^{1}, Teruyuki Ikeda

^{2,3}, G. Jeffrey Snyder

^{2}and Chris Dames

^{1,a)}

### Abstract

A model has been established for the effective thermal conductivity of a bulk polycrystal made of randomly oriented superlattice grains with anisotropicthermal conductivity. The in-plane and cross-plane thermal conductivities of each superlattice grain are combined using an analytical averaging rule that is verified using finite element methods. The superlatticeconductivities are calculated using frequency dependent solutions of the Boltzmann transport equation, which capture greater thermal conductivity reductions as compared to the simpler gray medium approximation. The model is applied to a nanobulk material to investigate the effects of period, specularity, and temperature. The calculations show that the effective thermal conductivity of the polycrystal is most sensitive to the in-plane conductivity of each superlattice grain, which is generally four to five times larger than the cross-plane conductivity of a grain. The model is compared to experimental measurements of the same system for periods ranging from 287 to 1590 nm and temperatures from 300 to 500 K. The comparison suggests that the effective specularity increases with increasing annealing temperature and shows that these samples are in a mixed regime where both Umklapp and boundary scattering are important.

This work is supported in part by the DARPA/DSO NMP program (Grant No. W911NF-08-C-0058) and the PRESTO program of Japan Science and Technology Agency. The views, opinions, and/or findings contained in this article are those of the authors and should not be interpreted as representing the official views or policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the Department of Defense. Approved for Public Release, Distribution Unlimited.

I. INTRODUCTION

II. AVERAGING RULES FOR THE EFFECTIVE THERMAL CONDUCTIVITY

A. General considerations

B. Averaging rule for a thin film

C. Averaging rule for a long wire

D. Averaging rule for nanobulk

E. Numerical analysis using finite element methods (FEM)

F. Sensitivity

III. MODELING AND OF A SINGLE SUPERLATTICE GRAIN

A. Approximating the dispersion relations

B. Phonon scattering mechanisms in bulk

C. Considerations for anisotropic constituent materials

D. In-plane thermal conductivity

E. Cross plane thermal conductivity

IV. NUMERICAL RESULTS AND DISCUSSION

A. Effect of period

B. Effect of specularity

C. Effect of temperature

D. Comparison of gray versus frequency dependent modeling

V. COMPARISON WITH EXPERIMENT

A. Comparison between model and experiment

VI. CONCLUSIONS

### Key Topics

- Thermal conductivity
- 69.0
- Superlattices
- 34.0
- Phonons
- 28.0
- Thermal models
- 22.0
- Polycrystals
- 21.0

## Figures

(a) Schematic of a nanobulk composite material made of randomly oriented superlattice grains, and the global coordinate system , which is aligned to the macroscopic temperature gradient. (b) A single superlattice grain with its local coordinate system , which is aligned to the superlattice planes.

(a) Schematic of a nanobulk composite material made of randomly oriented superlattice grains, and the global coordinate system , which is aligned to the macroscopic temperature gradient. (b) A single superlattice grain with its local coordinate system , which is aligned to the superlattice planes.

Averaging rules for the effective thermal conductivity of polycrystalline thin films (red squares), wires (blue circles), and nanobulk materials (green triangles). Points: FEM simulations of Table I. Lines: analytical results from Eqs. (11), (15), and (18). Inset: typical FEM simulation of a nanobulk configuration.

Averaging rules for the effective thermal conductivity of polycrystalline thin films (red squares), wires (blue circles), and nanobulk materials (green triangles). Points: FEM simulations of Table I. Lines: analytical results from Eqs. (11), (15), and (18). Inset: typical FEM simulation of a nanobulk configuration.

Theoretical thermal conductivity of PbTe using only the high-temperature Umklapp expression of Eq. (32) (blue solid line), as compared to experimental data from Greig in Ref. 35 and Devyatkova in Ref. 36 (red squares). Above 80 K, the data and model follow a relation almost perfectly (dashed green line). Inset: experimental data (Ref. 37) (points) for the thermal conductivity of in both -axis and -plane directions also follows a relation around room temperature.

Theoretical thermal conductivity of PbTe using only the high-temperature Umklapp expression of Eq. (32) (blue solid line), as compared to experimental data from Greig in Ref. 35 and Devyatkova in Ref. 36 (red squares). Above 80 K, the data and model follow a relation almost perfectly (dashed green line). Inset: experimental data (Ref. 37) (points) for the thermal conductivity of in both -axis and -plane directions also follows a relation around room temperature.

Thermal conductivity as a function of period for four different values of the specularity parameter , for a nanobulk system at with thickness ratio . Solid lines: and are the in-plane and cross-plane values for a single superlattice grain, while is the value for a bulk polycrystal with randomly oriented grains. Dashed lines: and are the classical Fourier-law values for a single superlattice grain, neglecting phonon size effects.

Thermal conductivity as a function of period for four different values of the specularity parameter , for a nanobulk system at with thickness ratio . Solid lines: and are the in-plane and cross-plane values for a single superlattice grain, while is the value for a bulk polycrystal with randomly oriented grains. Dashed lines: and are the classical Fourier-law values for a single superlattice grain, neglecting phonon size effects.

Bulk effective thermal conductivity as a function of specularity for four different periods , for a nanobulk system at with thickness ratio .

Bulk effective thermal conductivity as a function of specularity for four different periods , for a nanobulk system at with thickness ratio .

Bulk effective thermal conductivity as a function of temperature for four different values of the specularity , for a system with thickness ratio and fixed period . The black dotted line is the bulk classical Fourier law value.

Bulk effective thermal conductivity as a function of temperature for four different values of the specularity , for a system with thickness ratio and fixed period . The black dotted line is the bulk classical Fourier law value.

Comparison of present frequency-dependent model (solid lines) and traditional gray media model (dashed lines) for the period-dependence of the cross-plane superlattice thermal conductivity . The calculations are for a superlattice system at with thickness ratio .

Comparison of present frequency-dependent model (solid lines) and traditional gray media model (dashed lines) for the period-dependence of the cross-plane superlattice thermal conductivity . The calculations are for a superlattice system at with thickness ratio .

Comparison of normalized theoretical (lines) and measured (points) thermal conductivity at . The theoretical results are normalized as and the experimental results as .

Comparison of normalized theoretical (lines) and measured (points) thermal conductivity at . The theoretical results are normalized as and the experimental results as .

Temperature dependence of normalized effective lattice thermal conductivities for periods of 287, 577, and 1590 nm. All thermal conductivities are normalized as .

Temperature dependence of normalized effective lattice thermal conductivities for periods of 287, 577, and 1590 nm. All thermal conductivities are normalized as .

## Tables

Parameters used in FEM simulations.

Parameters used in FEM simulations.

Properties for PbTe and at 300 K. The adjustable parameters for were fitted using the room temperature bulk phonon thermal conductivities in Ref. 39. Densities of primitive unit cells are calculated using the lattice constant and crystal structure. The sound velocities of are estimated from those of using the scaling arguments explained in Appendix B.

Properties for PbTe and at 300 K. The adjustable parameters for were fitted using the room temperature bulk phonon thermal conductivities in Ref. 39. Densities of primitive unit cells are calculated using the lattice constant and crystal structure. The sound velocities of are estimated from those of using the scaling arguments explained in Appendix B.

Fraction transformed and interlamellar spacing of the samples used for lattice thermal conductivity measurements.

Fraction transformed and interlamellar spacing of the samples used for lattice thermal conductivity measurements.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content