Two-channel model for nonequilibrium thermal transport in pump-probe experiments
Abstract
We present an analytic solution for heat flow in a multilayer two-channel system for the interpretation of time-domain thermoreflectance (TDTR) experiments where nonequilibrium effects are important. The two-channel solution is used to analyze new room temperature TDTR measurements of Al/Cu and Al/Si${}_{0.99}$Ge${}_{0.01}$ systems. Cu and Si${}_{0.99}$Ge${}_{0.01}$ are examples of materials well suited for analysis with a two-channel model because thermal excitations responsible for the vast majority of the heat capacity in these solids contribute little to their thermal conductivity. Nonequilibrium effects are found to be unimportant for the interpretation of the Al/Cu TDTR data but dramatic for the Al/Si${}_{0.99}$Ge${}_{0.01}$ TDTR data. The two-channel model predicts a significant reduction in the effective thermal conductivity of Si${}_{0.99}$Ge${}_{0.01}$ in a region within 150 nm of the Al/Si${}_{0.99}$Ge${}_{0.01}$ interface. The extra thermal resistance in this region is a result of the disparate heat flux boundary conditions for low- and high-frequency phonons in combination with weak coupling between low- and high-frequency phonons. When the experimental data are analyzed with a single-channel model, both the conductance and thermal conductivity appear to depend on pump-modulation frequency, consistent with the two-channel model's predictions. Finally, we compare the results of our diffusive two-channel model to a nonlocal description for steady-state heat flow near a boundary and show they yield nearly identical results.
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Both the two-channel model and the ballistic/diffusive model reveal the critical influence of the interface on frequency dependent thermal transport.
“Importantly, both theories reveal the critical influence of the interface on frequency dependent thermal transport, thus underscoring the significance of the transducer/sample boundary condition.”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [17]
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“electron temperature will aid the interpretation of pump/probe measurements of nonequilibrium phenomena [17,54,65,68,70-72].”
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Heat transport in semiconductor crystals: Beyond the local-linear approximation
Ezzahri, Y., Joulain, K., & Ordoñez-Miranda, J. (2020) · Journal of Applied Physics · doi:10.1063/5.0017188 · Cited by 9 · see citing works
Compact stochastic models for multidimensional quasiballistic thermal transport
Vermeersch, B. (2016) · Journal of Applied Physics · doi:10.1063/1.4965866 · Cited by 9 · see citing works
Saurav, S., & Mazumder, S. (2024) · International Journal of Heat and Mass Transfer · doi:10.1016/j.ijheatmasstransfer.2024.125698 · Cited by 8 · see citing works
On the solenoidal heat-flux in quasi-ballistic thermal conduction
Ramu, A. T., & Bowers, J. E. (2015) · Journal of Applied Physics · doi:10.1063/1.4931610 · Cited by 8 · see citing works
“With mounting evidence of significant room temperature ballistic phonon transport in silicon and other materials [1-3], several theories of non-Fourier heat transfer that account for this effect have arisen [4-8].”
BackgroundMazhukin, V. I., Koroleva, O. N., Shapranov, A. V., Дeмин, М. М., & Алексашкина, А. А. (2022) · Mathematical Models and Computer Simulations · doi:10.1134/s2070048222040068 · Cited by 8 · see citing works
Hotspot Size-Dependent Thermal Boundary Conductance in Nondiffusive Heat Conduction
Ma, Y. (2015) · ASME Journal of Heat and Mass Transfer · doi:10.1115/1.4030170 · Cited by 8 · see citing works
Interfacial thermal transport in spin caloritronic material systems
Angeles, F., Sun, Q., Ortiz, V. H., Shi, J., Li, C. W., & Wilson, R. B. (2021) · Physical Review Materials · doi:10.1103/physrevmaterials.5.114403 · Cited by 8 · see citing works
Self-citationA Generalized Enhanced Fourier Law
Ramu, A. T., & Bowers, J. E. (2016) · ASME Journal of Heat and Mass Transfer · doi:10.1115/1.4034796 · Cited by 8 · see citing works
Thermal conductivity of low-dimensional materials: Recent progress, prospects, and challenges
Sheraz, A., Korotchenkov, O. A., Nasiri, M. A., de la Torre, M. A. L., & Cantarero, A. (2025) · Applied Physics Reviews · doi:10.1063/5.0274620 · Cited by 7 · see citing works
Unraveling intrinsic relationship of thermal properties in thermoreflectance experiments
Chen, T., & Jiang, P. (2024) · Acta Physica Sinica · doi:10.7498/aps.73.20241369 · Cited by 7 · see citing works
Yao, J., Maliyov, I., Gardner, D. J., Woodward, C. S., & Bernardi, M. (2025) · npj Computational Materials · doi:10.1038/s41524-025-01738-8 · Cited by 6 · see citing works
“In the absence of fi rst-principles simulations, the two-temperature model 62 is commonly used to describe the energy transfer from excited carriers to the lattice during the relaxation.”
