Phonon quarticity induced by changes in phonon-tracked hybridization during lattice expansion and its stabilization of rutile ${\mathrm{TiO}}_{2}$

Phonon quarticity induced by changes in phonon-tracked hybridization during lattice expansion and its stabilization of rutile ${TiO}_{2}$

Tian Lan, C. W. Li, O. Hellman, D. S. Kim, J. A. Muñoz, H. Smith, D. L. Abernathy, and B. Fultz

Phys. Rev. B 92, 054304 – Published 11 August 2015

Abstract

Although the rutile structure of ${TiO}_{2}$ is stable at high temperatures, the conventional quasiharmonic approximation predicts that several acoustic phonons decrease anomalously to zero frequency with thermal expansion, incorrectly predicting a structural collapse at temperatures well below 1000 K. Inelastic neutron scattering was used to measure the temperature dependence of the phonon density of states (DOS) of rutile ${TiO}_{2}$ from 300 to 1373 K. Surprisingly, these anomalous acoustic phonons were found to increase in frequency with temperature. First-principles calculations showed that with lattice expansion, the potentials for the anomalous acoustic phonons transform from quadratic to quartic, stabilizing the rutile phase at high temperatures. In these modes, the vibrational displacements of adjacent Ti and O atoms cause variations in hybridization of $3 d$ electrons of Ti and $2 p$ electrons of O atoms. With thermal expansion, the energy variation in this “phonon-tracked hybridization” flattens the bottom of the interatomic potential well between Ti and O atoms, and induces a quarticity in the phonon potential.

Received 20 January 2015
Revised 29 April 2015

DOI:https://doi.org/10.1103/PhysRevB.92.054304

Authors & Affiliations

Tian Lan^1,*, C. W. Li², O. Hellman¹, D. S. Kim¹, J. A. Muñoz¹, H. Smith¹, D. L. Abernathy³, and B. Fultz¹

¹Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA
²Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
³Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

^*tianlan@caltech.edu

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Vol. 92, Iss. 5 — 1 August 2015

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Figure 1
$S (Q, E)$ spectra of rutile ${TiO}_{2}$ measured at (a) 300 K and (b) 1373 K, respectively. The incident energy of neutron beam is 30 meV.
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Figure 2
(a) Neutron weighted phonon DOS of rutile ${TiO}_{2}$ from measurements at temperatures from 300 to 1373 K with an incident energy of 75 meV. Five vertical lines are aligned to peak centers at 300 K. (b) Phonon DOS of rutile ${TiO}_{2}$ at temperatures of 300 and 1373 K calculated with first-principles MD simulations and the Fourier transformed velocity autocorrelation (FTVAC) method.
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Figure 3
(a) Neutron weighted phonon DOS of rutile ${TiO}_{2}$ from measurements at temperatures from 300 to 1373 K with an incident energy of 30 meV (so data above 26 meV are less reliable). The dashed spectrum corresponds to the experimental result at 300 K, shifted vertically for comparison at each temperature. (b) Simulated total phonon DOS from the QHA at 300 K (black) and 1373 K (red).
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Figure 4
Diffuse curves are TDEP phonon dispersions below 25 meV at (a) 300 K and (b) 1373 K, compared with the results from the FTVAC method (red circles). The white curves are phonon dispersions for the quasiharmonicity plus quartic anharmonicity calculated with all $ψ_{i j k}$ set to zero in Eq. (3). In (b), the dispersions are compared to the quasiharmonic dispersions (orange dashed curve) and the single quartic oscillator model (orange triangles). (c) Frozen phonon potential (black) of TA mode at $R$ point with $q = (0.5, 0, 0.5)$ at 1373 K, showing the harmonic component (red) and quartic component (blue). The low-temperature potential surface is also shown (dashed black).
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Figure 5
(a) Displacements of atoms for the TA mode at the $R$ point in the (1-10) plane. Light blue spheres are Ti atoms, and O atoms are red. Arrows depict distortions of the structural units (dashed rhombuses). The rotational movements of structures, or the “ring” patterns, are indicated with circled arrows. (b) ELF isosurface of a “ring” shown in (a) with the value of 0.3. The ELF increase is apparent in the bond of shorter distance owing to the ring displacement. The ELF is the probability measure of finding an electron at a location given the existence of neighboring electrons. It ranges from 0 (no electron) to 1 (perfect localization). (c) COHP analysis of Ti-O bonds for equilibrium lattice parameter at $T = 0$ K (0%), and for linear expansions of 1% and 2%. Shown in color are $- COHP$ results for the same structures with the phonon of panel (a) having 0.14 $Å$ normal displacements of Ti atoms. Curves for 1% and 2% expansion are offset by 0.6 and 1.2.
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Figure 6
Phonon potentials for the “ring” pattern of Fig. 5. The solid curve is the potential of the $R$ -point mode at 1373 K as in Fig. 4. The dashed curve is the potential of the same mode 1373 K, but with the “ring” pattern broken by immobilizing the Ti atoms along the $A$ direction in Fig. 5. The shape of dashed curve is quadratic.
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Physical Review B

covering condensed matter and materials physics

Phonon quarticity induced by changes in phonon-tracked hybridization during lattice expansion and its stabilization of rutile ${TiO}_{2}$

Tian Lan, C. W. Li, O. Hellman, D. S. Kim, J. A. Muñoz, H. Smith, D. L. Abernathy, and B. Fultz

Phys. Rev. B 92, 054304 – Published 11 August 2015

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Physical Review B

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Phonon quarticity induced by changes in phonon-tracked hybridization during lattice expansion and its stabilization of rutile TiO2

Tian Lan, C. W. Li, O. Hellman, D. S. Kim, J. A. Muñoz, H. Smith, D. L. Abernathy, and B. Fultz

Phys. Rev. B 92, 054304 – Published 11 August 2015

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Phonon quarticity induced by changes in phonon-tracked hybridization during lattice expansion and its stabilization of rutile ${TiO}_{2}$