Identifying s-wave pairing symmetry in single-layer FeSe from topologically trivial edge states

Determining the pairing symmetry of single-layer FeSe on SrTiO3 is the key to understanding the enhanced pairing mechanism. It also guides the search for superconductors with high transition temperatures. Despite considerable efforts, it remains controversial whether the symmetry is the sign-preserving s- or the sign-changing s±-wave. Here, we investigate the pairing symmetry of single-layer FeSe from a topological point of view. Using low-temperature scanning tunneling microscopy/spectroscopy, we systematically characterize the superconducting states at edges and corners of single-layer FeSe. The tunneling spectra collected at edges and corners show a full energy gap and a substantial dip, respectively, suggesting the absence of topologically non-trivial edge and corner modes. According to our theoretical calculations, these spectroscopic features can be considered as strong evidence for the sign-preserving s-wave pairing in single-layer FeSe.

Recently, we have developed a promising way from a topo-logical point of view to settle the long-lasting debate between the sand s ± -wave pairing [30].Specifically, as shown in Figure 1(b), the inversion center in Se-Fe-Se triple-layer is at the nearest Fe-Fe bond center rather than the Fe site, which naturally generates the Rashba-type SOC between the nextnearest-neighbor Fe sites [15,[30][31][32][33]. Accompanied by the unique lattice structure and SOC, anomalous band degeneracies along the BZ boundary develop.As a consequence, a sign-changed s ± -wave pairing leads to a second-order superconducting state which supports two Dirac cones at the (01) edge and a pair of Majorana zero-energy modes at the corner between the (11) and (11) edges [Figure 1(b)] [30].In contrast, sign-preserving s-wave states remain topologically trivial even in the presence of inversion symmetric Rashba SOC.Therefore, the gapless edge modes and zero-energy corner modes, which can be directly probed by STM/S [Figure 1(c)], serve as the smoking-gun evidence to distinguish the sand s ± -wave pairing.
In this work, we investigate the superconducting states at isolated edges and corners of single-layer FeSe.Our spectroscopic investigations demonstrate that the superconductivity gets suppressed with moving close to the isolated edges but remains fully gapped along with both (01) and (11) edges.Furthermore, there is no evidence for corner Majorana modes at the intersection between the (11) and (11) edges.Based on our recent theoretical calculation [30], these topologically trivial superconducting states support the sign-preserving swave pairing in single-layer FeSe.

METHODS
High-quality FeSe thin films were grown on Nb-doped SrTiO 3 (001) (0.05 wt.%) substrates by molecular beam epitaxy.The surface is TiO 2 terminated after heating to 1150 °C for 15 minutes.Then FeSe films were fabricated by coevaporating Fe (99.995%) and Se (99.999%) from standard Knudsen cells onto SrTiO 3 kept at 450 °C.The growth rate was ∼ 0.024 unit cell (uc) per minute.By controlling the coverage to be less than 1 uc, well-defined edges with various orientations form between FeSe and vacuum.After annealing at 480 -490 °C for 4.5 hours, the sample was transferred in situ to the STM chamber.Topographic images were obtained by the constant-current mode.The tunneling spectra were measured using a standard lock-in technique with a sample bias (V s ) of 50 mV, a tunneling current (I t ) of 500 pA, and a bias modulation of 0.5 mV at 937.2 Hz.The differential conductance dI/dV (V ) of all spectra is calibrated proportionally by scaling dI/dV (V s ) to I t /V s .Commercial Pt/Ir tips were calibrated on Ag films before performing STM/S experiments.All STM/S experiments were carried out at 4.8 K.

