$Ab$ $initio$ low-energy effective Hamiltonians for high-temperature superconducting cuprates Bi$_2$Sr$_2$CuO$_6$, Bi$_2$Sr$_2$CaCu$_2$O$_8$, HgBa$_2$CuO$_4$ and CaCuO$_2$

Abstract

We derive $ab$ $initio$ low-energy effective Hamiltonians (LEH) for high-temperature superconducting (SC) copper oxides Bi$_2$Sr$_2$CuO$_6$ (Bi2201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 10$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212, $N_{\ell}=2$, $T_c^{\rm exp} \sim 84$ K), HgBa$_2$CuO$_4$ (Hg1201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 90$ K) and CaCuO$_2$ (Ca11, $N_{\ell}=\infty$, $T_c^{\rm exp} \sim 110$ K), with different experimental optimal SC transition temperature $T_c^{\rm exp}$ and number $N_{\ell}$ of laminated CuO$_2$ planes between the two neighboring block layers. We apply the latest methodology of the multiscale $ab$ $initio$ scheme for correlated electron systems (MACE), and focus on the LEH consisting of one antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$ orbital centered on each Cu atom. We discuss prominent features of this LEH: (1) The ratio $U/|t_1|$ between the onsite effective Coulomb repulsion (ECR) $U$ and amplitude of nearest neighbour hopping $t_1$ increases with $T^{\rm exp}_c$ and $N_{\ell}$, consistently with the expected increase in $d$-wave SC correlation function $P_{dd}$ with $U/|t_1|$. One possible cause of the increase of $U/|t_1|$ is the replacement of apical O atoms by Cu atoms from neighbouring CuO$_2$ planes when $N_{\ell}$ increases. Furthermore, we show that the increase in distance between Cu and apical O atoms decreases the effective screening (ES) by electrons outside of the LEH and increases $U/|t_1|$. (2) For Hg1201 and Ca11, we show that $U/|t_1|$ decreases when hole doping per AB orbital $\delta$ increases, which may partly account for the disappearance of SC when $\delta$ exceeds the optimal value in experiment. (3) For $N_{\ell} \geq 2$, off-site inter-CuO$_2$ plane ECR is comparable to off-site intra-CuO$_2$ plane ECR. We discuss contributions of inter-CuO$_2$ plane ECR to both $P_{dd}$ and the stability of the SC state.