Exotic s-wave superconductivity in alkali-doped fullerides

Figure 1 shows the crystal structure of the ${{A}_{3}}{{\text{C}}_{60}}$ systems. ${{\text{K}}_{3}}{{\text{C}}_{60}}$ and $\text{R}{{\text{b}}_{3}}{{\text{C}}_{60}}$ have the face-centered-cubic (fcc) structure [22–24]. $\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$ can be synthesized into either an fcc or an A15 structure [7, 9, 10]. In the fcc structure, the buckyballs are located at the fcc positions and the alkali atoms are intercalated in between the ${{\text{C}}_{60}}$ molecules. In the A15 structure, the ${{\text{C}}_{60}}$ molecules are located on the body-centered-cubic (bcc) lattice, while a unit cell contains two ${{\text{C}}_{60}}$ molecules with different orientations. In both structures, the intercalated alkali atoms donate electrons to the fullerene bands turning the semiconducting undoped ${{\text{C}}_{60}}$ solid into a metal [25].

Zoom In Zoom Out Reset image size

Figure 2 shows the most refined experimental phase diagrams for the (a) fcc and (b) A15 structures [7–10, 26]. In both cases, the horizontal axis is labelled by the volume ${{V}_{\text{C}_{60}^{3-}}}$ occupied per $\text{C}_{60}^{3-}$ anion in the actual solid structure. In the fcc case, ${{V}_{\text{C}_{60}^{3-}}}$ is controlled by physical and/or chemical pressure, where the latter is induced by changing the alkali species, with larger ions leading to larger lattice spacing between molecules. The phase diagram for the A15 structure was obtained by applying the physical pressure to $\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$ solid [8].

Zoom In Zoom Out Reset image size

Both phase diagrams show a superconducting phase with a high critical temperature, whose maximum reaches  ∼35 K and  ∼38 K for the fcc and A15 systems, respectively. Interestingly, the fcc and A15 phase diagrams share a similar shape of the ${{T}_{\text{c}}}$ dome as a function of ${{V}_{\text{C}_{60}^{3-}}}$ despite the lack of magnetic ordering in the Mott state for the frustrated fcc lattice. The symmetry of the superconducting order parameter was found to be of s-wave by various different experimental techniques [27–37] as in the whole family of alkali-doped fullerides. Also the suppression of the spin fluctuation in the superconducting state suggests a singlet pairing [38].

What makes the system more remarkable is the existence of the Mott insulating phase next to the s-wave superconducting phase [7–10]. This adjacency of the superconductivity and the Mott insulator is reminiscent of the cuprates [41], where the symmetry of the order parameter is established to be d-wave [42, 43]. In fullerides, this proximity is even more surprising because the s-wave superconductivity is believed to be fragile to the strong correlations, as opposed to the d-wave symmetry.

In the Mott phase, at low temperature, the antiferromagnetism has been observed in both fcc and A15 systems [8, 10, 26]. However, the Néel temperatures T_N are very different between the two systems. In the A15 systems, which are less frustrated than the fcc systems, the transition into the antiferromagnetic long range order with the wave vector $\mathbf{q}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ occurs at around 46 K [44]. In the fcc case, the magnetic instability is drastically suppressed to ${{T}_{\text{N}}}\sim 2$ K. Even below T_N, the magnetism is not purely long-range ordered: a specific heat measurement suggests the coexistence of the glass-like disordered magnetism and the antiferromagnetic order below T_N [45]. The suppression of magnetism and the complicated magnetic structure in the fcc systems are ascribed to the geometrical frustration and to disorder in the superexchange interactions driving the coupling between localized spins [45].

In 1990's (well before the discovery of the Mott phase and magnetism in $\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$ in 2008), the pairing mechanism had been often discussed based on the conventional phonon mechanism [14, 46–49]. Indeed, within the Migdal-Eliashberg theory [50], the theoretically and experimentally estimated electron–phonon coupling constant $\lambda \sim$0.5–1 [46–49, 51–67] and the high phonon frequency  ∼0.1 eV [46–49, 51–59, 61–63, 66–72] led to the prediction of a transition temperature comparable to the experimental value under the assumption that the Coulomb repulsion is reasonably weak. The phonon mechanism seemed consistent with the experimental observation of the Hebel–Slichter peak [31, 32] and the isotope effect with the exponent  ∼0.2–0.3 [73, 74], too⁶. The positive correlation between ${{T}_{\text{c}}}$ and the lattice constant [5, 78–81] also supported this scenario: when the lattice is expanded, the bandwidth decreases and hence the density of states (DOS) at the Fermi level increases. According to the Bardeen–Cooper–Schrieffer (BCS) theory, it leads to the increase of ${{T}_{\text{c}}}$ [4, 5].

