Observation of magnetic adatom-induced Majorana vortex and its fusion with eld-induced Majorana vortex in an iron-based superconductor

Braiding Majorana zero modes is essential for fault-tolerant topological quantum computing. Iron-based superconductors with nontrivial band topology have recently emerged as a surprisingly promising platform for creating distinct Majorana zero modes in magnetic vortices in a single material and at relatively high temperatures. The magnetic eld-induced Abrikosov vortex lattice makes it dicult to braid a set of Majorana zero modes or to study the fusion of a Majorana doublet due to overlapping wave functions. Here we report the observation of the proposed quantum anomalous vortex with integer quantized vortex core states and Majorana zero mode induced by magnetic Fe adatoms deposited on the surface and the realization of its fusion with a nearby eld-induced Majorana vortex in iron-based superconductor FeTe 0.55 Se 0.45 . We also observe vortex-free Yu-Shiba-Rusinov bound states at the Fe adatoms with a weaker coupling to the substrate, and discover a reversible transition between Yu-Shiba-Rusinov states and Majorana zero mode by manipulating the exchange coupling strength. The dual origin of the Majorana zero modes, from magnetic adatoms and external magnetic eld, provides a new single-material platform for studying their interactions and braiding in superconductors bearing topological band structures.


Background
The band structure of iron-based superconductor FeTe 0.55 Se 0.45 has a nontrivial Z 2 topological invariant and supports helical Dirac fermion topological surface states (TSS) 1, 2 , which was con rmed recently by spin-resolved and angle-resolved photoemission spectroscopy 3 . Remarkably, below the bulk transition temperature T c , superconducting (SC) TSS were observed with a pairing gap 3 . This makes FeTe 0.55 Se 0.45 a novel single-material platform for generating Majorana zero modes (MZMs) at the ends of a vortex line 2,4,5 . Recently, strong evidence of MZMs inside the magnetic eld induced vortices have been observed by scanning tunneling microscope/spectroscopy (STM/S) on the vortex lattice of this type-II superconductor [6][7][8][9] .
This Majorana platform also presents new challenges for the basic understanding of defect excitations in superconductors with a topological nontrivial band structure, and new possibilities for creating MZMs under different physical conditions. In general, superconductors can host two kinds of defect excitations as in-gap bound states: the Yu-Shiba-Rusinov (YSR) states [10][11][12][13][14][15][16][17] localized at a magnetic impurity and the Caroli-de-Gennes-Matricon (CdGM) states [18][19][20][21][22][23][24][25][26] inside a magnetic vortex core. To date, these excitations have appeared distinctly in nature; a magnetic impurity induces the YSR states carrying spin, whereas an external magnetic eld creates vortices of the whirling supercurrents. In stark contrast to the traditional YSR states, robust zero-bias peaks (ZBPs), sharing the quintessential spectroscopic properties of MZMs, were observed by STM/S at the magnetic interstitial Fe impurities in FeTe 0.55 Se 0.45 27 , but without applying a magnetic eld. A recent theoretical proposal 28 attributes the observed ZBP to a MZM bound to a quantum anomalous vortex (QAV) nucleated spontaneously at the magnetic Fe atom. The role of the magnetic eld is played by the exchange coupling of the spin and orbital moment of the Fe impurity located at the C 4 symmetric sites, which generates circulating supercurrents by the spin-orbit coupling and modulates the phase of the superconducting order parameter. When the exchange coupling is strong enough, a condition favored by the very small Fermi energy (~ 5meV) in FeTe 0.55 Se 0.45 3 , a transition from the vortex-free YSR states to the QAV was predicted to take place 28 . A MZM emerges inside the QAV core from the superconducting TSS, since the Berry phase of the Dirac fermions transforms the total angular momentum quantum number of the CdGM vortex core states 18 into integers, naturally supporting a zeroenergy bound state 8,28 .

Results
To probe the remarkable nature of defect excitations in the superconducting TSS, we deposit magnetic Fe adatoms on the cleaved (001) surface of a single crystal of FeTe 0.55 Se 0.45 ( Fig. 1a- The STM image of a surface region after the atomic deposition (Fig. 1c) shows scattered Fe adatoms as the bright spots with a coverage of ~0.04 %. The zero-energy dI/dV map in the same area of Fig. 1c, displays relatively high density of states at the locations of the Fe adatoms (Fig. 1d), consistent with the physical picture that magnetic Fe impurities generate in-gap states. From the statistics of more than one hundred measurements (Supplementary Note I, Fig. S2), we identify two types of in-gap states localized around the Fe adatoms with distinct dI/dV spectra exempli ed in Fig. 1e. The type-I adatoms, which represent about 10% of our measurements, exhibit a sharp ZBP reminiscent of a MZM coexisting with other in-gap states in the dI/dV spectrum. In contrast, the conductance spectrum of the type-II adatoms, which represent about 90% of our measurements, shows the YSR states, featuring a pair of in-gap states at particle-hole symmetric peak energy positions, but with asymmetric peak weights. In comparison, the typical dI/dV spectrum on the clean surface without the adatom shows a hard superconducting gap without in-gap states (Fig. 1e).
