Fulde-Ferrell-Larkin-Ovchinnikov state induced by antiferromagnetic order in $\kappa$-type organic conductors

We theoretically investigate superconductivity under a spin-split band structure owing to a collinear-type antiferromagnetic order in quasi-two-dimensional organic compounds $\kappa$-(BEDT-${\mathrm{TTF})}_{2}X$. We find that the magnetic order can induce a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, where the Cooper pair possesses a finite center-of-mass momentum. We show this from two types of analyses: (1) an effective model where simple intraband attractive interactions are assumed and (2) many-body calculations of the repulsive Hubbard model based on the fluctuation-exchange approximation and the linearized Eliashberg equation. Our results show the possibility of realizing the FFLO state without applying an external magnetic field.

Figure 1

Symmetry properties and schematic energy band structures in (a) magnetic quadrupole, (b) FM (or Zeeman field), and (c) magnetic octupole states. The red and blue dispersions represent (pseudo)spin up and down bands, respectively. The green dash-dotted lines are the Fermi surface in the paramagnetic state.

Figure 2

(a) Schematic view of the molecular arrangement in the conducting layer of $\kappa$-(${\mathrm{ET})}_{2}X$. The unit cell (green rectangle) contains four ET molecules labeled by $A$–$D$. The bonds with large hoppings are shown: $a$ (orange bold), $b$ (dotted), $p$ (solid), and $q$ (dashed). The red and blue arrows represent up and down spin moments, respectively, in the AFM phase. The noninteracting band structures for (b) $h=0$ and (c) $h=0.2$ eV. Energy bands of up- and down-spin electrons are represented by the red and blue lines, respectively. The green dashed lines show the Fermi levels for the electron density $n=1.3$, 1.5, and 1.7 per ET molecule. The inset in (b) represents the path in the 2D Brillouin zone. Note that, in the orthorhombic wallpaper group $p2gg, k_x=k_y=\pi$ is labeled by $S$, which corresponds to the $M$ point in previous studies [15, 16, 17, 77].

Figure 3

Fermi surfaces and $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ for (a) $(n,h)=(1.7,0)$, (b) $(n,h)=(1.7,0.2)$, (c) $(n,h)=(1.3,0)$, and (d) $(n,h)=(1.3,0.2)$. The temperature $T$ is set to 1 meV. The red solid (blue dashed) lines represent Fermi surfaces of the up-spin (down-spin) electrons, which are degenerate for $h=0$. The $A_2$-type ($d_{xy}$-wave) interaction parameters are used in the calculations of $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ for all cases. In the paramagnetic phase [(a), (c)], the maximum of $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ is located at $\mathit{Q}=\mathit{0}$. In the AFM phase [(b), (d)], $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ has peaks at finite momenta indicating the FFLO state.

Figure 4

Phase diagrams obtained from the analysis of the SC susceptibility for (a) $A_1$ and (b) $A_2$ interaction parameters. The temperature $T$ is set to 1 meV. The closed (open) circles indicate that the largest eigenvalue of $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ is larger (smaller) than unity. The color of each circle represents the norm of the optimal COM momentum $Q_{\text{opt}}=\left|{\mathit{Q}}_{\text{opt}}\right|$. The direction of ${\mathit{Q}}_{\text{opt}}$ is represented by gray arrows in the circles.

Figure 5

The renormalized Fermi surfaces and ${\chi }_{\text{LAFM}}\left(\mathit{q}\right)$ or ${\chi }_{\text{TAFM}}\left(\mathit{q}\right)$ for (a) $(n,h)=(1.7,0)$, (b) $(n,h)=(1.7,0.2)$, (c) $(n,h)=(1.3,0)$, and (d) $(n,h)=(1.3,0.2)$. The orange solid (green dashed) lines represent the Fermi surfaces of the up-spin (down-spin) electrons taking into account the self-energy, while the red solid (blue dashed) lines show the up-spin (down-spin) noninteracting Fermi surfaces. The gray arrows in (b) and (d) are the $\mathit{q}$ vector where the susceptibility reaches maximum, which corresponds to the nesting of the Fermi surfaces.

Figure 6

(a) Phase diagram on the $(h,n)$ plane obtained from the analysis of the Hubbard model on the $\kappa$-type structure. The temperature $T$ is set to 1 meV. The squares (circles) indicate that the stable order parameter belongs to the $A_1$ ($A_2$) irrep. The color of each point represents the norm of the optimal COM momentum $Q_{\text{opt}}=\left|{\mathit{Q}}_{\text{opt}}\right|$. (b), (c) The momentum dependence of the eigenvalue ${\lambda }_{\mathit{Q}}$ of the Eliashberg equation in the hole-doped regime ($n=1.3$) for $h=0$ and $h=0.2$, respectively. The eigenvalue has peaks at finite COM momenta in (c).

Figure 7

The LFM spin susceptibility defined for $n=1.5$ and $h=0$ in (a) convention 1 [Eq. (A18)] and (b) convention 2 [Eq. (A19)]. In (b), the susceptibility breaks the glide symmetries $G_x$ and $G_y$.

Figure 8

Results in the undoped case ($n=1.5$) for (a) $h=0$ eV and (b) $h=0.1$ eV. The temperature $T$ is set to 1 meV. Left panels: The renormalized Fermi surfaces (orange solid and green dashed lines) and the noninteracting Fermi surfaces (red solid and blue dashed lines). Middle and right panels: $X_{\text{SC}}^{\left(0\right)}\left(\mathit{Q}\right)$ and the LAFM spin susceptibility.

Figure 9

The molecular field $h$ dependence of the AFM fluctuations for each filling. The maxima of the LAFM and TAFM spin susceptibilities are shown. Both are degenerate for $h=0$ due to the absence of the spin–orbit coupling.