Strongly overdoped ${\mathrm{HgBa}}_{2}{\mathrm{Ca}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{8+\mathrm{\ensuremath{\delta}}}$ (Hg-1223) samples with superconducting transitions as low as 97 K were successfully obtained by using HgO, ${\mathrm{Ca}}_{2}{\mathrm{CuO}}_{3},$ CuO, and mixtures of highly oxidized ${\mathrm{BaCuO}}_{2+\mathrm{\ensuremath{\epsilon}}}$ $(\ensuremath{\epsilon}\ensuremath{\approx}0.13)$ and/or ${\mathrm{BaO}}_{2}$ powders as starting materials for the high-pressure synthesis at 5 GPa and 950 \ifmmode^\circ\else\textdegree\fi{}C. The overdoped state was confirmed by observing negative values for the Seebeck coefficient throughout the temperature range from ${T}_{c}$ to 320 K in a thermoelectric power measurement. Also, both of the cell parameters $a$ and $c$ were found to decrease with decreasing ${T}_{c},$ reflecting, respectively, an increase in hole concentration in the ${\mathrm{CuO}}_{2}$ planes and the incorporation of oxygen into the ${\mathrm{HgO}}_{\mathrm{\ensuremath{\delta}}}$ layer. The oxygen excess \ensuremath{\delta} as determined by the $\mathrm{Cu}(+\mathrm{I})/\mathrm{C}\mathrm{u}(+\mathrm{I}\mathrm{I})$ coulometric titration method, was $\ensuremath{\sim}0.19$ in the overdoped sample with ${T}_{c}=107\mathrm{K}.$ Subsequent reducing annealing in an Ar atmosphere at 280 \ifmmode^\circ\else\textdegree\fi{}C increased the ${T}_{c}$ to 131 K. Consistently, only positive Seebeck coefficient values were observed up to 320 K for the Ar-annealed sample. A clearly underdoped material with ${T}_{c}=118\mathrm{K}$ was obtained by annealing the same sample in Ar at a higher temperature $(\ensuremath{\sim}400\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}).$ Finally, high-pressure synthesis starting from less-oxidized ${\mathrm{BaCuO}}_{2+\mathrm{\ensuremath{\epsilon}}}$ $(\ensuremath{\epsilon}\ensuremath{\approx}0.06)$ yielded a Hg-1223 material with $\ensuremath{\delta}\ensuremath{\approx}0.10$ and ${T}_{c}=132\mathrm{K}.$
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