Thermal transport in phononic Cayley-tree networks

We analytically investigate the heat current $\mathcal{I}$ and its thermal fluctuations $\Delta$ in a branching network without loops (Cayley tree). The network consists of two types of harmonic masses: vertex masses $M$ placed at the branching points where phononic scattering occurs and masses $m$ at the bonds between branching points where phonon propagation takes place. The network is coupled to thermal reservoirs consisting of one-dimensional harmonic chains of coupled masses $m$. Due to impedance mismatch phenomena, both $\mathcal{I}$ and $\Delta$ are non-monotonic functions of the mass ratio $\mu =M/m$. Furthermore, we find that in the low-temperature limit the thermal conductance approaches zero faster than linearly due to the small transmittance of the long-wavelength modes.

Figure 1

(a) The schematic figure is a Cayley tree of generation $N=2$ with $N_b=2$ ‘small’ masses $m$ on each branch. The basic unit, is indicated with a dashed square and it consists of a branch and a vertex attached to the right end of the branch. A monochromatic incident wave $e^{ikz-i\omega t}$ is reflected back as $r^{\left(N\right)}{e}^{-ikz-i\omega t}$, where $r^{\left(N\right)}$ is the total reflection amplitude. For the study of thermal transport, the Cayley tree is connected to two heat baths at different temperatures, $T_H$ and $T_L$, respectively. (b) A scattering process occurring at a basic unit associated with the $l\mathrm{th}$ vertex. The basic unit is connected to three semi-infinite leads (blue boxes). Three incoming waves from the leads are transformed to three outgoing waves at each of the leads attached to the basic unit.

Figure 2

Transmittance ${\mathcal{T}}^{\left(N\right)}=1-{\left|r^{\left(N\right)}\right|}^2$ vs. wave number $k$ for various $N$-values obtained using Eq. (8). For comparison, we also plot the mean value approximation $\overline{\mathcal{T}}$ and the band-gap parameter $y$ (scaled by 160 to fit the figure). Here $m=1,\kappa =1,\mu =1.5,{\kappa }_0=0$, and $N_b=3$.

Figure 3

(a) The steady-state thermal current $\mathcal{I}\left[\hslash \frac{\kappa }{m}\right]$ vs. the mass ratio $\mu$ for $N_b=0$ (black line) and $N_b=3$ (red line). We have used Eq. (1) where the total transmittance has been calculated numerically using Eq. (8) for $N=50$. Black circles and red diamonds indicate the theoretical results for $N_b=0$ and $N_b=3$ when we use the mean value approximation for the transmittance. Inset: The exponential convergence of $\mathcal{I}$ with generation $N$ to its asymptotic value ${\mathcal{I}}_{\infty }$. Two typical cases with $\mu =0.05$ for the $N_b=0$ (black line) and $N_b=3$ (red line). (b) The same as in (a) but now for the thermal fluctuations $\Delta \left[{\hslash }^2{\frac{\kappa }{m}}^{3/2}\right]$. The parameters used are $T_H=0.41\frac{\hslash }{k_B}\sqrt{\frac{\kappa }{m}},T_R=0.39\frac{\hslash }{k_B}\sqrt{\frac{\kappa }{m}},{\kappa }_0=0$.

Figure 4

Numerical values of conductance ${\sigma }^{\infty }\left[k_B\sqrt{\frac{\kappa }{m}}\right]$ vs. temperature $T_{\mathrm{mean}}=(T_H+T_L)/2\left[\frac{\hslash }{k_B}\sqrt{\frac{\kappa }{m}}\right]$ for $N_b=0$ (black solid line) and $N_b=3$ (red dashed line). The horizontal lines (of the same type and color) are the classical results of Eq. (13). Inset: Numerical values (black circles) and classical values Eq. (13) (solid black line) of conductance versus $N_b$. The numerical results corresponds to $N=50$ and we have used Eq. (1) where the total transmittance has been calculated using Eq. (8). Other parameters are $\mu =1.5,{\kappa }_0=0$. For the inset we have used $T_{\mathrm{mean}}=3\left[\frac{\hslash }{k_B}\sqrt{\frac{\kappa }{m}}\right]$.