Creating Tunable Quantum Corrals on a Rashba Surface Alloy

Artificial lattices derived from assembled atoms on a surface using scanning tunneling microscopy present a platform to create matter with tailored electronic, magnetic, and topological properties. However, artificial lattice studies to date have focused exclusively on surfaces with weak spin–orbit coupling. Here, we illustrate the creation and characterization of quantum corrals from iron atoms on the prototypical Rashba surface alloy BiCu2, using low-temperature scanning tunneling microscopy and spectroscopy. We observe very complex interference patterns that result from the interplay of the size of the confinement potential, the intricate multiband scattering, and hexagonal warping from the underlying band structure. On the basis of a particle-in-a-box model that accounts for the observed multiband scattering, we qualitatively link the resultant confined wave functions with the contributions of the various scattering channels. On the basis of these results, we studied the coupling of two quantum corrals and the effect of the underlying warping toward the creation of artificial dimer states. This platform may provide a perspective toward the creation of correlated artificial lattices with nontrivial topology.

The resulting FT images revealed a set of pronounced rings. Discussing from the center outward, first we found a pronounced circular inner ring that we attribute to the isotropic q1. This is followed by two additional rings whose shapes resemble a hexagon with rounded vertices along the Γ " M $ directions (corresponding to the 〈100〉 directions in real space); we attribute these rings to q2 and q3, respectively.
The fourth feature in the FTs is a faint and partly interrupted ring with a hexagonal shape, which we attribute to q4. Finally, we observed six outer spots that belong to the atomic lattice of BiCu2, forming the )√3 × √3-30° superstructure on Cu(111). In total, the standing wave patterns inside the QCs consist of four wave vectors q1-q4. For a quantitative comparison, we extracted their magnitudes along both high-symmetry directions (Γ " M $ and Γ " K $ ) for all maps acquired inside the QC with R = 7.3 nm as well as the QC with R = 6.15 nm. Their dispersive behavior is displayed in Fig. S2(i). We note that the error based on the QPI FT analysis of the bare BiCu2 surface by Steinbrecher et al. 5 As can be seen, the wave vector dispersions of the QCs are in good agreement with the literature. Also, the anisotropic character of q2-q4 is seen in our data, while q1 remains isotropic. From this, we conclude that the underlying BiCu2 band structure determines the magnitudes of the possible scattering vectors within the QCs throughout the energy range probed in this study. We note that we found no difference in the dispersions between the R = 6.15 nm QC and the R = 7.3 nm QC. Furthermore, while the dispersions   of the four scattering wave vectors show good agreement both in the occupied and unoccupied states, respectively, this is not the case for the standing waves in real space, which deviate strongly from our model in the occupied states. These findings point to an energy-dependent change in the scattering potential, which varies the phase of the standing waves, not their wavelength. Due to a lack of additional input on how the scattering potential varies with energy, we did not include this in our model.
For the particle-in-a-box modelling, we took the average of the two scattering wave vectors measured for each voltage along Γ " M $ and Γ " K $ , and we fit the dispersion of these scattering wave vectors using parabolic fits, based on the effective masses and Rashba splitting reported for the BiCu2 surface. 5 We note that we did not include the hexagonal anisotropy in our model in order to be able to analytically solve the Schrödinger equation (the solutions being Bessel functions). This simplification is justified when only comparing cross sections of the spatially dependent LDOS vs. energy, because we found that the hexagonal anisotropy has a relatively small impact on the eigenstates (cf. Fig. S8). This leads to the approximated isotropic effective band structure of BiCu2, shown in Fig. S2(j). These effective bands are used to simulate the confined states of BiCu2. The red band stems from the inner spz-type Rashba-split surface state (q1), the green (q2) and dark blue (q3) bands simulate interband scattering wave vectors, and the light blue band stems from the outer pxpy-band (q4). With the parabolic dispersions, we treated each band independently using a particle-in-a-box model.

