<i>Ab initio</i> low-energy effective Hamiltonians for the high-temperature superconducting cuprates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CaCu</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>HgBa</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>CaCuO</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

Abstract

We derive ab initio low-energy effective Hamiltonians (LEH) for high-temperature superconducting (SC) copper oxides ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{6}$ (Bi2201, ${N}_{\ensuremath{\ell}}=1$, ${T}_{c}^{\mathrm{exp}}\ensuremath{\sim}10\phantom{\rule{0.16em}{0ex}}\mathrm{K}$), ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ (Bi2212, ${N}_{\ensuremath{\ell}}=2$, ${T}_{c}^{\mathrm{exp}}\ensuremath{\sim}84\phantom{\rule{0.16em}{0ex}}\mathrm{K}$), ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}$ (Hg1201, ${N}_{\ensuremath{\ell}}=1$, ${T}_{c}^{\mathrm{exp}}\ensuremath{\sim}90\phantom{\rule{0.16em}{0ex}}\mathrm{K}$), and ${\mathrm{CaCuO}}_{2}$ (Ca11, ${N}_{\ensuremath{\ell}}=\ensuremath{\infty}$, ${T}_{c}^{\mathrm{exp}}\ensuremath{\sim}110\phantom{\rule{0.16em}{0ex}}\mathrm{K}$), with substantially different values of experimental optimal SC transition temperature ${T}_{c}^{\mathrm{exp}}$ and number ${N}_{\ensuremath{\ell}}$ of laminated ${\mathrm{CuO}}_{2}$ 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) $\mathrm{Cu}3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$/$\mathrm{O}2{p}_{\ensuremath{\sigma}}$ 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-neighbor hopping ${t}_{1}$ increases with ${T}_{c}^{\mathrm{exp}}$ and ${N}_{\ensuremath{\ell}}$, consistently with the expected increase in $d$-wave SC correlation function ${P}_{dd}$ with $U/|{t}_{1}|$. One possible cause of the increase of $U/|{t}_{1}|$ with ${N}_{\ensuremath{\ell}}$ is the replacement of apical O atoms by Cu atoms from neighboring ${\mathrm{CuO}}_{2}$ planes when ${N}_{\ensuremath{\ell}}$ increases. Furthermore, we show that the increase in distance between Cu and apical O atoms decreases the effective screening (ES) defined as the screening by electrons outside of the LEH and increases $U/|{t}_{1}|$. (2) For Hg1201 and Ca11, we examine the variation in $U/|{t}_{1}|$ with hole doping per AB orbital $\ensuremath{\delta}$, and show that $U/|{t}_{1}|$ decreases when $\ensuremath{\delta}$ increases, which may partly account for the disappearance of SC when $\ensuremath{\delta}$ exceeds the optimal value in experiment. (3) For ${N}_{\ensuremath{\ell}}\ensuremath{\ge}2$, offsite inter-${\mathrm{CuO}}_{2}$ plane ECR is comparable to off-site intra-${\mathrm{CuO}}_{2}$ plane ECR. We discuss contributions of inter-${\mathrm{CuO}}_{2}$ plane ECR to both ${P}_{dd}$ and the stability of the SC state.

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