Coupling of Yu-Shiba-Rusinov states in one-dimensional chains of Fe atoms on Nb(110)

The hybridization of Yu-Shiba-Rusinov (YSR) states in dimers of magnetic impurities leads to molecularlike bonding and antibonding modes. In many-impurity systems, the interaction gives rise to YSR bands and can even result in the formation of a topologically nontrivial superconducting state, characterized by Majorana fermions at the edges of the system. To obtain a more detailed understanding of these interactions, we investigate the coupling of YSR states in short one-dimensional Fe chains on clean Nb(110). We observe a splitting of the single-atom YSR peaks into multiple states with even or odd spatial symmetry and identify a peculiar dependence of the even and odd states' energy position on the chain length.

Figure 1

Topographic images of the same Nb(110) surface area (scan range: $15\times 15$ ${\mathrm{nm}}^2$) recorded with (a) the CO-functionalized and (b) the bare metal tip. The CO-functionalized tip uncovers the Nb lattice and the individual Fe atoms in the self-assembled clusters. Scan parameters: (a) $U=-10\mathrm{mV}$, $I=5\mathrm{nA}$; (b) $U=-1\mathrm{V}$, $I=400\mathrm{pA}$.

Figure 2

(a) Tunneling spectra of a single Fe atom and the three different dimers with the shortest interatomic distance. The gray hatched line represents the spectrum on clean Nb(110). Spectra are vertically shifted for clarity. The gap width of the superconducting tip amounts to ${\mathrm{\Delta }}_{\mathrm{tip}}=1.31\mathrm{meV}$ and is indicated by dotted lines. (b)–(e) High-resolution STM images (scan range: $2\times 2{\mathrm{nm}}^2$) of the respective dimers (left) and their ball model representation (right). In each panel the dimer length and orientation is indicated. Stabilization parameters: (a) $U=-7\mathrm{mV}$, $I=400\mathrm{pA}$; (b)–(e) $U=-10\mathrm{mV}$, $I=5\mathrm{nA}$.

Figure 3

(a) Topographic image of a dimer along the nearest-neighbor $\left[1\overline{1}1\right]$ direction. (b) Spectra taken at the center and on the end of the dimer. (c) Numerically calculated second derivative of the $dI/dU$ signal measured along the dimer. White arrows highlight the low-intensity states. (d),(e) $dI/dU$ maps at indicated bias voltages corresponding to the peak positions marked by black arrows in (b). Black dots symbolize the positions of the Fe atoms. ${\mathrm{\Delta }}_{\mathrm{tip}}=1.33\mathrm{meV}$. Stabilization parameters: (a) $U=-10\mathrm{mV}$, $I=3\mathrm{nA}$; (b)–(e) $U=-7\mathrm{mV}$, $I=1\mathrm{nA}$. (f)–(j) and (k)–(o) are same as (a)–(e), but for a trimer and tetramer, respectively. The surface orientation of all scans is given in (f) and (i). Scale bars are $1\mathrm{nm}$.

Figure 4

(a)–(c) Calculated energy splitting for electronlike states as a function of the nearest-neighbor hopping term $k_1$ for a dimer, trimer, and tetramer, respectively. Hopping between next-nearest neighbor atoms is neglected. The symmetry is reflected by the color of the lines with blue indicating the even and red the odd states. (d)–(f) Energy of YSR states at positive bias extracted from the data in Figs. 3, 3 and 3 by subtracting ${\mathrm{\Delta }}_{\mathrm{tip}}$. The data is shifted to align the center energy of the experimental results with “0” in the calculations (see main text for details). Length of the $y$ axis is $2{\mathrm{\Delta }}_{\mathrm{Nb}}$, the same as in (a)–(c). The dimer is used as a reference to fix $k_1$ in the calculations, such that the calculations and experiment match perfectly for the dimer. As indicated by the arrows, the calculation results at the extracted value of $k_1$ reproduce the experimental results for the trimer and tetramer.