Superconductivity of K‐Intercalated Epitaxial Bilayer Graphene

Graphene‐based materials are among the most promising candidates for studying superconductivity arising from reduced dimensionality. Apart from doping by twisted stacking, superconductivity can also be achieved by metal‐intercalation of stacked graphene sheets, where the properties depend on the choice of the metal atoms and the number of graphene layers. Many different and even unconventional pairing mechanisms and symmetries are predicted in the literature for graphene monolayers and few‐layers. However, those theoretical predictions have yet to be verified experimentally. Here, it is shown that potassium‐intercalated epitaxial bilayer graphene is a superconductor with a critical temperature of Tc = 3.6 ± 0.1 K. By scanning‐tunneling microscopy and angle‐resolved photoelectron spectroscopy, the physical mechanisms are analyzed in great detail, using laboratory equipment. The data demonstrate that electron–phonon coupling is the driving force enabling superconductivity. Although the consideration of an s‐wave pairing symmetry is sufficient to explain the experimental data, evidence is found for the existence of multiple energy gaps. Furthermore, it is shown that low‐dimensional effects are most likely the cause of a gap ratio of 6.1 ± 0.2 that strongly exceeds the Bardeen‐Cooper‐Schrieffer (BCS) value of 3.52 for conventional superconductors. These results highlight the importance of reduced dimensionality yielding unusual superconducting properties of K‐intercalated epitaxial bilayer graphene.


Introduction
Superconductivity in the ultimate 2D limit of monoatomic layers is an important issue of fundamental research that focuses on understanding unconventional physical phenomena. [1,2] microscopy (STM) and angle-resolved photoelectron spectroscopy (ARPES), respectively.
A recently published study reports on a similar structural transition from epitaxial MLG on SiC(0001) to epitaxial BLG upon Ca intercalation, although the Ca atoms form a different ( 3 3) × R30° registry. [29] In particular, Toyama et al. show that Ca-intercalated epitaxial BLG is a superconductor with critical temperatures up to T c = 5.7 K. In contrast to K-intercalated epitaxial BLG, [28] filling of only one interlayer band is observed upon Ca intercalation. The detailed characterization of the superconducting state via transport measurements in Ref. [29] allows the conclusion that the interface between graphene, i.e., the former buffer layer, and the SiC substrate is crucial for enabling superconductivity in Ca-intercalated epitaxial BLG. However, the actual pairing mechanism leading to the superconducting state, which is essential for understanding the physics of superconductivity in metal-intercalated graphene-based systems, remains an open question.
In this study, we first prove superconductivity of K-intercalated epitaxial BLG on SiC(0001) with a critical temperature of T c = 3.6 ± 0.1 K by investigating the temperature-dependent energy gap using STM. We find that conventional s-wave pairing is sufficient to describe our data. We further identify electron-phonon coupling as the driving mechanism enabling superconductivity in this material in thin films as well. By a systematic analysis of our ARPES data, we even determine the electron-phonon coupling strength. However, we quantify an unconventionally high gap ratio of 2Δ 0 /k B T c = 6.1 ± 0.2, and we exclude strong coupling effects as a reason for this high value. Instead, it is most likely a result of the reduced film thickness compared to the bulk counterparts.

Analysis of the Energy Gap
We fabricated K-intercalated epitaxial bilayer graphene from an epitaxial monolayer graphene sample, where the covalently bound buffer layer decouples upon potassium intercalation from the SiC(0001) substrate (see Figure 1c). The K atoms form a (2 × 2) registry with respect to the graphene lattice below the uppermost graphene layer [28] that is mapped in the STM image in Figure 1a. We further examined the electronic structure of the Fermi level region as a function of temperature using high-resolution scanning tunneling spectroscopy (STS) shown exemplarily for the data acquired at the lateral position marked by the red cross in Figure 1a. The temperaturedependent differential conductance spectra in Figure 2a reveal a dip at the Fermi level (V = 0) in the spectrum recorded at 1.2 K (blue line). This dip is no longer present in the measurement at 4.0 K (black line), which already suggests the closure of the superconducting gap (compare Section S1, Supporting Information). Similar measurements of the pristine epitaxial MLG sample and of a pristine epitaxial BLG sample do not exhibit a dip that would suggest the existence of an energy gap at the Fermi level (see Section S2, Supporting Information).
The standard approach that is usually applied for the quantitative analysis of energy gaps of a superconductor presumes a sufficiently flat density of states (DOS) in the normal state of the sample (and tunneling tip) in the Fermi level region, which would result in flat differential conductance G = dI/dV spectra above the critical temperature T c . However, even though the dI/dV spectrum acquired at 4.0 K (black line in Figure 2a) does not show a narrow dip at the Fermi level, it is nevertheless not flat, which is the result of quasiparticle interference effects discussed in detail in Ref. [28]. Consequently, the standard approach that assumes flat densities of states in the region of the Fermi level is not applicable here. Instead, a deconvolution of the spectrum recorded at 4.0 K is performed by modeling the different contributions to the total tunneling current and 500 pA, T = 1.2 K) of K-intercalated epitaxial BLG that exhibits a contrast modulation caused by the K atoms forming a (2 × 2) superstructure with respect to the graphene lattice below the topmost graphene layer. The red cross indicates the lateral position where the STS spectra of Figure 2 are recorded. b) Top view of K-intercalated graphene. K atoms (red) are located above or below the empty sites of the graphene lattice (gray), forming a (2 × 2) registry. c) Tentative side view of K-intercalated graphene. The structure and amount of K atoms between the former buffer layer and the SiC substrate are unknown (cf. Ref. [28]).

