Abstract
Adsorption of magnetic transition metal atoms on a metal surface leads to the formation of Kondo states at the atom/metal interfaces. However, the significant influence of surrounding environment presents challenges for potential applications. In this work, we realize a novel strategy to regularize the Kondo states by moving a CoPc molecular mold on an Au(111) surface to capture the dispersed Co adatoms. The symmetric and ordered structures of the atommold complexes, as well as the strong d_{π}–π bonding between the Co adatoms and conjugated isoindole units, result in highly robust and uniform Kondo states at the Co/Au(111) interfaces. Even more remarkably, the CoPc further enables a fine tuning of Kondo states through the molecularmoldmediated superexchange interactions between Co adatoms separated by more than 12 Å. Being highly precise, efficient and reproducible, the proposed molecular mold strategy may open a new horizon for the construction and control of nanosized quantum devices.
Introduction
Magnetic atoms are important building blocks of nanodevices for applications in quantum information and quantum computation. For realizing ondemand design and fabrication of nanodevices, it is crucial to achieve precise manipulation and tuning of local quantum states in these magnetic nanostructures^{1,2,3,4,5,6,7}. This thus requires the characteristic quantum states stay robust upon variation of surrounding environment, and respond regularly and consistently under external manipulation.
Adsorption of a transition metal (TM) atom on a metal surface could lead to the formation of Kondo states at the atom/metal interface under sufficiently low temperatures, which has been affirmed by the substantially enhanced electric conductance across the interface^{8,9}. Physically, the Kondo states originate from the screening of the local spin of d electrons on the TM atom by the spins of itinerant electrons in the metal substrate. Recently, the precise control and tuning of Kondo states have become a focus of experimental efforts^{10,11,12,13,14,15,16,17}. However, it is generally observed that the Kondo features vary greatly with the local chemical environment surrounding the TM atom, such as the coordinating ligands, the chemical dopants, the metal substrate, etc. This makes it difficult to produce Kondo states at atom/metal interfaces with regular sizes and shapes.
It is wellknown that Kondo states emerge at a Co atom adsorbed on the fcc or hcp domain of Au(111) surface, whereas a Co atom adsorbed on the domain wall does not exhibit Kondo signature^{8}. This indicates that the interaction between the Co atom and the gold substrate depends sensitively on the atomistic structure of the surface. Moreover, the Kondo states centered at a Co adatom can be significantly influenced by another Co adatom in its vicinity, because of the direct^{9} or indirect^{14,18} interatomic interactions. For instance, the substratemediated Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between two Co adatoms can change from antiferromagnetic to ferromagnetic as the interatomic distance varies, leading to enhanced or suppressed Kondo screening^{4,19}.
In many experiments, a TM adatom is bound to organic ligands to form an organometallic compound^{20,21,22,23}, and its local electronic configuration is significantly influenced by the surrounding. However, the related Kondo states do not necessarily have regular shapes, because the ligand field and the local spin distribution are highly susceptible to variations in the surrounding environment. For instance, for an iron phthalocyanine (FePc) molecule adsorbed on the Au(111) surface, the dI/dV spectra measured at the ontop adsorption site exhibit Kondo signatures that are distinctly different from that at the bridge site^{20}. In addition, the presence of a nearby FePc molecule can lead to conspicuous broadening or splitting of the Kondo conductance peak, which is again caused by the substratemediated RKKY interaction between the two local spin moments^{21}.
The above experimental observations indicate that the local chemical environment has a profound influence on the Kondo states formed at atom/metal interfaces. However, firstprinciplesbased theoretical study on the Kondo states has remained rather scarce. This poses a serious challenge for potential practical applications—is there a way to regularize the Kondo states, so that they could exhibit uniform features that are insensitive to the variation of environment? To solve this problem, in this paper we propose a novel strategy, with which we successfully realize the regularization of Kondo states centered at the Co adatoms on an Au(111) surface.
