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 Rb$_3$C$_{60}$ films against merohedral disorder, non-magnetic single impurities and step edges at the atomic scale. Also observed have been Yu-Shiba-Rusinov (YSR) states induced by deliberately incurred Fe adatoms that act as magnetic scatters. The bound states display abrupt spatial decay and vary in energy with the Fe adatom registry. Our results and the universal optimal superconductivity at half-filling point towards local electron pairing in which the multiorbital electronic correlations and intramolecular phonons together drive the high-temperature superconductivity of doped fullerenes.

Disorder, impurities in an otherwise homogeneous superconductor, are often undesired aliens because they may hinder observations of intrinsic properties of the host material [1][2][3] . Yet dopant impurities could also be a double-edged sword by leading not only to emergent hightemperature (Tc) superconductivity in cuprates and iron pnictides 4,5 but also to uncovering the underlying mechanism of unconventional superconductivity [6][7][8][9][10] , especially as multiple unusual states are complexly intertwined in these correlated electron materials 11,12 . Whereas it has been well documented that non-magnetic impurities little affect Cooper pairs in conventional superconductors 13,14 , they turn out to induce local bound states inside the superconducting gap and suppress superconductivity via pair breaking for unconventional pairing symmetries, for example, in a d-wave or s± wave superconductor 7,[14][15][16][17] . Recently, anomalous enhancement of superconductivity by disorder is another example of impurities revealing their fundamental significance for the low-dimensional supercondcutors [18][19][20] . It is therefore tempting to consider impurities as a blessing in disguise to understand the physics of candidate superconductors [4][5][6][7][8][9][10][11][12][13][14][15][16][17] , to strive for optimal superconductivity 21 , and to create electronic states that never emerge from pure superconducting systems 22 .
Unlike atom-based superconductors, an organic superconductor is a synthetic moleculebased compound that uniquely exhibits additional degrees of freedom related to its molecular orientation. Consequently, inequivalent molecular orientations might unavoidably take place.
Such orientational (merohedral) disorder has been seen early in pure and doped fullerenes 23,24 , but its impact on superconductivity of fullerenes is highly controversial [25][26][27][28] . Furthermore, the fullerides represent an unusual category of organic superconductors in which the multiorbital electronic correlations and electron-phonon interactions are both suggested to be essentially 4 significant to reach high-Tc superconductivity 29,30 . In this context, a systematic investigation of impurity effects on superconductivity of doped fullerenes would provide justification of the previously advocated s-wave pairing symmetry [31][32][33] , as well as advance the understanding of their exotic superconducting state. However, such experiment was unexplored and the roles played by magnetic and non-magnetic impurities remain lacking in fulleride superconductors.
In this work, we use a state-of-the-art 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 both the superconducting K3C60 films without merohedral disorder 33 and insulating Cs3C60 ones with great merohedral disorder 34 , the merohedrally disordered Rb3C60 films are superconducting.
This allows for atomic-scale visualization of merohedral disorder impact on superconductivity of fullerenes, which, together with a detailed STS investigation on magnetic and non-magnetic impurities, shows that superconductivity of fullerenes is entirely consistent with local s-wave pairing. By exploring the thickness and electron filling variations of superconducting gap , we establish a unified phase diagram of fullerenes in which the optimal superconductivity always develops at half-filling.

Merohedral disorder and its impact on superconductivity
Figure 1a depicts a representative STM topography of 9-monolayer (ML) Rb3C60 thin films epitaxially grown on graphitized SiC(0001) substrates. Evidently, not all C60 molecules have the same orientation, although one three-fold symmetry axis for every C60 is always perpendicular 5 to the surface. More specifically, many nanoscale domains with two distinct C60 orientations, related by 44.48 o rotation 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 C60 that occur randomly and cause great merohedral disorder in face-centered cubic (fcc)-structured trivalent fullerides [24][25][26] . The C60 orientations are more disordered in regions between adjacent merohedral domains. In order to quantify the merohedral disorder, we calculate the averaged orientational correlation function <cos(ij)> 35 , in which ij = i -j denotes the angle between nearest neighbor C60 molecules (i.e. i and j), and summarize them in Fig. 1c. As the alkali metals increase in atomic radius, a decrease of orientational correlation means increased merohedral disorder. This is primarily caused by the weaking of Coulomb repulsions between neighboring trivalent C60 ions associated with a lattice expansion 36 , which otherwise stabilize a long-ranged merohedral order in the K3C60 films 33 , to wit <cos(ij)> = 1.
