Abstract
Chains of magnetic atoms with either strong spinorbit coupling or spiral magnetic order which are proximitycoupled to superconducting substrates can host topologically nontrivial Majorana bound states. The experimental signature of these states consists of spectral weight at the Fermi energy which is spatially localized near the ends of the chain. However, topologically trivial YuShibaRusinov ingap states localized near the ends of the chain can lead to similar spectra. Here, we explore a protocol to disentangle these contributions by artificially augmenting a candidate Majorana spin chain with orbitallycompatible nonmagnetic atoms. Combining scanning tunneling spectroscopy with abinitio and tightbinding calculations, we realize a sharp spatial transition between the proximitycoupled spiral magnetic order and the nonmagnetic superconducting wire termination, with persistent zeroenergy spectral weight localized at either end of the magnetic spiral. Our findings open a new path towards the control of the spatial position of ingap end states, trivial or Majorana, via different chain terminations, and the realization of designer Majorana chain networks for demonstrating topological quantum computation.
Introduction
Candidate Majorana platforms based on magnetic chains^{1,2,3,4,5,6,7} have been experimentally realized by selfassembled growth of chains of Fe^{8,9,10,11,12} or Co atoms^{13} on superconducting Pb(110), and by the controlled atombyatom assembly of Fe chains on superconducting Re(0001) using the tip of a scanning tunneling microscope (STM)^{14}. For both substrates, scanning tunneling spectroscopy (STS) supplied experimental evidence for topological superconductivity in Fe chains via the observation of zeroenergy spectral weight localized at the ends of such chains, which was interpreted as the signature of a Majorana bound state^{15,16}. For the Pb system^{12}, it was shown theoretically that the Majorana bound state can have a strong spectral weight in the superconducting substrate close to the chain’s end, which was experimentally supported by the detection of zeroenergy spectral weight localized in Pb overlayers covering the Fe chains close to their ends. However, for all investigated systems so far, the experiments suffer from a concurrence of the expected location of the Majorana bound state with the location of the termination of the chain. This termination is intrinsically different from the inner part of the chain, as the end atoms have a different coordination^{17}. Thereby, these atoms tend to have a different bonding length and, thus, a different hybridization with the substrate as compared to those in the interior of the chain. Moreover, the spinrelated properties at the ends of magnetic chains can differ drastically from their interior^{18,19,20}. All this could lead to a localization of topologically trivial Yu–Shiba–Rusinov (YSR) states at the chain’s ends^{9,13,17,21,22} and also accidentally close to the Fermi energy, and, therefore, hinder the identification of the spectral weight stemming from the soughtafter Majorana bound states. A solution to these issues would be the termination of the magnetic chain by a nonmagnetic superconducting wire made from a material with a similar orbital structure: such a termination ensures a smooth electronic continuation of the magnetic chain, which we expect to have a stronger impact on the properties of the YSR states than on those of the Majorana bound state, due to the robustness inherent to the topological character of the latter. In order to investigate the possibilities to realize such terminations, we study artificial chains of three different species from the 3d transitionmetal series (Mn, Fe, and Co), as well as hybrid chains of Co and Fe atoms, assembled by STMtip induced manipulation on superconducting Re(0001).
