Impact of temperature and mode polarization on the acoustic phonon range in complex crystalline phases: A case study on intermetallic clathrates

The low and weakly temperature-varying lattice thermal conductivity, ${\kappa }_L\left(\mathrm{T}\right)$, in crystals with a complex unit cell such as type-I clathrates is assumed to originate from a reduced momentum and energy space available for propagative lattice vibrations, which is caused by the occurrence of low-energy optical phonon modes. In the context of ab initio self-consistent phonon (SCP) theory, it has been shown that the cubic and quartic anharmonic interactions result in a temperature-induced energy renormalization of these low-lying optical branches which contributes to the anomalous behavior of ${\kappa }_L\left(\mathrm{T}\right)$ in structurally ordered type-I clathrates [T. Tadano and S. Tsuneyuki, Phys. Rev. Lett. 120, 105901 (2018)]. By means of inelastic neutron scattering, we provide evidence for this energy renormalization in temperature, which has been resolved for transversely and longitudinally polarized phonons in the single crystal type-I clathrate ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$. By mapping the neutron intensity in the momentum space, we demonstrate the coherent character of the low-lying optical phonons. The overall phonon spectrum and dynamical structure factors are satisfactorily reproduced by ab initio harmonic calculations using density functional theory with the meta-GGA SCAN functional and a fully ordered structure. However, a polarization-dependent cutoff energy with opposing temperature shifts for longitudinal and transverse acoustic dispersions is experimentally observed which is not reproduced by the simulations. Anharmonicity affects the energies of the low-lying optical phonons in the transverse polarization, which compares quantitatively well with available results from SCP theory, whereas differences are observed for the longitudinal polarization.

Figure 1

(a) Crystallographic structure of the type-I clathrate ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ (BGA) The cubic unit cell (space group $\mathrm{P}m\overline{3}n$) contains two tetrakaidecahedral ($5^{12}{6}^2$) and five dodecahedral ($5^{12}$) host cages formed by Ge atoms (light gray) with guest Ba (green) atoms encapsulated inside. One to three Au atoms (gold) substitute Ge atoms at the Wyckoff site 6c, which results in a slight distortion of the tetrakaidecahedron and an off-centering of the Ba atoms inside [13]. (b) The lattice thermal conductivity, ${\kappa }_L$, for BGA (black circles) [13] is compared to different experimental measurements of ${\kappa }_L$ for ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ (BGG), including those by Avila et al. [22], Sales et al. [23], May et al. [24], and the theoretical calculations of Tadano and Tsuneyuki [43]. Black and blue dashed lines show the deviation of ${\kappa }_L$ from $1/T$ for both BGA and BGG, respectively, at higher temperatures.

Figure 2

(a) The rate of thermal expansion, $\mathrm{\Delta }a/a$, for ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ has been measured with Larmor diffraction on TRISP@FRM-II (black circles) and compared to that of Falmbigl et al. (empty red squares) [50]. The numerical derivative of $\mathrm{\Delta }a/a$ is the linear thermal expansion coefficient, ${\alpha }_L$. The volumetric thermal expansion coefficient, ${\alpha }_V=3{\alpha }_L$, is plotted in (b).

Figure 3

(a) Transverse and longitudinal phonon dispersion curves extracted from IN5@ILL at 150 (blue circles), 300 (black triangles), and at 530 K (red stars) near the (006) Bragg peak. Gray solid lines depict the calculated dispersion curves. The wave vector $q$ is referenced from the zone center (006) (${\mathrm{\Gamma }}_{006}$). Acoustic phonons with $q$ increasing towards the zone center (116) (${\mathrm{\Gamma }}_{116}$) are mostly transversely polarized (left panel). Those with $q$ increasing towards the zone center (007) (${\mathrm{\Gamma }}_{007}$) are mostly longitudinal (right panel). Also shown in gray is the simulated phonon spectrum. (b) Similar experimental phonon dispersions near the (222) Bragg peak, at the same three temperatures, and compared to simulations. Starting from the zone center (222) (${\mathrm{\Gamma }}_{222}$), the transverse (left panel) polarization can be traced until zone center (114) (${\mathrm{\Gamma }}_{114}$), while the longitudinal (right panel) polarization propagates towards the ${\mathrm{\Gamma }}_{333}$. (c) Generalized vibrational density of states (GVDOS) obtained on IN5@ILL at 150 and 530 K, along with the calculated phonon DOS (intensity has been scaled down by a factor of 3.5). All simulations correspond to harmonic ab initio DFT calculations using the meta-GGA SCAN functional.

Figure 4

Calculated partial phonon density of states (pDOS), using harmonic ab initio DFT calculations and the meta-GGA SCAN functional. The site-weighted partial pDOS lines of Wyckoff positions Ge($16i$), Ge($24k$), Ba($2a$), Ba($6d$), and Au($6c$) show their respective contributions in energy as compared to the total pDOS.

Figure 5

Two-dimensional inelastic neutron scattering intensity distribution at fixed energy transfer in the momentum plane $\left(\left[110\right];\left[001\right]\right)$ for ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ at 300 K, recorded on IN5@ILL. The first three energy integrations (a)–(c) were taken at $3.5\pm 0.15$, $4.8\pm 0.15$, and $6.5\pm 0.15$ meV, respectively. (d) represents a larger integration interval of $9.5\pm 2.5$ meV. The color bar reflects a normalized intensity scale. Dashed white grids show the borders of the Brillouin zones.

