PHYSICAL REVIEW B 93, 014303 (2016)

Soft phonon modes in rutile TiO2

Björn Wehinger,1,2,* Alexeı̈ Bosak,3 and Paweł T. Jochym41Department of Quantum Matter Physics, University of Geneva, 24, Quai Ernest Ansermet, CH-1211 Genève, Switzerland

2Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland3European Synchrotron Radiation Facility, 71, Avenue des Martyrs, F-38000 Grenoble, France

4Department of Computational Materials Science, Institute of Nuclear Physics PAN, Cracow, Poland(Received 1 September 2015; revised manuscript received 9 November 2015; published 8 January 2016)

The lattice dynamics of TiO2 in the rutile crystal structure was studied by a combination of thermal diffusescattering, inelastic x-ray scattering, and density functional perturbation theory. We experimentally confirm theexistence of a predicted anomalous soft transverse acoustic mode with energy minimum at q = ( 12 12 14 ). Thephonon energy landscape of this particular branch is reported and compared to the calculation. The harmoniccalculation underestimates the phonon energies but despite this the shape of both the energy landscape andthe scattering intensities are well reproduced. We find a significant temperature dependence in the energy ofthis transverse acoustic mode over an extended region in reciprocal space, which point to a strong role ofanharmonicity in line with a substantially anharmonic mode potential-energy surface. The reported low-energybranch is quite different from the ferroelectric mode that softens at the Brillouin zone center and may helpexplain anomalous convergence behavior in calculating TiO2 surface properties and is potentially relevant forreal behavior in TiO2 thin films.

DOI: 10.1103/PhysRevB.93.014303

I. INTRODUCTION

The transition metal oxide TiO2 with its mixed ionic andcovalent bonding is a challenging system in condensed matterphysics with many interesting applications. Its low surfaceenergy with a particular energy landscape is, for example, veryattractive for surface chemistry and photocatalysis. In the rutilecrystal structure TiO2 is an incipient ferroelectric and shows alarge and strongly temperature-dependent dielectric constant.The consequent high refractive index is valuable for a diverserange of technological applications such as dielectric mirrors,antireflecting coatings, and pigments. Similarly to ferroelectricperovskites, the dielectric constant increases with decreasingtemperature. The incipient ferroelectric behavior and the highdielectric constant are directly linked to the lattice dynamicsand they were shown to be a consequence of a low-frequencytransverse optical phonon mode A2u [1]. Experimentally, thephonon dispersion relation could be measured using neutronspectroscopy [2]. Effects of temperature and pressure on thestatic dielectric constants, frequencies of Raman active modesand Grüneisen parameters were addressed by capacitancemeasurements and Raman scattering and a particularly strongdependence on both pressure and temperature was foundfor the A2u mode and related to structural and anharmoniccontributions [3,4]. Ab initio lattice dynamics calculationsusing density functional perturbation theory (DFPT) were usedto calculate phonon frequencies, Born effective charges and thedielectric permittivity [1,5–7]. These calculations emphasizethe importance of the mixed covalent ionic bonding of sand d orbitals, which is linked to its large polarizability dueto long-range Coulomb interactions between the ions. Thelarge Born effective charges obtained could be explained bydynamical electron transfers during atomic displacements.

*bjorn.wehinger@unige.ch

Some of the phonon modes show a pronounced stressdependence: a significant softening of the ferroelectric A2uphonon mode was found [5]. A recent theoretical workfound an anomalously soft transverse acoustic (TA) modequite separate from the A2u mode [6]. This mode shows aremarkable stress dependence in a region of the reciprocalspace unexplored by experiment and softens at around q =( 12

12

14 ) under isotropic expansion. The atomic displacements

of all atoms for this mode at this specific momentum transferare perpendicular to the (110) surface. They allow for longranged propagation of a displacement arising at the surfacedeep into the bulk. Under uniaxial strain, the mode is predictedto soften for both expansion and compression. The phononfrequencies are very sensitive to slight changes in the geometryand thus to the applied exchange-correlation functionalsand accuracy of the pseudopotentials [8]. Very recently, theimportance of anharmonicity in the lattice dynamics of TiO2rutile has been addressed by inelastic neutron scattering andab initio molecular dynamics simulations [9]. Temperaturedependent measurements of the phonon density of statesrevealed a significant increase in energy of the TA phononswith temperature. The phonon potentials of this mode werereported to develop remarkable quartic terms due to a dynamicchange of the hybridization between titanium and oxygenatoms.

