Universality and Diversity in a Phonon-Transmission Histogram of Isotope-Disordered Carbon Nanotubes

Universal fluctuations in phonon transmission and other features of phonon-transmission histograms are investigated by performing numerical simulations of coherent-phonon transport in isotope-disordered carbon nanotubes. Interestingly, the phonon-transmission fluctuation in the diffusive regime is universal, irrespective of the average phonon transmission, the tube chirality, and the concentrations, and masses of isotopes. We also find that the histogram, which has a Gaussian distribution in the diffusive regime, has a log-normal distribution in the localization regime.

Figure 1

Schematic of an isotope-disordered SWNT. The central region with a length $L$ and containing isotopes is connected to semi-infinite pristine SWNT leads.

Figure 2

(a) The average phonon transmission $⟨\zeta (\omega )⟩$ of the (5, 5) SWNT with 15% ${}^{13}\mathrm{C}$ for $L=0.005\mu \mathrm{m}$ (circles), $0.05\mu \mathrm{m}$ (rectangles), $0.5\mu \mathrm{m}$ (triangles), and $5\mu \mathrm{m}$ (crosses). (b) The $L$-dependence of $M(\omega )/⟨\zeta (\omega )⟩-1$ for several frequencies, where $M(\omega )$ is the number of phonon modes of pristine SWNT. The dashed lines indicate linear fits for estimating the mean free path. (c) The $L$ dependence of $⟨\ln \zeta (\omega )⟩$ for several frequencies. The dashed lines represent linear fits to estimate the localization length. (d) The mean free path (circles) and the localization length (rectangles) as functions of $\omega$.

Figure 3

The root-mean-square phonon transmission for (a) a 625 nm-long (5,5) SWNT with 15% ${}^{13}\mathrm{C}$ and (b) a 210 nm-long (8,0) SWNT with 9.4% ${}^{14}\mathrm{C}$. The triangles, circles, rectangles represent numerical data for $\Delta \zeta (\omega )$ in the ballistic, diffusive, and localization regimes, respectively. The dashed lines are the universal phonon-transmission fluctuation.

Figure 4

Phonon-transmission histograms $P(\zeta )$ for several frequencies in the diffusive regime for (a) the (5,5) SWNT with 15% ${}^{13}\mathrm{C}$ and $L=625\mathrm{nm}$ and (b) the (8,0) SWNT with 9.4% ${}^{14}\mathrm{C}$ and $L=210\mathrm{nm}$. $P(\ln \zeta )$ for several frequencies in the localization regime for the same SWNTs as (c) Figs. 4a, 4b, 4d. The insets show $\mathrm{Var}[\ln \zeta ]$ as a function of $⟨\ln \zeta ⟩$.