Temperature-driven α-to-β phase transformation in Ti, Zr and Hf from first-principles theory

combined with lattice dynamics

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December 2011

EPL, 96 (2011) 66006 www.epljournal.org

doi: 10.1209/0295-5075/96/66006

Temperature-driven α-to-β phase transformation in Ti, Zr and Hffrom first-principles theory combined with lattice dynamics

P. Souvatzis1(a), S. Arapan

1,2, O. Eriksson

1 and M. I. Katsnelson3

1Department of Physics and Astronomy, Division of Materials Theory Uppsala UniversityBox 516, SE-751210, Uppsala, Sweden, EU2 Institute of Electronic Engineering and Nanotechnologies, Academy of Sciences of MoldovaAcademiei 3/3, MD-2028 Chisinau, Moldova3 Radboud University Nijmegen, Institute for Molecules and Materials - NL-6525 AJ Nijmegen,The Netherlands, EU

received 30 September 2011; accepted in final form 9 November 2011published online 14 December 2011

PACS 64.70.kd – Metals and alloysPACS 63.20.kg – Phonon-phonon interactionsPACS 63.20.Ry – Anharmonic lattice modes

Abstract – Lattice dynamical methods used to predict phase transformations in crystals typicallydeal with harmonic phonon spectra and are therefore not applicable in important situations whereone of the competing crystal structures is unstable in the harmonic approximation, such as the bccstructure involved in the hcp-to-bcc martensitic phase transformation in Ti, Zr and Hf. Here wepresent an expression for the free energy that does not suffer from such shortcomings, and we showby self-consistent ab initio lattice dynamical calculations (SCAILD), that the critical temperaturefor the hcp-to-bcc phase transformation in Ti, Zr and Hf, can be effectively calculated from thefree-energy difference between the two phases. This opens up the possibility to study quantitatively,from first-principles theory, temperature-induced phase transitions.

Copyright c© EPLA, 2011

Introduction. – Martensitic phase transformationsare common, both in alloys frequently used in industry,such as shape memory alloys [1], and in the the elemen-tal group 3 to 4 transition metals [2], not to mentionmartensitic transformation in iron and iron-based alloys,a crucial phenomenon for metallurgy [3]. Thus there existsa substantial interest both from an industrial, applied andan academic point of view to develop accurate and effec-tive methods to understand and even predict martensiticphase transformations.The hcp-to-bcc (or α-to-β) transition in Ti, Zr and

Hf is a martensitic phase transformation that has beenthoroughly investigated both from an experimental [2,4]and a theoretical [5–8] perspective. Recently Henniget al. [5] developed and used a classical potential of themodified embedded atom method (MEAM) [9] to accu-rately reproduce the phase boundary between the hcp andbcc structure in Ti. However, there is up to this date alack of first-principles theoretical studies made of the mar-tensitic hcp-to-bcc phase transformation in Ti, Zr and Hf.

(a)E-mail: petros.souvatzis@fysik.uu.se

The problem is that anharmonic effects in lattice dynam-ics [10] are of crucial importance for finite-temperaturestructural phase transitions, and their quantitativefirst-principle treatment is a real challenge.A straightforward calculation using DFT molecular

dynamics (DFT-MD) [11] should in principle be able toreproduce the bcc-to-hcp phase transformation in Ti andsimilar materials, since DFT-MD implicitly include anhar-monic effects. However, DFT-MD is a computationallyvery demanding task which makes its use problematic.Instead we will here exploit the method of self-consistentab initio lattice dynamical calculations (SCAILD) [12].Here, we further develop this method in order to be ableto calculate thermodynamic properties, such as structuralfree-energy difference (before we were restricted by thecalculations of temperature-dependent phonon frequen-cies only [12–16]). The SCAILD scheme is a constrainedsampling method, in that it only samples the latticedynamical phase space along the normal mode directionsof commensurate phonons, and only at atomic displace-ments corresponding to the mean square atomic devia-tions dictated by thermodynamics [12,13]. Furthermore,

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P. Souvatzis et al.

in the SCAILD scheme thermal equilibrium is seta priori [12,13], instead of reached a posteriori, as inDFT-MD. Hence, by virtue of not being forced to estab-lish thermal equilibrium and due to the limited phasespace sampling, SCAILD calculations are much fasterand, thus, much more practical than the correspondingDFT-MD calculations.We propose here a new expression of the free energy of

the SCAILD scheme, and we show that from the atomicconfigurations and the phonon density of states producedby the SCAILD calculations, an accurate measure of thefree energy for the different phases can be obtained.

