Evidence for Helical Hinge Zero Modes in an Fe-Based Superconductor

TL;DR

This study provides experimental evidence that FeTe$_{0.55}$Se$_{0.45}$ is a higher order topological superconductor hosting helical hinge zero modes, using a novel junction fabrication method and detailed zero-bias anomaly measurements.

Contribution

The paper introduces a new method for creating normal metal/superconductor junctions with 2D atomic crystal heterostructures and demonstrates the presence of helical hinge zero modes in FeTe$_{0.55}$Se$_{0.45}$.

Findings

Zero-bias anomaly observed in hinge contacts consistent with helical hinge zero modes.

Suppression of the anomaly with temperature and magnetic field along the c-axis.

Evidence supports FeTe$_{0.55}$Se$_{0.45}$ as a higher order topological superconductor.

Abstract

Combining topology and superconductivity provides a powerful tool for investigating fundamental physics as well as a route to fault-tolerant quantum computing. There is mounting evidence that the Fe-Based Superconductor FeTe$_{0.55}$Se$_{0.45}$ (FTS) may also be topologically non-trivial. Should the superconducting order be s$^{\pm}$, then FTS could be a higher order topological superconductor with Helical Hinge Zero Modes (HHZM).To test the presence of these modes we developed a new method for making normal metal/superconductor junctions via 2D atomic crystal heterostructures. As expected,junctions in contact with the hinge reveal a sharp zero-bias anomaly whose suppression with temperature and magnetic field only along the c-axis are completely consistent with the presence of HHZM. This feature is completely absent when tunneling purely into the c-axis, and its characteristics are also inconsistent with other origins of zero bias anomalies. Furthermore, additional measurements with soft-point contacts in bulk samples with various Fe interstitial contents demonstrate the intrinsic nature of the observed mode. Thus we provide evidence that FTS is indeed a higher order topological superconductor as well as a new method for probing 2D atomic crystals.

Full text



Keywords: Higher-order topology, 2D superconductor, Hinge modes, Andreev Reflection

1 Introduction

New particles can be a convincing signature of emergent phases of matter, from spinons in quantum spin liquids1 to the Fermi arcs of Weyl semimetals2, 3. Beyond potentially indicating a broken symmetry or topological invariant, they can be put to use in future topological quantum computers4. Until recently it was believed the non-trivial topology of the bulk would lead to new states in one lower dimension at the boundary with a system of differing topology. However, higher order topological insulators (HOTI) have been realized5, 6, 7, 8, 9, 10, where the resulting boundary modes exist only at the intersection of two or more edges, producing 1D hinge or 0D bound states. One route to creating these higher order states is through the combination of a topological insulator and a superconductor with anisotropic pairing11, 12, 13, 14. Usually, this is done by combining two separate materials and inducing superconductivity into the TI via proximity15, 16, 17, 18, 19. However, this method requires long coherence lengths and extremely clean interfaces, making experimental realization of devices quite difficult. For studying HOTI, as well as the combination of strong correlations and topology, the material FeTe0.55Se0.45 (FTS) may be ideal, as it is a bulk, high-temperature superconductor with anisotropic pairing that also hosts topologically non-trivial surface states20, 21, 22.

FTS is part of the FeTe1-xSex family of Fe-based superconductors, which ranges from an antiferromagnet in FeTe to a bulk superconductor in FeSe23. These generally have the same Fermiology as the other Fe-based superconductors in that there are hole pockets at the $\Gamma$-point and electron pockets at the M-points24, 25, 26, 27, 20. The relative strengths of the interband vs intraband scattering in principle should determine the superconducting symmetry, however, there is a complex interplay between the spin-fluctuation exchange, intraband Coulomb repulsion, and the doping level that all contribute to the symmetry of the superconducting order parameter28, 29. Indeed, experiments performed on FeTe0.55Se0.45 find no evidence for a node with strong signatures of s*±* order,25, 26, 30 while experiments on other alloys suggest nodal s*±*, anisotropic s-wave, and even p-wave31, 32, 33, 34, 27. Interestingly, tuning away from FeSe leads to enhanced spin-orbit coupling and bandwidth. As a result, the p-orbital is shifted down in energy, crossing the d-orbitals with opposite parity along the $\Gamma$ to $Z$ direction (See Figure 1a and b). The first two crossings are protected by crystalline-symmetry resulting in bulk Dirac states above the Fermi energy. However, the lowest energy crossing is avoided resulting in a spin-orbit coupled gap, resembling those typically found in topological insulators21, 35. While the Fermi level falls into this gap, the original hole and electron Fermi surfaces at $\Gamma$ and $M$, respectively, are retained20, 21. ARPES measurements have observed the resulting spin-momentum locked surface states, as well as their gaping out in the superconducting state20, 36. Additionally, there is evidence from STM that this results in apparent Majorana zero-modes inside magnetic vortices22, 37, 38.

