Magnetic and thermodynamic properties of Cu\(_x\)TiSe\(_2\) single crystals

We present a detailed study of the phase diagram of copper intercalated TiSe\(_2\) single crystals, combining local Hall-probe magnetometry, tunnel diode oscillator technique (TDO), specific-heat, and angle-resolved photoemission spectroscopy measurements. A series of the Cu\(_x\)TiSe\(_2\) samples from three different sources with various copper content \(x\) and superconducting critical temperatures \(T_c\) have been investigated. We first show that the vortex penetration mechanism is dominated by geometrical barriers enabling a precise determination of the lower critical field, \(H_{c1}\). We then show that the temperature dependence of the superfluid density deduced from magnetic measurements (both \(H_{c1}\) and TDO techniques) clearly suggests the existence of a small energy gap in the system, with a coupling strength \(2\Delta_s \sim [2.4-2.8]k_BT_c\), regardless of the copper content, in puzzling contradiction with specific heat measurements which can be well described by one single large gap \(2\Delta_l \sim [3.7-3.9]k_BT_c\). Finally, our measurements reveal a non-trivial doping dependence of the condensation energy, which remains to be understood.