Merohedral disorder and impurity impacts on superconductivity of fullerenes

Local quasiparticle states around impurities provide essential insight into the mechanism of unconventional superconductivity, especially when the candidate materials are proximate to an antiferromagnetic Mott-insulating phase. While such states have been reported in atom-based cuprates and iron-based compounds, they are unexplored in organic superconductors which feature tunable molecular orientation. Here we employ scanning tunneling microscopy and spectroscopy to reveal multiple forms of robustness of an exotic s-wave superconductivity in epitaxial Rb3C60 films against merohedral disorder, non-magnetic single impurities and step edges at the atomic scale. Yu-Shiba-Rusinov (YSR) states, induced by deliberately incurred Fe adatoms that act as magnetic scatterers, have also been observed. The YSR bound states show abrupt spatial decay and vary in energy with the Fe adatom registry. These results and a doping-dependent study of superconductivity point towards local electron pairing in which the multiorbital electronic correlations and intramolecular phonons together drive the high-temperature superconductivity of doped fullerenes. Doped-fullerenes are a class of organic superconductors where disorder can be used to tune the superconducting temperature as well as the presence of subgap excitations such as Yu-Shiba-Rusinov states. Here, the authors investigate how structural disorder and non-magnetic impurities affect the superconductivity of Rb-doped fullerenes and what information this can provide about the underlying mechanisms.

Unlike atom-based superconductors, an organic superconductor is a synthetic molecule-based compound that uniquely exhibits additional degrees of freedom related to its molecular orientation. Consequently, inequivalent molecular orientations take place. Such orientational (merohedral) disorder has been seen early in pure and doped fullerenes 27,28 , but its impact, either harmful 29,30 or irrelevant 31,32 , to superconductivity of fullerenes, is highly controversial. In addition, the fullerides represent an unusual category of organic superconductors in which the multiorbital electronic correlations and electron-phonon interactions are both suggested to be significant to reach high-T c superconductivity 33,34 . Under this context, a systematic study of impurity effects on superconductivity of doped fullerenes would provide justification of the previously advocated s-wave pairing symmetry [35][36][37] , as well as advance the understanding of the superconducting state. However, such an experiment is unexplored and the roles played by magnetic and non-magnetic impurities remain unknown in fulleride superconductors.
In this work, we use a molecular beam epitaxy (MBE) technique to grow epitaxial films of rubidium (Rb)-doped fullerenes with thickness and filling tunability, and probe the local quasiparticle states in the vicinity of various impurities at the atomic scale by means of cryogenic scanning tunneling microscopy (STM) and spectroscopy (STS). Distinct from the superconducting K 3 C 60 films without merohedral disorder 37 and the insulating Cs 3 C 60 ones with great merohedral disorder 38 , merohedrally disordered Rb 3 C 60 films are superconducting. This allows for atomic-scale visualization of merohedral disorder impact on superconductivity of fullerenes, which, together with a detailed STS study of magnetic and non-magnetic impurities, shows that superconductivity of fullerenes is entirely consistent with local s-wave pairing. By studying the thickness and electronfilling dependence of superconducting gap Δ in Rb x C 60 , we further establish a unified phase diagram of fullerenes in which the optimal superconductivity always develops at half-filling (x = 3).

Results
Merohedral disorder and its impact on superconductivity. Figure 1a depicts a representative STM topography of nine monolayer (ML) Rb 3 C 60 thin films epitaxially grown on graphitized SiC(0001) substrates. Evidently, not all C 60 molecules have the same orientation, although one threefold symmetry axis for every C 60 is perpendicular to the surface. Specifically, nanoscale domains with two distinct C 60 orientations, related by 44.48°r otation about the [111] axis (Fig. 1b), develop and are partially opacified in red and orange, respectively. This is reminiscent of the two standard orientations of C 60 that randomly occur and cause merohedral disorder in face-centered cubic-structured trivalent fullerides [28][29][30] . The C 60 orientations are more disordered in regions between adjacent merohedral domains. In order to quantify the merohedral disorder, the averaged orientational correlation functions <cos(θ ij )> 39 , in which θ ij = θ i −θ j denotes the angle between nearest-neighbor C 60 molecules (i.e., i and j), are calculated and summarized in Fig. 1c. The orientational correlation decreases with alkali metal radius, indicating increasing merohedral disorder. This most probably arises from a weakening of Coulomb repulsion between adjacent trivalent C 60 ions associated with the lattice expansion 40 , which otherwise stabilizes a long-ranged merohedral order in the K 3 C 60 films 37 , to wit <cos(θ ij )> = 1. For a specific K 3 C 60 or Rb 3 C 60 compound, it turns out that the orientational correlation and thus merohedral disorder change little with the film thickness (Supplementary Figure 1).