Results
Construction of the chains via atom manipulation
By depositing the three transitionmetal elements onto the cold Re(0001) substrate and subsequent STMtip induced manipulation (see “Methods”), we place single Fe and Co atoms on two different adsorption sites: the hollow site that continues the hexagonal closepacked (hcp) stacking of the substrate, and the hollow site that corresponds to the stacking of a facecentered cubic (fcc) crystal. For single Mn atoms, only the fcc site is accessible^{23}. Due to increasing Kondo coupling with increasing dstate filling and a transition from outofplane to easyplane magnetic anisotropy, the energy of the YSR states, which the five species induce in the energy gap of the superconducting substrate, varies systematically from Mn^{fcc} over Fe^{fcc}, Fe^{hcp}, and Co^{fcc} to Co^{hcp}: the YSR states of Mn^{fcc} are located close to the superconducting substrate’s gap edge, the ones of Fe^{hcp} close to the center of the gap, and the ones of Co^{fcc} again at the gap edge. Interestingly, single Co^{hcp} atoms are found to have fully quenched magnetic moments and thus do not induce any ingap state^{23}. Moreover, artificial chains of Fe^{hcp} atoms on neighboring sites (Fig. 1a) form spin spirals and reveal strong indications for topological superconductivity by zeroenergy spectral weight localized at the ends of chains that are longer than ten atoms^{14}. Motivated by these previous results, we explore the possibilities to build chains of the other two elements, Mn and Co, on neighboring sites using STMtip induced manipulation (Fig. 1b, c).
We find that it is possible to manipulate straight chains of Co atoms on neighboring hcp sites (Fig. 1c) and zigzagshaped chains of Mn atoms adsorbed on neighboring sites alternating between fcc and hcp (see Fig. 1b, d, Supplementary Note 1, and Supplementary Fig. 1). The chains are up to more than 100 atoms long limited by the widths of the terraces of the substrate, residual substrate defects, and the number of available single atoms in the surrounding of the building area. However, it was impossible to build any other chain of closedpacked atoms of one of these three elements, e.g., straight chains of Mn atoms on neighboring fcc or hcp sites, or straight chains of Co atoms on neighboring fcc sites. This is most probably a result of an energetically unfavorable bond length in such configurations.
Ingap electronic structure of homoatomic chains
Next, we study the lowenergy electronic properties of three manipulated chains, as shown in Fig. 1, in the energetic region of the energy gap 2Δ = 0.51 meV of the superconducting substrate (Fig. 2). The spectral intensity (Fig. 2b, f, j) is symmetric with respect to the center of the chain. This is in particular true for the ends of the chains (see below for Fe and Supplementary Note 2 and Supplementary Fig. 2 for Mn). The Fe_{20} chain (Fig. 2b) reveals a zeroenergy spectral weight with a maximum localized on the two atoms that terminate the chain, which decays in an oscillatory fashion in intensity toward the center of the chain. Spectra taken at the ends of the chain in comparison to spectra taken at the center of the chain (Fig. 2c, d) show that in addition to this zeroenergy spectral weight, also the spectral intensity stemming from YSR bands at a nonzero energy of about +0.1 meV is increased toward the chain’s ends (see arrows in Fig. 2b, c, and d). This reproduces the data of a previous publication, which was taken in a different STM facility using a different STM tip and sample^{14}. Note that the spectra in Fig. 2d, h, l are normalized by subtraction of a spectrum averaged over a sufficiently large length along the chain’s interior in order to approximate the difference of the spectral distribution at the chain’s ends with respect to that of an infinite chain. With the help of ab initio and tightbinding model calculations, the zeroenergy spectral weight was interpreted as a signature for a Majorana bound state localized at each end of the Fe_{20} chain^{14}.
In stark contrast, the Mn_{101} chain’s ends do not show any zeroenergy spectral weight (Fig. 2f). In the interior of the chain, there is a YSR band whose energy is slightly smaller than Δ, which is visible as a shoulder of the coherence peak on the negativebias side (Fig. 2g, h). These results let us conclude that the relatively weak Kondo coupling^{23} of Mn compared to Fe prevents the development of a topologically superconducting phase via the YSR bands^{4}. Notably, the energy of the YSR band is slightly decreased at the chain’s ends, and thus somewhat approaches the Fermi level (see arrows in Fig. 2f, g). The changes to the YSR band close to the ends of the Fe chain (see above) and to the YSR band energy close to the ends of the Mn chain could be explained by the reduced coordination number of the chainterminating atoms. Such changes can, therefore, be reduced by attaching nonmagnetic atoms with a similar orbital structure as the ones in the magnetic chain to both of its ends. Figure 2i–l show that this possibility is provided by chains of Co^{hcp} atoms. The spectral intensity of these chains does not show any change when the tip moves along a line starting from the substrate and then across the entire Co chain. This implies that a closepacked linear chain made from the initially nonmagnetic individual Co^{hcp} atoms on the Re(0001) surface^{23} still has a completely quenched magnetization. The superconductivity from the substrate can, thus, penetrate this chain of atoms. Because both Fe and Co atoms occupy hcp adsorption sites, and because of their identical orbital structure on Re, the Co chain might represent an ideal termination for the Fe chain where the latter shows signatures of a topological superconductor.