On the other hand, the importance of the electron correlations has been argued since the early stage of the study. Auger spectroscopy measurements for the undoped ${{\text{C}}_{60}}$ solid lead to an estimate of  ∼1.6 eV for the effective intramolecular Coulomb interaction [82], which is bigger than the typical bandwidth of the low-energy bands  ∼0.5 eV. Furthermore, it has been argued that the retardation effect might be inefficient [17, 83–85]; because the electronic bands of the fullerides are distributed sparsely in energy, the density of states does not spread continuously any more [86–88] (section 3). Based on this fact, the typical electronic energy scale has been argued to be given by the bandwidth of the low-energy t_1u bands (∼0.5 eV), which are energetically isolated from the other bands. This leads to a large Coulomb pseudopotential, which challenges the conventional pairing mechanism [85]. There are also suggestions of purely electronic mechanism based on the resonating valence bond scenario [89–91]. Several works have taken into account both the electron correlations and the electron–phonon interactions to explain the superconductivity [18, 92–96].

The discovery of the Mott insulating phase in $\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$ strikingly confirmed the strength and the relevance of the electron–electron correlations [7–10, 26]. Through the metal-insulator transition, there is no structural transition [7, 8, 10]. Therefore, the alkali-doped fullerides provide a unique playground to study the s-wave superconductivity under the strong correlation.

This discovery has triggered both the experimental and theoretical studies [11, 19–21, 26, 88, 97–112]. As a result, various unusual properties in both metallic and insulating phases have been revealed. In the Mott phase, the size of the local spin per ${{\text{C}}_{60}}$ molecule was found to be S  =  1/2 (low-spin state) [8, 10], not S  =  3/2 (high-spin state) expected from Hund's rule. The analysis of the infrared (IR) spectrum revealed the presence of the dynamical Jahn–Teller distortion of the ${{\text{C}}_{60}}$ molecules [97, 98, 112]. The nuclear magnetic resonance (NMR), which probes a slower dynamics than the IR spectroscopy, observed a gradual freezing of the Jahn–Teller dynamics as the temperature decreases [99].

The metallic and superconducting states also show interesting behaviors near the metal-insulator transition. In the superconducting state, NMR measurements observed a deviation of the ratio between the gap($\Delta$) and ${{T}_{\text{c}}}$ ($2\Delta /{{k}_{\text{B}}}{{T}_{\text{c}}}$) from the BCS value of 3.53 to a larger value, while $2\Delta /{{k}_{\text{B}}}{{T}_{\text{c}}}\sim 3.53$ holds in the region of small ${{V}_{\text{C}_{60}^{3-}}}$ [101, 113]. The spin susceptibility in the normal phase also shows a larger value than that expected from the smooth extrapolation from the values in the small ${{V}_{\text{C}_{60}^{3-}}}$ region [26]. Moreover, an anomalous metallic region has been identified close to the metal-insulator boundary, which has been dubbed 'Jahn–Teller metal'. In this state, the IR spectrum is similar to that of the Mott insulating phase [11]. This result can be interpreted in terms of a slowing down of the dynamical distortion of the ${{\text{C}}_{60}}$ molecules when ${{V}_{\text{C}_{60}^{3-}}}$ increases. When the Mott localization is approached, the distortion timescale becomes eventually so long that the IR experiment probes the system in a distorted state on its characteristic timescale.

The clear fingerprints of the electron–phonon coupling in the superconducting state and the very existence of the Mott insulating state suggest the importance of considering both electron correlations and phonons for the understanding of the surprising phase diagram. In particular, it is a great challenge to understand why the s-wave superconductivity is robust against (or even benefits from) the strong correlations.

Another challenge is an ab initio calculation of ${{T}_{\text{c}}}$ of strongly-correlated unconventional superconductivity. A nonempirical calculation of ${{T}_{\text{c}}}$ is necessary for predicting/designing new high-temperature superconductors. Even for the conventional superconductors, the ${{T}_{\text{c}}}$ calculation usually relies on empirical parameters such as the Coulomb pseudopotential [114]. The recent development of the density-functional theory for superconductors (SCDFT) has enabled a ${{T}_{\text{c}}}$ calculation without empirical parameters [115–118]. By assuming the phonon-mechanism, the SCDFT has succeeded in reproducing ${{T}_{\text{c}}}$ of conventional superconductors in the accuracy of several tens percent. However, ${{T}_{\text{c}}}$ of ${{\text{C}}_{60}}$ superconductors is largely underestimated by the phonon-mechanism-based SCDFT [59]. This is another indirect indication that the alkali-doped fullerides are unconventional superconductors. While there have been several attempts to generalize the SCDFT to include e.g. the plasmon [119–121] and the spin-fluctuation [122, 123] as a pairing glue, there exists no established method to calculate ${{T}_{\text{c}}}$ of unconventional superconductors in which strong-correlation effects are important.

