The Mott to Kondo transition in diluted Kondo superlattices

In condensed matter, a tremendous effort has been generated to realise Kondo lattices both experimentally and theoretically. The pursuit of independent magnetic moments, via charge localization, is paramount for applications in nanotechnology. Particularly, systems with simultaneous charge/spin degrees of freedom can manifest both Kondo spin quenching and Mott–Hubbard charge localization. Experimental frameworks illuminating pathways between the two are physically and technologically significant, and hardly observed in reality. Recent developments in controlling densities/temperatures of strongly correlated impurities on surfaces has opened up new possibilities. Such systems introduce mechanisms to study Kondo/Mott-physics interplay methodically. However, the pathway between Kondo physics and charge localization remains elusive. In this work, we investigate the phase diagram of superlattice structures of f-elements on substrates, assessing required conditions for obtaining Kondo superlattices. We unveil pathways between Kondo quenching and Mott localization, and identify non-trivial charge density waves emerging from the competition of charge localization and Kondo physics. The experimental realisation of a Kondo lattice and the interplay with Mott–Hubbard charge localisation is one of the many challenges in condensed matter physics. The authors deposit f-elements onto a metallic substrate to elucidate the conditions required to obtain a Kondo lattice on a superlattice and investigate the interplay with the Mott physics.

C orrelated adatoms expand upon the plethora of remarkable properties already existent in two-dimensional (2D) metallic surfaces. Stemming from the competition of localization and itinerant characters, such properties include: hybridization, superlattice self formation [1][2][3][4][5][6] and long range interactions [7][8][9][10] . Moreover, due to the comparatively low coordination between adatoms and substrate, single atom magnets in the form of Pt/Ho [11][12][13] have been debated and magnetic hysteresis of Er trimers on Cu have been observed 14 . These phenomena have attracted a wide academic interest, but also offer pathways towards industrial applications, for example as candidates for future atomic scale memory storage [15][16][17] .
While the limits of Mott [18][19][20] superlattice and Kondo impurity behaviour are well documented, little is known about the transition between the densely packed (Mott) and the diluted (Kondo) limits of superlattice systems. Furthermore, although physical cases of 3d transition adatom 2,21-23 and rare-earth adatom [24][25][26] self-organization have been reported and justified with tight-binding models and kinetic Monte Carlo simulations 1,3,[27][28][29] , the quantum mechanism responsible for the formation of the superlattice, and the adatom-substrate requirements for self-formation are so far elusive.
To paint a picture of the phenomena observed in this work, an overview of impurity lattice physics is due: a single Kondo adatom within a material does not conserve momentum when scattering electrons. However, in this work, we discuss lattices whose unit cell contains Kondo impurities: Kondo lattices. Due to translational symmetries, Kondo scattering within such lattices conserve momentum, leading to coherent scattering and a decrease in resistivity below the Kondo temperature 30,31 . Interpreting requirements for constructing Kondo lattices will be essential in further understanding of correlated low-temperature research. Friedel oscillations 32 are another form of impurity-impurity communication. They describe electronic screening of an impurity and charge distribution on lattice sites. Different mechanisms have been proposed as the mediator for self-assembly. One such mechanism, which theorises how Friedel oscillations from two adatoms interact was proposed by Hyldgaard and Persson 33 . Lastly, Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions [34][35][36] provide a mechanism for spin-spin coupling between f and/or d band electrons.
2D systems promote surface electron states. These states are formed with lower electron densities in comparison to the bulk electron states observed in three-dimensional systems. Therefore, electrons become localized under the effects of impurities, and the system becomes devoid of conduction band electrons. Since there is no scattering of conduction band electrons, this leads to an atomic limit behaviour at the lattice impurities. Moreover, substrate electrons are further localized by the impurity induced potential. Under these circumstances, a spin-spin interaction can occur between the impurities electrons and electrons localized near the impurities, forming singlet/triplet bound states. Lastly, due to the periodic structure of both the lattice and the impurity ionic potential, localization can occur around both the adatoms and on the substrate. Localization on substrate sites might be expected when inter-adatom distances become sufficiently large, and the conduction band of the substrate sufficiently high, permitting localization effects and realizing a charge density wave concomitant with a pseudogap formation. These effects have so far not been investigated, due to the complexity of the problem, which involves dealing with large unit-cells and the presence of strong correlations in the f-shell of the adatoms. Typically, such systems are beyond the reach of density functional theory, and even more so, from many body perturbative techniques like GW and its extensions.
