Embedded YSR impurities below the La(0001) surface
We have studied bulk-like α-lanthanum films epitaxially grown on a Re(0001) single crystal [see “Methods” and Supplementary Note 1 and Supplementary Fig. 1]. STM/STS measurements were carried out at T = 1.65 K, which is well below the superconducting transition temperature of La films, Tc ~ 5–6 K20. The magnetic impurities were introduced during the La deposition, being contaminants in the 99.9+ % pure La material used for the experiments [see “Methods”].
The magnetic impurities are clearly identifiable in differential tunneling conductance (dI/dV) maps taken above the atomically flat La(0001) surface at a bias voltage corresponding to states inside the superconducting gap of La, as shown in Fig. 1a. Since there are no quasiparticle excitations in the bulk superconductor at this energy, the local modulation of the LDOS can solely be attributed to the YSR bound states formed by the impurities, which are embedded below the La(0001) surface [see Supplementary Note 2 and Supplementary Fig. 2]. The various shapes of the modulations (YSR1, YSR2, YSR3, and YSR4 in Fig. 1a, b) presumably originate from the various depths of the magnetic impurities [see Supplementary Note 3 and Supplementary Fig. 3]. However, it is also possible that different magnetic elements being present as natural impurities even in high-purity La material may contribute to the different spatial and spectroscopic signatures of the observed YSR impurities. In the following, we mainly focus on YSR1-type impurities, characterized by the highest modulation intensity.
Figure 1b, c shows zoomed-in dI/dV maps for the YSR1 impurity obtained at positive and negative bias voltages, or electron- and hole-like bound states, respectively. The spatial extension of the YSR bound state resembles a star-shaped pattern, with six beams protruding along the directions parallel to the reciprocal lattice vectors (see the inset of Fig. 1a), and an oscillatory modulation characteristic of YSR states6,7. The oscillations are observable up to ~30 nm away from the impurity along the beams, and up to ~10 nm distance along the directions halfway between the beams, both being considerably longer than in previously reported STM/STS studies of YSR bound states12,13,14,16,18.
Spectroscopic signature of spatially extended YSR states
In Fig. 1d, the YSR quasiparticle excitations show up as a pair of resonances in the dI/dV spectra inside the superconducting gap of La (ΔLa,surface ~ 1 meV). Taking into account the convolution of the Bardeen–Cooper–Schrieffer-type DOS of the sample and the La-coated superconducting tip in the measured tunneling spectrum21, the binding energies of the YSR states are found at EYSR = ±0.55 meV (red and blue arrows). Their symmetric position with respect to the Fermi level reflects the particle–hole symmetry of the superconductor.
The spatial extension of the YSR states shown in Fig. 1b, c has been quantitatively analyzed by taking differential tunneling conductance profiles (Fig. 1e) and spectroscopic maps (Fig. 1f) along lines crossing the center of the magnetic impurity. The period of the oscillations, corresponding to half of the Fermi wavelength (λF/2), is found to be ~0.99 nm along the beams (Q2 direction in Fig. 1e) and ~0.90 nm along the direction halfway between the beams (Q1 direction), respectively, for both positive and negative bias voltages. The phase shift between the spectroscopic maps at ±EYSR, determined by the binding energy17,18, is clearly conserved as far as 30 nm away from the impurity along the Q2 direction, confirming that even at this distant point, the LDOS modulation can be attributed to the coherent particle–hole symmetric YSR states.
Quasi-2D surface electronic structure of La(0001)
To address the origin of the anisotropic long-range extension of the YSR states, we turn our attention toward the surface electronic structure of La(0001). The dI/dV spectrum averaged over a flat terrace shows a sharp resonance peak at E = +110 meV, which reflects the band edge of the surface state of La(0001) as shown in the left panel of Fig. 2a [Supplementary Note 1]22. The atomic defects close to the surface create oscillatory quasiparticle interference (QPI) patterns in the measured dI/dV maps [see Supplementary Note 4 and Supplementary Fig. 4]. The Fourier transform (FT) of an STM image obtained at a bias voltage just outside the superconducting gap (Fig. 2c) provides direct information about the FS23, possessing a hexagonal shape with flat parts along the Q2 direction and rounded vertices along the Q1 direction. By quantitatively analyzing the FT maps as a function of bias voltage, the dispersion relation along the Q1 and Q2 directions can be derived from the varying size of the hexagon (right panel of Fig. 2a [see Supplementary Note 4]).
Since both the quasiparticle scattering at the Fermi energy and the YSR states find their origin in the FS, it is reasonable to consider the analogies between them. They share the same period of oscillation along the two directions Q1 and Q2. Due to the quasiparticle-focusing effect, both types of oscillations are extended to a longer range along the flat directions of the FS than along the vertices, as can be seen when comparing Fig. 2 with Fig. 1.