Recent theoretical work on FTS has suggested that the combination of an s*±* order parameter and topological surface states could give rise to higher order topological superconductivity12. In short, the changing superconducting phase causes the surface states to gap out anisotropically. Depending on the relative strength of the isotropic versus the anisotropic term, this could lead to the [001] and the [100] or [010] face having superconducting order parameters with opposite phase. As shown in Figure 1d), this is predicted to produce a pair of 1D Helical Majorana Hinge Modes emerging at the 1D interface of the top/side surfaces12. Whether or not the modes we observe are indeed Majorana modes, the appearance of HHZM requires both s*±* superconductivity as well as strong 3D TI surface states. Thus observing Helical Hinge Zero Modes in FTS would provide strong evidence that it is an s*±* topological superconductor.

To search for the HHZM it is tempting to rely on methods previously exploited to reveal the unconventional nature of the cuprates39. Specifically, normal-metal/superconductor junctions demonstrated Andreev Bound States resulting from the d-wave order only on [110] surfaces40, 41, 42, 19. In the case of FTS, this approach is more challenging as one must tunnel into the hinge between [001] and [010] and the modes are nominally charge neutral, thus requiring an Andreev process to be observed43. To achieve this, we created 2D atomic crystal heterostructures with thick hBN covering half of the FTS. By draping contacts over the side of the FTS or atop the hBN we can separately probe conductance into the hinge from the c-axis. As expected for modes protected from back-scattering, we find a cusp-like zero-bias peak only on the hinge contacts that is absent from the c-axis junctions. The mode is well-described by a Lorentzian, consistent with other studies on one-dimensional zero-energy bound states44. Confirmation that the mode does not result from our fabrication method or defect density is provided by soft-point contact measurements on facets of various bulk crystals (See Supplemental Fig S3). Taken together these data strongly suggest the presence of the HHZM in FTS resulting from its higher order topological nature and the presence of s*±* superconductivity45, 19. The helical hinge zero mode in FTS should only exist in the superconducting state. As such we expect a sharp zero-bias conductance feature below Tc on the hinges between the [001] and side surfaces as compared to purely on the [001] face. Alternatively, Majorana zero modes on the hinge should give quantized conductance, revealed through nearly perfect Andreev reflection.12 However, as discussed later, observing this quantized conductance may be challenging as the coherence length in FTS is $\approx 3nm$34, 33. To test this we used 2D atomic crystal heterostructures to simultaneously fabricate Normal Metal/Superconductor (NS) low-barrier junctions on various crystal facets (See Figure 2a and 2d). The first type of NS junction is a standard lithographically-defined contact that drapes over the edge of the exfoliated flake. This contact will form a junction with the [001] and [100] surfaces as well as the hinge between them. The second type of contact is fabricated by first transferring hexagonal Boron Nitride (hBN) over half of the FTS flake, insulating the side and edge from electrical contact. We then drape a contact over the side of the hBN, forming a junction primarily on the [001] face (See depiction of the side view in Fig 2d). The entire fabrication process, from exfoliation to device, is performed in an inert argon atmosphere or vacuum. Patterns for mesoscale contacts were defined using standard photolithography techniques and our Heidelberg $\mu$PG101 direct-write lithography system. Contact areas are then cleaned with an argon plasma at high vacuum immediately before thermal deposition of 5nm of Cr then 45nm of Au. Full fabrication details can be found in the Supplementary.

2 Results and Discussion

We first established that our control contacts are only tunneling into the c-axis by studying their base temperature differential conductance. Specifically, we sourced current between a top contact (5 or 6 in Fig. 2a) to one of the current leads (#1 or #4), while measuring the resulting voltage between the same top contact and the other current contact. This three-point experiment ensures the conductance results primarily from the interface of the top contact. As shown in Fig 2b, we observe a small zero bias conductance peak that is $\approx 20\%$ higher than the background. The shape and height are consistent with previous point contact Andreev reflection measurements along the c-axis of FeTe0.55Se0.45,46 and confirms the contacts are in the low-bias, Andreev regime. We note these previous works were performed at temperatures below our base temperature, and as such could resolve the rather small gap. At higher bias, we observe an enhancement in the conductance at $|V|\geq 20~{}meV$, consistent with spin-orbit induced gap. Above this value, we observe a series of conductance dips that are fully consistent with McMillan-Rowell Oscillations (MRO)47, 48. These MRO result from Fabry-Perot like interference of quasiparticles in the normal layer undergoing AR at the interface and reflecting off the back surface of the metal. The MRO are linearly spaced by voltages47 defined by the equation $\Delta(V)=n\cdot\frac{ev_{F}}{hd}$ where n is the dip number, $v_{F}$ is the Fermi velocity at the contact, and $d$ is the thickness of the metal which we set to 50 nm (See Figure 2e). From this fit, we extract a renormalized Fermi velocity of approximately $1.7\times 10^{5}m/s$. We note that similar behavior was observed if the current/Voltage was reversed between contacts #1 & #4, we measure from contact #6, or measuring between contacts #6 and #5 exclusively (see Supplemental Fig S4a). This shows the robustness of these results and combined with the detailed spectra, confirm the contacts over the hBN are Andreev tunneling only into the c-axis.