We begin with the type-I Fe adatoms. A high-resolution topographic image (Fig. 2a) shows an isolated type-I adatom.
A circular pattern appears in the zero-energy dI/dV map in the vicinity of the Fe site ( Fig.  2b), with the zero-energy intensity center slightly offset from the Fe site. The waterfall-like dI/dV spectra To explore the nature of the discretized in-gap states, we extract several dI/dV spectra from Fig. 2c and show them as a stacking plot in Fig. 2e. Doing so accounts for the spatial distributions of the in-gap states and the sample inhomogeneity 6, 7 due to Te/Se alloying, which has proven to be useful for studying the core states of magnetic eld-induced vortices in FeTe 0.55 Se 0.45 8 . The sequence of discretized bound states, including the zero-energy state, is clearly visible as the pronounced peaks labeled by L 0 , L ±1 , L ±2 . The energy positions of the conductance peaks (L n ) are plotted in Fig. 2f.
Intriguingly, the average energies (solid lines) of the discrete quantum states bound to the Fe adatom follow closely a sequence of integer quantization , with the minigap meV, the same integer sequence (with a minigap ~ 0.6 meV) followed by the quantized vortex core states observed recently 8  It is necessary to check the temperature and magnetic eld dependence of the ZBP, since a vortex MZM would respond differently than vortex-free defect states. The temperature evolution of the ZBP on the Fe adatom turns out to be very similar to that of the MZM in a eld-induced vortex 6 . The ZBP intensity of the center spectrum pointed by the black arrow in Fig. 2c decreases with increasing temperatures, and becomes almost invisible at 4.2 K (Fig. 2g). The magnetic eld dependence is also very similar to that of a vortex MZM as the ZBP does not split or broaden for elds up to 8 T (Fig. 2h), provided that no eldinduced vortices enter the region near the adatom when the magnetic eld is applied. The ZBP is however sensitive to the location of the adsorbed Fe adatom. We found that all type-I adatoms producing integer quantized bound states anchored by the ZBP are adsorbed at the high-symmetry sites in the center of four Te/Se atoms. To test the robustness of this nding, we manipulate a type-I adatom by the STM tip to a different location away from the C 4 symmetric site. The ZBP disappears and a pair of in-gap state at nonzero energy emerges (Supplementary Note III and Fig. S4). After annealing the sample to 15 K and perform the measurement again at 0.4 K, the adatom diffuses back to its original high-symmetry site and the ZBP reappears. Thus, the high-symmetry site is a prerequisite for the induced ZBP, which agrees with the proposed theory that the orbital magnetic moment of the Fe adatoms at C 4 symmetric locations plays an important role for the nucleation of the QAV 28 .
Next, we turn to study the type-II Fe adatoms (Supplementary Note IV). In contrast to the type-I adatoms, the conductance exhibits predominantly a pair of in-gap peaks at nonzero energies without the ZBP.
Applying an external magnetic eld, we observe that the peaks shift approximately linearly to higher energies ( Supplementary Fig. S5d) away from the Fermi level, consistent with a pair of spin-polarized YSR in-gap states. We nd that the type-II Fe adatoms are adsorbed at myriad locations on or off the high-symmetry axis and induce YSR states at different energies, indicative of broadly varying exchange couplings to the superconducting quasiparticles. In special cases, we also observe YSR states located very close to zero energy, which nevertheless split under the magnetic eld (Supplementary Note IV and These observations motivate us to manipulate the exchange coupling between the magnetic adatoms and the substrate by tuning the tip to sample distance 29,30 . The electrostatic force of an approaching tip can prod and move the Fe adatom in directions parallel and perpendicular to the surface (Fig. 3a), which can affect the atomic orbital moment of the Fe adatom and the spin-orbit exchange coupling to the superconductor. In STM/S, the tunnel-barrier conductance G N , where I t is the tunneling current and V s is the bias voltage, governs the tunnel coupling and changes with the tip-sample distance. Performing tunneling conductance measurements as a function of G N , we nd that the energies of the YSR states are modulated ubiquitously when the tip approaches the type-II Fe adatoms (Supplementary Note V). The observed crossing and reversal of the in-gap states ( Supplementary Fig. S7), a trademark of the YSR states, con rms that reducing the tip to sample distance monotonically increases the exchange coupling between the Fe atom and the superconductor.
