Thermal conductivity in the doped two-leg ladder antiferromagnet   Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$
  Within the t-J model, the heat transport of the doped two-leg ladder material Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$ is studied in the doped regime where superconductivity appears under pressure in low temperatures. The energy dependence of the thermal conductivity $\kappa_{c}(\omega)$ shows a low-energy peak, while the temperature dependence of the thermal conductivity $\kappa_{c}(T)$ is characterized by a broad band. In particular, $\kappa_{c}(T)$ increases monotonously with increasing temperature at low temperatures $T<0.05J$, and is weakly temperature dependent in the temperature range $0.05J<T<0.1J$, then decreases for temperatures $T>0.1J$, in qualitative agreement with experiments. Our result also shows that although both dressed holons and spinons are responsible for the thermal conductivity, the contribution from the dressed spinons dominates the heat transport of Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$. 
Jihong Qin, Shiping Feng, Feng Yuan and Wei Yeu Chen