Our strategy is motivated by the experience that a regular shape can be attained by using a rigid mold. Following this idea, we propose to use a planar molecule with conjugated rings to capture the dispersed TM adatoms on the metal surface. Each conjugated ring also serves as a mold, which regularizes the local spin distribution as well as the resulting Kondo states centered at the captured TM adatom. Consequently, the influence of environment beyond the range of the planar molecule is substantially weakened.
In this work, we choose to use a metal phthalocyanine (MPc) molecule, which possesses four isoindole units. Each isoindole unit can capture one Co adatom through the strong bonding interaction between the conjugated πorbitals on the isoindole and the d orbitals on the Co adatom. More interestingly, if there is more than one isoindole unit hosting a Co adatom, longrange superexchange interactions may emerge between these spatially separated Co adatoms, which allows for finetuning of Kondo states in a controllable manner.
Results
Construction of K_{n}(CoPc) on Au(111) surface
Figure 1 illustrates the experimental procedure using an MPc molecule to capture the Co atoms dispersed on the Au(111) surface. As shown in Fig. 1a, a CoPc molecule exhibits a clear fourlobe pattern in the scanning tunneling microscopy (STM) image, with each lobe representing an isoindole unit. Pushed by the atomically sharp STM tip, the CoPc molecule can be quite freely moved on the surface. When the CoPc is pushed towards and finally in contact with a Co adatom, one of its four lobes captures the Co adatom and thus forms an atommolecule complex (referred to as K_{1} hereafter). In K_{1} the Co atom is located underneath the molecular plane, and hence it interacts directly with both the isoindole unit and the gold surface. The K_{1} complex can also be moved around on the surface, while its structure remains stable (see Fig. 1b). This also indicates that the Co atom is bound more tightly to the isoindole unit than to the surface.
Manipulated by the STM tip (see Supplementary Movie 1), a CoPc can capture up to four Co adatoms. Figure 1c demonstrates the sequential formation of the K\({}_{2^{\prime} }\), K_{3}, and K_{4} complexes from K_{1}. Here, K_{n} denotes the complex with n Co adatoms captured by a CoPc. K_{2} and K\({}_{2^{\prime} }\) are isomers, in which the two Co adatoms locate at the ortho and para positions, respectively. During the process, the CoPc molecule plays the role of PacMan, while the Co adatoms are “devoured” like PacDots. The atomistic structures of these complexes are sketched in Fig. 1d, and the presence of the Co adatoms are verified by the topographic images depicted in Fig. 1e. Figure 1f depicts the Kondo maps of these complexes (see Supplementary Figs. 1 and 2). Clearly, a Kondo cloud emerges at the outer region of each occupied isoindole unit, while the cobalt ion at the center is Kondo inactive. It is remarkable that all the Kondo clouds have a similar crescent shape, except in K\({}_{2^{\prime} }\) where the two clouds appear more isotropic. This thus highlights the strong regularization effect of the CoPc molecular mold.
Experimental measurement of dI/dV spectra and T _{K}
We now examine more closely the Kondo characteristics by investigating the measured differential conductance (dI/dV versus V) spectra. As shown in Fig. 2a, the measured spectra of a bare Co adatom exhibit a Fano line shape depending sensitively on the adsorption domain. The Fano line shape originates from the quantum interference between two electron conduction channels^{24}: the throughspace and throughatom channels. In contrast, the measured spectra of a K_{1} complex exhibit a singlepeak structure independent of the adsorption domain. This is because the CoPc molecule blocks the throughspace tunneling channel and thus quenches the Fano interference^{24,25}. The Kondo temperature T_{K} is extracted from the width of the measured Kondo conductance peak (Γ), followed by deconvolution of thermal and instrumentinduced broadening (see Methods).
