Abstract
By using scanning tunneling microscopy (STM) we find and characterize dispersive, energysymmetric ingap states in the ironbased superconductor FeTe_{0.55}Se_{0.45}, a material that exhibits signatures of topological superconductivity, and Majorana bound states at vortex cores or at impurity locations. We use a superconducting STM tip for enhanced energy resolution, which enables us to show that impurity states can be tuned through the Fermi level with varying tipsample distance. We find that the impurity state is of the YuShibaRusinov (YSR) type, and argue that the energy shift is caused by the low superfluid density in FeTe_{0.55}Se_{0.45}, which allows the electric field of the tip to slightly penetrate the sample. We model the newly introduced tipgating scenario within the singleimpurity Anderson model and find good agreement to the experimental data.
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Introduction
The putative s_{±} superconductor FeTe_{0.55}Se_{0.45} is peculiar because it has a low Fermi energy and an unusually low and inhomogeneous superfluid density^{1,2,3,4,5,6,7}. It has been predicted to host a topological superfluid and Majorana zeromode states^{8,9,10}. These predictions have been supported by recent experiments: photoemission has discovered Diraclike dispersion of a surface state^{11} while tunneling experiments have concentrated on ingap states in vortex cores, which have been interpreted as Majorana bound states^{12,13} since the low Fermi energy allows to distinguish them from conventional lowenergy CaroliMatriconde Gennes states^{14}.
Ingap states have a long history of shining light into the properties of different host materials, and have allowed to bring insight into gap symmetry and structure, symmetry breaking, or the absence of scattering in topological defects, to name a few^{15,16,17,18,19,20,21,22,23,24}. Impurity bound states have also been investigated in chains or arrays of magnetic impurities on superconducting surfaces where they can lead to Majorana edgestates^{25,26,27,28}. In the case of FeTe_{0.55}Se_{0.45}, zerobias ingap resonances have become a primary way to identify Majorana bound states at magnetic impurity sites or in vortex cores. At impurity sites, robust zerobias peaks have been reported at interstitial iron locations which suggest the presence of Majorana physics^{29}. In addition, very recently STM experiments reported signatures of reversibility between magnetic impurity bound states and Majorana zero modes by varying the tipsample distance on magnetic adatoms^{30}. Interestingly, there have also been signatures of spatially varying ingap impurity states^{31,32} which are not yet understood.
Here we report the detection of ingap states at subsurface impurities, which are spatially dispersing, i.e., they change energy when moving away from the impurity site by a distance of Δy. The energy can also be tuned by changing the tipsample distance (Δd). We argue that the most likely explanation of our observations involves a magnetic impurity state of the YSR type affected by the electric field of the tip. We show good agreement between our experimental findings and the single impurity Anderson model.
Results
Detection of a particlehole symmetric ingap state in FeTe_{0.55}Se_{0.45}
We use FeTe_{0.55}Se_{0.45} samples with a critical temperature of T_{C} = 14.5 K. They are cleaved at ~30 K in ultrahigh vacuum, and immediately inserted into a modified Unisoku STM at a base temperature of 2.2 K, for preventing surface reconstruction and contamination. To increase the energy resolution, we perform all tunneling experiments using a superconducting tip, made by indenting mechanically ground PtIr tips into a clean Pb(111) surface. With the superconducting tip and to leading order in the tunnel coupling, the currentvoltage (I − V) characteristic curves are proportional to the convolution of the density of states of Bogoliubov quasiparticles in the tip and the sample
where D_{s(t)} is the density of states of the quasiparticles in the sample (tip), f(ω, T) is the FermiDirac distribution at temperature T and e is the electron charge. In such a superconducting tunnel junction the coherence peaks in the conductance spectra, dI/dV(r, V), appear at energies: ± (Δ_{t} + Δ_{s}), where Δ_{s(t)} is the quasiparticle excitation gap of the sample (tip). In addition, the energy resolution is far better than the conventional thermal broadening of ~3.5k_{B}T (k_{B} is the Boltzmann constant) since it is enhanced by the sharpness of the coherence peaks of D_{t}^{33,34,35}. To obtain the intrinsic local density of states (LDOS) of the sample, D_{s}(r, ω), we numerically deconvolute our measured dI/dV(r, V) spectra while retaining the enhanced energy resolution (for more details see Supplementary Note 1). For this, we use our knowledge of the density of states of the tip with a gap of Δ_{t} = 1.3 meV from test experiments on the Pb(111) surface using the same tip.
