A fully gapped superconducting state in small cuprate islands

We present a spectroscopic technique, based on an high-Tc superconducting nanoscale device that allows an unprecedented energy resolution thanks to Coulomb blockade effects, a regime practically inaccessible earlier in these materials. We found that the energy required to add an extra electron depends on the parity (odd/even) of excess electrons on the island and increases with magnetic field. This is inconsistent with a pure $d_{x^2-y^2}$ wave symmetry and demonstrates a complex order parameter component that needs to be encompassed in any theoretical model for high-Tc superconductivity. To address this inconsistency, we investigate subdominant order parameters stabilizing at low temperatures in nano-scale high-T$_c$ cuprate islands. Using complementary quasi-classical and tight-binding Bogoliubov-de Gennes methods, we show on distinctly different properties dependent on the symmetry being $d_{x^2-y^2}+i s$ or $d_{x^2-y^2}+i d_{xy}$. We find that a surface-induced $d_{x^2-y^2}+i s$ phase creates a global spectroscopic gap which increases with applied magnetic field, consistent with experimental observation.