Strong renormalization of Ba vibrations in thermoelectric type-IX clathrate ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$

We report a combined experimental and theoretical study of the lattice dynamics of the type IX clathrate ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ by inelastic neutron scattering (INS) experiments and density functional theory (DFT) calculations. We observe low-energy optical modes at $\sim 2–3\mathrm{meV}$ due to the motion of the heavy Ba atoms along the high-symmetry axis and the largest dimension of the open Ge@20 cages present in the compound. Even though the phonon participation ratio indicates that these low-energy modes are localized, their $Q$ dependence shows that the dynamics of the Ba guests are correlated. We observe a strong change in the spectral weight of these modes when the compound undergoes a temperature-induced structural transformation in the temperature range 190–230 K. In the high-temperature phase, the low-energy optical modes show high intensities in the INS data and frequencies rather insensitive to temperature changes up to ∼550 K. In the low-temperature structural modification, the low-energy mode intensities are strongly depleted and apparently shifted to higher energies; this behavior is in line with an off-centering of the Ba atoms at low temperatures. Our DFT calculations successfully approximate the essential features in the dynamics of the high-temperature ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ structure.

Figure 1

Total and atom-projected phonon density of states (DOS) of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ obtained from density functional theory (DFT) calculations. Contribution from Ba is discriminated according to their Wyckhoff positions. Vertical bars indicate computed Γ-point eigenfrequencies; open squares report Raman frequencies measured by Shimizu et al. [31].

Figure 2

Generalized density of states (GDOS) of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ (empty circles) measured on IN5 at 300 K with 5 Å and neutron-weighted DOS $Z^{\prime }\left(E\right)$ (black solid lines) computed from density functional theory (DFT) results and convoluted with a Gaussian mimicking the IN5 energy resolution. Red dashed line corresponds to $Z^{\prime }\left(E\right)$ data with energies renormalized by a factor 1.1 to match the position of the highest energy peak.

Figure 3

(left) Phonon dispersion and (right) participation ratios of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ based on density functional theory (DFT) calculations. For clarity, the corresponding phonon density of states (DOS) is reported in red with the participation ratio.

Figure 4

Intensity maps of the dynamic structure factor of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ measured on IN5 with an incident wavelength of 2.5 Å at the indicated temperatures. Note the strong dispersionless excitation band centered at ∼3.3 meV formed at $T$ above the structural transformation at 190–230 K.

Figure 5

(a) Density functional theory (DFT) calculated atomic potentials $\mathrm{\Delta }E$ vs $\mathrm{\Delta }x$ displacement of the Ba atoms in ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$. The red solid lines represent results from a polynomial fit of fourth order. (b) Directional atom-projected phonon densities of states of the Ba3 atom of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ obtained from DFT calculations.

Figure 6

(a) Debye plot of the total and partial heat capacity as a function of temperature. (b) Atomic displacement parameters of barium atoms in ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ from density functional theory (DFT) calculations compared with experimental results of Fukuoka et al. [11].

Figure 7

(a) Generalized density of states (GDOS) of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ as a function of temperature derived from inelastic neutron scattering (INS) experiments at IN5 with the incident wavelength of 5 Å. (b) Same datasets in a Debye plot $\mathrm{GDOS}/E^2$ vs $E$. (c) GDOS from INS measurements at IN6 with an incident wavelength of 4.14 Å. (d) Same datasets in a Debye plot $\mathrm{GDOS}/E^2$ vs $E$.

Figure 8

(a) GDOS of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ as function of temperature derived from IN5 data recorded with the incident wavelength of 1.8 Å. (b) Variation of the strong peak at 10 meV as a function of temperature identified in all our inelastic neutron scattering (INS) experiments as well as temperature dependence of the lattice parameter of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ at low (Ref. [20]) and high temperature (our data) upon heating.

Figure 9

Constant energy cuts through the dynamic susceptibility divided by the energy for 3.5 ± 0.5 meV from IN5 experiments with $\lambda =2.5\AA$. (left) Anti-Stokes line, and (right) Stokes line. 20 K data not shown due to enhanced background contribution. Wave vector transfers with peaking signal (2.3 and $3.5{\AA }^{–1}$) correspond to points with strong Bragg peak intensities and densities the origin of acoustic phonon signal. Note the overall sudden reduction of signal below the temperature of structural transition.

Figure 10

Constant wave vector cuts through the dynamic susceptibility divided by the energy from IN6 experiments with $\lambda =4.14\AA$ at (a) 100 K and (b) 300 K. Only anti-Stokes line intensity is shown. Note that, at 300 K, the well-defined low-energy peak ∼3 meV persists at $Q$ points away from high Bragg peak intensity $\sim 2.3{\AA }^{–1}$. This feature is not detectable in the inelastic response below the structural phase transition at 100 K.

Figure 11

(a) LeBail refinement of the x-ray diffraction (XRD) pattern of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ at 300 K. (b) Thermal variation of the magnetic susceptibility of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$.

Figure 12

Electronic density of states of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ from density functional theory (DFT) calculations.

Figure 13

Atom-projected phonon density of states of the Ge atoms of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ obtained from density functional theory (DFT) calculations.

Figure 14

Isotropic atomic displacements parameters ${\mathrm{U}}_{\mathrm{iso}}$ of germanium atoms of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ from density functional theory (DFT) calculations compared to experimental results of Fukuoka et al. [13].

Figure 15

Intensity maps of the dynamic structure factor of ${\mathrm{Ba}}_{24}{\mathrm{Ge}}_{100}$ measured on IN5 with an incident wavelength of 1.8 Å at the indicated temperatures. Positive energies denote the Stokes line, negative energies the anti-Stokes line signals.

Figure 16

$Q$ cut through the dynamic susceptibility divided by the energy for 2.3 ± 0.2, 2.8 ± 0.2, and $3.5\pm 0.2{\AA }^{–1}$, from IN5 experiments with $\lambda =2.5\AA$. (left) Anti-Stokes line, and (right) Stokes line. 20 K data not shown due to enhanced background.