Inspired by the work of Mitev et al. [6] and motivated by thefact that soft modes are very sensitive to the approximationsused in theory, we experimentally address the TA mode byinelastic and diffuse x-ray scattering and compare the resultsto harmonic DFPT calculations. We explore an extendedregion of reciprocal space by thermal diffuse scattering (TDS),measure the energy of the TA phonons with mode levelresolution by temperature dependent inelastic x-ray scattering(IXS), and carefully compare the phonon energy landscapeand scattering intensities to the calculation. Finally, we discussdetails of the softening and the importance of anharmonicity.

2469-9950/2016/93(1)/014303(7) 014303-1 ©2016 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.93.014303

WEHINGER, BOSAK, AND JOCHYM PHYSICAL REVIEW B 93, 014303 (2016)

II. METHODS

A. Experiments

Synthetic TiO2 single crystals from CrysTec GmbH (Berlin,Germany) grown by the Verneuill process were used in thisstudy. A rectangular bar with dimensions of approximately0.2 × 0.2 × 1.5 mm3 was produced from a commerciallyavailable crystal by cutting and polishing. The crystal qualitywas checked by high-resolution x-ray diffraction and is visiblein the reconstructed planes of diffuse scattering, see Sec. III.

Single crystal IXS was carried out at beamline ID28 at theEuropean Synchrotron Radiation Facility (ESRF) in Grenoble,France. The spectrometer was operated at 17.794 keV incidentenergy, providing an energy resolution of 3.0 meV full widthhalf maximum (FWHM). The momentum resolution was set to0.3 (horizontal) × 0.9 (vertical) nm−2 FWHM. Energy transferscans were performed in transmission geometry at constantmomentum transfer (Q), selected by appropriate choices ofscattering angle and sample orientation. Further details ofthe experimental setup can be found elsewhere [10]. Thetemperature dependent study combines data from experimentsusing a closed cycle helium cryostat (20, 50, 75, 122.5, 150,200, and 250 K), a cryostream cooling system (105 K), and nocooling device at all for the room-temperature measurement.

X-ray diffuse scattering was performed on beamline ID29at the ESRF. Monochromatic x rays with 16 keV energy werescattered form the single crystal in transmission geometry.The sample was incrementally rotated orthogonal to the beamdirection by 0.1◦ over a full cycle while diffuse scatteringpatterns were collected in a shutterless mode with a single-photon counting pixel detector with no readout noise andpixel size of 172 μm (PILATUS 6M detector from DectrisAG., Baden, Switzerland). The orientation matrix was definedusing the CrysAlis software package (Agilent technologies,Santa Clara, USA). Reciprocal space reconstructions wereprepared using locally developed software. Corrections forpolarization of the incoming x rays and for solid angleconversion associated with the planar projection were appliedto the scattering intensities of individual pixels.

B. Calculation

Lattice dynamics calculations were performed in theharmonic approximation using density functional perturba-tion theory (DFPT) [11] as implemented in the CASTEPcode [12,13]. In the calculations, we employed local den-sity approximation (LDA) with the exchange correlationfunctional by Perdew and Zunger [14]. We used the plane-wave formalism and norm-conserving pseudopotentials ofthe optimized form [15]. The LDA exchange correlationfunctional was chosen because the derived lattice parametersand vibrational properties are correctly described. Withinthe generalized-gradient approximations (PBE and PW91),the equilibrium crystal structure of TiO2 is predicted to beunstable, which leads to imaginary phonon frequencies [8,16].We have used the pseudopotentials from Mitev et al. [6] whichtreat five reference orbitals as valence states for titaniumand two for oxygen. A plane wave cutoff of 1100 eV andan electronic grid sampling on a 4 × 4 × 6 Monkhorst-Packgrid ensured convergence of forces to

SOFT PHONON MODES IN RUTILE TiO2 PHYSICAL REVIEW B 93, 014303 (2016)

FIG. 2. IXS intensity of TiO2 crystal as a function of energy transfer along Q = (2.5 2.5 0.5) to (2.5 2.5 1.5) (A-M-A), obtained fromexperiment at 295 K (a) and calculation (b). Positions of q = ( 12 12 14 ) are indicated by arrows. The experimental map consists of eleven linearlyspaced spectra with energy steps of 0.7 meV. The calculation was performed at 200 q points.

