Abstract
Kondo lattices are ideal testbeds for the exploration of heavyfermion quantum phases of matter. While our understanding of Kondo lattices has traditionally relied on complex bulk felectron systems, transition metal dichalcogenide heterobilayers have recently emerged as simple, accessible and tunable 2D Kondo lattice platforms where, however, their ground state remains to be established. Here we present evidence of a coherent ground state in the 1T/1HTaSe_{2} heterobilayer by means of scanning tunneling microscopy/spectroscopy at 340 mK. Our measurements reveal the existence of two symmetric electronic resonances around the Fermi energy, a hallmark of coherence in the spin lattice. Spectroscopic imaging locates both resonances at the central Ta atom of the charge density wave of the 1T phase, where the localized magnetic moment is held. Furthermore, the evolution of the electronic structure with the magnetic field reveals a nonlinear increase of the energy separation between the electronic resonances. Aided by ab initio and auxiliaryfermion meanfield calculations, we demonstrate that this behavior is inconsistent with a fully screened Kondo lattice, and suggests a ground state with magnetic order mediated by conduction electrons. The manifestation of magnetic coherence in TMDbased 2D Kondo lattices enables the exploration of magnetic quantum criticality, Kondo breakdown transitions and unconventional superconductivity in the strict twodimensional limit.
Introduction
The complexity of the Kondo lattice problem is best appreciated in comparison with the relative simplicity of that involving a single Kondo impurity. At temperatures below T_{K}, a magnetic impurity coupled to a metal with Kondo exchange coupling J_{K} starts developing singlet correlations with the conduction electrons and eventually becomes completely screened at T = 0 K. In a periodic array of such impurities, the magnetic moments also develop independent singlet correlations around T_{K}. However, as the temperature is further lowered below a coherence scale T*, the competition between the Kondo exchange and Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions between moments can drive the system to different ground states like a Kondo paramagnet or magnetically ordered state, as first proposed by Doniach^{1} (Fig. 1). A microscopic understanding of the ground state of this coherent Kondo lattice remains a central problem in condensed matter physics, especially as more complex scenarios than those envisioned by Doniach are also possible, where Kondo screening coexists with magnetic order^{2}. Understanding the nature and possible types of zerotemperature quantum critical points between such phases and their influence in phenomena like nonFermi liquid behavior, the Kondo breakdown and fluctuationmediated superconductivity remains a great challenge in the field. Experimentally, the understanding of these fascinating problems has traditionally been hindered by the complexity and lack of tunability of the felectron compounds like those based in Ytterbium^{3} or Cerium^{4}.
The observation of the Kondo effect in transition metal dichalcogenide (TMD) heterobilayers formed by vertical stacks of T and Htype monolayers has recently opened a new, simple, and accessible platform to design artificial Kondo lattices^{5, 6} in strict twodimensions. In these systems, the 1T monolayer develops a √13 × √13 CDW known as the StarofDavid (SoD) pattern, which leaves a single unpaired electron near the Fermi energy (E_{F}) that forms an effective impurity flat band. The residual Hubbard interaction in this flat band leads to an array of local moments and to a magnetic Mott insulating state. When the 1T layer is stacked onto a metal, the hybridization of the impurity band with the Fermi surface enables the Kondo effect. A narrow Kondo peak has been observed in these T/H heterobilayers of TaSe_{2}, TaS_{2}, and NbSe_{2} with T_{K} in the range 18–57 K, providing compelling evidence of the formation of local moments^{5,6,7,8,9}. However, lowertemperature evidence for any coherence behavior of the Kondo lattice remains sorely lacking, and fundamental questions such as the ground state of these compounds in the Doniach diagram remain unanswered.
In this work, we address this problem by performing highresolution scanning tunneling microscopy/spectroscopy (STM/STS) experiments at 340 mK in the prototype 1T/1Hheterostructure and demonstrate coherent behavior beyond single moment physics, in the form of a split Kondo peak whose separation increases nonlinearly upon an outofplane magnetic field (B_{z}). Using a periodic Anderson model with parameters extracted from ab initio calculations to compare the signatures of different candidate ground states, we conclude that a magnetic phase of the coherent Kondo lattice with inplane magnetic order is the most likely scenario consistent with our experiments.