From Fig. 2a, an isolated K_{1} has an almost constant T_{K} of about 122 K. By manipulating the STM tip, a K_{1} can be moved towards a bare Co adatom. Throughout this process, the spectral line shape of the K_{1} remains largely unchanged, while the T_{K} exhibits a weak oscillation around the constant value (see Fig. 2b and Supplementary Fig. 3). Such an oscillation is supposed to originate from the substratemediated RKKY interaction between the Co adatom in the K_{1} and the nearby bare Co adatom^{4,19}, and it is greatly suppressed with the nearby bare Co adatom captured by a CoPc mold. This affirms that the strong interaction between a Co adatom and the associated isoindole unit on the CoPc gives rise to highly regular and robust local spin distribution, so that the Kondo states are hardly affected by the variation of environment beyond the phthalocyanine ring.
For a K_{n} complex hosting two or more Co adatoms, the dI/dV spectra measured at each of the Co adatoms are almost identical (see Supplementary Fig. 2). While the spectral line shape of a K_{n} (n ≥ 2) much resembles that of a K_{1}, the Kondo peak of the former is somewhat wider, suggesting a higher T_{K} (see Fig. 2c). Figure 2d gives an overview of T_{K} for all members of the K_{n} family. Two main features are easily recognized: (1) T_{K} increases linearly with n for n = 1, 2, 3, 4. (2) T_{K} of K\({}_{2^{\prime} }\) is distinctly higher than that of its isomer K_{2}, and lies outside the above linear relationship. These features indicate that the Kondo state centered at a captured Co adatom is affected subtly but regularly by the other Co adatoms confined within the same phthalocyanine ring. This lays the foundation for finetuning the Kondo states formed at atom/metal interfaces through precise arrangement of the TM adatoms.
Theoretical calculations of dI/dV spectra and T _{K}
The above experimental manipulations and measurements affirm the efficacy of CoPc molecular mold for regularizing the Kondo states at Co/Au(111) interfaces. For practical purposes, it is crucial to elucidate the regularization mechanisms, e.g., what is the origin of the regular spectral line shapes, and what leads to the intriguing T_{K} versus n relationship. To provide theoretical insights into these questions, we carry out firstprinciplesbased calculations by a combination of density functional theory (DFT)^{26,27} and hierarchical equations of motion (HEOM)^{28,29,30} approach. In Fig. 2d, the T_{K} of K_{n} extracted from the calculated dI/dV spectra are compared against the experimental data (to be elaborated later).
Figure 3a depicts the optimized geometries and spin density distribution of a bare Co atom and a K_{n} complex (\(n=1,2^{\prime} ,3\)) adsorbed on the Au(111) surface. Apparently, the planar CoPc mold is always parallel to the surface, and the captured Co adatom is right beneath the sixmember ring of an isoindole unit. Such a geometry remains unchanged upon moving the K_{n} to different sites on the surface (Supplementary Figs. 4–6). The spinunpaired electrons reside predominantly on the Co adatoms with only a small fraction on the central cobalt ion. Thus, each Co adatom becomes the center of a Kondo cloud, whereas the cobalt ion on the CoPc mold is Kondo inactive.
From Fig. 3a it is also noted that the local spin density of a bare Co adatom exhibits an appreciable change at different adsorption sites. In contrast, the local spin of a captured Co adatom in a K_{n} is hardly affected by the surrounding environment (see also Supplementary Figs. 7 and 8). These results are consistent with the experimental findings as displayed in Fig. 2a, b. Therefore, the CoPc mold indeed regularizes and preserves the local spins of the Co adatoms, and thus facilitates the formation of uniform Kondo states at the K_{n}/Au(111) interfaces.
By analyzing the projected density of states (PDOS) of captured Co adatoms, we obtain a schematic diagram showing the relative energy, broadening and electron occupancy of each d orbital; see the upper right inset of Fig. 3b. The local electron configuration of a captured Co adatom is determined to be \({d}_{xy}^{1.6}{d}_{{x}^{2}{y}^{2}}^{\ 1.6}{d}_{{z}^{2}}^{1.8}{d}_{xz}^{1.3}{d}_{yz}^{1.2}\) (see Supplementary Table 1), and the local spin moment originates mainly from the d_{π} (d_{xz} and d_{yz}) orbitals. Thus, a captured Co adatom is considered to be in a local \(S=\frac{1}{2}\) state, in clear contrast to the local S = 1 state of a bare Co adatom^{31}.