Importantly, we experimentally confirm the existence of thepreviously predicted low-energy feature close to a reducedmomentum transfer q = ( 12 12 14 ). IXS scans along M-A aresummarized in an intensity map, see Fig. 2. The dispersionclearly shows dips at Q = (2.5 2.5 0.75) and (2.5 2.5 1.25),which both correspond to q = ( 12 12 14 ).1 The shape of thedispersion is reasonably reproduced within the harmonic (0K)

1Scattering intensities depend on the absolute momentum transfer.We thus report both absolute (Q) and reduced (q) values.

DFPT calculation in spite of a significant underestimation inphonon energies.

The variation of the TA mode energy across the HHLplane is investigated by IXS and measured at a grid of Qpoints on that plane. The extracted phonon energies of thisparticular mode are presented on a color map and comparedto the calculation in Fig. 3. We note that the phonon energylandscape forms an elongated valley. The energy landscape isnicely described by the calculation along [ξξ l]. It is also wellreproduced along [hhξ ], except in the center of the image,where the valley appears to be a little broader (along [00ξ ]). Acloser look at the absolute energies (note the different absolute

FIG. 3. Phonon energy landscape of the soft TA mode on the HHL plane. The phonon energies depicted in color were extracted from IXSmeasurements taken at 105 K on 90 Q points (a) and calculated on 5150 Q points (b). Please note the different energy scales of the subpanels.Positions of q = ( 12 12 14 ) are over-plotted by dots.

014303-3

WEHINGER, BOSAK, AND JOCHYM PHYSICAL REVIEW B 93, 014303 (2016)

FIG. 4. TDS of TiO2 crystal. (a) and (b) Intensity distribution across high symmetry planes in reciprocal space, reconstructed fromexperiment (left part of individual panels), are compared to the calculation (right part of individual panels). (c) and (d) Isointensity distributionof TDS in 3D around Q = (2.5 2.5 1) as obtained from experiment and calculation, respectively. The isovalue is chosen arbitrarily for bestvisualization. Colors indicate the distance from the (2.5 2.5 1) point.

color scales in the figure) reveals that the calculated phononenergies differ by about 5 meV across the plane. Only the redareas have comparable phonon energies.

The variation in 3D reciprocal space is investigated byTDS. Scattering intensities on reconstructed high-symmetryreciprocal space sections and the 3D isointensity distributionaround Q = (2.5 2.5 1) are shown in Fig. 4 and compared tocalculations. On the HK0 plane, we note that the intensitydistribution of TDS, which contains contributions from allphonon modes, is well reproduced by the calculation acrossthe plane. Indeed, in this plane the calculated phonon dis-persion is quite accurate. The intensity of diffuse scatteringis also sensitive to the phonon eigenvectors. The excellentagreement gives high degree of confidence that the calculationproperly reproduces both energies and eigenvectors across thisreciprocal space section. The diffuse scattering from the TAmode along �-M is particularly strong and forms streaksof high intensity for example between (0 2 0) and (2 0 0).

On the HHL plane one can identify the diffuse scatteringfrom the soft TA mode. The shape of this feature is verysimilar to the phonon energy landscape (Fig. 3). In fact,this mode is well separated from higher energy modes andcontributes to 98% to the total TDS at Q = (2.5 2.5 0.75). Theeffect of the calculation underestimating the phonon energyis clearly noticeable: calculated diffuse scattering intensitiesare significantly stronger than experimentally observed. Anillustration if the isointensity distribution gives insight to theenergy landscape in 3D, see Figs. 4(c) and 4(d). We note anelongated shape in HHL with very little variation orthogonalto it.

The variation of selected phonon energies with temperatureis shown in Fig. 5. A significant temperature dependence ofthe soft TA mode at q = ( 12 12 14 ) and Z is visible. We note analmost linear increase of phonon energies with temperature.Interestingly, the dependence on Z point and q = ( 12 12 14 ) isvery similar. For both q points, the calculated phonon energies

014303-4

SOFT PHONON MODES IN RUTILE TiO2 PHYSICAL REVIEW B 93, 014303 (2016)

FIG. 5. Temperature dependence of the phonon energies of the TA mode at selected q values. (a) Energy transfers measured by IXS(connected blue symbols) and calculated values (black symbols at zero temperature) at q = ( 12 12 14 ), Z and q = (0 14 0). (b) Dispersion along�-Z, measured by IXS (connected symbols, 20 K data at Z point only) and calculation (solid black line).

are significantly too large. At q = (0 14 0), the temperaturedependence is less pronounced and the agreement with thecalculation is rather good.