Results
Our experiments were carried out in highquality 1T/1H heterobilayers of TaSe_{2} grown on epitaxial bilayer graphene (BLG) on 6HSiC(0001), as sketched in Fig. 2a. The vertical 1T/1H heterobilayer is naturally formed during the growth of fewlayer TaSe_{2} as both polytypes coexist due to their similar formation energies^{10} (see Supplementary Information (SI) for growth details and morphology). The robust commensurate charge density wave (CDW) developed in bulk 1TTaSe_{2} is also easily distinguished in STM images in the monolayer limit as a (√13 × √13)R13.9° triangular superlattice (yellow rhombus in Fig. 2a). Each bright spot in the superlattice corresponds to a cluster of 13 Ta atoms (the SoD) where an individual magnetic moment develops due to the presence of an unpaired electron.
To study the electronic structure of 1TTaSe_{2} on 1HTaSe_{2}, we carried out STS measurements at 4.2 K and 0.34 K. dI/dV spectra (dI/dV ∝ LDOS) taken on this heterostructure at 4.2 K (Fig. 2b) are dominated by a prominent single peak centered at the Fermi level (E_{F}) with typical width of 10 meV. The existence of this peak feature, previously reported for singlelayer 1TTaS_{2} and 1TTaSe_{2}, has been ascribed to a Kondo resonance due to the screening of the SoD magnetic moments by the underlying metal substrate (1H phase)^{5, 6,8,9}. This peak is flanked by two broader peaks that can be identified as the lower (upper) Hubbard bands, which yield an experimental measure of the Hubbard repulsion of U = 208 ± 4 meV (see SI). As the temperature of the 1T/1H heterostructure is lowered below 4.2 K, the sharp peak feature at E_{F}, however, is better resolved spectroscopically and two symmetric peaks with respect to E_{F} are observed, as shown in the representative dI/dV spectrum taken at our base temperature of T = 0.34 K in Fig. 2c. It is worth mentioning that the gradual emergence of these two peaks is not due to a Tdependent process but rather due to an improvement of energy resolution in STS as T is lowered (see SI for the full evolution with T). This doublepeak feature is in contrast with the 1TTaS_{2} case, where only one peak appears at subkelvin temperatures at zero magnetic field^{6}.
To gain further knowledge about the lowlying electronic structure of the 1T/1H heterostructure, we carried out spatially resolved measurements at 0.34 K in several nmsized regions, one of which is shown in Fig. 3a. Figure 3b shows three dI/dV spectra taken at the center of three neighboring clusters (colored dots in Fig. 3a), which reveal that the separation between the two peaks (Δ) presents significant variations. We have quantified these variations by spatially mapping Δ in this region (Fig. 3c) from a 40 × 40 dI/dV grid. As seen, Δ mostly varies between SoD clusters, being largely constant inside each of them. Figure 3d shows the resulting histogram of Δ, which shows a mean value of \(\bar{\Delta }=1.2\pm 0.2\,{{{{{\rm{mV}}}}}}\). Very similar values were found in all the regions studied.
We have also characterized the spatial extension of the two electronic peaks by measuring atomically resolved conductance maps (constant height) of the 1T/1H heterostructure (Fig. 3e–g). Figures 3e, g show two conductance maps taken consecutively at ± 1 mV in the same region shown in Fig. 3f. Both conductance maps are nearly identical and reveal that the intensity of these two peaks is mostly localized on the three Se atoms directly bonded to the Ta atom at the center of the SoD, although some residual intensity lies on the second Se neighbors. This is in good agreement with the expected location of the unpaired electron that gives rise to a net magnetic moment.
