Thermal Conductivity due to Spins in the Two-Dimensional Spin System Ba$_2$Cu$_3$O$_4$Cl$_2$

We have measured the temperature dependences of the thermal conductivity of Ba$_2$Cu$_{3-x}M_x$O$_4$Cl$_2$ ($M$ = Pd, Ni, Co; $x$ = 0, 0.03) single crystals including two-dimensional (2D) Cu$_3$O$_4$ planes consisting of a strong 2D spin network of CuA$^{2+}$ spins and a weak 2D spin network of CuB$^{2+}$ spins. It has been found that the thermal conductivity due to spins, $\kappa_{\mathrm{spin}}$, exists in the thermal conductivity parallel to the Cu$_3$O$_4$ plane owing to the strong 2D spin network of CuA$^{2+}$ spins and exhibits a broad peak around room temperature. The maximum value of $\kappa_{\mathrm{spin}}$ is ~7 W/Km and comparable with that in Nd$_2$CuO$_4$ with almost the same 2D spin network of Cu$^{2+}$ spins. The $\kappa_{\mathrm{spin}}$ has been found to be suppressed by 1% impurities on account the decrease in the mean free path of magnetic excitations, suggesting that $\kappa_{\mathrm{spin}}$ is expected to be enhanced in 2D quantum spin systems such as Ba$_2$Cu$_3$O$_4$Cl$_2$ by reducing the amount of impurities in a single crystal. Moreover, it has concluded that the frustration between CuA$^{2+}$ and CuB$^{2+}$ spins little affects the existence of $\kappa_{\mathrm{spin}}$.