Abstract
The interaction among magnetic moments screened by conduction electrons drives quantum phase transitions between magnetically ordered and heavyfermion 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 regularlyspaced 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 subatomic resolution. For sufficiently 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.
Introduction
The interaction of a localized magnetic moment with a metallic host gives rise to the Kondo manybody state (KS)^{1,2}. The Kondo temperature defines the characteristic energy scale of the singleion KS, k_{B}T_{K}. For T ≲ T_{K}, the conduction electrons align antiferromagnetically to the localized moment, resulting in a complete magnetic screening of a spin1/2 impurity, and forming a singlet in the KS (Fig. 1b). In this way, the KS can be understood as the coupling of the quantum numbers of the conduction electrons and the localized moment to form a hybrid quasiparticle.
A Kondo lattice is a set of localized magnetic impurities arranged in a regular pattern, which interacts with a bath of delocalized conduction electrons. When the two subsystems are weakly coupled, the Kondo lattice falls in the paramagnetic or antiferromagnetic (AFM) regimes (Fig. 1a, c), both characterized by a finite impurity spin. The AFM coupling is driven by the RudermanKittleKasuyaYosida (RKKY) indirect exchange interaction. In the limit of a large coupling strength, Jρ, between localized moments and extended states of the conduction electrons, the two subsystems cannot be treated separately. The Kondo lattice introduces a new energy scale, k_{B}T^{*}, which plays the role of T_{K} in the sense that below T^{*} the magnetic susceptibility starts being anomalously reduced due to partial screening^{3}. But T^{*} and T_{K} are not trivially related^{4}. In an extended Kondo lattice, for \(T \ll T^ \ast\), the system falls in the heavy Fermi liquid (HFL) phase (Fig. 1d). The HFL state is nonmagnetic, because the conduction electrons fully screen the lattice spins. As in the case of the singleion KS, the localized magnetic states contribute to the quasiparticle band developed by the coupled Kondo states, becoming in this way a part of the Fermi volume or, equivalently, delocalized in space^{5}. For this reason, this process is often called spin deconfinement.
The Kondo screened state and the AFM interimpurity order are separated by a quantum critical point (QCP, Fig. 1c, d)^{4,6,7,8}, which makes the ground state very sensitive to external parameters such as temperature, pressure or magnetic field. Using scanning tunnelling microscopy (STM) and spectroscopy (STS), the atomistic perspective of this transition has been studied in dimers^{9,10,11,12,13,14,15,16}. It was found that the KS develops magnetic correlations for sufficiently small interimpurity distances (d), indicating that the dimers are in the range of small Jρ, where the HFL phase cannot set in (Fig. 1). Since Jρ cannot be tuned in situ, the HFL state could not be addressed in this kind of atomic dimers. Finite size lattices composed by spin moments of molecular selfassemblies were also reported, but here again the AFM coupling dominates over Kondo screening due to either RKKY^{17} or superexchange^{18} coupling.
Here, we fabricate chains of coupled singleion Kondo resonances by atomic manipulation in order to characterize in real space the crossover between the KS and the finite size analogue to a HFL (Fig. 1b, d). To this end, we choose a sample with a strong Jρ coupling, so that Kondo screening overcomes any intersite interaction. The 3d orbitals of individual Co atoms adsorbed on a clean Ag(111) surface play the role of localized spins and metal host, respectively. Although Co on Ag(111) displays a similar Kondo strength as on Cu(111) and Au(111)^{19}, the near Fermi level electronic structures of these surfaces are markedly different. In particular, the onset of the surface conduction band of Ag(111) is right below the Fermi level, which gives rise to strong fluctuations of the density of screening electrons and an anomalously large Fermi wavelength^{20}. By means of an analysis of the multiimpurity Anderson model (MIAM), we link the Ag(111) surface electronic structure to the experimental fingerprints of coherent and collective screening of the artificial Kondo lattice. We probe experimentally the deconfinement of Co magnetic moments upon formation of a collective KS by imaging the amplitude of the Kondo resonance in tunnelling conductance spectra, which delocalizes away from the position of the impurity centres. The delocalization occurs in a gradual manner as the interimpurity distance in dimers and chains is reduced, which is a way of tuning the ratio T^{*}/T_{K}. Note that we use the lattice geometry as the control parameter to induce collective screening right from the singleion KS. Such transitionis not accessible in temperature dependent studies of spectral features in extended Kondo lattices^{21,22,23,24}, because the large impurity density impedes the formation of a pure singleion KS.