Because of their common symmetry, the d_{π} orbitals on a Co adatom and the conjugated πorbitals on an isoindole unit of the CoPc mold interact strongly with each other. The resulting d_{π}–π bonds further consolidate the atommold complex, and give rise to the dumbbellshaped spin density distribution (see Fig. 3a). The physical origin of the regularization effect is thus clear. Confined by the CoPc mold, the local electronic structure of a Co adatom is significantly reshaped and regularized by the strong d_{π}–π bonds, leading to a local spin state distinctly different from that of a bare Co adatom.
The strength of Kondo correlation depends critically on how strongly the local spin on a Co adatom is screened by the electronic spins in the surrounding environment. For the K_{n}/Au(111) composites, the Kondo screening is realized via the hybridization of the Co d_{π} orbitals with the s orbitals of the nearby gold atoms. It is such s–d hybridization that leads to the broadening and splitting of the Co d_{π} orbitals as shown in Fig. 3b and Supplementary Fig. 9. The hybridization strength is characterized by Δ_{s}, which is taken as the halfwidth of the split PDOS peaks that constitute the main contribution to the local spin moment.
With more Co adatoms captured by a same CoPc, the split peaks of PDOS are slightly more broadened. This signifies a perturbation by the other Co adatoms confined within the same CoPc mold, possibly through the substratemediated RKKY interactions. The magnitude of such a perturbation is thus roughly proportional to the number of Co adatoms inside the atommold complex. Intriguingly, the split peaks of a K\({}_{2^{\prime} }\) is somewhat broader than a K_{2}, implying the symmetry of the complex has a subtle influence on the strength of RKKY interaction. Accordingly, Δ_{s} increases linearly with n except for \(n=2^{\prime}\) (see the lower right inset of Fig. 3b). The apparent resemblance between the Δ_{s} versus n and T_{K} versus n relationships (cf. Fig. 2d) affirms the key importance of Δ_{s} to the characteristic features of the Kondo states^{32}.
To quantitatively describe the Kondo effect, we employ quantum impurity models which explicitly include the electron–electron interactions. We first assume the Kondo clouds centered at different Co adatoms are independent of each other, and thus a tip/K_{n}/Au(111) junction can be represented by a singleorbital Anderson impurity model (AIM)^{33}. The calculated dI/dV spectra agree remarkably with the experimental measurements (see Supplementary Fig. 10). The T_{K} of K_{n} are determined based on the height or the width of the Kondo conductance peaks (see Methods and also Supplementary Figs. 11 and 12), and the theoretical values accurately and consistently reproduce the experimental data (see Fig. 2d).
With the singleorbital AIM, the T_{K} of a K\({}_{2^{\prime} }\) is predicted to be 157 K (labeled by K\({}_{2^{\prime} }^{* }\) in Fig. 2d), only slightly higher than that of a K_{2}, yet considerably lower than the experimental value of 191 K (Supplementary Fig. 13 and Supplementary Table 3). This indicates that the use of a singleorbital AIM is inadequate, because the two Kondo clouds at the K\({}_{2^{\prime} }\)/Au(111) interface are not independent to each other.
Influence of longrange superexchange interaction on T _{K}
To gain deeper insights into the unusually high T_{K} of a K\({}_{2^{\prime} }\), we perform additional experiments by replacing CoPc with CuPc and H_{2}Pc molecules (see Supplementary Note 12 for details). Similar atommold complexes are constructed by manipulating the STM tip. Unlike the case of CoPc, the measured T_{K} of K\({}_{2^{\prime} }\)(CuPc) and K\({}_{2^{\prime} }\)(H_{2}Pc) are only slightly higher than their isomers K_{2}(CuPc) and K_{2}(H_{2}Pc) by 4–5 K (see Supplementary Figs. 14 and 15). These findings highlight the unique role of the central cobalt ion in the CoPc, which may serve as a hub to assist the two Co adatoms in the para positions to interact with each other.