Method. – In order to properly describe temperature-driven phase transformations in general, one must includethe interaction between phonons [10]. As a result, phononfrequencies turn out to be temperature dependent whichwe explore numerically in this study by means of theSCAILD method [12–16].The SCAILD method is based on the calculation of

Hellman-Feynman forces of atoms in a supercell. Themethod can be viewed as an extension of the frozenphonon method [17], in which all phonons with wavevectors q commensurate with the supercell are excitedtogether in the same cell by displacing atoms situated atthe undistorted positions R+bσ, according to R+bσ→R+bσ +URσ, where the displacements are given by

URσ =1√N

∑q,s

Aσqsεσqseiq(R+bσ). (1)

Here R represent the N Bravais lattice sites of thesupercell, bσ the position of atom σ relative to thissite, εσqs are the phonon eigenvectors corresponding tothe phonon mode, s, and the mode amplitude Aσqs iscalculated from the different phonon frequencies ωqsthrough

Aσqs =±√

�

Mσωqs

(12+nqs

), (2)

where nqs = n(ωqskBT), with n(x) = 1/(ex− 1), are the

phonon occupational numbers, Mσ the atomic massesand T is the temperature of the system. Thus, due to theanharmonic nature of the force constants, and throughthe Bose-Einstein occupational factors, nqs, appearing inthe mode amplitudes Aqs, the temperature dependence isintroduced to the renormalized phonon frequencies. Thephonon frequencies, ωqs, are obtained through projectingthe Fourier-transformed atomic forces, Fσq, along theeigenvectors, εσqs, of the phonon modes [12].Since in the SCAILD scheme, the atomic displacements

present in the calculation of the forces, are the results ofsuperpositions of all the commensurate phonon modes (asopposed to displacing the atoms along one single mode,which would correspond to the frozen phonon method),the interaction between different lattice vibrations aretaken into account1 and the phonon frequencies are thus

1See eq. (16), p. 890 and the appendix of ref. [13].

renormalized by the very same interaction. For moredetails on the SCAILD method we refer to refs. [12–14].It should be mentioned that we do not consider here thephonon decay processes (see, e.g., ref. [18] and referencestherein).The free energy as a function of volume, V, and temper-

ature for the bcc and hcp structures can be calculatedthrough the expression

F (T, V ) =U0(V )+Fph(V, T )+Fel(V, T ), (3)

where U0 is the static ground-state energy of the respectivestructures at T = 0K (i.e. without any phonons excitedand temperature excitations of electronic states ), Fph isthe free energy of the phonons and Fel is the free energyof the electrons. The temperature dependent parts of thefree energy can be found as

Fph(V, T )+Fel(V, T ) =1

NI

∑{UR}

∆F ∗({UR}, V, T )

+3

2kBT −TSph(V, T ). (4)

Here ∆F ∗ is the change in free energy relative to theground-state energy U0, caused by the phonon-inducedatomic displacements described by eq. (1), and thermalexcitations of the electronic states. Here it should benoted that included in the term ∆F ∗ is also the freeenergy of the electrons Fel. The sum on the right-handpart of eq. (4) is over the NI number of different atomicconfigurations, {UR}, generated throughout the SCAILDself-consistent run. Since ∆F ∗ are calculated at atomicconfigurations accommodating the different frozen phononsuperposition of eq. (1), ∆F ∗ not only contains the finite-temperature contribution to the electronic free energyfor a given atomic configuration, but also, the potentialenergy provided by the frozen lattice waves, i.e. the poten-tial energy of the phonons at a particular phonon super-position [19]. The phonon kinetic energy is given by3kBT/2 per atom which means that atomic motion isconsidered as classical; typically, temperatures of struc-tural phase transformations are higher than the Debyetemperature, thus, this approximation is well justified.In practice, the sum of the finite-temperature elec-

tron free energy and phonon potential energy, ∆F ∗, wasobtained by calculating the total free energy of the corre-sponding atomic configuration, {UR}, using a Fermi-Diractemperature smearing of the Kohn-Sham occupationalnumbers [20], and then subtracting the static potentialenergy U0 (here U0 is calculated with the tetrahedronmethod to provide a good reference to the temperatureexcited electronic states). The number of atomic config-urations, NI , used for each volume and temperature wastypically 400.Another problem is how to calculate the phonon

entropy. We will assume that it depends on the phononoccupation numbers nqs in the same way as for

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Temperature-driven α-to-β phase transformation

-0.6

-0.5

-0.4

-0.3

-0.2

F [e

V]

HCPBCC -0.35

-0.3

-0.25

-0.2

-0.15

F [e

V]

HCPBCC -6

-5.9

-5.8

-5.7

-5.6

F [e

V]

HCPBCC

800 1000 1200

T [K]

-30

-20

-10

0

10

∆F [

meV

]

