Superdiffusive heat conduction in semiconductor alloys. I. Theoretical foundations
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Semiconductor alloys exhibit a strong dependence of effective thermal conductivity on measurement frequency. So far this quasiballistic behavior has only been interpreted phenomenologically, providing limited insight into the underlying thermal transport dynamics. Here, we show that quasiballistic heat conduction in semiconductor alloys is governed by L\'evy superdiffusion. By solving the Boltzmann transport equation (BTE) with ab initio phonon dispersions and scattering rates, we reveal a transport regime with fractal space dimension $1<\ensuremath{\alpha}<2$ and superlinear time evolution of mean-square energy displacement ${\ensuremath{\sigma}}^{2}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}(1<\ensuremath{\beta}<2)$. The characteristic exponents are directly interconnected with the order $n$ of the dominant phonon scattering mechanism $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}n}(n>3)$ and cumulative conductivity spectra ${\ensuremath{\kappa}}_{\ensuremath{\Sigma}}(\ensuremath{\tau};\ensuremath{\Lambda})\ensuremath{\sim}{(\ensuremath{\tau};\ensuremath{\Lambda})}^{\ensuremath{\gamma}}$ resolved for relaxation times or mean free paths through the simple relations $\ensuremath{\alpha}=3\ensuremath{-}\ensuremath{\beta}=1+3/n=2\ensuremath{-}\ensuremath{\gamma}$. The quasiballistic transport inside alloys is no longer governed by Brownian motion, but instead is dominated by L\'evy dynamics. This has important implications for the interpretation of thermoreflectance (TR) measurements with modified Fourier theory. Experimental $\ensuremath{\alpha}$ values for InGaAs and SiGe, determined through TR analysis with a novel L\'evy heat formalism, match ab initio BTE predictions within a few percent. Our findings lead to a deeper and more accurate quantitative understanding of the physics of nanoscale heat-flow experiments.
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The majority of experimental work in the literature deduced thermal properties based on Fourier's law of heat conduction with an effective thermal conductivity.
“Similarly, while TDTR can be used to study non-equilibrium phenomena such as quasi-ballistic phonon transport 90, 91, 109-111 and electronphonon coupling 55 due to its ability to detect temperature evolutions at micrometer-scale and picosecond-scale resolutions, the majority of the experimental work in the literature deduced thermal properties based on Fourier's”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [110]
The Implications for transient laser thermoreflectometry section (p. 7) says many observations of quasi-ballistic heat flow are interpreted with modified Fourier theory using a reduced effective thermal conductivity. The Introduction (p. 1) calls such models conventional for experimental analysis, but the paper does not establish that a majority of experimental work in the literature uses them. (LLM, conf=0.82)
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Heat transport in semiconductor crystals under large temperature gradients
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Boundary conditions dictate frequency dependence of thermal conductivity in silicon
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Lévy walk of quasiballistic phonons in nanowires
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Circuit modeling of the electro-thermal behavior of nanoscale bulk-FinFETs
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Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation
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Green's function solution for dual-phase-lag heat conduction model in electronic nanostructures
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Determining Parameters of the Dual-Phase-Lag Model of Heat Flow
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Relativistic stable processes in quasiballistic heat conduction in semiconductors
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Self-citationThermal Conductivity of Nanostructured Semiconductor Alloys
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Phonon Spectrum and Transient Regimes in the KCM
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Nonlocal thermal transport modeling using the thermal distributor
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Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics
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High-order perturbation expansion of hydrodynamic phonon theory
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- Authors
- Vermeersch, B., Carrete, J., Mingo, N., & Shakouri, A.
- Affiliations
- Purdue University West Lafayette (US)
- Commissariat à l'Énergie Atomique et aux Énergies Alternatives (FR)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2015-02-10
- Volume
- 91
- Issue
- 8
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W1997447750
- Cited by
- 96
- References
- 37
- Citations by year
- 2026: 4 · 2025: 4 · 2024: 5 · 2023: 4 · 2022: 10 · 2021: 9 · 2020: 7 · 2019: 5 · 2018: 19 · 2017: 7 · 2016: 9 · 2015: 13
- FWCI
- 6.49
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (DE-SC0001009)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
- Physical Sciences › Materials Science › Materials Chemistry › Advanced Thermoelectric Materials and Devices
- Keywords
- semiconductor alloys, Boltzmann equation, phonon scattering mechanisms, thermoreflectance, modified Fourier law, InGaAs, SiGe
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