In this review, among the various studies on the ${{\text{C}}_{60}}$ superconductors, we mainly focus on the most recent ab initio studies [19–21], which aimed at (i) the unified description of the phase diagram including the s-wave superconductivity and the Mott phase and (ii) the nonempirical calculation of ${{T}_{\text{c}}}$. The nonempirical calculations have elucidated that, in the fullerides, the phonon-mediated negative exchange interaction surpasses the positive Hund's coupling and thereby realizes an inverted Hund's rule, as predicted by Capone et al [18, 93–95]. On the other hand, the intramolecular Hubbard-type interaction is strongly repulsive because the strong local Coulomb interaction far exceeds the attraction mediated by phonons. The value of this repulsion is larger than that of the t_1u bandwidth, which brings about the Mott physics in the system.

By analyzing a realistic low-energy Hamiltonian with the above-mentioned unusual interactions, the theoretical phase diagram was derived without empirical parameters. Remarkably, it shows a good agreement with the experimental phase diagram at a quantitative level. In particular, the calculated ${{T}_{\text{c}}}$'s agree with the experimental data within a difference of 10 K. It indicates that the scheme employed in [19–21] properly captures the essence of the fulleride superconductivity. Based on the success of our approach, we argue that the unusual intramolecular interaction is the key to explain the phase diagram in a unified manner: it allows electrons to form a pair on the molecules in contrast to the naïve expectation that the electron correlations drastically suppress the pair formation [96]. Surprisingly, the strong correlations are found to even help the formation of the pair. We show that this leads to a surprising cooperation between the phonons and Coulomb interactions to realize an exotic high-${{T}_{\text{c}}}$ pairing next to the Mott phase.

The outline of this review is as follows. Throughout the review, we mainly focus on the fcc systems. In section 2, we describe the methods, which were employed in [19–21]. The methods construct, from first principles, a realistic three-band Hamiltonian with including the phonon degrees of freedom from only the information of the crystal structure. We solve it accurately by means of a many-body method. Through the derivation of the realistic Hamiltonian, we discuss the electronic structure of the fullerides (section 3) and the detail of the above-mentioned unusual intramolecular interaction (section 4). The analysis of the derived model follows in section 5, 6 and 7. First, we show the theoretical phase diagram in section 5. Then, we discuss the properties of the metal-insulator transition in section 6 and finally the superconducting mechanism in section 7. We give a summary of the review and future perspectives in section 8.

4.1.1. Formulation.

Using the Wannier basis, we define the Coulomb interaction part

$$\begin{eqnarray}&&{{\hat{\mathcal{H}}}_{\text{el}-\text{el}}}=\underset{i,j,k,l}{\sum}\underset{\sigma,{{\sigma}^{\prime}}}{\sum}{{V}_{ij,kl}}~\hat{c}_{i\sigma}^{\dagger}\hat{c}_{l{{\sigma}^{\prime}}}^{\dagger}{{\hat{c}}_{k{{\sigma}^{\prime}}}}{{\hat{c}}_{j\sigma}}\end{eqnarray} \tag{ 3 }$$

in the low-energy Hamiltonian in equation (1). The Coulomb interaction parameters V_ij,kl are calculated by the cRPA [141]. The cRPA provides partially screened Coulomb interactions which are screened only by the polarization processes involving the high-energy bands (the bands other than the t_1u bands). The partially screened interactions can be considered as effective Coulomb interactions within the t_1u manifold [141]. Because the screening processes within the low-energy subspace are considered by the E-DMFT, they are excluded in the cRPA calculation to avoid the double counting of them.

4.1.2. Results.

Table 1 lists the cRPA interaction parameters. The intramolecular Hubbard U is about 1 eV, which is larger than the t_1u bandwidth W. Since the ratio U/W exceeds 1, the alkali-doped fullerides can be regarded as strongly-correlated materials. The previous estimates of U in the literature give $U\sim 1$–1.5 eV [82, 143–145]. Compared to them, the cRPA values are slightly small. Despite that the size of the maximally localized Wannier orbitals depends on materials only weakly [21], the material dependence in the Hubbard U is nonnegligible. This is because the screening strength by the high-energy bands depends on materials [21]. While U is larger than W, Hund's coupling J is small  ∼34 meV. This is explicable by the fact that the Wannier orbitals spread over the molecules, which makes the exchange Coulomb matrix element small.

Table 1. The cRPA interaction parameters taken from [21].

Material (fcc structure)		U (eV)	${{U}^{\prime}}$ (eV)	J (meV)	V (eV)	W (eV)
${{\text{K}}_{3}}{{\text{C}}_{60}}$	(722)	0.82	0.76	31	0.24–0.25	0.50
$\text{R}{{\text{b}}_{3}}{{\text{C}}_{60}}$	(750)	0.92	0.85	34	0.26–0.27	0.45
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(762)	0.94	0.87	35	0.27–0.28	0.43
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(784)	1.02	0.94	35	0.28–0.29	0.38
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(804)	1.07	1.00	36	0.30	0.34

Note: The values in the parentheses after the material denote ${{V}_{\text{C}_{60}^{3-}}}$ in ${\mathring{\text{A}}^{3}}$. U, ${{U}^{\prime}}$, and J are intramolecular interactions (see figure 7 for their definition), for which the relation ${{U}^{\prime}}\sim U-2J$ holds well. There is no orbital dependence in U, ${{U}^{\prime}}$, and J. V is the nearest neighbor intermolecular interaction. For comparison, the DFT bandwidth W of the t_1u bands is also listed.