800 900 1000

T [K]

-40

-30

-20

-10

0

10

20

∆F [

meV

]

1500 1600 1700 1800

T [K]

-20

0

20

40

60

∆F [

meV

]

(a)

(b) (d) (f)

(c) (e)

Ti Zr Hf

Tc = 1107 K T

c = 916 K

Tc = 1660 K

Fig. 1: (Color online) In (a), (c) and (e) the calculated free energy of the bcc and hcp phase, in Ti, Zr and Hf, respectively. In(b), (d) and (f) the calculated free-energy difference, ∆F = Fhcp−Fbcc, in Ti, Zr and Hf, respectively.

noninteracting bosons:

Sph = kB∑qs

[(1+nqs) ln (1+nqs)−nqs lnnqs] . (5)

This is the only entropy expression consistent with eq. (2),which can be proved in the exact same manner as wasdone for fermions in the Landau theory of a normal Fermiliquid [21]. Thus the SCAILD scheme, together with afree energy defined through eqs. (4) and (5), constitutesnothing but a theory of a “normal Bose liquid”. Expression(5) can be written in terms of the phonon density of states,g(ω), produced by a converged SCAILD calculation, andis given by

TSph(V, T ) =

∫ ∞0

dωg(ω, V, T )�ω

[n

(�ω

kBT

)

−kBT�ωln(1− e− �ω

kBT

)]. (6)

Here the phonon frequencies used to calculate the phonondensity of states, g(ω), are the normal mode configura-tional mean values

〈�ωqs〉= 1NI

∑{UR}

δEtot({UR}, V, T )δnqs

. (7)

It should be stressed that the partitioning of the freeenergy through eqs. (3), (4) and (6) has been chosento maximize both the accuracy of the phonon potentialenergy, which in the form of eq. (4) take into accountanharmonicity up to infinite order, and the phononentropy, which in the form given by eq. (6) is accurate toleading order in anharmonic perturbation theory [22].

The phonon density of states and the corresponding freeenergies for the hcp and bcc structures were calculated atup to five different temperatures, and at each temperature,SCAILD calculations were performed at up to five differentvolumes.As regards the other computational details of the force

calculation we used the VASP package [20], within thegeneralized gradient approximation (GGA). The projectoraugmented-wave (PAW) potentials required energy cutoffsof 232 eV. The Ti(4s,3d), Zr(4s,4p,5s,4d) and Hf(6s,5d)levels were treated as valence electrons. The k -pointmesh was a 5× 5× 5 Monkhorst-Pack grid in the bccphase calculations. In the hcp phase calculations 6× 6× 6gamma centered mesh was used. In order to include theelectron entropy in the calculations, Fermi-Dirac temper-ature smearing were applied to the Kohn-Sham occupa-tional numbers. The bcc and hcp supercells used wereobtained by increasing the bcc primitive cell 4 times andthe hcp primitive cell 3 times, along the respective bccand hcp primitive lattice vectors. The sizes of the super-cells where chosen such that to ensure a sufficient decayof the interatomic force constant within the supercell,permitting a proper sampling of the lattice dynamicalphase space [23]. For the calculations of the static poten-tial energy, U0 in eq. (3), the all-electron full-potentiallinearized augmented-plane wave (FP-LAPW) packageELK [24] was used within the GGA approximation. Thiswas found necessary to ensure a high accuracy of thezero temperature part of the bcc-hcp energy difference.An energy cutoff of 270 eV together with a 24× 24× 24k-point mesh and a 24× 24× 15 k-point mesh were usedfor the bcc and hcp structures, respectively, in these calcu-lations the Methfessel-Paxton integration scheme was used

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P. Souvatzis et al.

with a 0.2 eV smearing of the Kohn-Sham eigenvalues. Itshould be noted that the electronic smearing used in thecalculations of the static energy, U0, was only introducedin order to achieve faster convergence of the energy withrespect to the number of k-points, and not introduced onphysical grounds, as in the case of the Fermi-Dirac temper-ature smearing employed in the calculations of ∆F ∗. Togauge the error introduced by the utilized 0.2 eV smearing,additional calculations were performed with the same k-point mesh but with the much more accurate linear tetra-hedron scheme [25]. The energies obtained with the lessaccurate smearing scheme differed only with ∼0.3meV,which is two orders of magnitude smaller than the energyof the zero point motion or the electronic entropy atT = 1100K.

