Low-temperature phonon thermal conductivity of cuprate single crystals

The effect of sample size and surface roughness on the phonon thermal conductivity $κ_p$ of Nd$_2$CuO$_4$ single crystals was studied down to 50 mK. At 0.5 K, $κ_p$ is proportional to $\sqrt{A}$, where $A$ is the cross-sectional area of the sample. This demonstrates that $κ_p$ is dominated by boundary scattering below 0.5 K or so. However, the expected $T^3$ dependence of $κ_p$ is not observed down to 50 mK. Upon roughing the surfaces, the $T^3$ dependence is restored, showing that departures from $T^3$ are due to specular reflection of phonons off the mirror-like sample surfaces. We propose an empirical power law fit, to $κ_p \sim T^α$ (where $α< 3$) in cuprate single crystals. Using this method, we show that recent thermal conductivity studies of Zn doping in YBa$_2$Cu$_3$O$_y$ re-affirm the universal heat conductivity of d-wave quasiparticles at $T \to 0$.

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