Zhao, B., Qu, Y., Zhao, W., & Deng, Y. (2024) · Materials Today Physics · doi:10.1016/j.mtphys.2024.101379 · Cited by 6 · see citing works
Segovia, M., & Xu, X. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.108.125202 · Cited by 5 · see citing works
Boundary conditions dictate frequency dependence of thermal conductivity in silicon
Liu, Y., Li, Q., Liu, F., Wang, X., & Sun, B. (2025) · Applied Physics Reviews · doi:10.1063/5.0254248 · Cited by 4 · see citing works
“…Wilson et al. interpreted the frequency dependent thermal transport using a two-channel model based on the phonon non-equilibrium, where weak interactions between low-and high-frequency phonons constitutes additional non-equilibrium resistance [17], dictated by the transducer/sample interface.”
InfluentialThermal model for time-domain thermoreflectance experiments in a laser-flash geometry
Peng, W., & Wilson, R. B. (2022) · Journal of Applied Physics · doi:10.1063/5.0082549 · Cited by 4 · see citing works
Self-citationFreedman, J. P., Davis, R. F., & Malen, J. A. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.054308 · Cited by 4 · see citing works
Angeles, F., Shi, X., & Wilson, R. B. (2022) · Journal of Applied Physics · doi:10.1063/5.0084573 · Cited by 4 · see citing works
Self-citationHow long, and why, do Photoexcited Electrons in a Metal Remain Hot?Preprint
Wilson, R. B., & Coh, S. (2019) · arXiv (Cornell University) · Cited by 4 · see citing works
Self-citationMachine learning-driven data analysis for three-layer time-domain thermoreflectance measurements
Wang, Z., Linghu, B., Li, D., Zhu, J., & Tang, D. (2025) · International Communications in Heat and Mass Transfer · doi:10.1016/j.icheatmasstransfer.2025.110398 · Cited by 3 · see citing works
Raman spectroscopy for thermal transport characterization: Principles, techniques, and applications
Kazan, M. (2026) · Journal of Applied Physics · doi:10.1063/5.0306485 · Cited by 3 · see citing works
Jensen, W. A., Liu, N., Donovan, B., Tomko, J. A., Hopkins, P. E., & Floro, J. A. (2017) · Materials Chemistry and Physics · doi:10.1016/j.matchemphys.2017.12.038 · Cited by 3 · see citing works
“In metallic systems, thermal transport is dominated by electrons, and thus, electron effects (such as electron-phonon coupling in interfacial regions) may not be easily dismissed [52] [53] [54] [55].”
BackgroundLi, H., Liu, X., Zhang, C., & Zheng, W. (2024) · International Journal of Thermal Sciences · doi:10.1016/j.ijthermalsci.2024.109307 · Cited by 3 · see citing works
Qu, Z., Wang, D., & Ma, Y. (2017) · AIP Advances · doi:10.1063/1.4973331 · Cited by 3 · see citing works
Multi-scale Thermal Conductivity Measurements for Cryobiological Applications
Natesan, H., Choi, J., Lubner, S., Dames, C., & Bischof, J. C. (2016) · Frontiers in nanobiomedical research · doi:10.1142/9789814733199_0005 · Cited by 3 · see citing works
Boundary conditions dictate frequency dependence of thermal conductivity in silicon
Liu, Y., Li, Q., Liu, F., Wang, X., & Sun, B. (2024) · arXiv · doi:10.48550/arxiv.2412.02227 · Cited by 2 · see citing works
“…Wilson et al. interpreted the frequency dependent thermal transport using a two-channel model based on the phonon non-equilibrium, where weak interactions between low-and high-frequency phonons constitutes additional non-equilibrium resistance [17], dictated by the transducer/sample interface.”
InfluentialMazhukin, V. I., Koroleva, O. N., Shapranov, A. V., Дeмин, М. М., & Алексашкина, А. А. (2021) · Matematicheskoe modelirovanie · doi:10.20948/mm-2022-01-05 · Cited by 2 · see citing works
Ramu, A. T., & Bowers, J. E. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1506.00668 · Cited by 2 · see citing works
“INTRODUCTION Reports of significant room-temperature quasi-ballistic phonon transport [1-3] have spurred modeling efforts [4-11] aimed at explaining observed experimental results and predicting new effects.”
BackgroundNanoscale Thermal Transport at Graphene-Soft Material Interfaces
Liu, Y. (2016) · VTechWorks (Virginia Tech) · Cited by 2 · see citing works
Dispersion relation and frequency-dependent thermal conductivity of the two-temperature systems
Sobolev, S. L., & Kudinov, I. V. (2025) · International Journal of Thermal Sciences · doi:10.1016/j.ijthermalsci.2025.109937 · Cited by 1 · see citing works
- Authors
- Wilson, R. B., Feser, J. P., Hohensee, G. T., & Cahill, D. G.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2013-10-15
- Volume
- 88
- Issue
- 14
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W2321018845
- Cited by
- 112
- References
- 32
- Citations by year
- 2026: 5 · 2025: 9 · 2024: 7 · 2023: 5 · 2022: 6 · 2021: 8 · 2020: 7 · 2019: 7 · 2018: 9 · 2017: 8 · 2016: 14 · 2015: 17 · 2014: 10
- FWCI
- 6.16
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review B (publishedVersion)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Engineering › Mechanics of Materials › Thermography and Photoacoustic Techniques
- Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
- Keywords
- two-channel model, time-domain thermoreflectance, thermal conductivity, phonon boundary scattering, low-frequency phonons, high-frequency phonons, phonon-phonon interaction, thermal resistance, Cu/Al interface, multilayer systems, nonlocal heat transport
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