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We have measured the temperature dependences of the thermal conductivity ofBa$_2$Cu$_{3-x}M_x$O$_4$Cl$_2$ ($M$ = Pd, Ni, Co; $x$ = 0, 0.03) singlecrystals including two-dimensional (2D) Cu$_3$O$_4$ planes consisting of astrong 2D spin network of CuA$^{2+}$ spins and a weak 2D spin network ofCuB$^{2+}$ spins. It has been found that the thermal conductivity due to spins,$\kappa_{\mathrm{spin}}$, exists in the thermal conductivity parallel to theCu$_3$O$_4$ plane owing to the strong 2D spin network of CuA$^{2+}$ spins andexhibits 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 onaccount the decrease in the mean free path of magnetic excitations, suggestingthat $\kappa_{\mathrm{spin}}$ is expected to be enhanced in 2D quantum spinsystems such as Ba$_2$Cu$_3$O$_4$Cl$_2$ by reducing the amount of impurities ina single crystal. Moreover, it has concluded that the frustration betweenCuA$^{2+}$ and CuB$^{2+}$ spins little affects the existence of$\kappa_{\mathrm{spin}}$.

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