Occupational disorder as the origin of flattening of the acoustic phonon branches in the clathrate ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$

In the search for high-performance thermoelectrics, materials such as clathrates have drawn attention due to having both glasslike low phonon thermal conductivity and crystal-like high electrical conductivity. ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (BGG) has a loosely bound guest Ba atom trapped inside rigid Ga-Ge cage structures. Avoided crossings between acoustic phonons and the flat guest atom branches have been proposed to be the source of the low lattice thermal conductivity of BGG. Ga-Ge site disorder with Ga and Ge exchanging places in different unit cells has also been reported. We used time-of-flight neutron scattering to measure the complete phonon spectrum in a large single crystal of BGG and compared these results with predictions of density functional theory to elucidate the effect of the disorder on heat-carrying phonons. Experimental results agreed much better with the calculation assuming the disorder than with the calculation assuming the ordered configuration. Although the atomic masses of Ga and Ge are nearly identical, we found that disorder strongly reduces phonon group velocities, which significantly reduces thermal conductivity. Our work points to a path towards optimizing thermoelectrics.

Figure 1

Summary of the lattice dynamical phenomena considered in this Research Letter. (a) Ba-$6d$ containing ${(\mathrm{Ga},\mathrm{Ge})}_{24}$ cage portion of the three-dimensional unit cells of perfectly ordered ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (top) and disordered ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ with scrambled Ge and Ga atom occupations (bottom). (b) and (c) Black curves show neutron-weighted densities of states (DOSs) [inelastic neutron scattering (INS) intensities integrated over the $3<H<6, 3<K<7, 4.5<L<6.5$ reciprocal lattice units (r.l.u.) region of reciprocal space] measured with incident energy $E_i=22$ meV (b) and $E_i=52$ meV (c). The green (purple) lines are the same quantity calculated from DFT using the ordered (disordered) unit cells and broadened with an approximate resolution function fit to the experimental intensity (see Supplemental Material [19]). The disordered calculation and the experimental curves for ${\mathrm{E}}_i=22$ meV are offset vertically by 350 and 400 counts, respectively. The experimental curve has an additional large ($\sim 250$ counts) background that is not present in the calculations. For $E_i=52$ meV, the disordered and experimental curves are offset by 500 and 700 counts, respectively. Here, the experimental background is $\sim 600$ counts. (d) and (e) The ordered (d) and disordered (e) phonon dispersions and phonon densities of states (PDOSs). The dash-dotted red curve is the projection onto the Ba atoms; the black curve is the total DOS (Tot.). (f) The average group velocities squared, $\langle v_g^2\rangle$, calculated from the dispersions in (d) and (e) by averaging over all modes and $\mathit{q}$ points. (g) Colored curves represent the spectral thermal conductivities $\kappa \left(\omega \right)$, and black curves represent the cumulative thermal conductivities ${\kappa }_{\mathrm{cum}}\left(\omega \right)$ at 300 K calculated from the group velocities and densities of states in (f) and in (d) and (e). Only umklapp processes are considered; $\tau \left(\omega \right)={\tau }_0{\omega }^{-2}$, with ${\tau }_0$ being the same for the ordered (ord.) and disordered (dis.) calculations. The data in (g) are in units of the total thermal conductivity of the ordered crystal.

Figure 2

Ordered (left column) and disordered (middle column) neutron spectra compared with experiment (right column). (a) Bragg peaks ($-1<E<1$ meV) in the ($H,K,L=6$) plane. In the ordered cell, coherent scattering from the ordered arrangement of atoms results in certain Bragg peaks [e.g., $\mathbf{Q}=(4,4,6)$] having no intensity. On the other hand, scattering from the disordered arrangement of atoms results in some remaining intensity at these Bragg peaks, in excellent agreement with the experimental Bragg pattern. (b)–(d) The remaining rows show inelastic scattering spectra $\mathrm{S}(\mathbf{Q},\omega )$ along the $\mathbf{Q}=(H,6,6)$ (b), $\mathbf{Q}=(H,2,8)$ (c), and $\mathbf{Q}=(H,5,3)$ (d) reciprocal lattice directions. The disordered calculation is averaged over all otherwise equivalent directions as explained in the text. Red ovals in the ordered calculation indicate intensity from excitations that is not visible in the disordered calculation and experiment as discussed below.

Figure 3

(a) LA and (b) TA phonon scattering intensities (color maps) at $\mathbf{Q}=(5.5+h,0,0)$ (a) and $\mathbf{Q}=(6,h,0)$ (b) and experimental phonon energies (circles) obtained from the multizone fit of experimental inelastic neutron scattering intensity $\mathrm{S}(\mathbf{Q},\omega )$ (see Fig. S5 of the Supplemental Material). The white-filled circles indicate acoustic phonons, and the open circles are optic modes. We do not include the small intensity near 2.5 meV in our phonon fits as it does not appear in any other zones and may be an artifact.