3 PRESTO, Japan Science and Technology Agency, Saitama 332â0012, Japan. ... bars show the standard deviation in the measurements on three identical ...
Supporting information Quasi-Ballistic Heat Conduction due to Lévy Phonon Flights in Silicon Nanowires Roman Anufriev1,*, Sergei Gluchko1,2, Sebastian Volz1,2, & Masahiro Nomura1,3 1
Institute of Industrial Science, the University of Tokyo, Tokyo 153–8505, Japan. Laboratory for Integrated Micro Mechatronic Systems / National Center for Scientific Research-Institute of Industrial Science (LIMMS/CNRS-IIS), the University of Tokyo, Tokyo 153–8505, Japan. 3 PRESTO, Japan Science and Technology Agency, Saitama 332–0012, Japan. 2
Supplementary Figure 1. Extraction of the thermal conductivity from the experimental data via FEM simulations. (a) The FEM model is built as a quarter of the actual structure with symmetry boundary conditions. The Gaussian-shaped heat flux is applied at the centre to simulate the pump laser. The temperature of the aluminium pad is monitored in time. The thermal conductivity of the NW (nw) is changed to sample several values. (b) The produced decay curves closely resemble the experimentally measured curve; one of them fits. Both experimental and simulated curves decay exponentially with τexp as the decay time. (c) Plotting the sampled thermal conductivities vs obtained decay times, we interpolate the (τ) dependence and thus can find a corresponding thermal conductivity for any experimentally measured decay time (τexp). For the sake of example, τexp corresponds exactly to the simulation for = 45 Wm-1K-1, but it could also be in-between the simulated points.
Supplementary Figure 2. TEM images of NW boundaries. (a) Image of a sample that has been thinned for TEM study. (b) The top view shows the crystallinity of silicon NWs and surface roughness of a few atomic layers. (c, d) The slightly inclined views of the NW boundaries show that the surface roughness remains low for tens of nanometers along both sides of the NW. (e) Close-up view of the panel (b) shows that the surface irregularity does not exceed 2 – 3 atomic layers and also shows the oxide layer formed on the surface. However, unlike the TEM samples, samples used for the TDTR measurements were exposed to air only for a short time and probably have a much thinner oxide layer.
Supplementary Figure 3. Length dependence of α coefficient in κ Lα dependence at 4 K, obtained as derivative of the experimental data.
Supplementary Figure 4. Length dependence of thermal conductivity on an additional set of samples. An additional set of samples was independently fabricated and measured at 4 and 300 K confirming the data obtained on the main set of samples from the main text. The error bars show the standard deviation in the measurements on three identical samples.
Supplementary Figure 5. FEM simulation of heat flux in the serpentine NW. The simulation shows that the heat flux follows the shortest path cutting corners at the turns.
Supplementary Figure 6. FEM simulations decay time in NW with direct and indirect detectors. The simulation shows that in diffusive regime samples with both detectors have identical decay curves.
Supplementary Figure 7. Schematic of the experiment probing directional phonon transport. In case of some directional transport, the passage through the direct detector should be faster than through the indirect detector. Otherwise, if probability to pass though either arm is equal, the decay times should be equal.
Supplementary Figure 8. Directionality of phonon transport on an additional set of samples. An additional set of samples was independently fabricated and measured. In the additional set of samples, the difference in the thermal decay time between the samples with “direct” and “indirect” detectors is rather small, similarly to the data in the main text. The error bars show the standard deviation in the measurements on three identical samples.
Supplementary Figure 9. Simulation of NWs of different shapes. Examples phonon trajectories in NWs of different shapes obtained using the Monte Carlo simulations
Supplementary Figure 10. Analysis of phonon directions in the corrugated NW. Fit with two Gaussian peaks shows that the peak of the directional phonons constitutes only 18.4% of all phonons.
Supplementary Note 1. The original paper by Heron et al. provides data points at many temperatures, but not exactly the same for straight and serpentine NWs. To obtain the relative values and the error bars, we digitised three points around a temperature each 0.5 K, averaged these three values and calculated the standard deviation. Thus, the error bars are probably overestimated because the values for each point are extracted at slightly different temperatures.