Next, we turn to the spectra measured in an identical manner, but with the hinge contact (#3 in Fig. 2a). Since the normal-state and high bias resistance of the hinge contact is nearly identical to the control contact we expect the spectra to be similar. However, as shown in Fig. 2c the zero-bias conductance in the hinge contact is quite distinct from the response observed in the control contact and previous point contact experiments. Specifically, we observe a cusp-like zero-bias conductance peak (ZBCP) in the hinge contact that reaches a value 17-times higher than the high bias or $T\approx T_{c}$ conductance. This rather large enhancement is also likely responsible for the absence of a clear observation of the gap, which would be far smaller. These results provide strong evidence for a zero mode that only exists on the hinge. The "cusp-like" shape and magnitude of the peak could result from an Andreev Bound State (ABS)39, 41, 42, however, this requires either a node in the superconducting gap or time-reversal symmetry breaking,49, 50 neither of which has been detected in FeTe0.55Se0.4532, 30, 33, 27, 26, 34, 25. As discussed later, direct evidence against the ABS interpretation is provided by the dependence of the peak on temperature, and near independence on the contact’s type (planar, point contact) or material (Ag, Au, Bi2Te2Se1). Interestingly, this behavior is also inconsistent with previous observations of standard Andreev Reflection(AR)40, Coherent Andreev Reflection (CAR)51, the Kondo Effect52, 53, and Joule heating54.

To ensure the zero bias conductance peak emerges at Tc and is not the result of an ABS, we directly analyzed its temperature dependence by fitting the data with a Lorentzian line shape. This is based on recent theoretical studies on one-dimensional superconducting wires showing that both Majorana Zero Modes and ABS produce a Lorentzian differential conductance spectra44. While this may not be the correct model for our case, to the best of our knowledge there are no calculations for the conductance spectra expected from hinge modes in a higher order topological superconductor. Nonetheless, the differential conductance spectra are generally well described by a Lorentzian (see Fig 3a). The temperature dependence of the height and width of the peak determined by the fits for the data presented in Fig. 2f are shown in Fig. 3b & c, respectively. These data provide direct evidence for the connection to the bulk superconductivity, though are inconsistent with an ABS. Indeed, we find that as the temperature is raised, the height of the ZBCP decreases exponentially until it is completely quenched at $T_{c}$ (see Fig2a and Fig3b), where we define $T_{C}$ as the temperature for which $\frac{dR}{dT}$ passes through zero. While lower temperature data are required to determine the exact functional form, it is clear from Fig. 3b & c that the mode is substantially different from the $1/T$ behavior typically expected from an ABS. Furthermore, we found a similar shape and temperature dependence in contacts of various barrier height, also inconsistent with standard Andreev reflection.55, 19, 56

Similar to the height of the peak, we find the width of the zero bias conductance peak grows exponentially with temperature (see Fig. 3). Interestingly the energy scale governing the peak height ($E_{H}\approx 0.08~{}meV$) and the width ($E_{\Gamma}\approx 0.1~{}meV$) are quite close. We note that comparable results were obtained from other contacts revealing the hinge mode. Nonetheless, the energy scales governing the temperature dependence of the mode are far smaller than either the superconducting gap of the bulk or the surface states.20 However, to the best of our knowledge, the size of the superconducting gap on the side surface has not been measured. As such we speculate this small apparent energy scale results from a much weaker proximity effect on the [010] and [100] surface states. Interestingly, extrapolating the width of the zero bias peak to zero temperature suggests an extremely narrow mode ($\approx 3.5~{}\mu eV$). While further studies at lower temperatures are required to confirm this extrapolation and the specific shape of the mode, if correct it points to the highly coherent nature of the excitation. As such the temperature dependence is consistent with our expectations for topologically protected 1D modes.

