Wave-Function Hybridization in Yu-Shiba-Rusinov Dimers

Magnetic adsorbates on superconductors induce local bound states within the superconducting gap. These Yu-Shiba-Rusinov (YSR) states decay slowly away from the impurity compared to atomic orbitals, even in 3D bulk crystals. Here, we use scanning tunneling spectroscopy to investigate their hybridization between two nearby magnetic Mn adatoms on a superconducting Pb(001) surface. We observe that the hybridization leads to the formation of symmetric and antisymmetric combinations of YSR states. We investigate how the structure of the dimer wave functions and the energy splitting depend on the shape of the underlying monomer orbitals and the orientation of the dimer with respect to the Pb lattice.

Figure 1

(a) $dI/dV$ spectrum at the center of the single Mn adatom shown in (b). Three subgap resonances ($\pm \alpha$, $\pm \beta$, $\pm \gamma$) and the tip gap ($\pm \mathrm{\Delta }$) are marked in the spectrum by dashed vertical lines (blue, orange, red, gray). For reference, a trace taken on the pristine substrate is superimposed (solid gray line). Set point: 300 pA, 5 mV. (c) $dI/dV$ maps of a monomer at $+\alpha$, $+\beta$, and $+\gamma$, covering the same area as in (b). (d) $dI/dV$ spectrum of the dimer shown in (e), which is oriented along the $[1\overline{1}0]$ direction and separated by $1.38\pm 0.08\mathrm{nm}$. Each subgap state is split into two resonances ${\alpha }_{s,a}$, ${\beta }_{s,a}$, and ${\gamma }_{s,a}$ (marked by arrows). Set point: 200   pA, 4  mV. Lock-in: 912 Hz, $15\mu {\mathrm{V}}_{\mathrm{rms}}$. (f) $dI/dV$ maps taken at the positive-energy YSR resonances as marked in the figure. The scale for the resonances $\beta$ in (c) and for ${\beta }_{a,b}$ in (f) is cut to emphasize the laterally extended intensity around the high intensity at the impurity center.

Figure 2

(a) $dI/dV$ spectrum at the center of the single Mn. Three subgap resonances ($\pm \alpha$, $\pm \beta$, $\pm \gamma$) and the tip gap ($\pm \mathrm{\Delta }$) are marked in the spectrum by dashed vertical lines (blue, orange, red, gray). For reference, a trace taken on the pristine substrate is superimposed (solid gray line). Set point: 300 pA, 5 mV. Lock-in: 912 Hz, $15\mu {\mathrm{V}}_{\mathrm{rms}}$. (b) Topography and (c) $dI/dV$ maps of three Mn adatoms, two of which form a dimer oriented along the [100] direction and separated by $1.47\pm 0.08\mathrm{nm}$. (d) $dI/dV$ spectrum of the dimer shown in (b). $\gamma$ is split into two resonances ${\gamma }_{\mathrm{s},\mathrm{a}}$ (marked by arrows). A faint signal at the energy of the monomer $\pm \gamma$ peaks hints at a third resonance. Set point: 200 pA, 4 mV.

Figure 3

(a) $dI/dV$ spectra of Mn dimers with different interatomic distances oriented along the $⟨110⟩$ direction. Spectra are recorded at the center of one of the adatoms of a pair. Setpoint: 200 pA, 4 mV. (b)–(d) The splitting in energy of the peaks $\alpha$, $\beta$, and $\gamma$ as a function of the interatomic distance for the $⟨110⟩$ direction. The same color and symbol indicate split pairs of resonances from the same dimer spectrum. Different symbols and colors correspond to data from different dimers except for yellow circles, which indicate data points where no splitting was observed for the respective resonance.

Figure 4

(a) $dI/dV$ spectra of Mn dimers with different interatomic distances oriented along the $⟨100⟩$ direction. Spectra are recorded at the center of one of the adatoms of a pair. Set point: 200 pA, 4 mV. (b)–(d) The splitting in energy of the peaks $\alpha$, $\beta$, and $\gamma$ as a function of the interatomic distance for this orientation. Same color code as in Fig. 3 3–3.