The cRPA Coulomb interactions have a long-range tail proportional to 1/r with respect to the distance r between the centers of the Wannier orbitals [21], whose nearest-neighbor part V is  ∼0.25–0.3 eV. The offsite Coulomb interactions effectively reduce the size of the onsite Coulomb interaction [146]. This effect is taken into account by the E-DMFT. While the offsite Coulomb interactions quantitatively shift the phase boundaries between e.g. the metal and the insulator, they do not play an essential role in driving the superconductivity. We find that it is the form of onsite (intramolecular) interactions that is important. In appendix, we discuss this point in more detail.

4.2.1. Formulation.

The electron–phonon coupling ${{\hat{\mathcal{H}}}_{\text{el}-\text{ph}}}$ and phonon one-body ${{\hat{\mathcal{H}}}_{\text{ph}}}$ terms in the low-energy Hamiltonian in equation (1) are written as

$$\begin{eqnarray}&&{{\hat{\mathcal{H}}}_{\text{el}-\text{ph}}}=\underset{i,j,\sigma}{\sum}\underset{\nu}{\sum}~g_{ij}^{\nu}~(\hat{c}_{i\sigma}^{\dagger}{{\hat{c}}_{j\sigma}}-\langle \hat{c}_{i\sigma}^{\dagger}{{\hat{c}}_{j\sigma}}\rangle )~{{\hat{x}}_{\nu}}\end{eqnarray} \tag{ 4 }$$

with the displacement operator ${{\hat{x}}_{\nu}}=\hat{b}_{\nu}^{\dagger}+{{\hat{b}}_{\nu}}$, and

$$\begin{eqnarray}&&{{\hat{\mathcal{H}}}_{\text{ph}}}=\underset{\nu}{\sum}{{\omega}_{\nu}}\hat{b}_{\nu}^{\dagger}\,{{\hat{b}}_{\nu}},\end{eqnarray} \tag{ 5 }$$

respectively. Here, $\hat{b}_{\nu}^{\dagger}$ (${{\hat{b}}_{\nu}}$) is a creation (annihilation) operator of the phonon and ν is a composite index for the momentum and the branch. In solids, the electron–phonon coupling $g_{ij}^{\nu}$ and the phonon frequency ${{\omega}_{\nu}}$ are subject to renormalization by the electrons: the electron–phonon coupling is screened by the electronic polarization; the phonons are dressed by electrons, which results in phonon softening. As in the case of the Coulomb interaction parameters in equation (3), in order to avoid the double counting of the renormalization, $g_{ij}^{\nu}$ and ${{\omega}_{\nu}}$ in the effective low-energy Hamiltonian should not include the renormalization effect originating from the low-energy electrons [20, 142, 147]. The recently developed cDFPT [20, 142] calculates such partially-renormalized phonon quantities by taking into account only the renormalization processes involving the high-energy electrons. It is an extension of the DFPT [148–151], which is a well-established ab initio scheme to calculate the phonon properties in solids. The expectation value $\langle \hat{c}_{i\sigma}^{\dagger}{{\hat{c}}_{j\sigma}}\rangle$ in equation (4) is calculated at the DFT level, where the subtraction of $\langle \hat{c}_{i\sigma}^{\dagger}{{\hat{c}}_{j\sigma}}\rangle$ corresponds to the double-counting correction with respect to the equilibrium positions of the ions [20].

In the action corresponding to the low-energy Hamiltonian in equation (1), the phonon fields are at most quadratic. Therefore, we can analytically integrate out the phonon degrees of freedom without introducing any approximation, which leads to an effective action involving only the electronic degrees of freedom. In this effective action, the effective interaction between the electrons is given by the sum of the Coulomb interactions and the retarded phonon-mediated interactions (figure 6). This retarded phonon-mediated interactions $V_{ij,kl}^{\text{ph}}(i{{\Omega}_{n}})$ at the Matsubara frequencies (${{\Omega}_{n}}=2\pi nT$ with the temperature T) are given by [20, 142]

$$\begin{eqnarray}&&V_{ij,kl}^{\text{ph}}(i{{\Omega}_{n}})=-\underset{\nu}{\sum}\frac{2g_{ij}^{\nu}g_{kl}^{\nu \ast}}{\Omega _{n}^{2}+\omega _{\nu}^{2}}.\end{eqnarray} \tag{ 6 }$$

An important point here is that the phonon-mediated interaction is given by the sum over the phonon modes. We do not assume any particular type of the vibration modes a priori to study the superconductivity. We call the onsite (intramolecular) part of these interactions ${{U}_{\text{ph}}}(=V_{\alpha \alpha,\alpha \alpha}^{\text{ph},\text{onsite}})$, $U_{\text{ph}}^{\prime}(=V_{\alpha \alpha,\beta \beta}^{\text{ph},\text{onsite}})$, and ${{J}_{\text{ph}}}(=V_{\alpha \beta,\beta \alpha}^{\text{ph},\text{onsite}}=V_{\alpha \beta,\alpha \beta}^{\text{ph},\text{onsite}})$, respectively, with the orbital indices α and β.