In order to elucidate the origin of this twopeak feature, we have performed a thorough characterization of the electronic structure subject to external magnetic fields (out of plane). Figure 4a shows a representative series of dI/dV spectra within ± 25 mV acquired consecutively as B_{z} is swept up to 11T. The most obvious effect of B_{z} on the LDOS is the gradual shift of both peaks from E_{F,} which leads to an increase of Δ with B_{z}. In parallel to this energy shift, the LDOS is gradually depleted around E_{F} and the intensity of the peaks diminish with B_{z}, as shown in the zoomin of Fig. 4b and plots in Fig. 4c. Beyond these changes, the electronic structure for larger energies seems to be nearly unperturbed by the magnetic field. To confirm that no further features appear in the lowenergy electronic structure upon the application of B_{z}, highresolution dI/dV spectra (~30 µV/point) were systematically taken around the peaks maxima in different regions in the 0–11 T range. As shown in Fig. 4d, no further structures appear within our energy resolution at T = 0.34 K (see SI and ref. ^{11}), and only two peaks remain visible at high magnetic fields.
From our Bdependent STS measurements such as the dI/dV sets of Figs. 4a, d, we have quantified the evolution of the peaks’ maxima with B_{z}. Figure 5a shows a typical plot of the peaks’ separation Δ as a function of B_{z}. We consistently find that the evolution of Δ with B_{z} has two differentiated regimes: a strict linear dependence of Δ for B_{z} ≳ 0.5 T (linear fit in red, see SI) preceded by a nonlinear regime at lower B_{z} fields (region shaded in green). The linear dependence of Δ with B_{z} is attributed to a Zeeman splitting with a Landé gfactor of g = 3.1 ± 0.2 (measured from the steepest slope of the dI/dV curves). Figure 5b shows a zoomin of the lowB field regime, where Δ(B_{z}) gradually deviates from the linear dependence (beyond the confidence band depicted in gray) below 0.5 T to ultimately reach a deviation of \(\leftS\right=0.17\,{{{{{{\rm{mV}}}}}}}\) at B_{z} = 0 T from the linear behavior. Another important observation regarding the Δ(B_{z}) dependence is that this behavior is independent from the polarity of the magnetic field (±B_{z}). This is shown in Fig. 5c (zoomin in Fig. 5d), where B_{z} was swept in between ±11 T. As seen, Δ shows an equal behavior with B_{z} at both polarities, which allow us to rule out hysteresis phenomena and nonsymmetric behavior. This symmetric tworegime behavior of Δ(B_{z}) has been found in most of the SoD clusters where the B_{z}dependent STS was measured, from which we can extract an average Δ deviation at zero field of \(\leftS\right=0.15\pm 0.02\,{{{{{{\rm{mV}}}}}}}\) (see histogram in Fig. 5e) and an average field for the nonlinear to linear transition at \(\bar{{B}_{T}}=0.45\pm 0.07\,{{{{{\rm{T}}}}}}\) (see SI for other dI/dV sets).
The observation of a split Kondo peak at zero magnetic field is inconsistent with the behavior of isolated impurities and reveals that the temperature has been lowered enough for lattice coherence effects to set in. Further analysis is nevertheless required to determine the nature of the ground state, and where it lies in the Doniach phase diagram. This analysis can be framed in terms of a periodic Anderson model, where a periodic array of Anderson impurities corresponding to the 1T SoD moments is coupled to the folded metallic bands of the 1H layer
Here \({c}_{i\sigma }^{+}\) creates an electron with spin \(\sigma\) in metal site \(i\) on the H layer, \({f}_{i\sigma }^{+}\) creates an electron on the SoD Wannier orbital \(i\) on the T layer with \({n}_{f\sigma,i}={f}_{i\sigma }^{+}{f}_{i\sigma }\tfrac{1}{2}\), \({t}_{{ij}}\) are the metal hoppings, \({\epsilon }_{0}\) is the flat band energy, V is the Kondo hybridization between the f orbital and the c orbital directly below it, and U is the effective Hubbard interaction for the flat band. Since there is one moment per 13 metal atoms, this is a dilute Kondo lattice^{12}. In the large U limit where charge fluctuations are frozen, the flevel can be represented as a spin ½ exchangecoupled to the metal spins via a SchriefferWolf transformation, giving rise to the Kondo lattice model
where J_{K} = 2 V^{2}[1/(U+\({\epsilon }_{0}\))+1/\({\epsilon }_{0}\)] is the Kondo coupling, \({\vec{S}}_{f,i}\) is the effective moment at SoD site i, and \({S}_{c,i}\) = ½\({c}_{i}^{+}{\vec{\sigma }c}_{i}\) is the metal spin directly below the SoD center. The predictions of this model are well known: a Kondo resonance emerges at each impurity at \({T}_{K}=W{exp}(1/{{\rho J}_{K}})\), where \(\rho\) is the Fermi level DOS and W the metal bandwidth^{11,12}. As temperature is further lowered, two types of ground states can be realized: a fully Kondo screened paramagnet is realized at high J_{K}, while different types of magnetic states emerge at low J_{K}, possibly coexisting with partial Kondo screening, with quantum critical points separating these phases at values of order \({J}_{K}\rho \sim 1\) (refs. ^{1,13,14}). Knowledge of the dimensionless parameter \({J}_{K}\rho\) is therefore of key importance to interpret our results.
