ARTICLE https://doi.org/10.1038/s41467-019-10103-5 OPEN Real space manifestations of coherent screening in atomic scale Kondo lattices María Moro-Lagares1,2,3, Richard Korytár4, Marten Piantek1,5, Roberto Robles 6, Nicolás Lorente7,8, Jose I. Pascual1,9,10, M. Ricardo Ibarra1,5 & David Serrate 1,5,11 1234567890():,; The interaction among magnetic moments screened by conduction electrons drives quantum phase transitions between magnetically ordered and heavy-fermion ground states. Here, starting from isolated magnetic impurities in the Kondo regime, we investigate the formation of the finite size analogue of a heavy Fermi liquid. We build regularly-spaced chains of Co adatoms on a metallic surface by atomic manipulation. Scanning tunneling spectroscopy is used to obtain maps of the Kondo resonance intensity with sub-atomic resolution. For suf- ficiently small interatomic separation, the spatial distribution of Kondo screening does not coincide with the position of the adatoms. It also develops enhancements at both edges of the chains. Since we can rule out any other interaction between Kondo impurities, this is explained in terms of the indirect hybridization of the Kondo orbitals mediated by a coherent electron gas, the mechanism that causes the emergence of heavy quasiparticles in the thermodynamic limit. 1 Laboratorio de Microscopias Avanzadas, Instituto de Nanociencia de Aragón, University of Zaragoza, E-50018 Zaragoza, Spain. 2 Institute of Physics, Academy of Sciences, Prague 16200, Czech Republic. 3 Regional Centre of Advanced Technologies and Materials, Faculty of Science, Department of Physical Chemistry, Palacky University, Olomouc 78371, Czech Republic. 4 Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, 121 16 Prague 2, Czech Republic. 5 Dpto.Física Materia Condensada, University of Zaragoza, E-50009 Zaragoza, Spain. 6 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain. 7 Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU), 20018 Donostia-San Sebastián, Spain. 8 Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain. 9 CIC NanoGUNE, E-20018 Donostia-San Sebastián, Spain. 10 IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain. 11 Instituto de Ciencia de Materiales de Aragón, CSIC - Universidad de Zaragoza, 50009 Zaragoza, Spain. Correspondence and requests for materials should be addressed to D.S. (email: serrate@unizar.es) NATURE COMMUNICATIONS | (2019)10:2211 | https://doi.org/10.1038/s41467-019-10103-5 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10103-5 T he interaction of a localized magnetic moment with a becoming in this way a part of the Fermi volume or, equivalently, metallic host gives rise to the Kondo many-body state (KS) delocalized in space5. For this reason, this process is often called 1,2. The Kondo temperature defines the characteristic spin de-confinement. energy scale of the single-ion KS, kBTK. For T ≲ TK, the con- The Kondo screened state and the AFM inter-impurity order duction electrons align antiferromagnetically to the localized are separated by a quantum critical point (QCP, Fig. 1c, d)4,6–8, moment, resulting in a complete magnetic screening of a spin-1/2 which makes the ground state very sensitive to external para- impurity, and forming a singlet in the KS (Fig. 1b). In this way, meters such as temperature, pressure or magnetic field. Using the KS can be understood as the coupling of the quantum scanning tunnelling microscopy (STM) and spectroscopy (STS), numbers of the conduction electrons and the localized moment to the atomistic perspective of this transition has been studied in form a hybrid quasiparticle. dimers9–16. It was found that the KS develops magnetic correla- A Kondo lattice is a set of localized magnetic impurities tions for sufficiently small inter-impurity distances (d), indicating arranged in a regular pattern, which interacts with a bath of that the dimers are in the range of small Jρ, where the HFL phase delocalized conduction electrons. When the two subsystems are cannot set in (Fig. 1). Since Jρ cannot be tuned in situ, the HFL weakly coupled, the Kondo lattice falls in the paramagnetic or state could not be addressed in this kind of atomic dimers. Finite antiferromagnetic (AFM) regimes (Fig. 1a, c), both characterized size lattices composed by spin moments of molecular self- by a finite impurity spin. The AFM coupling is driven by the assemblies were also reported, but here again the AFM coupling Ruderman-Kittle-Kasuya-Yosida (RKKY) indirect exchange dominates over Kondo screening due to either RKKY17 or interaction. In the limit of a large coupling strength, Jρ, between superexchange18 coupling. localized moments and extended states