EXPERIMENTAL RESULTS
Figure 2(a) shows a typical topographic image of singlelayer FeSe.The dark area in the upper right corner is the exposed SrTiO 3 substrate.The apparent height of the step between FeSe and SrTiO 3 is ∼ 660 pm at a V s of 1 V, which is slightly larger than the out-of-plane lattice constant of FeSe (550 pm) due to the varied density of states [34].This edge is along the (11) direction, as judged from the atomically resolved Se lattice structure shown in the inset image.Figure 2(b) shows two sets of tunneling spectra far from (blue) and near (red) the edge.There is no significant difference in the zero-bias conductance of these tunnel spectra.We also record tunneling spectra over the area indicated by the black dashed box in Figure 2(a).Figure 2(c) and Figure S1 in Supplemental Material [35] display the position-dependent superconducting energy gap (∆), where ∆ is extracted by half the distance between coherence peaks and Dynes formula [36], respectively.Obviously, ∆'s near the (11) edge are smaller than those in bulk, which may be due to the lattice discontinuity.Nevertheless, the zero-bias conductance, our main concern, is uniform in real space [Figure 2 01) edge.This specular edge extends more than 12 nm, providing an excellent opportunity to study the edge modes.As shown in Figure 3(b), the coherence peak shrinks near the (01) edge while the zerobias conductance remains unchanged.For further verification, we collect a spectroscopy map over the same field of view as shown in Figure 3(a).The evolutions of ∆ derived from coherence peaks spacing and Dynes formula are presented in Figure 3(c) and Figure S1 in Supplemental Material [35], respectively.Similar to the results obtained near the (11) edge, ∆ decreases when approaching the (01) edge.To capture the trace of the edge modes, we check the shape of tunneling spectra and find that almost all spectra near the (01) edge remain fully gapped.Figure 3(d) shows the zero-bias conductance map, and Figure 3(e) presents the column-averaged values.It is obvious that zero-bias conductance is position-independent.The difference in values of zero-bias conductance in Figures 2(e) and 3(e) could be due to the different sample batches and tip apexes.We also collect the spectra at other (01) edges and get the same results repeatedly (see section B of Supplemental Material [35]).Therefore, the superconducting states at the (01) edge are topologically trivial.
To further confirm the topological properties of single-layer FeSe, the corner states are studied.Figure 4(a) shows a topographic image containing a corner formed by the ( 11) and ( 11) edges, which can be verified by the Se lattice shown in Figure 4(b).Due to the epitaxial growth of FeSe, intersected ( 11) and ( 11) edges are rarely observed and generally extend only a few nanometers.Figure 4(d) presents a set of tunneling spectra measured along the grey arrow in Figure 4(a), with the top two spectra taken on the SrTiO 3 (001) surface.The superconductivity is gradually suppressed when approaching the corner, but without anomalies around 0 mV, that is, zero-bias conductance peak (ZBCP) as supposed for s ± -pairing is missing.For careful verification, we perform differential conductance mapping at 0 mV over intersections between the (11) and ( 11) edges.As exemplified in Figure 4(c), the differential conductance map corresponding to the area outlined by the black box in Figure 4(a) shows that the conductance is relatively uniform.We emphasize that the results are highly reproducible (see section C of Supplemental Material [35]), confirming the topologically trivial superconductivity in single-layer FeSe.