Tunneling spectroscopy of fullerides probes the local density of quasiparticle states (DOS) and measures the superconducting energy gap at the Fermi level (EF). In Fig. 1d, we compare the tunneling dI/dV spectra on 9 ML trivalent fulleride films A3C60 (A = K, Rb, Cs) doped with different alkali metals. A noticeable observation is that the merohedral disorder blurs the two sharp DOS peaks around -0.4 eV and 0.1 eV, which happen exclusively for the merohedrally ordered K3C60 films (top curve) 33 . This results in a generally smooth variation of electronic DOS in both Rb3C60 and RbCs2C60 films, as theoretically anticipated 37 . It is, however, worth noting that the low-lying DOS width roughly estimated as the spacing between the two conductance minima below and above EF (> 1.2 eV, see the two dashed lines in Fig. 1d) is significantly larger than the commonly argued t1u bandwidth of  0.5 eV 37,38 . Such a discrepancy should originate 6 from the Jahn-Teller (JT) instabilities and Coulomb interactions omitted by three-band firstprinciples calculations 37,38 . In consideration of JT-induced subband splitting and electronic correlations, the t1u bandwidth would be substantially increased 30,39,40 and accords with our observations. A further enhancement of electronic correlations U pushes the t1u-derived DOS toward higher energy and concurrently opens a Mott insulating gap in the most expanded Cs3C60 films (see the bottom curve in Fig. 1d) 34 .
Despite significant variation in low-lying electronic DOS, superconductivity develops well in Rb3C60 films and has little to do with the nanoscale merohedral disorder. 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 regions between nearest neighbor merohedral domains, the superconducting gaps exhibit apparent coherence peaks (blue curves) and are immune to the local merohedral disorder (Fig. 1a). This is more compellingly confirmed in Cs and Rb co-doped RbCs2C60 films, which imprint a somewhat stronger merohedral disorder but still exhibit a superconducting transition temperature up to Tc = 23 K ( Supplementary Fig. 1).
Nevertheless, the dI/dV spectra present some spatial electronic inhomogeneities, especially in the superconducting coherence peak. A careful examination of the superconducting Rb3C60 films at varied thicknesses and spatial locations intriguingly reveals that the coherence peak amplitude scales inversely with the gap size  (Supplementary Fig. 2). This is unexpected by the conventional wisdom of Bardeen-Cooper-Schrieffer (BCS) picture, and mostly ascribed to a coexistence of competing order, e.g. the ubiquitous pseudogap phase (Supplementary Fig.   3) 33 . Similar behavior and pseudogap phenomenology have been well documented in cuprate superconductors 41 .

Thickness and filling dependence of superconductivity
Having established the merohedral disorder-independent superconductivity in trivalent fullerides, we then explore its dependence on thickness and electron filling. Plotted in Fig. 2ac is the temperature variation of spatially-averaged tunneling spectra measured on 3 ML, 6 ML and 9 ML Rb3C60, respectively. Again, the fully gapped superconductivity with an isotropic s-wave gap function is consistently demonstrated in the superconducting Rb3C60 films and gets smeared out at elevated temperatures. By examining the temperature dependence of the gap depth in Fig. 2d, the critical temperature Tc is determined and increases from 23 K for 3 ML, to 26 K for 6 ML, and 28 K for 9 ML Rb3C60. Such a Tc evolution with film thickness stands in marked contrast to K3C60, where the maximum Tc occurs in 3 ML films 33 . This hints at another factor, possibly linking with the alkali metal ion size-dependent electronic correlations (U) of In what follows, we explore the superconductivity of RbxC60 by tuning the stoichiometry and thus electron filling x. Figure 2e summarizes the superconducting gap  (top panel) and averaged orientational correlation (bottom panel) as a function of Rb doping level x. Clearly,  increases with the film thickness, in good accordance with Tc (Fig. 2d). The extracted reduced gap ratio 2/kBTc = 6.0  0.4 is comparable to that of KxC60 films 33 , but appreciably exceeds the 8 canonical BCS value of 3.53. More interestingly,  invariably reaches its peak at half-filling irrespective of film thickness, and declines more quickly below half-filling for thin Rb3C60 films.