Figure 1a shows a topography of the cleaved surface of FeTe_{0.55}Se_{0.45} obtained with a Pb coated tip (see inset). Brighter (darker) regions correspond to Te (Se) terminated areas of the cleaved surface which has a tetragonal crystal structure. Our samples exhibit no excess Fe atoms or clusters on the cleaved surface. Spatially resolved scanning tunneling spectroscopy shows that most locations have a flat gap, as shown in Fig. 1b, c. However, when we acquire spectra at specific points indicated by black circles (r_{2} and r_{3}) in Fig. 1a, we find sharp ingap states. Figure 1d–e and f–g shows such states, both in the raw data as well as in the deconvoluted results. The measured ingap state is symmetric in energy, i.e., it is visible at ± E_{ig} (Fig. 1g), or at the Fermi level, E_{ig} = 0 (Fig. 1e). In the raw data (before numerical deconvolution) the states are located at energies ± (Δ_{t} ± E_{ig}) (see arrows in Fig. 1d and f) due to the use of the superconducting tip.
Spatial dispersion of the ingap state
In order to characterize the impurity in more detail we acquire a spatially resolved dI/dV(r, V) map in the area shown in Fig. 2a. Three energy layers of the deconvoluted map depicting the LDOS variations are shown in Fig. 2b–d. The impurity exhibits a clear ringshaped feature which eventually becomes a disk with smaller radius at the Fermi level. A spatial line cut profile along the red dashed line shown in Fig. 2b reveals two symmetric resonances around zero energy that extend over ~10 nm in space (Fig. 2e). Importantly, the energies of the ingap states vary spatially as shown in the spatial cuts (Fig. 2e–g) obtained from the same conductance map. The dispersion of the ingap states shows an Xshaped profile where the crossing point is indicated with r_{0} (Fig. 2a). In more detail, the state is at zero energy at r_{0}, and then moves away from the Fermi level, before fading out slightly below the gap edge. We will show later that the character of this dispersion is dependent on the tipsample distance, and that there can also be zero or two crossing points. By inspecting the topography at r_{0} we find no signature of irregularities, which points towards a subsurface impurity defect as the cause of the observed ingap peaks in the spectra. We note that these impurities are sparse; we found a total of 5 in a 45 × 45 nm^{2} fieldofview. These all show the same characteristic dispersions, but the X point is estimated at different tip heights. For details, see Supplementary Note 7. Similar observations have been reported previously on FeTe_{0.55}Se_{0.45}, but without a clear energy cross at the Fermi level^{31,32}.
YSR impurity states
Our observations are reminiscent of YSR states caused by magnetic impurities in conventional superconductors^{35,36,37,38,39}. When a single magnetic impurity is coupled to a superconductor with energy gap Δ via an exchange coupling J then the ground state of the manybody system depends on the interplay between superconductivity and the Kondo effect (described by the Kondo temperature T_{K}). For Δ ≳ k_{B}T_{K} the superconducting ground state prevails (unscreened impurity) whereas for Δ < k_{B}T_{K} the Kondo ground state dominates (screened impurity). In each case, quasiparticle excitations above the ground state give rise to resonances symmetrically around the Fermi level inside the superconducting gap. In an STM experiment, this results in peaks in the conductance spectrum at the energy of the two YSR excitations which is determined by the product ν_{F}JS, where S is the impurity spin and ν_{F} the normal state density of states in the superconducting host (FeTe_{0.55}Se_{0.45} in our case).
It is important to note that the s_{±} symmetry of the order parameter in FeTe_{0.55}Se_{0.45} can lead to a very similar phenomenology between magnetic and potential scatterers. While in conventional swave superconductors, magnetic impurities are required to create ingap (YSR) states, in s_{±} superconductors, subgap resonances can also occur for nonmagnetic scattering centers. This can be shown using different theoretical techniques, including Tmatrix method^{40,41}, Bogoliubovde Gennes equations^{42,43} and Green’s functions^{44,45,46} applied to multiband systems with s_{±} symmetry. The similarity of magnetic and potential scatterers makes a distinction between these cases more challenging (but possible, with an external magnetic field^{30}). In either case, theory predicts energysymmetric ingap states with particlehole asymmetric intensities.
The Xshaped phenomenology of the ingap states shown above shares also similarities with boundstates that have been observed in Pb/Co/Si(111) stacks^{47}, where they have been interpreted as topological^{47,48}. However, as we will show here, in our experiments the single point of zero bias is just one particular case of a manifold of dispersions that depend on the tipsample distance.