Figure 6

Simulated two-dimensional neutron scattering function, S(Q,$\omega$), at fixed energy transfer in the momentum plane $\left(\left[110\right];\left[001\right]\right)$ for an ideal ordered model of a type-I clathrate with the composition ${\mathrm{Ba}}_8{\mathrm{Ge}}_{40}{\mathrm{Au}}_6$. The first three energy integrations (a)–(c) were taken at $3.5\pm 0.15$, $4.8\pm 0.15$, and $6.5\pm 0.15$ meV, respectively. (d) represents a larger integration interval of $9.5\pm 2.5$ meV. The intensity in each map has been normalized to the same intensity scale, which is reflected in the color bar. Dashed white grids show the borders of the Brillouin zones.

Figure 7

(a)–(d) Inelastic neutron scattering (INS) intensity for constant $\mathbf{Q}=\left(113\right)$ scans as a function of energy (black circles) recorded on IN12@ILL between 200–500 K for ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$. The solid red lines are Gaussian fits of the three optical bands whose energies are referred to as $E_{\text{Ba}}$ and $E_{3,4}$. The vertical dotted black lines indicate their values at 200 K. (e) Temperature dependence of $E_{\text{Ba}}$ and $E_{3,4}$ obtained from the fits shown in panels (a)–(d) (black circles) and from the $E_{\text{Ba,Au}}$ and $E_3$ peaks in the generalized vibrational density of states shown in Fig. 3 (purple crosses). Blue triangles show the temperature dependence of $E_{\text{Ba}}$ in ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ [59]. In (f), the temperature dependence of $E_{\text{Ba}}$ for ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ (black circles) and ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ (blue triangles), corrected for the thermal expansion (TE), are shown and compared to that of the ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ self-consistent phonon theory calculation that includes cubic and quartic terms (SCPB) [43] (green stars). The rates of TE of ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ and ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ by INS and SCPB (same symbols/colors) are also plotted.

Figure 8

The energy resolution, given our specific experimental conditions, of the cold-neutron time-of-flight spectrometer IN5@ILL is plotted in the energy range of interest for results discussed in this article. The calculation is made by considering the incident neutron wavelength, chopper speed, and other instrumental parameters specific to IN5. The equation for calculating energy resolution is derived in Ref. [70].

Figure 9

The temperature-dependent Grüneisen parameter ($\gamma$) in (d) for ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$ is calculated using the rate of thermal expansion ($\mathrm{\Delta }a/a$) in (a), the volumetric coefficient of thermal expansion (${\alpha }_V$) in (b), the constant volume specific heat in (c), and the bulk modulus calculated in Appendix pp2. The rate of thermal expansion has been measured by Larmor diffraction on TRISP@FRM-II (black circles), and compared to that of Falmbigl et al. (empty red squares) [50]. Its numerical derivative is used to find ${\alpha }_L={\alpha }_V/3$. The measurement of $C_P$ (black circles) from Ikeda et al. [19] was used to calculate $C_V$ (solid blue line).

Figure 10

Calculations from Table S2 of Ref. [43] are plotted in order to obtain $(\frac{\partial E_j}{\partial V}{)}_T$ for the $E_{\text{Ba}}$ optical band, namely, the rate of change in volume of ${\mathrm{Ba}}_8{\mathrm{Ge}}_{30}{\mathrm{Ga}}_{16}$ with energy for “SCP” (a) and “$\mathrm{SCP}+\mathrm{Bubble}$” (b).

Figure 11

Generalized vibrational density of states obtained from the data collection recorded on IN5@ILL at 150 and 530 K. The energy axis of the data at 530 K has been scaled by 3%, leading to an alignment of all but the lowest peaks.

Figure 12

Two-dimensional inelastic neutron scattering intensity reported as function of the energy transfer and along the [110] [(a), (c), (e)] and [001] [(b), (d), (f)] directions in ${\mathrm{Ba}}_{7.81}{\mathrm{Ge}}_{40.67}{\mathrm{Au}}_{5.33}$, recorded on IN5@ILL. The data have been taken at three temperatures: 530 K [(a), (b)], 300 K [(c), (d)], and 150 K [(e), (f)]. Measurements were centered at the zone center ${\mathrm{\Gamma }}_{006}$ ($Q=3.5{\AA }^{-1}$), from which the acoustic phonon polarization is longitudinal when the propagation direction is [001] [panels (b), (d), (f)], and transversely polarized [001] when the propagation direction is [110] [panels (a), (c), (e)]. The Brillouin zones are indicated with vertical solid white lines.

Figure 13

Energy scans taken on IN5@ILL near the (006) Bragg peak with $\lambda =3.2\AA$ at 530 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and their fit as a solid red line.

Figure 14

Energy scans taken on IN5@ILL near the (006) Bragg peak with $\lambda =3.2\AA$ at 300 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and their fit as a solid red line.

Figure 15

Energy scans taken on IN5@ILL near the (006) Bragg peak with $\lambda =3.2\AA$ at 150 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and the fit as a solid red line.

Figure 16

Energy scans taken on IN5@ILL near the (222) Bragg peak with $\lambda =3.2\AA$ at 530 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and their fit as a solid red line.

Figure 17

Energy scans taken on IN5@ILL near the (222) Bragg peak with $\lambda =3.2\AA$ at 300 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and their fit as a solid red line.

Figure 18

Energy scans taken on IN5@ILL near the (222) Bragg peak with $\lambda =3.2\AA$ at 150 K. Plots reflect a normalized intensity scale, with data points as empty black circles, and the fit as a solid red line.