The effect of anharmonicity is investigated by frozenphonon calculations at q = ( 12 12 14 ). The resulting phononenergy potential surfaces for different isotropic cell expansionsand the corresponding displacement pattern are shown inFig. 6 together with quadratic (x2) and quartic (x2 and x4

terms) fits. The potential energy curves show clear evidenceof anharmonicity. All curves require x4 terms for a successfulfit. A double-well potential forms for one percent expansion.

IV. DISCUSSION

The combined use of the experimental techniques TDS andIXS enables to probe extended regions in reciprocal space andthus allowed for a detailed measurement of the low-energy TAphonon mode. A predicted dip structure around q = ( 12 12 14 )could be confirmed. It is worth noting, that this phenomenonwas entirely missed by measuring the phonon dispersion alonghigh-symmetry directions. Modern techniques, including thecalculations, allow scanning across the Brillouin zone and

FIG. 6. Frozen-phonon potential-energy curves for differentisotropic cell expansion (symbols [6]) for the soft TA mode at q =( 12

12

14 ). The mode amplitude is given for Ti ion displacement. The

dashed and full lines in the corresponding color show, respectively,quadratic and quartic fits to the DFT calculation. Eigenvectors arerepresented within the unit cell.

are thus sensitive to such anomalies. Experimental resultson phonon energies and scattering intensities show that theharmonic DFPT calculation reproduces dispersion and eigen-vectors of most phonon branches, including its rich varietyof features. The shape of phonon energy surfaces is correctlydescribed across the Brillouin zone in spite of significantlyunderestimating the phonon energy of the TA mode.

The theoretical description of the lattice dynamics of TiO2in the rutile structure from first principles methods is ratherchallenging. The mixed covalent bonding of s-d orbitals isresponsible for long-ranged force constants. In order to capturesubtle features in the lattice dynamics, including the incipientferroelectric behavior and the anomaly descried here, thecalculations require careful convergence tests. The presentcalculation profits from accurate pseudopotentials and con-sequently precise wave functions providing lattice parameterswithin 0.5%–1.0% compared to the limit in LDA [6].

Some optical phonon branches, and most importantly tothis study, the lowest energy TA mode are substantiallyunderestimated in energy. Worthy of comment is the particularsensitivity of the TA phonon mode with anomalous lowenergy to convergence issues and external stress. An isotropiccompression of the relaxed structure by 0.5% brings thephonon frequencies closer to the low temperature experimentalvalues in LDA calculations [6].

The temperature dependent experimental data and thefrozen phonon calculations show the importance of anhar-monic contributions to the lattice dynamics in TiO2. We findthat the phonon energies at Z point and q = ( 12 12 14 ) increasealmost linearly with temperature. Measurements of the phonondensity of states as function of temperature by Lan et al. [9]show that the entire TA branch stiffens with temperature. Thisobservation is inline with our results. The change in energyat q = (0 14 0) is less pronounced and potentially provides im-portant details how the phonon density of states broadens withtemperature. The mode level resolution of our measurementswill allow for direct comparison with temperature dependentdispersion relations. A proper anharmonic treatment, whichis beyond the scope of this work, is required for a correctdescription of the TA mode. Nevertheless, even without such

014303-5

WEHINGER, BOSAK, AND JOCHYM PHYSICAL REVIEW B 93, 014303 (2016)

treatment we can extract some information from the datapresented in Fig. 6. The phonon mode potential has strongpositive fourth order components. This leads to an increase ofthe phonon frequency for large atomic displacements (i.e.,at higher temperatures), which may explain at least partof the discrepancy between purely harmonic calculation inFig. 5(b) and nonzero temperature measurements in the samefigure. Furthermore, a rather strong dependence of fourth ordercomponents of the potential on unit cell volume indicatesthat thermal expansion is an important factor in the phononfrequency - temperature relationship of the investigated TAmode.

A full quantum mechanical description of the latticedynamics including anharmonic effects would require thecomputation of the multi-dimensional potential-energy surfaceand solving Schrödingers equation for the relevant degrees offreedom. Ab initio molecular dynamics simulation followedby a mapping onto a model Hamiltonian that describes thelattice dynamics [23] could be an option. This so called’temperature dependent effective potential technique’ wassuccessfully used to describe the temperature dependence ofthe phonon density of states in TiO2 rutile [9] and phonondispersions in VO2 rutile [24], which is similarly affected byanharmonic contributions. The employed method should bereliable for classical nuclear motions but quantum effects needto be considered at low temperatures. Alternative methodsfor studying anharmonicity have been proposed, for examplein Refs. [25–27]. The presented data provide importantinformation for benchmarking such approaches.

