A new regime of nanoscale thermal transport: Collective diffusion increases dissipation efficiency

Abstract

Understanding thermal transport from nanoscale heat sources is important for a fundamental description of energy flow in materials, as well as for many technological applications including thermal management in nanoelectronics and optoelectronics, thermoelectric devices, nanoenhanced photovoltaics, and nanoparticle-mediated thermal therapies. Thermal transport at the nanoscale is fundamentally different from that at the macroscale and is determined by the distribution of carrier mean free paths and energy dispersion in a material, the length scales of the heat sources, and the distance over which heat is transported. Past work has shown that Fourier's law for heat conduction dramatically overpredicts the rate of heat dissipation from heat sources with dimensions smaller than the mean free path of the dominant heat-carrying phonons. In this work, we uncover a new regime of nanoscale thermal transport that dominates when the separation between nanoscale heat sources is small compared with the dominant phonon mean free paths. Surprisingly, the interaction of phonons originating from neighboring heat sources enables more efficient diffusive-like heat dissipation, even from nanoscale heat sources much smaller than the dominant phonon mean free paths. This finding suggests that thermal management in nanoscale systems including integrated circuits might not be as challenging as previously projected. Finally, we demonstrate a unique capability to extract differential conductivity as a function of phonon mean free path in materials, allowing the first (to our knowledge) experimental validation of predictions from the recently developed first-principles calculations.

Keywords: high harmonic generation; mean free path spectroscopy; nanoscale thermal transport; nondiffusive transport; ultrafast X-rays.

Fig. 1.

Nanoscale heat transport is determined by the interplay between three length scales: the size of the heat sources L, the spacing of the heat sources P, and the MFPs Λ_i of heat-carrying phonons. Materials support a broad distribution of MFPs, represented here by short (black)- and long (white)-MFP phonons. (A) When all MFPs are smaller than L, heat dissipation is fully diffusive. (B) As L shrinks, long-MFP phonons travel ballistically, decreasing the rate of heat dissipation relative to diffusive predictions. Short-MFP phonons remain diffusive. (C) When both L and P shrink, long-MFP phonons originating from neighboring heat sources interact as they would if they originated from a single, large heat source, enabling more efficient diffusive-like heat transfer.

Fig. 2.

Effective thermal boundary resistivities are extracted from dynamic EUV diffraction. (A) Dynamic diffraction from 60-nm-wide nickel lines on sapphire (Top) and silicon (Bottom) display a sudden rise due to impulsive thermal expansion following laser heating, a long decay due to thermal relaxation, and oscillations due to surface acoustic waves. Dashed black lines plot the diffusive prediction, which significantly underestimates the thermal decay time. Green lines plot the decay using a best fit to the effective thermal boundary resistivity. (B) Extracted effective resistivities for each line width L on both substrates increase with decreasing line width until the periods (equal to 4L) are comparable to the average phonon MFP. For smaller periods (spacing), the effective resistivity decreases and approaches the diffusive limit (black dashed line). The error bars represent the SD among multiple datasets for the same line width samples. Dotted red lines: predictions for isolated heat sources based on the gray model. Dash-dot blue lines: gray model including the onset of the collectively diffusive regime. Solid purple lines: more complete model that includes contributions from multiple phonon MFPs.

Fig. 3.

Line width and period define a suppression filter for phonon MFP spectra. (A) The observed increase in effective thermal boundary resistivity for small line widths L is due to the suppression of the contribution to thermal conductivity of phonon modes with MFP larger than L. Decreasing the period P can reactivate modes with MFP larger than P, decreasing the effective resistivity. In the limiting case of a uniformly heated layer, P approaches L and all phonon modes participate in thermal transport. We use as an example the smoothed differential conductivity distribution for silicon (top graphs, green line), calculated from first-principles DFT (SI Text, section S4). (B) A comparison of the thermal decay of small line width gratings for two different periods directly validates the prediction of the suppression filter model, i.e., small line widths spaced far apart (red lines) exhibit a slower initial thermal decay than small line widths spaced closer together (blue lines). The dashed lines provide a guide to the eye for the thermal decay through the center of the acoustic oscillations.

Fig. 4.

By fitting r_eff with multiple bins of phonon modes, the weights k(Λ_i) assigned to those bins give the average relative contribution to the differential thermal conductivity (purple shading). Both differential (distributions) and cumulative (lines) conductivities are normalized to the total bulk conductivity. For sapphire (Top), our data (solid purple line) and first-principles DFT calculations (dashed green line) indicate there are no significant contributions from long-MFP phonons, so the cumulative curves approach unity at 1 μm. For silicon (Bottom), our data are consistent with large contributions from longer MFPs.