Tunneling spectroscopy of fullerides probes the local density of quasiparticle states (DOS) and measures the superconducting energy gap at the Fermi level (E F ). In Fig. 1d, we compare tunneling dI/dV spectra on various trivalent fulleride films A 3 C 60 (A = K, Rb, Cs) with the same thickness of 9 ML. Although two sharp DOS peaks develop~−0.4 eV and 0.1 eV in the merohedrally ordered K 3 C 60 films 37 , they are largely smoothed out in both Rb 3 C 60 and RbCs 2 C 60 with the great merohedral disorder. This observation is consistent with the theoretical calculation that the merohedral disorder would blur t 1u -derived DOS peaks in A 3 C 60 41 . It is, however, worth noting that the low-lying DOS width estimated as the spacing between the two conductance minima below and above E F (>1.2 eV, see the two dashed lines in Fig. 1d) is significantly larger than the commonly argued t 1u bandwidth of ∼0.5 eV 41,42 . Such a discrepancy might originate from the Jahn-Teller (JT) instabilities and Coulomb interactions omitted by three-band first-principles calculations 41,42 . In consideration of JT-induced subband splitting and electronic correlations, the t 1u bandwidth could be substantially increased 34,43,44 and accords with our observation. A further enhancement of electronic correlations U in the most expanded Cs 3 C 60 films pushes the t 1u -derived DOS toward higher energy and opens a Mott-insulating gap 38 , in contrast to the superconducting ground state in K 3 C 60 and Rb 3 C 60 (Fig. 1d).
Despite nanoscale merohedral disorder, superconductivity develops well in Rb 3 C 60 films. We unambiguously reveal this by measuring the spatial dependence of superconducting gaps at the atomic scale via STS, as exemplified in Fig. 1e. Even on the boundaries between adjacent merohedral domains, the superconducting gaps exhibit clear coherence peaks (blue curves) and are immune to the local merohedral disorder (Fig. 1a). This is further confirmed in Cs and Rb co-doped RbCs 2 C 60 films, which imprint a comparably large merohedral disorder but exhibit a superconducting transition temperature up to T c = 23 K (Supplementary Figure 2). Nevertheless, the superconducting spectra present some spatial electronic inhomogeneities, especially for the coherence peaks. A careful examination of Rb 3 C 60 films at varied thicknesses and spatial locations reveals that the coherence peak amplitude scales inversely with Δ (Supplementary Figure 3). This is unexpected by the conventional wisdom of Bardeen-Cooper-Schrieffer (BCS) picture, and we ascribe to coexistence of competing order, e.g., the ubiquitous pseudogap phase (Supplementary Figure 4) 37 . Similar behavior and pseudogap phenomenology have been documented in cuprate superconductors 45 .
Thickness and filling dependence of superconductivity. Having established the merohedral disorder-independent superconductivity in fullerides, we then explore its dependence on thickness and electron filling. Elaborated in Fig. 2a-c is the temperature dependence of spatially averaged dI/dV spectra measured on 3 ML, 6 ML, and 9 ML Rb 3 C 60 , respectively. Again, the fully gapped superconductivity with an isotropic s-wave pairing is consistently confirmed in Rb 3 C 60 and gets smeared out at elevated temperatures. By examining the temperature dependence of the gap depth in Fig. 2d, the critical temperature T c is determined and increases from 23 K for 3 ML, to 26 K for 6 ML, and 28 K for 9 ML Rb 3 C 60 . Such a T c evolution with film thickness stands in marked contrast to K 3 C 60 , where the maximum T c occurs in 3 ML films 37 . This hints at other factor, possibly linking with alkali metal-dependent electronic correlations (U) 36 , to consider for a unified understanding of superconductivity of A 3 C 60 films at varied thicknesses. Note that due to the significantly increased U monolayer and bilayer Rb 3 C 60 films are non-superconducting at all (Supplementary Figure 5), in analogy to the K 3 C 60 counterparts 37 . A residual DOS depletion around E F , hence pseudogap, is also observed in all superconducting Rb 3 C 60 films above T c (see the red curves in Fig. 2a-c) and within vortices (Supplementary Figure 4b).
