Kondo-free mirages in elliptical quantum corrals

The quantum mirage effect is a fascinating phenomenon in fundamental physics. Landmark experiments on quantum mirages reveal atomic-scale transport of information with potential to remotely probe atoms or molecules with minimal perturbation. Previous experimental investigations are Kondo-effect based; the quantum mirages appear only near the Fermi energy. This strongly limits the exploration of the mechanism and potential application. Here we demonstrate a Kondo-free quantum mirage that operates in a wide energy range beyond Fermi energy. Together with an analytical model, our systematic investigations identify that the quantum mirage is the result of quantum interference of the onsite electronic states with those scattered by the adatom at the focus of elliptical quantum corrals, where two kinds of scattering paths are of critical importance. Moreover, we also demonstrate the manipulation of quantum mirages with pseudo basic logic operations, such as NOT, FANOUT and OR gates.

, which presents a resonance peak at 130  meV . The resonance is assigned to the adatom-induced bound state 1-3 , resulting from the coupling of the Fe adatom's s state with the surface states 4,5 . It was suggested that the resonance is irrelevant with the d-state magnetism of the Fe adatom 4, 5 .
To confirm this, we performed experiments with a Ag adatom placed on a wide Ag(111) terrace, and observed a similar bound state. Supplementary Fig. 1c shows the dI/dV map of a single Ag adatom at a bias voltage of 100  mV. The spatial distribution demonstrates it a localized state. The corresponding topographic image is inserted at the left bottom. The Ag atom was obtained by means of the atom transfer technique 6 . Firstly, a W tip was decorated with Ag atoms by soft indentations into the surface. Secondly, a feedback loop was open after the tip was stabilized at V bias = 120 mV, I = 1 nA. Thirdly, the tip was driven towards the surface by 0.5 nm with rate ~0.1 nm•s -1 and retracted back at the same rate. Finally, we scanned the area to check if there was tip-apex atom transfer.
The Ag atom was further identified by the bound state 3 . The dI/dV spectrum over the Ag adatom is presented in Supplementary Fig. 1d, which is consistent with the work of Ref. 3 . Supplementary Note 2: Inversion effect of CQCs of different radii 5 Here, we clarify the inversion effect for circular quantum corrals of different radii.
Supplementary Fig. 2a shows a typically topographic image of an empty corral built with Fe adatoms. When the corral is small (radius r = 3.5 nm), there is only one quantum well state (peak I´) probed at the center of the empty corral in the measured voltage range ( Supplementary Fig. 2b). When enlarging r to 5.5 nm, three quantum well states emerge as peaks I´, II´ and IIII´ ( Supplementary Fig. 2c). After placing single Fe adatoms at the centers of the corresponding corrals ( Supplementary Fig. 2d), the spectra obtained on top of the Fe adatom at the corral center are shown in Supplementary Figs. 2e and 2f, respectively. These spectra display one-to-one inverted features of those of the empty corrals (peak to dip and dip to peak). This demonstrates that the observed effect is indeed an inversion effect. Fe off focus, and empty corral, respectively. h, j, l are the corresponding dI/dV maps at 24 mV. 8 We also performed the dI/dV spectral analysis at high symmetry points of the ellipse.

Supplementary
More specifically, we placed the adatom at different points, either at the long axis or the short axis, and made spectroscopy measurements. The results are shown below in

Supplementary Note 4: Symmetry check for Fe adatom placed at left or right focus
To explore the influence of the symmetry on the quantum mirage, we performed control experiments for Fe adatom at left or right focus. Supplementary Fig. 5a shows the topographic image of an EQC (e = 0.5 and a = 7.6 nm) with an Fe adatom placed at the left focus. The measured dI/dV spectra of both the left focus Fe adatom (blue curve) and right focus (red curve) are presented in Supplementary Fig. 5b. The spectrum on the right focus (red curve in Supplementary Fig. 5b) mimics that of the Fe adatom at left focus (blue curve in Supplementary Fig. 5b), indicating the quantum mirage effect.
When moving the extra Fe adatom to the right focus ( Supplementary Fig. 5c), the spectrum on the right focus (red curve in Supplementary Fig. 5d) also resembles that of the Fe adatom at left focus. The peak 4 still has the strongest intensity (blue curve in Supplementary Fig. 5d). Notice the similarity of the spectra for the Fe adatom placed at the left and the right foci, indicating that the symmetry of the Fe adatom positioning (at the left or right focus) has little influence on the quantum mirage. Any difference is attributed to slight differences of background spectra between left and right focus ( Supplementary Fig. 5f), which is caused by the imperfect symmetry for the atom positioning in constructing the EQC (Supplementary Fig. 5e).
To obtain the influence of the extra Fe adatom on the quantum mirage, we subtract the corresponding spectrum of the empty corral and normalize it by the spectrum on the Fe adatom. Supplementary Fig. 6 shows the transfer functions for Fe adatom at left focus ( Supplementary Fig. 6a) and right focus (Supplementary Fig. 6b). They exhibit oscillations, where labels 3 to 6 denote the peaks. The peaks of Supplementary Fig. 6a 10 have one-to-one correspondence with that of Supplementary Fig. 6b