In this work, we use a combination of dynamical mean field theory (DMFT) and tight binding calculations for a triangular lattice substrate, techniques which have been utilised on previous works concerning other superstructures 19,37,38 . We present our investigations into the generic properties of single orbital correlated adatom structure upon a triangular metallic substrate. This model is a canonical model for the description of a 4f1 orbital hybridized to a Ag(111) substrate. We systematically investigate the phase diagram as a function of correlation strength, substrate electronic density and adatom inter-distances, and outline the different phases obtained. We study the strength of correlations, quantified by the quasi-particle weight factor (Z), and examine the electron localization at the adatom and its impact on the correlated energy bands and Fermi surfaces.
We report a sharp Mott to Kondo transition as adatom spacing varies from a tightly packed solid to a diluted solid. We also report the stabilization of a charge density wave competing with Kondo physics at large inter-adatom distances. For any given superlattices, large mass renormalization are obtained at low and high-density regimes, corresponding to the limit of a weakly hybridized impurity, and the limit of a screened impurity respectively. These two phases are separated by the formation of a tightly bound triplet between the adatom and the substrate that exhibits a weak mass renormalization. In our view, the experimental realization of the Ce/Ag(111) superlattice falls in the latter regime, and our result shed light on the lack of evidence for Kondo physics as previously observed by spin-polarized scanning tunneling microscopy. Our work predicts that Kondo Lattices are obtained at intermediate adatom distances (5-9 substrate lattice sizes, L).

Results
Strength of correlations and emergent quasi-particles in the superlattice. First, we report the systematic study of the superlattice correlation strength, as function of the substrate electronic density and adatom distances, obtained by paramagnetic DMFT calculations (permitting fluctuating moments and correlations across adatoms, but not for long-range magnetic order). Correlation strength is measured by the adatom quasi-particle weight factor Z, obtained with the low energy imaginary component of Matsubara self-energy Σ(iω n ), reported in Fig. 1a. The results outlined below correspond to Hubbard repulsion U/t = 6, where t is the tunnelling amplitudes across the substrate, typically t = 0.75 eV for the case of the Ag/Ce superlattice (we discuss results for this Coulomb repulsion thereafter, extended the calculations to other values of the Coulomb repulsion, see Supplementary  Fig. 1).
We report a phase diagram in terms of the adatom electronic occupation n f (all occupation refer to the single spin component only, as the calculations are paramagnetic). Although in our calculations we systematically change the chemical potential, we report the phase diagram in terms of the adatom occupancy. The total occupation is indeed a monotonic function of the adatom density at any finite temperature. We find two dominant regimes where correlation effects are large, at two nearly horizontal minima near the single spin adatom (f band) electron density n f = 0.3 and 0.7, separated by a weakly correlated regime at halffilled adatom n f = 0.5 across all values of U (see also Supplementary Figs. 1 and 2).
We report the electronic density of the substrate in Fig. 1b (blue denotes an empty substrate, and red the band insulating limit).
The energy resolved spectral weight ρ(ω), obtained respectively at adatom density n f = 0.7 (see Fig. 1c), at half-filled adatom density n f = 0.5 (see Fig. 1d, which also is concomitant with a filling of two electron in total per unit-cell, meaning there is one electron in the substrate), and at low density n f = 0.3 (see Fig. 1e).
In Fig. 2a, b, we report the spectral function A(k, ω) resolved in real frequencies ω and momentum k obtained for the same density. Note that, as the band structure is trivially folded due to the unit-cell construction of the periodic superlattice, we reconstruct the unfolded band structure of the original unit-cell of the substrate, which allows for comparison across different superlattices.
