Topological Fermi-arc surface state covered by floating electrons on a two-dimensional electride

Two-dimensional electrides can acquire topologically non-trivial phases due to intriguing interplay between the cationic atomic layers and anionic electron layers. However, experimental evidence of topological surface states has yet to be verified. Here, via angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM), we probe the magnetic Weyl states of the ferromagnetic electride [Gd2C]2+·2e−. In particular, the presence of Weyl cones and Fermi-arc states is demonstrated through photon energy-dependent ARPES measurements, agreeing with theoretical band structure calculations. Notably, the STM measurements reveal that the Fermi-arc states exist underneath a floating quantum electron liquid on the top Gd layer, forming double-stacked surface states in a heterostructure. Our work thus not only unveils the non-trivial topology of the [Gd2C]2+·2e− electride but also realizes a surface heterostructure that can host phenomena distinct from the bulk.

Although the floating electrons on the surface of [Gd2C] 2+ ×2e -were experimentally captured with sizable electron correlation strength evidenced by its liquid nature 9 , the existence of topological surface states has not been demonstrated experimentally.To complete the presence of these unique surface states at the edge of 2D electrides, together with unique stacking formation, we investigated the electronic structure of [Gd2C] 2+ ×2e -using angle-resolved photoemission spectroscopy (ARPES), scanning tunnelling microscopy (STM) and density functional theory (DFT) calculations, focusing on the topological aspects of Weyl fermions in the bulk states and Fermi-arc states at the surface.