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fitting the model to the actual data as described in detail in Section S3 (Supporting Information). The result of this procedure is shown as a red line in Figure 2a.
The detailed knowledge of the different contributions to the total tunneling current enables normalization of the measured data G(T i ) at each temperature to the spectrum G n (T i ) at the very same temperature that would be expected without the existence of an energy gap (see Section S4, Supporting Information). This procedure avoids artifacts that would result from unequal thermal broadening if G n were chosen for a different temperature than G. The temperature-dependent data normalized as G(T i )/G n (T i ) are depicted as color-coded map in Figure 2b and show that the size of the energy gap decreases with increasing temperature. Finally, the energy gap disappears at ≈3.6 K.
The modified approach introduced here allows a quantitative analysis of the data at each temperature based on the contributions to the total tunneling current determined from the spectrum at 4.0 K. Thereby, the contribution of the electronic sample DOS at each temperature is incorporated using a modified Dynes equation [30] (see Section S4, Supporting Information).
Using this approach, the temperature-dependent tunneling data (black lines in Figure 2c) can be described adequately (red lines) by a fitting procedure that varies the gap parameter Δ(T i ) and the lifetime parameter Γ(T i ) shown in Figure 2d,e, respectively. We emphasize that the temperature dependence of the gap parameter follows the gap equation for conventional superconductors [31,32] as depicted by the red line in Figure 2d. The numerical fit yields a critical temperature of T c = 3.60 ± 0.05 K and a gap parameter Δ 0 = 0.94 ± 0.02 meV, resulting in a gap ratio of 2Δ 0 /k B T c = 6.1 ± 0.2. Notably, the energy gap is sufficiently described by the modified Dynes equation [30] that includes lifetime broadening effects and non-flat densities of states in the normal-conducting state. Thus, there is no evidence in the spectral shape of the energy gap that would suggest a mechanism deviating from a conventional s-wave pairing of the Cooper pairs. This finding is in agreement with the spectroscopic data of Ca-intercalated bulk graphite (CaC 6 ), [33,34] which are -to the best of our knowledgethe only available tunneling data for a comparable system.
While the unusually high Γ values in Figure 2e may be linked to the existence of multiple gaps (for a detailed discussion see Figure 2. Analysis of the temperature-dependent energy gap around the Fermi level. a) Tunneling spectra at 1.2 K (blue), 4.0 K (black) and fit of the 4.0 K data (red), explained in detail in the Sections S3 and S4 (Supporting Information). b,c) Temperature-dependent dI/dV spectra (black, y-shifted by 0.25) normalized using the modified approach (G n determined at 4.0 K) and corresponding fitted spectra (red). The total gap size 2Δ(T i ) is depicted in panel (c) by green markers. d,e) Gap Δ and lifetime parameter Γ vs. temperature determined from the fitting procedure. Δ(T i ) is further fitted (red curve) using the gap equation [31] to determine T c and Δ 0 .

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Section S5, Supporting Information), it is still an open question why the determined gap ratio of 6.1 ± 0.2 drastically exceeds the expected Bardeen-Cooper-Schrieffer (BCS) value of 3.52 for conventional superconductors and even the already larger value reported for CaC 6 (gap ratio 4.6 [34] ). High gap ratios are commonly observed for strongly coupled superconductors, [35] which was also suggested in Ref. [34] for the elevated gap ratio of Ca-intercalated graphite. Thus, the EPC strength will be determined in the following by a systematic analysis of related ARPES data.