The harmonic DFPT calculation by Mitev et al. [6]predicted the TA mode to become soft at q = ( 12 12 14 ) un-der isotropic expansion or uniaxial strain. Our temperaturedependent measurements show the opposite under thermalexpansion. This is caused by the fact that even at substantial(1%) expansion the energy difference between the globalminimum and the local maximum in the potential is smallin comparison with thermal energy. It suggests that theanharmonic contribution to the phonon energies at highertemperatures is much more important than the pure volumeeffect arising from thermal expansion. The open question tobe addressed is what happens under application of strain.In fact, from the harmonic calculations we expect the TAmode to soften under application of external stress at smalldisplacements. The application of uniaxial expansion could berealized by bending thin films or using tensile load. TDS andIXS studies can be performed in such conditions in grazingincidence [28] or transmission geometry, respectively.

The experimental proof of the existence of the TA modewith an anomalous low-energy minimum at q = ( 12 12 14 ) is

important for the explanation of why perturbations on cleavedTiO2 surfaces have an effect deep into the bulk. Our resultssupport the proposition of Mitev et al. [6] for the anomalousslow and oscillatory convergence of thin film (“slab”) modelsin TiO2. The eigenvectors of the soft TA mode at q = ( 12 12 14 )are very similar to layer displacements perpendicular to the(110) surface, see Fig. 6. As the second layer below the surfacehas a structure similar to bulk TiO2 [29,30], the high compli-ance of the eigenvectors allow for long-ranged propagation ofsurface displacements. The surface oxygen lone-pair orbitalsare undercoordinated and result in strengthened Ti-O bondingand shorter Ti-O distances between the topmost pair of planesand larger distances between this plane and the next lowerplane. Depending whether a slab model consists of odd oreven numbers of layers, the displacements are in or out ofphase, respectively.

From an experimental point of view this study is abenchmark for the sensitivity of diffuse x-ray scattering toparticular phonon branches. In TiO2, the scattering from aspecific branch contributes up to 98% to the total TDS notonly close to �. The shape of TDS can be rapidly exploredover a large volume in reciprocal space with good momentumresolution.

V. CONCLUSIONS

The lattice dynamics of TiO2 exhibits a rich structureincluding an TA mode with anomalous low energy with aminimum at q = ( 12 12 14 ). With help of inelastic and diffusex-ray scattering we were able to measure the energy of thismode over an extended region in reciprocal space and validateits eigenvectors by careful analysis of scattering intensities.DFPT correctly described the shape of the dispersion relationsincluding many subtle features. Absolute phonon energies areunderestimated for several branches including the TA modeof interest. The lattice dynamics shows a significant stressdependence and is thus very sensitive to the approximationsused in the calculations. The temperature dependent studyrevealed a strong temperature variation of the soft TA modeover an extended region in reciprocal space which showsthe importance of anharmonic contributions to the latticedynamics in TiO2. Our experimental observation of the softTA mode with a minimum at q = ( 12 12 14 ) finally supportsthe explication by Mitev et al. [6] of anomalous convergencebehavior in slab model calculations and real behavior in thinfilms.

ACKNOWLEDGMENT

We would like to thank Keith Refson, Barbara Montanari,and Peter Blaha for fruitful discussions and comments.

[1] C. Lee, P. Ghosez, and X. Gonze, Lattice dynamics and dielectricproperties of incipient ferroelectric TiO2 rutile, Phys. Rev. B 50,13379 (1994).

[2] J. G. Traylor, H. G. Smith, R. M. Nicklow, and M. K. Wilkinson,Lattice dynamics of rutile, Phys. Rev. B 3, 3457 (1971).

[3] R. A. Parker, Static dielectric constant of rutile (TiO2), 1.6-1060◦ K, Phys. Rev. 124, 1719 (1961).

[4] G. A. Samara and P. S. Peercy, Pressure and temperaturedependence of the static dielectric constants and Raman spectraof TiO2 (rutile), Phys. Rev. B 7, 1131 (1973).