of the conduction elec- Here, we fabricate chains of coupled single-ion Kondo reso- trons, the two subsystems cannot be treated separately. The nances by atomic manipulation in order to characterize in real Kondo lattice introduces a new energy scale, kBT*, which plays space the crossover between the KS and the finite size analogue to the role of TK in the sense that below T* the magnetic suscept- a HFL (Fig. 1b, d). To this end, we choose a sample with a strong ibility starts being anomalously reduced due to partial screening3. Jρ coupling, so that Kondo screening overcomes any intersite But T* and TK are not trivially related4. In an extended Kondo interaction. The 3d orbitals of individual Co atoms adsorbed on a lattice, for T  T  , the system falls in the heavy Fermi liquid clean Ag(111) surface play the role of localized spins and metal (HFL) phase (Fig. 1d). The HFL state is non-magnetic, because host, respectively. Although Co on Ag(111) displays a similar the conduction electrons fully screen the lattice spins. As in the Kondo strength as on Cu(111) and Au(111)19, the near Fermi case of the single-ion KS, the localized magnetic states contribute level electronic structures of these surfaces are markedly different. to the quasiparticle band developed by the coupled Kondo states, In particular, the onset of the surface conduction band of Ag(111) Small J Large J a b d Paramagnetic Single ion kondo state (T 4 and d = 2a, the two outermost end atoms Still, AK decreases smoothly in a distance range (d ≤ 5a) well display a prominent edge lobe of AK with maximum signal in above the spatial extent of Co atomic orbitals, pointing out that between them. Enhanced edge lobes in the AK intensity are also some type of indirect interaction (other than RKKY) mediated by found in chains with d = 3a, although in this case they coincide the conduction electrons comes into play. More interestingly, in the with the end atoms, and disappear for N = 10 (Fig. 4a). dimer at d = 2a, the amplitude of the ZBF is larger at the midpoint between the Co atoms than over their centres (Fig. 2b). This suggests that the observed AK decrease is not simply reflecting an overall The MIAM. To rationalize the Kondo redistribution pattern, we weakening of the KS, but rather a spatial redistribution of its employ a MIAM where each Co atom is represented by a single associated many-body DoS. In order to study the spatially resolved Anderson impurity (d-state), which hybridizes with a single band intensity of the KS wave function, we constructed AK(x, y) maps of conduction electrons (k-states), see Methods. We include from constant height dI/dV images at the two applied biases marked hopping terms between each k-state and any of the i = 1, …, N in Fig. 2a. These measurements are performed at constant height, impurities, which give rise to a hybridization function Γij(ω), a and thereby our data are free of feedback artefacts, in such a way matrix in the impurities sub-space. The quantity Γij(ω) accounts that the AK image is totally independent from the DOS in the entirely for the effect of the conduction band on the impurities. In absence of Kondo fluctuations, and portrays exclusively the Kondo- the single impurity limit, Γ11(ω = 0) determines the value of TK related intensity (for further details on the method see Supplemen- along with U, ϵd and the bandwidth (see Methods). In the multi- tary Note 1 and Supplementary Fig. 1). The map of the single atom impurity case, the off-diagonal terms of Γij(ω) represent hybri- is shown in Fig. 3a, b, yielding a circular region with non-vanishing dization between i and j impurities mediated by the conduction AK of diameter ≃5a around the centre. band. We disregard in the MIAM a Heisenberg term between Figure 3c compares constant height tunnelling current images impurity spins because our low bias dI/dV spectra are not com- and AK(x, y) maps of Co dimers for several d values. In the most patible with any kind of magnetic exchange interactions (see diluted case (d = 5a), each atom possesses an individual Kondo Fig. 2d, and Supplementary Figs. 4 and 7). We calculate the resonance located right at the atom centre. As d decreases, the AK impurity Green’s function Gij(ω) by a self-consistent many-body intensity deviates from the atomic cusps and shifts towards the perturbation theory (see Supplementary Note 3) and obtain the centre of the dimer. At d = 2a (dense regime), the two KS collapse local spectral function Aimp/subs(r, ω) of the impurities/substrate in a single one with maximum AK in between the atoms and (representing the local DoS) as explained in Methods. approximately halved intensity at the atoms position, as shown