_{ x }Ba

_{8−}

_{ x }Zn

_{ y }Ge

_{46−}

_{ y }clathrates

^{1,a)}, Yuta Sasaki

^{1}, Tsuyoshi Koyanagi

^{1}, Kenji Ohoyama

^{2}and Koji Akai

^{3}

### Abstract

Polycrystalline samples of degenerate *n*-type K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } () with the type-I clathrate structure (No. 223, ) were prepared by powder metallurgy to obtain a high-efficiency Ge-based clathrate. Their Zn atoms preferred to exist at the 6*c* site in the framework, and consequently, the samples with *x* around 4, such as K_{4}Ba_{4}Zn_{6}Ge_{40}, possessed highly ordered Zn/Ge atom frameworks whose 6*c*, 16*i*, and 24*k* sites were occupied almost solely by Zn,Ge, and Ge atoms, respectively. In spite of such ordered structures and small numbers of substituting Zn atoms, these samples exhibited carrier mobilities lower than those of Ba_{8}Zn_{8}Ge_{38} and Ba_{8}Ga_{16}Ge_{30}. Band structure calculations implied that the combination of the rattler K and Ba atoms in the cages considerably modified the conduction band edge of the corresponding clathrates; such a modification is considered to strengthen alloy disorder scattering, which reduces carrier mobility. The maximum dimensionless figure-of-merit *ZT* was 0.51 at 1000 K for the K_{2}Ba_{6}Zn_{7}Ge_{39} sample, which is similar to that of 0.50 at 900 K for the Ba_{8}Zn_{8}Ge_{38} sample.

The authors would like to thank M. Ohkawara for his support with the neutron diffraction measurements, which were carried out under the Joint-use Research Program for Neutron Scattering, Institute for Solid State Physics (ISSP), the University of Tokyo, at the Research Reactor JRR-3, JAEA (No. 10749). The authors would also like to acknowledge the PC cluster resources of the Media and Information Technology Center, Yamaguchi University, for calculations of the electronic structure. This work was partly supported by a Seeds-Discovery Grant No. H21-1521 from the Japan Science and Technology Agency.

I. INTRODUCTION

II. EXPERIMENTAL

III. RESULTS AND DISCUSSION

A. Crystal structures

B. Mobility and effective mass

C. Thermoelectric properties

IV. CONCLUSIONS

### Key Topics

- Germanium
- 105.0
- Zinc
- 87.0
- Clathrates
- 43.0
- Elemental semiconductors
- 43.0
- Carrier mobility
- 35.0

## Figures

Crystal structure of the type-I clathrate Ba_{8}(X,Ge)_{46}. Large balls represent Ba atoms and smaller balls represent X and Ge atoms; black, gray, and white balls are at the 6*c*, 16*i*, and 24*k* sites, respectively. This figure was drawn using the vesta program (Ref. 10).

Crystal structure of the type-I clathrate Ba_{8}(X,Ge)_{46}. Large balls represent Ba atoms and smaller balls represent X and Ge atoms; black, gray, and white balls are at the 6*c*, 16*i*, and 24*k* sites, respectively. This figure was drawn using the vesta program (Ref. 10).

(a) Powder neutron diffraction pattern of the K_{4}Ba_{4}Zn_{6}Ge_{40} sample and (b) powder x-ray diffraction pattern of the K_{8}Zn_{4}Ge_{42} sample, refined by Rietveld analysis.

(a) Powder neutron diffraction pattern of the K_{4}Ba_{4}Zn_{6}Ge_{40} sample and (b) powder x-ray diffraction pattern of the K_{8}Zn_{4}Ge_{42} sample, refined by Rietveld analysis.

Band structures and densities-of-state for (a) Ba_{8}Zn_{8}Ge_{38} and (b) K_{8}Zn_{4}Ge_{42} clathrates calculated in the virtual crystal approximation. The solid curve in the right-hand side of the figure indicates the total DOS, and the black, dark gray, and light gray areas indicate partial DOSs of the K/Ba atoms at the 2*a* site, of the K/Ba atoms at the 6*d* site, and of all the Zn/Ge atoms, respectively.

Band structures and densities-of-state for (a) Ba_{8}Zn_{8}Ge_{38} and (b) K_{8}Zn_{4}Ge_{42} clathrates calculated in the virtual crystal approximation. The solid curve in the right-hand side of the figure indicates the total DOS, and the black, dark gray, and light gray areas indicate partial DOSs of the K/Ba atoms at the 2*a* site, of the K/Ba atoms at the 6*d* site, and of all the Zn/Ge atoms, respectively.

Room-temperature Hall mobility as a function of carrier concentration for the K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } samples and some related clathrates: Ba_{8}(Zn,Ge)_{46} (Ref. 14), Ba_{8}(Ga,Ge)_{46} (Ref. 11), Ba_{8}(Ni,Ga,Ge)_{46} (Ref. 8), and Ba_{8}(Cu,Ga,Ge)_{46} (Ref. 13). The lines fit (: constant).

Room-temperature Hall mobility as a function of carrier concentration for the K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } samples and some related clathrates: Ba_{8}(Zn,Ge)_{46} (Ref. 14), Ba_{8}(Ga,Ge)_{46} (Ref. 11), Ba_{8}(Ni,Ga,Ge)_{46} (Ref. 8), and Ba_{8}(Cu,Ga,Ge)_{46} (Ref. 13). The lines fit (: constant).