014303-6

http://dx.doi.org/10.1103/PhysRevB.50.13379http://dx.doi.org/10.1103/PhysRevB.50.13379http://dx.doi.org/10.1103/PhysRevB.50.13379http://dx.doi.org/10.1103/PhysRevB.50.13379http://dx.doi.org/10.1103/PhysRevB.3.3457http://dx.doi.org/10.1103/PhysRevB.3.3457http://dx.doi.org/10.1103/PhysRevB.3.3457http://dx.doi.org/10.1103/PhysRevB.3.3457http://dx.doi.org/10.1103/PhysRev.124.1719http://dx.doi.org/10.1103/PhysRev.124.1719http://dx.doi.org/10.1103/PhysRev.124.1719http://dx.doi.org/10.1103/PhysRev.124.1719http://dx.doi.org/10.1103/PhysRevB.7.1131http://dx.doi.org/10.1103/PhysRevB.7.1131http://dx.doi.org/10.1103/PhysRevB.7.1131http://dx.doi.org/10.1103/PhysRevB.7.1131

SOFT PHONON MODES IN RUTILE TiO2 PHYSICAL REVIEW B 93, 014303 (2016)

[5] B. Montanari and N. M. Harrison, Pressure-induced instabilitiesin bulk TiO2 rutile, J. Phys.: Condens. Matter 16, 273 (2004).

[6] P. D. Mitev, K. Hermansson, B. Montanari, and K. Refson, Softmodes in strained and unstrained rutile TiO2, Phys. Rev. B 81,134303 (2010).

[7] E. Shojaee and M. R. Mohammadizadeh, First-principles elasticand thermal properties of TiO2: A phonon approach, J. Phys.:Condens. Matter 22, 015401 (2010).

[8] K. Refson, B. Montanari, P. D. Mitev, K. Hermansson, and N. M.Harrison, Comment on “First-principles study of the influenceof (110)-oriented strain on the ferroelectric properties of rutileTiO2”, Phys. Rev. B 88, 136101 (2013).

[9] T. Lan, C. W. Li, O. Hellman, D. S. Kim, J. A. Muñoz,H. Smith, D. L. Abernathy, and B. Fultz, Phonon quarticityinduced by changes in phonon-tracked hybridization duringlattice expansion and its stabilization of rutile TiO2, Phys. Rev.B 92, 054304 (2015).

[10] M. Krisch and F. Sette, Inelastic X-ray Scattering from Phonons.Light Scattering in solids, Novel Materials and Techniques,Topics in Applied Physics 108 (Springer-Verlag, 2007).

[11] X. Gonze and C. Lee, Dynamical matrices, Born effectivecharges, dielectric permittivity tensors, and interatomic forceconstants from density-functional perturbation theory, Phys.Rev. B 55, 10355 (1997).

[12] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. J.Probert, K. Refson, and M. C. Payne, First principles methodsusing CASTEP, Zeitschrift für Kristallographie - CrystallineMaterials 220, 567 (2009).

[13] K. Refson, P. R. Tulip, and S. J. Clark, Variational density-functional perturbation theory for dielectrics and lattice dynam-ics, Phys. Rev. B 73, 155114 (2006).

[14] J. P. Perdew and A. Zunger, Self-interaction correction todensity-functional approximations for many-electron systems,Phys. Rev. B 23, 5048 (1981).

[15] A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos,Optimized pseudopotentials, Phys. Rev. B 41, 1227 (1990).

[16] B. Montanari and N. Harrison, Lattice dynamics of TiO2rutile: Influence of gradient corrections in density functionalcalculations, Chem. Phys. Lett. 364, 528 (2002).

[17] B. G. Pfrommer, M. Côté, S. G. Louie, and M. L. Cohen,Relaxation of crystals with the quasi-newton method, J. Comput.Phys. 131, 233 (1997).

[18] X. Gonze, J.-C. Charlier, D. Allan, and M. Teter, Interatomicforce constants from first principles: The case of α -quartz, Phys.Rev. B 50, 13035 (1994).

[19] K. Parlinski, Z. Q. Li, and Y. Kawazoe, First-PrinciplesDetermination of the Soft Mode in Cubic ZrO2, Phys. Rev. Lett.78, 4063 (1997).

[20] R. Xu, and T. C. Chiang, Determination of phonon dispersionrelations by x-ray thermal diffuse scattering, Zeitschrift fürKristallographie - Crystalline Materials 220, 1009 (2009).

