Precursors of Majorana modes and their length-dependent energy oscillations probed at both ends of atomic Shiba chains

Abstract

Isolated Majorana modes (MMs) are highly non-local quantum states with non-Abelian exchange statistics, which localize at the two ends of finite-size 1D topological superconductors of sufficient length. Experimental evidence for MMs is so far based on the detection of several key signatures: for example, a conductance peak pinned to the Fermi energy or an oscillatory peak splitting in short 1D systems when the MMs overlap. However, most of these key signatures were probed only on one of the ends of the 1D system, and firm evidence for an MM requires the simultaneous detection of all the key signatures on both ends. Here we construct short atomic spin chains on a superconductor—also known as Shiba chains—up to a chain length of 45 atoms using tip-assisted atom manipulation in scanning tunnelling microscopy experiments. We observe zero-energy conductance peaks localized at both ends of the chain that simultaneously split off from the Fermi energy in an oscillatory fashion after altering the chain length. By fitting the parameters of a low-energy model to the data, we find that the peaks are consistent with precursors of MMs that evolve into isolated MMs protected by an estimated topological gap of 50 μeV in chains of at least 35 nm length, corresponding to 70 atoms.

Effect of substrate spin-orbit coupling on the topological gap size of Shiba chains

“…The shortcomings of these systems regarding atom manipulation and spectroscopic resolution, respectively, fueled the search for new platforms with the possibility to construct artificial chains. Subsequently fundamental phenomena were observed, including the Shiba bandformation in chains [20,21] by the hybridization of Yu-Shiba-Rusinov (YSR) states of single atoms [12][13][14][15][16][17][18][19], Shiba band structures in k-space revealing a topological gap [22] and length-dependent energy-oscillations of precursors of MBS [23]. However, even though spin-orbit coupling (SOC) is considered to be a key ingredient for the realization of MBS and a large topological gap, an experimental comparison of the in-gap band structures of systems with strongly different SOC but otherwise similar properties is lacking, so far.…”

“…They share the body-centered cubic crystal structure, their lattice constants are only 0.3 % apart [26], their work functions differ by only 1.5 % [27], they have almost identical Fermi surfaces [28,29] and both have an occupied d z 2 -like surface state with similar effective masses and binding energies [30][31][32][33]. One major difference, however, is the strength of SOC effects which is increased by a factor of ∼ 3 for Ta(110) compared to Nb(110) [23,30,32]. This allows us to directly compare new results using the Ta(110) substrate with previous ones using a Nb(110) substrate, which is particularly well-characterized in combination with Cr [34], Mn [22,23,35] and Fe [36] adatom structures.…”

Correlation of Magnetism and Disordered Shiba Bands in Fe Monolayer Islands on Nb(110)

“…As outlined in the introduction, excluding very specific and rare values of the chemical potential or specific types of disorder, 17,35,38 the strong disorder on the order of the pairing energy present in the investigated lattices would destroy any topological gap as well as topological edge modes. 33,35,36 We, therefore, conclude that besides a specific design of the multi-orbital Shiba bands in ferromagnetic layers in direct contact to elemental superconductors, 23 a pseudomorphic growth with the least amount of disorder is probably best suited for the design of two-dimensional topological superconductivity and the related Majorana edge modes. This yet adds another constraint and further narrows down the number of experimental systems which are suitable for the realization of this intriguing quantum state.…”

“…50,54 However, as this shift continues, the spectral weight will move towards the coherence peaks on the other side of E F , which would probably balance the former effect. In the spin-lattice, in addition to these effects, an increasing exchange coupling between Yu-Shiba-Rusinov states will widen the Shiba bands, 23 and this exchange coupling most probably is also different for the three ML types as it might be linked to the one that determines the coercivity. Most importantly, our study reveals that there are neither indications for a gap at E F , nor of any edge states for all three types of disordered ferromagnetic spin-lattices investigated here, neither at nor off the Fermi energy.…”

“…However, it was theoretically predicted that the Majorana edge modes have a dispersion, which can be experimentally resolved. 22 The realization of Shiba-lattices directly on a surface of a metal usually enables atomic manipulation 23,24 and, thereby, would permit the building of more complex structures, as, e.g. 1D Shiba-chains attached to 2D Shiba-lattices allowing for additional experiments in order to prove or disprove the Majorana edge mode origin of enhanced densities of states.…”