Dome structure in pressure dependence of superconducting transition temperature for HgBa${}_2$Ca${}_2$Cu${}_3$O${}_8$ -- Studies by $ab$ $initio$ low-energy effective Hamiltonian

Abstract

The superconducting (SC) cuprate HgBa${}_2$Ca${}_2$Cu${}_3$O${}_8$ (Hg1223) has the highest $T_c^{\mathrm{opt}}\simeq 138$ K (the experimental SC transition temperature at optimal hole doping) among cuprates at ambient pressure $P_{\mathrm{amb}}$. $T_c^{\mathrm{opt}}$ increases under pressure $P$ and reaches $164$ K at $P_{\mathrm{opt}}\simeq 30$ GPa, then decreases with increasing $P>P_{\mathrm{opt}}$. To understand the microscopic origin of this dome-like $P$ dependence of $T_c^{\mathrm{opt}}$, we consider the $ab$ $initio$ low-energy effective Hamiltonian (LEH) for the antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma }$ band. In the AB LEH for cuprates with $N_{\ell }\le 2$ laminated CuO${}_2$ planes between block layers, it was proposed that $T_c^{\mathrm{opt}}\simeq 0.16|t_1|F_{\mathrm{SC}}$, where $t_1$ is the nearest neighbor hopping and the SC order parameter $F_{\mathrm{SC}}$ at optimal hole doping mainly depends on $u=U/|t_1|$ ($U$ is the onsite effective Coulomb repulsion): $F_{\mathrm{SC}}$ is maximal at $u_{\mathrm{opt}}\simeq 8.5$ and decreases sharply with decreasing $u$ for $u<u_{\mathrm{opt}}$. In this paper, we show that (I) $|t_1|$ increases with $P$, whereas (II) $u$ decreases with $P$ in the AB LEH of Hg1223. Based on (I,II), the dome-like $P$ dependence of $T_c^{\mathrm{opt}}$ emerges if we assume (A) Hg1223 with $N_{\ell }=3$ obeys $T_c^{\mathrm{opt}}\simeq 0.16|t_1|F_{\mathrm{SC}}$, and (B) $u\gtrsim u_{\mathrm{opt}}$ at $P_{\mathrm{amb}}$ and $u\simeq u_{\mathrm{opt}}$ at $P_{\mathrm{opt}}$ by including expected corrections to our $ab$ $initio$ calculation. The consequence of (A,B) is the following: At $P<P_{\mathrm{opt}}$, the increase in $T_c^{\mathrm{opt}}$ is accounted for by (I) whereas $F_{\mathrm{SC}}$ is insensitive to (II) and hence to $P$. At $P>P_{\mathrm{opt}}$, the decrease in $T_c^{\mathrm{opt}}$ is accounted for by (II): The rapid decrease in $F_{\mathrm{SC}}$ with decreasing $u$ for $u<u_{\mathrm{opt}}$ dominates over (I). We support (A,B) based on studies of other cuprates.

23 pages, 15 figures, 5 tables. The Supplemental Material contains the complete list of parameters for selected effective Hamiltonians that are presented in the main text and in Appendix C. This Supplemental Material may be obtained from the authors upon request

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