[21] B. Wehinger, A. Bosak, G. Piccolboni, K. Refson, D.Chernyshov, A. Ivanov, A. Rumiantsev, and M. Krisch, Diffusescattering in metallic tin polymorphs, J. Phys.: Condens. Matter26, 115401 (2014).

[22] D. T. Cromer and J. B. Mann, X-ray scattering factors computedfrom numerical Hartree–Fock wave functions, Acta Crystallogr.Sect. A 24, 321 (1968).

[23] O. Hellman, P. Steneteg, I. A. Abrikosov, and S. I. Simak,Temperature dependent effective potential method for accuratefree energy calculations of solids, Phys. Rev. B 87, 104111(2013).

[24] J. D. Budai, J. Hong, M. E. Manley, E. D. Specht, C. W. Li, J. Z.Tischler, D. L. Abernathy, A. H. Said, B. M. Leu, L. A. Boatner,R. J. McQueeney, and O. Delaire, Metallization of vanadiumdioxide driven by large phonon entropy, Nature (London) 515,535 (2014).

[25] P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P. Rudin,Entropy Driven Stabilization of Energetically Unstable CrystalStructures Explained from First Principles Theory, Phys. Rev.Lett. 100, 095901 (2008).

[26] B. Monserrat, N. D. Drummond, and R. J. Needs, Anharmonicvibrational properties in periodic systems: energy, electron-phonon coupling, and stress, Phys. Rev. B 87, 144302 (2013).

[27] I. Errea, M. Calandra, and F. Mauri, Anharmonic free energiesand phonon dispersions from the stochastic self-consistent har-monic approximation: Application to platinum and palladiumhydrides, Phys. Rev. B 89, 064302 (2014).

[28] J. Serrano, A. Bosak, M. Krisch, F. J. Manjón, A. H. Romero,N. Garro, X. Wang, A. Yoshikawa, and M. Kuball, InN ThinFilm Lattice Dynamics by Grazing Incidence Inelastic X-rayScattering, Phys. Rev. Lett. 106, 205501 (2011).

[29] G. Cabailh, X. Torrelles, R. Lindsay, O. Bikondoa, I. Joumard,J. Zegenhagen, and G. Thornton, Geometric structure of TiO2(110)(1×1): Achieving experimental consensus, Phys. Rev. B75, 241403 (2007).

[30] R. Lindsay, A. Wander, A. Ernst, B. Montanari, G. Thornton, andN. M. Harrison, Revisiting the surface structure of TiO2 (110):A quantitative low-energy electron diffraction study, Phys. Rev.Lett. 94, 246102 (2005).

