Superconducting dome in doped quasi-two-dimensional organic Mott insulators: A paradigm for strongly correlated superconductivity

Layered organic superconductors of the BEDT family are model systems for understanding the interplay of the Mott transition with superconductivity, magnetic order, and frustration, ingredients that are essential to understand superconductivity also in the cuprate high-temperature superconductors. Recent experimental studies on a hole-doped version of the organic compounds reveals an enhancement of superconductivity and a rapid crossover between two different conducting phases above the superconducting dome. One of these phases is a Fermi liquid, the other not. Using plaquette cellular dynamical mean field theory with state-of-the-art continuous-time quantum Monte Carlo calculations, we study this problem with the two-dimensional Hubbard model on the anisotropic triangular lattice. Phase diagrams as a function of temperature $T$ and interaction strength $U/t$ are obtained for anisotropy parameters $t^{\prime }=0.4t,t^{\prime }=0.8t$ and for various fillings. As in the case of the cuprates, we find, at finite doping, a first-order transition between two normal-state phases. One of theses phases has a pseudogap while the other does not. At temperatures above the critical point of the first-order transition, there is a Widom line where crossovers occur. The maximum (optimal) superconducting critical temperature $T_c^m$ at finite doping is enhanced by about 25% compared with its maximum at half filling and the range of $U/t$ where superconductivity appears is greatly extended. These results are in broad agreement with experiment. Also, increasing frustration (larger $t^{\prime }/t)$ significantly reduces magnetic ordering, as expected. This suggests that for compounds with intermediate to high frustration, very light doping should reveal the influence of the first-order transition and associated crossovers. These crossovers could possibly be even visible in the superconducting phase through subtle signatures. We also predict that destroying the superconducting phase by a magnetic field should reveal the first-order transition between metal and pseudogap. Finally, we predict that electron doping should also lead to an increased range of $U/t$ for superconductivity but with a reduced maximum $T_c$. This work also clearly shows that the superconducting dome in organic superconductors is tied to the Mott transition and its continuation as a transition separating pseudogap phase from correlated metal in doped compounds, as in the cuprates. Contrary to heavy fermions for example, the maximum $T_c$ is definitely not attached to an antiferromagnetic quantum critical point. That can also be verified experimentally.

Figure 1

Schematic generalized normal-state phase diagram for the cuprates and the layered organics, in the limit of low temperature neglecting broken-symmetry phases. The yellow surface represents the Mott-insulating phase at zero hole doping $\delta =0$, or half filling $n=1$. Leaving the Mott-insulating phase by increasing doping yields a second-order transition to a pseudogap phase up to the magenta surface where a first-order transition to a correlated metal occurs (at least close to half filling). That first-order transition extends from the first-order Mott metal-insulator transition that occurs at the boundary of the yellow region. Cuprate superconductors are found in the red region at negative $t^{\prime }/t$. The half-filled layered organics are found for different $t^{\prime }/t$ along the blue regions in the zero-hole-doping plane. They are strongly influenced by pressure, whose effect is represented vertically, although the ratio $t^{\prime }/t$ will generally also be influenced by pressure. A doped layered organic is found in the green region. It allows one to expand the analogy between organics and cuprates. The organic and cuprate lattices are different, as illustrated respectively by the left and right insets. Nevertheless, the physics of interactions, frustration, and doping is present in both types of compounds. Superconductivity and antiferromagnetism are broken-symmetry phases that are strongly influenced by the underlying normal state illustrated by this figure. In particular, the superconducting phase has a maximum $T_c$ in the vicinity of the boundary of the magenta region.

Figure 2

Periodic partitioning of the anisotropic triangular lattice into $2\times 2$ frustrated square clusters for this work using CDMFT.

