Anisotropic low-energy vibrational modes as an effect of cage geometry in the binary barium silicon clathrate $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$

The low lattice thermal conductivity in inorganic clathrates has been shown recently to be related to the low-energy range of optical phonons dominated by motions of guest atoms trapped in a network of host covalent cages. A promising route to further reduce the heat conduction, and increase the material efficiency for thermoelectric heat waste conversion, is then to lower the energy of these guest-weighted optical phonons. In the present work, the effect of the host cage geometry is explored. The lattice dynamics of the binary type-IX clathrate, $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$, has been investigated experimentally by means of inelastic neutron scattering as a function of temperature between 5 and 280 K, and computed by ab initio density functional techniques. It is compared with the lattice dynamics of $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$, the simplest representative of the well-known type-I clathrate structure. The binary $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ and $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ materials have both a cubic unit cell made of different Si cages. The energies, the degree of anharmonicity as well as the anisotropy of the optical phonon modes weighted by Ba motions are found to depend strongly on the size and shape of the cages. The lowest optical phonon energies in $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ are found around 2.5–4 meV, while those in $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ have higher energies around 7–9 meV. The low-lying optical phonons in $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ are mainly weighted by the motion of Ba in the opened $\mathrm{S}{\mathrm{i}}_{20}$ cage, which does not exist in $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$. Moreover, the Ba vibrations within the opened $\mathrm{S}{\mathrm{i}}_{20}$ cages are found intrinsically anisotropic, strongly dispersionless in some directions, and exhibit a significant anharmonicity, which is not observed for any optical phonon modes in $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$.

Figure 1

Crystal structures of type-I clathrate $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ (a) and of type-IX clathrate $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ (b) plotted using the Vesta software [31]. The top row displays the Si (blue spheres) polyhedra (light green) centered by Ba atoms (red spheres) in the cubic unit cell (black line). In the bottom row, the building units of each clathrate structures are shown: type-I clathrate is made of $\mathrm{S}{\mathrm{i}}_{20}$ ($V\sim 104.9{\AA }^3$) and $\mathrm{S}{\mathrm{i}}_{24}$ ($V\sim 148.5{\AA }^3$) polyhedra occupied by $\mathrm{B}{\mathrm{a}}_{\mathrm{I}}\left(2\mathrm{a}\right)$ and $\mathrm{B}{\mathrm{a}}_{\mathrm{I}}\left(6\mathrm{d}\right)$, type-IX clathate is made of closed ($V\sim 68.2{\AA }^3$) and open $\mathrm{S}{\mathrm{i}}_{20}$ ($V\sim 102.2{\AA }^3$), $\mathrm{S}{\mathrm{i}}_8/\mathrm{S}{\mathrm{i}}_{11}$ ($V\sim 99.5{\AA }^3$) polyhedra occupied by $\mathrm{B}{\mathrm{a}}_{\mathrm{IX}}\left(8\mathrm{c}\right)$, $\mathrm{B}{\mathrm{a}}_{\mathrm{IX}}\left(12\mathrm{d}\right)$ and $\mathrm{B}{\mathrm{a}}_{\mathrm{IX}}\left(4\mathrm{b}\right)$ atoms, respectively.

Figure 2

(Left column) Ab initio calculated atomic potential $\mathrm{\Delta }E\mathrm{vs}\mathrm{\Delta }x$ displacement of the Ba atoms in $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ (a) and $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ (b). The red solid lines represent results from a polynomial fit up to fourth order. Blue dotted lines represent fit results with a harmonic, squared potential. (Right column) Temperature dependence of the diagonal ${\mathrm{U}}_{\mathrm{ii}}$ and isotropic ${\mathrm{U}}_{\mathrm{iso}}$ atomic displacement parameters of $\mathrm{B}{\mathrm{a}}_{\mathrm{IX}}\left(12\mathrm{d}\right)$ in open $\mathrm{S}{\mathrm{i}}_{20}$ cages of $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ (c) and of $\mathrm{B}{\mathrm{a}}_{\mathrm{I}}\left(6\mathrm{d}\right)$ in $\mathrm{S}{\mathrm{i}}_{24}$ cages of $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ (d). Lines correspond to DFT derived results and open symbols with $\mathrm{B}{\mathrm{a}}_{\mathrm{I}}\left(6\mathrm{d}\right)$ in $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ to experimental data [54]. The insets in (c) and (d) show an open $\mathrm{S}{\mathrm{i}}_{20}$ cage of $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ projected in the $ab$ plane and an $\mathrm{S}{\mathrm{i}}_{24}$ cages of $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ projected in the $ab$ plane, respectively.

Figure 3

Inelastic response of the type-IX clathrate $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ measured by INS in the low-energy range of Ba-dominated modes. (a) Dynamic structure factor $\mathrm{S}(E,T=280\mathrm{K})$, (b) generalized density of states ${\mathrm{G}}^{\exp }(E,T=280\mathrm{K})$ obtained from INS experiments (solid symbols) compared to the powder averaged lattice dynamics computed density of states ${\mathrm{G}}^{\mathrm{PALD}}(E,T=280\mathrm{K})$.

Figure 4

(a) Temperature dependence of ${\mathrm{G}}^{\exp }(E,T)/E^2$ of $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$. Temperatures are listed in the figure. Right and left sides correspond to the Stokes and anti-Stokes lines, respectively. Arrows are guides for the eye indicating the peak shift upon cooling. (b) Temperature dependence of the peak energies identified and fitted in ${\mathrm{G}}^{\exp }(E,T)/E^2$. Lines correspond to calculations with the DUM [68] characterizing the anharmonicity of the fitted peaks by a dimensionless parameter β whose value is 0.015 for dashed lines and 0.025 for dotted lines.

Figure 5

Phonon dispersion curves (a), phonon density of states (PDOS) (b) and phonon participation ratio (c) calculated for the type-IX $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ clathrate in the energy range 0–12 meV. The phonon dispersion curves have been computed along the high symmetry directions in the Brillouin zone. The color code with the dispersion curves indicates the symmetry of the modes at the Γ point: ${\mathrm{A}}_1$ (orange), ${\mathrm{A}}_2$ (red), E (green), ${\mathrm{T}}_1$ (black), ${\mathrm{T}}_2$ (blue).

Figure 6

(a) Partial phonon density of states (p-DOS) for the different Ba atoms and for all the Si atoms; (b) partial phonon density of states for the $\mathrm{B}{\mathrm{a}}_{\mathrm{IX}}\left(12\mathrm{d}\right)$ atoms projected along the high symmetry direction (here [100]) and the other directions (here [010] and [001]) calculated for the type-IX $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ clathrate in the energy range 0–12 meV.

Figure 7

Total and partial heat capacity $C_v\left(T\right)$ derived from DFT calculations plotted in a Debye plot $C/T^3$ vs T and compared to experimental data $C_p\left(T\right)$ [16] for $\mathrm{B}{\mathrm{a}}_{24}\mathrm{S}{\mathrm{i}}_{100}$ (a) and $\mathrm{B}{\mathrm{a}}_8\mathrm{S}{\mathrm{i}}_{46}$ (b).