014303-7

http://dx.doi.org/10.1088/0953-8984/16/3/008http://dx.doi.org/10.1088/0953-8984/16/3/008http://dx.doi.org/10.1088/0953-8984/16/3/008http://dx.doi.org/10.1088/0953-8984/16/3/008http://dx.doi.org/10.1103/PhysRevB.81.134303http://dx.doi.org/10.1103/PhysRevB.81.134303http://dx.doi.org/10.1103/PhysRevB.81.134303http://dx.doi.org/10.1103/PhysRevB.81.134303http://dx.doi.org/10.1088/0953-8984/22/1/015401http://dx.doi.org/10.1088/0953-8984/22/1/015401http://dx.doi.org/10.1088/0953-8984/22/1/015401http://dx.doi.org/10.1088/0953-8984/22/1/015401http://dx.doi.org/10.1103/PhysRevB.88.136101http://dx.doi.org/10.1103/PhysRevB.88.136101http://dx.doi.org/10.1103/PhysRevB.88.136101http://dx.doi.org/10.1103/PhysRevB.88.136101http://dx.doi.org/10.1103/PhysRevB.92.054304http://dx.doi.org/10.1103/PhysRevB.92.054304http://dx.doi.org/10.1103/PhysRevB.92.054304http://dx.doi.org/10.1103/PhysRevB.92.054304http://dx.doi.org/10.1103/PhysRevB.55.10355http://dx.doi.org/10.1103/PhysRevB.55.10355http://dx.doi.org/10.1103/PhysRevB.55.10355http://dx.doi.org/10.1103/PhysRevB.55.10355http://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1103/PhysRevB.73.155114http://dx.doi.org/10.1103/PhysRevB.73.155114http://dx.doi.org/10.1103/PhysRevB.73.155114http://dx.doi.org/10.1103/PhysRevB.73.155114http://dx.doi.org/10.1103/PhysRevB.23.5048http://dx.doi.org/10.1103/PhysRevB.23.5048http://dx.doi.org/10.1103/PhysRevB.23.5048http://dx.doi.org/10.1103/PhysRevB.23.5048http://dx.doi.org/10.1103/PhysRevB.41.1227http://dx.doi.org/10.1103/PhysRevB.41.1227http://dx.doi.org/10.1103/PhysRevB.41.1227http://dx.doi.org/10.1103/PhysRevB.41.1227http://dx.doi.org/10.1016/S0009-2614(02)01401-Xhttp://dx.doi.org/10.1016/S0009-2614(02)01401-Xhttp://dx.doi.org/10.1016/S0009-2614(02)01401-Xhttp://dx.doi.org/10.1016/S0009-2614(02)01401-Xhttp://dx.doi.org/10.1006/jcph.1996.5612http://dx.doi.org/10.1006/jcph.1996.5612http://dx.doi.org/10.1006/jcph.1996.5612http://dx.doi.org/10.1006/jcph.1996.5612http://dx.doi.org/10.1103/PhysRevB.50.13035http://dx.doi.org/10.1103/PhysRevB.50.13035http://dx.doi.org/10.1103/PhysRevB.50.13035http://dx.doi.org/10.1103/PhysRevB.50.13035http://dx.doi.org/10.1103/PhysRevLett.78.4063http://dx.doi.org/10.1103/PhysRevLett.78.4063http://dx.doi.org/10.1103/PhysRevLett.78.4063http://dx.doi.org/10.1103/PhysRevLett.78.4063http://dx.doi.org/10.1524/zkri.2005.220.12.1009http://dx.doi.org/10.1524/zkri.2005.220.12.1009http://dx.doi.org/10.1524/zkri.2005.220.12.1009http://dx.doi.org/10.1524/zkri.2005.220.12.1009http://dx.doi.org/10.1088/0953-8984/26/11/115401http://dx.doi.org/10.1088/0953-8984/26/11/115401http://dx.doi.org/10.1088/0953-8984/26/11/115401http://dx.doi.org/10.1088/0953-8984/26/11/115401http://dx.doi.org/10.1107/S0567739468000550http://dx.doi.org/10.1107/S0567739468000550http://dx.doi.org/10.1107/S0567739468000550http://dx.doi.org/10.1107/S0567739468000550http://dx.doi.org/10.1103/PhysRevB.87.104111http://dx.doi.org/10.1103/PhysRevB.87.104111http://dx.doi.org/10.1103/PhysRevB.87.104111http://dx.doi.org/10.1103/PhysRevB.87.104111http://dx.doi.org/10.1038/nature13865http://dx.doi.org/10.1038/nature13865http://dx.doi.org/10.1038/nature13865http://dx.doi.org/10.1038/nature13865http://dx.doi.org/10.1103/PhysRevLett.100.095901http://dx.doi.org/10.1103/PhysRevLett.100.095901http://dx.doi.org/10.1103/PhysRevLett.100.095901http://dx.doi.org/10.1103/PhysRevLett.100.095901http://dx.doi.org/10.1103/PhysRevB.87.144302http://dx.doi.org/10.1103/PhysRevB.87.144302http://dx.doi.org/10.1103/PhysRevB.87.144302http://dx.doi.org/10.1103/PhysRevB.87.144302http://dx.doi.org/10.1103/PhysRevB.89.064302http://dx.doi.org/10.1103/PhysRevB.89.064302http://dx.doi.org/10.1103/PhysRevB.89.064302http://dx.doi.org/10.1103/PhysRevB.89.064302http://dx.doi.org/10.1103/PhysRevLett.106.205501http://dx.doi.org/10.1103/PhysRevLett.106.205501http://dx.doi.org/10.1103/PhysRevLett.106.205501http://dx.doi.org/10.1103/PhysRevLett.106.205501http://dx.doi.org/10.1103/PhysRevB.75.241403http://dx.doi.org/10.1103/PhysRevB.75.241403http://dx.doi.org/10.1103/PhysRevB.75.241403http://dx.doi.org/10.1103/PhysRevB.75.241403http://dx.doi.org/10.1103/PhysRevLett.94.246102http://dx.doi.org/10.1103/PhysRevLett.94.246102http://dx.doi.org/10.1103/PhysRevLett.94.246102http://dx.doi.org/10.1103/PhysRevLett.94.246102