End States and Subgap Structure in Proximity-Coupled Chains of Magnetic Adatoms

A recent experiment [Nadj-Perge et al., Science 346, 602 (2014)] provides evidence for Majorana zero modes in iron (Fe) chains on the superconducting Pb(110) surface. Here, we study this system by scanning tunneling microscopy using superconducting tips. This high-resolution technique resolves a rich subgap structure, including zero-energy excitations in some chains. We compare the symmetry properties of the data under voltage reversal against theoretical expectations and provide evidence that the putative Majorana signature overlaps with a previously unresolved low-energy resonance. Interpreting the data within a Majorana framework suggests that the topological gap is smaller than previously extracted from experiment. Aided by model calculations, we also analyze higher-energy features of the subgap spectrum and their relation to high-bias peaks which we associate with the Fe $d$ bands.

Figure 1

$dI/dV$ spectra recorded at the end of six different chains (small arrows mark the position on the chain). Chains in (a)–(d) are terminated by a small cluster; in (e),(f) they have a sharp cutoff. The energy resolution in (e),(f) is reduced (width of BCS resonance: $\simeq 330\mu \mathrm{V}$) due to nonbulklike superconductivity of the tip. ${\mathrm{\Delta }}_{\mathrm{tip}}$ in meV: (a) 1.36; (b) 1.42; (c),(d) 1.38; (e),(f) 1.24. Chain lengths measured between the chain end and the cluster onset in nm: (a) 13.9, (b) 9.5, (c) 6.2, (d) 6.0, (e) 7.7, (f) 4.0. Scale bars correspond to 4 nm. For the full spectra of (b)–(f), see the Supplemental Material [39].

Figure 2

$dI/dV$ spectra of different Fe entities on Pb(110). (a) Two types of single adatoms: both spectra exhibit a single Shiba resonance close to the gap edge. The adatoms differ in apparent height by $\simeq 20\mathrm{pm}$ at 50 mV, 50 pA. (b) Fe dimer: a variety of Shiba states with the lowest energy resonance at $ϵ\simeq 150\mu \mathrm{eV}$ is observed. ${\mathrm{\Delta }}_{\mathrm{tip}}=1.39\mathrm{meV}$. (c) Chain end (the blue line) as in Fig. 1, and bare Pb(110) (the gray line) for comparison. (d) Zoom on (c): a manifold of Shiba resonances is resolved. A peak at $+{\mathrm{\Delta }}_{\mathrm{tip}}$ may be the fingerprint of a Majorana bound state. Low-energy states lie at $ϵ\simeq 80\mu \mathrm{eV}$ (the black solid arrow) and $ϵ\simeq 270\mu \mathrm{eV}$ (the blue dashed arrow). At 1.1 K, low intensity thermal resonances are expected at $(\pm {\mathrm{\Delta }}_{\mathrm{tip}}-ϵ)$ [38]. Albeit not resolvable, they contribute to the tail in intensity at biases between $\pm {\mathrm{\Delta }}_{\mathrm{tip}}$. Scale bars in the topography insets correspond to 1 nm.

Figure 3

Spatially resolved $dI/dV$ spectra along the chain in Fig. 1, going from the onset of the Fe cluster (No. 1) to the bare surface (No. 35). At the chain end [No. 29, same as in Fig. 1 and Fig. S2(b)], a zero-energy resonance at $-{\mathrm{\Delta }}_{\mathrm{tip}}$, and at the Fe cluster (No. 1), a resonance at $+{\mathrm{\Delta }}_{\mathrm{tip}}$ are visible as local maxima. At the opposite bias, a shoulder is observed. The lowest nonzero energy resonance is found at $ϵ\simeq 80\mu \mathrm{eV}$ in No. 29 and is modulated along the chain. As a guide for the eye, local maxima or shoulders with energies $0<ϵ<100\mu \mathrm{eV}$ are marked by ticks. The high intensity peaks at positive bias shift along the chain with $ϵ$ ranging from 700 to $1000\mu \mathrm{eV}$. Offset for clarity: $-30\mathrm{nS}$/spectrum. Distance between spectra: 0.33 nm. ${\mathrm{\Delta }}_{\mathrm{tip}}=1.42\mathrm{meV}$.

Figure 4

False-color plot of the $dI/dV$ spectra of (a) the subgap (the same spectra as in Fig. 3) and (c) the $d$-band structure, aligned with the topography in (b). The subgap structure (a) exhibits variations of the Shiba peak energies and intensities, as well as modulations of the low-energy resonance at $ϵ\simeq 80\mu \mathrm{eV}$ along the chain. A zero-energy resonance is found at both chain terminations. Resonances linked to the $d$-band structure (c) vary around $-380\mathrm{mV}$ ($\alpha$), and 450 mV (${\alpha }^{\prime }$) along the chain. (d) Sketch of a spin-polarized $d$ band (the full line) and its hole complement (the dashed line) crossing the Fermi level of the superconductor.

Figure 5

Numerical results for a chain of 30 sites. (a) Color plot of the differential conductance along the chain at subgap energies for a superconducting tip. (b) Spatially varying on-site energies of impurity levels. (c) Bulk subgap band structure of an impurity chain on the surface of an $s$-wave superconductor. We used ${\mathrm{\Delta }}_{\text{sample}}/{\mathrm{\Delta }}_{\mathrm{tip}}=0.958$. See Ref. [39] for other parameters.