbor Born-von K´arm´an model withbond-length dependent force-constant tensors obtainedby computing reaction forces on crystalline structuresperturbed with strain and atomic displacements.
Configurational entropy has been estimated from Boltz-mann’s expression by accounting for all possible ways ofarranging atoms in a cell given its composition. Com-puted configurational, electronic, and vibrational contri-butions have been combined and temperature-dependentECI have been generated, allowing finite-temperaturefree energy simulations to be performed in larger cellswith dimensions on the order of hundred of nanometerswith with relatively low computational effort.The phase boundary between ordered and disorderedstates with varying temperature has been obtained acrossthe composition range via Monte Carlo (MC) simula-tions. MC simulations employing the described latticemodel and cluster expansion ECI have been performedon simulation cells with dimensions of at least 150 nmin order to include long-range interactions and accu-rately trace the boundary between low-temperature or-dered states and high-temperature random alloys usingthe procedure and implementation described in ref. .We complement our cluster expansion results by di-rectly exploring the properties of random alloys using DFT through the use of Special Quasirandom Structures(SQS). SQS represent a periodic supercell approximationto random alloys by targeting multisite correlations char-acteristic of the disordered state, thus enabling ab-initio simulations on such systems. We have employed a setof 64-atom SQS generated by targeting disordered-statepair and triplet correlations with ranges up to a nanome-ter; structures covering the ful composition range weregenerated using the algorithm and implementation de-scribed in refs. . III. RESULTS AND DISCUSSIONA. Bulk alloys
The configurational dependence of the random alloy’senergy across the composition range is estimated in a firstapproximation within the Bragg-Williams model by ob-taining nearest-neighbor interaction energies as the for-mation energy per bond of germanium and tin’s α phase,and an x = 0 . V GeSn = 11 meV, indicating the alloy’s preferenceto segregate at zero temperature.A more accurate description of alloy energetics hasbeen obtained by fitting the energy of 47 crystalline struc-tures with up to 8 atoms per cell to a cluster expansionincluding pairs with atoms up to 1 nm apart, and tripletswith ranges up to 0.52 nm. Figure 1(a) shows fittedvalues of effective cluster interactions, where we observecoefficients to decay rapidly with cluster diameter andnumber of sites in the cluster from around 20 meV/atomfor nearest-neighbor pairs down to less than 5 meV/atomfor longer range pairs and even lower values associated totriplets included in the fit. The CE cross-validation score–a measure of its predictive power analogous to the rootmean square error– is 5 meV/atom, indicating a levelof accuracy similar to that of DFT simulations used inthe construction of the fit. In accordance with observedexperimental behavior and results from the BW model,CE results predict germanium and tin to be immiscibleat zero temperature as no ordered structures were foundto be energetically favorable with respect to segregationinto each of the component’s α phase across the entirecomposition range, as shown in fig. 1(b).Results obtained with lattice models are complementedwith directly computed quantities employing ab-initio simulations of 64-atom SQS. Atomic positions and latticevectors in generated SQS have been relaxed using DFTunder a scheme that constrains the cell shape whilst al-lowing cell volume and atomic positions to vary; this pro-cedure provides the most accurate representation of thedisordered state by not allowing macroscopic anisotropyin cell shape relaxations. The difference in atomic radiiresults in structures where relaxed atomic positions ex-hibit root-mean-square deviations from ideal lattice sitesof up to 0 .
157 nm away for x = 0 .
5. Their structural pairs 5 10 triplets 5 10Cluster diameter (˚A) − − E n e r g y ( m e V ) . . . . . . ∆ E c o n f ( m e V /a t o m ) fittedcalculated (a)(b) FIG. 1. (a) Magnitude of effective cluster interactions (ECI)obtained in the cluster expansion fit for bulk alloys and (b)predicted and directly computed formation energies per atomfor all structures included in the fit. . . . . . . ∆ E c o n f ( m e V /a t o m ) (a) (b) (c) (d) FIG. 2. Formation energy of mixing at zero temperature of(a) random alloys and (b) quasi -random alloys as predictedby the cluster expansion; (c) random alloys as predicted bythe BW model, and (d) 64-atom SQS as predicted by DFTsimulations. properties are compared to available literature by fittingtheir equilibrium lattice constant across the compositionrange to a Ge − x Sn x = a Ge0 (1 − x) + a Sn0 x + b a x(1 − x) , (6)where we have obtained a bowing parameter value of b a = 0 .
