First-principles phonon lifetimes and rattling-mode suppression of the lattice thermal conductivity in thermoelectric sodium cobaltate Na$_x$CoO$_2$

Thermoelectric Na$_{0.8}$CoO$_2$ suppresses its lattice thermal conductivity by a factor of about six through anharmonic rattling vibrations of sodium ions inside ordered multi-vacancy clusters, as established by inelastic X-ray/neutron scattering and density-functional phonon calculations (Voneshen et al., 2013). What remains open is the quantitative microscopic theory behind that observation: (i) a first-principles calculation of phonon linewidths/lifetimes that decomposes the measured widths into anharmonic phonon-phonon and electron-phonon contributions and predicts $\kappa^L(T)$; (ii) identification of the mechanism behind the ~50% linewidth broadening and slight dispersion softening that appear when long-range vacancy order is lost at elevated temperature; and (iii) a systematic first-principles and experimental map of how vacancy-cluster arrangement and sodium concentration control rattler energies, anharmonicity, and $\kappa^L$ suppression, yielding predictive design rules.

Background

A thermoelectric material converts heat into electricity, and its efficiency is gauged by the dimensionless figure of merit $ZT = S^2\sigma T/\kappa$, where $S$ is the Seebeck coefficient, $\sigma$ the electrical conductivity, and $\kappa$ the thermal conductivity (the sum of an electronic part and a lattice part $\kappa^L$ carried by phonons, the quantized lattice vibrations). Good thermoelectrics therefore need high $\sigma$ but low $\kappa^L$, a design conflict that motivates the phonon-glass electron-crystal concept: crystals in which charge carriers move freely while loosely bound guest atoms rattle inside oversized structural cages and scatter heat-carrying phonons. A rattling mode is a weakly dispersive, Einstein-like vibration (i.e., behaving like an isolated oscillator at a nearly fixed energy) of such a loosely bound atom.

Layered sodium cobaltate Na$_x$CoO$_2$ consists of conducting triangular CoO$_2$ sheets separated by layers of sodium ions. At thermoelectric compositions the Na layer contains ordered arrays of vacancies: in Na$_{0.8}$CoO$_2$ below about 290 K the vacancies form a square lattice of tri-vacancy clusters (clusters of three neighboring missing Na ions) with a large-period superstructure, and Na ions bordering these clusters are loosely bound. Voneshen et al. (Nature Materials 12, 1028, 2013) combined inelastic X-ray scattering (IXS) and inelastic neutron scattering (INS) - momentum-resolved spectroscopies that measure phonon dispersions $E(\mathbf{Q})$ and, through the energy linewidth $\Gamma$ of a phonon peak, its lifetime $\tau = \hbar/\Gamma$ - with density-functional-theory (DFT) phonon calculations on a 114-atom supercell of the tri-vacancy-ordered structure. They directly observed an Einstein-like Na rattling mode near 13 meV with predominantly cubic anharmonicity, measured phonon lifetimes as a function of temperature by INS (resolution-limited below about 80 K), and, inserting the measured lifetimes into the kinetic expression $\kappa^L = \sum_{j\mathbf{Q}} v_{jx}^2\,\tau_j(\mathbf{Q})\, c_j(\mathbf{Q})$ (group velocity, lifetime, and heat capacity per mode), estimated the lattice thermal conductivity: rattling modes suppress $\kappa$ by a factor of six relative to vacancy-free NaCoO$_2$.

Three issues were left open by that study. First, the measured linewidths were used empirically but not explained: a first-principles calculation of the phonon linewidths themselves - the contributions from anharmonicity (in practice, three-phonon scattering and, where relevant, four-phonon scattering, obtained from third- and fourth-order force constants) and from electron-phonon scattering (phonons damping by creating electron-hole pairs in this metallic oxide) - remained to be done. Second, IXS at 320 K (disordered stripe phase) and 400 K (fully disordered phase) showed the dispersion softening slightly and linewidths broadening by about 50% once long-range vacancy order is lost; whether this is caused mainly by increased configurational disorder of the vacancies, by thermal lattice expansion, or by increased phonon-phonon scattering is undetermined, and the absence of long-range periodicity in those phases blocks the straightforward supercell DFT treatment used at low temperature. Third, because multi-vacancy clusters exist over a wide range of Na concentrations and most likely host rattling modes throughout the thermoelectric composition range, translating these insights into materials design requires a systematic first-principles and experimental mapping of how vacancy-cluster arrangements and sodium concentrations set the rattler energies, their anharmonicity, and the resulting $\kappa^L$ suppression.

Problem Statement

For thermoelectric Na$_x$CoO$_2$, whose Na$_{0.8}$ member suppresses its lattice thermal conductivity by roughly a factor of six through anharmonic rattling of sodium ions in ordered multi-vacancy clusters, establish the quantitative microscopic theory of phonon damping and its consequences for thermal transport, specifically:

1. Compute, from first principles, the phonon linewidths (lifetimes) of the vacancy-ordered Na$_{0.8}$CoO$_2$ superstructure - including anharmonic phonon-phonon scattering (three-phonon and, where relevant, four-phonon processes) and electron-phonon scattering - and decompose the measured INS/IXS linewidths into these contributions, quantitatively accounting for their observed temperature dependence and enabling a first-principles prediction of $\kappa^L(T)$ to replace the semi-empirical kinetic estimate that inserted measured lifetimes into $\kappa^L = \sum_{j\mathbf{Q}} v_{jx}^2\,\tau_j\, c_j$.

2. Determine the dominant mechanism of the changes observed on heating above the loss of long-range vacancy order - the renormalization of the dispersion to slightly lower energies and the broadening of linewidths by about 50% - by distinguishing quantitatively among increased configurational disorder of the vacancies, thermal lattice expansion, and increased phonon-phonon scattering.

3. Map, systematically and jointly by first-principles calculation and experiment, how different vacancy-cluster arrangements and sodium concentrations across the thermoelectric composition range $0.5 \le x \le 0.85$ set the rattler mode energies, their anharmonicity, and the resulting suppression of $\kappa^L$, and derive from this map predictive design rules for lowering $\kappa^L$ (at preserved electronic transport) in Na$_x$CoO$_2$ and analogous layered oxides. The map is to name explicitly the vacancy-cluster arrangements it covers - including the low-$x$ row (zigzag), stripe, and square tri-vacancy-cluster arrangements and any others that occur in this range - and the design rules are to relate the controllable variables (Na content $x$ and vacancy-cluster arrangement) to the resulting rattler energies, anharmonicity, and $\kappa^L$ suppression.

These three components jointly serve one objective: replacing the empirical observation of rattler-induced thermal-conductivity suppression with a predictive, first-principles account linking vacancy superstructure, rattler anharmonicity, phonon lifetimes, and $\kappa^L$, valid at thermoelectric operating temperatures and across compositions. Each item must be resolved for Na$_x$CoO$_2$ itself: item 1 must deliver the linewidth decomposition together with the resulting first-principles $\kappa^L(T)$ prediction, item 2 the quantitative attribution among the three candidate mechanisms, and item 3 a composition- and arrangement-dependent account across the thermoelectric composition range; the extension to analogous layered oxides is optional.

References

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