Colloquium: Modeling the unconventional superconducting properties of expanded $A_3{\mathrm{C}}_{60}$ fullerides

The trivalent alkali fulleride solids of generic composition $A_3{\mathrm{C}}_{60}$, where ${\mathrm{C}}_{60}$ is the fullerene molecule and $A=\mathrm{K}$, Rb, and Cs, are a well-established family of molecular superconductors. The superconductive electron pairing is of regular $s$-wave symmetry and is accounted for by conventional coupling of electrons to phonons, in particular by well-understood Jahn-Teller intramolecular ${\mathrm{C}}_{60}$ vibrations. A source of renewed interest in these systems is the surprising indication of strong electron-electron repulsion phenomena, which has emerged in compounds where the ${\mathrm{C}}_{60}\text{−}{\mathrm{C}}_{60}$ distance is expanded, by either a large cation size or other chemical or physical means. Several examples are now known where this kind of expansion, while leading to a high superconducting temperature at first, gradually or suddenly causes a decline of superconductivity and its eventual disappearance in favor of a Mott insulating state. This type of insulating state is the hallmark of strong electron correlations in cuprate and organic superconductors, and its appearance suggests that fullerides might also be members of that family. Our approach to fullerides is theoretical, and based on the solution of a Hubbard-type model, where electrons hop between molecular sites. In a Hubbard model of fullerides, unlike models for the strongly correlated cuprates, all important electron correlations occur within the molecular site, so it is efficiently soluble in the dynamical mean-field theory (DMFT) approximation. DMFT solutions confirm that superconductivity in this model fulleride, although of $s$-wave symmetry rather than $d$-wave, shares many of the properties that are characteristic of high-$T_c$ cuprates. The calculations are heavy, and while the working model used is several years old, the new results presented pertain to the interesting case of three electrons per ${\mathrm{C}}_{60}$ molecule, appropriate to $A_3{\mathrm{C}}_{60}$, and have become possible only recently due to a stronger computational effort. The zero-temperature phase diagram is calculated as a function of the ratio of intramolecular repulsion parameter $U$ to the electron bandwidth $W$, the increase of $U∕W$ representing the main effect of lattice expansion. The phase diagram is close to that of actual materials, with a dome-shaped superconducting order parameter region preceding the Mott transition for increasing cell volume. Unconventional properties of expanded fulleride superconductors predicted by this model include (i) an energy pseudogap in the normal phase; (ii) a gain of electron kinetic energy and of conducting Drude weight at the onset of superconductivity, as in high-$T_c$ cuprates; (iii) a spin susceptibility and a specific-heat behavior that are not drastically different from those of a regular phonon superconductor, despite strong correlations; and (iv) the emergence of more than one energy scale governing the renormalized single-particle dispersion, electronic entropy, and specific-heat jump. These predictions, which if confirmed should establish fullerides as members of the wider family of strongly correlated superconductors, are discussed in light of existing and foreseeable experiments.

Figure 1

(Color online) Schematic representation of a planar projection of the crystal structure of $\mathrm{N}{\mathrm{H}}_3{\mathrm{K}}_3{\mathrm{C}}_{60}$. Dots are potassium atoms, other dots surrounded by three gray dots are $\mathrm{N}{\mathrm{H}}_3$ molecules, and ${\mathrm{C}}_{60}$ molecules are shown according to their actual spatial orientation. Courtesy of Kosmas Prassides.

Figure 2

(Color online) Superconducting solution (triangles) and antiferromagnetic solution (squares) zero-frequency anomalous self-energies $\Delta \left(0\right)$ as a function of $U∕W$. Solid symbols are used when the corresponding symmetry-broken phase is stable, while open symbols are used when it is metastable, i.e., has higher energy than the other phase. The first-order transition between the two phases is indicated by a vertical line separating the superconductor from the antiferromagnetic Mott insulator. The case in the presence of a frustrating next-nearest-neighbor hopping $t^{\prime }=0.3t$, absent in (a).