To estimate the values of the model parameters, we performed ab initio calculations of the band structures for the 1H, 1T, and 1H/1TTaSe_{2} structures (see SI). The 1H polytype presents a single halffilled dorbital band at the Fermi level, with a bandwidth of W = 1.2 eV and a Fermi level DOS of \(\rho\) = 2.5 eV^{−1} (we neglected the known 3 × 3 reconstruction of this band for simplicity, see SI). The 1T polytype displays the √13 × √13 reconstruction with a halffilled flatband of width 25 meV lying within the CDW gap of 0.55 eV and with its real space spectral weight concentrated in the central atom of the SoD pattern, all consistent with previous reports^{15,16,17}. Our calculation of the 1T/1H heterostructure reveals a set of bands that correspond to those of 1T and 1H sublayers, with a weak hybridization and weak charge transfer from the T to the H layer. This band structure can be fitted with the periodic Anderson model discussed above at U = 0 to obtain the metal dispersion and Kondo hybridization V. To do so, we first fitted the uncoupled 1H metal band, and then coupled it to a single flat band representing the 1T layer with constant hybridization V which was left as a fitting parameter (see SI for details). Good agreement with the ab initio bands was obtained for V in the range V = 15–20 meV. With the value of U obtained experimentally U = 208 ± 4 meV, which is consistent with ab initio estimates for the effective Hubbard U of the flat band in other 1T TMDs^{18,19}, we finally obtain assuming the symmetric limit of the Anderson model \(({\epsilon }_{0}=U/2)\), J_{K} = 8 V^{2}/U = 8–15 meV. Taken at face value, the product \({J}_{K}\rho=0.037\) then clearly indicates we should be deep into the magnetic side of the Doniach diagram. An order of magnitude estimate for \({J}_{K}\) can also be obtained from the measured Kondo temperatures^{5,6} \({T}_{K}=W{e}^{\tfrac{1}{{\rho J}_{K}}}\), which yields somewhat larger values \({J}_{{K\cdot }}\rho\) ~ 0.1, but still in the magnetic side.
Our Kondo lattice model can next be used to rationalize the magneticfield dependence of the split Kondo peak, which we will argue is also consistent with magnetic order, but not with a Kondo insulator. The essential lowenergy features in tunneling to the localized magnetic orbitals are known from meanfield^{20,21,22} and quantum Monte Carlo (QMC) studies of the Kondo lattice^{13,14,23,24}. In the meanfield picture, for energies much lower than U, the localized spin is represented by an auxiliary lowenergy fermion \(\widetilde{f}\) as \({\vec{S}}_{f}={\widetilde{f}}^{+}\vec{\sigma }\widetilde{f}\), and the Kondo exchange is decoupled as \({J}_{K}\vec{{S}_{c}}\cdot \vec{{S}_{f}}\to {J}_{K}[\widetilde{V}({\widetilde{f}}^{+}c+{c}^{+}\widetilde{f})+{\vec{S}}_{f}\cdot \langle \vec{{S}_{c}}\rangle+{\vec{S}}_{c}\cdot \langle \vec{{S}_{f}}\rangle ]\) with \(\widetilde{V}=\langle {c}^{+}\widetilde{f}\rangle\) the renormalized Kondo hybridization. This auxiliary fermion band is pinned to E_{F} and represents the lowenergy Kondo peak, which emerges at \(T \sim {T}_{K}\). Further lowering the temperature leads to a splitting of the Kondo peak of different origins, due to the different finite order parameters \(\widetilde{V},\langle \vec{{S}_{c}}\rangle,\langle \vec{{S}_{f}}\rangle\). When \({J}_{K} > {J}_{C}\) this splitting originates from the hybridization \(\widetilde{V}\) between the auxiliary fermion band and the metal and leads to a paramagnetic Kondo insulator. If \({J}_{K} < {J}_{C}\) the splitting rather originates from ferromagnetic (FM) or antiferromagnetic (AFM) order in \(\langle \vec{{S}_{c}}\rangle\)^{25,26}. A splitting of the Kondo peak is therefore not sufficient to make a claim about either ground state.