The Mott to Kondo crossover in the diluted solid. To investigate the correlated regions of the phase diagram (Fig. 1a), a polynomial regression was preformed on the adatom self-energy to obtain the electron scattering rate and the Kondo temperature (T K ) in the two correlated phases obtained at respectively n f = 0.3 ( Fig. 3a) and n f = 0.7 (Fig. 3b). In both phases, we obtain a significant mass renormalization, concomitant with a large magnetic moment (see Supplementary Fig. 3), a required but not sufficient condition to obtain Kondo physics.
The Kondo temperature T K , extracted from the low frequency hybridization (see Method section) was used here as a marker of the formation of a Kondo singlet. Note that T K as extracted in this approximation provides the right trend, but is known to overestimate T K . We obtain (see Fig. 3c, d) that a crossover occurs for both regions as the adatom distances reaches L = 3-4 for n f = 0.3 and L = 5-6 for n f = 0.7, respectively.
The calculation of the Kondo temperature has been validated by the measurement of the static local magnetic susceptibility χ (ω = 0) of the adatom, obtained from calculations carried out at different temperatures, this is presented in Fig. 4.
Weakly correlated triplet bound state. Finally, we investigated the region of the phase diagram that divides the two correlated regions obtained in the phase diagram (obtained at n f = 0.3 and n f = 0.7, respectively). At n f = 0.5, which corresponds to the case For the latter, the adatom is in the atomic limit (AL), as the conduction band remains weakly populated, as proposed in the discussion of a half-filled f level, we obtain the weakest mass renormalization and a regime of nearly free electrons (NFE) (see Fig. 1a). In Fig. 5, we report the integrated occupation over the supercell (both adatom and substrate). In particular, we find that the half-filled adatom f shell is obtained with a concomitant total charge of 2 electrons, and hence an occupation of a single electron per unitcell on the substrate. The real space average charge (not shown) also reveals that the electron pair is localized in the unit cell around the adatom. This can be rationalized in a simple toy model, where 2 electrons are forming a bound state between the adatom and the three substrate neighbours (see Fig. 5). This toy model can be used as an approximation to capture the physics of the superlattice locally.
Mott localization driven by adatom interactions mediated by the substrate. We also extended our calculations to larger and smaller values of the correlation U. Whilst most results remain identical, we note that for the strong coupling regime (U/t = 10), the systems undergoes a charge localization for L = 9 for n f = 0.7, followed by the formation of a pseudogap and the suppression of the Kondo resonance (see Supplementary Fig. 4). The localization is directly observed in the real space electronic density (compare Fig. 6a, b with Fig. 6c). Albeit observed in all calculations that electrons localized naturally around the adatoms, due to the attractive ionic potential, the localization induced by the Coulomb repulsion in this particular regime of parameters happens in-between the adatoms.

Discussion
In Fig. 1a, low n f , we obtain here that the quasi-particle state is localized at the adatom. As n f increases, however, Z also increases, reaching a maximum when the adatom is exactly half-filled.
Although we might naively expect that the reduction of the mass renormalization is associated with a delocalization of the charge, we observe that the electron remain tightly bound to the adatom location. In our view, this is associated with the creation of a twoelectron tightly bound triplet state between the adatom and the substrate (see Fig. 5). Finally, at densities larger than two-electron per unit-cell, we obtain another localization process near n f = 0.7 with a concomitant increase of the mass renormalization. Local magnetic susceptibility. χ(ω = 0) of the adatom for L = 6 and n f = 0.7, as a function of inverse temperature β. At high temperature (T > T C ), the susceptibility is proportional to 1/T, indicating that the system is above the Curie temperature, whereas at low temperature (T < T K ), the susceptibility is temperature independent, showing that the system has entered into the Kondo regime Moving onto Fig. 1b. As expected, at low occupation of the adatom, the substrate is empty, and we obtain the limit of an impurity in vacuum (atomic limit, AL). This limit is not physical and provides a lower bound on the adatom occupation for a given superlattice structure. Additionally, at small adatom distances L, we observe a rapid filling of the substrate upon increasing the adatom occupation, an effect related to the Coulomb repulsion U at the adatom site, that prevents an occupation of more than one electron and induces a charge transfer to the substrate.