Electron-Phonon Coupling Strength
For the determination of the EPC strength, we acquired ARPES data for the same sample already examined with tunneling spectroscopy. The lowest sample temperature accessible in the ARPES setup is ≈35 K, which is significantly above the critical temperature. Thus, a direct ARPES analysis of the superconducting gap around the Fermi level is not possible. The red line in Figure 3d illustrates the reciprocal-space section at the Fermi level where the data were collected. It corresponds to a slightly bent path nearly perpendicular to the ΓK direction. The ARPES data in Figure 3a exhibit a kink at about E B = 160 meV in the π* band accompanied by an increased intensity for low binding energies. Previous studies on alkali-metal-intercalated graphene-based systems attribute this observation to renormalization of the π* band upon electron-phonon coupling. [36,37] The magnitude of this renormalization depends on the adatom species [36] and the number of graphene layers, as the comparison of the particular case of K-intercalated graphite [15] and MLG on Au/Ni [19] reveals.
The ARPES data are analyzed using the standard three-step procedure as for instance in Ref. [37]. A detailed description of the procedure to extract a 2 F(w) and the EPC strength l(w) is provided in Section S4, Supporting Information.
The comparison of a 2 F(w) with the phonon DOS F(w) (Figure 3e) shows that the peaks centered at ≈60, 80, 160, and 185 meV relate to phonon modes of graphene, whereas the slight shift of the highest-energy feature is probably a result of the modified environment upon K intercalation. However, the peak centered at ≈130 meV does not possess a direct counterpart in F(w). Remarkably, a similar feature has also been observed for K-intercalated MLG on Au/Ni. [19] Furthermore, a theoretical study of heavily n-doped graphene predicts such a contribution to a 2 F(w) due to renormalization of the phonon  Supporting Information). b,c) Data (black) and fit function (red) of the real and imaginary part of the electron-phonon contribution to the self energy, respectively. d) Surface Brillouin zone of graphene (gray) and the (2 × 2) superstructure (red) as well as the Fermi surfaces at both K points. The red solid arc (highlighted by an arrow) represents the cut in k space at E F where the ARPES data in panel (a) were determined. e) Eliashberg function a 2 F(w) (red) and mass enhancement factor l(w) (blue) as a function of phonon energy ℏw. For comparison, the calculated phonon DOS F(w) of a graphene monolayer (MLG) and a graphene bilayer (BLG) is shown (from Ref. [39]).

