We investigate the extended $t\text{−}J$ model on honeycomb lattices with next-nearest-neighbor electron hopping $t^{\prime }$ and superexchange coupling $J^{\prime }={(t^{\prime }/t)}^{2}J$ using large-scale density-matrix renormalization group (DMRG) simulations and slave-boson mean-field theory (SBMFT). By systematically varying $t^{\prime }$ and cylinder geometries, our DMRG results reveal several competing phases with distinct charge and superconducting (SC) properties. On YC4-0 cylinders possessing bonds lying along the ${\vec{e}}_y$ direction, the ground state of doped models exhibits pronounced quasi-long-range $d$-wave SC with coexisting armchair-oriented stripes across a broad range of $t^{\prime }$. Notably, the SC Luttinger exponent has a nonmonotonic dependence on $t^{\prime }$, showing the strongest pairing tendency near $t_{\text{op}}^{\prime }\approx 0.4$ among the doping levels and parameter values studied. Conversely, XC cylinders host a competing long-range zigzag-stripe phase with exponentially decaying SC correlations for $t^{\prime }>0.5$, highlighting the role of boundary geometry in stabilizing distinct competing phases in DMRG. Complementary SBMFT calculations compare the relative energies of the competing charge patterns without finite-cylinder constraints and find the armchair stripe to have the lowest energy among the selected charge-density-wave candidates.

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• Superconductivity
• Strongly correlated systems
• Density matrix renormalization group