Results
Singleion Kondo state
The KS produces a resonant peak in the density of states (DoS) around the Fermi level, which manifests in STM as a zerobias feature (ZBF) in differential conductance dI/dV spectra^{25,26}. On metals, the ZBF can be generically described by a Fano lineshape^{25,27,28,29}. Figure 2a shows the fit of the ZBF measured over a single Co atom to a Fano function. We describe the resonance as \(\mathrm{d}I/\mathrm{d}V(\epsilon ) = \rho _{\mathrm{0}}(\epsilon )\rho _{\mathrm{K}}(\epsilon )\), where ρ_{0} is a polynomial background and the Fano function enters ρ_{K} linearly (see Supplementary Note 1):
Here \(\epsilon = eV\), V the sample bias voltage, A_{K} the Kondo resonance amplitude, and Γ_{0} the resonance linewidth. Care must be taken that there are not tip resonances contributing to the ZBF, which could affect the intrinsic values of the fitting parameters (see Supplementary Fig. 2). Γ_{0} is related with the Kondo temperature as \(2k_{\mathrm{B}}\sqrt {(\pi T)^2 + 2T_{\mathrm{K}}^2} = \Gamma _0\) if k_{B}T/Γ_{0} → 0^{30} (Supplementary Note 1). We obtain k_{B}T_{K} = 3.2 ± 0.6 meV for isolated single atoms far from defects and impurities^{20}. This, together with the negligible unixial aniostropy energy of 0.14 meV compared with k_{B}T_{K} that we obtained by density functional theory (DFT) calculations (see Methods), is consistent with a fully screened KS at the working temperature (k_{B}T = 0.09 meV).
Interaction of two Kondo states
We are interested in a dense system, with interatomic distances below the length scale in which the singleion KS picture does not suffice to describe the Physics. To determine such characteristic length scale, we probe variations in the Kondo ZBF when a second Co atom is approached (Supplementary Fig. 3). The interatomic distance, d, is expressed as a multiple of the Ag(111) lattice parameter along the [\(1\bar 10\)] close packed directions, a = 2.89 Å. As shown in Fig. 2c, the Kondo amplitude measured on top of the Co atom drops gradually as the separation decreases from d = 6a to d = 2a, whereas the lineshape (given by q and Γ_{0}) of the ZBF remains constant (see Fig. 2d and raw data in Supplementary Fig. 4) within our experimental uncertainty (q = 0.16 ± 0.1 in this dataset).
For the compact dimer (d = a), the ZBF is totally suppressed in the dI/dV spectrum (Supplementary Fig. 5). This distance is within the regime where orbital overlap can occur, triggering direct magnetic exchange that leads to a new entangled spin state^{8,9,10,31}, which presumably is not compatible with the Kondo effect. On the other hand, indirect RKKY coupling of the order of k_{B}T_{K} does not suppress the ZBF, but it is known to have a stark effect on it, producing either a splitting^{13,17,32,33,34} or a broadening^{11,12,14,15} of the resonance. In contrast, the lineshape (q, Γ_{0}) of the ZBF does not change for d ≥ 2a in our Co dimers. Consequently, we conclude that there are not direct or indirect magnetic interactions, in line with the negligible exchange coupling of Co atoms on the surface at d = 2a calculated by DFT (ΔE_{ex} = 0.64 meV \(\ll k_{\mathrm{B}}T_{\mathrm{K}}\)). In the context of Kondo lattices, this implies that our system lies in the region of large Jρ of the phase diagram shown in Fig. 1, where Kondo screening overcomes indirect RKKY magnetic coupling. Note that there are other interactions with the immediate environment that would also produce noticeable changes of the lineshape, which can thereby be discarded also for our Co chains. Examples are magnetocrystalline anisotropy^{35}, shift of atomic states^{36,37}, adsorption geometry^{38,39}, chemical bonding^{16,18}, or charge doping^{40}.