At high electron density n f = 0.7 (see Fig. 1c), we observe the emergence of a resonance, that starts to develop at L = 5, and becomes resonant at L = 7, and ultimately vanishes at L = 10. We associate this feature to the presence of a screened Kondo singlet (see discussion thereafter). We note that at the same density, we observe at large adatom distances L > 9 the formation of a charge density wave, where electron on the substrate localize half-way inbetween adatoms, with the opening of a pseudogap in the spectral density (see Supplementary Figs. 1 and 4). This phase is denoted as CDW in the phase diagram.
The spectral feature of the weakly correlated region at n f = 0.5 (see Fig. 1d) exhibits a large degree of hybridization between the adatom and substrate. We associate this state to a Triplet Bound State (TBS) formation.
The low density regime exhibits (see Fig. 1e) sharp features that corresponds to the localized states at the adatom. We obtain a transition from a broad feature, reminiscent the correlated metal, to a narrow spectral feature at the Fermi level obtained at adatom distances L > 4. This is fingerprint of a localized electron at the adatom location, which we denote at a Kondo impurity (KI). Ultimately, the KI localized quasi-particle states converges to the atomic limit (AL) at large adatom separation.
In Fig. 2, we observe that the adatom are strongly hybridized to the substrate for n f = 0.3, L = 5 (Fig. 2a), as shown by the fluctuations induced in the substrate bands by the adatom, but dehybridize at larger adatom distance L = 10 (Fig. 2b), a fingerprint that the atomic limit has been reached (AL).
Regarding Fig. 3c, for the limit of the Kondo impurity (n f = 0.3), the system enters into the Kondo regime abruptly via an abrupt transition (see Supplementary Fig. 2) between a non Fermi liquid at short adatom distances to a Fermi liquid at intermediate adatom distances (a linear behaviour of the self-energy as a function of the Matsubara frequency is obtained in the selfenergy, a signature of a Fermi liquid, see Supplementary Fig. 2).
For n f = 0.7 (Fig. 3d), the crossover is continuous and, unlike the abrupt change obtained between the NFL to FL regime obtained at n f = 0.3, an increase in the mass renormalization occurs simultaneously with the large increase of T K at n f = 0.7.
In Fig. 4, we observe that the system manifests true Kondo behaviour in the large density regime n f = 0.7, for temperatures lower than T = 0.007 × t, which corresponds to ≈ 60°K (the tunneling amplitude for the silver substrate is t = 0.75 eV). We, therefore, identify the region of the phase diagram L > 5 and n f = 0.7 as a Kondo lattice (KL): a lattice of independent screened magnetic moments. At large temperature T > 0.02 × t, we recover the usual Curie Weiss law, which describes a fluctuation moment.
Moreover, in the low density limit (n f = 0.3), we find that the local susceptibility remains small at all temperatures with weak temperature dependence (χ~0.02), as expected for a system which exhibits a large degree of localization due to correlation effects. We also note that a larger Coulomb repulsion U/t = 10 opens a gap and suppresses the resonance obtained in this regime (see Supplementary Fig. 4), gaping in turn the fluctuations associated with the magnetic moment by Mott physics. Due to the low occupation of the conduction band, we consider the region L > 3 and n f = 0.3 as a system of Kondo impurities (KI).
The solution of the 2 electron model in Fig. 5 gives a bound triplet ground state for any realistic values of the screened Coulomb interaction U (U < 11 eV). For U > 11 eV, we obtain a singlet ground state. It is worth noting that this limit was obtained by renormalizing the hopping parameters between the substrate atoms to recover the bandwidth of the conduction band, as those have been integrated out in this simple picture (without this renormalization, the U separating the singlet from the triplet is larger). We, therefore, conclude that for this region of the phase diagram, we obtain triplet bound electron pairs localized around the adatoms. Note that the triplet can be seen as an uncorrelated state, as the Pauli principle excludes the double occupations. In our view, this phenomena explains the nearly free electron regime, separating the correlated phases at low and high n f .