Supplementary Note 5: Quantum mirage for a Ag adatom in an EQC
The inversion effect results from the hybridization of the adatom's s state with the surface states. Therefore, the inversion effect induced quantum mirage is irrelevant to magnetism. To test this statement experimentally, we performed the following experiments with a (nonmagnetic) Ag atom. Firstly, we built the corral atom-by-atom with Fe atoms. Supplementary Fig. 7a shows the topograph of an EQC (e = 0.5, a = 7.6 nm) with a Ag adatom at the left focus. Note that a Ag atom was transferred to the corral by the aforementioned atom-transfer technique and crosschecked with the identified bound state. The measured spectra over the left focus Ag adatom (blue curve) and right focus (right focus) are presented in Supplementary Fig. 7b. The peaks at the left focus Ag and right focus (labelled 3 -6) line up with one-to-one correspondence. This indicates the transfer of the electronic structure of the left focus Ag adatom to the right focus. Interestingly, peak 4 of the right focus is much stronger than that of the left focus Ag ( Supplementary Fig. 7b), which is the same behavior we observed with an Fe adatom. Supplementary Fig. 7c shows the corresponding transfer function. It is almost the same with that of Fe adatom ( Supplementary Fig. 6), which further attests to the generality of the quantum mirage concept.

Supplementary Figure 7 | Quantum mirage and transfer function for Ag adatom
in an EQC. a, An EQC with a = 7.6 nm, e = 0.5. b, The spectra on the left focus Ag adatom (blue curve) and right focus (red curve). c, The transfer function from the left focus Ag adatom to the right focus.

Supplementary Note 6: Quantum mirages of different-sized EQCs
In this part, we further demonstrate the generality of quantum mirages. For an EQC, there are two-independently controllable parameters. First, we performed experiments for EQCs with a fixed a and different e values. Then, we performed the experiments for EQCs with different a values and a fixed e. 13 Supplementary Fig. 8 presents typical dI/dV spectra for EQCs with a fixed a and different e values. Supplementary Fig. 8a shows the topographic image of an EQC (a = 7.6 nm, e = 0.5) with an Fe atom located at the left focus. Supplementary Fig. 8b is the dI/dV spectrum obtained on top of the Fe adatom located at the left focus. It exhibits peaks labelled 3, 4 and 5. The spectrum obtained at the right focus is present in Supplementary Fig. 8c. It also shows several peaks with the peak positions lining up with those in Supplementary Fig. 8b. Peak 4 in Supplementary Fig. 8c is much stronger than that of Supplementary Fig. 8b, confirming the results found in Fig. 1 of our paper.
They are similar with those of Figs. 1f and 1i, while the peaks shift left. The stronger intensity in peak 4 reflects the high signal intensity, which differs from that of an attenuated Kondo resonance 7 . We also find that other peaks, like 3 and 5, are not enhanced. Thus, the enhancement is energy-dependent. Supplementary Fig. 8d  The factor  describes the sum term of single scattering. Supplementary Fig. 10a shows the value of  versus e for different N. The number of atoms N used to build the corrals is varied from 24 to 26 to keep the mean distance between adjacent adatoms unchanged. Note that  is almost around 0.9 in the range of e experimentally studied. Supplementary Fig. 10b shows the ratio of the scattering amplitudes for the first three terms with k = 0.83 nm -1 (Fermi wave number) and a = 7.6 nm. Note that A 2 /A 3 is comparable, while A 1 /A 3 is very small. Thus, the dominant terms are A 2 and A 3 . We used Eq. (2) to fit the experimental results of Fig. 2

Supplementary Note 12: Pseudo OR logic gate
When the input and output are swopped, the confocal EQC can also be used to construct another basic pseudo logic operation-the OR gate as illustrated in Supplementary Fig. 14. In it, we used both foci B and C as the inputs and the joint focus A as the output. When both B and C are empty, namely "0" (Supplementary Fig. 14a), the corresponding dI/dV at joint focus A (Supplementary Fig. 14b) has a low intensity of 29.2 nS. When one of the foci B and C are configured as "1" (Supplementary Figs. 14c & 14e), the output at A has an intensity of ~34 nS ( Supplementary Figs. 14d & 14f).
When both foci B and C are configured as "1" (Supplementary Fig. 14g), the output at A has the intensity of 38.2 nS. Even though this intensity is slightly higher than that obtained when either B or C is configured as "1", the device functions as an OR logic 22 gate when the threshold is set to ~32 nS. (Note that if the threshold is set to ~36 nS, this logic operation can be viewed as an AND gate). In the above we used the energy at the constructive phase of interference. When the energy is chosen to be at the destructive phase, a pseudo NOR logic gate (not shown) can also form, similar to a pseudo NOT logic gate.
Supplementary Figure 14 | Pseudo OR logic gate. The left column shows the 23 topographical images (a, c, e, g) and the right column is the corresponding dI/dV maps with bias voltage +12 mV (b, d, f, h). Both foci B and C are inputs and the joint focus A is used as the output. If one or more of the left and right foci are occupied, the output is 1. Otherwise, the output is 0. The EQCs used to build the confocal EQC have a = 6.6 nm and e = 0.6.