Figure 8: Ballistic versus diffusive predictions for the heat current. | Nature Communications

Figure 8: Ballistic versus diffusive predictions for the heat current.

From: Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments

Figure 8

(a) Amplitude of temperature oscillations predicted by the heat diffusion equation for 80 nm Al/Si sample undergoing sinusoidal surface heating at f=10 MHz and w0=1 μm. (bd) Qualitative comparison between the first-order Taylor series approximation of the temperature profile used in Fourier’s law (red lines), −2T, and the temperature profile actually traversed by 2 μm MFP phonons (black lines) with the trajectories illustrated by the arrows in a. The first-order Taylor series that Fourier’s law uses (equation (8)) is a poor approximation for the temperature profile that is traversed by phonons that are (b) travelling in the in-plane direction across r=0 and (c) travelling in the through-plane direction after reflecting from the interface at z=0. (e) Average magnitude of Jr and Jz predicted by Fourier’s law for the temperature profile in a. The colour axis is normalized to the maximum value predicted by Fourier’s law, which occurs in the z direction at r=z=0 and in the r direction at z=0 and rw0. TDTR measurements are most sensitive to transport properties in the regions where the heat current is largest. (f,g) Ratio of the average magnitude of Jz and Jr predicted by equation (9), to the predictions of Fourier’s law for 1 μm MFP phonons. The heat current profiles shown in f assume all 1 μm MFP phonons reflect at the interface, R=1 (adiabatic limit). The heat current profiles shown in g assume 1 μm MFP phonons transmit at the interface, R=0 (radiation limit). (h) Magnitude of the heat current predicted by equation (9) relative to Fourier’s law at two representative points. The red curves are for Jr at r=1 μm and z=150 nm. The blue curves are for the Jz at r=0 and z=500 nm. These two points are selected because they are representative of how Fourier’s law over-predicts the heat current in the regions the measurement is most sensitive to the effective thermal properties. Solid lines were calculated assuming R=1 (adiabatic limit), while the dashed lines were calculated assuming R=0 (radiation limit).

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