Phonon-glass electron-crystal thermoelectric clathrates: Experiments and theory

Type-I clathrate compounds have attracted a great deal of interest in connection with the search for efficient thermoelectric materials. These compounds constitute networked cages consisting of nanoscale tetrakaidecahedrons (14-hedrons) and dodecahedrons (12-hedrons), in which the group-1 or -2 elements in the periodic table are encaged as so-called “rattling” guest atoms. It is remarkable that, although these compounds have a crystalline cubic structure, they exhibit glasslike phonon thermal conductivity over the whole temperature range depending on the states of rattling guest atoms in the tetrakaidecahedron. In addition, these compounds show unusual glasslike specific heats and terahertz-frequency phonon dynamics, providing a remarkable broad peak almost identical to those observed in amorphous materials or structural glasses, the so-called boson peak. An efficient thermoelectric effect is realized in compounds showing these glasslike characteristics. In this decade, a number of experimental works dealing with type-I clathrate compounds have been published. These are diffraction, thermal, and spectroscopic experiments in addition to those based on heat and electronic transport. These form the raw materials for this review based on advances from this decade. The subject of this review involves interesting phenomena from the viewpoint not only of physics but also of the practical problem of elaborating efficient thermoelectric materials. This review presents a survey of a wide range of experimental investigations of type-I clathrate compounds, together with a review of theoretical interpretations of the peculiar thermal and dynamic properties observed in these materials.

Figure 1

Thermoelectric devices contain many couples of $n$-type and $p$-type thermoelectric elements wired electrically in series and thermally in parallel.

Figure 2

Crystal structures of type-I, type-II, type-III, type-VIII, and type-IX intermetallic clathrates.

Figure 3

Crystal structures and single crystals of type-I and type-VIII ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$. The units of rulers are given in millimeters. From [262].

Figure 4

(a) Tetrahedral bonds of framework atoms, and (b) guests in dodecahedral and tetrakaidecahedral cages for type-I clathrates.

Figure 5

Split $24j$ and $24k$ sites for guest (2) atoms around $6d$ sites in tetrakaidecahedral cages for a type-I clathrate.

Figure 6

The specific-heat data plotted as $C_V(T)/T^3$ vs $T$ (open circles) for type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30-x}{\mathrm{Ge}}_x$. The dotted, dashed, and dash-dotted lines are from the Einstein model, the Debye model, and the electronic components, respectively. From [263].

Figure 7

The specific-heat data plotted as $C_V/T^3$ vs $T$ for $n$-type type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and $n$-type and $p$-type type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ single crystals. Adapted from [12], [13], and [262].

Figure 8

The temperature-linear factor $\gamma$ of low-temperature specific heat for type-I ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Sr}$ and Ba) plotted as a function of the carrier concentration $n^{1/3}$. Adapted from [296].

Figure 9

Temperature dependences of the thermopower $S$ [$\mu \mathrm{V}/\mathrm{K}$] for ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Ba}$, Sr, Eu) single crystals. From [241].

Figure 10

Temperature dependences of electrical resistivity $\rho$ for ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ polycrystalline samples with various carrier densities (see text). From [209].

Figure 11

Temperature dependences of the thermopower $S$ [$\mu \mathrm{V}/\mathrm{K}$] for type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ polycrystalline samples with various carrier densities (see text). From [209].

Figure 12

Thermoelectric power factor at room temperature as a function of Ga composition $x$ for $n$-type and $p$-type ${\mathrm{Ba}}_8{\mathrm{Ga}}_x{\mathrm{Ge}}_{46-x}$ polycrystalline samples. Adapted from [8].

Figure 13

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}$ for polycrystalline samples of type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. The dashed line represents ${\kappa }_{\text{ph}}$ for vitreous ${\mathrm{SiO}}_2$. From [62].

Figure 14

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}(T)$ for type-I ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Ba}$, Sr, Eu) single crystals. Adapted from [241].

Figure 15

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}$ for type-I and type-VIII ${\mathrm{Eu}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ polycrystalline samples. From [220].

Figure 16

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}(T)$ vs $T$ on a log-log scale for type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30-x}{\mathrm{Ge}}_x$ single crystals. From [263].

Figure 17

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}$ for $n$-type and $p$-type ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and type-VIII ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ single crystals on a log-log scale. From [12].