The Zeeman field dependence of the split peak is however very different in each case. In the Kondo insulator, the mean field bands are spindegenerate, and the Zeeman field shifts the spin up and down bands rigidly in opposite directions^{27,28} (see Fig. 1b). This gives rise to four peaks and an eventual closing of the gap^{29}. Our resolution clearly allows us to discard this scenario. In the magnetic phase, the main features in the \(\widetilde{f}\)fermion spectrum derive from the interplay between the exchange field derived from the metal magnetization \(\langle \vec{{S}_{c}}\rangle\) and the Zeeman coupling
where \({\mu }_{B}\) is the Bohr magneton and g = 4.3 is the measured gfactor. The gap of the \(\widetilde{f}\)fermion is obtained as \(\Delta={J}_{K}\langle \vec{{S}_{c}}\rangle {\mu }_{B}g\vec{B}\), from which several general statements can be made, even without knowing the full B_{z}dependent meanfield solution for \(\widetilde{V},\langle \vec{{S}_{c}}\rangle,\langle \vec{{S}_{f}}\rangle\). First, the Zeeman coupling polarizes \(\langle \vec{{S}_{f}}\rangle\), which leads to \(\langle \vec{{S}_{c}}\rangle\) of opposite sign due to the AFM Kondo coupling. If \(\vec{{S}_{c}}\parallel \vec{B}\), then the gap depends on B_{z} as \(\Delta={J}_{K}{S}_{c}({B}_{z}){\mu }_{B}g{B}_{z}\), (with \({S}_{c}({B}_{z}) < 0\) for \({B}_{z} > 0\)). Regardless of the form of \({S}_{c}({B}_{z})\), it is clear that \(\Delta\) will not be symmetric in B_{z} and will be zero when the Zeeman field flips the spin polarization, represented in Fig. 6a for the simplest case of constant \({S}_{c}\). However, if \(\vec{{S}_{c}}\perp \vec{B}\), \(\Delta=\sqrt{({J}_{K}^{2}{S}_{C}^{2}({B}_{z})+{\mu }_{B}^{2}{g}^{2}{B}_{z}^{2})}\), and Δ grows symmetrically with B_{z} (Fig. 6b, again for constant \({S}_{c}\)). Our observations are therefore consistent with inplane magnetic order (FM or AFM since these cannot be distinguished without dispersion for the \(\widetilde{f}\)fermion).
The nonlinear lowfield behavior of the gap can also be understood assuming a smooth interpolating function \(\langle \vec{{S}_{c}} \rangle ({B}_{z})= ({\cos (b)\,S}_{c}^{0},0,\, {\sin (b)S}_{c}^{\max })\) with \(b=\tfrac{\pi }{2}\tfrac{B}{{B}_{c}}\), which interpolates between inplane and outofplane order that saturates at \({B}_{c}=0.5\) T. \({S}_{c}^{0}\) can be obtained from the measured \(\Delta\) = 1.2 meV at zero field and \({J}_{K}=15\) meV as \({S}_{c}^{0} \sim 0.066\), while \({S}_{c}^{\max }\) is the intercept of the linear fit with B = 0 T (see Fig. 5b), which is on average \(\Delta+S\) = 1.35 meV and gives the value of \({S}_{c}^{\max }\,\)= 0.077. With these parameters, we obtain Δ(B_{z}) shown in Fig. 6c for different values of \({S}_{c}^{0}\). The qualitative nonlinear behavior of the experiment is well reproduced with this model when \({S}_{c}^{0} \, < \, {S}_{c}^{\max }\), with a faster growth below \({B}_{c}\) and a slower one above it. In some instances, we have also observed the opposite behavior (see SI), which is obtained for \({S}_{c}^{0} > {S}_{c}^{\max }\) in the model. The nonlinear behavior is therefore supportive of a Zeemaninduced transition from inplane to outofplane order.
