Nonequilibrium phonon transport induced by finite sizes: Effect of phonon-phonon coupling

In heat conduction through a homogenous nanomaterial, various phonons may exhibit diverse temperatures even at the same location at a steady state, known as the local phonon nonequilibrium phenomenon. Different phonons are often considered to behave independently, and the phonons with longer mean free paths (MFPs) have smaller temperature gradients. That is, the temperature gradient exhibits the following order: ballistic phonons < semiballistic phonons ≈ lattice (average) temperature gradient < diffusive phonons, where ballistic phonons have MFPs much larger than the characteristic length, semiballistic phonons have MFPs like the characteristic length, and diffusive phonons have MFPs much smaller than the characteristic length. However, in this paper, we reveal that the effect of phonon-phonon coupling leads nonequilibrium phonon temperature gradients to the following trend: diffusive phonon temperature gradients will decrease to the lattice temperature, and temperature gradients of some semiballistic phonons even surpass that of diffusive phonons. The diffusive phonon temperature is merged onto the lattice temperature since they have large scattering rates and can be equilibrated quickly to the lattice temperature after traveling for a short distance away from the boundaries into the nanomaterial. The semiballistic phonons have large scattering rates but not large enough to bring them down to the lattice temperature. To obtain a further understanding of the nonequilibrium phonon temperatures, we have also derived a simple analytical model which can accurately predict the temperature profiles of all individual phonons given their MFPs. Using this model, we find that, near the boundary, phonon temperatures decay with position exponentially (instead of linearly), with a rate inversely proportional to their MFPs. Our findings offer insight for the understanding and prediction of phonon nonequilibrium temperatures within nanodevices.

Figure 1

The dimensionless phonon temperature $T^{*}=(T_{\lambda }-T_2)/(T_1-T_2)$ for silicon thin film ($L=20\mathrm{nm}$) vs dimensionless position $x/L$ for the cases (a) without and (b) with the phonon-phonon coupling effect.

Figure 2

Selected nonequilibrium phonon temperatures for the validation of our analytical model on silicon thin film with a thickness (a) $L=10\mathrm{nm}$, (b) $L=100\mathrm{nm}$, and (c) $L=500\mathrm{nm}$. The gray shaded regions represent the temperatures of other phonon modes.

Figure 3

(a) The dimensionless phonon temperature jumps at the boundary $\mathrm{\Delta }{T}_{\lambda }/(T_1-T_2)$ vs the mode-level Knudsen number $K{n}_{\lambda ,x}={\mathrm{\Lambda }}_{\lambda ,x}/L$ for film thicknesses $L=10$, 100, and 500 nm. The inset of (a) shows that the $\mathrm{\Delta }{T}_{\lambda }$ is extracted by linear fitting of the central part of the temperature profile. (b) The correspondence coefficient β between the nonequilibrium phonon temperature and phonon mean free path (MFP) spectra vs the Knudsen number $K{n}_{\lambda ,x}$.

Figure 4

Comparison of the correspondence coefficient β between our analytical model and the finite volume method (FVM) Boltzmann transport equation (BTE) results for (a) $L=10\mathrm{nm}$, (b) $L=100\mathrm{nm}$, and (c) $L=500\mathrm{nm}$. The dashed lines represent $\beta =2$, i.e., without the phonon-phonon coupling effect.