056 ˚A, in agreement with values reported in re-cent theoretical and experimental works.
Figure 2shows the formation energy per atom of random alloyswith respect to spinodal decomposition across the con-centration range as predicted with the Bragg-Williamsmodel, the cluster expansion, and as directly computedwith DFT simulations of structurally optimized SQS. Weobserve that although the BW model correctly predictsimmiscibility, it significantly underestimates the forma-tion energy of random alloys compared to results ob-tained with CE and SQS. The higher accuracy and pre-dictive power of the CE is reflected by its significantlysmaller deviations away from formation energies directlycomputed with DFT using SQS, where an RMS error of2 meV/atom between both datasets has been computed.We additionally plot the CE predicted formation energiesfor quasi -random structures with site correlations corre-sponding to those of generated SQS; the closer tracking ofSQS formation energies by this curve –especially aroundintermediate compositions, where larger deviations arefound– highlights the impact of imposed periodicity onSQS energetics.We extend our model to finite temperatures by es-timating the magnitude of configurational, electronic,and vibrational entropy contributions to alloy free en-ergy. Electronic entropy has been estimated employingthe electronic density of states computed with DFT forSQS across the full composition range: contributions toalloy free energy have been found to be negligible withcomputed values below 1 meV/atom at temperatures upto 1000 K across the entire concentration range, as ex-pected for non-metallic alloys. Electronic entropy contri-butions to free energy have not been included in resultspresented in the remainder of this work.Free energy contributions arising from vibrational de-grees of freedom have been estimated by fitting the com-ponents of a nearest-neighbor force constant tensor to re-action forces on crystalline structures with small atomicdisplacements and for varying degrees of strain. Vibra-tional entropy contributions for random alloys across thecomposition range have then been computed employingphonon frequencies associated with bond length distribu-tions in structurally optimized SQS cells.The relative magnitude of entropy contributions in-cluded are explored through results directly obtained forSQS cells. Figure 3 shows the temperature dependenceof the free energy of mixing for structures at composi-tions of x = 0 .
25, 0 .
5, and 0 .
75: configurational con-tributions dominate alloy stabilization while vibrationalentropy of mixing contributes to a lesser degree with val-ues around an order of magnitude lower at temperaturesabove 200 K in compositions shown in the figure. We − ∆ F conf ∆ F vib ∆ F elec ∆ F − E n e r g y ( m e V /a t o m ) − .
25x = 0 .
50x = 0 . FIG. 3. Temperature dependence of the total free energyof mixing and considered contributions as estimated for 64-atom SQS for (a) x = 0 .
25, (b) x = 0 .
