Spin gap and dynamics in ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ comprising hole-doped two-leg spin ladders: Cu NMR study on single crystals

We report comprehensive Cu NMR studies on single crystals of ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},$ which contain simple ${\mathrm{CuO}}_2$ chains and two-leg ${\mathrm{Cu}}_2{\mathrm{O}}_3$ ladders. From measurements of the ${}^{63}\mathrm{Cu}$ NMR shift, it is clear that the spin gap in the ladders decreases with isovalent Ca substitution from $\Delta =550\mathrm{}\pm 30$ K for ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (Sr14) to $350\pm 30$ K, $280\pm 30$ K, and $270\pm 30$ K for $x=6\mathrm{}$ (Ca6), $x=9\mathrm{}$ (Ca9), and $x=11.5\mathrm{}$ (Ca11.5), respectively. The exponential decrease of the nuclear spin-lattice relaxation rate ${1/T}_1$ below ∼130 K is consistent with the presence of the spin gap in the spin excitation spectrum. In the $T$ range higher than $\sim 200$ K, we observed the following dependences: ${1/T}_1=\mathrm{const}$ and the square of Gaussian spin-echo decay time, $T_{2G}^2\propto T$ which are consistent with the scaling theory for the $S=1/2\mathrm{}$ one-dimensional (1D) Heisenberg model. The value of $T_{2G}{/T}_1\sqrt{T}$ is compatible with the theoretical prediction of an exchange constant along the leg $J_{\parallel }\sim 1800$ K for Ca6 and $J_{\parallel }\sim 1500$ K for Ca9 and Ca11.5. A notable finding is that the magnitude of the spin gap remains nearly constant and characteristics of novel 1D-like spin dynamics are maintained in the content varying from Ca9 to Ca11.5. On the other hand, the charge transport changes with increasing Ca content so that the more conductive Ca11.5 exhibits pressure-induced superconductivity exceeding 3.5 GPa. We have found that $T_{2G}^2$, which is proportional to the inverse spin correlation length ${\xi }^{-1}$, deviates from a linear $T$ dependence upon cooling and is described by $A+BT\exp (-\Delta /T)$, regardless of the Ca substitution. We point out that the value of