Figure 3

(Color online) Superconducting solution (SC, left) and antiferromagnetic solution (AFM, right) anomalous self-energies (top) and order parameters (bottom) as functions of $U∕W$. The top panels also show (diamonds) the spectral gaps obtained on multiplying $\Delta$ by the quasiparticle weight $Z$ of each solution. The gaps are in units of the bandwidth $W$ (the order parameters are by definition dimensionless).

Figure 4

(Color online) Quasiparticle superconducting energy gap in units of the bandwidth $W\simeq 0.6\mathrm{eV}$ and on a larger scale than Fig. 3, computed through the anomalous self-energy [Fig. 2a] multiplied by the quasiparticle residue $Z\left(U\right)$. Note that, above $U∕W\simeq 1.1$, the superconducting solution is metastable since the antiferromagnetic one has lower energy; see Fig. 2. We note that the maximum gap $Z{\Delta }_c\simeq 0.045W\simeq 27\mathrm{meV}$.

Figure 5

(Color online) Anomalous self-energy $\Delta \left(0\right)$ (related to the superconducting gap by a factor $Z^{-1}$) for the superconducting solution and different values of the coupling parameter $J=0.02$, 0.05, 0.1, and 0.2, which correspond to $\lambda =0.09$, 0.21, 0.42, and 0.85, respectively. Note that for large $J$, corresponding to a case in which the Jahn-Teller coupling is not canceled by Hund’s rule exchange, superconductivity is strongest at $U=0$. For increasing cancellation (decreasing $J$), two separate superconducting pockets emerge: a BCS pocket near $U=0$ and a SCS pocket near the Mott insulator. When the cancellation is nearly complete, the SCS pocket is many orders of magnitude stronger than the BCS. This is the situation we propose in our model of fullerides. Similar physics was described for a simpler model by 13.

Figure 6

(Color online) Impurity spectral function with a rigid bath characterized by a flat density of states (no DMFT self-consistency) across the critical point: the curves from top to bottom correspond to the evolution from the Kondo screened regime toward the pseudogap regime. The dotted line is the spectral function deep inside the non-Fermi-liquid phase $J⪢T_K$. Inset: The same curves plotted on a larger energy range, where the pseudogap of the dotted line is visible. The Fermi-liquid behavior corresponds to the normal state of the SCS superconducting phase for $J<J_{\mathrm{cp}}$ (analogous to the overdoped regime of cuprates), whereas the pseudogap behavior corresponds to the SCS normal state for $J>J_{\mathrm{cp}}$ (analogous to the underdoped cuprates). From 22.

Figure 7

(Color online) Calculated weights of the zero-frequency delta-function contribution to the optical conductivity for the superconducting phase (superfluid stiffness, squares) and the normal phase (Drude weight, circles). The zero-frequency anomalous self-energy is also plotted. Note the higher values in the SCS superconductor relative to the nonsuperconducting metal phase.

Figure 8

(Color online) Energetic balance underlying superconductivity. $\Delta E_{\mathrm{pot}}=E_{\mathrm{pot}}^S-E_{\mathrm{pot}}^N$ is the difference between the potential energies of the superconducting and normal solutions, while $\Delta E_{\mathrm{kin}}=E_{\mathrm{kin}}^S-E_{\mathrm{kin}}^N$ is the same difference between the kinetic energies of the two solutions.

Figure 9

(Color online) Specific-heat jump, in units of $T_c{\gamma }_0$, as a function of $U∕W$. For comparison we also show the behavior of $1∕Z\left(U\right)$, which should correspond to the same quantity if Fermi-liquid theory is valid.

Figure 10

(Color online) Normal-phase uniform magnetic susceptibility $\chi$ normalized to the noninteracting value ${\chi }_0$. For $U=0$, $\chi \simeq {\chi }_0$, the difference being extremely small since $J⪡W$. The plateau between $U∕W=0.5$ and 1 signals the effective crossover from a Fermi liquid with $S=3∕2$ per site to a non-Fermi-liquid with $S=1∕2$ per site.

Figure 11

(Color online) Simulated photoemission spectra elaborated from the DMFT result in the normal phase. The left panel refers to $U∕W=1.1$ (corresponding to unexpanded fullerides), while the right panel is for $U∕W=1.3$ (corresponding to expanded fullerides).