Doping dependence of the superconducting gap in ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ from heat transport

We present low-temperature thermal conductivity measurements on the cuprate ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ throughout the overdoped regime. In the $T\to 0$ limit, the thermal conductivity due to $d$-wave nodal quasiparticles provides a bulk measurement of the superconducting gap $\Delta$. We find $\Delta$ 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 $\simeq 0.1$ 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.

Figure 1

(Color online) Thermal conductivity of one Tl-2201 sample, measured at two different hole concentrations, as indicated by the respective $T_c$ values, plotted as $\kappa ∕T$ vs $T$ below $1\mathrm{K}$. The lines are fits to Eq. (2). A threefold decrease in $T_c$ corresponds to a sixfold increase in the residual linear term [which is inversely proportional to the superconducting gap, via Eq. (5)], hence a twofold decrease in the gap-to-$T_c$ ratio with doping.

Figure 2

(Color online) Thermal conductivity of four Tl-2201 samples, each with a different hole concentration, as indicated by their respective $T_c$ values, plotted as $\kappa ∕T$ vs $T$ below $1\mathrm{K}$. The lines are fits to Eq. (2).

Figure 3

(Color online) Upper panel: the gap maximum ${\Delta }_0$ vs hole concentration $p$ derived from thermal conductivity measurements on Tl-2201 (circles) and YBCO (square, Refs. 8, 21). The solid (red) circles denote the same Tl-2201 sample measured at two different doping levels, as discussed in the text. Spectroscopic measurements of the gap from ARPES on Tl-2201 (red down-triangle, Refs. 9, 22) and YBCO (blue down triangle, Ref. 23) and SIS tunneling on Bi-2212 (black up-triangles, Ref. 24) are also shown. The dashed line gives the BCS doping dependence of the gap based on the measured $T_c$—namely, ${\Delta }_0=\mu k_B{T}_c$—using a strong-coupling value of $\mu =3$ and $T_c\left(p\right)$ from Eq. (1). Lower panel: the gap-to-$T_c$ ratio ${\Delta }_0∕k_B{T}_c$ vs $p$ for Tl-2201 from thermal conductivity data. The solid (red) line is a fit through the two data points (solid red circles) obtained on a single sample, for which the relative uncertainty is minimal $(\sim 3\%)$, given the fixed geometric factor. The dashed (red) lines on either side reflect the uncertainty on the absolute value ($\pm 17\%$; see Table ). The horizontal (black) dashed line is the weak-coupling $d$-wave BCS prediction: namely, ${\Delta }_0∕k_B{T}_c$ = 2.14 (Ref. 25).

Figure 4

(Color online) Thermal conductivity in an applied field $(H\Vert c)$ for the Tl-2201 sample with $T_c=76\mathrm{K}$ plotted as $\kappa ∕T$ vs $T$. ${\kappa }_0∕T$ increases by a factor of $\sim 3$ in an applied field of $11.5\mathrm{T}$. The lines are fits to Eq. (2). Inset: the in-field electronic thermal conductivity ${\kappa }_0\left(H\right)∕T$ normalized to the zero-field value ${\kappa }_0\left(0\right)∕T$ as a function of magnetic field in a Tl-2201 sample with $T_c=89\mathrm{K}$ (not used in the main study). The solid line is a fit to the semiclassical model [Eq. (A1)], which yields $\gamma ∕{\Delta }_0\simeq 0.1$ (see text).