The experimental QPI results are remarkably well reproduced by the Bloch spectral function (BSF) calculated based on the screened Korringa–Kohn–Rostoker method within density functional theory, providing information on the electronic band structure [see “Methods”]. Two high-intensity spectral features are observed close to the Γ point (Fig. 2b): the bulk band (B) is present in all layers, while the surface band (S) appears only in the top few atomic layers, with a flat dispersion around E = 0.1 eV, characteristic of localized d-type surface states. The BSF at the Fermi energy in the surface Brillouin zone is displayed in Fig. 2d. Since the momentum transfer vectors Q1 and Q2 are twice as long as the Fermi wave vectors along the relevant directions, it can be concluded that the hexagonal QPI pattern in Fig. 2c arises from intraband scattering of quasiparticles originating from the surface band (inner hexagon in Fig. 2d). The QPI pattern and the YSR states appear to be insensitive to the bulk band (outer hexagon present in the BSF calculations in Fig. 2d), which is rotated by 30° with respect to the surface band. This implies that the surface band of La(0001) plays a major role in the quasiparticle scattering at the surface, while the contribution of the bulk band is weak or lacking. Note that similarly to the present case, only one of the two FSs contributes to the observed YSR states in Pb(111)17 as well as in NbSe218.
Quasiparticle focusing of YSR states via the FS
The QPI patterns and the first-principles calculations thereby reveal two major contributions to the anisotropic long-range extension of the YSR states. The first is the quasi-2D electronic structure, because primarily electrons in the surface band contribute to the scattering processes. The second is the quasiparticle-focusing effect, causing beam-like extensions perpendicular to the flat parts of the anisotropic FS. In order to quantify the role of the shape of the FS on the spatial extension, we performed 2D atomistic model calculations parameterized by the first-principles results to determine the energy and the LDOS modulation of the YSR states [see Methods]. As shown in Fig. 3a, an ideal circular FS leads to an isotropic decay of the YSR intensity, following a 1/r power law at intermediate distances where r is the distance from the impurity17,18. For the hexagonal FS in Fig. 3b, the LDOS along the beam directions perpendicular to the flat sides of the FS decays slower than 1/r, effectively reducing the dimensionality due to the strong focusing effect [see Supplementary Note 5, Supplementary Fig. 5, and Supplementary Movie 1]3,24. By considering the realistic FS of La(0001) obtained from the first-principles calculations, which is slightly rounded from the ideal hexagon as shown in Fig. 2c, d, our numerical results (Fig. 3c, left) are consistent with the experimentally obtained dI/dV map for the YSR1 impurity (Fig. 3c, right) with regard to the shape and the range of the extension, as well as the oscillation period.
It is worth mentioning that the focusing due to the anisotropic FS and the dimensionality of the system cannot rigorously be distinguished, because they are essentially different manifestations of the same mechanism. For example, an isotropic two-dimensional FS (circle) corresponds to a cylindrical three-dimensional FS, which is strongly anisotropic since the quasiparticles cannot propagate along the axis of the cylinder. Incidentally, the innermost sheet of the FS of bulk La is almost perfectly cylindrical according to our calculations [see Supplementary Fig. 6]. Conversely, the contribution of the anisotropic FS may be understood as an effectively reduced dimensionality [see Supplementary Note 5 and Supplementary Fig. 5]. However, we find that a circular two-dimensional or cylindrical three-dimensional FS could not fully account for the long-range extension, since they would lead to a characteristic 1/r decay that is faster than the spatial decay observed in the experiments (~1/r0.92).
Interacting YSR impurities at large distances
Finally, we demonstrate that the YSR impurities on La(0001) exhibit a significant long-range coupling between them. The interaction between YSR impurities results in a variation of EYSR compared to the case of a single impurity19,25,26,27,28. Figure 4a presents four YSR1-type impurities: two of them (A, B) at about 3.6 nm distance roughly along the Q2 direction, and two isolated ones (I1, I2) being significantly further away. Interestingly, due to the interference between the YSR states of the impurities A and B, the LDOS shows a twin-star pattern with inhomogeneous modulations around the paired YSR1-type impurities. As shown in Fig. 4b, the binding energy EYSR = 0.56 meV for the isolated I1 and I2 impurities is consistent with the one for the YSR1 impurity in Fig. 1d. However, on the A and B impurities, a sizeable shift of around 90 μeV is observed, being identical for the two atoms. This clearly indicates that the extended YSR bound states give rise to mutual long-range interactions between the magnetic impurities.