In what follows, we explore the superconductivity of Rb x C 60 by tuning the stoichiometry and thus electron filling x. Figure 2e summarizes the superconducting gap Δ (top panel) and averaged orientational correlation (bottom panel) as a function of Rb doping level x. Clearly, Δ increases with the film thickness, in good accordance with T c (Fig. 2d). The extracted reduced gap ratio 2Δ/k B T c = 6.0 ± 0.4 is comparable to that of K x C 60 films 37 , but appreciably exceeds the canonical BCS value of 3.53. Interestingly, Δ reaches its peak at half-filling irrespective of film thickness, and declines more quickly below half-filling for thin Rb 3 C 60 films. Notwithstanding a dome-shaped variation of Δ, the merohedral disorder remains unchanged with electron filling x and film thickness (bottom panel of Fig. 1e). This not only corroborates the above claim that superconductivity is little influenced by merohedral disorder 31,46 , but also hints that the dome-shaped superconducting phase diagram does not correlate from any x-dependent merohedral disorder effects.
Robust superconductivity against non-magnetic impurities. As the electron doping of Rb x C 60 deviates slightly from half-filling, subsurface tetragonal Rb vacancies emerge as dark windmills as x < 3, whereas excess K adatoms appear and occupy the octahedral sites as x > 3. Analogous to K x C 60 ( Supplementary Figure 6a, b) 37 , they do not alter profoundly the orientation of nearby fullerene molecules and thus serve as intrinsically non-magnetic impurities to test the fully gapped superconductivity in fullerides. Figure 3a-d shows the STM topographies of a single Rb vacancy and an excess Rb adatom, as well as linecut dI/dV spectra taken across both impurities. No in-gap bound state is revealed (red curves), although Δ shrinks by ∼25% on Rb excess impurity (Supplementary Figure 7a, b). Similar responses of the superconducting gap to K impurities have been observed in K x C 60 as well (Supplementary Figure 6c). Here the Δ reduction possibly arises from a local doping variation, namely a deviation of x from 3. The vacancies are located beneath the top C 60 molecules, rendering the local Δ reduction invisible for surface-sensitive STS. On the other hand, step edges could be seen as one-dimensional perturbations and bring about Andreev bound states as they are normal to the possible sign-changing direction in Δ 45,47 . The spectroscopic signature of these bound states, e.g., a zero-bias conductance peak (ZBCP), has been observed experimentally in a few cuprate and iron-pnictide superconductors 45,48,49 . Figure 3e depicts a topographic STM image of one monomolecular step edge separating Rb 3 C 60 epitaxial films between 8 ML (lower terrace) and 9 ML (upper terrace). Note that all step edges run along the close-packed directions of C 60 molecules. Figure 3f shows dI/dV spectra taken along a trajectory approaching the step edge (solid line in Fig. 3e). The superconducting gap remains undisturbed at the step edge and nearby, and no evidence of Andreev bound states is found (Supplementary Figure 7c). Some random variations in the coherence peak, including strong coherence peaks near Rb excess impurity in Fig. 3d, might be related to the slight electronic inhomogeneity of superconducting Rb 3 C 60 films (Fig. 1e).
Local probe of Yu-Shiba-Rusinov states. To fully understand the impurity impact on fulleride superconductivity, we intentionally deposited Fe atoms on Rb 3 C 60 surface at low temperature (∼100 K). Single Fe adatoms formed (bright protrusions) and occupied top or hollow sites of the surface C 60 lattice, dubbed as Fe(I) and Fe (II) in Fig. 4a. Figure 4b represents the dI/dV spectra on both Fe impurities and defect-free regions. Note that multiple Fe(I) and Fe(II) impurities have been measured and averaged to eliminate the spatial inhomogeneity effects on dI/dV spectra. Evidently, both Fe(I) and Fe(II) adatoms act as magnetic scatterers and significantly suppress the superconducting coherence peaks, whereas a prominent ZBCP is clearly observed on Fe(I). They are hallmarks of the Yu-Shiba-Rusinov (YSR) states induced by coupling of magnetic impurity to an s-wave superconductor 10,12,16,17,26,[50][51][52][53] . Figure 4c shows a series of tunneling spectra across an isolated Fe (I) impurity. The ZBCP intensity decreases quite abruptly and gets barely visible at a spatial distance of 1.4 nm from the impurity site.