We first confirm that the delocalization of the Kondo by the horizontal profiles in Fig. 3d. This redistribution of amplitude in two-impurity systems is supported by our MIAM the Kondo amplitude explains the decrease of AK for atom calculations. To this end, we set the impurity parameters (ϵd ; U) pairs closer than d = 5a. Furthermore, it suggests that the and Γii(0) into the Kondo limit and plot Aimp(r, ω0) maps of screening of the magnetic moments is a collective effect involving dimers for varying Γ12(0), with ω0 the energy at which the Kondo the two atoms and the metallic host. Finally, the amplitude resonance is centred. For independent impurities, i.e., when Γ12 redistribution observed for Co dimers is genuinely due to an = 0, the map displays an incoherent sum of the two Co wave interaction between two single-ion KS, because approaching a functions (φi in Eq. (6)) used to reproduce the experimental AK non-magnetic Ag atom to a KS does not give rise to any for the single atom case (cf. Figs. 3a and 5a). In the opposite limit systematic variation of AK (Fig. 2c). In addition, the AK map of of strong cross-talk between impurities, we set Γ12 = Γ11 = Γ22 = the Ag-Co dimer at d = 2a (Supplementary Fig. 6) is the same as Γ21. In line with the experimental results of Fig. 3c, the spectral for the isolated Co. function map is now a single cloud with maximum amplitude at 4 NATURE COMMUNICATIONS | (2019)10:2211 | https://doi.org/10.1038/s41467-019-10103-5 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10103-5 ARTICLE a b High High 20 pA 0.1 Current (pA) AK Low Low –0.8 0.0 0.8 Distance (nm) c d 2a 10 pA 0.1 3a Current (pA) AK 4a 5a –3 –2 –1 0 1 2 3 Distance/a Fig. 3 Mapping of the experimental Kondo resonance amplitude in Co dimers. a Simultaneous constant height images of the tunnelling current (left) and Kondo amplitude, AK (right) of an individual Co atom. AK is the depth of the Kondo resonance, see Eq. (1) and Fig. 2a. b Line profile across the atom (see lines in (a)) of the current (thin black line) and AK (thick blue line). c The same as (a) for a Co dimer at d = 2a, 3a, 4a, and 5a modified by atomic manipulation (a = 2.89 Å). d Horizontal profiles of the tunnelling current (black thin lines), and AK (blue thick line) from (c). Data were vertically offset for clarity. All tunnelling current images (greyscale) were acquired at Vref = 17 mV sample bias. Bottom scale bar corresponds to 1 nm the dimer centre (Fig. 5b), reflecting a coherent superposition of conduction electrons due to the impurities, δAsubs(r, ω0). We the original Kondo clouds. begin with D = 2 and kFd = 0.18π, very close to the experimental For longer chains, we express explicitly the dependence of scenario with d = 2a and kF = 0.085 Å−1 41. Selected maps of Γij(ω) on the distance between impurities i and j, which for our δAsubs(r, ω0) are shown in the insets of Fig. 5c for N = 9, 12, and evenly spaced chain amounts to δr = d|i − j|. We assume that the 15. For the shortest chains up to N ≤ 9, the map consists of a inter-impurity hybridization is mediated by a translationally single cloud. In the range 10 ≤ N ≤ 14, two distinct maxima invariant substrate. Then, the asymptotic distance dependence of appear at the edges of the chain. For even longer chains, a new Γij(ω) must be that of a spherical wave-function emanating from peak emerges in the chain centre merging with the edge features. one impurity and reaching the other. Its envelope, for a coherent Remarkably, the source of the edges and other patterns can be D dimensional Fermi gas, should get reduced by the factor δr−(D traced to the off-diagonal elements of the impurity Green’s −1)/2 and, owing to the Bloch phase picked at the Fermi surface, it function Gij(ω) (see Supplementary Fig. 9). In contrast to D = 2, should oscillate periodically with the Fermi wavelength (λF). The we do not observe the edge enhancement for D = 3. This is due to impurities are now considered point-like, and the analytic a faster decay of the substrate-mediated hybridization across the expressions of Γij(ω) in D = 2 and D = 3 for a homogeneous chain. Hence, the presence of edge lobes is a fingerprint of the D electron gas with bandwidth 2E0 are, as derived in the = 2 character of the conduction band, which suggests that the 2D Supplementary Note 3: Shockley surface state governs the substrate-mediated impurity hybridization. This is in agreement with recent experimental and Γij ðωÞ ¼ Γ0 J0 ðkF dji  jjÞ  ΘðE02  ω2 Þ in D ¼ 2 ð2Þ theoretical studies demonstrating that the surface state plays a primary role in the Kondo effect of Co/Ag(111)20,42. sinðkF dji  jjÞ The role of λF is neatly demonstrated by the general behaviour