Tuning the energy of the ingap state with the tip
Figure 3a shows an intensity plot of a series of spectra above r_{0}, the location showing the zerobias impurity state, with changing tipsample distance (see inset for a schematic). We normalized each spectrum by the normal state resistance R_{N} = V_{set}/I_{set}. In order to reduce the distance, we control the tip in constant current feedback and increase the setpoint for the current while keeping the voltage bias constant. In addition, we measure the tipsample distance relative to the setpoint: V_{set} = 5 mV, I_{set} = 0.4 nA. Strikingly, we observe a shift in the energy of the ingap state with varying the tipsample distance Δd. When the tip is brought closer to the sample surface, the subgap resonances shift towards the Fermi energy (Fig. 3b) where they cross and split again. We also point out that there is a strong particlehole asymmetry in the intensity of the ingap resonances. It can be clearly seen in Fig. 3b that the relative intensity between the positive (p) and negative (n) resonances (I_{p} − I_{n}) changes sign after the cross at the Fermi level. Further, we note that the energy shift for varying Δd is stronger than the spatial dispersion. This can likely be explained by the tip shape, which can be approximated as a sphere of roughly 20 nm. As the tunneling current falls off exponentially with distance, one always tunnels to the point closest to the surface. However, the field is algebraic in the distance, and thus a change horizontally has less of an effect than a change vertically, as shown in Supplementary Note 5.
To obtain a more complete picture of the tuning of the ingap states as we vary Δd, we measured five dI/dV(r, V) maps (each at different tipsample distance) and analyzed azimuthallyaveraged radial profiles through the impurity center (Fig. 3c–d shows two of these profiles. See Supplementary Note 3 for the other 3). We extract the energy of the resonances by Lorentzian fits (Fig. 3e), to observe that they cross the Fermi level at the impurity center when being close to the sample. This is the first time that such a crossing has been observed in an unconventional superconductor.
Microscopic origin
The important question that arises is: what tunes the impurity resonances that we observe? In previous experiments with magnetic adatoms or admolecules on conventional superconductors^{49,50}, it has been shown that the force of the tip changes the coupling between moment and substrate, and that the coupling J and the YSR energy could be tuned in this way. In this case, when the energy crosses the Fermi level at the critical coupling J_{C}, a firstorder quantum phase transition between the singlet (screened) and the doublet (unscreened) ground state is expected^{49,51}. Very recently, a similar forcebased scenario has been reported in different systems involving magnetic adatoms on top of superconductors^{15}, including Fe(Te,Se)^{30}. As discussed in Supplementary Note 2, a similar scenario can in principle explain the subgap dispersion discovered here. However, as the impurity is not loosely bound on top of the surface in the present case, a movement between a subsurface impurity and the superconductor due to the tip force as the cause for the tuning, seems unlikely. Therefore, we pursue alternative mechanisms. Motivated by the phenomenology of semiconductors^{52} or Mott insulators^{53}, where the tip can act as a local gate electrode (mediated by the poor screening), we propose a similar gating scenario for YSR states in the present case: the electric field of the tip can tune the energy of the impurity state and thus lead to a dispersing YSR state.
First, we note that there can be a significant difference between the work functions of the tip and the sample. Typical work functions are in the range of a few electronvolts, and differences between chemically different materials of the order of an electronvolt are common (see Supplementary Note 5). Hence, it is possible to have a voltage drop between them that is larger than the applied bias. Secondly, the low carrier density in FeTe_{0.55}Se_{0.45} leads to a nonzero screening length giving rise to penetration of the electric field of the tip inside the sample. An estimation of the penetration depth in the sample can be made in the ThomasFermi approximation (cf. e.g.^{54}). In this framework, the screening length is given by \({\lambda }_{{\rm{TF}}}={(\pi {a}_{0}/4{k}_{{\rm{F}}})}^{1/2}\), where a_{0} is the Bohr radius and k_{F} the Fermi wavevector. Using reported parameters^{11,12}, this yields λ_{TF} = 0.5 nm, which is comparable to the interlayer distance^{55}. This implies that in principle, an impurity residing between the topmost layers can be affected by the electric field of the tip. In Supplementary Note 6 we further test this possibility by performing an estimate of the potential shift in the impurity, when the tipsample distance is varied using a simple model for screening (image charges method) to estimate that the shift is comparable to the charging energy of the impurity. We note that this estimate is approximative, as some key parameters are unknown for FeTe_{0.55}Se_{0.45}.