Still, A_{K} decreases smoothly in a distance range (d ≤ 5a) well above the spatial extent of Co atomic orbitals, pointing out that some type of indirect interaction (other than RKKY) mediated by the conduction electrons comes into play. More interestingly, in the 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 A_{K} decrease is not simply reflecting an overall weakening of the KS, but rather a spatial redistribution of its associated manybody DoS. In order to study the spatially resolved intensity of the KS wave function, we constructed A_{K}(x, y) maps from constant height dI/dV images at the two applied biases marked in Fig. 2a. These measurements are performed at constant height, and thereby our data are free of feedback artefacts, in such a way that the A_{K} image is totally independent from the DOS in the absence of Kondo fluctuations, and portrays exclusively the Kondorelated intensity (for further details on the method see Supplementary Note 1 and Supplementary Fig. 1). The map of the single atom is shown in Fig. 3a, b, yielding a circular region with nonvanishing A_{K} of diameter ≃5a around the centre.
Figure 3c compares constant height tunnelling current images and A_{K}(x, y) maps of Co dimers for several d values. In the most diluted case (d = 5a), each atom possesses an individual Kondo resonance located right at the atom centre. As d decreases, the A_{K} intensity deviates from the atomic cusps and shifts towards the centre of the dimer. At d = 2a (dense regime), the two KS collapse in a single one with maximum A_{K} in between the atoms and approximately halved intensity at the atoms position, as shown by the horizontal profiles in Fig. 3d. This redistribution of the Kondo amplitude explains the decrease of A_{K} for atom pairs closer than d = 5a. Furthermore, it suggests that the screening of the magnetic moments is a collective effect involving the two atoms and the metallic host. Finally, the amplitude redistribution observed for Co dimers is genuinely due to an interaction between two singleion KS, because approaching a nonmagnetic Ag atom to a KS does not give rise to any systematic variation of A_{K} (Fig. 2c). In addition, the A_{K} map of the AgCo dimer at d = 2a (Supplementary Fig. 6) is the same as for the isolated Co.
We exclude interference between tunnelling channels through both atoms as responsible for the enhanced A_{K} signal when the tip is at the middle point. First, the experimental measure of interference is the qfactor of the Kondo dip, which together with Γ_{0} remains constant across the dimer in our fits to Eq. (1). Second, we have confirmed that the intensity redistribution towards the dimer centre is more intense the sharper the tip is (sharpness is quantified by the corrugation of the tunnelling current profiles at V_{ref} in dimers at d = 2a).
Kondo redistribution in finite Kondo lattices
In order to find out how these results translate into a periodic Kondo lattice, we have built atomic chains of N atoms length (3 ≤ N ≤ 14) with spacing of either d = 2a or d = 3a. Selected images of A_{K} and tunnelling conductance are shown in Fig. 4. The resonance width (∝T_{K}) of an atom remains constant as the chain is constructed sideways, as well as in all atoms within a chain of arbitrary length (see Supplementary Fig. 7). This indicates again that there is no magnetic exchange among impurities, in contrast to the case of a molecular the Kondo chain^{18}. For all values of N at d = 3a, Fig. 4a exhibits a set of peaks in A_{K}(x, y) coinciding with the atoms positions. However, when the interatomic distance is reduced to d = 2a in Fig. 4b, the Kondo amplitude spreads over the chains and displays only a weak modulation (below the experimental uncertainty) along all inner atoms. The difference between d = 2a and d = 3a is consistent with our observation on the dimers. In addition, the end atoms of the chains show a striking feature. For the chains with N > 4 and d = 2a, the two outermost end atoms display a prominent edge lobe of A_{K} with maximum signal in between them. Enhanced edge lobes in the A_{K} intensity are also found in chains with d = 3a, although in this case they coincide with the end atoms, and disappear for N = 10 (Fig. 4a).
The MIAM
To rationalize the Kondo redistribution pattern, we employ a MIAM where each Co atom is represented by a single Anderson impurity (dstate), which hybridizes with a single band of conduction electrons (kstates), see Methods. We include hopping terms between each kstate and any of the i = 1, …, N impurities, which give rise to a hybridization function Γ_{ij}(ω), a matrix in the impurities subspace. The quantity Γ_{ij}(ω) accounts entirely for the effect of the conduction band on the impurities. In the single impurity limit, Γ_{11}(ω = 0) determines the value of T_{K} along with U, \(\epsilon _{\mathrm{d}}\) and the bandwidth (see Methods). In the multiimpurity case, the offdiagonal terms of Γ_{ij}(ω) represent hybridization between i and j impurities mediated by the conduction band. We disregard in the MIAM a Heisenberg term between impurity spins because our low bias dI/dV spectra are not compatible with any kind of magnetic exchange interactions (see Fig. 2d, and Supplementary Figs. 4 and 7). We calculate the impurity Green’s function G_{ij}(ω) by a selfconsistent manybody perturbation theory (see Supplementary Note 3) and obtain the local spectral function A_{imp/subs}(r, ω) of the impurities/substrate (representing the local DoS) as explained in Methods.
