Doping dependence of the superconducting gap in Tl 2 Ba 2 CuO 6 + δ from heat transport

We present low-temperature thermal conductivity measurements on the cuprate ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ throughout the overdoped regime. In the $T\ensuremath{\rightarrow}0$ limit, the thermal conductivity due to $d$-wave nodal quasiparticles provides a bulk measurement of the superconducting gap $\ensuremath{\Delta}$. We find $\ensuremath{\Delta}$ to decrease with increasing doping, with a magnitude consistent with spectroscopic measurements (photoemission and tunneling). This argues for a pure and simple $d$-wave superconducting state in the overdoped region of the phase diagram, which appears to extend into the underdoped regime down to a hole concentration of $\ensuremath{\simeq}0.1$ hole/Cu. As hole concentration is decreased, the gap-to-${T}_{c}$ ratio increases, showing that the suppression of the superconducting transition temperature ${T}_{c}$ (relative to the gap) begins in the overdoped regime.