Within the Kondo lattice model, we have argued that our observed peak splitting indicative of lattice coherence, and more consistent with a magnetically ordered state than with a Kondo paramagnet. As a further argument that coherence has been reached in our system, we now discard a last possible scenario. If the isolated substrate 1HTaSe_{2} was magnetically ordered by itself, isolated moments at SoD sites would display a splitting due to the exchange coupling with a ferromagnet, as observed with isolated spins^{30,31}. To put this hypothesis to test, we have measured the Kondo resonance in isolated CoPC impurities placed on top of the 1HTaSe_{2} substrate (see SI). A sharp Kondo peak is observed without any splitting at 340 mK, the same temperature at which the 1T/1H structure does show a splitting. This clearly rules out a magnetism in 1HTaSe_{2} (consistent with previous Xray magnetic circular dichroism measurements^{5}), and leaves a coherent lattice state as the most likely explanation for the split Kondo peak.
Discussion
While 1HTaSe_{2} is not magnetic, isolated 1HTMD metals are predicted to be close to a magnetic instability with anomalously large spin susceptibilities, which may contribute to the mechanism selecting inplane vs out of plane order. For example, isoelectronic NbSe_{2} shows two leading instabilities, an inplane AFM (or spinspiral) with period 45 lattice constants^{12,32,33}, and a subleading fully ferromagnetic state^{33,34}. The periodicity of the inplane AFM state is very close to that of the SoD periodicity, which makes the emergence of inplane order natural. Nevertheless, the subleading ferromagnetic instability makes the moments relatively soft for tilting out of the plane. Our experiment therefore reveals an interesting connection between the magnetic states of the 1T/1H Kondo lattice and the magnetic fluctuations of the isolated H layer.
Our work demonstrates that the coherence regime in the artificial 2D Kondo lattice realized in 1T/1H TMD heterobilayers can be accessed experimentally. Our observations place the 1T/1HTaSe_{2} system likely on the magnetic side of the Doniach phase diagram. The realization of a coherent 2D Kondo lattice with magnetic order enables a model platform where soughtafter Kondo phenomenology can be studied with unprecedented resolution and tunability. Several avenues are now opened to fully characterize and tune this versatile Kondo platform, for example by using probes with magnetic sensitivity such as spin susceptibility or spinresolved STS, which can offer detailed information about the exact magnetic ground state. Furthermore, we expect that the dimensionless parameter \({J}_{{K\cdot }}\rho\) can be tuned by either electrostatic or chemical doping (which can induce large changes in \(\rho\) due to the proximity to a van Hove singularity) or by a displacement field, which introduces an interlayer bias and changes J_{K} through its dependence on \({\epsilon }_{0}\), so that the critical point can be approached. This offers an unprecedented window to access the quantum criticality regime, and potentially to induce unconventional superconductivity^{35,36}.
Methods
Growth of 1TTaSe_{2}/1HTaSe_{2} heterobilayers
1TTaSe_{2}/1HTaSe_{2} heterobilayers were grown on BLG/SiC (0001) substrates in a twostep process in our homemade ultrahigh vacuum molecular beam epitaxy (UHVMBE) system under base pressure of ~2 × 10^{−10} mbar. First, uniform bilayer graphene was prepared by direct annealing 4HSiC (0001) at a temperature around 1400 °C for 35 min. Second, the asgrown BLG/SiC (0001) substrates were maintained around 550 °C to grow monolayer 1HTaSe_{2} by coevaporation of highpurity Ta (99.95%) and Se (99.999%) with a flux ratio (Ta:Se) of ~1:30. The growth rate for TaSe_{2} was 2.5 ho/monolayer. Third, the temperature of the substrate was then increased to 640 °C while keeping evaporation parameters unchanged to obtain 1TTaSe_{2}. After the growth, the samples were kept annealed in Se environment for 2 min, and then immediately cooled down to room temperature. The growth was monitored in situ by reflection highenergy electron diffraction. A ~10nmthick Se layer was deposited on the prepared sample before taking it out of UHV conditions for further exsitu UHVSTM measurements^{37,38}. The Se capping layer was subsequently removed in the UHVSTM by annealing at ~300 °C for 40 min. After this process, the typical morphology of our samples, as seen in the STM, is shown in the Supplementary Figure 1.