We first confirm that the delocalization of the Kondo amplitude in twoimpurity systems is supported by our MIAM calculations. To this end, we set the impurity parameters (\(\epsilon _{\mathrm{d}},U\)) and Γ_{ii}(0) into the Kondo limit and plot A_{imp}(r, ω_{0}) maps of dimers for varying Γ_{12}(0), with ω_{0} the energy at which the Kondo resonance is centred. For independent impurities, i.e., when Γ_{12} = 0, the map displays an incoherent sum of the two Co wave functions (φ_{i} in Eq. (6)) used to reproduce the experimental A_{K} for the single atom case (cf. Figs. 3a and 5a). In the opposite limit of strong crosstalk between impurities, we set Γ_{12} = Γ_{11} = Γ_{22} = Γ_{21}. In line with the experimental results of Fig. 3c, the spectral function map is now a single cloud with maximum amplitude at the dimer centre (Fig. 5b), reflecting a coherent superposition of the original Kondo clouds.
For longer chains, we express explicitly the dependence of Γ_{ij}(ω) on the distance between impurities i and j, which for our evenly spaced chain amounts to δr = di − j. We assume that the interimpurity hybridization is mediated by a translationally invariant substrate. Then, the asymptotic distance dependence of Γ_{ij}(ω) must be that of a spherical wavefunction emanating from one impurity and reaching the other. Its envelope, for a coherent D dimensional Fermi gas, should get reduced by the factor δr^{−(D−1)/2} and, owing to the Bloch phase picked at the Fermi surface, it should oscillate periodically with the Fermi wavelength (λ_{F}). The impurities are now considered pointlike, and the analytic expressions of Γ_{ij}(ω) in D = 2 and D = 3 for a homogeneous electron gas with bandwidth 2E_{0} are, as derived in the Supplementary Note 3:
Apart from the dimensionality, the only remaining parameter is the dimensionless product k_{F}d, where k_{F} = 2π/λ_{F}. As in the twoimpurity case, we tune the system parameters into the Kondo limit. We analyze the change of the local spectral function of the conduction electrons due to the impurities, δA_{subs}(r, ω_{0}). We begin with D = 2 and k_{F}d = 0.18π, very close to the experimental scenario with d = 2a and k_{F} = 0.085 Å^{−1}^{41}. Selected maps of δA_{subs}(r, ω_{0}) are shown in the insets of Fig. 5c for N = 9, 12, and 15. For the shortest chains up to N ≤ 9, the map consists of a single cloud. In the range 10 ≤ N ≤ 14, two distinct maxima appear at the edges of the chain. For even longer chains, a new peak emerges in the chain centre merging with the edge features. Remarkably, the source of the edges and other patterns can be traced to the offdiagonal elements of the impurity Green’s function G_{ij}(ω) (see Supplementary Fig. 9). In contrast to D = 2, we do not observe the edge enhancement for D = 3. This is due to a faster decay of the substratemediated hybridization across the chain. Hence, the presence of edge lobes is a fingerprint of the D = 2 character of the conduction band, which suggests that the 2D Shockley surface state governs the substratemediated impurity hybridization. This is in agreement with recent experimental and theoretical studies demonstrating that the surface state plays a primary role in the Kondo effect of Co/Ag(111)^{20,42}.
The role of λ_{F} is neatly demonstrated by the general behaviour observed for various impurity spacings d. Figure 5c shows the chain lengths for which the two edge lobes exist as a function of d, both theoretically and experimentally. Upon increasing N, the edge lobes appear, then remain up to a critical length, and finally are replaced by more structured patterns in very long chains. Remarkably, we observe theoretically a robust scenario in which the edge lobes show up in chains with lengths of at least L ≈ 0.75λ_{F} and merge when \(L\gtrapprox 1.1\lambda _{\mathrm{F}}\). Thus, the control parameter is not the spacing, but the total length L = d(N − 1). The scattering of the boundary region is just d, i.e., one impurity of the chain. We note that the Kondo resonance needs to be fully developed (T < T_{K}) since, by increasing the temperature, the patterns get weaker and disappear.