Disentangling the scattering wave vector contributions
To be able to attribute features present in the experimental and theoretical spectrum to a certain confined state, we plot in Fig. S3  to the state that is responsible for the ring-like shape near the center of the map at 150 mV in Fig. 2(b).
From the contribution of q1 alone, one would expect a maximum in the center. For the quantum corral with R = 3.1 nm, we found a ring-like state at 200 mV (see Fig. 5(a)), which is also found in the simulations of q3, see the yellow arrows in Fig. S3(c). Overall, we found multiple instances where details in the observed results can only be understood by inclusion of q3. This is very different in comparison to our calibration study using QCs on Cu(111). Figure S4 compares the STS spectra measured across the QC with R = 7.3 nm with two simulations without ( Fig. S4(b)) and with (Fig. S4(c)) the contribution of q4, respectively. Additional modulations are found in the simulation that includes q4. For example, the confined state close to the band onset (near 250 mV) shows a pronounced beating. An extra maximum is found close to 110 mV in the center of the QC. These features were, however, not found in the experiment. In addition, we did not see standing waves above the onset of the inner spz-type Rashba-split surface state in the experiments, where only features related to q4 could contribute. From these findings, we conclude that the impact of q4 is negligible, which is why we excluded all contributions from q4.   We show here the cross section along [110] (Fig. S6(b)), complimentary to that shown along [100] in Fig. 4(d). There was no change in the energies of the QC eigenstates when comparing both directions.

Additional electronic characterization of the smaller QCs
The corresponding simulation for this QC is shown in Fig. S6(c) for the sake of completeness. Again, the comparability with the experiment is better above EF, and unsatisfactory below EF. Hexagonal anisotropies are evident in the dI/dV maps (g-i), leading to slight differences of the spatial distribution of the eigenstates (see Fig. S8). 8 An STM image of the QC with R = 6.15 nm (see Fig. 2(b) in the main text for corresponding dI/dV maps) is presented in Fig. S6(d). The QC states found therein are plotted in Fig. S6(e), together with the corresponding simulation in Fig. S6(f).
To complement the dI/dV maps of the coupled QC pairs, Fig. S6(g-i) presents dI/dV maps of the QC with R = 3.1 nm at the voltages labeled state "A", "B" and "C" in Fig. 5 and Fig. S11(a), respectively, i.e., displaying the LDOS distributions of the uncoupled QC states.