From the calculated electronic structures of K\({}_{2^{\prime} }\)(CoPc)/Au(111) and K_{4}(CoPc)/Au(111) composites (see Supplementary Figs. 16 and 17), a delocalized Kohn–Sham orbital, which links the d_{π} orbitals on the two distantly separated Co adatoms via the πorbitals on the isoindole units and a d_{π} orbital on the central cobalt ion, is recognized (see Fig. 4a, b). This thus provides a channel for the superexchange (SE) interaction^{34,35} between the two separated local spins. However, such a channel breaks down in a K\({}_{2^{\prime} }\)(H_{2}Pc) or a K\({}_{2^{\prime} }\)(CuPc) on the Au(111) surface, because the bridging orbital at the central hub is absent or energetically incompatible (see Fig. 4c, d). Consequently, the longrange SE interaction is strong in a K\({}_{2^{\prime} }\)(CoPc), but either absent or rather weak in a K\({}_{2^{\prime} }\)(H_{2}Pc) or K\({}_{2^{\prime} }\)(CuPc).
To assess the influence of SE on the Kondo states in a K\({}_{2^{\prime} }\), a twoorbital AIM is employed, which explicitly includes the molecularmoldmediated SE interaction between the two Co adatoms. Accurate computation of the strength of SE interaction (Δ_{SE}) by ab initio quantum chemistry methods is highly desirable yet rather challenging. Instead, Δ_{SE} is determined by exploring the variation of T_{K}. To retrieve the experimental value of T_{K} = 191 K (see the inset of Fig. 4), we have Δ_{SE} = 123 meV for a K\({}_{2^{\prime} }\)(CoPc), which amounts to an effective ferromagnetic coupling^{36} of J_{eff} ≈ 4 meV between the two local spins separated by as far as 12.5 Å (see Supplementary Notes 6 and 15 for details).
As shown in Fig. 4, the T_{K} of a K\({}_{2^{\prime} }\)(CoPc) exhibits a nonmonotonic dependence on Δ_{SE}. Such a phenomenon can be understood as follows. As Δ_{SE} increases from zero, the local spin on a Co adatom starts to feel the ferromagnetic interaction from the other Co adatom, and thus it is screened less strongly by the surrounding environment, which leads to the attenuated Kondo correlation. However, when Δ_{SE} reaches a certain value (see Supplementary Fig. 18), the whole K\({}_{2^{\prime} }\)(CoPc) complex favors an S = 1 state, resulting in a substantially enlarged spin moment for the total complex. Consequently, the Kondo temperature rises again with further increasing of Δ_{SE}.
Among all the other K_{n}, only K_{4}(CoPc) possesses a delocalized SE channel. This indicates that the structural symmetry of the whole complex is vital for the existence of SE. Nevertheless, the SE in a K_{4} is much weaker than in a K\({}_{2^{\prime} }\) (see Supplementary Fig. 19), because the delocalized orbital involves also the two disconnected Co adatoms at the ortho positions. It is estimated that the SE could lead to a minor change in T_{K} by ~4 K for a K_{4}.
Besides the strength of Kondo correlation, the SE also affects the spatial distribution of Kondo states. As noted earlier, the Kondo clouds at the K\({}_{2^{\prime} }\)/Au(111) interface appear to be more isotropic than in the other composites, which is possibly due to the strong longrange SE.
