Coexistence of superconductivity and antiferromagnetism in a self-doped bilayer $t\text{−}{t}^{\prime }\text{−}J$ model

A self-doped bilayer $t\text{−}{t}^{\prime }\text{−}J$ model of an electron- and a hole-doped planes is studied by the slave-boson mean-field theory. In our model, a renormalized interlayer hopping connects the differently doped planes, which is generated by a site potential. We find coexistent phases of antiferromagnetic (AFM) and superconducting orders, although the magnitudes of order parameters become more dissimilar in the bilayer away from half-filling. Fermi surfaces (FS’s) with the AFM order show two pockets around the nodal and the antinodal regions. These results can be interpreted as a coexistence of electron- and hole-doped Fermi pockets in the bilayer. In the nodal direction, the FS splitting is absent even in the bilayer system, since one band has a gap due to the AFM order.

Figure 1

(Color online) The order parameters in the undoped case with $t^{\prime }/t=-0.4$. The antiferromagnetic and the superconducting orders coexist in both planes. The following properties in electron- and hole-doped planes are plotted as functions of holon (doublon) density; (a) site potential, $W$, (b) $d$-wave pairing amplitude, ${\Delta }^{\left(l\right)}$, (c) the uniform bond order, ${\chi }^{\left(l\right)}$, (d) AFM order parameter, $m^{\left(l\right)}$, and (e) magnitude of interlayer hopping, ${\Delta }_p$. The superscript (1) refers to the hole-doped layer and (2) is the electron-doped layer. Four lines in each figure correspond to different value of interlayer hopping given by, $t_{\perp }/t=0.0$ (full square), $t_{\perp }/t=0.2$ (empty circle), $t_{\perp }/t=0.5$ (empty triangle), $t_{\perp }/t=0.8$ (full triangle).

Figure 2

(Color online) The order parameters in the doped case with $t^{\prime }/t=-0.4$ and $W/t=0.05$. The following properties in electron- and hole-doped planes are plotted as functions of total electron number $n$; (a) holon (doublon) density, (b) $d$-wave pairing amplitude, ${\Delta }^{\left(l\right)}$, (c) the uniform bond order, ${\chi }^{\left(l\right)}$, and (d) AFM order parameter, $m^{\left(l\right)}$. Four lines in each figure correspond to different value of interlayer hopping given by, $t_{\perp }/t=0.0$ (full square), $t_{\perp }/t=0.2$ (empty circle), $t_{\perp }/t=0.5$ (empty triangle), $t_{\perp }/t=0.8$ (full triangle).

Figure 3

(Color online) The phase diagram in the $W\text{−}n$ plane for $t_{\perp }/t=0.5$. Each abbreviation means the followings: AFM, the antiferromagnetic phase; SC, $d$-wave superconducting phase; $\text{AFM}+\text{SC}$, the coexistent phase of AFM and SC. For large $n$, both planes are hole doped. Different lines correspond to different values of $t^{\prime }/t$ as, $-0.2$, $-0,3$ and $-0.4$ from bottom to top.

Figure 4

(Color online) FS’s and dispersion relations of self-doped bilayer system; (a) and (e) $n=0.98$, ${\delta }_h^{\left(1\right)}=0.130$, ${\delta }_d^{\left(2\right)}=0.090$; (b) and (f) $n=0.95$, ${\delta }_h^{\left(1\right)}=0.180$, ${\delta }_d^{\left(2\right)}=0.080$; (c) and (g) $n=0.90$, ${\delta }_h^{\left(1\right)}=0.225$, ${\delta }_d^{\left(2\right)}=0.025$; (d) and (h) $n=0.85$, ${\delta }_h^{\left(1\right)}=0.25$, ${\delta }_h^{\left(2\right)}=0.05$. In (e)–(h), the band dispersions are along the high-symmetry directions of the Brillouin zone, $\left(0,0\right)\text{−}\left(\pi ,0\right)\text{−}\left(\pi /2,\pi /2\right)\text{−}\left(0,0\right)$. In these figures, $W/t=0.05$ and $t_{\perp }/t=0.5$ are fixed, and SC order parameters are imposed to be zero. In (d) and (h), both two planes are in hole-doped regions.