Γij ðωÞ ¼ Γ0  ΘðE02  ω2 Þ in D ¼ 3 ð3Þ observed for various impurity spacings d. Figure 5c shows the kF dji  jj chain lengths for which the two edge lobes exist as a function of d, Apart from the dimensionality, the only remaining parameter is both theoretically and experimentally. Upon increasing N, the the dimensionless product kFd, where kF = 2π/λF. As in the two- edge lobes appear, then remain up to a critical length, and finally impurity case, we tune the system parameters into the Kondo are replaced by more structured patterns in very long chains. limit. We analyze the change of the local spectral function of the Remarkably, we observe theoretically a robust scenario in which NATURE COMMUNICATIONS | (2019)10:2211 | https://doi.org/10.1038/s41467-019-10103-5 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10103-5 a d = 3a N=2 N=3 N=5 N=6 High N=7 N = 10 Low b d = 2a N=2 N=3 N=5 N=6 High N=7 N = 14 Low Fig. 4 Spin de-confinement in 1-D artificial Kondo lattices. Simultaneous constant height tunnelling current (greyscale) and Kondo amplitude AK (red-yellow scale) maps of N Co atoms chains with spacing d = 3a (a) and d = 2a (b). White lines are guides to highlight the position of the AK maxima with respect to the atoms’ position. a At d = 3a, all atoms retain an associated Kondo resonance, with maximum AK at end atoms for N < 10. b At d = 2a, the AK edge maxima are found in between the two end Co atoms of each side. The bottom scale bar represents 2 nm and applies to the entire figure. All tunnelling current images (greyscale) were acquired at Vref = 17–20 mV sample bias the edge lobes show up in chains with lengths of at least L ≈ theory predicts a critical chain length, Lc, for the appearance of 0.75λF and merge when L]1:1λF . Thus, the control parameter is Kondo amplitude maxima near the edge atoms, which scales with not the spacing, but the total length L = d(N − 1). The scattering the Fermi wavelength of the conduction band. Moreover, it also of the boundary region is just d, i.e., one impurity of the chain. predicts the suppression of the edge lobes above a maximum We note that the Kondo resonance needs to be fully developed length, LM. As illustrated in Fig. 5c, both predictions are experi- (T < TK) since, by increasing the temperature, the patterns get mentally observed in chains with two different spacings d = 2a weaker and disappear. and 3a. The model approximations lead to quantitative differ- ences in the threshold lengths: Lc−M/λF = 0.75–1.15 and 0.33–1.06 for theory an experiment, respectively. Despite of this, Discussion the excellent qualitative agreement between the experimental AK For non-interacting impurities, the spectral function Aimp (see maps and the MIAM calculation implies that our Co atoms Fig. 5a) reproduces quite accurately the circular objects described cannot be viewed as a chain of local scatterers, because they give by the Kondo amplitude that are observed experimentally rise to a non-local Kondo resonance. The non-local origin of the (Fig. 3a, b). When the substrate-mediated inter-impurity hybri- conductance patterns in the Kondo lattices constructed in this dization Γ12(ω) is turned on, the spectral function displays the work contrasts with the patterns found in quantum mirages, same delocalized pattern as in the experimental Kondo amplitude which were explained by purely local scattering44. Non-local of the dimer at d = 2a (cf. Figs. 3c and 5b). Note that Γ12(ω) Kondo screening has also been observed in MnxFe chains31, originates from coherent hoppings of electrons from a single where the spin of the constituent atoms is strongly entangled by conduction band to either Anderson impurity. The fact that the exchange coupling. Here we explore the opposite limit with impurity spins are more coupled to conduction electrons away negligible coupling between the atomic spins of the chain. The from the magnetic orbitals is the signature of spin de- MnxFe chains act as a single Kondo impurity with a complex confinement43. The latter has been attributed to the intersite pattern of Kondo fluctuations spreading over several atoms in the RKKY interaction in ref. 17. As we show, the spin de-confinement chain, whereas for Co/Ag(111) chains we observe manifestations in Co dimers is linked to the effect of the overlap of Kondo of collective screening of a set of individual impurities. clouds. Formally, this is expressed by Γ12(ω), which is second The single-ion picture clearly fails for our chains and dimers order in Vki, whereas the RKKY appear only in the fourth order in with the shortest impurity spacings, which suggests a substantial Vki (see Supplementary Note 4). increase in T* 3,4 above the