Notwithstanding a dome-shaped variation of , the merohedral disorder remains essentially unchanged with electron filling x and film thickness (bottom panel of Fig. 1e). This not only corroborates the above claim that superconductivity is unaffected by merohedral disorder 27,42 , 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 RbxC60 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 KxC60 ( Supplementary Fig. 5a,b) 33 , they do not alter profoundly the orientation of the nearby fullerene molecules and thus serve as intrinsically non-magnetic impurities to test the fully gapped superconductivity in fullerides. 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  43 . 9 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-pniticide superconductors 41,44,45 .
Figure 3e depicts a topographic STM image of one monomolecular step edge separating Rb3C60 epitaxial films between 8 ML (lower terrace) and 9 ML (upper terrace). We note that the step edge runs nearly along the close-packed directions of C60 molecules. Figure 3f shows a series of dI/dV spectra taken along a trajectory approaching the step edge (solid line in Fig. 3e). The superconducting gap proves to be undisturbed at the step edge and nearby, and no evidence of Andreev bound states is found. Some random variations in the coherence peak, including the pronounced coherence peaks near Rb excess impurity in Fig. 3d, might be related to the slight electronic inhomogeneity of superconducting Rb3C60 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 Rb3C60 surface at low temperature ( 100 K). Single Fe adatoms formed (bright protrusions) and occupied top or hollow sites of the surface C60 lattice, dubbed as Fe(I) and Fe(II) in Fig. 4a. Figure 4b represents the dI/dV spectra on both Fe impurities and defectfree regions. Note that multiple Fe(I) and Fe(II) impurities have been measured and averaged to eliminate the spatial inhomogeneity effects of dI/dV spectra. Evidently, both Fe(I) and Fe(II) adatoms act as magnetic scatterers and significantly suppress the superconducting coherence peaks, while a prominent zero-bias conductance peak (ZBCP) develops on Fe(I). They are the hallmarks of Yu-Shiba-Rusinov (YSR) states induced by a coupling of magnetic impurity to an s-wave superconductor 8,10,13,14,22,[46][47][48][49] . Figure 4c shows a series of tunneling spectra across an isolated Fe(I) impurity. The ZBCP intensity decreases quite abruptly and becomes barely visible 10 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 Rb3C60 films 50 . 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 solely on single magnetic Fe adatoms, in conjunction with the multiple forms of robustness of superconductivity against non-magnetic merohedral disorder and impurities, unambiguously confirm a sign-unchanged s-wave pairing state in fulleride superconductors. Distinct from a conventional superconductor, however, in charged fullerenes the t1u-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 [51][52][53] . As a result, superconductivity with local nonretarded attractive interactions 29,30 would be less sensitive to distribution of the electronic DOS in conduction band 54 , and instead determined by some ensemble averaged DOS 55 . This differs from the usual BCS superconductor where Tc is essentially governed by the DOS at EF, and happens to match the non-correlated superconductivity with the merohedral disorder that significantly change the t1u-derived DOS distribution (Fig. 1d). Moreover, such local electron pairing 29,30 , mediated by intramolecular JT phonons 56,57 , has also been reinforced by a short coherence length. As estimated from the vortex core radius (Supplementary Fig. 3a,c), the coherence length of a Cooper pair is 1.5  0.2 nm in Rb3C60 and 2.6  0.5 nm in K3C60 33 , respectively, which amount to only about twice the separation between nearest neighbor fullerene molecules.