Discussion
To summarize, unlike all the previous studies in which the Kondo states formed around an adsorbed TM atom always vary sensitively to the surrounding environment, we demonstrate that, the proposed strategy of using a symmetric and conjugated molecular mold gives rise to highly robust and regularized Kondo states. Adjusting the chemical composition of the molecular mold further allows for a finetuning of the Kondo characteristics. The resulting atommold complexes can move freely on the gold surface and exhibit remarkably uniform Kondo features. Therefore, they may serve as standard building blocks for the design and fabrication of novel quantum devices.
Methods
Scanning tunneling microscopy experiments
The experiments are carried out with a low temperature STM (Omicron) using Au(111)/mica substrates in vacuum under a base pressure of 3 × 10^{−11} Torr. The Au(111) surface is cleaned by repeat cycles of Ar ion sputtering at 800 V for 15 min and annealed at 600 K. Submonolayers of Co atoms and CoPc molecules are coevaporate by ebeam evaporation and by sublimation, respectively, on an Au(111) substrate. The substrates are placed in situ on the cryostat of the microscope with a temperature of about 6.7 K. The samples are then investigated in situ at 5 K. The chemically etched tungsten tip is carefully cleaned by circular Argon ion sputtering at 800–1000 V for about 5–10 min and by field emission lasting for about 30 s under a sample bias of around −100 V with emission current of 2 μA in each cycle^{37,38}.
In the consecutive formation of K_{n} complexes, the manipulation conditions are 50 mV and 10 nA, while the imaging conditions are 1 V and 0.2 nA for Fig. 1a–c. For the measurement of topographic images and simultaneously acquired Kondo maps in Fig. 1e–f, the imaging conditions are −15 mV and 2 nA with a sinusoidal modulation of 2 mV and 730 Hz. The dI/dV spectra in Fig. 2 are recorded with a lockin method. The acquisition conditions are −150 mV and 2 nA with a sinusoidal modulation of 2 mV and 730 Hz. The Kondo maps and dI/dV spectra of the K_{n} complexes are acquired at different domains of the Au(111) reconstruction surface, including hcp and fcc domains and the domain walls.
The Kondo line shapes in the dI/dV spectra are fit to a single Fano function^{39} as follows:
where ε = (eV − ε_{K})/Γ, ε_{K} is the center of the Kondo resonance, Γ is the halfwidth at halfmaximum (HWHM) of the Kondo resonance peak, and q is the asymmetry parameter. The Kondo temperature T_{K} is extracted from Γ by deconvolution of the thermal and instrumentinduced broadening as follows^{40,41,42,43,44}:
Here, k_{B} is the Boltzmann’s constant, T = 5 K is the environmental temperature, the coefficient λ assumes an empirical value of 2.7 as adopted in previous experiments^{42}, and V_{m} = 1 mV is the amplitude of the modulation voltage (corresponding to a peaktopeak value of 2 mV).
It is worth pointing out that in principle the thermal and instrumentinduced broadening of any spectroscopic feature should be examined by a convolution of the expected signal with a proper broadening kernel^{40,41,42,43}. In practice it is often more convenient to use the approximate formula of Eq. (2).
Density functional theory method
Firstprinciples calculations are performed by using the spinpolarized density functional theory (DFT) method implemented in the Vienna ab initio simulation package (VASP)^{45}. The generalized gradient approximation developed by Perdew, Burke and Ernzerholf (PBE)^{46} is chosen for the exchangecorrelation functional. The projected augmented wave method is adopted with the energy cutoff of 400 eV. The zero damping DFTD3 method of Grimme^{47} is used to improve the description of van der Waals interactions.
During the structural optimizations, a slab model including three layers of Au atoms is used to represent the Au(111) surface, with each layer containing 56 Au atoms, A vacuum space of 16 Å is used to eliminate the interactions between different slabs. All the atoms except those in the bottom two Au layers are fully relaxed until the residual force on every atom was <0.02 eV Å^{−1}. The convergence criterion is set to 1 × 10^{−5} eV for the total energy. Because of the large size of the supercell, the Γpoint approximation is adopted. Additional numerical tests have been carried out, which have verified that a larger slab model consisting of four atomic layers does not change the main results qualitatively.
