Heat capacity of single-crystal Cu${}_x$TiSe${}_2$ superconductors

We present heat capacity measurements on a series of superconducting Cu${}_x$TiSe${}_2$ single crystals with different Cu content down to 600 mK and up to 1 T performed by ac microcalorimetry. The samples cover a large portion of the phase diagram from an underdoped to a slightly overdoped region with an increasing superconducting critical temperature and the charge density wave (CDW) order gradually suppressed. The superconducting electronic heat capacity as a function of normalized temperature $T/T_c$ shows no difference regardless of the concentration of copper, i.e., regardless of how much the CDW order is developed in the samples. The data analysis reveals consistently a single $s$-wave gap with an intermediate coupling strength $2\Delta /k_B{T}_c\approx 3.7$ for all samples.

Figure 1

Total heat capacity of sample A plus addenda in 0 and 1 T. Inset: Temperature dependence of the heat capacity after subtraction of the normal-state background measured in different magnetic fields for a field in the $c$ direction (upper panel, applied fields 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.075, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, and 0.45 T) and parallel to the $ab$ plane (lower panel, applied fields 0, 0.05, 0.1, 0.18, 0.27, 0.36, 0.45, 0.54, 0.63, 0.72, and 0.81 T).

Figure 2

Superconducting electronic heat capacity of sample A (open symbols) and corresponding single-gap $\alpha$-model fit (thick line), compared to that of NbS${}_2$ (Ref. 12) (solid symbols) and $\alpha$-model fit corresponding to the presence of two energy gaps or one anisotropic gap (thin line). Upper inset: Angular dependence of $T_c$ at 0.15 T (symbols) and a Ginzburgh-Landau fit for a single-gap superconductor (line). Lower inset: Deviation function—($T_{c2}$/$T_{c2}^{\text{GL}}$)${}^2$ for sample A (open symbols) and ($H_{c2}$/$H_{c2}^{\text{GL}}$)${}^2$ for NbS${}_2$ (Ref. 19) (solid symbols); the line is a guide to the eye.

Figure 3

The temperature dependence $\Delta C/T$ of samples A, B, C, and D in a temperature scale normalized to the $T_c$ of each sample. Inset: The phase diagram as proposed by Morosan et al. (Ref. 2) and the critical temperature of our samples with respect to their copper content (large symbols).

Figure 4

Circles: Upper critical field of the samples for field orientation parallel (solid circles) and perpendicular to the $ab$ planes (open circles) and corresponding theoretical curves from the WHH model (lines). Squares: Superconducting anisotropy $\Gamma =H_{c2}^{ab}/H_{c2}^c$; the right $y$ axis applies.