Fermi surface splittings in multilayered high-$T_{\rm c}$ cuprates with charge imbalance

Cuprate superconductors have layered structure of CuO$_2$ planes, which makes conducting blocks separated by an charge- reservoir block. Multilayered high-$T_{\rm c}$ cuprates, e.g., Ba$_2$Ca$_3$Cu$_4$O$_8$(O$_{1-y}$F$_y$)$_2$ and HgBa$_2$Ca$_4 $Cu$_5$O$_y$, have two kinds of CuO$_2$ planes in a unit cell; the outer-pyramidal-coordinated-planes (OP's) and the inner- square-coordinated-planes (IP's). The carrier density in the OP is generally different from that in the IP. We call such an inhomogeneous charge-distribution $\lq$charge imbalance'. We study doping dependence of interlayer hoppings, $t_{\perp}$, in such a charge-imbalance system in the Gutzwiller approximation. When the double occupancy is forbidden in the CuO$_2$ plane, an effective amplitude of $t_{\perp}$ is shown to be proportional to the square root of the product of doping rates in adjacent two planes. Therefore, the charge imbalance in more than three-layered cuprates results in two different values of $t^{\rm eff}_{\perp} $, i.e., $t^{\rm eff}_{\perp1}\propto t_{\perp}\sqrt{\delta_{\rm IP} \delta_{\rm IP}}$ between IP's, and $t^{\rm eff}_{\perp2}\propto t_{\perp}\sqrt{\delta_{\rm IP} \delta_{\rm OP}}$ between IP and OP, where $\delta_{\rm IP}$ ($\delta_{\rm OP}$) is the doping rates in IP (OP). Fermi surfaces are calculated in the four-layered $t$-$t'$- $t''$-$J$ model by the mean-field theory. The order parameters, the renormalization factor of $t_{\perp}$, and the site- potential making the charge imbalance between IP and OP are self-consistently determined for several doping rates. We show the interlayer splitting of the Fermi surfaces, which may be observed in the angle resolved photoemission spectroscopy measurement. *cond-mat/0511249.