Based on these considerations, we conclude that it is possible that the tip acts as a local gate electrode that influences the energy levels of the impurity, which in turn influences the energy of the ingap states, as we will demonstrate in the modelling carried out below. By adjusting the tipsample distance the field penetration is modulated leading to an energy shift of the ingap resonances. The spatial dependence can be explained similarly: when moving the tip over the impurity location, we change the local electric field, which is at a maximum when the tip is right above the impurity. We emphasize that, similar to experiments on semiconductors and Mott insulators, we expect that the details of the gating process depend on the tip shape Supplementary Note 5.
Gatetunable single impurity Anderson model
We model the subgap state as a YSR state arising from the magnetic moment of a subsurface impurity level, whose energy is effectively gated by the tipinduced electric field. It should be noted that the subgap states arising in an s_{±}wave superconductor from a simple nonmagnetic impurity can produce a dispersive cross in the ingap energies as a function of the impurity potential. However, this is only true for a particular range of potentials, and will not generally trace out a single dispersive cross as a function of the impurity strength^{42,44}. Therefore, we are led to conclude that the impurity at hand involves a finite magnetic moment. Local impurityinduced magnetic moments may indeed be particularly prominent in correlated systems like FeSe where even nonmagnetic disorder, in conjunction with electron interactions, can generate local moments^{56}. Because of the magnetic nature of the impurity site, the results of our calculations are qualitatively independent on whether we treat the system as an s or s_{±}wave superconductor. For simplicity, we perform our calculations assuming standard swave pairing.
The superconducting single impurity Anderson model^{57} involves an impurity level ϵ_{0} with charging energy U coupled via a tunneling rate Γ_{s} to a superconducting bath with energy gap Δ_{s}^{58,59,60}. We represent the sample by a simple swave Bardeen–Cooper–Schrieffer (BCS) superconductor, and use the zerobandwidth approximation, including only a single spindegenerate pair of quasiparticles at energy Δ_{s}^{59,61}. We further assume that the gating from the tip changes the impurity level ϵ_{0} linearly with distance. We then obtain the YSR states by calculating the local impurity spectral function, D_{I}(ω, ϵ_{0}), as a function of ϵ_{0} (and thus of gating) using the Lehmann representation (see Supplementary Note 2 for details). The result is plotted in Fig. 4a, where the observed crossing of the subgap states indicates a change between singlet, and a doublet ground state^{62}. From the spectral function we can determine the current using leadingorder perturbation theory in the tunnel coupling connecting the impurity to the tip, t_{t}:
here D_{t}(ω, Δ_{t}, γ_{t}) is the spectral function of the superconducting tip with a finite quasiparticle broadening incorporated as a Dynes parameter^{63}, γ_{t} and ℏ the reduced Planck’s constant. A phenomenological relaxation rate, Γ_{r}, is incorporated into the Lehmann representation, (see Supplementary Note 2), to construct D_{I}(ω, ϵ_{0}). This parameter accounts for quasiparticle relaxation of the YSR resonances at ω = ± E_{ig}. The validity of the expansion in Γ_{t} = πν_{F}∣t_{t}∣^{2}, which captures only single electron transport and omits Andreev reflections, rests on the assumption that the subgap state thermalizes with rate Γ_{r} between each tunneling event. In the opposite limit, Γ_{t} ≫ Γ_{r}, transport takes place via resonant Andreev reflections, and the subgap conductance peaks at eV = ± (Δ_{t} + E_{ig}) display a bias asymmetry that is reversed compared to the bias asymmetry of the single electron transport regime^{39}. In principle, these two regimes can be differentiated by varying Γ_{t}, since the conductance peaks increase linearly with Γ_{t} in the single electron regime, and sublinearly in the resonant Andreev regime^{39,64}. The experimental data shown in Fig. 3 display a marked asymmetry, consistent with our assumption of relaxation dominated transport where conductance asymmetry will follow the asymmetry of the underlying spectral function.
Next, we investigate the situations where the tip moves over the impurity along the surface, or towards the impurity as a function of tipsample distance Δd. These situations are marked with blue and red arrows in Fig. 4a, respectively. In Fig. 4b, c we then plot subgap conductance as a function of level position, ϵ_{0}, corresponding to the red/blue traces, assuming a linear dependence of ϵ_{0} with tipsample distance. The agreement between our model and the data is good, both in terms of the energy dispersion and the asymmetry. Also, in both experiment and theory, additional conductance peaks at eV = ± (Δ_{t} − E_{ig}) are visible close to the singletdoublet phase transition. We interpret these lines as the additional single electron processes shown in Fig. 4d, which arise from thermal population of the excited state close to the phase transition where E_{ig} ≲ k_{B}T. The conductance peaks at Δ_{t} ± E_{ig} meet at the point where the YSR states cross zero energy, signaling the change between singlet, and doublet ground states, and the asymmetry in intensity between the conductance peaks at eV = ± (Δ_{t} + E_{ig}) switches around.
