Effect of interlayer processes on the superconducting state within the $t-J-U$ model: Full Gutzwiller wave-function solution and relation to experiment

The Gutzwiller wave-function solution of the $t-J-U$ model is considered for the bilayer high-$T_C$ superconductor by using the so-called diagrammatic expansion method. The focus is on the influence of the interlayer effects on the superconducting state. The chosen pairing symmetry is a mixture of $d_{x^2-y^2}$ symmetry within the layers and the so-called $s^{\pm }$ symmetry for the interlayer contribution. The analyzed interlayer terms reflect the interlayer electron hopping, the interlayer exchange coupling, and the interlayer pair hopping. The obtained results are compared with selected experimental data corresponding to the copper-based compound Bi-2212 with two Cu-O planes in the unit cell. For the sake of comparison, selected results for the case of the bilayer Hubbard model are also provided. This paper complements our recent results obtained for the single-plane high temperature cuprates [cf. J. Spałek, M. Zegrodnik, and J. Kaczmarczyk, Phys. Rev. B 95, 024506 (2017)].

Figure 1

(a), (b) Dimensionless intra- (${\mathrm{\Delta }}_G^{\left|\right|}$) and interlayer (${\mathrm{\Delta }}_G^{\perp }$) correlated gap parameters, both as a function of doping for $t_z=0$, and $t_{\mathrm{bs}}$ varying between 0.0 and $-0.75$, with the step $-0.25$. For $t_{\mathrm{bs}}=0$ the interlayer gap is zero in the whole doping range. (c), (d) the intra- and interlayer correlated gap parameters, respectively, as a function of doping for $t_{\mathrm{bs}}=0$ and $t_z$ varying between 0.0 and $-0.45$ with the step $-0.15$. Similarly as above, the interlayer gap is zero for the case with $t_z=0$. For nonzero $t_z$ both gaps vanish at the same doping. The results correspond to $J^{\perp }=0$.

Figure 2

(a) Intra- and interlayer correlated gap parameters as a function of doping for two selected values of $J^{\perp }$, for $t_z=-0.3$ and $t_{\mathrm{bs}}=0$. (b), (c) The intra- and interlayer correlated gap parameters, respectively, as a function of the interlayer exchange coupling for the selected value of doping $\delta =0.2$ which is close to optimal doping. The effect of $J^{\perp }$ is minor and thus the principal single-plane features of the SC state are preserved.

Figure 3

Intralayer (a) and interlayer (b) correlated gaps as a function of doping for $t_z=-0.3,t_{\mathrm{bs}}=0,J^{\perp }=0$ and for selected values of the pair hopping tunelling magnitude $U^{″}$ varying between 0.0 and $-0.9$ with the step $-0.3$, assuming $U^{\prime }=-2U^{″}$. The substantial value of $U^{″}\le -0.3\left|t\right|$ does not reproduce the experimental phase diagram with ${\mathrm{\Delta }}_G=0$ for $\delta \gtrsim 0.4$.

Figure 4

(a) Fermi surface sheets close to the optimal doping. (b) Theoretical (solid lines) and experimental (circles) dispersion relations in the underdoped regime. The experimental data have been taken from Ref. [2]. (c) The nodal bilayer splitting in energy (top) and in the reciprocal space (bottom), both as a function of doping. The shaded regions correspond roughly to the measured values reported in Ref. [2]. (d) Nodal Fermi velocity as a function of doping corresponding to the bonding and antibonding bands. The gray (solid) line corresponds to the measured value taken from Ref. [2] which refers to the bonding band. (e) Fermi momenta corresponding to the bonding and antibonding bands, both as a function of doping. (f) Intralayer correlated gap magnitude as a function of doping. The presented fittings have been obtained for the following values of the model parameters: $t^{\prime }=0.36,t_z=-0.15,t_{\mathrm{bs}}=-0.06,U=13.5,J=0.25$ (all in units of $\left|t\right|$).

Figure 5

(a) Nodal bilayer splitting in energy as a function of doping for different values of the interlayer exchange coupling varying between $J^{\perp }=0.0$ and $J^{\perp }=0.25$ with the step 0.05. (b) Nodal Fermi velocity as a function of doping corresponding to the bonding and antibonding bands for two values of the interlayer exchange coupling.

Figure 6

Intra- (a) and interlayer (b) correlated gap vs doping for $t_z=0$ for $t_{\mathrm{bs}}$ varying between 0.0 and 0.75 with the step 0.25. For $t_{\mathrm{bs}}=0$ the interlayer gap is zero in the whole doping range. The data correspond to the Hubbard model with $U=14$ and ($U^{\prime }=U^{″}=0$).

Figure 7

Correlated gaps for the bilayer Hubbard model in the first (red solid line) and second (blue solid line) layers vs doping for different values of Coulomb repulsion in the second layer, $U_2$ (the explicit values are provided in the figures). The value $U_1$ corresponding to the Coulomb repulsion in the first layer remains constant and equal to $U_1=14$ while the interlayer electron hopping parameters are set to $t_{\mathrm{bs}}=-0.1$ and $t_z=0$. In (a) we have $U_1=U_2$ and both correlated gaps have the same values marked by blue line, while in (f) the value of $U_2$ is so low that the correlated gap in the second layer is practically zero. The data correspond to no interlayer pair hopping term ($U^{\prime }=U^{″}=0$).

Figure 8

Intralayer correlated gap (a), the number of carriers per atomic site (b), and double occupancy probability (c) in the first (blue line) and second (red line) atomic layer, all as a function of Coulomb repulsion in the second layer. For (a), (b), and (c) the interlayer pair hopping is set to zero $U^{″}=U^{\prime }=0$. (d) Intralayer correlated gaps as functions of interlayer pair hopping for $U_2=0$. We keep $U^{\prime }=-2U^{″}$ fulfilled. For (a), (b), (c), and (d) the Coulomb repulsion in the first layer is set to $U_1=14$ and the interlayer electron hopping parameters are $t_{\mathrm{bs}}=-0.1$ and $t_z=0$, while the selected value of doping is $\delta =0.21$.