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The superconducting phase of the $\mathrm{HgBa}_2\mathrm{CuO}_{4+\delta}$(Hg-1201) and $\mathrm{HgBa}_2\mathrm{Ca}_2\mathrm{Cu}_3\mathrm{O}_{8+\delta}$(Hg-1223) cuprates has been investigated by Raman spectroscopy underhydrostatic pressure. Our analysis reveals that the increase of $T_c$ withpressure is slower in Hg-1223 cuprate compared to the Hg-1201 due to a chargecarrier concentration imbalance (accentuated by pressure) between the$\mathrm{CuO}_2$ layers of Hg-1223. We find that the energy variation underpressure of the apical oxygen mode from which the charge carriers aretransferred to the $\mathrm{CuO}_2$ layers, is the same for both the Hg-1223and Hg-1223 cuprates and it is controlled by the inter-layer compressibility.At last, we show that the binding energy of the Cooper pairs related to themaximum amplitude of the $d-$ wave superconducting gap at the anti-nodes, doesnot follow $T_c$ with pressure. It decreases while $T_c$ increases. In theparticular case of Hg-1201, the binding energy collapses from 10 to 2 $K_B T_c$as the pressure increases up to 10 GPa. These direct spectroscopic observationsjoined to the fact that the binding energy of the Cooper pairs at theanti-nodes does not follow $T_c$ either with doping, raises the question of itslink with the pseudogap energy scale which follows the same trend with doping.

Exploration of the Hg-based cuprate superconductors by Raman spectroscopy under hydrostatic pressure