Results
To understand the non-trivial surface states of [Gd2C] 2+ ×2e -, we studied the band structure of [Gd2C] 2+ ×2e -with DFT slab calculations.Figures 1b-d show the calculated band contribution of surface floating electrons, topmost Gd atomic orbitals, and interstitial anionic electrons (IAEs) localized between the layers along the high symmetry directions, respectively.Red and blue colours represent spin up and down states, split by the bulk ferromagnetic order of [Gd2C] 2+ ×2e -14,15 .It is noteworthy that the bands with dominant contribution from the electrons in the topmost Gd atoms are distinct from previously observed floating bands and bulk IAE bands 8 , implying the existence of additional surface states at topmost atomic layer.In detail, the pair of parabolic bands along Γ " − K % point ascribes to spin up/down bands of floating electrons.Distinguished from the floating electrons, the additional surface character is captured for the band along K % − M % direction (purple arrow in Fig. 1c), which follows the predicted dispersion of Fermi-arc states in the previous report 3 .
Figure 1e displays a Fermi surface (FS) of [Gd2C] 2+ ×2e -measured by ARPES.Apparently, all the band structures of IAEs, floating electrons and Fermi arc are captured in the FS, indicating that they are located within the probing depth of ARPES.The circular FS centered at Γ " point originates from the floating electrons 9 , and their Fermi liquid-like dispersion is confirmed in Figure 1f.The complex FS around Γ " point is mostly from [Gd2C] 2+ layers and IAEs.Noticeably, the clue of the predicted Fermi-arc states is found near K % points, as marked by the dashed curves in Figure 1e, which will be discussed in detail.
Among the key band features measured by ARPES, we first investigate the floating electrons in real space using STM. Figure 2a shows an STM image measured on the as-cleaved [Gd2C] 2+ ×2e -sample, capturing the floating electrons on the surface, as the nearest states to the STM tip. Figure 2b depicts the height profile obtained along the dotted line in Figure 2a.The step height is approximately 6 Å, agreeing with the atomic structure of [Gd2C] 2+ ×2e -(ref.14).The ARPES measurement reveals that the band onset of the floating electrons lies 0.25 eV below the Fermi energy (the inset in Fig. 2c), which is represented as a peak in the STM spectrum (Fig. 2c).Owing to the delocalized nature of the floating electrons, quasiparticle interference (QPI) patterns are expected to be observed around the impurities and step edges.
Figures 2d and 2e depict an STM image obtained near the Fermi energy (-50 meV) and its differential conductance (dI/dV) map, respectively.The QPI patterns in the dI/dV map are isotropic, and their Fourier transformation (FT) shows that the modulation vectors (qs) form a ring shape around the Γ " point (Fig. 2h).
The size of the modulation vectors agrees well with the scattering vectors taken within the surface electron pocket measured by ARPES, as indicated by the red arrows in Figures 2g and 2h.As the STM bias voltage is decreased, the size of q decreases, resulting in a longer wavelength in the QPI pattern (Figs.2f,i).The decreasing trend of q by the bias voltage aligns with the scattering vectors extracted from the surface electron pocket measured by ARPES (Fig. 2j and Supplementary Fig. 1).
Turning our focus to the band topology of [Gd2C] 2+ ×2e -, Weyl points (WP) accessible with ARPES (W13 and W14 points) are investigated.They are expected to be located near kz = π plane along Γ " − M % high symmetry line as represented with blue and red spheres in Figure 3a.The dispersion measured along Γ " − M % direction at kz = π plane is given in Figure 3c.In the red box, the cone-like feature is captured, which is spreading from W13 or W14 points in the momentum space.Since Weyl cones have 3D dispersions, i.e., gapless only at the WP, the kz modulation of band dispersions should be examined.The upper panels of Figure 3d are band dispersions along Γ " − M % direction at different kz positions, and the lower panels are 2D curvature plots of the upper panels 16 .In both plots, the cone-like dispersions are clearly shown, although the intensity of one band is dominating among the two intersecting bands.The uneven intensity observed between two intersecting bands is frequently observed in cases of Dirac/Weyl crossings, resulting from the differing symmetries of the corresponding wave functions of each band concerning the experimental geometry of light polarization and incident light direction [17][18][19][20] .The smooth intensity profile, lacking a true dip along the dominant band, can be attributed to the band crossing occurring without hybridization with the minor band.Hybridization typically encodes the character of the minor band into the dominant band.
By considering the intensity profile of the dominant band, and by tracing the dispersion of the minor band against the dominant band, the intersecting point for W13/14 + was determined to be located at kz = π, as predicted.This proves that the cone-shaped features are Weyl cones of [Gd2C] 2+ ×2e -.
Next, the photon energy dependence of the FS near K % point was diagnosed to capture the signature of Fermi-arc states.While keeping the in-plane momentum window as in the top plot, FSs are accumulated with different photon energies ranging from 90 eV to 110 eV which covers half of the Brillouin zone (BZ) along kz direction from 0 to π (Fig. 4).The schematics of the expected Fermi-arc states near the K % points are overlaid on the measured FSs.Blue and red dots are WPs with opposite chirality, and black dashed line segments correspond to the Fermi-arc states.Left and right panels in Figure 4 are raw images and their curvature plots, respectively.Remarkably, the shape of segments tightly follows the Fermi-arc states that are predicted in the calculations 3 .Furthermore, the segments of FS indicated with dashed lines are found to be intact across the photon energy.Compared to the apparent photon energy dependence of the adjacent bulk bands, this negligible photon energy dependence further shows the 2D nature of the Fermi-arc states.
Therefore, we conclude that Fermi-arc states exist in the [Gd2C] 2+ ×2e -, which completes the presence of Weyl fermions, together with the captured Weyl cones.
As the last step, the Fermi-arc states are investigated in real space by STM.Since the floating electrons cover the surface of Gd2C sample (Fig. 2a), STM cannot directly access the topmost Gd layer which harbours the Fermi arc.To remove such floating electrons, the sample was heated to 80 K and cooled to 4 K after an hour.Figure 5a shows the STM image of the [Gd2C] 2+ ×2e -sample after the heating cycle.
Remarkably, the floating electrons have vanished on the surface, exposing the Gd layer in the [Gd2C] 2+ .
Only some amounts of floating electron residue remain, primarily near the step edges.Figure 5b illustrates the height profile taken along the dotted line in Figure 5a.The 3 Å height of floating electrons is consistent with the theoretical calculation 8 .The removal of floating electrons is confirmed by the ARPES measurements, wherein the band of floating electrons is absent in the sample subjected to the heating cycle (inset of Fig. 5c and Supplementary Fig. 3).Accordingly, the floating electron peak is also missing in the STM spectrum (Fig. 5c).
Given that no floating electrons are present on the surface, the electrons under the surface must be accommodated to satisfy the electrostatic conditions.Within [Gd2C] 2+ •2e -, IAEs are squeezed between [Gd2C] 2+ layers, and their electron densities will change to screen the surface charges.Nevertheless, the ARPES data measured on Gd2C, which underwent the same heating process as in STM, clearly show the presence of Fermi-arc states on the surface (Supplementary Fig. 3), although there is a slight Fermi level shift.This demonstrates that the charge redistribution does not alter the bulk topology of the system.
Figure 4d shows the larger area of the Gd surface measured at Vbias = 25 mV.The dI/dV map obtained simultaneously with Figure 5d is presented in Figure 5e, where plentiful QPI patterns are developed.To understand the QPI patterns alongside the Fermi-arc states, we conducted FT analysis on the STM image and its dI/dV map.In the FT of the STM image (Fig. 6a), the lattice peaks (qBragg) of the Gd layer are identified, helping to define the BZ.The presence of the Gd lattice peaks indicates that there is no surface modification upon the removal of the floating electrons.Figure 6b displays the FT of the dI/dV map, revealing two distinct modulation vectors, q1 along the Γ " − M % direction and q2 along the Γ " − K % direction.
The magnitude of q1 extends up to 1.3 Å -1 beyond the BZ, while q2, slightly smaller than q1, also reaches the BZ boundary.These large modulation vectors can be constructed by the nesting of scattering wavevectors between the Fermi-arc states positioned at the K % and K′ % points, represented by the red and green arrows for q1 and q2 in Figure 6d, respectively.The depiction of the Fermi-arc states in Figure 6d is reproduced based on the ARPES data.
To further understand the nesting condition of the scattering wavevectors, we investigate the Fermiarc states associated with the q1 and q2 vectors.In Figure 6e, the q1 vector (red arrow) joins the parallel segments of Fermi-arc states, highlighted by the dotted lines, across the BZ. Figure 6f shows that the q2 vector (green arrow) connects the parallel segments of Fermi-arc states within the BZ.The spin degeneracy of the Fermi arc is already lifted by the bulk ferromagnetism and thus does not significantly affect the nesting conditions of q1 and q2 21 , although the spin-momentum locking property imposes an additional helical spin texture on the Fermi-arc states 3,22 .Notably, the parallel segments associated with the q1 and q2 vectors are closely tied to the WPs that are fixed in the momentum space.Therefore, simply speaking, the q1 and q2 vectors are linking the WPs.Altering the STM bias voltage could modify the detailed structure of the Fermi arc.However, as long as the WPs maintain their positions as topological objects, the sizes of q1 and q2 can remain constant by the bias voltage.
To check whether the q1 and q2 vectors are affected by the bias voltage, we lowered the bias voltage to -100 mV and obtained the QPI patterns (Fig. 5f).The FT of the QPI patterns is displayed in Figure 6c.When compared to Figure 6b, the positions of q1 and q2 remain relatively unaltered, although their intensities diminish.Figures 6g and 6h demonstrate that the nesting conditions for q1 and q2 can be loosened by lowering the bias voltage.However, the magnitudes of q1 and q2 do not change significantly since the Fermi-arcs states are anchored to the Weyl cone.This is further seen in the ARPES data, where the Fermi-arc states are mildly dispersive near the Weyl cone and thus the nesting vectors maintain their magnitudes (Supplementary Fig. 4).The overall trend of q1 and q2, depending on the bias voltage, is illustrated in Figure 6i, confirming that the modulation vectors q1 and q2 are only weakly affected by the minimal changes in the Fermi-arc states near the Weyl cone.This robust dispersion of QPI modulation vectors, which is different from the dispersion of the Fermi liquid shown in Figure 2j, again supports the topological nature of the Fermi-arc states of [Gd2C] 2+ ×2e -.

