Anharmonicity in the High-Temperature $Cmcm$ Phase of SnSe: Soft Modes and Three-Phonon Interactions

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Anharmonicity in the High-Temperature $C m c m$ Phase of SnSe: Soft Modes and Three-Phonon Interactions

Jonathan M. Skelton, Lee A. Burton, Stephen C. Parker, Aron Walsh, Chang-Eun Kim, Aloysius Soon, John Buckeridge, Alexey A. Sokol, C. Richard A. Catlow, Atsushi Togo, and Isao Tanaka

Phys. Rev. Lett. 117, 075502 – Published 10 August 2016

Abstract

The layered semiconductor SnSe is one of the highest-performing thermoelectric materials known. We demonstrate, through a first-principles lattice-dynamics study, that the high-temperature $C m c m$ phase is a dynamic average over lower-symmetry minima separated by very small energetic barriers. Compared to the low-temperature $P n m a$ phase, the $C m c m$ phase displays a phonon softening and enhanced three-phonon scattering, leading to an anharmonic damping of the low-frequency modes and hence the thermal transport. We develop a renormalization scheme to quantify the effect of the soft modes on the calculated properties, and confirm that the anharmonicity is an inherent feature of the $C m c m$ phase. These results suggest a design concept for thermal insulators and thermoelectric materials, based on displacive instabilities, and highlight the power of lattice-dynamics calculations for materials characterization.

Received 13 January 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.075502

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anharmonic lattice dynamics Lattice dynamics Thermoelectric effects

Semiconductors

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jonathan M. Skelton, Lee A. Burton, Stephen C. Parker, and Aron Walsh^*

Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

Chang-Eun Kim and Aloysius Soon

Department of Materials Science and Engineering, Yonsei University, Seoul 120-749, Korea

John Buckeridge, Alexey A. Sokol, and C. Richard A. Catlow

University College London, Kathleen Lonsdale Materials Chemistry, Department of Chemistry, 20 Gordon Street, London WC1H 0AJ, United Kingdom

Atsushi Togo and Isao Tanaka

Elements Strategy Initiative for Structural Materials, Kyoto University, Kyoto Prefecture 606-8501, Japan

^*a.walsh@bath.ac.uk

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Issue

Vol. 117, Iss. 7 — 12 August 2016

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Images

Figure 1
Lattice dynamics of SnSe. Plots (a) and (b) show the phonon dispersions and densities of states of the optimized equilibrium $P n m a$ and $C m c m$ structures illustrated in (c) and (d), respectively. The dispersion of the high-temperature $C m c m$ phase shows imaginary modes at the symmetry points $Γ$ and $Y$ , marked $λ_{1}$ and $λ_{2}$ , respectively. The black lines in the $C m c m$ dispersion are segments computed from supercells chosen to have a larger number of commensurate points along the corresponding reciprocal-space directions (Ref. [35]). The 2D potential-energy surface along the two corresponding normal-mode coordinates ( $Q_{1} / Q_{2}$ ) is shown in plot (e). The left- and right-hand snapshots in (c) and (d) are taken through the ab and ac planes, respectively, and the Sn and Se atoms in the models are colored silver and green. The images were prepared using vesta (Ref. [39]).
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Figure 2
Renormalization of the imaginary harmonic-phonon modes in the equilibrium $C m c m$ SnSe structure. Plots (a) and (c) show the anharmonic double-well potentials along the two modes labeled $λ_{1}$ and $λ_{2}$ in Fig. 1 as a function of the corresponding normal-mode coordinates $Q_{λ = 1, 2}$ . The markers show the calculated points, the solid lines are fits to a 20-power polynomial, and the black lines inside the potentials show the eigenvalues obtained by solving a 1D Schrödinger equation. Plots (b) and (d) show the effective renormalized frequencies of the two modes ( ${\tilde{v}}_{λ = 1, 2}$ ) as a function of temperature, calculated as the harmonic frequencies which reproduce the contributions of the modes to the vibrational partition function. Plot (e) compares phonon dispersions and densities of states calculated without renormalization (black) and with the imaginary modes renormalized to the calculated 0 (blue), 300 (red), and 800 K (orange) frequencies.
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Figure 3
Thermal-transport properties of the $P n m a$ (a),(d) and $C m c m$ (b),(c),(e),(f) phases of SnSe. Plots (a)–(c) show the averaged three-phonon interaction strengths, $P_{q j}$ (as defined in Ref. [28]), overlaid on the phonon densities of states. The error bars show the standard deviation on the average in each histogram bin. Plots (d)–(e) show the calculated lattice thermal conductivity ( $κ_{L}$ ) along each of the three crystallographic axes, together with the isotropic average. The $C m c m$ -phase data shown in (b) and (e) was modeled without renormalization of the imaginary modes, while that shown in (c) and (f) was modeled with the second-order (harmonic) force constants corrected using the 800 K renormalized frequencies.
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