Uniform hole doping in HgBa$_{2}$Ca$_{2}$Cu$_{3}$O$_{8+\delta}$ studied by $^{63}$Cu NMR

We studied local magnetic properties of triple-CuO$_2$-layer superconductors HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ with rich $^{63}$Cu isotope (underdoped $T_\mathrm{c}$ = 124 K and optimally doped $T_\mathrm{c}$ = 134 K) by $^{63}$Cu NMR spin-echo techniques. The temperature dependences of the $^{63}$Cu nuclear spin-lattice relaxation rate $^{63}$(1/$T_1$) and the spin Knight shift $^{63}K_\mathrm{spin}$ of the inner plane Cu(1) site were nearly the same as those of the outer plane Cu(2) site in the normal states. The pseudogap temperature $T^*$ defined as the maximum temperature of $^{63}$(1/$T_1T$) was 190 K for the underdoped $T_\mathrm{c}$ = 124 sample and 143 K for the optimally doped $T_\mathrm{c}$ = 134 K sample. The NMR results indicate the uniform hole distribution in the three layers, which is consistent with a coalescence of three-derived Fermi surfaces predicted by a band theory. The room temperature in-plane $^{63}K_\mathrm{spin}^{ab}$ depends on whether the number of CuO$_2$ planes per unit cell is even or odd for the Hg-based multilayer cuprate superconductors.