Discussion
For noninteracting impurities, the spectral function A_{imp} (see Fig. 5a) reproduces quite accurately the circular objects described by the Kondo amplitude that are observed experimentally (Fig. 3a, b). When the substratemediated interimpurity hybridization Γ_{12}(ω) is turned on, the spectral function displays the same delocalized pattern as in the experimental Kondo amplitude of the dimer at d = 2a (cf. Figs. 3c and 5b). Note that Γ_{12}(ω) originates from coherent hoppings of electrons from a single conduction band to either Anderson impurity. The fact that the impurity spins are more coupled to conduction electrons away from the magnetic orbitals is the signature of spin deconfinement^{43}. The latter has been attributed to the intersite RKKY interaction in ref. ^{17}. As we show, the spin deconfinement in Co dimers is linked to the effect of the overlap of Kondo clouds. Formally, this is expressed by Γ_{12}(ω), which is second order in V_{ki}, whereas the RKKY appear only in the fourth order in V_{ki} (see Supplementary Note 4).
Further evidence of collective screening is provided by the comparison of A_{K} maps of the long chains (Fig. 4) and its closest theoretical analogue, δA_{subs}. As a result of the coherent interference of the electron gas coupled to the impurity lattice, the theory predicts a critical chain length, L_{c}, for the appearance of Kondo amplitude maxima near the edge atoms, which scales with the Fermi wavelength of the conduction band. Moreover, it also predicts the suppression of the edge lobes above a maximum length, L_{M}. As illustrated in Fig. 5c, both predictions are experimentally observed in chains with two different spacings d = 2a and 3a. The model approximations lead to quantitative differences in the threshold lengths: L_{c−M}/λ_{F} = 0.75–1.15 and 0.33–1.06 for theory an experiment, respectively. Despite of this, the excellent qualitative agreement between the experimental A_{K} maps and the MIAM calculation implies that our Co atoms cannot be viewed as a chain of local scatterers, because they give rise to a nonlocal Kondo resonance. The nonlocal origin of the conductance patterns in the Kondo lattices constructed in this work contrasts with the patterns found in quantum mirages, which were explained by purely local scattering^{44}. Nonlocal Kondo screening has also been observed in Mn_{x}Fe chains^{31}, where the spin of the constituent atoms is strongly entangled by exchange coupling. Here we explore the opposite limit with negligible coupling between the atomic spins of the chain. The Mn_{x}Fe chains act as a single Kondo impurity with a complex pattern of Kondo fluctuations spreading over several atoms in the chain, whereas for Co/Ag(111) chains we observe manifestations of collective screening of a set of individual impurities.
The singleion picture clearly fails for our chains and dimers with the shortest impurity spacings, which suggests a substantial increase in T^{*}^{3,4} above the working temperature of 1.1 K when d decreases below 3a. In this regime, we observe the defining characteristics of the HFL phase in a Kondo lattice^{2,5}: spin deconfinement and collective screening. The HFL state has been investigated by STM in 2D^{43} or 3D^{22,23,24} dense selfassembled Kondo lattices. In these examples, the transition between KS (independent impurity behaviour with energy scale ~T_{K}) and the HFL state (energy scale ~T^{*}) could not been reached, unlike the case of our Co chains built atombyatom where the origin of the Kondo features can be traced down to the singleion state. The conditions to obtain both states in the same sample cannot be met either in the case of bulk crystalline heavy fermion compounds^{3,4,45,46,47}.
The possibility of observing the real space structures in the adatom Kondo lattice is intimately linked to the fact that the chains are finite. This grants access to quantum size effects of the Kondo lattice Physics. In exchange, some more common features of the Kondo lattice problem in bulk heavy fermion compounds are missing. For instance, the opening of a hybridization gap upon formation of the HFL phase^{23,48,49}, which should manifest as a new ZBF in the dI/dV spectra^{21,22,50}. We do not observe any additional feature on top of the Kondo resonance upon the formation of the Kondo lattice. The reason for this absence remains an open question for further theoretical investigations: the finite size of the Co lattice, its coupling to a metallic host with lower dimensionality, or even an eventual opening of the gap at temperatures lower than experimentally available.