Results. – For each temperature the free energyobtained through eq. (4) was fitted to a first-order poly-nomial in V . Then by using these first-order fits togetherwith eq. (3), the total free energy at each temperature wasobtained through minimization with respect to volume.In fig. 1(a), (c) and (e), the minimized free energy at eachtemperature is displayed for the bcc and hcp structure inTi, Zr and Hf, respectively. In fig. 1(b), (d) and (f) thefree-energy difference between the structures is displayedfor Ti, Zr and Hf, respectively. Due to the computationalintensity involved in the free-energy calculations, onlyin the case of Ti the free energy was calculated for 5temperatures, whereas in the case of Zr and Hf, thecalculations were limited to two temperatures.The temperature-driven hcp-to-bcc phase transforma-

tion in Ti can be seen to occur at T ∼1100K, whichis reasonably close to the experimentally observed phasetransition temperature of 1155K [26], whereas in Zr andHf, the transition is predicted to occur at T ∼920K andT ∼1660K, respectively. The theoretical estimates of thetransition temperature in Zr and Hf, with their respec-tive experimental data of 1135K and 2015K [2], do notcorrespond to experimental data as well as in the caseof Ti, however, given that these transition temperatureswere achieved by means of interpolating the free-energydifference on a twice as sparse temperature mesh ascompared with the temperature mesh employed in the Tifree-energy calculations, and that first-principles calcula-tions (at T = 0), with current exchange and correlationsfunctionals, have a problem in resolving energy differ-ences between different crystallographic phases betterthan ∼10meV/atom, one may not expect from any first-principles–based theory, like the SCAILD method usedhere, to reproduce temperature-induced phase transitionswith an accuracy better than a few hundred kelvin.It should be noted here that the presently used free-

energy expression differs significantly from the previouslyused expression from quasi-harmonic theory (e.g. usedin ref. [13]) in that it takes into account the anhar-monic contributions to the lattice dynamical potentialenergy without projecting it down on a quasi-harmonic

Fig. 2: The calculated phonon dispersions for Ti in the bccand hcp phases, here displayed as full drawn curves. In (a)the phonon dispersion for Ti in the bcc phase calculated atthe finite temperature T = 1293K, here displayed togetherwith experimental data [26] obtained at the same temperature(down-pointing triangles). In (b) the phonon dispersion ofhcp Ti calculated at T = 0K, here displayed together withexperimental data [27] measured at T = 295K (filled blackcircles). In (c) the calculated phonon dispersion for hcp Tiat T = 1000K, here displayed together with experimentaldata [27] measured at T = 1054K (up-pointing triangles).

description as in ref. [13]. In fact, using the old free-energyexpression together with the first-principles interatomicforces calculated in this work, results in the hcp phase inTi being > 90meV lower in free energy compared to thefree energy of the bcc phase throughout the entire temper-ature interval 800K<T < 1200K.It may also be of interest to mention the importance of

the different energy contributions appearing in the presentcalculations. Here the static potential energy, U0, of thehcp phase was in the order of ∼100meV lower than thecorresponding energy in the bcc phase. The most impor-tant contribution to the decrease of the free-energy differ-ence between the two phases at elevated temperatures wasthe phonon-related free energy, where the bcc phase freeenergies were of the order ∼100meV lower compared tothe corresponding energies in the hcp phase. The elec-tronic free-energy contribution related to the thermal exci-tations was one order of magnitude smaller compared tothe corresponding phonon contribution.

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Temperature-driven α-to-β phase transformation

Furthermore, to show that the phonon part of thecalculation is comparable to experimental data, we herealso show in fig. 2, as an example, the calculated Tiphonon dispersions for three representative cases. Apartfrom the complete absence of imaginary modes in thephonon dispersion of the Ti bcc structure, see fig. 2(a),one of the more striking temperature effects can beseen by studying the low- and high-temperature phonondispersions of the Ti hcp phase, shown in fig. 2(b), (c).Here the hardening of the high-frequency mode betweenthe Γ and A high-symmetry points, as well as betweenthe Γ and M symmetry points, is captured, resultingin an increased splitting between the longitudinal andtransverse optic modes, not only along the [001] direction,but also along the [100] direction close to the Γ point, asthe temperature is increased to 1000K, which is also theobservation made in experiment.

Conclusion. – In summary, we introduce here anexpression for the free energy which is applicable also forhighly anharmonic crystals. The free-energy expressioncan readily be interfaced with lattice dynamic methods,for instance the SCAILD technique, and we show thatthe temperature-induced hcp → bcc transition of Ti, Zrand Hf can be reproduced by theory. Theory puts thetransition temperature in Ti, Zr and Hf to 1100K, 920Kand 1660K, respectively.

∗ ∗ ∗

We would like to thank the Swedish National Infrastruc-ture for Computing (SNIC) for the allocation of compu-tational time at NSC, HPC2N and C3SE that madethis work possible. Support from VR, ERC (grant247062 - ASD), the KAW foundation and ESSENCE isacknowledged.

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