Probing Atomic Structure and Majorana Wavefunctions in Mono-Atomic Fe-chains on Superconducting Pb-Surface

Motivated by the striking promise of quantum computation, Majorana bound states (MBSs) in solid-state systems have attracted wide attention in recent years. In particular, the wavefunction localization of MBSs is a key feature and crucial for their future implementation as qubits. Here, we investigate the spatial and electronic characteristics of topological superconducting chains of iron atoms on the surface of Pb(110) by combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM). We demonstrate that the Fe chains are mono-atomic, structured in a linear fashion, and exhibit zero-bias conductance peaks at their ends which we interprete as signature for a Majorana bound state. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends (below 25 nm), with two localization lengths as predicted by theory. Our observation lends strong support to use MBSs in Fe chains as qubits for quantum computing devices.

Motivated by the striking promise of quantum computation, Majorana bound states (MBSs) [1] in solid-state systems [2,3] have attracted wide attention in recent years [4][5][6][7][8][9][10][11][12][13][14][15][16]. In particular, the wavefunction localization of MBSs is a key feature and crucial for their future implementation as qubits [2,3]. Here, we investigate the spatial and electronic characteristics of topological superconducting chains of iron atoms on the surface of Pb(110) by combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM). We demonstrate that the Fe chains are mono-atomic, structured in a linear fashion, and exhibit zero-bias conductance peaks at their ends which we interprete as signature for a Majorana bound state. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends ( 25 nm), with two localization lengths as predicted by theory [17]. Our observation lends strong support to use MBSs in Fe chains as qubits for quantum computing devices.

Majorana fermions are real solutions of the Dirac equation and by definition fermionic
particles which are their own antiparticles [1]. While intensely searched in particle physics as neutrinos, Majorana fermions have recently been predicted to occur as quasi-particle bound states in engineered solid-state systems [2]. Such systems not only offer the possibility to observe the exotic properties of such MBSs, but also open up an interesting playground for topological quantum computing [2,3]. The fundamental ingredients to generate MBSs in semiconductor-superconductor heterostructures is to combine a spin texture with an s-wave superconductor (SC) allowing to create a superconducting state with effective p-wave pairing, giving birth to a new state of matter-topological superconductivity [2,3]. MBSs arise as zero-energy states lying in the superconducting gap and being spatially localized at the interfaces.
Experimentally, the induced proximity gap probed in the Fe chain is found to be very small (≈ meV), while the exchange interaction is in the eV range, implying on theoretical grounds a large localization length of the MBS wavefunction, in contrast to the observation [9] have triggered interesting discussions on the physical origin of the ZBP [18,19]. It further implies that possible MBS in such chains may easily hybridize into conventional fermions, raising the question as to whether MBS in such Fe/Pb hybrid system will exhibit non-Abelian braiding statistics [3].
In this work, we go a decisive step further and combine STM and AFM measurements at low temperatures to characterize the electronic properties and structure, respectively, of superconducting Fe chains self-assembled on the surface of Pb(110), see Fig. 1a. To engineer such systems, atomically cleaned surfaces of an s-wave superconductor, Pb(110), were prepared by a few sputtering/annealing cycles (see Supp. Info Fig. S1) on top of which Fe atoms were deposited on the sample kept at ≈ 400 K. In such way, topographic STM images reveal the formation of long chains extending up to 70 nm (Fig. 1b). In agreement with [9], chains are partially covered by small clusters mainly localized around their centers (see Supp. Info Fig. S2). Careful real-space observations reveal that numerous Pb vacancies are also created in the vicinity of the chains. We assume that the relative high temperature required for the self-assembly also promotes the diffusion of Pb atoms and their nucleation at the chain sides. During our study, several chains having lengths from 40 to 70 nm (Fig. 1b) were investigated and showed two distinct topographic signatures by means of STM at their ends ( Fig. 1 c and d). They correspond to a variation of the apparent STM height of about 10 pm. Along the chains and independent of the termination types, no clear atomic periodicity has been observed by STM (see Fig. S3) which could unambiguously be related to the structure of the chain. We then systematically acquired conductance point-spectra at the two chain ends to identify the presence of a ZBP which we interprete as signatures for MBS. Infos Fig. S4). We note that the lowest temperature of our experiments is 4.7 K which limits our spectral resolution to about 1.4 meV. Although the energy gap in our spectra is broadened compared to [9] due to the higher measurement temperature, a superconducting proximity gap of about = 1.1 meV is observed in the Fe chain, as well as a clear ZBP which we interprete as a signature of a MBS that has been predicted to occur in such systems due to various mechanisms [7][8][9][10][11][12].