Room-temperature weighted mobility as a function of carrier concentration for K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } and some Ba-containing Ge clathrates (Refs. 8, 11, 13, and 14). The line is a guide for the eyes.

Room-temperature weighted mobility as a function of carrier concentration for K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } and some Ba-containing Ge clathrates (Refs. 8, 11, 13, and 14). The line is a guide for the eyes.

Temperature dependences of (a) Hall mobility and (b) the ) value for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2} Ge_{38+} _{ x } _{/2} and Ba_{8}Ga_{16}Ge_{30} samples. The thin lines in proportion to and represent the temperature dependences of mobilities dominated by alloy disorder scattering and acoustic phonon scattering, respectively. The thick solid and dotted curves fit the relation (, : constant).

Temperature dependences of (a) Hall mobility and (b) the ) value for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2} Ge_{38+} _{ x } _{/2} and Ba_{8}Ga_{16}Ge_{30} samples. The thin lines in proportion to and represent the temperature dependences of mobilities dominated by alloy disorder scattering and acoustic phonon scattering, respectively. The thick solid and dotted curves fit the relation (, : constant).

Band structures and densities-of-state for hypothetical (a) Na_{8}Zn_{4}Ge_{42} and (b) Cs_{8}Zn_{4}Ge_{42} clathrates calculated in the virtual crystal approximation. The solid curve in the right-hand side of the figure indicates the total DOS, and the black, dark gray, and light gray areas indicate partial DOSs of the Na/Cs atoms at the 2 site, of the Na/Cs atoms at the 6 site, and of all the Zn/Ge atoms, respectively.

Band structures and densities-of-state for hypothetical (a) Na_{8}Zn_{4}Ge_{42} and (b) Cs_{8}Zn_{4}Ge_{42} clathrates calculated in the virtual crystal approximation. The solid curve in the right-hand side of the figure indicates the total DOS, and the black, dark gray, and light gray areas indicate partial DOSs of the Na/Cs atoms at the 2 site, of the Na/Cs atoms at the 6 site, and of all the Zn/Ge atoms, respectively.

Temperature dependences of (a) thermal conductivity , (b) lattice thermal conductivity , (c) Seebeck coefficient , (d) electrical conductivity , (e) power factor , and (f) dimensionless figure-of-merit for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2} Ge_{38+} _{ x } _{/2} and Ba_{8}Ga_{16}Ge_{30} samples. The curves in (a) and (f) are guides for the eyes; the solid curves in (b) fit the relation (, : constant), and the dotted curve in (b) represents the theoretical minimum value of Ge (Ref. 38).

Temperature dependences of (a) thermal conductivity , (b) lattice thermal conductivity , (c) Seebeck coefficient , (d) electrical conductivity , (e) power factor , and (f) dimensionless figure-of-merit for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2} Ge_{38+} _{ x } _{/2} and Ba_{8}Ga_{16}Ge_{30} samples. The curves in (a) and (f) are guides for the eyes; the solid curves in (b) fit the relation (, : constant), and the dotted curve in (b) represents the theoretical minimum value of Ge (Ref. 38).

## Tables

Structural parameters for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2}Ge_{38+} _{ x } _{/2} clathrate (No. 223, ) samples, refined by the Rietveld method using powder neutron and x-ray diffraction under the following conditions: (1) no vacancy exists; (2) K and Ba atoms occupy 2 (0, 0, 0) and 24 (1/4, , 1/2 + ) sites, and Zn and Ge atoms occupy 6 (1/4, 0, 1/2), 16 (, , ), and 24 (0, , ) sites; (3) both the K/Ba and Zn/Ge ratios are the same as those determined by electron probe microanalysis; (4) equivalent atomic displacement parameters are equal at each site.

Structural parameters for the K_{ x }Ba_{8−} _{ x }Zn_{8−} _{ x } _{/2}Ge_{38+} _{ x } _{/2} clathrate (No. 223, ) samples, refined by the Rietveld method using powder neutron and x-ray diffraction under the following conditions: (1) no vacancy exists; (2) K and Ba atoms occupy 2 (0, 0, 0) and 24 (1/4, , 1/2 + ) sites, and Zn and Ge atoms occupy 6 (1/4, 0, 1/2), 16 (, , ), and 24 (0, , ) sites; (3) both the K/Ba and Zn/Ge ratios are the same as those determined by electron probe microanalysis; (4) equivalent atomic displacement parameters are equal at each site.

Room-temperature properties of the K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } clathrate samples: (g/cm^{3}), density; (10^{20} cm^{−3}), carrier concentration; (cm^{2} V^{−1} s^{−1}), Hall mobility; (), effective mass; = (cm^{2} V^{−1} s^{−1}), weighted mobility; (mW cm^{−1} K^{−1}), lattice thermal conductivity; = (10^{3} cm^{3} K V^{−1} J^{−1}), materials factor.

Room-temperature properties of the K_{ x }Ba_{8−} _{ x }Zn_{ y }Ge_{46−} _{ y } clathrate samples: (g/cm^{3}), density; (10^{20} cm^{−3}), carrier concentration; (cm^{2} V^{−1} s^{−1}), Hall mobility; (), effective mass; = (cm^{2} V^{−1} s^{−1}), weighted mobility; (mW cm^{−1} K^{−1}), lattice thermal conductivity; = (10^{3} cm^{3} K V^{−1} J^{−1}), materials factor.

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