Effects of pressure and disorder on superconductivity in Tl$_{\mathrm{2}}$Ba$_{\mathrm{2}}$Ca$_{\mathrm{n-1}}$Cu$_{\mathrm{n}}$O$_{\mathrm{2n+4+\delta \thinspace }}$(n$=$2,3)

The structural, vibrational, and superconducting properties of nearly optimally doped Tl$_{\mathrm{2}}$Ba$_{\mathrm{2}}$Ca$_{\mathrm{n-1}}$Cu$_{\mathrm{n}}$O$_{\mathrm{2n+4+\delta }}$ (n$=$2,3) single crystals are~studied~at high pressures. While the superconducting transition temperature $T_{\mathrm{c}}$ of the bilayer system exhibits a parabolic behavior with a maximum around an optimal pressure level, its increased path has a kink in the trilayer compound, indicating a joint effect from the inner CuO$_{\mathrm{2}}$ plane. The latter is further supported by the appearance of the two additional Raman modes. At higher pressures, we observe the sudden increase of the full width at half maximum of some lattice modes as well as the anomaly of the lattice parameters at certain pressure. These together contribute the enhanced disorder and the~$T_{\mathrm{c}}$ reduction accordingly. We thus are able to distinguish the contribution to $T_{\mathrm{c}}$ from the intrinsic pressure variables, disorder, and the CuO$_{\mathrm{2}}$ plane number and type in this layered family.