Lastly, regarding the Coulombic localization effects in Fig. 6. In our view, this effect is driven by the two-body potential induced . Across all lattice size L, the graph follows the same linear increase from an empty system to (n f = 0.5, N tot = 1). We conclude that (taking both spins into account) when the adatom is singly occupied, the conduction band must also be singly occupied, allowing singlet/ triplet states between the adatom and substrate to form 6 Charge density wave. Real space electronic density n CB of the substrate conduction band for the strongly coupled regime and n f~0 .7 for a L = 3, b L = 7 and c L = 9. Black (white) colours indicate large (small) electronic density. For L = 3, we observe localized states around the adatom (the adatom, not shown, is located at the centre of the black circles). As L increases, we observe a delocalization, which is concomitant with Kondo physics (see Discussion hereafter). For the limit of diluted solids, corresponding to L = 9 and larger, we observe the formation of a charge density wave (CDW), where localized states develop in between adatoms. The CDW is concomitant with the formation of a pseudogap (see Discussion hereafter) by the interference energies between Friedel oscillations. This two-body adatom interaction, mediated by surface state electrons, is an oscillating function of the adatom distance (similarly to the RKKY interaction), and is parametrized by the electronic mass. In the free electron model, its oscillations are involving distances larger than 11 lattice spacings, and therefore does not play a role in our calculations for L < 11. Furthermore, this interaction is also suppressed when the states at the Fermi level are absent (as it is the case in the low density regime n f = 0.3 at large U/t = 10 that opens a gap). However, at large density n f = 0.7, the adatom mediated interaction is modified by the large mass renormalization induced by the local Hubbard U term, and produces an attractive potential in-between the adatoms.
We emphasize that this CDW is fundamentally different from Friedel oscillations (FO), where the amplitude decreases as distance from impurity increases. Secondly, as discovered by Chatterjee et al. 39 . the amplitude of FO decreases as U increases, in contradiction to our observation in the Kondo superlattice. Furthermore, the charge localization, in our measurements, is obtained in the large U limit (U/t = 10), and is concomitant with a renormalization of the electronic mass (see Supplementary  Fig. 1). We associate hence the CDW to a charge ordering induced by the narrowing of the conduction band induced by correlation. The electronic localization is associated with the opening of a pseudogap.

Methods
In this paper, we have used the following computational setup in order to model the deposition of Ce adatoms on Ag substrate: the Ag substrate was modelled by a 2D triangular lattice of Ag 5s orbitals, while the Ce adatom 4f orbital was represented by an additional site connected with its three nearest neighbour Ag atoms. The heavy deposited Ce adatom hybridize with the substrate, and also induce an attractive ionic potential on the three neighbours of the substrate.
In this work, we use the dynamical mean-field theory (DMFT) approximation. Heavy f-materials have been extensively studied by DMFT, which captures well both the itinerant and localized behaviour of the electrons in f-materials [40][41][42][43][44] . In the DMFT formalism, the adatom (Ce) sites with the on-site U are treated as strongly correlated orbitals, while the conduction band of the substrate (Ag sites) are considered as uncorrelated. More formally, the Hamiltonian of the system considered here reads in real-space as: where t is the inter-substrate hopping (in our calculation t is the energy scale, for the case of a Ag(111) substrate, t = 0.75 eV), ε imp is the adatom energy level, U the adatom Hubbard repulsion, μ the chemical potential and h.c. denotes Hermitian conjugate. c ðyÞ iσ and f ðyÞ iσ are the Fermionic annihilation (creation) operators of the site i, spin σ, surface state and the adatom respectively. V i describes the on-site potential asserted upon the substrate by the adatom (as described in ref. 1 ). And lastly, the adatom/substrate hopping parameter V if ¼ V Ã fi ¼ t Ad is assumed finite for three of the substrate sites, otherwise it is zero. This assumption can be justified through consideration of the triangular substrate 1,4 . The schematic view of our computational setup is shown in Fig. 7.
In analogy with DFT+DMFT, the calculation requires convergence with respect to the number of k-points in the Brillouin zone of substrate lattice. Calculations carried out in this work were computationally intensive, convergence in some cases required 100 × 100 sixfold reduced k-points (triangular substrate symmetry). Without any simplifications, calculations of each k-point scales as n freq N 3 , for number of sites N and number of Green's function frequency points n freq ¼ Oð1000Þ. For the largest lattice size considered, L = 15, of the order of 1000*225 3 calculations was needed to invert the Hamiltonian to calculate the Green's function for a single DMFT iteration.