Anderson impurity models
The AIMs^{33} which explicitly include the electron–electron Coulomb interaction and the Kondo effects are adopted to represent the tip/K_{n}/Au(111) junctions. The total AIM Hamiltonian is comprised of three parts:
where H_{imp} and H_{env} are the Hamiltonian of the magnetic impurity and that of its environment, respectively.
In particular, the environment is modeled by reservoirs of noninteracting electrons, i.e., \({H}_{{\rm{env}}}={H}_{{\rm{tip}}}+{H}_{{\rm{sub}}}={\sum }_{\alpha = {\rm{t,s}}}{\sum }_{\sigma }{\epsilon }_{\alpha k}{\hat{c}}_{\alpha k\sigma }^{\dagger }{\hat{c}}_{\alpha k\sigma }\), where H_{tip} and H_{sub} represent the STM tip (α = t) and the gold substrate (α = s), respectively. \({\hat{c}}_{\alpha k\sigma }^{\dagger }\)\(({\hat{c}}_{\alpha k\sigma })\) is the creation (annihilation) operator for a spinσ electron on the kth orbital in the αth reservoir. The impurityenvironment coupling has the form of \({H}_{{\rm{coup}}}={\sum }_{\alpha \sigma ki}{t}_{\alpha ki}\ {\hat{c}}_{\alpha k\sigma }^{\dagger }{\hat{d}}_{i\sigma }+{\rm{H.c.}}\), where {t_{αki}} are the coupling strengths between the ith impurity orbital and the kth reservoir orbital. The influence of electron reservoirs on the impurity is accounted for through the hybridization functions defined by \({\Lambda }_{ij\alpha }(\omega )\equiv \pi {\sum }_{k}{t}_{\alpha ki}{t}_{\alpha kj}^{* }\delta (\omega {\epsilon }_{\alpha k})\).
In a singleorbital AIM, the spinpolarized d_{π} orbitals on each captured Co adatom are represented by a single impurity orbital, which is described by
Here, \({\hat{n}}_{\sigma }={\hat{d}}_{\sigma }^{\dagger }{\hat{d}}_{\sigma }\), where \({\hat{d}}_{\sigma }^{\dagger }\) (\({\hat{d}}_{\sigma }\)) creates (annihilates) an electron of spinσ (σ = ↑ or ↓) on the impurity orbital of energy ϵ_{d}; and U is the intraorbital electron–electron interaction energy. The reservoir hybridization functions assume a Lorentzian form of \({\Lambda }_{\alpha }(\omega )=\frac{{\Delta }_{\alpha }}{2}\frac{{W}_{\alpha }^{2}}{{(\omega {\Omega }_{\alpha })}^{2}+{W}_{\alpha }^{2}}\), where Δ_{α} is the effective coupling strength between the impurity orbital and the αth reservoir (α = s or t), and Ω_{α} (W_{α}) is the band center (width) of the αth reservoir.
The K\({}_{2^{\prime} }\)/Au(111) composite is also represented by a twoorbital AIM, in which the d_{π} orbitals on each of the two Co adatoms are modeled by an impurity orbital, and the total impurity is described by
The coupling between the ith orbital and αth reservoir is characterized by the hybridization function \({\Lambda }_{ii\alpha }(\omega )=\frac{{\Delta }_{i\alpha }}{2}\frac{{W}_{\alpha }^{2}}{{\left(\omega {\Omega }_{\alpha }\right)}^{2}+{W}_{\alpha }^{2}}\). The CoPcmoldmediated SE interaction between the two remotely separated local spins is quantified by the offdiagonal hybridization functions \({\Lambda }_{12{\rm{s}}}(\omega )={\Lambda }_{21{\rm{s}}}(\omega )=\frac{{\Delta }_{{\rm{SE}}}}{2}\frac{{W}_{{\rm{s}}}^{2}}{{\left(\omega {\Omega }_{{\rm{s}}}\right)}^{2}+{W}_{{\rm{s}}}^{2}}\).