Electronic and Structural Characterization along the Mono-atomic Fe chain. A ZBP localized at the end of the chains is one of the hallmarks of a MBS [9]. Alternatively, however, such a ZBP can arise from magnetic impurities such as Shiba (near) mid-gap states in the vicinity of single adatoms [20,21], molecules [22,23] or disorder effectss. To further identify the origin of the observed zero-energy subgap states and attribute it to a MBS, we compared STM and AFM imaging obtained at the atomic scale below and above the superconducting transition temperature (Fig. 2). a weak perturbation of the electronic density along the chain. This last feature might be 5 caused by the presence of defects below or within the chain. Howewer, since STM reflects the electronic density between tip and sample, the 'true' atomic structure can be masked by delocalized electronic states of the system [24,25]. As we noticed for the present system, the STM topographic data can lead to important misinterpretations of the atomic structure of the chain.
To unambiguously address this issue, we employed the AFM imaging technique which is rather insensitive to the delocalization of electronic states close to the Fermi level.  (Fig. 2c) shows a perfect round-shape halo of 0.8 nm in diameter as a spatial signature of this peculiarity. Within the halo, the last atoms are visible and perfectly aligned with the rest of the chain thus excluding atomic disorder or corrugation effects. The decrease of the attractive forces at such close tip-sample distances implies additional contributions of repulsive forces between tip and sample. Tuning fork AFM is known to be sensitive to such forces which usually originate from Pauli exclusion principle between the electronic wavefunctions of the tip and the surface [24]. Given that two end MBSs can host at most one fermion, it is plausible that a similar repulsive Pauli effect is responsible for the observed halo in the force maps. In comparison, site 2 shows a less pronounced ∆f variation compared to site 1 which coincides with slightly misaligned Fe atoms with respect to the chain structure. The ∆f is less negative by ≈ -10 Hz compared to the chain. We further point out the absence of any halo at site 2 and conclude that the ∆f variation is imputed to atomic disorders, most likely due to local topographic variations. We note that we also investigated chains without 6 end-states and no such force contributions were observed (see Supp. Infos Fig. S5).
To further support our observation of MBS, we compared normalized conductance maps, i.e. (dI/dV )/(I/V ) ∝ local density of state (LDOS) between tip and sample at the Fermi level, obtained at the same locations and at about 5 K and 10 K, respectively. Fig. 2d shows the LDOS(x, y) map at 5 K and reveals a clear ZBP, ascribed to the MBS, localized at the end of the chain as determined by the AFM data. A weak modulation of the LDOS is also observed along the chain which we attribute to the decay of the MBS wavefunction. At site 2, the LDOS is almost zero and might be due to weak magnetic perturbations induced by lattice disorder [20,22,23]. In order to suppress the superconducting state of the system and thus to enforce the disappearance of the MBS, we measured the same chain above the critical temperature of lead (T ≥ T c = 7.2 K). The corresponding normalized LDOS(x, y) map (see inset Fig. 2d) shows a homogenous LDOS along the chain which drastically contrasts to the one at 5 K. By comparing the LDOS profiles taken along the chain at 5 and 10 K (black and grey dots in Fig. 2e), the ZBP has completely disappeared at 10 K (site 1) as well as the oscillations along the chain due to the suppression of topological superconductivity. Since no external magnetic field is applied [9], we clearly address the interplay between superconductivity and the ZBP observed in our data which provides a strong evidence of the MBS presence in this system.