Magnetic properties of Coterminated Fe chains
To follow the idea of terminating the magnetic Fe chain by the nonmagnetic Co chain, we first investigate the magnetic properties at the material transition in hybrid Co^{hcp}–Fe^{hcp} chains in the normal metallic state of the substrate. This is done via ab initio calculations using the Korringa–Kohn–Rostoker (KKR) Green’s function method based on an embedding scheme, together with an effective spin model (see “Methods” and Supplementary Note 3). Our calculations reveal that the exchange interactions between the nearest and nextnearest neighbors within the Fe_{20} chain are strongly antiferromagnetic (Supplementary Fig. 3), which leads to spin frustration. This frustration is resolved by the formation of a cycloidal spin spiral of wavelength between three and four lattice constants (Fig. 3), in agreement with previous experimental results^{14}. Consequently, in this system, spin spirals originate from spin frustration rather than from Dzyaloshinskii–Moriya (DM) interaction. The DM interaction only sets the plane of rotation, which is at an angle of 30° to the surface plane for the Fe_{20} chain (Fig. 3c), and the rotational sense, but it has only a minor effect on the spinspiral wavelength. Note that for the pure Fe_{20} chain, the magnetic moments and interactions of the Fe atoms at both ends differ from the Fe atoms in the interior of the chain (Fig. 3d and Supplementary Figs. 3 and 4). However, when terminating the Fe_{20} chain with Co_{5} chains, both the magnetic moments and the interactions of these Fe atoms become similar to those of the Fe atoms in the interior of the chain (Fig. 3b, d and Supplementary Figs. 3 and 4). Moreover, the magnetic moments of the Co atoms in the Co_{5} chain attached to the Fe_{20} chain are essentially zero. Only the first Co atom at the transition to the Fe chain has a considerably induced magnetic moment (Fig. 3b, d). As a result, the impact of the Co chains on the magnetic structure in the interior of the Fe chain is negligible. Therefore, already fiveatomlong Co chains realize a perfect termination of the Fe_{20} chain with an atomically sharp transition between the Fe chain’s spinspiral state and the nonmagnetic d states of the Co chain.
Ingap electronic structure of Coterminated Fe chains
Keeping these results in mind, we experimentally investigate the ingap electronic structure in the superconducting state of the substrate along hybrid Fe_{20}–Co_{5} and Co_{5}–Fe_{20}–Co_{5} chains that have been built by successively attaching Co atoms first to the right (Fig. 4c, d) and then to the left side (Fig. 4e, f) of the pure Fe_{20} chain (Fig. 4a, b). Indeed, the ingap electronic structure measured on the last few Fe atoms close to the Co termination is considerably different from those measured on the last few Fe atoms at the open ends in the hybrid chains and in the pure Fe_{20} chain. In particular, the spectral intensity of the YSR band at +0.1 meV, which is increased at the open ends of the pure Fe_{20} chain and at the open end of the Fe_{20}–Co_{5} chain (see arrows in Fig. 4b, d), is almost completely moved out of the gap region, both at the single and at the two Coterminated ends of the Fe_{20}–Co_{5} and Co_{5}–Fe_{20}–Co_{5} chains, respectively (see also the spectra in Fig. 4g, h for comparison with Fig. 2c, d). Most notably, the zeroenergy spectral weight maximum is persistent at the Coterminated Fe chains. Its position is only slightly shifted toward the interior of the Fe part of the hybrid chain by about two atomic lattice constants after having attached the Co termination. This is visible in the zeroenergy spectral weight extracted from Fig. 4b, d, f as shown in Fig. 4i. Simultaneously, the five local maxima and minima of the oscillations of the zeroenergy spectral weight in the interior of the chain shift slightly toward the center. This experimental observation is consistent with the interpretation of the zeroenergy spectral weight as a signature of a Majorana bound state localized at each end of the Fe_{20} chain, which is expected to be protected against the local perturbation of the Fe_{20} chain by the Co termination. Such a perturbation, which is not affecting the internal spin structure