4.2.2. Electron-phonon coupling.

4.2.3. Phonon frequencies.

Here, we look at the partially-dressed phonon frequencies needed to calculate the phonon-mediated interaction in equation (6). Figure 8 shows the partially-dressed phonon frequencies (red curves) calculated by the cDFPT for the frequency range from 1100 to 1400 cm⁻¹. In the cDFPT calculation, the renormalization of the phonon frequencies coming from the t_1u electrons is excluded. These partially-dressed frequencies are defined in the low-energy Hamiltonian. Thus, they cannot be directly compared to the experimental data. To compare with experiments, we need to solve the low-energy Hamiltonian to obtain fully-dressed phonon frequencies.

Zoom In Zoom Out Reset image size

In figure 8, for comparison, we also show the fully-dressed phonon frequencies (blue-dotted curves) calculated by the conventional DFPT, where the renormalization effect of the t_1u electrons is further incorporated within the DFPT framework. The dispersion of both the partially and fully dressed frequencies is tiny, which reflects the intramolecular nature of the modes (almost perfectly described as the Einstein phonons). We see that the red curves agree with the blue-dotted curves for the majority of the bands, and that the main difference is in the bands around the frequency of 0.14 and 0.16 eV. These two correspond to the two H_g modes out of the eight H_g modes. Because of the cubic symmetry, the frequencies of the H_g modes, which are fivefold degenerate in the molecular limit, are split into threefold- and twofold-degenerate frequencies. The A_g modes are located at the energies beyond the range of figure 8. Experimentally, the two A_g modes are observed at 496 and 1470 cm⁻¹for undoped C₆₀ [68].

The difference between the partially- and fully-dressed phonon frequencies in figure 8 originates from the renormalization effect of the t_1u electrons. Therefore, the frequencies of the H_g modes, which couple to the t_1u electrons, are further renormalized from the partially-dressed frequencies in the red curves, while the other modes are not⁹.

Because the C-C bonds are rather stiff and the mass of C atom is light, the maximum frequency of the intramolecular phonons can be rather large, reaching about 0.2 eV, which is comparable to the t_1u bandwidth  ∼0.5 eV. It indicates that we cannot ignore the vertex corrections, i.e. the Migdal theorem [160] does not hold any more. The low-energy vertex corrections within a molecule can be captured by the E-DMFT, while the nonlocal vertex corrections are not.

4.2.4. Results.

Table 2 shows the phonon-mediated interactions at zero frequency (${{\Omega}_{n}}=0$) calculated with the partially-renormalized phonon frequencies and electron–phonon interactions [19, 20]. The frequency dependence of these interactions is discussed in section 4.3. The value of the intraorbital interaction ${{U}_{\text{ph}}}(0)$ lies between  −0.15 eV and  −0.1 eV, which are small compared to the Hubbard repulsion U. On the other hand, an unusual situation is realized in the exchange channel: the phonon-mediated exchange interaction ${{J}_{\text{ph}}}(0)$ is about  −51 meV, and its absolute value is larger than the value of Hund's coupling $J\sim 34$ meV. As discussed in section 4.2.2, only the Jahn–Teller phonons contribute to ${{J}_{\text{ph}}}(0)$. Thus, this means that the Jahn–Teller phonons surpass the Coulomb exchange interactions and realize an effectively negative exchange interaction. However, the size of this negative exchange interaction (∼−17 meV) is small.

Table 2. The phonon-mediated interactions at zero frequency calculated by the cDFPT.

Material (fcc structure)		${{U}_{\text{ph}}}(0)$ (meV)	$U_{\text{ph}}^{\prime}(0)$ (meV)	${{J}_{\text{ph}}}(0)$ (meV)
${{\text{K}}_{3}}{{\text{C}}_{60}}$	(722)	−152	−53	−50
$\text{R}{{\text{b}}_{3}}{{\text{C}}_{60}}$	(750)	−142	−42	−51
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(762)	−114	−13	−51
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(784)	−124	−22	−51
$\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$	(804)	−134	−31	−52

Note: The values are taken from [19]. The values in the parentheses after the material compositions denote ${{V}_{\text{C}_{60}^{3-}}}$ in ${\mathring{\text{A}}^{3}}$. See figure 7 for the definition of U_ph, $U_{\text{ph}}^{\prime}$, and J_ph. There is no orbital dependence in U_ph, $U_{\text{ph}}^{\prime}$, and J_ph. The relation $U_{\text{ph}}^{\prime}\sim {{U}_{\text{ph}}}-2{{J}_{\text{ph}}}$ holds well.