Figure 18

Temperature dependences of phonon thermal conductivity ${\kappa }_{\text{ph}}(T)$ of $n$-type and $p$-type type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ single crystals on a log-log scale. From [12], [13], and [262].

Figure 19

Polarization dependence of Raman spectra for type-I ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Ba}$, Sr, Eu) at room temperature. “G” are the symmetry-allowed Raman-active phonons, and the bars and open circles denote the assigned peak. The peaks marked by $*$ are additional modes. From [267].

Figure 20

Illustration of fundamental modes of the guest atom $\mathcal{R}(2)$. See the text for the notations for the additional $A_{1g}$, $E_g(\mathrm{A})$, and $T_{2g}(1)$ modes. Adapted from [268].

Figure 21

Temperature dependences of Raman spectra for type-I ${\mathrm{Eu}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. From [267].

Figure 22

Temperature dependences of mode energies of guest atoms, $T_{2g}(1)$ and $E_g(\mathrm{A})$, for type-I ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Ba}$, Sr, Eu). From [267].

Figure 23

Temperature dependences of the mode energies $T_{2g}(1)$ and $E_g(\mathrm{A})$ for $n$-type and $p$-type type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ and ${\mathcal{R}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ ($\mathcal{R}=\mathrm{Sr}$, Eu). Adapted from [267], and [264].

Figure 24

Temperature dependences of conductivity $\sigma (\omega )={\sigma }_1(\omega )+i{\sigma }_2(\omega )$ for type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. From [184].

Figure 25

(a) Temperature dependences of the real part of the conductivities ${\sigma }_1$ for type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$. (b), (c) Experimental data (open circles) and fitting results (solid lines) at 296 and 7 K, respectively (see text). From [183].

Figure 26

${}^{71}\mathrm{Ga}$ NMR linewidth (square root of the second moment) vs temperature for type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. Adapted from [95].

Figure 27

The definition of the position vector of the $\ell$th cage at time $t$ as ${\mathit{R}}_{\ell }+{\mathit{r}}_{\ell }(t)$, where ${\mathit{R}}_{\ell }$ is the equilibrium position of the $\ell$th cage center. The vector ${\mathit{r}}_{\ell }(t)$ represents a small deviation from ${\mathit{R}}_{\ell }$. The position of the $\mathcal{R}(2)$ guest atom from the center of the $\ell$th cage ${\mathit{R}}_{\ell }$ is defined by the vector ${\mathit{U}}_{\ell }+{\mathit{u}}_{\ell }(t)$, where ${\mathit{U}}_{\ell }$ is the equilibrium position of the $\mathcal{R}(2)$ guest atom from ${\mathit{R}}_{\ell }$, and ${\mathit{u}}_{\ell }(t)$ is a small deviation from ${\mathit{U}}_{\ell }$ at time $t$.

Figure 28

Neutron scattering intensity corresponding to PDOSs for type-I $n\text{−}{\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, $p\text{−}{\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, and $n\text{−}{\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. Gaussian fits are shown for the first curves. From [54].

Figure 29

Phonon dispersion curves for quasi-on-center ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ along $[hh0]$ around the positions $G=(222)$, $G=(004)$, and $G=(330)$. The error bars represent the standard deviation of the fitted energies. From [50].

Figure 30

The PDOSs of type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ obtained from INS measurements. The dashed line is the contribution of the modes associated with Ba(2) guest atoms and Sr(2), the dot-dashed line is the parabolic Debye contribution, and the dotted line is a Gaussian fit to the optical phonon density of states below 15 meV. Because the neutron scattering lengths of Ba and Sr are different, the measured PDOSs have been arbitrarily scaled. Insets: The Gaussian peaks at $E_R$ associated with Ba(2) and Sr(2). Deviations in the inset for type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ are related to the Ba(1) mode. From [106].

Figure 31

INS data as a function of temperature 300, 200, and 100 K for type-I ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and $p$-type ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. These are given by the squares, circles, and triangles, respectively. From [53].

Figure 32

Temperature dependences of elastic stiffness constants $C_{11},C_{44}$, and $(C_{11}-C_{12})/2$ in $n$-type type-I ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$. The insets represent the data in an expanded scale below 5 K. From [120].

