Electron and phonon temperatures: Application to the thermal-shock propagation in nanowires

Abstract

Depending on the material properties of the system at hand, in some circumstances the application of a short pulse laser may initially bring a system into a highly non-equilibrium state wherein either the electron temperature may rise up while the lattice still remains cold, or conversely, because the heat capacity of electrons is usually different from that of lattice. In order to correctly describe the heat transport in those situations, therefore, one should properly split the non-equilibrium temperature into two different contributions: the electron temperature and the phonon temperature. By means of a recent two-temperature model (which fully agrees with the Maxwell-Cattaneo theory), in this paper we show that the values of those two temperatures may be always predictable during the thermal-shock propagation in nanowires, after having illustrated both that the employed theoretical model agrees with the basic tenets of continuum mechanics, and that it is mathematically well posed in the case of particular initial and boundary conditions. The main results of this paper could be used to deepen the concept of non-equilibrium temperature.

Keywords:

• Electron temperature
• nanoscale heat transfer
• non-equilibrium temperature
• phonon temperature
• thermal-shock propagation
• two-temperature model

PACS numbers:

• 02.30.Jr,02.90.+p,05-70.Ln,44.04.+e

Acknowledgments

This work has been carried out under the auspices of the GNFM (Italian National Group of Mathematical Physics, INdAM). The authors warmly acknowledge the anonymous Reviewers of this paper for all the suggestions, comments and observations received. The following research projects are acknowledged:

• Transport Phenomena in Low Dimensional Structures: Models, Simulations and Theoretical Aspects (NextGenerationEU - PRIN2022)

• The Mathematics and Mechanics of Nonlinear Wave Propagation in Solids (NextGenerationEU - PRIN2022)

• Non Linear Models for Magma Transport and Volcanoes Generation (National Recovery and Resilience Plan/NRRP - NextGenerationEU)

• SUSTBUILD–SUSTainable composite structures for energy-harvesting and carbon-storing BUILDings (National Recovery and Resilience Plan/NRRP - NextGenerationEU)

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Phonons are quanta of the vibrational mechanical energy arising from oscillating atoms within a lattice.

2 We remark, valid under the general ID in Eqs. (29)(29a) $$\begin{array}{cccc}{\mathsf{T}}^e\left(\mathbf{x},0\right)={\mathsf{T}}_0\left(\mathbf{x}\right)& & & \forall \mathbf{x}\in \Omega \end{array}$$(29a) and the mixed BCs in Eqs. (30)(30a) $$\begin{array}{cccccc}{\mathsf{T}}^e\left(\mathbf{x},\mathsf{t}\right)={\mathcal{T}}_1\left(\mathbf{x},\mathsf{t}\right)& & & & & \forall \mathbf{x}\in {\overline{\partial \Omega }}_1,\forall \mathsf{t}\in {\mathbb{R}}^{+}\end{array}$$(30a) .

3 According with Eqs. (12)(12a) $$\underset{\alpha \to 0^{+}}{\mathrm{\text{lim}}} \frac{G}{\alpha }=0$$(12a) , we note that from Eq. (59c)(59c) $$\xi =\frac{\mathsf{G}}{\alpha \left(1-\alpha \right)}$$(59c) in the present model one always has ξ= 1, as well as ξ = ω_e and ξ = ωp.