Figure 9 summarizes the ${{V}_{\text{C}_{60}^{3-}}}$ dependence of the Coulomb interactions and the phonon-mediated interactions. As ${{V}_{\text{C}_{60}^{3-}}}$ increases, the Hubbard U increases, while the DFT t_1u bandwidth W decreases. Thus, the change in U/W is steeper than that expected from the change in the bandwidth. Hund's coupling J and the phonon-mediated exchange interaction ${{J}_{\text{ph}}}(0)$ are almost constant throughout the ${{V}_{\text{C}_{60}^{3-}}}$ range. In figure 9, only the phonon attraction ${{U}_{\text{ph}}}(0)$ has nonmonotonic ${{V}_{\text{C}_{60}^{3-}}}$ dependence. This can be ascribed to the contribution from the alkali-ion vibrations [20]. We find that the contribution from the intramolecular modes is nearly ${{V}_{\text{C}_{60}^{3-}}}$ independent [20]. Like the A_g modes, the alkali-ion phonons couple to the total t_1u occupations. Therefore, they do not contribute to ${{J}_{\text{ph}}}(0)$. The couplings to the alkali modes seem to be nonnegligible, however, one has to pay attention to the fact that they do not include the screening from the t_1u electrons. When the metallic screening is considered by solving the model, these couplings are efficiently screened and the fully-screened couplings become very small. Thus, these alkali modes do not play an important role in the superconductivity. On the other hand, the Jahn–Teller-type couplings are poorly screened [20] and contribute essentially to the superconductivity.

Zoom In Zoom Out Reset image size

Figure 10 shows the real-frequency dependence of the effective interactions between the t_1u electrons, which are given by the sum of the Coulomb and the retarded phonon-mediated interactions. In the frequency region below 0.2 eV, the attractions from the phonons work since the frequencies of the intramolecular phonons exist up to  ∼0.2 eV. Several peak-like structures reflect that several phonon modes with different frequencies contribute to the effective interactions. By the phonon contribution, the effective exchange interaction ${{J}_{\text{eff}}}\left(\omega \right)$ becomes negative. On the other hand, the density-type interactions ${{U}_{\text{eff}}}\left(\omega \right)$ and $U_{\text{eff}}^{\prime}\left(\omega \right)$ are strongly repulsive. However, in the region where ${{J}_{\text{eff}}}\left(\omega \right)$ becomes negative, $U_{\text{eff}}^{\prime}\left(\omega \right)$ becomes slightly larger than ${{U}_{\text{eff}}}\left(\omega \right)$ because of the relation $U_{\text{eff}}^{\prime}\left(\omega \right)\sim {{U}_{\text{eff}}}\left(\omega \right)-2{{J}_{\text{eff}}}\left(\omega \right)$. In the following sections, we discuss the surprising consequence of this unusual relationship between the interactions $U_{\text{eff}}^{\prime}>{{U}_{\text{eff}}}$ and ${{J}_{\text{eff}}}<0$.

Zoom In Zoom Out Reset image size

By solving the Hamiltonian in equation (1) by the E-DMFT¹⁰, we derive the theoretical phase diagram [19], which is shown in figure 11. Here we study superconductiviy by directly allowing the symmetry breaking in the E-DMFT calculation. Within the single-site E-DMFT calculation for the fcc lattice (non-bipartite lattice), we also allow for ferro-orbital/magnetic order, while we do not consider a possibility of the antiferromagnetic order, which, in the actual fcc lattice, is observed only at very low temperature, significantly smaller than the lowest temperature (10 K) we used in our E-DMFT.

Zoom In Zoom Out Reset image size

As a result, we obtain three different phases: the paramagnetic metal, the paramagnetic Mott insulator, and the s-wave superconducting phase. The s-wave superconductivity is characterized by a nonzero superconducting order parameter $\Delta =\sum\nolimits_{\alpha =1}^{3}\langle {{c}_{\alpha s\downarrow}}{{c}_{\alpha s\uparrow}}\rangle$, which describes intraorbital Cooper pairs for the t_1u electrons. Here, α and s are the orbital and site (= molecule) indices, respectively. We omitted a site (= molecule) index in $\Delta$, because $\Delta$ does not depend on a site (the solution is homogenous in space). In the Mott insulating phase, the self-energy diverges and hence the Mott gap opens in the spectral function (the blue-dotted curve in figure 12). Throughout the phase diagram above 10 K, a solution with a ferro-orbital order is not stabilized. In the region between the blue curve and the black dotted curve in figure 11, both metallic and insulating solutions can be stabilized depending on the initial conditions of the E-DMFT calculations. This suggests that a first-order transition, where the global minimum of the free energy changes from the metallic to insulating solution, should take place between these two curves. According to general entropic arguments, we expect the transition to be close to the blue curve [164].