The MBS Localization Lengths. To further exclude other explanations of the ZBP [26], we next investigate another strong fingerprint of a MBS, namely its wavefunction and associated localization lengths [17]. For this we assume that the Fe chain with a proximity gap is driven into the topological phase by a spin texture which gives rise to a helical field (see Fig.   1a). Although Nadj-Perge et al. [9] measured the chain magnetization and concluded to a ferromagnetic behavior, the precise value of the magnetization is however not known and could be substantially away from full ferromagnetic order. In turn, this does not exclude helical order since the helix can be around the magnetization axis with some small angle.
Moreover, we wish to point out that, recently, such a helical state has been experimentally reported in a similar 1D-system [27]. Such helical fields can either follow from spin orbit interaction combined with Zeeman fields [5,6,8] or exchange fields [7,9] or from a 'self-tuning' RKKY interaction between the Fe spins [10][11][12]. These two mechanisms are related by a spin-dependent gauge transformation and thus mathematically equivalent [28]. We further assume that the pitch of the helix is much larger than the lattice constant of the Fe chain and that the helical field and the spin orbit interaction is sufficiently strong. In this case, the MBS wavefunction is a superposition of contributions coming from different extrema in the spectrum, which results in two different localization lengths for a single MBS [17]. Assuming a semi-infinite chain with one MBS at each end (without overlap), these localization lengths are determined by the corresponding gaps and given by The MBS probability density is then defined as 8 where N is the normalization constant, ξ 1 and ξ 2 are the localization lengths, ∆ and ∆ m are the proximity gap and the helical field gap of the chain (which depends on the exchange coupling constant), respectively, and k F is the Fermi wavevector of the chain. The topological phase of the chain is reached when ∆ m > ∆ [10].
The black curve shown in Fig. 2e show the LDOS profile extracted along the Fe chain at 5 K. The orange curve represents a tentative fit with the theoretical wavefunction |ψ| 2 giving ξ 1 ≈ 110 nm and ξ 2 ≈ 0.75 nm. We remark that ξ 2 , constituting the short localization of the MBS, is about the same as the diameter of the halo observed by AFM in Fig. 2c.
However, the second localization length ξ 1 is only indicative as it exceeds the chain length and is presumably affected by the defect states present in this chain.
To improve on such issues, we systematically investigated defect-free chains hosting MBS identified by both AFM and conductance measurements. Figures 3a and b show the zerobias constant-height AFM maps and the normalized LDOS(x, y) of such defect-free chains below T c . Both data were acquired at slightly larger tip-sample separations (≈ +50 pm) compared to Fig. 2c. For that reason, the AFM image do not resolve the atomic lattice along the chain. However, both the dark halo (Fig. 3a) and a ZBP in the conductance map shows the experimental LDOS(x) profile extracted along the chain (black dots). As shown by the superimposed blue curve, a remarkable agreement with the raw data periodicity is obtained allowing the extraction of the two localization lengths ξ 1 and ξ 2 . These localization lengths are then found to be 22 nm and 0.75 nm, respectively, and correspond to about 59 and 2 atomic sites with respect to the chain lattice. Several data sets were analyzed this way always showing the same values for these localization (see Fig. S5). In addition, the proximity gap associated with ξ 2 is found to be ∆ m ≈ 33 meV.
To account for the effect of the tip on the wavefunction measurement, we modify the formula for the probability density |ψ| 2 by including a broadening effect resulting from a tip size effect (see additional text in Supp. Info.). This approximation considers a symmet-9 ric apex having metallic character and giving the modulus squared of the wavefunctions at the Fermy energy. The orange curve of Fig. 3b shows the result of such an approximation using a tip apex of 0.17 nm and demonstrates that such a broadening effect is sufficient to reasonably reconstruct the experimental data. We think that the use of lower measurement temperatures and p-wave STM tips [29] in future experiments might help to resolve more precisely the MBSs wavefunction. In