For additional confirmation that the ZBCP does not result from fabrication, exfoliation, impurities or the specific metal used in the contact, we performed a series of additional control experiments, summarized in Fig (4). First, the topological gap in FTS closes with reduced tellurium levels, thus we expect the hinge mode is absent from FeSe. To confirm this as well as the irrelevance of contact type or normal metal used, we employed soft-point contact measurements. For FeSe we observe no evidence of an increase in conductance at zero bias below Tc (see Fig (4a). However, performing the same soft-point contact spectroscopy across multiple different FeTe0.55Se0.45 crystals always produces an increase in conductance at zero-bias when cooled below Tc consistent with the data on contacts made via photolithography (see Supplemental Fig S3). The soft-point contacts revealed a smaller enhancement of the zero bias conductance in the superconducting state. However this is expected since the quasi-particle lifetime in the Ag paint contact is likely lower, which smears the spectra and reduces the height at zero bias. Similarly, we used planar junctions with Bi2Te2Se1 via a method that has previously enabled spectroscopic studies with low barriers in van der Waals materials.15 As shown in Fig. 4b, these junctions also resulted in nearly identical spectra near zero bias. Here the lower zero bias conductance is expected as it contains contributions from the normal material being in series with the contact. Another extrinsic explanation for the peak is the interstitial Fe-atoms known to be present in these materials. However, we excluded this explanation by measurements on annealed samples where the Fe impurity content is dramatically reduced (see Supplemental Fig S3a), though the topology and Tc are only mildly affected.

An alternate mechanism for producing a ZBCP is Joule heating at the contact. We took a number of steps to rule this out. First, similar results were obtained regardless of the exact contact configuration (e.g. swapping contacts employed for current versus voltage in point contact or three-point measurements). In addition, we compared the voltage and temperature data by inverting the $\frac{dI}{dV}$ spectra and comparing it to the resistance versus temperature data taken on the same contact configuration (see Fig 4c). To align the two curves, we translate the $\frac{dV}{dI}$ curve such that zero voltage coincides with the temperature at which it was recorded (7 K). Next, we assume the voltage where the maximum resistance is measured is equivalent to heating to TC, as this is the temperature where a peak in resistance is typically observed (see Fig 1d). While the exact voltage dependence due to heating could be more complex, it is clear the $\frac{dV}{dI}$ versus voltage spectra are far in excess of the resistance measured at Tc, though at high bias they do return to the value measured at Tc. This further excludes voltage induced heating as the origin of the zero bias conductance peak. In addition, the background conductances in the c-axis, hinge, and point contacts are nearly identical. Therefore the heating across all of them should be approximately the same. However, they reveal quite distinct spectra (i.e. strong ZBCP in the hinge contact vs. nearly none in the c-axis) which, combined with the emergence of the zero-bias conductance peak (ZBCP) at Tc in numerous contacts (see Figure 2 and Supplemental Figure S2), eliminates heating.

In summary, via a variety of contact methods, we reveal helical hinge zero modes in the topological superconductor FeTe0.55Se0.45. Specifically, contacts to the [001] surface made using hBN reveal standard Andreev reflection, while those draped over the hinge contain a cusp-like, zero-energy feature in the differential conductance. By combining with measurements using soft-point contacts on various crystals, we further confirm the intrinsic nature of this new mode. Furthermore, the appearance of an HHZM in FTS helps to establish both the topological and s*±* nature of the superconductivity. An important question raised by these results is the large size and the temperature dependence of the HHZM. It is possible that the large ratio of contact area to coherence length at the measured temperature ($\approx 1000x$), makes the measurement essentially many point-like contacts in parallel, leading to an apparently large conductance. The contact size may also play a role in the temperature dependence, as could the unknown size of the superconducting gap on the side surface. Thus future theoretical and experimental efforts must be made to better separate out the contact effects from the intrinsic response of the hinge mode we observe.

3 Acknowledgments

K.S.B., R.O., and M.J.G. acknowledge support from the National Science Foundation, Award No. DMR-1709987. Work at Brookhaven National Laboratory was supported by the Office of Science, U.S. Department of Energy under Contract No. DE-SC0012704. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. The work at CCES is supported by the Institute of Basic Science (IBS) in Korea (Grants No.IBS-R009-G2). We are grateful for numerous discussions with V. Yakovenko, V. Galitski, K.T. Law, R.-X. Zhang, W. Cole, K. Jiang and Z. Wang.

ASSOCIATED CONTENT: Supporting information available. Supporting information includes details regarding: exfoliation and fabrication of devices, experimental measurement setup, additional crystal measurements, and additional controls and checks performed on the devices.