Figure 33

Temperature dependences of elastic stiffness constants $C_{11},C_{44}$, and $(C_{11}-C_{12})/2$ in type-I ${\mathrm{K}}_8{\mathrm{Ga}}_8{\mathrm{Sn}}_{38}$. From [120].

Figure 34

Schematic illustration of the anharmonic potential for the $\mathcal{R}(2)$ guest atom in oversized cages in type-I clathrate compounds. From [13].

Figure 35

The emergence of 2 degrees of freedom of motion: stretching $h_{\ell }(t)$ and libration $U_0\delta {\theta }_{\ell }(t)$.

Figure 36

(a) Phonon dispersion curves along the [001] direction for BGG on-center systems. Dispersion relations for longitudinal modes are plotted with solid lines and for transverse modes with dashed lines. Dotted linear lines from the origin represent the long-wave limit of the acoustic dispersion relations. The degeneracy of optical modes at the $\mathrm{\Gamma }$ point is observed at 4.2 meV arising from the on-centered symmetric potential. (b) Phonon dispersion curves along the [001] direction for $\beta$-BGS off-center systems. Dispersion relations for longitudinal (transverse) modes are illustrated by the region bounded by solid (dashed) lines. Dotted linear lines from the origin again represent the long-wave limit of acoustic dispersion relations. Note that optic modes observed at the $\mathrm{\Gamma }$ point are not separated at 2.2 and 4.1 meV. Adapted from [195].

Figure 37

Illustration showing the coupling mechanism between acoustic phonons and $\mathcal{R}(2)$ guest atoms. (a) The case in which the off-center guest atom takes the far-side position for incident longitudinal ($\parallel$) acoustic phonons with the wave vector ${\mathit{k}}_{\text{ph}}$; (b) the $\mathcal{R}(2)$ guest atom is perpendicular to the same longitudinal acoustic phonons. The coupling to parallel components of relevant displacements becomes effective regardless of whether the mode is transverse or longitudinal.

Figure 38

Phonon densities of states for both type-I (a) ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (BGG) and (b) ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ ($\beta$-BGS) obtained by means of coherent INS experiments in the temperature range from 5 to 290 K. The data provide integrated densities of states from 0.4 to $0.7{\AA }^{-1}$ in $\mathit{k}$ space. See text for the assignments of the spectra. From [190].

Figure 39

Schematic illustration of a double-well potential representing the two-level tunneling state.

Figure 40

Schematic illustration of the configuration of the guest atoms in the oversized cages in $\beta$-BGS. The fourfold inversion axes are directed along $x$, $y$, and $z$. (a) The deviation from the on-center position (filled circles) induces the electric dipoles (arrows). Here two nearest dipoles are depicted. The electric dipoles rotate in the plane perpendicular to the axis linking the nearest dipoles. (b) The configuration of the on-center positions of the guest atoms. The filled circles represent the positions of off-center guest atoms, around which the electric dipoles are induced. The sites $A$, $B$, and $C$ in (a) are seated on the chains parallel to $x$, $y$, and $z$, respectively: $A=(a/4,0,a/4)$, $B=(0,a/4,3a/4)$, and $C=(a/2,a/2,a/2)$. The distance between the next-nearest neighbors (dashed lines) is $\sqrt{3/8}a$. Note that these three dipoles constitute an equilateral triangle and easily generate a frustrated situation. (c) The 3D configuration of the dipole chains is illustrated.

Figure 41

(a) Illustration of two 14-hedron cages along the $z$ direction consisting of anions and guest cations. (b) The cage (outer circle) and the symmetry-broken off-center guest atom compose an effective electric dipole moment (thick arrow), which is the vector sum of each dipole (thin arrow).

Figure 42

Schematic illustration of free-energy landscape representing nonequilibrium states.

Figure 43

A first-order process of local-mode hopping. The local mode (double lines) interacts with the acoustic phonon (wavy line) with the eigenfrequency ${\omega }_{\mathit{k}}$, hopping from the initial state with ${\omega }_{\lambda }$ to the final state with ${\omega }_{{\lambda }^{\prime }}$, where ${\omega }_{{\lambda }^{\prime }}>{\omega }_{\lambda }>{\omega }_{\mathit{k}}$. The set of arrowheads in the opposite directions on the double lines indicate that the local modes consist of the superposition of plane waves with opposite directions.