Zoom In Zoom Out Reset image size

Comparing the theoretical phase diagram in figure 11 with the experimental one in figure 2(a), we find a good agreement between them. The theoretical phase diagram reproduces the s-wave superconductivity next to the Mott phase. Theoretically calculated ${{T}_{\text{c}}}$'s with the maximum of  ∼28 K agree with the experimental ${{T}_{\text{c}}}$'s within 10 K difference. This agreement is remarkable since our ${{T}_{\text{c}}}$ calculation does not rely on any empirical parameters: it starts from the DFT calculation using only the information of the crystal structure and all the parameters used in the E-DMFT analysis are calculated from first principles.

Furthermore, the slope between the paramagnetic metal and insulator is consistent between the theory and experiment. As temperature increases, the insulating region expands, which indicates that the insulator has a larger entropy than the metal. The position of the metal-insulator boundary is also consistent. It is known that the DMFT often overestimates the stability of the metallic phase [165, 166] because it neglects non-local correlation effects and it becomes exact only in the limit of large coordination number. However, in the present case, the large coordination number (12) of the fcc lattice makes the DMFT reliable [167]. Furthermore, the ${{V}_{\text{C}_{60}^{3-}}}$ dependence of the U/W ratio is rather steep so that even if the critical U/W for the metal-insulator transition changes, it does not lead to a drastic change in the critical ${{V}_{\text{C}_{60}^{3-}}}$. We think that the above two factors lie behind the nice quantitative agreement between the theory and experiment as far as the metal-insulator boundary is concerned.

While we see a good agreement, there are minor discrepancies between the theory and experiment. For example, the metal-insulator boundary looks like a crossover in the experiment, while the theory shows a clear first order transition. While the reason of the discrepancy is not yet clear, one of the possible reasons is the disorder in the orientations of the buckyballs [22, 23, 40], which is neglected in the calculation. Another discrepancy can be seen in the shape of the ${{T}_{\text{c}}}$ curves. In the experiment, the ${{T}_{\text{c}}}$ curve shows a dome-like shape, while the theoretical ${{T}_{\text{c}}}$ curve increases toward the Mott transition. We come back to this point in section 7.

Figure 12 shows the spectral functions of the t_1u bands for three different ${{A}_{3}}{{\text{C}}_{60}}$ systems, derived by the analytic continuation based on the maximum entropy method [168, 169]. The DFT density of states for ${{\text{K}}_{3}}{{\text{C}}_{60}}$ with ${{V}_{\text{C}_{60}^{3-}}}=722$ ${\mathring{\text{A}}^{3}}$ is also shown for comparison. In the metallic phase (red and green curves), as ${{V}_{\text{C}_{60}^{3-}}}$ increases, i.e. as the correlation strength increases, the width of the quasiparticle part becomes narrower and the incoherent peaks become more prominent. We note that the position of the incoherent part and the renormalization factor of the quasiparticle band for ${{\text{K}}_{3}}{{\text{C}}_{60}}$ are consistent with the ARPES (angle-resolved photo-emission spectroscopy) measurements for the ${{\text{K}}_{3}}{{\text{C}}_{60}}$ monolayer [170]¹¹.

Figure 13(a) shows the dependence of several observables on ${{V}_{\text{C}_{60}^{3-}}}$ calculated at 40 K [19]. Here, we focus on the metallic solution in the coexistence region (the region delimited by the blue and black-dotted curves in figure 11). The blue curve shows the size of spin S per molecule, which decreases as ${{V}_{\text{C}_{60}^{3-}}}$ increases. In the insulating phase, the value of S becomes  ∼0.5, i.e. the low-spin state with S  =  1/2 is realized. This is because an effectively negative exchange interaction is realized (section 4.3), which favors the low-spin state rather than the high-spin state.

Zoom In Zoom Out Reset image size

Another interesting observation is that the double occupancy $D=\langle {{n}_{\alpha \uparrow}}{{n}_{\alpha \downarrow}}\rangle$ on each molecule increases toward the Mott transition. This is in contrast with the intuition and the behavior of the single-band Hubbard model and of multiorbital models with the standard exchange interaction, where D is suppressed by the repulsive Hubbard U. While the fullerides also have a strongly repulsive Hubbard U, the interorbital repulsion ${{U}^{\prime}}$ is effectively larger than the Hubbard U (section 4.3). Therefore the electrons (3 electrons per molecule on average) prefer to sit in the same orbital despite they pay an energy cost of U [96], which is clearly seen in the inequality $\langle {{n}_{\alpha \uparrow}}{{n}_{\alpha \downarrow}}\rangle >\langle {{n}_{\alpha \uparrow}}{{n}_{\beta \downarrow}}\rangle$ in figure 13(a).

We can gain more insight by looking at the histogram of the weight of the intramolecular electronic configurations (figure 13(b)). The histogram shows which intramolecular electronic state is dominantly realized on the molecule within the E-DMFT [171]. It is obtained by the continuous-time quantum Monte Carlo simulation [126, 172] of the E-DMFT impurity problem, which consists of the single correlated molecule and the dynamical bath.