working temperature of 1.1 K when d Further evidence of collective screening is provided by the decreases below 3a. In this regime, we observe the defining comparison of AK maps of the long chains (Fig. 4) and its closest characteristics of the HFL phase in a Kondo lattice2,5: spin de- theoretical analogue, δAsubs. As a result of the coherent inter- confinement and collective screening. The HFL state has been ference of the electron gas coupled to the impurity lattice, the investigated by STM in 2D43 or 3D22–24 dense self-assembled 6 NATURE COMMUNICATIONS | (2019)10:2211 | https://doi.org/10.1038/s41467-019-10103-5 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10103-5 ARTICLE a b 0.0006 (nV) 0.002 (nV) Γ12 = 0 Γ12 = Γ11 0 0 (L\F)exp. c 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.35 0.30 2 3 4 5 6 10 14 0.25 kFd = 0.18 kFd = 0.18 kFd = 0.18 kFd L/F = 0.72 L/F = 0.99 L/F = 1.26 0.20 F F F 2 3 4 5 6 7 8 14 0.15 0.10 0.8 1.0 1.2 1.4 (L\F)theo. Fig. 5 Spectral function maps of the MIAM in the Kondo regime. a, b Theoretical map of the impurities’ spectral function evaluated at the energy of the Kondo peak of a dimer with spacing d = 2.6a = 7.5 Å. Scale bars represent 4 Å. The two figures (a, b) differ only in the off-diagonal element of the hybridization function (Γ11 = 10 meV, Γ12 = 0 and Γ11 = Γ12 = 10 meV, respectively). The other model parameters are given in Methods. The impurities’ positions are marked by yellow circles. c Diagram for the appearance of edge lobes in spectral function map of conduction electrons (see Methods, Γ0 = 60 meV), constructed by analyzing chains of different spacing d and length L = (N − 1)d. The pale blue area corresponds to the d − L region where edge lobes appear. Insets represent selected spectral function maps of chains with N = 9, 12, 15 atoms and spacing d = 2.2a, exemplifying the presence or absence of edge lobes. The spectral function is again evaluated at the energy of the Kondo dip. The impurities’ positions are marked by white crosses. The chain length for theoretical results is given in the bottom abscissa in units of λF. Green(red) empty circles are experimental data points from AK maps with (without) edge lobes in chains as deduced from AK maps in Fig. 4 for d = 2a and d = 3a (the images of the chains not shown there are available in Supplementary Fig. 8). The number enclosed by the circles is the number of atoms, N. The experimental chain length is given in the upper abscissa Kondo lattices. In these examples, the transition between KS dimensionality, or even an eventual opening of the gap at tem- (independent impurity behaviour with energy scale ~TK) and the peratures lower than experimentally available. HFL state (energy scale ~T*) could not been reached, unlike the To sum up, we have visualized in real space the onset of HFL case of our Co chains built atom-by-atom where the origin of behaviour as the redistribution of the Kondo amplitude (and the Kondo features can be traced down to the single-ion state. consequently spin de-confinement). In dimers, it appears as a The conditions to obtain both states in the same sample cannot delocalization of the Kondo amplitude towards its centre. In be met either in the case of bulk crystalline heavy fermion chains, it manifests as patterns at length scales of the order of λF. compounds3,4,45–47. Apart from the Kondo temperature, the theoretical model The possibility of observing the real space structures in the ad- reproduces the essential experimental observations without atom Kondo lattice is intimately linked to the fact that the chains adjustable parameters: the dimensionless product kFd is fixed by are finite. This grants access to quantum size effects of the Kondo the experimental conditions, and the leading dimension of the lattice Physics. In exchange, some more common features of the electron gas is two. Therefore, our microscopic description should Kondo lattice problem in bulk heavy fermion compounds are apply for different material parameters, and for arbitrary lattice missing. For instance, the opening of a hybridization gap upon site and dimensions, laying foundations to further investigations formation of the HFL phase23,48,49, which should manifest as a of correlated nanostructures engineered with atomic precision. new ZBF in the dI/dV spectra21,22,50. We do not observe any additional feature on top of the Kondo resonance upon the for- mation of the Kondo lattice. The reason for this absence remains Methods Experimental. The experiments have been performed in a SPECS Joule-Thompson an open question for further theoretical investigations: the finite STM51 in ultra-high vacuum (chamber base pressure <1 × 10−10 