# Size Effect on the Thermal Conductivity of a Type-I Clathrate

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## Abstract

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## 1. Introduction

**Figure 1.**(

**a**) Crystal structure [24] of a type-I clathrate, with the guest atoms (green) sitting inside cages formed by essentially covalently bonded host atoms (blue and orange). In the case of La${}_{1.2}$Ba${}_{6.8}$Au${}_{5.8}$Si${}_{38.8}{\square}_{1.4}$ (La-BAS), the green spheres denote Ba and La, the blue ones Si and the orange ones Au. The vacancies □ are distributed between Si and Au; (

**b**) sketch of the phonon dispersion of type-I clathrates, as proposed in [13]. An acoustic phonon mode (the dashed line shows the dispersion assumed in the Debye model) hybridizes with a flat rattling mode (blue), resulting in a severe flattening of the former at large wave vectors q (red). This shifts the characteristic energy scale from the Debye energy $\hslash {\omega}_{\mathrm{D}}$ to the Einstein energy $\hslash {\omega}_{\mathrm{E}}$ of the lowest-lying rattling mode.

## 2. Materials and Methods

#### 2.1. Samples

#### 2.2. Nanofabrication Process

#### 2.3. Measurement Method

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Laue X-ray diffraction pattern of the used La-BAS single crystal; (

**b**) oriented single crystal.

**Figure 3.**Scanning electron microscopy images of (

**a**) the bulk single crystal with cut out lamellas; (

**b**) a lamella; (

**c**) mesowires cut from the lamella; and (

**d**) a single mesowire on the measuring platform.

**Figure 5.**(

**a**) Frequency dependence of the third harmonic voltage signal of a Pt test wire at 300 K, measured with 3.5 mA; (

**b**) current dependence of the third harmonic of a 920 nm La-BAS mesowire at different temperatures. Prior to measuring the $V\left(I\right)$ dependencies, frequency sweeps were performed to select a measuring frequency at which the $3\omega $ signal is nearly frequency independent and thus Equation (2) is valid. For better visibility, not all measured temperatures are included in this plot.

**Figure 6.**Temperature-dependent (

**a**) phonon thermal conductivity (full symbols, left axis) and electron thermal conductivity (open symbols, right axis), as estimated from the Wiedemann–Franz law and (

**b**) electrical resistivity.

**Figure 7.**Temperature-dependent phonon thermal conductivity of a bulk La-BAS single crystal and mesowires fabricated from the same batch (symbols), the best fit to the bulk single crystal (full blue line) and simulated curves (dashed lines) for which the boundary scattering length is fixed to the mesowire diameters and all other parameters are kept unchanged, using (

**a**) the modified Callaway model with ${\theta}_{\mathrm{E}}=82.3$ K and (

**b**) the original Callaway model with ${\theta}_{\mathrm{D}}=348.3$ K. Data for the bulk single crystal are taken from [34]. Only the modified Callaway model can capture the overall high-temperature behavior. Deviations at low temperatures are attributed to small changes in the defect scattering rate due to the FIB processing.

**Figure 8.**Phonon thermal conductivity as a function of boundary scattering length at 300 K, simulated using the modified (black) and the original Callaway model (red). Squares denote mesowire data, the circle the bulk data point taken from [34].

**Figure 9.**$ZT$ as a function of T. The behavior of the mesowires with a 630 nm diameter at high temperatures points towards an improvement as a result of mesostructuring.

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**MDPI and ACS Style**

Lužnik, M.; Lientschnig, G.; Taupin, M.; Steiger-Thirsfeld, A.; Prokofiev, A.; Paschen, S.
Size Effect on the Thermal Conductivity of a Type-I Clathrate. *Crystals* **2023**, *13*, 453.
https://doi.org/10.3390/cryst13030453

**AMA Style**

Lužnik M, Lientschnig G, Taupin M, Steiger-Thirsfeld A, Prokofiev A, Paschen S.
Size Effect on the Thermal Conductivity of a Type-I Clathrate. *Crystals*. 2023; 13(3):453.
https://doi.org/10.3390/cryst13030453

**Chicago/Turabian Style**

Lužnik, Monika, Günther Lientschnig, Mathieu Taupin, Andreas Steiger-Thirsfeld, Andrey Prokofiev, and Silke Paschen.
2023. "Size Effect on the Thermal Conductivity of a Type-I Clathrate" *Crystals* 13, no. 3: 453.
https://doi.org/10.3390/cryst13030453