STM/STS measurements and tip calibration
STM/STS data were acquired in a commercial UHVSTM system equipped with perpendicular magnetic fields up to 11T (Unisoku, USM1300). The measurements were carried out at temperatures between 0.34 K and 4.2 K with Pt/Ir tips. To avoid tip artifacts in our STS measurements, the STM tips were calibrated using a Cu(111) surface as reference. We also performed careful inspection of the DOS around E_{F} to avoid the use of functionalized tips showing strong variations in the DOS. The typical lockin a.c. modulation parameters for low and largebias STS were 20–50 µV and ~1–2 mV at f = 833 Hz, respectively. The energy resolution of the STM instrument at 0.34 K has been tested in bulk Pb(111) before carrying out this experiment. We used Pt/Ir tips for the STM/STS experiments. STM/STS data were analyzed and rendered using WSxM software^{39}.
Density functional calculations
We performed ab initio calculations using densityfunctional theory (DFT) within the planewave Quantum Espresso package^{40,41} using the Perdew–Burke–Ernzerhof (PBE)^{42} parametrization of the exchangecorrelation. We have used an ultrasoft pseudopotential with 13 electrons in the valence for Ta and a norm conserving pseudopotential with 6 electrons in the valence for Se. We used a kinetic energy cutoff of 45 Ry and a charge density one of 450 Ry. The structure adopted has a lattice parameter of 3.48 Å in the unit cell of the high symmetry phase and for that phase we used a 40 × 40 × 1 k grid and a MethfesselPaxton smearing of 0.005 Ry for the Brillouin zone integrals^{43}. For the supercell 1T in the CDW phase, we used a denser 24 × 24 × 1 k grid to have a good accuracy for the flat band close to the Fermi energy. We assumed the same lattice constant for the H and the T polytypes, and for simplicity, we neglect the 3 × 3 CDW of the H phase. The 3 × 3 CDW in TaSe_{2} is weak and does not lead to a dramatic change in the band structure or DOS. Including it in a commensurate cell with the √13 × √13 CDW of the T layer would require a prohibitively large unit cell.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
M.M.U. acknowledges support by the ERC Starting grant LINKSPM (Grant 758558) and by the grant no. MAT201788377C21R funded by MCIN/AEI/10.13039/501100011033. R.H. acknowledges funding support for project MAGTMD from the European Union’s Horizon 2020 research and innovation program under the Marie SklodowskaCurie grant agreement No 101033538. F.J. acknowledges funding from the grant PGC2018101988BC21 by MCI/AEI/10.13039/501100011033 and by the European Union, and from the Diputación de Gipuzkoa through Gipuzkoa Next (grant 2021CIEN00007001). H.G. acknowledges funding from the EU NextGenerationEU/PRTRC17.I1, as well as by the IKUR Strategy under the collaboration agreement between Ikerbasque Foundation and DIPC on behalf of the Department of Education of the Basque Government.
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M.M.U. conceived the project. W.W. carried out the MBE growth, the morphology characterization of the samples with the help of R.H., P.D. and S.S. W.W. measured and analyzed the STM/STS data with the help of R.H., P.D., S.S. and M.M.U. S.S. and H.G. measured and analyzed the CoPC/1HTaSe_{2} system. A.M. and I.E. carried out the ab initio calculations. M.M.U. supervised the project. F.J. provided the theoretical support and participated in the interpretation of the experimental data. M.M.U. and F.J. wrote the paper with help from the rest of the authors. All authors contributed to the scientific discussion and manuscript revisions.
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Wan, W., Harsh, R., Meninno, A. et al. Evidence for ground state coherence in a twodimensional Kondo lattice. Nat Commun 14, 7005 (2023). https://doi.org/10.1038/s41467023428034
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DOI: https://doi.org/10.1038/s41467023428034
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