50, and (c) x = 0 . observe predicted critical temperatures –i.e. tempera-tures at which the disordered phase becomes energeti-cally favorable– to decrease for larger tin compositions ina result attributable to the asymmetry observed in the al-loy’s configurational energy of mixing ∆ E conf (see fig. 2).While this asymmetry in formation energy with composi-tion is also predicted by the CE fit, the BW model’s lackof triplet interactions fails to reproduce it. We furtherinvestigate the origin of this feature in alloy energeticsby decomposing the computed energy of mixing as:∆ E conf = ∆ E V D + δE chem UR + δE int , (7)where the volume deformation energy ∆ E V D corre-sponds to the energy required to hydrostatically strain
Composition ∆ E conf ∆ E V D δE chemUR δE int (x) (meV/atom)0.25 36 192 -88 -670.50 41 207 -90 -760.75 28 128 -52 -48TABLE II. Formation energy of mixing decomposition intocontributions listed in eq. (7) for bulk alloys with varyingcomposition. each of the constituents to the alloy’s equilibrium latticeparameter, the chemical or spin-flip energy δE chem UR corresponds to the energy gained when both componentsalready strained bond together to form the alloy, andthe internal relaxation energy δE int is the energy gainedwhen atomic positions in the alloy are allowed to relax.Figure 4 shows the magnitude of each contribution inquasi-random alloys as computed for 64-atom SQS withcompositions x=0.25, x=0.50, and x=0.75. The largedifference between both components’ equilibrium latticeparameter results in the volume deformation energybeing the single largest contribution to mixing enthalpyand main source of alloy destabilization. Contributionsarising from chemical interactions between differentatomic species and the relaxation of internal coordinatesare of similar magnitude and act to stabilize the alloys,partially counteracting the effects of volume deformation.Table II lists the the magnitude of each contribution forthree alloy compositions: the difference in components’bulk moduli introduces an asymmetry in alloy energeticsand results in larger volume deformation contributions–and thus decreased stability– in germanium-rich alloyswhen compared to tin-rich alloys. While stabilizingchemical and internal relaxation contributions alsoexhibit larger magnitudes for germanium-rich alloystheir smaller magnitude does not offset the destabilizingasymmetry, resulting in an increased energetic stabil-ity of alloys on the tin-rich side of the composition range.We explore the effects of this asymmetry on the sys-tem’s critical temperature by tracing the phase boundarybetween the disordered phase and decomposition into el-emental α phases by combining the obtained CE fit witha lattice model Monte Carlo (MC) simulation. MC sim-ulations of lattice models represent an accurate methodto compute thermodynamic properties of substitutionalalloys. These methods extend beyond the capabilities offirst-principles calculations by allowing the simulation ofcells with sizes on the order of hundreds of nanometers,thus enabling the inclusion of long-range interactions im-practical to include in smaller ab-initio simulations cells.We include the effects of vibrational degrees of freedomby fitting contributions obtained for SQS cells across thefull composition range to temperature-dependent ECIsincorporating both the effects of configurational and vi-brational degrees of freedom. . . . . . . − E n e r g y ( m e V ) ∆ E conf ∆ E V D δE chemUR δE int FIG. 4. Formation energy of mixing decomposition obtainedfrom DFT simulations of bulk alloys employing SQS. . . . . . . T e m p e r a t u r e ( K ) FIG. 5. Phase diagram of bulk alloys as calculated employ-ing CE fits constructed including (dashed line) and excluding(solid line) vibrational degrees of freedom. Regions above thelines represent stability of random alloys.
The phase diagram for the Ge − x Sn x system calcu-lated via MC lattice model is shown in fig. 5. Criticaltemperatures obtained with this method by only includ-ing configurational degrees of freedom are significantlyhigher than those predicted using SQS cells. This resultis presumably due to a combination of the latter’s devi-ations of site correlations from those of random alloys,and the omission of long-range interactions inherent inthe use of smaller cells required to maintain tractabilityfor ab-initio simulations. Inclusion of vibrational degreesof freedom reduces predicted critical temperatures witha more pronounced effect on the germanium-rich side ofthe composition range. This indicates the stabilizationassociated with the softening of phonon modes in thematerial by addition of tin partially counteracts thedecreased stability associated with the compression ofgermanium bond lengths. . . . . . a S (nm)0 . . . . . . . . . q ( a S , ˆ G ) . . . . . . . . . . . . q ( a S , ˆ G ) h ih ih i GeSn(a)(b)
FIG. 6. Epitaxial softening functions for (a) Ge and (b) Snas computed employing DFT simulations.