To sum up, we have visualized in real space the onset of HFL behaviour as the redistribution of the Kondo amplitude (and consequently spin deconfinement). In dimers, it appears as a delocalization of the Kondo amplitude towards its centre. In chains, it manifests as patterns at length scales of the order of λ_{F}. Apart from the Kondo temperature, the theoretical model reproduces the essential experimental observations without adjustable parameters: the dimensionless product k_{F}d is fixed by the experimental conditions, and the leading dimension of the electron gas is two. Therefore, our microscopic description should apply for different material parameters, and for arbitrary lattice site and dimensions, laying foundations to further investigations of correlated nanostructures engineered with atomic precision.
Methods
Experimental
The experiments have been performed in a SPECS JouleThompson STM^{51} in ultrahigh vacuum (chamber base pressure <1 × 10^{−10} mbar) at 1.14 K. Some control experiments were repeated at 4.7 K. All images and spectroscopy data are acquired at constant tip sample distance unless stated otherwise (feedback opened over the Ag(111) surface at regulation set point of −100 mV and 40 pA). In our set up, the tip is grounded and bias voltage is applied to the sample. The Ag(111) surface was cleaned by repeated cycles of Ar^{+} sputtering at 1 kV, 1 × 10^{−6} mbar and annealing at 450 °C. Co atoms were deposited in situ onto the Ag(111) substrate from an ebeam evaporator loaded with a high purity Co rod focused at the STM sample stage held at T < 5 K to prevent atoms diffusion. Ag atoms were extracted from the substrate by means of controlled tip indentation. We used lateral atomic manipulation to move individual atoms to designated positions and construct artificial atomic ensembles. Atom manipulation was performed exerting a lateral force on the atom with the STM tip in the attracting regime (pulling mode, manipulation parameters I_{0} ≥ 40 nA and V_{0} = 3 mV). The tip is an electrochemically etched W wire, which is in situ ion milled by means of Ar^{+} sputtering at 5 kV. Afterwards, the tip apex is gently dipped in the Ag(111) surface until it becomes structurally stable against atomic manipulation. Co atoms can be adsorbed on the two types of hollow sites of the Ag(111) surface: fcc or hcp. Our data are fairly independent on the site type, with the exception that in one of them the resonance amplitude is systematically lower than in the other. In order to overcome this duality, we ensure that all atoms are in equivalent sites by carefully assembling all atomic structures along [\(1\bar 10\)] atomic closed packed directions of the Ag(111) substrate. All differential conductance dI/dV data are obtained using a lockin amplifier at 911 Hz and modulation voltage of 0.5 mV rms for spectra and 2 mV rms for imaging. The whole image processing has been carried out with WSxM^{52}.
Ab initio calculations
The electronic structure of Co/Ag(111) has been evaluated using the VASPDFT code, including spinorbit coupling. In the two types of hollow adsorption sites, we obtain a spin magnetic moment of 2.04 μ_{B} with negligible orbital moment and very small uniaxial anisotropy of 0.14 meV. Therefore, the leading interaction is Kondo screening. In absence of electronic correlations (U = 0), Co atoms exhibit two unoccupied antibonding dorbitals and thereby they can be described by an isotropic S = 1 spin system. The interimpurity exchange energy in the dimer was calculated as the energy difference between the parallel and antiparallel spin alignment.
Theoretical DOS in the MIAM
The Hamiltonian of the MIAM reads
where the first term describes a single conduction electron band (substrate), the second term stands for impurities (adatoms) m = 1, …, N with equal onsite energy \(\epsilon _{\mathrm{d}}\), the third term is a Hubbardtype repulsion on each impurity orbital and the last term is impuritysubstrate hybridization. We do not assume any direct interimpurity hoppings. We take the approximation U = ∞ and calculate the multiimpurity Green’s function G_{ij}(ω) by selfconsistent manybody perturbation theory, see Supplementary Note 3 for details.
Upon integrating out the conduction electrons, the impurities acquire an embedding selfenergy \(\Sigma _{ij}(\omega )\). For noninteracting electrons (U = 0), the diagonal terms Σ_{ii} shift and broaden the impurity levels and the offdiagonal terms provide substratemediated hybridization. The central role is played by the hybridization function \(\Gamma _{ij}(\omega ) = \frac{{\mathrm{i}}}{{2\pi }}\left[ {\Sigma _{ij}(\omega )  \Sigma _{ji}^ \ast (\omega )} \right]\), given by the antiHermitian part of the embedding selfenergy.