Here the distinct behaviors of YSR states on Fe(I) and Fe(II) may be caused by the varied coupling strength between them and the Rb 3 C 60 films 54 . In other words, the exchange coupling of Fe(II) adsorbed at the hollow sites to Copper pairs might be so significantly weak that the YSR states nearly merge into the superconducting gap edges and are little discernible. Further theoretical analysis is needed to comprehensively understand the Fe registry site-dependent YSR bound states in fulleride superconductors.

Discussion
Our atomic-scale observations of short-range YSR bound states on magnetic Fe adatoms, robust superconductivity against nonmagnetic merohedral disorder and impurities compellingly confirm a sign-unchanged s-wave pairing state in fulleride superconductors 17 . Distinct from a conventional superconductor, however, in charged fullerenes the t 1u -derived conduction band of ∼0.5 eV is narrow and comparable to the electron-vibron interactions, thereby causing a breakdown of the Migdal's theorem [55][56][57] . As a result, superconductivity with local nonretarded attractive interactions 33,34 is less sensitive to the distribution of the electronic DOS in conduction band 58 , and instead determined by some ensemble-averaged DOS 59 . This differs from the classic BCS superconductors where T c is essentially governed by the DOS at E F , and happens to match our finding, i.e., the merohedral disorder considerably modifies the t 1u -derived DOS distribution but never affect superconductivity (Fig. 1d, e). Such local electron pairing 33,34 , mediated by intramolecular JT phonons 60,61 , has also been reinforced by a short coherence length in fullerides. As estimated from   Fig. 2 Thickness dependence of superconductivity. a-c Spatially averaged and normalized conductance dI/dV spectra as a function of temperature and thickness of Rb 3 C 60 films as indicated. The normalization was performed by dividing the raw tunneling spectrum by its background, which was extracted from a cubic fit the conductance for |V | > 10 mV. Setpoint: V = 30 mV and I = 200 pA. The red curves denote the residual pseudogap justly above the superconducting transition temperature (T c ). d Dependence of the superconducting gap depth on temperature, yielding a gradual increase of T c with film thickness (see the guided solid lines and arrows). Here the gap depth denotes the difference between unity and the normalized zero-energy conductance. e Electronic phase diagram showing the evolution of superconducting energy gap Δ (empty circles) and averaged orientational correlation (diamonds) as a function of Rb doping x. Note that the orientational correlation function <cos(θ ij )> is only averaged over the trivalent C 60 molecules to minimize any disruption by Rb impurities and excess atoms away from half-filling.  the vortex core radius (Supplementary Figure 4a,c), the coherence length of a Cooper pair is 1.5 ± 0.2 nm in Rb 3 C 60 and 2.6 ± 0.5 nm in K 3 C 60 37 , respectively, which are only about twice the separation between nearest-neighbor fullerene molecules.
In the local pairing mechanism, the key ingredients for high-T c superconductivity are the strong coupling of the t 1u electrons to intramolecular JT phonons in trivalent fullerides [59][60][61] . The phonon-mediated unusual multiorbital (attractive) interactions lead to an effectively inverted Hund's coupling (S = 1/2) 36 and a local spin-singlet s-wave pairing on the same orbital 34 , further enhanced via coherent tunneling of pairs between orbitals (the Suhl-Kondo mechanism) 62,63 . On the other hand, the multiorbital electronic correlations suppress electron hopping-induced charge fluctuations and effectively bind electrons into intraorbital pairs 33 . In this sense, the Coulomb interactions actually help the local pairing, until they are strong enough to drive a transition from the superconductivity to Mott-insulating phase 37 . Such a local pairing scenario naturally accounts for the dome-shaped dependence of T c on C 60 packing density-controlled U 31,36,46 as well as the conflicting variation of superconductivity with film thickness in K 3 C 60 and Rb 3 C 60 . In K 3 C 60 , U is relatively small and its enhancement at reduced film thicknesses stabilizes the local pairing and thus enhances superconductivity 37 , whereas the opposite holds true owing to the already large U in Rb 3 C 60 . A further enhancement of U pushes thin Rb 3 C 60 films closer to a Mott transition and suppresses superconductivity, as observed. In Fig. 5, we show the phase diagram of charged fullerides and discover universal optimal superconductivity at half-filling, no matter how the electronic correlations U change with the alkali metal and film thickness. This finding is unusual and probably stems from a decrease in the dynamical JT-related pair binding energy (U x , negative) away from half-filling 61,62 , until the superconductivity vanishes as the U x changes its sign. For the evenly charged fullerenes, U 2 and U 4 are positive and the JT coupling instead stabilizes two correlated insulating ground states 61,62 . It is also important to note that an asymmetry of Δ versus x phase diagram relative to half-filling (x = 3) occurs as U becomes strong. This is related to a monotone shrinkage of U with the doping x in view of the enhanced Coulomb screening from itinerant electron carriers 37 . In strongly correlated regimes, a small increase of U below half-filling can suppress superconductivity significantly and leads to the observed dome asymmetry.