Ruling out artifacts in STS
All dI/dV maps presented in the main manuscript were measured in constant-current mode, i.e., the tipsample separation was varied during data acquisition to keep the current constant. This can introduce artifacts in the data, especially when measuring close to the Fermi energy. 10 Two methods can effectively reduce these artifacts: (i) point spectra stabilized at larger bias voltage and (ii) constant height maps. 10 In case of the former, the tunneling current, given by the integrated LDOS between EF and the applied voltage, does not vary spatially, and the measurement is essentially like a constant-height measurement.
To rule out that artifacts entered our data, we first compared the features observed in constant-current mode with maps measured in constant-height mode. Fig. S7 shows such a comparison for the QC with R = 7.3 nm. The signal-to-noise (S/N) ratio of the constant-height maps is rather low, as we had to take the map at a relatively large tip-sample separation to prevent manipulating the Fe atoms. In constantcurrent mode, we were able to reduce the tip-sample distance inside the QC without the risk to crash into the Fe atoms, yielding a much higher S/N. Nevertheless, all features are found in the maps for both modes. For example, we found a clear hexagonal symmetry in the constant-current map at 184 mV, To further underline that the change in rotation of hexagonal features is not caused by artifacts of the measurement mode, we compared the constant-current map measured at 125 mV with point spectra measured across the QC with R = 7.3 nm, see Fig. S8(a-c). Prior to opening the feedback loop and acquiring the point spectra, the tip was stabilized at a high bias voltage of 500 mV to ensure that the tipsample separation was the same at each point within the QC (i.e. the measurement is identical to a constant-height measurement). The set of spectra in Fig. S8(a) was measured along a line running horizontally across the QC (i.e. along the [100] direction), while the set in Fig. S8(b) was taken along a line running perpendicular to it (i.e. along [110]). To obtain a direct comparison, Fig. S8(c) shows the position-dependent dI/dV intensity at 125 mV (highlighted by black/red dashed lines in Fig. S8(a,b)) along both directions. While the outer maximum has similar intensity in both directions, the signal of the A similar direction-dependent comparison for the 3.1 nm QC is presented in Fig. S8(d-f). Here the effect of the hexagonal anisotropy shown for the linecut at -300 mV (f) clearly shows that the peripheral maximum along the [110] direction (red) is further away from the QC center, which is consistent with the dI/dV map at this energy ( Fig. S6(i)) which shows hexagonal vertices along the 〈110〉 directions.
Finally, we compared the constant-current maps of the R = 3.1 nm QC with the spatial dependence of dI/dV intensity at the respective voltages, extracted from point spectra measured across the QC, see Fig. S9. An STM image of the QC is shown in Fig. S9(a). The dashed line across the QC represents the path of data acquisition, plotted in Fig. S9(b). To directly compare the constant-current maps in Fig.   S9(c) with point spectra, we evaluated the dI/dV signal at the energies highlighted with black lines in Fig. S9(b). The results are shown in Fig. S9(d). Starting at 216 mV, we found a ring-like state inside the QC. Consistent with this finding, the dI/dV signal measured across the QC shows two maxima. At 200 mV, we found again a ring in the constant-current map. While the cross-sectional dI/dV signal of the point spectra does not reflect the node in the center, it still shows two maxima. Further analysis revealed that the dI/dV signal at 200 mV is not an eigenstate but a superposition of the ring-like state at 216 mV We note that, from a simple particle-in-a-box picture, it may be unexpected that the first eigenstate observed in the 3.1 nm QC is a ring, but this is merely a consequence of the multi-band nature of BiCu2.
As shown in Fig. S9(c), at Vs ≈ 0.22 V we observe the QPI from the interband scattering channels q2 and q3, whereas there is no eigenstate from q1, yet. Contributions from the first eigenstate (i.e. an antinode in the center, which is usually referred to as the "1S" state) of q1 is observed further away from the band onset, at Vs ≈ 0.17 V. A "1S" state of q2-q4 cannot be observed due to the dispersions of the BiCu2 bands ( Fig. 1(d)). As the band onset of the outer band is at a much higher energy (E4 ≈ 1 eV, see section S2), the "1S" state of q4 is expected to be far above the band onset (E1 = 0.26 eV) of the inner Rashba-split bands. Furthermore, as the inner band does not exist above this onset, interband scattering does not exist above Vs = 0.26 V. This is why low q values (and hence "1S" states) do not exist for q2 and q3. 13

Additional data on coupled QC pairs
For completeness, Fig. S10 presents the same data as shown in Fig. 4 of the main text, but displayed here at the full energy range, including the dominating broad peak at Vs ≈ 0.2 V. We could not identify a significant splitting of this peak in the coupled QC pairs. We assume that this is due the fact that this peak is due to a superposition of a hexagonal ring-like state at Vs ≈ 0.21 V (originating from q2 and q3 scattering) and an isotopic state with a broad central antinode at Vs ≈ 0.17 V (q1 scattering), as shown in Fig. S9. The point spectra of the single 3.1 nm QC are dominated by the 0.17 V state, but spectra taken off-center show a slight shift to higher energy by ca. 4 mV (Fig. S11(a)). For the QC pairs, an energy shift of ca. 8 mV can be seen for spectra taken at the outer edge vs. the center of the pairs (b,c).
While this is indicative of a bonding/antibonding splitting, it is not reliably quantifiable for this state. 14 comparison of the two spectra. However, the dI/dV maps in Fig. 5 clearly show nodes and antinodes in the center of the QC pairs.

Structural models of the QCs
Figures S12 presents structural models of the R = 7.3 nm and R = 6.15 nm QCs, including the exact locations (i.e. adsorption sites) of the Fe adatoms on the BiCu2 surface. Fig. S13

Maps close to the Fermi energy
Figure S14 presents two LDOS maps for the QC with R = 7.3 nm of the two confinement states closest to the Fermi energy. The map of the unoccupied state (Vs = 54 mV) qualitatively differs significantly from that of the occupied state (Vs = -25 mV), which is evidence that these are two different confinement states.