Lévy walk of quasiballistic phonons in nanowires

Phonon transport in square-cross-section nanowires is studied using spectral Monte Carlo simulations. Our results show the evolution of the different transport regimes described by Lévy statistics as a function of the surface roughness–to–thermal wavelength ratio $\sigma /\lambda$. More precisely, the relationship between the Lévy index $\gamma$ describing the mean free path distribution $\mathrm{\Psi }\left(\mathrm{\Lambda }\right)$ and $\sigma /\lambda$ is established for the classical diffusive regime, the superdiffusive regime, and the ballistic regime in the nanowire. Besides the conventional superdiffusive regime that is marked by $\mathrm{\Psi }\left(\mathrm{\Lambda }\right)$ with a single heavy-tailed peak, we reveal an unconventional superdiffusive subregime featuring $\mathrm{\Psi }\left(\mathrm{\Lambda }\right)$ with sawtooth oscillations when $\sigma /\lambda \sim 0.01$. Investigation of the direction of propagation of phonons shows a significant narrowing of the angular distribution around the long axis of the nanowire due to the diffuse scattering at rough boundaries when $\sigma /\lambda >0.1$. These results shed light on the transport mechanisms of quasiballistic phonons and will help in nanowire design for specific applications.

Figure 1

(a) Schematics of square-cross-section nanowire illustrating the top and bottom roughness and the side roughness. (b) Representation of Lambert's cosine law, which gives the probability of the scattered angle upon a diffuse scattering event on a rough surface.

Figure 2

(a) Specular scattering rates at the smooth top and bottom walls. Gray lines are obtained with Eq. (2) using a roughness of 0.5 nm and different incident angles as indicated. (b) Specular scattering rate at the rough sidewall boundaries. The dashed lines were also obtained with Eq. (2). (c) Total specular rate in the nanowire. The dotted lines represent the isotropic rate $S_{\text{tot}}^{\text{iso}}=(S_s+S_t)/2$. The inset shows $S_{\text{tot}}$ at $\tau =0.5$. (d) Transmission rate of phonons to the cold reservoir. The inset shows the linear relation between the wavelength at $\tau =0.5$ and the sidewall roughness.

Figure 3

Spectral evolution of the fraction of scattering events at the sidewall boundaries $N_s/N_{\text{tot}}$.

Figure 4

Distribution of in-plane ${\theta }_{\text{out}}$ and out-of-plane ${\varphi }_{\text{out}}$ exit angles at the cold end of the nanowire for different wavelengths: (a) and (b) $\lambda =1$ nm, (c) and (d) $\lambda =10$ nm, (e) and (f) $\lambda =100$ nm, and (g) and (h) $\lambda =1000$ nm.

Figure 5

ATC $\alpha \left(\mathrm{\Lambda }\right)$ and PDF $\mathrm{\Psi }\left(\mathrm{\Lambda }\right)$ in nanowires with roughnesses $\sigma =1$, 2, 3, and 4 nm at selected wavelengths. (a) and (b) $\lambda =1$ nm. The ATCs are identical for each roughness and form one sharp peak in the PDF. Our ATC is in good agreement with DFT calculations on Si nanowire with roughness $\sigma =1$ nm from Ref. [36]. (c) and (d) $\lambda =10$ nm. The ATCs at different roughnesses separate for intermediate MFPs. In the PDF, mainly one peak appears with a heavy tail. (e) and (f) $\lambda =100$ nm. The ATCs at different roughnesses separate further and sawtooth oscillations appear in the PDF. (g) and (h) $\lambda =1000$ nm. The ATCs collapse again. One single broad peak in the PDF appears when the MFP reaches the length of the nanowire.

Figure 6

Lévy index $\gamma$ obtained by fitting the first peak in the MFP distribution: $\gamma >2$, diffusive regime; $1<\gamma <2$, superdiffusive regime; and $\gamma <1$, ballistic regime. Two subregimes are distinguished within the superdiffusive regime: the conventional superdiffusive regime characterized by a single heavy-tailed peak and the unconventional superdiffusive regime characterized by sawtooth oscillations. The black line is a guide for the eye.

Figure 7

Lévy index $\gamma$ obtained for the first peak, second peak, and third peak of the MFP distribution at $\lambda =100$ nm. The inset is a close-up of Fig. 5 around the first three peaks.

Figure 8

Probability to obtain $n$ successive specular scattering events.