of the Fe_{20} chain, can merely shift the lateral position of a Majorana bound state, but cannot completely remove it. In contrast, it can strongly influence the topologically trivial YSR bands. In order to further support this interpretation, we also investigated the topological properties of infinite Fe chains and the spatially resolved ingap electronic structure of the pure and Coterminated chains with a tightbinding model using the parameters extracted from the above KKR calculations (see “Methods” and Supplementary Notes 4 and 5). For an appropriately tuned superconducting energy gap Δ, the tightbinding model reproduces the zeroenergy spectral weight localized at both ends of the pure Fe chains, whose spatial localization is slightly increased when attaching the Co terminations, as shown in Fig. 4j. Using the same Δ, the tightbinding model for the infinite Fe chain displays the topologically superconducting phase. These results corroborate that the experimentally observed zeroenergy spectral weight localized at the ends of the Fe chain is a signature of a Majorana bound state that persists when terminating the chain with topologically trivial Co chains.
Our results, thus, suggest that appropriate terminations of topologically superconducting chains realized by artificial hybrid transitionmetal atom chains can be used to tune the properties of Majorana bound states and trivial YSR bands. We thereby establish essential next steps toward the atombyatom design of hybrid networks of spin chains and nonmagnetic superconducting chains, and toward the controlled manipulation of Majorana bound states, which are desired for Majorana braiding and the demonstration of topological quantum computation.
Methods
Experimental procedures
All measurements were performed in a homebuilt ultrahighvacuum STM setup at T = 0.3 K^{24}. We used electrochemically etched tungsten tips that were flashed to T = 1500 K before inserting them into the STM. The Re(0001) crystal was cleaned by Ar ion sputtering, followed by multiple cycles of O_{2} annealing at T = 1530 K and flashing to T = 1800 K. Mn, Fe, and Co atoms were successively deposited keeping the substrate at T < 10 K. The biasdependent differential tunneling conductance dI/dV was measured using a LockIn amplifier by modulating the bias voltage V with V_{mod} = 20–40 µV at a frequency of f_{mod} = 4142 kHz, and at constant tip height stabilized at a bias voltage V_{stab} and tunnel current I_{stab} before opening the feedback loop for measurement. The bias voltage is applied to the sample and zero bias corresponds to E_{F}. Single atoms were manipulated using STMtip induced atom manipulation by lowering the bias voltage and increasing the setpoint current to the manipulation parameters V = 1 mV and I = 100 nA.
Ab initio calculations
The density functional theory (DFT) calculations were performed employing the fullpotential KKR Green function method with spin–orbit coupling added to the scalar relativistic approximation (see Supplementary Note 3)^{25}. The exchange and correlation potential is treated within the local spindensity approximation using the parameterization of Vosko, Wilk, and Nusair^{26}. The Re(0001) substrate is modeled by 22 layers of Re augmented by two vacuum regions corresponding to four interlayer distances. A kmesh of 150 × 150 and an angular momentum cutoff for the scattering problem of l_{max} = 3 are used. The magnetic chains are deposited in the hcpstacking position on the Re(0001) surface with a relaxation of 20% of the interlayer distance toward the Re surface, using an embedding technique. Three systems are investigated: in addition to an Fe_{20} chain, we added five Co atoms to one end of the chain (Fe_{20}–Co_{5}) and five Co atoms to each end of the chain (Co_{5}–Fe_{20}–Co_{5}). The realspace clusters that are embedded on the Re(0001) surface contain the nearestneighbor Re atoms (and vacuum sites), resulting in cluster sizes of 146, 181, and 216 sites, respectively. The magnetic exchange interactions were obtained using the magnetic force theorem in the frozenpotential approximation and the infinitesimal rotation method^{27,28}. The siteresolved magnetic onsite anisotropy is obtained using the method of constraining fields^{29}.
