Structural Optimization at finite temperature based on anharmonic phonon theory
R. Masuki1, T. Nomoto2, R. Arita2,3, T. Tadano4
1Department of Applied Physics, The University of Tokyo, 2Research Center for Advanced Science and Technology, The University of Tokyo
3RIKEN Center of Emergent Material Science, 4CMSM, National Instituite of Material Science (NIMS)
R. Masuki, T. Nomoto, R. Arita, T. Tadano Phys. Rev. B 106, 224104
Highlights
ü We have developed the theory and the efficient computational scheme of structural optimization at finite temperature based on self-consistent phonon (SCP) theory.
ü We applied the method to BaTiO3 and succeeded in reproducing its three-step structural phase transition and the 𝑃 − 𝑇 diagram.
ü The presented methodology is applicable to a wide range of materials. It is expected to be an essential tool in exploring exotic properties near the structural phasetransition or in computational material discoveries at finite temperature.

Background and Introduction

Calculation scheme

The structural optimization based on DFT (density functional theory)ü The effect of lattice vibrations (finitetemperature effect) and zero point motions(nuclear quantum effect) are disregarded.
→ zero-temperature methodü The finite-temperature method (moleculardynamics, ...) are computationally expensive.
→ cannot be applied to complex crystals

IFC renormalization
ü Calculate the set of IFCs in the updated structure from the set of IFCs in
the reference structure [5].

Calculation scheme,)&V
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Crystal structures at finite temperatureü structural phase transitionsof BaTiO3 [1]ü No need for expensive DFT calculationsin the structural optimization loop.
→ efficient!

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ü Charge density wave (CDW)
in 1T-TaSe2 [2]ü Both the atomic positions and the shape ofthe unit cell are optimized.

ü We have implemented the calculation
scheme to the ALAMODE package [4]→ will be made public in future release

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LQWHUQDOFRRUGLQDWH&U\VWDOVWUXFWXUHDWILQLWHWHPSHUDWXUHVThe purpose of this study

1RResults and Discussion
Free energy of the four phases
0.05

F1 ° F1,cubic [meV]ü A wide range of materials show structural phase transition by varying
temperature and pressure.
ü Exotic phenomena are observed near the structural phase transition.

&RQYHUJHQW"

<HV

0.00

°0.05

ü Development of efficient first-principles method of crystal structures atfinite temperatures that takes into account the phonon anharmonicity andthe nuclear quantum effect.

cubictetraorthorhombo°0.10
°0.15
°0.20

rhombo°0.25
350

Theory

ortho

400

450

tetra
500

550

ü Three-step structural phase transitionis successfully reproduced [5].

cubic
600

650

Temperature [K]Temperature-dependent crystal structure(a)Interatomic force constants (IFCs)
ü We start from the Taylor expansion of the potential energy surface.
Interatomic force constants (IFCs)

4.04
4.03

a4.02
a4.01
4.00
350

c
ortho

rhombo
400

450

tetra
500

550

cubic
600

3.98
200

250

300

350

5400

450

500

550

600

650

1050400

150

Temperature [K]

200

250

300350Temperature [K]400Experiment data istaken Ref [6].

Temperature-dependent crystal structure

Self-consistent phonon theory(a) Calculation [5]

(b)

700

450

ü The ”mean field theory” of phonon anharmonicity.
ü Based on variational principle of the free energy.

cubic-tetra
tetra-ortho
ortho-rhombo

600[3,4].

Temperature T [K]with respect totetra
ortho

200

rhomboü Calculate the crystal structure (internal coordinates and the shape ofthe unit cell) that minimizes the SCP free energy.
ü The gradient of the SCP free energy with respect to a structural degreeof freedom is250

tetra

200

ortho

150

rhombo5001234Pressure p [GPa]Structural optimization based on SCP theory

3001001000ü The three phase boundary
curves reach 0 K as pressure
is applied.
ü Cubic phase is stable at 0 K
above ~6 GPa.

cubic-tetra
tetra-ortho
ortho-rhombo

cubic

350

400
300

Experiment [7]

400

cubic

500

Temperature T [K]ü SCP equation is obtained by minimizingof the trial harmonic Hamiltonian

cubic

Temperature [K]Pz (exp) [µC/cm2]

Lattice constant (exp) [Å]

4.00tetra

10(b)

4.02

150

ortho

150350650

Temperature [K](b)𝑛 ≥ 3 : anharmonic termsrhombocPz (calc) [µC/cm2]

Lattice constant (calc) [Å](a)56001234567

Pressure p [GPa]Reference[1] M. Acosta, et al. Appl. Phys. Rev. 4.041305 (2017)[2] Y. Chen, et al. Nat. Phys. 16.218 (2020)[3] D. J. Hooton. Phil. Mag. 3.25 (1958)[4] T. Tadano, S. Tsuneyuki. Phys. Rev. B 92.054301 (2015)[5] R. Masuki, et al. Phys. Rev. B 106, 224104[6] H. F. Kay, P. Vousden. Phil. Mag. 40.309 (1949)[7] T. Ishidate, et al. Phys. Rev. Lett. 78.2397 (1997)