DISCUSSION
We first clarify the effects of the substrate on the detection of the edge/corner modes.The coupling between SrTiO 3 and FeSe may lead to the non-observation of topological modes in two ways: (i) The edge/corner states leak into the substrate.(ii) The non-uniform strain arising from structural instability of SrTiO 3 [37,38] introduces local mirror symmetry breaking, which is detrimental to the topological superconductivity.Case (i) is unlikely because the SrTiO 3 (001) surface contributes no electronic states near the Fermi energy [Figure 4(d) and Figures S2-S3 in Supplemental Material [35]], which in- dicates that the edge/corner modes will exponentially decay in the substrate and have to be localized around the edge/corner.For case (ii), the local mirror symmetry breaking, if any, is relatively weak as confirmed by topographic images [Figures 2(a), 3(a), 4(a) and Figures S2-S5 in Supplemental Material [35]].In addition, even if the two Dirac edge modes at the (01) edge could hybridize and be gapped out under weak mirror symmetry breaking, the single Majorana Kramers' pair located at the isolated corner can exist stably [39].On the other hand, the spectra collected under I t /V s = 10 nS with our equipment can capture anomalies in differential conductance of meV-scaled energy [Figure 2(d) and Ref. [21,40,41]].Therefore, the spectra measured at the edge/corner intrinsically reflect the topological property of single-layer FeSe.
The topological property of single-layer FeSe provides essential information on its pairing symmetry [30].In the case of s ± -wave pairing, the second-order topological superconductivity arises in centrosymmetric single-layer FeSe with the help of additional glide-plane and mirror symme- tries [30].Specifically, two Dirac cones and one single Majorana Kramers' pair are expected respectively at the (01) edge and the corner between the (11) and (11) edges.The Dirac cones contribute finite energy-independent density of states within the superconducting gap, and the Majorana Kramers' pair leads to a quantized ZBCP in the tunneling spectrum.In the case of s-wave pairing, however, the superconducting states at all edges and corners are topologically trivial, i.e., fully gapped.Therefore, as summarized in Table I, studying the spectroscopic features at edges/corners is a practical way to distinguish between the sign-preserving sand signchanged s ± -wave states.The tunneling spectra shown in Figures 2 ∼ 4 definitively exclude the existence of edge modes and Majorana modes.Consequently, we unambiguously conclude that the pairing symmetry of single-layer FeSe is the sign-preserving s-wave rather than the sign-changed s ± -wave.It is worthy to note that previous works, Refs.[16] and [42], have also investigated edge states of single-layer FeSe.Ref. [16] reports the fully gapped spectra along two kinds of (01) edges.The first kind of edge is formed by the 1 uc and 2 uc FeSe, and the second kind of edge consists of 1 uc FeSe situated on either side of the SrTiO 3 step.In the former case, the superconducting bottom layer of the 2 uc FeSe side [43] smoothly extends to the 1 uc FeSe side, indicating that such configuration is actually the edge of the nonsuperconducting upper layer of the 2uc FeSe side.In the latter case, FeSe on adjacent SrTiO 3 terraces is non-separate, since the thickness of FeSe (550 pm) is larger than the step height of SrTiO 3 (390 pm).The physics near such edge is elusive, and whether there is a response in electronic states to the topological superconductivity needs more detailed study.Under a simplest assumption that FeSe films on both sides are weakly linked, the tunneling spectra at the edge are expected to be fully gapped, regardless of the pairing symmetry.In contrast, all edges and corners investigated in our work are constructed of 1uc FeSe and vacuum, which is an ideal condition for detecting the edge/corner modes.Ref. [42] reports a pair of emergent conductance peaks located near the superconducting gap at the (01) edge.However, the conductance peak, which can only be resolved after a normalization done by subtracting the spectrum far from the edge, is most likely due to the reduction in superconducting gap near the edge and therefore are not related to topological properties (see section D of Supplemental Material [35]).

CONCLUSION
In conclusion, we fabricate high-quality single-layer FeSe by molecular beam epitaxy and investigate the electronic structures at different edges and corners by STM/S.We neither observe gapless edge modes at the (01) edges nor detect ZBCP at the corners between the (11) and (11) edges.The topologically trivial superconducting states are solid evidence supporting the sign-preserving s-wave pairing symmetry of single-layer FeSe.More delicate experiments are to be designed to identify the pairing glue supporting s-wave state such as orbital fluctuation, phonon, etc. [44][45][46].Finally, our achievements also pave a promising way to determine the pairing symmetry of other iron-based superconductors such as single-layer Fe(Se,Te) [47], K x Fe 2 Se 2 [48] and (Li 1−x Fe x )OHFeSe.