B. Epitaxial alloys
Predicted critical temperatures for bulk alloys at in-termediate compositions shown in fig. 5 are well abovetin’s melting temperature and even near germanium’s,indicating growth of bulk Ge − x Sn x alloys is not possi-ble at ambient pressure. Since the main contribution tosuch high critical temperatures has been found to arisefrom the large difference in volume between components,we explore the effects of coherent epitaxial growth on al-loy energetics as a way of reducing forces driving desta-bilization by constraining the lattice spacing of the al-loy –and thus of the potentially segregating components–along the plane defined by the substrate surface in a phe-nomenon known as epitaxial stabilization . To investigate how tin-germanium alloys can be sta-bilized by epitaxial growth of thin films, we simu-late SQS cells with a crystallographic plane constrainedto lattice spacing corresponding to substrates whichhave been previously employed for growing epitaxi-ally stable films of α -Sn and/or germanium-tin al-loys: Ge (5 . / GaSb (6 . / InSb(6 . In order to select a crystallographic plane along whichto simulate growth we compute the epitaxial softeningfunction of both germanium and tin as q ( a S , ˆ G ) = ∆ E epi ( a S , ˆ G )∆ E bulk ( a S ) , (8)which gives the ratio between the epitaxial increase inenergy due to biaxial strain to a particular substrate . . . . . . − − − − − − − CE-SQSCEDFT-SQS c eq -CET c eq -DFT . . . . . . . . . . − − − − − − − ∆ E e p ( m e V /a t o m ) . . . . c e q ( ˚A ) . . . . . . − − − − − − −
505 5 . . . . FIG. 7. Formation energy of mixing with respect to coher-ent decomposition and out-of-plane equilibrium cell parame-ter c eq for alloys grown on (a) Ge, (b) ZnTe/GaSb, and (c)CdTe/InSb. lattice constant a S along a plane perpendicular todirection ˆ G , and the hydrostatic increase in energy dueto triaxial strain to the same a S . This dimensionless pa-rameter quantifies the degree of out-of-plane relaxationexhibited by a material grown epitaxially; it is desirableto minimize q ( a S , ˆ G ) (and thus ∆ E epi ( a S , ˆ G )) for agiven substrate in order to avoid or reduce dislocationsand other strain-induced film/surface defects. Figure 6shows the epitaxial softening functions of germaniumand tin as computed with DFT, where we observe h i to be the softest direction for both constituents aroundtheir equilibrium lattice constant, and for alloys acrossthe composition range if we approximate their epitaxialsoftening function as the corresponding weighted sum.We thus find that (100)-oriented substrates minimizestrain energy and thus structural defects on epitaxiallycoherent Ge − x Sn x films, in agreement with experimen-tal literature. We simulate epitaxial growth on (100)-oriented sub- strates by constraining 64-atom SQS along two of their h i directions to match the lattice spacing of each ofthe proposed substrates while allowing the cell to relaxalong the perpendicular direction. While this schemeneglects the potentially large role of surfaces and inter-faces inherently present around thin films, it provides anenergetic baseline and thus insight into the stability ofepitaxially alloys independent of surface effects and forportions of the material deep into coherently grown filmswith thicknesses greater than a few tens of nanometer,once surface effects have been screened. Figure 7 showsthe computed epitaxial formation energy relative tocoherent decomposition of its constituents; we observehow the strain-induced destabilization of constituentsegregation imposed by epitaxial coherency results inthe stabilization of alloys even at zero temperature.Structural relaxation simulations of alloys constrainedto Ge(100) substrates with compositions above x = 0 . ab-initio simulations have been fitted toCEs with the corresponding symmetry in order to inves-tigate the case of random alloys and eliminate variationsdue to deviations in site correlations springing fromthe use of periodic cells. The CE fits predict all alloysto be energetically favorable with respect to coherentdecomposition, with results showing a trend where sta-bility decreases along the (ZnTe/GaSb)-(CdTe/InSb)-Gesequence, indicating growth on germanium to be theleast favorable choice. Additionally, Figure 7 also showsthe magnitude of the out-of-plane lattice parameter c eq for each substrate as computed with SQS-DFT, andas predicted by continuum elasticity theory (CET) where good agreement between both methods is ob-tained across most of the composition range in Geand ZnTe/GaSb substrates, and significant deviationsare observed for low tin content alloys coherent withCdTe/InSb substrates.Decompositions of epitaxial formation energies akin tothose defined for bulk alloys in eq. (7) are presented intable III for alloys with composition x = 0 .
