Enhancement of Pairing Correlation byt′in the Two-Dimensional Extendedt−JModel

We investigate the effects of the next-nearest-neighbor ($t^{\prime }$) and the third-nearest-neighbor ($t^{\prime \prime }$) hopping terms on superconductivity correlation in the 2D hole-doped extended $t-J$ model based on the variational Monte Carlo, mean-field calculation and exact diagonalization method. Despite the diversity of the methods employed, the results all point to a consistent conclusion: While the $d$-wave superconductivity correlation is slightly suppressed by $t^{\prime }$ and $t^{\prime \prime }$ in underdoped regions, it is greatly enhanced in the optimal and overdoped regions. The optimal $T_c$ is a result of the balance of these two opposite trends.