# Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films

## Abstract

Cross-plane heat transport in thin films with thicknesses comparable to the phonon mean free paths is of both fundamental and practical interest for applications such as light-emitting diodes and quantum well lasers. However, physical insight is difficult to obtain for the cross-plane geometry due to the challenge of solving the Boltzmann equation in a finite domain. Here, we present a semi-analytical series expansion method to solve the transient, frequency-dependent Boltzmann transport equation that is valid from the diffusive to ballistic transport regimes and rigorously includes the frequency-dependence of phonon properties. Further, our method is more than three orders of magnitude faster than prior numerical methods and provides a simple analytical expression for the thermal conductivity as a function of film thickness. Our result enables a straightforward physical understanding of cross-plane heat conduction in thin films.

- Authors:

- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125 (United States)

- Publication Date:

- OSTI Identifier:
- 22403011

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Applied Physics

- Additional Journal Information:
- Journal Volume: 117; Journal Issue: 17; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-8979

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 77 NANOSCIENCE AND NANOTECHNOLOGY; ANALYTICAL SOLUTION; BOLTZMANN EQUATION; COMPARATIVE EVALUATIONS; FREQUENCY DEPENDENCE; LASERS; LIGHT EMITTING DIODES; MEAN FREE PATH; PHONONS; QUANTUM WELLS; THERMAL CONDUCTION; THERMAL CONDUCTIVITY; THICKNESS; THIN FILMS; TRANSIENTS

### Citation Formats

```
Hua, Chengyun, and Minnich, Austin J., E-mail: aminnich@caltech.edu.
```*Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films*. United States: N. p., 2015.
Web. doi:10.1063/1.4919432.

```
Hua, Chengyun, & Minnich, Austin J., E-mail: aminnich@caltech.edu.
```*Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films*. United States. doi:10.1063/1.4919432.

```
Hua, Chengyun, and Minnich, Austin J., E-mail: aminnich@caltech.edu. Thu .
"Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films". United States. doi:10.1063/1.4919432.
```

```
@article{osti_22403011,
```

title = {Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films},

author = {Hua, Chengyun and Minnich, Austin J., E-mail: aminnich@caltech.edu},

abstractNote = {Cross-plane heat transport in thin films with thicknesses comparable to the phonon mean free paths is of both fundamental and practical interest for applications such as light-emitting diodes and quantum well lasers. However, physical insight is difficult to obtain for the cross-plane geometry due to the challenge of solving the Boltzmann equation in a finite domain. Here, we present a semi-analytical series expansion method to solve the transient, frequency-dependent Boltzmann transport equation that is valid from the diffusive to ballistic transport regimes and rigorously includes the frequency-dependence of phonon properties. Further, our method is more than three orders of magnitude faster than prior numerical methods and provides a simple analytical expression for the thermal conductivity as a function of film thickness. Our result enables a straightforward physical understanding of cross-plane heat conduction in thin films.},

doi = {10.1063/1.4919432},

journal = {Journal of Applied Physics},

issn = {0021-8979},

number = 17,

volume = 117,

place = {United States},

year = {2015},

month = {5}

}