mbar) at 1.14 K. size of the Co lattice, its coupling to a metallic host with lower Some control experiments were repeated at 4.7 K. All images and spectroscopy data NATURE COMMUNICATIONS | (2019)10:2211 | https://doi.org/10.1038/s41467-019-10103-5 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10103-5 are acquired at constant tip sample distance unless stated otherwise (feedback the impurity Green’s function opened over the Ag(111) surface at regulation set point of −100 mV and 40 pA). In 1 X our set up, the tip is grounded and bias voltage is applied to the sample. The Ag Aimp ðr; ωÞ ¼  Im Gij ðωÞφi ðrÞφj ðrÞ: ð6Þ (111) surface was cleaned by repeated cycles of Ar+ sputtering at 1 kV, 1 × 10−6 π ij mbar and annealing at 450 °C. Co atoms were deposited in situ onto the Ag(111) We use the wave functions φi(r) = C exp(−|Ri − r|/α) with radius α = 1.5 Å. The substrate from an e-beam evaporator loaded with a high purity Co rod focused at two orbital centres are separated by |R1 − R2| = 2.6a. For such small length scales the STM sample stage held at T < 5 K to prevent atoms diffusion. Ag atoms were ( λF ), we neglect the spatial dependence of Γij, and parametrize its matrix extracted from the substrate by means of controlled tip indentation. We used elements as given in the main text with the following frequency dependence: lateral atomic manipulation to move individual atoms to designated positions and construct artificial atomic ensembles. Atom manipulation was performed exerting a Γij ðωÞ ¼ Γ0 expððω=2E0 Þ2 Þ: ð7Þ lateral force on the atom with the STM tip in the attracting regime (pulling mode, We note that the actual Co d-orbitals are quite localized (orbital radius ≈0.5 Å). manipulation parameters I0 ≥ 40 nA and V0 = 3 mV). The tip is an electro- Therefore, the φi is to be viewed rather as a superposition of non-magnetic local chemically etched W wire, which is in situ ion milled by means of Ar+ sputtering at states in the vicinity of each Co. These atomistic details become less important as L 5 kV. Afterwards, the tip apex is gently dipped in the Ag(111) surface until it approaches λF and the description by means of the homogeneous electron gas (cf. becomes structurally stable against atomic manipulation. Co atoms can be adsor- Eq. (5)) is more appropriate. bed on the two types of hollow sites of the Ag(111) surface: fcc or hcp. Our data are We choose parameters that fall well within the Kondo regime in all cases: fairly independent on the site type, with the exception that in one of them the U = ∞, ϵd ¼ 0:4, E0 = 4 and temperature kBT = 10−4 (all in eV). We take Γ0 = resonance amplitude is systematically lower than in the other. In order to overcome 0.06 and 0.01 eV for long chains and dimers, respectively. The qualitative features this duality, we ensure that all atoms are in equivalent sites by carefully assembling of the local DoS are unchanged as long as we stay in the Kondo regime. all atomic structures along [110] atomic closed packed directions of the Ag(111) substrate. All differential conductance dI/dV data are obtained using a lock-in amplifier at 911 Hz and modulation voltage of 0.5 mV rms for spectra and 2 mV Data availability rms for imaging. The whole image processing has been carried out with WSxM52. The spectroscopic data and microscopy images in binary format that support the findings of this study are available in DIGITAL.CSIC with the following persistent identifier: Ab initio calculations. The electronic structure of Co/Ag(111) has been evaluated http://hdl.handle.net/10261/178494. Additional information can be obtained from the using the VASP-DFT code, including spin-orbit coupling. In the two types of authors. hollow adsorption sites, we obtain a spin magnetic moment of 2.04 μB with neg- ligible orbital moment and very small uniaxial anisotropy of 0.14 meV. Therefore, Code availability the leading interaction is Kondo screening. In absence of electronic correlations Computer code used to solve the Anderson Hamiltonian and generate the profiles and (U = 0), Co atoms exhibit two unoccupied antibonding d-orbitals and thereby they images of the spectral functions can be shared by the authors upon request. can be described by an isotropic S = 1 spin system. The inter-impurity exchange energy in the dimer was calculated as the energy difference between the parallel and anti-parallel spin alignment. Received: 21 December 2018 Accepted: 18 April 2019 Theoretical DOS in the MIAM. The Hamiltonian of the MIAM reads P P ^y ^ P y ^ ^y ^ ^ ¼ H diσ diσ þ U d^m" ϵk^cykσ ^ckσ þ ϵd dm" dm# dm# þ kσ mσ m PP   ð4Þ References þ Vkm^cykσ d^iσ þ h:c: 1. Kondo, J. 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