The good agreement between this simple model (Fig. 4b, c) and the data presented in Fig. 2e and a, supports our interpretation that the tip exerts an effective gating of the impurity. We discuss alternative scenarios further in Supplementary Note 2, but the fact that our impurity is below the surface and the excellent agreement between the model and the data lead us to conclude that the gating scenario is most likely in the present case.
In summary, we have reported on the properties of energy symmetric ingap states in FeTe_{0.55}Se_{0.45} that can be tuned through the Fermi level. These states extend over a large (~10 nm) area around the center locations. Our data point towards a subsurface magnetic impurity embedded in a lowdensity superfluid with large screening length that leads to YSRlike ingap states. We propose a novel tipgating mechanism for the dispersion and perform calculations within the single impurity Anderson model that show excellent agreement with the data. Such a mechanism could also play a role in previous experiments on elemental superconductors or heterostructures. How such states are related to the topological superconductivity in FeTe_{0.55}Se_{0.45} remains an open question. Our work further shows that one needs to be careful when interpreting zerobias peaks in putative topological states, and junction resistance dependent experiments are a necessary—ideally combinded with other techniques such as noise spectroscopy^{65,66,67} (see also Supplementary Note 4), spinpolarized STM^{68}, or photonassisted tunneling^{69} will allow for better understanding. Independent of this, tunable impurity states like the one we report here could offer a platform to study quantum phase transitions, impurity scattering, and the screening behavior of superfluids.
Methods
Sample preparation
The FeTe_{0.55}Se_{0.45} single crystal samples were grown using the Bridgman method and show a critical temperature of T_{C} = 14.5 K. We cleave them at ultrahigh vacuum (P_{base} ~ 1 × 10^{−10} mbar) and low temperature (~30 K) and immediately insert them to our precooled STM (USM1500, Unisoku Co., Ltd). The STM tips used in this work are mechanically sharpened PtIr wires. They are Pbcoated by indenting them on a Pb(111) surface which was first cleaned by repetitive cycles of Ar sputtering (P_{base} ~ 5 × 10^{−5} mbar) followed by thermal annealing.
Measurement
Standard lockin technique is employed for the tunnelling conductance measurements at 887 Hz. All measurements were performed at 2.2 K.
Data availability
The data of this work are available from the corresponding author upon request.
Code availability
The code used in this work is available from the corresponding author upon request.
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Acknowledgements
We acknowledge J. de Bruijckere, J.F. Ge, M.H. Fischer, P.J. Hirschfeld, D.K. Morr, P. Simon, J. Zaanen, and H.S.J. van der Zant for fruitful discussions. This work was supported by the European Research Council (ERC StG SpinMelt) and by the Netherlands Organization for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience programme, as well as through a Vidi grant (68047536). G.G. is supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, US Department of Energy (DOE) under contract number desc0012704. B.M.A. acknowledges support from the Independent Research Fund Denmark grant number DFF802100047B. The Center for Quantum Devices is funded by the Danish National Research Foundation. D. Cho was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1007895 and 2017R1A5A1014862) and the Yonsei University Research Fund of 2019220209. A.A. acknowledges the support of the Max PlanckPOSTECHHsinchu Center for Complex Phase Materials, and financial support from the National Research Foundation (NRF) funded by the Ministry of Science of Korea (Grant No. 2016K1A4A01922028).
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D. Chatzopoulos, D. Cho, and K.M.B. performed the experiments and data analysis. D. Chatzopoulos, G.O.S., D.B., A.A., J.P., and B.M.A. performed the simulations. G.G. grew and characterized the samples. All authors contributed to the interpretation of the data and writing of the manuscript. M.P.A. supervised the project.
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Chatzopoulos, D., Cho, D., Bastiaans, K.M. et al. Spatially dispersing YuShibaRusinov states in the unconventional superconductor FeTe_{0.55}Se_{0.45}. Nat Commun 12, 298 (2021). https://doi.org/10.1038/s4146702020529x
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DOI: https://doi.org/10.1038/s4146702020529x
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