Finally, we note that the large ratio of energy gap Δ to critical temperature (e.g., 2Δ/k B T c > 6.0) seems to be a generic trait of high-T c superconductivity in narrow-band systems 10,20,35,37,49 , including the copper-oxide superconductors 45 . Such a large deviation from the canonical BCS value of 3.53 could be straightforward to understand theoretically in the framework of local nonretarded superconductivity 58 . Experimentally, a similar local pairing mechanism, assisted cooperatively by a dynamic interfacial polaron, has been recently proposed to be responsible for the high-T c superconductivity in monolayer FeSe epitaxial films grown on SrTiO 3 substrate 64 . A question naturally arises as to whether the local pairing mechanism is applicable to other narrow-band cuprates and multi-band iron pnictides 65 . Another interesting issue is to unravel the nature of pseudogap that ubiquitously emerges from the doped fullerene films with no spatially modulated electronic charge density (Fig. 2a-c). This excludes a possible origin of the pseudogap from charge orders. Whether the pseudogap shares the same mechanism as that of cuprates and how it interplays with high-T c superconductivity remain unsolved issues that merit further investigations. In any case, our experimental results of fulleride superconductors shed important light on the electron pairing in narrow-band high-T c superconductors.

Methods
Sample preparations. Our experiments were conducted in a commercial Unisoku 1500 ultra-high vacuum STM facility, connected to an MBE chamber for in situ film preparation. The base pressure of both chambers is lower than 2.0 × 10 −10 Torr. C 60 molecules were evaporated from a standard Knudsen diffusion cell and grew layer-by-layer on nitrogen-doped SiC(0001) wafers (0.1 Ω cm) at 473 K, which were pre-graphitized by thermal heating (up to 1600 K) to form bilayer graphene. Desired alkali metal atoms (Rb or Cs) were then deposited from thoroughly outgassed SAES getters on C 60 films at a low temperature of ∼200 K step by step, followed by >3 h of post-annealing at room temperature. The layer index n of Rb x C 60 multilayers (n ≤ 3) is determined from the step height, STM topographies and tunneling dI/dV spectra, which varies significantly with n. The flux rate of C 60 is therefore calculated by dividing the coverage of fulleride films by growth time, and is used as a reference for the determination of the nominal thickness for thicker fullerides, e.g., n = 6 and 9 in the main text.
Away from half-filling, the electron doping x is calculated directly from the areal density of Rb vacancies (Fig. 3a) or excess Rb dopants (Fig. 3c). For x ∼ 3, there exist little defect that leads to trivalent fulleride films (Fig. 1a). A single Rb vacancy (excess) is reasonably considered as one missing (additional) Rb dopant relative to Rb 3 C 60 . For higher doping, x is estimated from the coverage of Rb clusters, since the excess Rb dopants are individually undistinguishable. By this method, the estimated x has a statistical error of <0.5%. STM measurements. After the sample growth, the fulleride epitaxial films were immediately transferred into our STM chamber for all STM and STS data collections at 4.6 K. A bias voltage was applied to the samples. To accurately characterize the superconductivity and electronic structure of fulleride films, special measures such as grounding and shielding were taken to optimize the stability and spectroscopic resolution of our STM facility. Polycrystalline PtIr tips were used after careful calibration on Ag films grown on Si(111). All STM topographic images were taken in a constant current mode. Tunneling dI/dV spectra and electronic DOS maps were acquired using a standard lock-in technique with modulation frequency f = 975 Hz, while the modulation amplitudes were 0.2 meV and 20 meV for measuring the superconducting gaps and wider-energy-range (±1.0 eV) dI/dV spectra, respectively. The empty circles and squares distinctively mark the experimental Δ measured in K 3 C 60 and Rb 3 C 60 films, respectively. Contour plots of Δ with a separation of 2 meV are shown in gray dashes. Note that the fulleride superconductivity is always peaked at half-filling at any specific U/W, with W denoting the t 1u bandwidth.