In the local pairing mechanism, the key ingredients for high-Tc superconductivity are the strong coupling of the t1u electrons to intramolecular JT phonons in trivalent fullerides [55][56][57] . The phonon-mediated unusual multiorbital (attractive) interactions lead to an effectively inverted Hund's coupling (S = 1/2) 32 and a local spin-singlet s-wave pairing on the same orbital 30 , further enhanced via a coherent tunneling of pairs between orbitals (the Suhl-Kondo mechanism) 58,59 .
On the other hand, the multiorbital electronic correlations suppress electron hopping-induced charge fluctuations and more effectively bind electrons into intraorbital pairs 29 . In this sense, the Coulomb interactions actually helps the local pairing, until they are strong enough to drive a transition from a superconductivity to Mott-insulating phase 33 . Such a local pairing scenario naturally accounts for the dome-shaped dependence of Tc on C60 packing density-controlled U 27,32,42 as well as the conflicting variation of superconductivity with film thickness in K3C60 and Rb3C60. In K3C60, U is relatively small and its enhancement at reduced film thicknesses stabilizes the local pairing and thus enhances superconductivity 33 , whereas the opposite holds true due to the already strong electronic correlations in Rb3C60. A further enhancement of U pushes thin Rb3C60 films closer to a Mott transitions and thus weakens superconductivity, as observed above. In Fig. 5, we schematically illustrate a unified phase diagram of the charged fullerides and discover a universal optimal superconductivity at half-filling, no matter how the electronic correlations U varies with the alkali metal and film thickness. This finding is unusual and most probably stems from a decrease in the dynamical JT-related pair binding energy (Ux, negative) away from half-filling 57,58 , until the superconductivity vanishes as the Ux changes its sign. For the evenly charged fullerenes, U2 and U4 are positive and the JT coupling instead stabilizes two correlated insulating ground states 57,58 . It is also important to note that an asymmetry of  versus x phase diagram relative to half-filling (x = 3) occurs as the electronic correlations are 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 33 . In strongly correlated regimes, a small increase of U below half-filling would suppress superconductivity significantly and leads to the observed dome asymmetry.
Finally, we note that the large ratio of energy gap  to critical temperature (e.g. 2/kBTc > 6.0) seems to be a generic trait of high-Tc superconductivity in narrow-band systems 8,17,31,33,45 , including the copper-oxide superconductors 41 . Such a large deviation from the canonical BCS value of 3.53 could be straightforward and easy to understand theoretically in the framework of superconductivity with local nonretarded attractive interactions 54 . Experimentally, a local pairing mechanism, assisted cooperatively by a dynamic interfacial polaron, has been recently proposed to be responsible for the high-Tc superconductivity in monolayer FeSe epitaxial films on SrTiO3 substrate 60 . A question naturally arises as to whether the local paring mechanism is applicable to other narrow-band cuprates and multi-band iron pnictides 61 . Another interesting issue is to unravel the nature of pseudogap phenomenology that ubiquitously emerges from doped fullerene films (Fig. 2a-c). Whether the pseudogap shares the same mechanism as that of cupartes and how it interplays with cuprate 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-Tc superconductors.

Sample preparations.
Our experiments were conducted in a commercial Unisoku 1500 ultra- 13 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. C60 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 a bilayer graphene. Desired alkali metal atoms (Rb or Cs) were then deposited from thoroughly outgassed SAES getters on C60 films at a low temperature of  200 K step by step, followed by > 3 hours of post-annealing at room temperature. The layer index n of RbxC60 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 C60 is therefore calculated by dividing the coverage of fulleride films by growth time, and is used as a reference for 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 Rb3C60. 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 14 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.  Note that the fulleride superconductivity is always peaked at half-filling at any specific U/W, with W denoting the t1u bandwidth.