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dispersion of in-plane optical modes upon n-doping, which is also responsible for the softening of both high-energy modes as observed here. [38] In comparison to this study, the contributions to a 2 F(w) discussed so far are well described by renormalization of the graphene phonon dispersion upon n-doping.
On closer inspection, a 2 F(w) exhibits a low-energy peak centered at ≈10 meV that is similar to the low-energy feature suggested from the analysis of the tunneling spectroscopy data (see Section S3, Supporting Information), but not predicted in the calculations of highly n-doped (single-layer) graphene. [38] The energy of this feature agrees well with the out-of-plane acoustic mode of AB-stacked BLG. [39] According to our structural analysis, [28] however, K atoms are located between the topmost graphene and the former buffer layer, so that i) adjacent layers are most likely arranged by AA stacking [7,40] and ii) the interlayer spacing is definitely increased compared to freestanding BLG [40] as it is also apparent in K-intercalated graphite. [41] Therefore, the calculated low-energy mode of AB-stacked BLG is most likely modified and does not explain this low-energy feature here. Thus, it is more likely assignable to a vibrational mode of the metal atoms originating from in-plane displacements that is theoretically predicted for freestanding, K-intercalated, AA-stacked BLG at ≈8 − 9 meV. [40] Similar low-energy modes were also reported in theoretical studies regarding freestanding K-intercalated MLG [42] and BLG, albeit with lower K content (C 6 KC 6 ). [43] Such low-energy inplane modes are known to be essential for superconductivity in graphite intercalation compounds. [40,44] From the Eliashberg function, we estimate an EPC constant of l ≈ 0.46. We emphasize that low-energy contributions have a crucial impact on this value due to the 1/ω weighting of a 2 F(w). As already mentioned, K-intercalated graphene-based systems all exhibit anisotropic EPC strengths, resulting in varying l values for every direction in k-space. [15,17,19,[45][46][47][48] The l values reported for K-intercalated MLG on Au/Ni range from ≈0.1 in ΓK direction to ≈0.2 in KM direction. [19] Although extracted from a direction where an even lower value is expected, the EPC constant identified here exceeds the reported maximum value by a factor >2. Apart from the low-energy mode of the K interlayer, this originates from an increased intensity of the peaks in a 2 F(w) that can be ascribed to a significantly enhanced intensity of the phonon DOS. Similar to the twofold degeneracy in AB-stacked pristine BLG, [39] this can be provided by an additional graphene layer. The enhanced intensity must originate from the former buffer layer that is not available in the MLG on Au/Ni sample studied in Ref. [19]. In turn, this proves the effective decoupling of the former buffer layer from the SiC substrate, since it acts like freestanding BLG regarding the phonon degeneracy. In contrast, Ref. [49] reports l ≈ 0.3 for K-intercalated epitaxial MLG on SiC, determined in the same direction in k-space as the data discussed here. There, the K atoms were deposited at low temperatures, [50] which resulted in a severely lower doping level (n = 0.56 × 10 14 cm −2 [49,50] ) than in the sample studied here (n = 4.46 × 10 14 cm −2 [28] ).
K-intercalated bulk graphite (KC 8 ) also exhibits an anisotropic EPC strength ranging from l ≈ 0.15 in ΓK direction to l ≈ 0.79 in KM direction, with an average of l = 0.45 determined at the angle bisector between the ΓK and the KM direction. [15] The average value reported in Ref. [15] and the EPC constant extracted here were determined in almost identical directions (cf. Figure 3d) and agree quantitatively. Thus, the EPC strength determined here is comparable to K-intercalated graphite rather than K-intercalated single-layer graphene because of the effective decoupling of the former buffer layer. The comparison to the data reported for KC 8 [15] further reveals that the determined l value in the given direction here represents a sufficient estimate for the average EPC strength.
It should be noted that the substrate and the K atoms intercalated beneath the graphene sheets most likely have an influence on the electron-phonon coupling as well. In particular, there is a feature in the Eliashberg function a 2 F(w) at ≈130 meV that has no direct counterpart in the phonon dispersion F(w), cf. Figure 3e. Such a feature is not predicted in theoretical studies of similar systems (e.g., C 6 KC 6 in Ref. [43]) where freestanding K-intercalated bilayer graphene is treated without considering a substrate. However, the effect of the SiC substrate on the phonon dispersion of K-intercalated epitaxial bilayer graphene could only be addressed by additional calculations that are computationally extremely expensive and thus currently not feasible. In fact, to the best of our knowledge, all theoretical studies on graphene mono-and bilayers decorated or intercalated with metal atoms consider freestanding systems, i.e., none of those studies takes the possible influence of an underlying substrate into account.
From the average l = 0.46 and the Eliashberg function a 2 F(w) the critical temperature can be estimated. [51] Considering the theoretical value of the screened Coulomb pseudopotential μ* = 0.14 [14] a critical temperature of T c ≈ 3.7 ± 0.1 K can be derived based on ARPES, which compares favorably with T c = 3.60 ± 0.05 K extracted from tunneling spectroscopy. The agreement between the independently determined critical temperature values is remarkable, given that the ARPES data were acquired significantly above T c and that for the analysis of those data an average value of the EPC strength is assumed.

Discussion of the Superconducting Properties
The Eliashberg function allows us to estimate the logarithmically weighted phonon energy as ℏω log = 68 meV according to the method proposed in Ref. [52]. On that basis, the influence of the EPC on the gap ratio can be evaluated. [35] For our critical temperature determined via STS (and ARPES), the impact of the EPC on the gap ratio is negligible. Thus, the extraordinarily high gap ratio of 6.1 ± 0.2 determined from the STS analysis cannot be attributed to strong-coupling effects. As already mentioned above, tunneling experiments of CaC 6 also reveal an elevated gap ratio of 4.6, which Kurter et al. attribute to strong-coupling effects. [34] Moreover, an average EPC constant of l = 0.53 was reported for that system. [17] Since l is directly related to the gap ratio, [35] the lower average EPC constant of l ≈ 0.46 determined here would suggest a gap ratio even smaller than 4.6. Nonetheless, the gap ratio determined in the present study exceeds that value by far. This comparison implies that a different mechanism must be responsible for the high gap ratio in K-intercalated epitaxial BLG.
The quantitative agreement between the critical temperature estimated from the EPC and the T c determined via STS www.advmatinterfaces.de suggests that the electron pairing leading to superconductivity takes place at the Dirac cone. Tunneling to bands centered at the K point can also be feasible and efficient for a low stabilization bias, due to the small tip-sample distance. [53,54] It is conceivable that a similar EPC on one (or both) of the interlayer bands is responsible for the electron pairing and the observation of an energy gap related to superconductivity via STS. As already discussed, different pairing processes might occur on one (or both) interlayer band(s) leading to the unusual temperature dependence of the unexpectedly high broadening parameter Γ.
High Γ values were also reported for nanometer-sized domains of conventional superconductors accompanied by drastically increased gap ratios. [55][56][57] In particular, for Pb islands with a diameter of 9 nm an elevated Γ ≈ 0.7 meV and a gap ratio >6 were reported. [56] Elevated Γ values (0.1 − 0.6 meV) were also reported for superconducting La films upon decreasing film thickness. [58] Since the thinnest film in Ref. [58] was ≈10 nm thick, a severe impact of finite size effects on the broadening parameter Γ, but also on the gap ratio of the sample investigated in this work (film thickness <1 nm), is expectable. Accordingly, the high Γ value and the extraordinarily high gap ratio of K-intercalated quasi-freestanding epitaxial BLG are most likely caused by low dimensional effects due to the limited thickness. Since the largest area, where the (2 × 2) superstructure could be imaged, was ≈15 × 15 nm 2 , lateral confinement in the nanometer-regime cannot be ruled out. This might also have an impact on the gap ratio. Due to the unusual temperature dependence of Γ, multiple energy gaps from additional pairing on the interlayer bands are probable, and the value can also be influenced upon anisotropic coupling.