Unexpectedly, as the STM tip approaches a signi cant number of the type-II Fe adatoms (~ 27%), the pair of YSR states modulates with increasing G N , but then coalesces in a captivating manner into a single ZBP in the water-fall plot of dI/dV spectra (Fig. 3c) and the intensity plot (Fig. 3d), which remains robust under further increase of the barrier conductance. Note that the emergence of the ZBP out of the YSR states is different from the transition point between the screened spin-singlet and doublet ground states 29,31 , where the two YSR states are approximately degenerate at zero energy as marked by the red arrow in Fig. 3d. To probe the change in the nature of the in-gap states with different barrier conductance, we repeated the entire process under an applied magnetic eld. The dI/dV spectra and the intensity plot obtained under 6 T ( Fig. 3e-f) show that the vortex-free YSR states no longer cross zero energy, due to the Zeeman splitting that removes the accidental degeneracy. However, the emergence of the unsplit ZBP at higher G N is unabridged even at such a high eld, indicating that ZBP corresponds to a single MZM robust against an applied magnetic eld. This identi cation is further corroborated by performing the measurements on type-I Fe adatoms that show the ZBP associated with the MZM upholds its integrity and does not shift or split with increasing G N in a eld as high as 6 T (Supplementary Note VI and Fig.   S8). The compelling evidence attributes the novel coalescence of in-gaps states to the ZBP as a topological transition (Fig. 3b) from the vortex-free YSR states to a vortex MZM, which is fully consistent with the theoretical prediction that increasing the exchange coupling of an Fe impurity induces a transition from the YSR states to the QAV states hosting a MZM in FeTe 0.55 Se 0.45 superconductors 28 . The transition between the YSR states and the MZM is even reversible, as the dI/dV spectra as a function of the barrier conductance retrace that shown in Fig. 3e-f upon withdrawing the tip (Fig. 3g-h) in a controlled manner. The transition is also replicable when the type-II Fe adatom under the tip in Fig. 3 is moved to a different location about 1 nm away ( Supplementary Fig. S9). These observations reveal the unprecedented nature of defect excitations in the superconducting TSS where local magnetic moment and screening currents are inextricably connected through the magnetoelectric effect.
The phase coherence of the MZMs is stored nonlocally and protected by the topological degeneracy against the ravages of environmental decoherence caused by local perturbations, which is the central to the idea of topological quantum computing 32,33 . The coupling of two MZMs su ciently close by annihilates of the nonabelian anyonic zero modes and creates a pair of fermionic states at nonzero energies. This fusion process usually requires two overlapping magnetic eld induced vortices, which is di cult to control on the Abrikosov lattice 7, 8, 34 . Our system allows a new possibility, i.e., the fusion between MZMs hosted in a QAV and a eld-induced vortex (Fig. 4). The zero-energy dI/dV map (Fig. 4a) shows a MZM in the QAV nucleated at a type-I Fe adatom in a magnetic eld of -0.2 T, also visible in the intensity plot (Fig. 4b) as sharp ZBPs along the line cut across the adatom. A eld-induced vortex is observed to enter the eld of view subsequently. The latter sits very close to the Fe site, thus enlarges signi cantly the region with spectral weight at zero-energy (Fig. 4c). Remarkably, acquiring the intensity plot along the same line cut shows that the ZBP splits into two peaks separated by an energy spacing 0 .25 meV (Fig. 4d). During the second round of the measurements, the vortex creeps away. The zeroenergy map recovers (Fig. 4e) and the ZBP reemerges in the intensity plot (Fig. 4f). Throughout the fusion process, the temperature and the magnetic eld are kept stable. It is necessary to point out that the creeping of the vortices is observed quite often when we detected MZMs in the led-induced vortices in our previous work. Three representative dI/dV spectra corresponding to the three conditions are extracted and displayed in Fig. 4g for a better comparison. Repeating the measurements on the same Fe adatom in a higher magnetic eld of -3 T reveals again the splitting of the ZBP caused by the presence of a nearby eld-induced vortex (Fig. 4h), with a larger energy spacing ~ 0.35 meV (Fig. 4i), possible due to the shorter distance and stronger overlap of the two MZMs as indicated by the smaller ring feature in the zero-energy map in Fig. 4h