Figure 3

Double occupancy $D_{occ}$ as a function of pressure (bottom horizontal axis) or interaction strength $U/t$ (top horizontal axis) for fixed filling, $n=0.99$. The value $t=0.044$ eV is used to convert to physical units [63]. The lower horizontal axis is labeled $t/U$ to suggest the pressure dependence, but the numbers on that horizontal axis are given by the value of $1/U$ expressed in electron volts using the above conversion factor. At $T=t/60$ there is a first-order hysteresis region: the brown squares are obtained from the insulating solution and the blue squares for the conducting solution. At $T=t/12$, there is no hysteresis, only an inflection point that determines the Widom line.

Figure 4

The imaginary part of the Matsubara Green's function $ImG\left(i{\omega }_n\right)$ plotted as a function of Matsubara frequency gives information about the density of states at the Fermi level, $-2ImG(i{\omega }_n\to i\eta )$ with $\eta \to 0$. The behavior differs depending on the value of $U/t$ . The decrease towards zero of the density of states at larger $U/t$ indicates a pseudogap, while its increase at smaller $U/t$ indicates a metallic phase.

Figure 5

Phase diagrams for the Hubbard model on the anisotropic triangular lattice with $t^{\prime }=0.4t$ for various fillings. The antiferromagnetic state has been studied for the half-filled case and for 10% doping. The value $t=0.044$ eV is used to convert to physical units [63]. The lower horizontal axis is labeled $t/U$ to suggest the pressure dependence, but the numbers on that horizontal axis are given by the value of $1/U$ expressed in electron volts using the above conversion factor. The same convention is used throughout the paper. Lines are guides to the eye. (a) Phase diagram for the half-filled case. In the blue region, the Mott insulator and the metallic state coexist. At the blue points, we find a first-order jump in the normal-state double occupancy. The lines between the points can be identified as the spinodal lines. $d$-wave superconductivity occurs in the orange region: Decreasing pressure, we find a first-order jump of the SC order parameter to zero at the orange points while upon increasing pressure we find a second-order transition. The AFM phase occurs below the red line that interpolates between the triangles where we find the Néel second-order transition. (b) Phase diagram for the 1%-hole-doped case. The colors have the same meaning as for the half-filled case, except that in the blue region two different conducting states are found instead of a metallic and an insulating state like at half filling. First-order jumps are observed at the blue points. There is coexistence in the blue region. Also, the red dots connected to the blue region indicate a strong crossover (Widom line) between a pseudogap state at small pressure (large $U/t)$ and a metallic state at large pressure (small $U/t)$. Orange points denote where we detect a second-order transition from the SC state to the normal state. (c) Phase diagram for the 10%-hole-doped case. The transitions between the normal and SC state (orange circles) are second order. The AFM phase is between the red lines. These lines interpolate between the second-order Néel transition that we find where the triangles are located. (d) Phase diagram for the 10%-electron-doped Hubbard model. The brown dot and line are extrapolations. The transitions between the normal and SC state (orange circles) are second order.

Figure 6

Superconducting $T_c^m$ as a function of filling and $U/t$ projected in the T-n plane. The actual values of $T,U,n$ are listed in Table 2.

Figure 7

Superconducting phase diagram combining in T-U-n space the constant-doping T-U planes of Fig. 5. The line of $T_c^m$ in Fig. 6 also appears on this plot.

Figure 8

Order parameter $\left(d_{SC}\right)$ and uniform susceptibility $\left({\chi }_z\right)$ for the superconducting phase of the Hubbard model on the anisotropic triangular lattice with $t^{\prime }=0.4t$. Each filling has its specific color and plot marker in the various panels of this figure and those of Fig. 11. Lines are guides to the eye. (a) $d_{SC}$ for the half-filled case. Two different temperatures are plotted. (b) $d_{SC}$ for various fillings. The temperature is $T/t=1/60$. The dashed purple line is an extrapolation. (c) $d_{SC}$ (green diamonds) and ${\chi }_z$ (blue squares) for the 1%-hole-doped case. ${\chi }_z$ is divided by 100 to fit on the same vertical axis. $U_{MMT}$ denotes the vicinity of interaction strengths where the pseudogap to metal first-order transition discussed in Sec. 4 is found. $U_c^m$ stands for the critical interaction strength associated with $T_c^m$, the maximum $T_c$. (d) $d_{SC}$ (orange triangles) and ${\chi }_z$ (blue squares) for the 10%-hole-doped case. ${\chi }_z$ is divided by 100 to fit on the same vertical axis. $U_c^m$ has the same significance as in (c). The orange curve for $d_{SC}$ is the same as in (b).