Effective spinmodel calculations
The magnetic exchange interactions and the onsite magnetic anisotropy obtained from the ab initio calculations are used to parameterize the following classical Heisenberg model (see Supplementary Note 3):
where the unit vectors e_{i} point along the magnetization of atom i in the chain, \(\overline{\overline {\cal{K}}} _i\) are the magnetic onsite anisotropy matrices, J_{ij} are the isotropic exchange interactions, D_{ij} are the Dzyaloshinskii–Moriya interaction vectors, and \(\overline{\overline J} _{ij}^{s{\mathrm{ym}}}\) are the symmetric anisotropic exchange interaction matrices. The magnetic ground states are obtained from numerically minimizing the Heisenberg model starting from several random initial configurations of e_{i}.
Tightbinding model calculations
The finite magnetic chains on superconducting Re(0001) are modeled by a tightbinding Hamiltonian with realistic parameters obtained from DFT (see Supplementary Notes 4 and 5). The model explicitly considers the d orbitals of the chain atoms, with parameters that account for the Re substrate via an effective Hamiltonian construction. The siteresolved onsite electronic structure is modeled by a chemical potential, a spin splitting generating the magnetic moments, a local spin–orbit coupling, and an orbitaldependent crystalfield splitting. The hoppings between the sites are assumed to be hermitian, spinindependent, and symmetric in the orbitals. Superconductivity is added in the swave approximation with the local pairing potential being an orbitalindependent parameter^{30}. The local density of states is obtained by assuming an artificial temperature broadening, which is half of the spectral gap of the superconducting state.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its supplementary information files.
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Acknowledgements
L.S., M.S., T.P., R.W., and J.W. gratefully acknowledge funding by the Cluster of Excellence “Advanced Imaging of Matter” (EXC 2056—project ID 390715994) of the Deutsche Forschungsgemeinschaft (DFG). L.S., M.S., R.W., and J.W. acknowledge support by the SFB 925 “Light induced dynamics and control of correlated quantum systems” of the Deutsche Forschungsgemeinschaft (DFG). L.S. and R.W. acknowledge funding by the ERC Advanced Grant ADMIRE (No. 786020). S.B., M.d.S.D., and S.L. acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERCconsolidator grant 681405—DYNASORE) and the computing time granted by the JARAHPC Vergabegremium and VSR commission on the supercomputer JURECA at Forschungszentrum Jülich. We thank H. Kim and L. Rózsa for helpful discussions.
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L.S. and J.W. conceived the experiments. L.S. did the measurements. M.S. and J.H. started the initial experiments together with L.S. L.S. and J.W. analyzed the experimental data. S.B. performed the KKRbased DFT calculations, the effective spinmodel calculations, and the tightbinding model simulations. T.P. contributed to the tightbinding model simulations. S.B., M.d.S.D., and S.L. conceived the theoretical framework and analyzed the results of all calculations. L.S. prepared the figures and J.W. wrote the paper. L.S., S.B., T.P., M.d.S.D., S.L., R.W., and J.W. contributed to the discussions and to correcting the paper.
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Schneider, L., Brinker, S., Steinbrecher, M. et al. Controlling ingap end states by linking nonmagnetic atoms and artificiallyconstructed spin chains on superconductors. Nat Commun 11, 4707 (2020). https://doi.org/10.1038/s41467020185403
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DOI: https://doi.org/10.1038/s41467020185403
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