The substrate local DoS (LDoS) A_{subs}(r, ω) is determined by the substrate Green’s function \({\cal{G}}({\mathbf{r}},{\mathbf{r}}\prime ,\omega )\) as \(A_{{\mathrm{subs}}}({\mathbf{r}},\omega ) =  {\mathrm{Im}}{\cal{G}}({\mathbf{r}},{\mathbf{r}}\prime ,\omega )/\pi\). The effect of the Anderson impurities on the substrate is expressed in a closed form^{2} by a Tmatrix relation. Assuming pointlike impurities at locations R_{i} and a homogeneous substrate, we get
In the last equation, the Green’s function of the unperturbed substrate is denoted with a superscript (0) and G_{ij}(ω) is the Green’s function of the impurities.
In two dimensions, \({\cal{G}}^{(0)}({\mathbf{r}}  {\mathbf{r}}\prime ;\;E_{\mathrm{F}}) = \frac{i}{2}H_0^{(2)}(k_{\mathrm{F}}{\mathbf{r}}  {\mathbf{r}}\prime )\) (Hankel function). The first term in the righthand side of Eq. (5) gives just the constant DoS and we discard it. The second term provides the change of the LDoS due to impurities, δA_{subs}(r, ω). It reflects the propagation of the nonlocal Kondo resonance in the substrate. Finally, γ is a constant proportional to Γ_{0}.
We use the latter approach to calculate the LDoS in long chains, as it captures well the interference effects at the length scales comparable with λ_{F} = 2π/k_{F}. However, in the twoimpurity systems, we want to describe features at smaller length scales (≈.5a, impurity spacing), where atomistic details matter. Accordingly, for the real space maps, we add to the homogeneous electron gas exponential wave functions φ_{i}(r) to calculate the LDoS of the Anderson impurities A_{imp}(r;ω) from the impurity Green’s function
We use the wave functions φ_{i}(r) = C exp(−R_{i} − r/α) with radius α = 1.5 Å. The two orbital centres are separated by R_{1} − R_{2} = 2.6a. For such small length scales (\(\ll \lambda _F\)), we neglect the spatial dependence of Γ_{ij}, and parametrize its matrix elements as given in the main text with the following frequency dependence:
We note that the actual Co dorbitals are quite localized (orbital radius ≈0.5 Å). Therefore, the φ_{i} is to be viewed rather as a superposition of nonmagnetic local states in the vicinity of each Co. These atomistic details become less important as L approaches λ_{F} and the description by means of the homogeneous electron gas (cf. Eq. (5)) is more appropriate.
We choose parameters that fall well within the Kondo regime in all cases: U = ∞, \(\epsilon _{\mathrm{d}} =  0.4\), E_{0} = 4 and temperature k_{B}T = 10^{−4} (all in eV). We take Γ_{0} = 0.06 and 0.01 eV for long chains and dimers, respectively. The qualitative features of the local DoS are unchanged as long as we stay in the Kondo regime.
Data availability
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: http://hdl.handle.net/10261/178494. Additional information can be obtained from the authors.
Code availability
Computer code used to solve the Anderson Hamiltonian and generate the profiles and images of the spectral functions can be shared by the authors upon request.
Change history
13 October 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41467021263415
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Acknowledgements
Financial support was provided by the Spanish Plan Nacional de I+D+i (grants MAT 201346593C63P, MAT201678293C66R, MAT201566888C32R, and FIS201564886C53P), Charles University (programme PRIMUS/Sci/09) and the European Union through programmes InterregPOCTEFA (grant TNSI/EFA194/16) and H2020EINFRA52015 MaX Center of Excellence (grant no. 676598). M.M.L., M.P., and D.S. acknowledge the use of SAI at Universidad de Zaragoza. R.R. acknowledges The Severo Ochoa Centers of Excellence Program (grant no. SEV20170706) and Generalitat de Catalunya (grant no. 2017SGR1506 and the CERCA Programme).
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D.S. and M.M.L. conceived the project. M.M.L., M.P., J.I.P., and D.S. performed the experiments and analyzed the data. R.K. developed the MIAM model and carried out the associated calculations. N.L. and R.R. performed the DFT and the MIAM model calculations for dimers. All authors discussed the results and participated in the drafting of the paper. D.S., R.K, M.M.L., and M.R.I. wrote the manuscript.
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MoroLagares, M., Korytár, R., Piantek, M. et al. Real space manifestations of coherent screening in atomic scale Kondo lattices. Nat Commun 10, 2211 (2019). https://doi.org/10.1038/s41467019101035
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DOI: https://doi.org/10.1038/s41467019101035
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