In figure 13(b), '(2 1 0)' denotes the set of the half-filled configurations in which one orbital is doubly occupied and the third electron occupies another orbital, namely the configurations where $\left\{{{n}_{1}},{{n}_{2}},{{n}_{3}}\right\}=$ {2, 1, 0}, {0, 2, 1}, {1, 0, 2}, {2, 0, 1}, {1, 2, 0}, and {0, 1, 2}. '(1 1 1)' denotes the set of the half-filled configurations with equal occupations on each orbital, i.e. $\left\{{{n}_{1}},{{n}_{2}},{{n}_{3}}\right\}=\left\{1,1,1\right\}$. '$N\ne 3$' indicates the total weight of all the configurations away from half filling.

In the non-interacting limit at half filling, all the 64 intramolecular electronic configurations have equal weight. Then, the weights for the (2 1 0), (1 1 1), and $N\ne 3$ configurations are 0.1875, 0.125, and 0.6875, respectively. In the presence of correlation effects, the (2 1 0) configurations acquire the largest weight. This is again because of the unusual molecular interactions with $U_{\text{eff}}^{\prime}>{{U}_{\text{eff}}}$ and ${{J}_{\text{eff}}}<0$, which prefer the $\left(2\,1\,0\right)$-type configurations to the other configurations. The increase of the weight of the (2 1 0) configurations explains the increase of D and the decrease of S with the increase of ${{V}_{\text{C}_{60}^{3-}}}$.

In the metallic phase, however, there exist charge fluctuations because the non-half-filled ($N\ne 3$) configurations have a nonnegligible weight. The non-half-filled ($N\ne 3$) configurations gradually lose their weight toward the Mott transition. In the Mott insulating phase, we see that $N\ne 3$ weight becomes tiny. It indicates that the charge degrees of freedom are frozen and the filling on each molecule is nearly fixed at half filling, which is nothing but the Mott physics induced by the strongly repulsive U_eff [18, 94].

However, the orbital and spin degrees of freedom are active even when the charge degrees of freedom are frozen and the balance of the interaction favors low-spin configurations [94]. As we discussed in section 5.1, the Mott insulating phase has no ferro-orbital/spin order. Thus, all the (2 1 0) configurations, which are dominant in the Mott phase, are degenerate, offering a room for the orbital and spin fluctuations [18, 94]. The absence of the orbital order is consistent with the experimentally observed dynamical Jahn–Teller effect [97, 98]: each (2 1 0) configuration can be regarded as a snapshot state where the orbital degeneracy is dynamically lifted, however, if we take a long-time average, the orbital degeneracy is recovered. We note that the orbital degeneracy plays a role of increasing the critical U for the Mott transition [173, 174], with stabilizing the metallic/superconducting state against the Mott phase.

We have reviewed the properties of alkali-doped fulleride superconductivity, starting from the basic electronic structure of the fcc ${{A}_{3}}{{\text{C}}_{60}}$ systems. The band structure of ${{A}_{3}}{{\text{C}}_{60}}$ fullerides is well described by a picture in terms of molecular orbital levels connected by relatively small hoppings between them. The half-filled low-energy t_1u bands are the main playground for the superconductivity.

We have elucidated that the effective intramolecular interactions between the t_1u electrons, which arise from the combination of the Coulomb repulsion and electron–phonon interactions, have an unusual structure characterized by a strongly repulsive Hubbard repulsion and a weakly negative exchange interaction, which leads to an inverted Hund's coupling. This unusual situation occurs since the fullerides are the degenerate multi-orbital systems having tiny Hund's coupling and strong coupling to Jahn–Teller modes.

The realistic Hamiltonian with this form of the intramolecular interaction comprehensively reproduces the experimental phase diagram including the adjacency of the Mott insulator and the s-wave superconductivity. Remarkably, the agreement is not only qualitative but also quantitative; the theoretically calculated ${{T}_{\text{c}}}$ using only the information of the crystal structure, agrees with the experimental ${{T}_{\text{c}}}$ within a difference of 10 K. The derived Mott insulating phase is characterized by a low-spin (S  =  1/2) state with orbital degeneracy, completely consistent with the experimental observations.

The analysis on the superconducting mechanism reveals that the high-${{T}_{\text{c}}}$ s-wave superconductivity is driven by an unusual cooperation between the strong correlations and the Jahn–Teller phonons. This confirms that the mechanism of the fulleride superconductivity is indeed unconventional and requires strong correlation effects, despite the crucial role of the electron–phonon interaction and the s-wave symmetry of the order parameter.

8.2.1. Study on A15 $\text{C}{{\text{s}}_{3}}{{\text{C}}_{60}}$.

8.2.2. Light-induced superconducting-like state in ${{\text{K}}_{3}}{{\text{C}}_{60}}$.