Thermal conductivity in intermetallic clathrates: A first-principles perspective

Inorganic clathrates such as ${\mathrm{Ba}}_8{\mathrm{Ga}}_x{\mathrm{Ge}}_{46-x}$ and ${\mathrm{Ba}}_8{\mathrm{Al}}_x{\mathrm{Si}}_{46-x}$ commonly exhibit very low thermal conductivities. A quantitative computational description of this important property has proven difficult, in part due to the large unit cell, the role of disorder, and the fact that both electronic carriers and phonons contribute to transport. Here, we conduct a systematic analysis of the temperature and composition dependence of low-frequency modes associated with guest species in ${\mathrm{Ba}}_8{\mathrm{Ga}}_x{\mathrm{Ge}}_{46-x}$ and ${\mathrm{Ba}}_8{\mathrm{Al}}_x{\mathrm{Si}}_{46-x}$ (“rattler modes”), as well as thermal transport in stoichiometric ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. To this end, we account for phonon-phonon interactions by means of temperature-dependent effective interatomic force constants, which we find to be crucial in order to achieve an accurate description of the lattice part of the thermal conductivity. While the analysis of the thermal conductivity is often largely focused on the rattler modes, here it is shown that at room temperatures modes with $\hslash \omega \gtrsim 10\mathrm{meV}$ account for 50% of lattice heat transport. Finally, the electronic contribution to the thermal conductivity is computed, which shows the Wiedemann-Franz law to be only approximately fulfilled. As a result, it is crucial to employ the correct prefactor when separating electronic and lattice contributions for experimental data.

Figure 1

Crystal structure of type I clathrates. The guest species (Ba) occupies Wyckoff sites of type $2a$ and $6d$, while the host species (Ga, Ge) occupy Wyckoff sites of type $6c, 16i$, and $24k$.

Figure 2

Temperature dependence of the lattice parameter for ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ obtained within the quasiharmonic approximation. Experimental data are from Ref. [11].

Figure 3

(a)–(c) Phonon dispersion of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ showing the low-frequency region along the $\mathrm{\Gamma }-\mathsf{R}$ direction derived (a), (b) from IFCs obtained in the static (0 K) limit by the finite-displacement (FD) method and (c) from temperature-dependent interatomic force constants (TDIFCs) corresponding to a temperature of 300 K. Black dashed lines indicate the result of a simple spring model fitted to experimental data [27]. The model underestimates the splitting at the zone boundary close to 5 meV, which is actually approximately 1 meV. (d) Total phonon densities of states. (e) Partial densities of states showing the contributions from Ba on 2a Wyckoff sites (blue line), Ba on 6d sites (red line), and contributions from the Ga/Ge cage structure (green).

Figure 4

(a), (b) Composition dependence of the lowest-lying rattler modes in (a) ${\mathrm{Ba}}_8{\mathrm{Ga}}_x{\mathrm{Ge}}_{46-x}$ and (b) ${\mathrm{Ba}}_8{\mathrm{Al}}_x{\mathrm{Si}}_{46-x}$ along with the variation of the lattice constant (in gray). Specifically, the red, yellow, and blue lines correspond to the average frequencies for modes 1–6, 7–12, and 13–18, respectively. The standard deviations, obtained by averaging over 20 representative configurations at each composition, are indicated by shaded filled curves. (c) Temperature dependence of the partial density of states associated with Ba atoms on 6d Wyckoff sites [shaded filled curves; compare Fig. 3] in comparison with experimental data from inelastic neutron scattering on powder (PINS) and single crystalline (SINS) samples [27], Raman scattering [27, 54, 55, 56, 57], and THz spectroscopy [58, 59, 60].

Figure 5

Lattice thermal conductivity ${\kappa }_l$ of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ as a function of temperature. (a) Comparison of ${\kappa }_l$ due to different included scattering channels, calculated using IFCs achieved with the vdW-DF-cx functional, as well as a comparison to the computations using temperature-dependent interatomic force constants (TDIFCs). (b) Comparison between computations, using IFCs from the vdW-DF-cx functional (red solid line), the PBE functional (solid blue line), temperature-dependent force constants (squares), and experimental data sets (dashed lines marked by numbers). The inset shows the same data on a linear scale. The experimental data are from Refs. [71] (1, $p$ type), [64] (2, $p$ type; 3, $n$ type), [12] (4, $n$ type), [72] (5, $n$ type), and [73] (6, 7). The authors of Ref. [72] (5) point out that due to the large surface to volume ratio, their measurements become unreliable above approximately 100 K due to thermal emission. The shaded areas and open squares represent the difference between the electronic contribution to the thermal conductivity from BTT ${\kappa }_e$ and the Wiedemann-Franz law $L\sigma T$ with $L=2.0{(k_B/e)}^2$.

Figure 6

(a) Phonon dispersion in ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ from static vdW-DF-cx IFCs. (b) Normalized accumulated lattice thermal conductivity, with respect to energy, as computed with static vdW-DF-cx IFCs at 3, 30, and 300 K (solid lines), compared to calculations based on the temperature-dependent interatomic force constants (TDIFCs) at 300 K. (c) Phonon-phonon limited lifetimes at 300 K computed with PBE IFCs (black markers), vdW-DF-cx IFCs (red markers), and TDIFCs (blue markers). The filled yellow curves indicate the overdamped region for a classical harmonic oscillator. (d) Comparison of lifetimes computed from static vdW-DF-cx IFCs at 3, 30, and 300 K.

Figure 7

Electronic contribution ${\kappa }_e$ to the thermal conductivity calculated for the ground state structure as well as for structures extracted from Monte Carlo simulations [21] representative of the chemical order at different temperatures. Data obtained using the Wiedemann-Franz law ${\kappa }_e=L\sigma T$ are shown by dashed lines, whereas the thermal conductivity obtained within the framework of Boltzmann transport theory [33, 86] is shown by solid lines. Note that below 900 K the two sets of data deviate by as much as 25%, whereas above approximately 900 K the BTT data indicate a sharp rise that is not predicted by the Wiedemann-Franz law.