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, ${\kappa }_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$, 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 ${\kappa }_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ 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 ${\kappa }_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ has been found to be suppressed by 1% impurities on account the decrease in the mean free path of magnetic excitations, suggesting that ${\kappa }_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ 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 ${\kappa }_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$.