Coulomb enhancement of superconducting pair-pair correlations in a $\frac{3}{4}$-filled model for $\kappa \text{−}\left(\text{BEDT-TTF}\right){}_{2}X$

We present the results of precise correlated-electron calculations on the monomer lattices of the organic charge-transfer solids $\kappa \text{−}(\text{BEDT-}{\mathrm{TTF})}_{2}X$ for 32 and 64 molecular sites. Our calculations are for band parameters corresponding to $X=\text{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]\mathrm{Cl}$ and ${\mathrm{Cu}}_2({\mathrm{CN})}_3$, which are semiconducting antiferromagnetic and quantum spin liquid, respectively, at ambient pressure. We have performed our calculations for variable electron densities $\rho$ per BEDT-TTF molecule, with $\rho$ ranging from 1 to 2. We find that $d$-wave superconducting pair-pair correlations are enhanced by electron-electron interactions only for a narrow carrier concentration about $\rho =1.5$, which is precisely the carrier concentration where superconductivity in the charge-transfer solids occurs. Our results indicate that the enhancement in pair-pair correlations is not related to antiferromagnetic order, but to a proximate hidden spin-singlet state that manifests itself as a charge-ordered state in other charge-transfer solids. Long-range superconducting order does not appear to be present in the purely electronic model, suggesting that electron-phonon interactions also must play a role in a complete theory of superconductivity.

Figure 1

Lattice structure of $\kappa -({\mathrm{ET})}_{2}X$, showing individual monomer molecules. In order of decreasing magnitude, $b_1,b_2,p$, and $q$ label the intermolecular hopping integrals. Superconducting pairs are constructed from the shaded molecules numbered $1\text{–}8$ as detailed in Section 3b. The a and c crystal axes for $\kappa$-Cl are indicated; for $\kappa$-CN the corresponding axes are conventionally labeled b and c. The x and y axes for the effective dimer model are indicated by dashed arrows.

Figure 2

Dimer spin structure factor $S\left(\mathbf{q}\right)$ as a function of wave vector calculated using PIRG for $\rho =1.5$. Wave vectors are defined in terms of the effective dimer lattice with axes x and y as shown in Fig. 1. Panels are (a) 32 sites, $\kappa$-Cl, (b) 32 sites, $\kappa$-CN.

Figure 3

Pairing symmetries considered in our calculations. Each ellipse surrounding two sites indicates the location of a singlet in the superposition for the pair operator ${\mathrm{\Delta }}^{†}$. Blue, solid (red, dashed) singlets have opposite signs.

Figure 4

The enhancement factor ${\mathrm{\Theta }}_P$ for the long-range component of the pair-pair correlation (${\mathrm{\Theta }}_P>0$ implies pair-pair correlations enhanced over their $U=0$ values, see text), as a function of $\rho$, for the $\kappa$-Cl system for $U$=0.5 eV. Pair symmetries are (a) ${\mathrm{d}}_1$, (b) ${\mathrm{d}}_2$, (c) ${\mathrm{d}}_3$, and (d) ${\mathrm{d}}_4$ as defined in Fig. 3. Shaded (striped) bars are for 32 (64) site lattices. The symbols ‘*’ and ‘#’ indicate densities not shown, for 32 and 64 sites, respectively; finite-size effects are particularly strong at these excluded $\rho$. Pair-pair correlations are suppressed by $U$ at these excluded $\rho$, precluding pairing; see Supplemental Material [44].

Figure 5

Same as in Fig. 4, but for parameters for $\kappa$-CN. As in Fig. 4, behavior of all pair-pair correlations against $U$, including those for the excluded $\rho$ are shown in the Supplemental Material [44]. See text regarding the peak in panel (b) at $\rho \approx 1.16$; we believe the apparent enhancement here is a finite-size effect.

Figure 6

$U$ dependence of $\overline{P}$ for densities showing significant enhancement of pair-pair correlations, for (a) $\kappa$-Cl, 32 sites, (b) $\kappa$-CN, 32 sites, (c) $\kappa$-Cl, 64 sites, and (d) $\kappa$-CN, 64 sites. all panels open (filled) symbols were calculated with PIRG (CPMC).