Influence of Magnetic Ordering between Cr Adatoms on the Yu-Shiba-Rusinov States of the $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ Superconductor

We show that the magnetic ordering of coupled atomic dimers on a superconductor is revealed by their intragap spectral features. Chromium atoms on the superconductor $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface display Yu-Shiba-Rusinov bound states, detected as pairs of intragap excitations in tunneling spectra. By means of atomic manipulation with a scanning tunneling microscope’s tip, we form Cr dimers with different arrangements and find that their intragap features appear either shifted or split with respect to single atoms. These spectral variations are associated with the magnetic coupling, ferromagnetic or antiferromagnetic, of the dimer, as confirmed by density functional theory simulations. The striking qualitative differences between the observed tunneling spectra prove that intragap Shiba states are extremely sensitive to the magnetic ordering on the atomic scale.

Figure 1

Cr atoms on the Bi-terminated $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface. (a) Atomic resolution of a Bi-terminated clean $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface in a constant height mode at 3 mV, 200 nA with the junction resistance of $15\mathrm{k}\mathrm{\Omega }$ (size $3\times 3{\mathrm{nm}}^2$). (b) STM image of Cr atoms deposited on the Bi-terminated $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface ($V=1\mathrm{V}$, $I=10\mathrm{pA}$, size $10\times 10{\mathrm{nm}}^2$). (c) Results of a full DFT relaxation showing that the most stable position of a Cr atom (green) is at a hollow site (Bi atoms in pink and Pd atoms in bronze). Differential-conductance spectra measured (d) on the bare surface and (e) on a Cr adatom, using a superconducting $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ tip (feedback loop opening condition 3 mV, 250 pA). Tunneling between coherent peaks of the tip and sample results in sharp $dI/dV$ peaks at $\pm 2\mathrm{\Delta }/e$ (dashed lines in the plots). (f) Density of states obtained by deconvoluting the tip DOS from the spectra on the Cr adatom (blue line) and on the bare surface (black line). Analysis of STM and scanning tunneling spectroscopy data was performed with the WSxM [27] and SpectraFox [28] software packages.

Figure 2

(a)–(d) STM image of Cr dimers at indicated spacings ($V=1\mathrm{V}$ and $I=10\mathrm{pA}$, size $3\times 3{\mathrm{nm}}^2$). The crossing of square mesh depicts the hollow sites of the $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface. (e)–(h) Differential conductance spectra on the Cr atom indicated by a white dot before (gray lines) and after (red lines) the second atom is moved to the dimer position (feedback loop opening condition 3 mV, 250 pA). As in Fig. 1, all spectra are measured using superconducting tips with the same $\mathrm{\Delta }$ as the substrate (dashed lines mark the $\pm 2\mathrm{\Delta }/e$ bias in the middle panels and the $\pm \mathrm{\Delta }/e$ bias in the bottom panels). The corresponding tip-deconvoluted DOS of the corresponding $\mathrm{Cr}/\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ system is shown in the lower panels.

Figure 3

Model calculation of the Shiba DOS of an antiferromagnetically or ferromagnetically ordered dimer separated by (a) $d=2a$ and (b) $d=\sqrt{2}a$ (see details [21]). For both dimer structures, FM order leads to split Shiba peaks, while AFM shows only a degenerate one. Panels (c) and (d) show the corresponding $dI/dV$ spectra obtained by convoluting the dimer’s DOS with a superconducting tip DOS.

Figure 4

Topography profiles and spectral ($dI/dV$ vs $V$) maps measured along the axes of the (a) $2a$ and (c) $\sqrt{2}a$ dimers (gray scale covers the $dI/dV$ range from 0 to $0.3\mu \mathrm{S}$). The top panels in (b) and (d) correspond to the dimer STM image, scanned at 3 meV and 10 pA. Below, we show the 2D spatial distribution of the differential conductance at the bias of the Shiba peaks for (b) the $2a$ dimer (size $3\times 3{\mathrm{nm}}^2$) and (d) the $\sqrt{2}a$ dimer (size $2.5\times 2.5{\mathrm{nm}}^2$), at the given energies.

Figure 5

Spin polarization (a) of a Cr adatom and of various dimers [$d=1a$ (b), $2a$ (c), and $\sqrt{2}a$ (d)] on the $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ surface. In each case, the results correspond to the minimum-energy configuration obtained from DFT simulations of the relaxed system. The plot shows 3D isosurfaces of constant electronic density difference between majority and minority spins, in the normal state. The surface is tilted differently in every case to picture the extension of the spin density to inner layers. Bi atoms are represented in pink and Pd atoms in bronze. The isocontours of spin are yellow and red for each of the two spin components (the isocontour is $0.005e/{\AA }^3$ in all graphs).