Near-degeneracy of extended $s+d_{x^2-y^2}$ and $d_{xy}$ order parameters in quasi-two-dimensional organic superconductors

The symmetry of the superconducting order parameter in quasi-two-dimensional bis-ethylenedithio-tetrathiafulvalene (BEDT-TTF) organic superconductors is a subject of ongoing debate. We report ab initio density-functional-theory calculations for a number of organic superconductors containing $\kappa$-type layers. Using projective Wannier functions, we derive the parameters of a common low-energy Hamiltonian based on individual BEDT-TTF molecular orbitals. In a random-phase approximation spin-fluctuation approach, we investigate the evolution of the superconducting pairing symmetry within this model, and we point out a phase transition between extended $s+d_{x^2-y^2}$ and $d_{xy}$ symmetry. We discuss the origin of the mixed order parameter and the relation between the realistic molecule description and the widely used dimer approximation. Based on our ab initio calculations, we position the investigated materials in the obtained molecule model phase diagram, and we simulate scanning tunneling spectroscopy experiments for selected cases. Our calculations show that many $\kappa$-type materials lie close to the phase-transition line between the two pairing symmetry types found in our calculation, possibly explaining the multitude of contradictory experiments in this field.

Figure 1

(a) Molecule model of the $\kappa \text{−}{\left(\mathrm{ET}\right)}_2^{+}$ layer. Individual ET molecules are represented by white and dark gray ellipsoids. The four dominant hopping integrals are $\left(t_1,t_2,t_3,t_4\right)$. Note the pronounced asymmetry in magnitude between $t_2$ and $t_4$, which is indicated here by different line thicknesses. (b) Dimer-approximated $\kappa \text{−}{\left(\mathrm{ET}\right)}_2^{+}$ layer. Two molecules are contracted into one dimer site indicated by a bold shaded circle. The intradimer hopping integral $t_1$ is integrated out, while $t_2$ and $t_4$ are averaged. Therefore, the dimer model is characterized by only two hopping parameters $\{t,t^{\prime }\}=\left\{\right(|t_2|+|t_4\left|\right)/2,|t_3|/2\}$. In both subfigures, the unit cell considered in our work is indicated by a bold magenta-colored line.

Figure 2

(a) The inner bold lines show the Brillouin zone and the Fermi surface of a generic $\kappa \text{−}{\left(\mathrm{ET}\right)}_{2}X$ material. The outer dashed lines show the Brillouin zone and the Fermi surface of the unfolded one-band dimer model. (b) $d_{xy}$ order parameter in the physical Brillouin zone. Nodes are located in the $x$ and $y$ directions. (c) $d_{x^2-y^2}$ order parameter in the unfolded Brillouin zone. Nodes are located along the Brillouin-zone diagonals. The different designation is only due to a rotation of the coordinate axes by ${45}^{\circ }$. (d) $d_{x^2-y^2}$ order parameter in the physical Brillouin zone. (e) $d_{xy}$ order parameter in the unfolded Brillouin zone.

Figure 3

Comparison of molecule models (a)–(d) with different ratios of $t_4/t_2$, which all correspond to the same dimer model (e) with $t^{\prime }/t=0.6$. The top panel shows the spin susceptibilities, where arrows ${\vec{q}}_1$ and ${\vec{q}}_2$ indicate the main features, while the bottom panel shows the leading eigenfunction of the superconducting gap equation on the Fermi surface. In the molecule models $t_3/t_1=0.333$ is fixed, while the ratio of $t_4/t_2$ is varied under the condition $t_3/(t_2+t_4)=t^{\prime }/t=0.6$.

Figure 4

(a) Spin susceptibility of the molecule model at $t_2/t_1=0.417,t_3/t_1=0.333$, and $t_4/t_1=0.139$, with arrows ${\vec{q}}_1$ and ${\vec{q}}_2$ indicating the main features. The data shown are the same as in Fig. 3. (b) Superconducting gap function on the Fermi surface for the same parameter values as in (a). Vectors ${\vec{q}}_1$ and ${\vec{q}}_2$ are the same as in (a) and connect parts of the Fermi surface with different signs of the gap. Note that the plot range of the Brillouin zone is shifted by a vector $(\pi ,\pi )$ compared to Fig. 3.

Figure 5

Hopping structure in the molecule model for the $\kappa \text{−}{\left(\mathrm{ET}\right)}_2^{+}$ layer. The dark gray arrows indicate the connection between hopping parameters and the symmetry functions appearing in the solution of the superconducting gap equation.

Figure 6

Superconducting phase diagram of the individual molecule model. Different symmetries of the superconducting order parameter are color-coded. A $d_{xy}$ symmetry of the pairing interaction is favored when the orthorhombicity of the system is small, i.e., when $t_4\lesssim t_2$ and $t_3\ll t_2$. In the rest of the phase diagram, an extended $s+d_{x^2-y^2}$ symmetry prevails. The numbered symbols correspond to the location of real materials in the phase diagram, enumerated as in Table 1. Materials were sorted into the subplot to which their true value of $t_4/t_1$ is closest.

Figure 7

Eigenvalues of the gap equation for $s_{\pm }+d_{x^2-y^2}$ and $d_{xy}$ pairing symmetries as a function of the in-plane anisotropy $t_4/t_2$. The isotropic case is realized for $t_4/t_2=1$. Only $t_4$ was varied. The other parameters were fixed to $t_2/t_1=0.5$ and $t_3/t_1=0.33$.

Figure 8

Gap function on the Fermi surface (top panel), magnitude of the gap function vs angle measured with respect to the $k_x$ direction (middle panel), and simulated quasiparticle density of states in the superconducting state (bottom panel). In all cases, we assumed an energy scale ${\mathrm{\Delta }}_0=10\mathrm{meV}$. Only $t_3/t_1$ is varied. Other parameters are fixed to $t_2/t_1=0.4375$ and $t_4/t_2=0.1$. Maxima of the superconducting gap magnitude are labeled with upper-case letters. Nodes of the superconducting order parameter are labeled with greek letters. All nodes and maxima not labeled explicitly are symmetry-equivalent to the labeled ones. Column (a) shows the case of $s_{\pm }+d_{x^2-y^2}$ symmetry with four nodes $\left(t_3/t_1=0.5\right)$. Column (b) shows the results for $s_{\pm }+d_{x^2-y^2}$ symmetry with eight nodes $\left(t_3/t_1=0.3475\right)$. Column (c) shows the $d_{xy}$ case $\left(t_3/t_1=0.25\right)$.