FIG. 1 .
FIG. 1.(a) Schematic illustration of nodeless d-, sign-changed s ±and sign-preserving s-wave pairings with band hybridization taken into account.Different colors denote the opposite sign of order parameters.(b) Definition of edges and the corner.The blue and red cones indicate the band structures at various edges in the case of sand s ± -wave pairing, respectively.The lower right panel indicates zero-energy modes for the s ± -wave case.(c) Supposed tunneling spectra at the (11) and (01) edges and the corner for the s-(blue) and s ± -wave (red) cases.The grey lines are spectra away from the edge/corner.
FIG. 2. (a) STM topographic image (V s = 1 V, I t = 50 pA) of the (11) edge.Inset: atomically resolved image (V s = 50 mV, I t = 500 pA) of FeSe.(b) Two sets of tunneling spectra measured along the blue and red arrows in (a).The black lines indicate individual zero conductance.(c)-(d) Energy gap map (c) and zero-bias conductance map (d) obtained from the spectroscopic mapping over the area outlined by the black dashed box in (a).The purple circles mark the origin point.(e) Column-averaged zero-bias conductance as a function of the distance to the (11) edge.

Figure 3 (
Figure3(a) depicts an STM image of (01) edge.This specular edge extends more than 12 nm, providing an excellent opportunity to study the edge modes.As shown in Figure3(b), the coherence peak shrinks near the (01) edge while the zerobias conductance remains unchanged.For further verification, we collect a spectroscopy map over the same field of view as shown in Figure3(a).The evolutions of ∆ derived from coherence peaks spacing and Dynes formula are presented in Figure3(c) and FigureS1in Supplemental Material[35], respectively.Similar to the results obtained near the (11) edge, ∆ decreases when approaching the (01) edge.To capture the trace of the edge modes, we check the shape of tunneling spectra and find that almost all spectra near the (01) edge remain fully gapped.Figure3(d)shows the zero-bias conductance map, and Figure3(e) presents the column-averaged values.It is obvious that zero-bias conductance is position-independent.The difference in values of zero-bias conductance in Figures2(e) and 3(e) could be due to the different sample batches and tip apexes.We also collect the spectra at other (01) edges and get the same results repeatedly (see section B of Supplemental Material[35]).Therefore, the superconducting states at the (01) edge are topologically trivial.To further confirm the topological properties of single-layer FeSe, the corner states are studied.Figure4(a) shows a topographic image containing a corner formed by the (11) and (11) edges, which can be verified by the Se lattice shown in Figure4(b).Due to the epitaxial growth of FeSe, intersected (11) and (11) edges are rarely observed and generally extend only a few nanometers.Figure4(d) presents a set of tunneling spectra measured along the grey arrow in Figure4(a), with the top two spectra taken on the SrTiO 3 (001) surface.The superconductivity is gradually suppressed when approaching the corner, but without anomalies around 0 mV, that is, zero-bias conductance peak (ZBCP) as supposed for s ± -pairing is missing.For careful verification, we perform differential conductance mapping at 0 mV over intersections between the (11) and (11) edges.As exemplified in Figure4(c), the differential conductance map corresponding to the area outlined by the black box in Figure4(a) shows that the conductance is relatively uniform.We emphasize that the results are highly reproducible (see section C of Supplemental Material[35]), confirming the topologically trivial superconductivity in single-layer FeSe.
FIG. 3. (a) STM topographic image (V s = 1 V, I t = 50 pA) of the (01) edge.Inset: atomically resolved image (V s = 50 mV, I t = 500 pA) of FeSe.(b) Two sets of tunneling spectra measured along the blue and red arrows in (a).(c)-(d) Energy gap map (c) and zerobias conductance map (d) obtained from the spectroscopic mapping over the same field of view as in (a).(e) Column-averaged zero-bias conductance as a function of the distance to the (01) edge.

FIG. 4 .
FIG. 4. (a) STM topographic image (V s = 1 V, I t = 50 pA) of the corner.Inset: zoom-in image.(b) Atomically resolved image (V s = 50 mV, I t = 500 pA) taken from the area outlined by the blue box in (a).(c) Zero-bias conductance map obtained from the spectroscopic mapping over the area indicated by the black dashed box in (a).The purple circles mark the origin point.(d) The tunneling spectra collected along the grey arrow in (a).

TABLE I .
Features of tunneling spectra of single-layer FeSe at various edges and corners.