Lattice thermal conductance in nanowires at low temperatures: Breakdown and recovery of quantization

The quantization of lattice thermal conductance $g$ normalized by $g_0={\pi }^2{k}_B^{2}T∕3h$ (the universal quantum of thermal conductance) was recently predicted theoretically to take an integer value over a finite range of temperature and then observed experimentally in nanowires with catenoidal contacts. The prediction of this quantization by Rego and Kirczenow [Phys. Rev. Lett. 81, 232 (1998)] relies on a study of only dilatational (longitudinal) vibrational mode in the wires. We study the thermal conductance in catenoidal wires by explicitly calculating the transmission rates of the six distinct vibrational modes (four acoustic and two low-lying optical modes) and applying the Landauer formula for the one-dimensional thermal transport in the ballistic regime. In a temperature range similar to the one predicted by Rego and Kirczenow, we find the presence of a plateau in $g∕g_0$. However, below this temperature range $g∕g_0$ is modified—that is, the quantization is broken—due to imperfect transmission of the acoustic modes of vibration. Our new observation is that, as temperature goes down further, the recovery of the quantization of $g∕g_0$ should occur. These results are found assuming GaAs as a wire material, but we also make similar calculations for silicon nitride wires used experimentally.

Figure 1

Diagram defining the heat flow $J$ in the system consisting of a catenoidal wire (thickness $h$ in $z$ direction is uniform). The wire is attached to the ideal leads (labeled $H$ and $C$) in which the wave number $k$ and vibrational modes $m$ and $m^{\prime }$ are defined. The ideal leads are also attached to the hot and cold heat reservoirs with temperatures $T_H$ and $T_C$, respectively.

Figure 2

(a) The transmission rate versus normalized frequency of the dilatational vibrations in a catenoidal GaAs wire. The solid line is the analytical result and dots are calculated with the transfer-matrix method. The parameters used are $\lambda =0.86\mu \mathrm{m}$, $d=2\mu \mathrm{m}$, and $h=0.05\mu \mathrm{m}$. ${\nu }_D^c$ is the cutoff frequency of the dilatational mode. The inset shows the definition of parameters characterizing the catenoidal wire (the only negative $x$ region is shown). (b) The calculated transmission rates of the dilatational mode (solid line) and torsional mode (dotted line) in the same GaAs wire as for (a). ${\nu }_T^c$ denotes the cutoff frequencies of the torsional mode.

Figure 3

The calculated transmission rates of the coupled flexural (thin line) and shear (bold line) mode (a) in the thickness direction and (b) in the width direction in the same GaAs wire as for Fig. 1. ${\nu }_{F,z}^c$ and ${\nu }_{F,y}^c$ (${\nu }_{S,z}^c$ and ${\nu }_{S,y}^c$) are the cutoff frequencies of the flexural (shear) modes in the thickness and width directions, respectively. The inset of (a) shows the transmission rates in the frequency range where the shear mode rates become finite. The thin line (bold line) is the shear mode in the thickness direction (in the width direction). The rates of four acoustic modes are practically unity in this range. ${\nu }_{S,\mathit{min}}$ is the threshold frequency of the shear modes.

Figure 4

(a) Calculated thermal conductance $g$ (normalized by the quantum of thermal conductance $g_0={\pi }^2{k}_B^{2}T∕3h$) versus temperature for a GaAs catenoidal wire with $\lambda =9.2\mu \mathrm{m}$, $d=2\mu \mathrm{m}$, and $h=0.05\mu \mathrm{m}$. The inset shows the contributions of the dilatational mode (bold solid line), torsional mode (dotted line), and the coupled flexural-shear modes (denoted conveniently as bending modes) in the thickness direction (dashed line) and in the width direction (thin solid line). (b) $g∕g_0$ for the catenoidal wires with various magnitude of $\lambda$—i.e., $\lambda =9.2\mu \mathrm{m}$ (dots), $\lambda =4.6\mu \mathrm{m}$ (squares), $\lambda =0.86\mu \mathrm{m}$ (triangles), and $\lambda =0.2\mu \mathrm{m}$ (open circles). The wire widths $d=2\mu \mathrm{m}$ and $h=0.05\mu \mathrm{m}$ are the same as (a). For $T_0$, $T_b$, and $T_c$ indicated in (a), see the text.

Figure 5

Calculated thermal conductance $g$ (normalized by the quantum of thermal conductance $g_0={\pi }^2{k}_B^{2}T∕3h$) versus temperature for catenoidal silicon nitride wires with $\lambda =1\mu \mathrm{m}$ (circles), $\lambda =3\mu \mathrm{m}$ (triangles), and $\lambda =5\mu \mathrm{m}$ (squares). For other parameters of the wires, see the text. The inset shows the log-log plot which compares the experimental data by Schwab et al. (dots) with the theoretical results for $\lambda =1\mu \mathrm{m}$ (circles).