The values of the involving energetic parameters, including ϵ_{d}, U, Δ_{α}, Ω_{α}, and W_{α}, are extracted from the DFT calculation results (see Supplementary Table 2)^{37,48,49}.
Hierarchical equations of motion method
The HEOM method for fermionic baths^{29,30} implemented in the HEOM–QUICK program^{50} is employed to solve the stationary states of the AIMs, and to compute the key physical observables to compare directly with the experimental measurements. The detailed form of the HEOM can be found in ref. ^{29}. Its basic variables are the system reduced density matrix and a hierarchical set of auxiliary density operators (ADOs).
To ensure the numerical accuracy for ultralow temperatures, a recently developed Fano spectrum decomposition scheme^{51} is adopted to accurately unravel the reservoir correlation functions. Limited by the computer resources at our disposal, the lowest temperature accessed by the HEOM method is 30 K for the singleorbital AIM, with the truncation tier set to L = 4. The tunneling current across a tip/K_{n}/Au(111) junction under a given bias voltage V is extracted from the firsttier ADOs^{29}, and then the dI/dV spectrum is obtained with a finite difference analysis.
We employ two approaches to theoretically determine the Kondo temperature T_{K} of K_{n}/Au(111) composites. The first one makes use of an empirical scaling relation between the zerobias conductance \(G\equiv {\left(\frac{dI}{dV}\right)}_{V = 0}\) of a Kondo impurity and the environmental temperature T as follows^{52,53}:
Here, G_{0} is the conductance at T → 0, g_{b} is the background conductance due to electron transport through the nonKondo states, and s is a parameter whose value depends on the local spin state of the impurity. Particularly, s = 0.22 is adopted as suggested by previous calculations for a spin\(\frac{1}{2}\) impurity^{52,53,54}. The Kondo temperature T_{K} can thus be determined by fitting the calculated G versus T to Eq. (6).
The second approach is based on the relation of Eq. (2) from the Fermi liquid theory^{55}. This requires the calculation of the full dI/dV spectra for all the tip/K_{n}/Au(111) junctions. Since the environmental temperature T adopted in the calculation is somewhat higher than the experimental counterpart, calculation needs to be done at a series of temperatures. The Kondo temperature T_{K} is then extracted by fitting the resulting Γ versus T to Eq. (2) while taking λ as a tunable parameter (see Supplementary Note 10 for more details).
Data availability
The data that support the findings of this study are available from the authors on reasonable request; see author contributions for specific data sets.
Code availability
The codes that were employed in this study are available from the authors on reasonable request.
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Acknowledgements
The support from the National Key Research and Development Program of China (Grant No. 2016YFA0200600), the National Natural Science Foundation of China (Grant Nos. 21973086, 21573202, 21633006, 21688102, and 21603205), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB36000000), and the Anhui Initiative in Quantum Information Technologies (Grant No. AHY090000) is greatly appreciated. The authors thank Xiaoli Wang, Yu Wang, Yao Wang, Xiangzhong Zeng, and Haoqi Chen for helpful discussions. The computational resources are provided by the Supercomputing Center of University of Science and Technology of China and Tianjin Supercomputer Center.
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J.Y., X.Z., and B.W. conceived the project. L.Z. and J.L. performed the experiments. X.L. conducted the theoretical calculations. X.L., X.Z., B.L., J.Y., B.W., Y.L., J.H., P.G., A.Z., and L.Y. analyzed the experimental and theoretical data. X.L., X.Z., J.Y., and B.W. cowrote the paper. All authors discussed the results and commented on the paper.
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Li, X., Zhu, L., Li, B. et al. Molecular molds for regularizing Kondo states at atom/metal interfaces. Nat Commun 11, 2566 (2020). https://doi.org/10.1038/s41467020164026
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DOI: https://doi.org/10.1038/s41467020164026
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