Figure 9

Phase diagrams for the Hubbard model on the anisotropic triangular lattice. $\beta$ is the inverse temperature in $1/t$ units. The value $t=0.044$ eV is used to convert to physical units [63]. Lines are guides to the eye. (a) Phase diagram at half filling for an anisotropy parameter $t^{\prime }=0.4t$. This figure is the same as Fig. 5 except that it shows the complete antiferromagnetic phase. (b) Phase diagram at half filling for an anisotropy parameter $t^{\prime }=0.8t$. This figure is the same as Fig. 10 except that it shows the complete magnetic phase.

Figure 10

Phase diagrams for the Hubbard model on the anisotropic triangular lattice with anisotropy parameter $t^{\prime }=0.8t$ and two fillings. The antiferromagnetic state was studied at half filling and also for $n=0.9$. In the latter case, no AFM was found for $T$ down to $1/60$ and $8<U/t<30$. The value $t=0.044$ eV is used to convert to physical units [63]. Lines are guides to the eye. (a) Phase diagram for the half-filled case. Metallic and insulating phases coexist in the blue region. A first-order jump in the normal-state double occupancy is found at the blue points. The value of the critical point for the Mott transition is $\left(U/t=7.93,\beta =10.00\right)$. With increasing pressure, we find at the red triangles that the Néel AFM order parameter disappears (to the right of the red line) through a first-order jump for $\beta >10$ and through a second-order transition for $\beta <10$. When pressure is decreased, the paramagnetic metal is unstable to the AFM insulator along the green line through a first-order jump $\left(\beta >10\right)$ or through a second-order transition $\left(\beta <10\right)$. The orange region is where superconductivity manifests itself. The SC state gives way to the insulating phase along a first-order jump upon decreasing pressure, and to the metallic phase along a second-order transition line upon increasing pressure. (b) Phase diagram for the 10%-hole-doped case. The transition from the SC state to the normal state is second order.

Figure 11

The order parameter $\left(d_{SC}\right)$ and uniform susceptibility $\left({\chi }_z\right)$ in the superconducting phase of the Hubbard model on the anisotropic triangular lattice with frustration value $t^{\prime }=0.8t$. Each doping has the same specific color and plot marker as in Fig. 8. Lines are guide to the eye. (a) $d_{SC}$ for the half-filled case. Two different temperatures are plotted. (b) $d_{SC}$ for various fillings and ${\chi }_z$ for the 10%-hole-doped case. The temperature is $T/t=1/60$. $U_c^m$ stands for the superconducting critical interaction strength associated with $T_c^m$. The spin susceptibility ${\chi }_z$ calculated with Eq. (6) in dimensionless units is divided by 200 to fit on the same vertical axis.

Figure 12

Schematic phase diagram for layered organics. Orange planes indicate the three-dimensional region where superconductivity is present. The antiferromagnetic three-dimensional dome is below the red lines. In the normal state, there is a surface of first-order transition whose coexistence region is delimited by the blue lines. This first-order transition separates a pseudogap phase with small double occupancy from a correlated metal with larger double occupancy, by analogy with Ref. [35]. At larger $U$ and doping, the sign problem prevents us from observing directly the first-order transition or its continuation as a different phenomenon. Frustration can appreciably displace and eventually completely eliminate the antiferromagnetic region [73, 107].