Heat transport in ultrathin dielectric membranes and bridges

Phonon modes and their dispersion relations in ultrathin homogeneous dielectric membranes are calculated using elasticity theory. The approach differs from the previous ones by a rigorous account of the effect of the film surfaces on the modes with different polarizations. We compute the heat capacity of membranes and the heat conductivity of narrow bridges cut out of such membranes, in a temperature range where the dimensions have a strong influence on the results. In the high-temperature regime we recover the three-dimensional bulk results. However, in the low-temperature limit the heat capacity $C_V$ is proportional to $T$ (temperature), while the heat conductivity $\kappa$ of narrow bridges is proportional to $T^{3∕2}$, leading to a thermal cutoff frequency $f_c=\kappa ∕C_V\propto T^{1∕2}$.

Figure 1

(a) A schematic drawing of a thin free standing membrane of the type discussed in this paper. The thickness $b$ is typically a few hundreds of $\mathrm{nm}$. The other dimensions $l_1$ and $l_2$ are of the order of $100\mu \mathrm{m}$. (b) Such a membrane (central part) suspended to the bulk by four bridges. The bridge width $w$ ranges from few $\mu \mathrm{m}$ to tens of $\mu \mathrm{m}$.

Figure 2

Reflection of elastic waves at the free surfaces of the membrane of thickness $b$. A horizontal shear wave reflects also as a horizontal shear wave (a), while a wave longitudinally or transversally polarized into the $xz$ plane reflects as a superposition of longitudinally and transversally polarized waves (b).

Figure 3

The branches of the dispersion relations of the phonon eigenmodes of a free standing thin membrane. (a) $h$ modes, (b) $s$ modes, and (c) $a$ modes.

Figure 4

The three lowest branches of the dispersion relations of the phonon eigenmodes of a free standing thin membrane shown in the limit of long wavelengths. The linear behavior of the $h$ and $s$ branches (labeled $h$ and $s$, respectively) as well as the quadratic behavior of the $a$ branch can be seen here.

Figure 5

The temperature exponents $p_f\equiv \partial \ln f∕\partial \ln T$ of the heat capacity $(f\equiv C)$ and heat conductivity $(f\equiv k)$ in narrow bridges of thin films. The insets show the behavior of the curves for small values of $T∕T_C$. (a) The temperature exponent of $C_V$, as it would be if the modes were not coupled. The dimensionality crossover around $T_C=(\hslash c_t∕2k_B)b^{-1}$ can be seen here quite nicely. For a $100\mathrm{nm}$ thin membrane the critical temperature of ${\mathrm{SiN}}_x$ is $237\mathrm{mK}$. (b) The temperature exponent of the heat capacity of a thin membrane or a narrow bridge. (c) The temperature exponent of the heat conductivity along a narrow bridge.

Figure 6

The cutoff frequency of an ac-heated membrane, which is connected to the bulk material by narrow bridges. Here we normalized the function to its maximum value, so that $f_c^{\prime }=f_c∕f_{c,0}\propto \kappa ∕\left(l{C}_V\right).$ For small values of $T∕T_C{f}_c^2$ becomes linear, which is shown in the inset.