Emergence of hydrodynamic heat transport in semiconductors at the nanoscale

Several recent experiments have demonstrated the failure of the Fourier heat equation in semiconductors at the nanoscale. We show that a generalized heat transport equation including a hydrodynamic term can explain three of these experiments on a silicon substrate: heater lines experiments both in stationary and nonstationary settings and the nonstationary response in grating experiments. To this end, we solve the hydrodynamic heat equation either numerically or analytically. The non-Fourier response observed in those experimental situations can be easily explained in terms of hydrodynamic concepts, such as friction and vorticity. For instance, the experimentally observed increase of thermal boundary resistance as size decreases can be simply understood as an increase of friction. We conclude that, contrary to common belief, hydrodynamics is a fundamental ingredient of semiconductor heat transport in the nanoscale at room temperature, providing physical insight and a unifying framework on the observed non-Fourier response. The relations between hydrodynamic heat transport with phonon momentum conservation and the Boltzmann transport equation are also provided.

Figure 1

(a) Calculated 3D temperature profile based on KCM (color code) and superposed heat flux vector (red arrows) with the temperature gradient (white arrows). According to the KCM model, the heat flux is not parallel to the temperature gradient due to the hydrodynamic nature of heat flowing around a corner. The dotted line represents the cross section on which the temperature profile of (b) and (c) are calculated. Details of the cross section are also provided. For more information of the experimental setup, see Refs. [14, 32]. (b) Temperature cross section comparing experimental thermoreflectance thermal imaging data (symbols) with Fourier and hydrodynamic (KCM) modeling results (solid lines) of a 400 nm nanoheater line on B-doped silicon substrate with a 20 nm interlayer of ${\mathrm{Al}}_2{\mathrm{O}}_3$. (c) Comparison of the KCM results for a 400 nm heater line on top of InGaAs and B-doped Si substrate, both with a 20 nm interlayer of ${\mathrm{Al}}_2{\mathrm{O}}_3$.

Figure 2

Effective thermal resistivity as function of the heater width $L$ with periodicity $4L$ on top of silicon. Black dots denote the effective resistivity needed by Fourier law to fit the experimental data (Ref. [1]). Red and blue dots correspond to the effective resistivity needed by Fourier law to reproduce the KCM decay curves for a single line and a periodic array of lines, respectively.

Figure 3

(a) Typical thermal grating temperature profile and decay curve. (b) Effective decay rate as a function of the square inverse grating length scale: experiments (symbols), Fourier predictions with the bulk heat conductivity (blue line), and hydrodynamic predictions, Eq. (4) (green line) with $\ell =190\mathrm{nm}$. The red line gives the hydrodynamic prediction for asymptotic long grating periods, i.e., thin films. Film width, $h=400\mathrm{nm}$.