Conclusion
Our local and area-averaged spectroscopic results provide a consistent picture of the superconducting properties of K-intercalated epitaxial bilayer graphene. We found that the superconductivity of this system with a T c of 3.6 ± 0.1 K is caused by the presence of interlayer bands in combination with electron pairing mediated by phonons of the K atom interlayer and both graphene layers. However, the temperature-dependent tunneling data exhibit evidence of multiple gaps that are possibly enabled by the presence of interlayer bands or anisotropic coupling. Moreover, an extraordinarily high gap ratio of 6.1 ± 0.2 was found that is most likely not a consequence of strong coupling, but can be attributed to low-dimensional effects. In comparison to K-intercalated few-layer graphene, which exhibits a T c of 4.5 K, [13] the critical temperature here is slightly lower. In the few-layer thickness regime, an increased interlayer spacing might still be present and affect the superconducting properties. Considering the T c of 0.55 K reported for K-intercalated graphite, [12] there seems to be a non-monotonic dependence of the critical temperature on the number of K-intercalated graphene layers and, thus, an optimal layer thickness for a maximal T c . Compared to few-layer graphene, however, the lower value determined here most likely arises from low-dimensional effects owing to a pronounced influence of the surface and/ or a non-negligible influence of the SiC substrate underneath.
Our study demonstrates the important role of reduced dimensionality yielding unusual superconducting properties of K-intercalated epitaxial bilayer graphene.

Experimental Section
Epitaxial MLG samples were prepared from 6H-SiC(0001) wafers according to the procedure reported in Ref. [28]. The graphene layer count was double-checked using X-ray photoelectron spectroscopy [59] and ARPES. [60] All experiments were carried out under ultrahigh vacuum conditions (base pressure ≈ 10 −10 mbar).
Potassium was deposited from dispensers (SAES Getters and Alvatec), while the sample was kept at 298 K.
Tunneling experiments were performed with a JT-STM/AFM system from SPECS Surface Nano Analysis by using a mechanically cut PtIr tip. The STM system was operated with a Nanonis SPM controller (Version 5) featuring an integrated lock-in amplifier that was used for the STS experiments. During the spectroscopic measurements, the feedback loop controlling the tip height above the sample was switched off, and a modulation voltage of V mod = 0.1 mV with a frequency of f mod = 1612 Hz was applied at the lock-in amplifier if not mentioned otherwise. If not further specified, the presented spectra were each the average of 10 accumulated spectra including forward and backward sweeps. A minimum temperature of 1.2 K was reachable. The temperatures were determined with Cernox sensors located close to the sample as a part of a resistance bridge, both from Lake Shore Cryotronics. In thermal equilibrium, the absolute temperature within the cryoshield could be determined with an accuracy of ΔT < 0.01 K.
ARPES measurements were performed with a surface analysis system from SPECS Surface Nano Analysis equipped with a microwave-heated light source (UVLS) using He(I)a excitation (21.2182 eV) in combination with a toroidal mirror monochromator (TMM 304, line width <1 meV adjusted for p-polarization). A hemispherical electron energy analyzer (PHOIBOS 150) equipped with a delay line detector (3D-DLD4040-150) was used to analyze the emitted photoelectrons. The sample was cooled with liquid helium in a continuous-flow cryostat to reach a sample temperature of ≈ 35 K.
The analysis of the tunneling spectra and the ARPES data was performed with software tools developed by the authors in Matlab specifically for that purpose. Details of the procedures are provided in Sections S3 and S4 (Supporting Information).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.