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Research Papers: Heat and Mass Transfer

Application of Houston's Method to the Calculation of the Direction-Dependent Thermal Conductivity in Finite Crystals at Low Temperatures

[+] Author and Article Information
M. Kazan

Department of Physics,
American University of Beirut,
P.O. Box 11-0236,
Riad El-Solh,
Beirut 1107-2020, Lebanon
e-mail: mk140@aub.edu.lb

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received November 8, 2016; final manuscript received April 21, 2017; published online June 1, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(10), 102004 (Jun 01, 2017) (9 pages) Paper No: HT-16-1727; doi: 10.1115/1.4036601 History: Received November 08, 2016; Revised April 21, 2017

This paper presents significant advances in the analytical calculation of the low-temperature lattice thermal conductivity in finite crystals. It shows that an accurate prediction of the direction-dependent lattice thermal conductivity can be obtained at low temperatures when Houston's method is used to account for the anisotropy of the Brillouin zone in the calculation of the phonon spectrum. It also provides an approach to predict from a spatial-dependent Boltzmann equation the rate at which phonons are scattered by the sample boundary in the presence of intrinsic scattering mechanisms, which is crucial for the calculation of the lattice thermal conductivity in finite crystals.

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Figures

Grahic Jump Location
Fig. 1

Longitudinal and transverse temperature-dependent Debye temperature in the six considered symmetry directions

Grahic Jump Location
Fig. 2

Phonon spectrum of Si. Symbols: first principles calculations. Solid lines: Debye spectrum calculated with the different expressions for the phonon spectrum in Eq. (9). Inset: the part of the density of states where Houston's method gives a satisfactory agreement with the density functional theory calculations. Because the various expressions for the density of states derived with Houston's method are to a large extent equivalent to each other, the difference between the plots of these expressions is not very clear even with a zoomed plot.

Grahic Jump Location
Fig. 3

Thermal conductivities of [001]-oriented Si and [113]-oriented Ge. Symbols: experimental data (data for Si are obtained from Ref. [31] and data for Ge are obtained from Refs. [11] and [32]). Solid lines: model results obtained by considering the different expressions for the phonon spectrum in Eq. (9).

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