Discussion
Present results not only confirm the existence of Weyl fermions and Fermi-arc states in [Gd2C] 2+ ×2e - but also complete the full characterization of possible states at the edge of the system -a realization of heterosurface states with correlated electron liquid states and topological Fermi-arc states, as schematically illustrated in Figure 1a.This resembles recent approaches of material engineering, combining correlated and topological materials into 2D heterostructure in order to add up the topological flavour to the correlated phenomena or vice versa.An example would be attaching superconductivity to topological materials to induce the topological superconducting state by injecting the cooper pairs into the topological state through proximity coupling [23][24][25][26][27][28][29] .Beyond that, this platform of double-stacked heterosurface states can lead to an emergent phenomenon that is hardly achieved in bulk heterostructures, by virtue of the unique characteristics of states at the surface.For instance, the non-trivial momentum dependence of Fermi-arc states, i.e., only the segment of FS exists, could provoke a non-trivial proximity effect that is limited for the electron within a specific momentum range where the Fermi arc exists, which has not been accounted for before.Therefore, the insights gained in this study will trigger to see how the two essential physical ingredients of non-trivial topology and pure electron-correlated systems mutually influence each other in exploring emergent quantum phenomena 30 .

STM measurements.
STM measurements were performed using a home-built cryogenic STM working at 4.3 K.The [Gd2C] 2+ •2e − single crystal was glued on the sample holder using a conducting epoxy (EPO-TEK, H20E) in a glove box filled with Ar gas.A cleaving post was subsequently glued on the sample using a nonconducting epoxy (EPO-TEK, H74F).The sample was then transferred to the UHV chamber and cleaved below 20 K under the vacuum pressure better than 5 × 10 −11 Torr.All the measurements were conducted at 4.3 K.The dI/dV spectra and dI/dV maps were acquired using a standard lock-in technique (Signal Recovery, Model 7265) with a modulation frequency f = 716.3Hz.

DFT calculations.
DFT calculations were performed using the generalized gradient approximation with the Perdew-Burke-Ernzerhof (PBE) functional 31 and the projector augmented plane-wave method 32 implemented in the Vienna Ab initio Simulation Program code 33 .The 4f, 5s, 5p, 5d and 6s electrons of Gd and the 2s and 2p electrons of C were used as valence electrons.The plane-wave-basis cut-off energy was set to 600 eV.We have chosen a sufficiently thick slab model of [Gd2C] 2+ ×2e -consisting of 18 atoms and a vacuum layer of 20 Å along the c axis was used for the surface calculation.The middle layer of the slab was fixed in bulk position.
The geometrical relaxation of the slab was performed using 8 × 8 × 1 k-point meshes until the Hellmann-Feynman forces were less than 10 −5 and 10 −3 eV/Å, respectively.The empty spheres with a Wigner-Seitz radius of 1.25 Å were placed both on the surface and at the interlayer space to obtain the projected density of states of the surface and interlayer positions, respectively.