5. The effectsof growth on lattice-matched substrates can be observedin the formation energy decomposition correspondingto ZnTe/GaSb substrates: while computed epitaxialspin-flip ( δE chem , epi ) and internal relaxation ( δE int , epi )energies remain the same as in the corresponding bulkalloy, volume deformation energy relative to epitaxiallycoherent decomposition ∆ E epi V D is reduced by 37%, re-sulting in significantly enhanced stability. By comparingresults across substrates we observe that while destabi-lization of component segregation results in an overallreduction of ∆ E epi V D with increasing substrate latticeparameter due to germanium’s larger bulk modulus, themagnitudes of stabilizing contributions δE chem , epi UR and δE int , epi are observed to also decrease with increasingbond lengths associated with alloys grown on substrateswith larger lattice spacings. Substrate ∆ E epiconf ∆ E epi V D δE chem, epi UR δE int, epi (meV/atom)Ge -10 195 -107 -97ZnTe/GaSb -32 131 -89 -75CdTe/InSb -24 114 -76 -62TABLE III. Epitaxial formation energy of mixing decomposi-tion for alloys with composition x = 0 . . . . . . . ∆ E e p i c o n f ( m e V /a t o m ) x = 0.25 x = 0.76 GeZnTe/GaSbCdTe/InSb
FIG. 8. Formation energy of mixing of epitaxially grown al-loys with respect to decomposition into bulk components.
Finally, we compare the relative stability of alloysgrown on different substrates in fig. 8 by computing epi-taxial alloys’ formation energies with respect to spinodaldecomposition into their bulk components. The substrate providing the lowest formation energy (and thus higheststability with respect to non-coherent decomposition) isobserved to depend on alloy composition. Within theconsidered set of substrates, Ge is preferred for composi-tions below x = 0 .
25, CdTe/InSb for compositions abovex = 0 .
76, and ZnTe/GaSb for 0 . < x < .
76, indicatingthe latter to be energetically favorable across most of thecomposition range where the alloy exhibits semimetal-lic behavior. The asymmetry discussed for bulk alloyswhereby tin-rich compositions exhibit increased stabilityis also present for alloys grown on ZnTe/GaSb, as can beclearly seen in fig. 8.
IV. CONCLUSION
The thermodynamics of tin germanium alloys havebeen studied using a cluster expansion (CE) approachparametrized from first principle DFT simulations. Thenearest neighbor version of the CE (the BW model) fora random alloy correctly predicts tin and germanium tobe immiscible across the full concentration range at zerotemperature. Inclusion of clusters with higher order and longer range into the expansion dramatically improvesagreement with first principle simulations, both in termsof magnitude and an asymmetric skew observed in alloyformation energies as a function of alloy composition. Itis found that including pair interactions with ranges ofup to a nanometer and triplet interactions up to aroundhalf a nanometer is sufficient to describe the energet-ics of bulk alloys. Temperature effects on free energyare included by adding configurational, electronic, andvibrational entropies. Electronic entropy contributionsare estimated from a one electron approximation and byconsidering band energies to be independent of temper-ature, and contributions arising from vibrational degreesof freedom to both entropy and internal energy (zero-point energy) are estimated from bond stiffness versusbond length approximation. Free energy contributionsarising from electronic degrees of freedom are found tobe negligible and are expected to only weakly influencethe alloys’ thermodynamics. It is shown that the criti-cal temperature for stability of random alloys generallydecreases for higher Sn concentrations as a result of anasymmetry in the mixing enthalpy; decomposition of thelatter into a volume deformation term related to eachelement’s elastic properties, a chemical energy term re-lated to bond strengths, and an internal relaxation en-ergy. The volume deformation term dominates the zero-temperature mixing enthalpy and severely reduces thestability of bulk alloys, although it is partially counter-acted by the stabilizing effects of the chemical and relax-ation energies. The large volume deformation energiesresults in predicted critical temperatures that are higherthan the melting temperature of Sn and are only slightlylower than the melting temperature for Ge.After identifying the volume deformation energy as thelargest contribution towards destabilization of bulk al-loys, the influence of biaxial strain due to coherent epi-taxial growth is considered for different substrates whichhave been previously employed to grow germanium-tinalloys. The role of epitaxial stabilization on the freeenergy is considered for three substrates: Ge (5 . / GaSb (6 . / InSb (6 . ACKNOWLEDGMENTS
This work was funded by Science Foundation Ire-land through a Principal Investigator award Grant No.13/IA/1956. ∗ [email protected] U. Mizutani,
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