In DMFT, we have employed an exact diagonalization (ED) solver improved by using the cluster perturbation theory (CPT). The ED solver [45][46][47] uses a finite discretization approximation of hybridization Δ(iω n ) function, which induces finite discretization errors 48 . As the size of the basis used to represent the Hamiltonian scales exponentially with the number of bath energies in the system, solving the eigenvalue problem is not computationally achievable if the number of bath orbitals is typically larger than 15. ED-CPT 49 bypasses this problem by utilizing a small number of bath systems and incorporating additional bath sites in order to mimic the effect of coupling to an infinite bath, which was used in our work to check that no remaining systematic error due to the bath discretization was obtained in our calculations. Finally, we have used Lichtenstein's double counting scheme 50 .
Sherman-Morrison. Calculation of the full lattice Green's function G(k) involves a matrix inversion at each k-point and ω. In the case of large supercell sizes this becomes a bottleneck of the formalism. An optimization is possible here due to the fact that the matrix to be inverted contains a large number of empty entries. This is due to the properties of the supercell: the self-energy is non-zero only at the adatom, and the hybridization between the adatom and the substrate occurs only on three sites of the substrate. Therefore, to reduce the computational load, the Sherman-Morrison algorithm 51 was implemented to update the full Green's function G(k) from the substrate Green's function G 0 (k): . . .
Here V if , ε f , t, and V i are defined in (1), iω n is the n's Matsubara frequency, while the substrate sites are ordered in such a way that the three sites connected to the adatom are listed first.
G 0 is obtained via matrix products for all n freq (N 2 operations), G is then updated from G 0 . Since only three substrate sites hybridizes with the adatom, the updating process scales linearly with N.
Kondo physics. In order to extract the Kondo physics fingerprints from our DMFT results, we have calculated the scattering rate Γ and Kondo temperature T K .
In order to estimate Γ, we performed the polynomial regression with the following low frequency, low-temperature expansion of Σ′′(iω n ) in the Kondo regime 52 : Γ is obtained from the above parameters a and Z as follows: We used 53 the following definition of the Kondo temperature T K after Hewson's renormalized perturbation theory 54 of the Anderson impurity model (AIM): where Δ′′ is the imaginary part of the resonant level scattering width as defined in ref. 54 .
Triplet bound state toy model. Here we briefly outline the simple toy model used to analyze the regime of the phase diagram, where we have two electrons per unit- Fig. 7 Cartoon picture of the dynamical mean field theory setup. The model includes the conduction band of the substrate (depicted by the blue circles) and the concomitant Hamiltonian parameters. The correlated adatom (red circle) sits in between three sites of the substrate (blue circles) cell. As we observed in the superlattice calculations that the electronic states are localized around the adatoms, we solve the isolated plaquette of three substrate atoms connected to the adatom site. This Hamiltonian includes a local Coulomb repulsion U on the adatom site, an hybridization between the adatom and the three substrate atoms, and hopping across the substrate atoms. The impurity energy and hybridization are the same as the one treated in the full DMFT calculations, whereas the hopping across the substrate elements is renormalized to account for the bandwidth of the full calculation. Upon diagonalizing this Hamiltonian and finding the ground state, we then evaluate the expectation value of the spin correlator K operator between the adatom and one of the substrate atom, which identifies a singlet (triplet) ground state if K < 0 (K > 0). We identify the first-order transition between the triplet state at small U and the singlet state at large U (see Supplementary Fig. 5). Interestingly, we find that the critical U to obtain a singlet state is larger than all realistic U considered in our calculation. This suggests that in the limit of low occupation, pairs of electrons will form a triplet state for realistic values of the Coulomb repulsion.
This triplet state is indeed a multiplet that also includes the polarized contribution with two electrons of same spin, and exhibit a lower degree of correlation. Indeed, in the S z tot ¼ ± 1 sectors, there are no contributions from the Coulomb energy, as observed in the phase diagram near the n f = 0.5 region, where the mass renormalization is weak. We attribute hence this region of the phase diagram to a triplet bound state.

Data availability
Selected input and output files that support the findings of this study have been deposited in figshare with the identifier: https://doi.org/10.6084/m9.figshare.7796636.