Molecular molds for regularizing Kondo states at atom/metal interfaces

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 atom-mold complexes, as well as the strong dπ–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 molecular-mold-mediated 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 nano-sized quantum devices.

depict the topographic image and the differential conductance (dI/dV ) map of a K 1 complex adsorbed on the Au(111) surface, respectively. In the former picture, the isoindole unit hosting a Co adatom is more protruded in comparison with those isoindole units without a captured Co adatom, and the center of the protrusion reflects the actual adsorption site of the captured Co adatom. However, in the simultaneously acquired dI/dV map at the condition of −15 mV and 2 nA, an area with lower value appears. The corresponding dI/dV curves are obtained by placing the STM tip at the positions marked by the white points, as shown in Supplementary Figure 1(C). The Kondo cloud centered at a captured Co adatom has a crescent shape distribution at the outer region of an isoindole unit. As illustrated in Supplementary Figure 1(E), the curve a denotes the typical I-V behavior around a captured Co adatom, while the curve b (or c) denotes other sites far from the captured Co adatom. When we took dI/dV map at lower bias voltages, say at point 1 (or point 2), the slope of the curve a around the point 1 and point 2 is much smaller than that of curve b (or c), hence the dI/dV map gives lower value around the captured Co adatom than the other site far from it. For a maximal contrast, the dI/dV map in Supplementary  Figure 1 For a K n complex hosting two or more Co adatoms, the dI/dV spectra measured by placing the STM tip on top of different captured Co adatoms almost overlap each other; see Supplementary Figure 2. The central Co ion in any K n complex is clearly Kondo inactive. In general, the spectral line shape of a K n (n ≥ 2) much resembles that of a K 1 , but the zero-bias Kondo peak becomes broader as n increases from 1 to 4. Intrinsically, the Kondo peak in the dI/dV spectrum of a K 2 complex is much wider than that of a K 2 or a K 3 .
Supplementary Figure 2: The dI/dV spectra of (A) a K 2 , (B) a K 2 , (C) a K 3 and (D) a K 4 measured by placing the STM tip at the marked points. The insets in the upper-left and upperright corners of each panel are the topographic image and simultaneously acquired dI/dV map, respectively. Image size: 2.5×2.5 nm 2 .

Supplementary Note 3: Manipulation of a K 1 complex on the Au(111) surface
By manipulating the STM tip, a K 1 can be moved to the proximity of a bare Co adatom or another K 1 . Supplementary Figure 3(A) and (B) demonstrate the process that a K 1 is moved toward a target Co adatom. The distance between the captured Co adatom in K 1 and a bare Co adatom can reach 1.05 and 0.85 nm, respectively. Similarly, Supplementary  Figure 3(C) and (D) demonstrate the process that a K 1 is moved toward another target K 1 complex. The distance between two captured Co adatom in K 1 complexes can reach 0.96 and 0.84 nm, respectively. The variation of T K of the target Co adatom and the target K 1 complex during the manipulation processes are exhibited in Fig. 2  In our experiments, the Kondo temperature of a K n /Au(111) composite is measured by taking the following procedure. First, the dI/dV versus V spectrum is measured at a fixed temperature T . Then, the differential conductance spectrum is fit to the Fano line shape, from which the half width at half maximum (HWHM) of the Kondo conductance peak centered at zero bias (Γ) is extracted. Finally, Kondo temperature is determined by using the relation of Fermi liquid theory 1 with the thermal and instrument-induced broadening deconvoluted from the measured spectra: 2-5 Here, T is the environmental temperature; the coefficient λ is equal to π in the original Fermi liquid theory, 6 while it has assumed an empirical value of 2.7 in previous experiments; 4 and the last term in the square-root accounts for the instrument-induced peak broadening caused by the sinusoidal modulation voltage of the amplitude V m . 7 The influence of instrument on the measured Kondo temperature has been examined carefully in the literature. 2,3 Since our experiments are conducted at a low temperature T = 5 K by using a small modulation voltage (peak-to-peak value is 2 mV and hence the amplitude is V m = 1 mV), the thermal fluctuations (k B T ) and the modulation voltage (V m ) give rise to a rather minor contribution to the total broadening Γ. For instance, the measured broadening is Γ = 10.6 meV for the K 1 /Au(111) composite, we thus have k B T K ≈ Γ 2 − (1.16 meV) 2 − (0.87 meV) 2 = 10.5 meV, i.e., the influence of thermal and instrument-induced broadening amounts to a small correction of T K of only about 1.2 K. Even so, we have explicitly accounted for these minor contributions in Fig. 2  An isolated CoPc molecule has a planar geometry with the D 4h symmetry. It contains a Co ion at its center, which is coordinated with four nitrogen atoms and enclosed by an aromatic macrocycle. In the ground state, the molecule has a net spin of S = 1 2 due to a spin-unpaired electron occupied the d z 2 orbital of the central Co ion. When the CoPc molecule is adsorbed on the Au(111) surface, the molecule structure does not undergo a significant change and remains almost planar.  Figure 5 displays the optimized structure of a K 1 complex adsorbed at different sites on the Au(111) surface. The molecular plane of CoPc is parallel to the gold surface, with the six-member ring of each isoindole unit located right above a captured Co adatom. Such a geometry remains almost unchanged upon moving the complex to different sites or domains on the Au surface. For all the K n (n = 1, 2, 2 , 3, 4) complexes, our density functional theory (DFT) calculations affirm that the spin-unpaired electrons reside predominantly on the captured Co adatoms, with only a tiny amount on the Co ion of the CoPc mold.
Supplementary Figure 6 shows the PDOS of different parts of a K 1 complex with the captured Co adatom adsorbed at a hollow site on the Au(111) surface. Clearly, the PDOS of the d orbitals on the captured Co adatom exhibits conspicuous spin polarization. Influenced by the captured Co adatom, the PDOS of the Co ion in the CoPc mold also exhibits weak spin polarization, and so does that of the Pc macrocycle.
Supplementary Table 1 lists the electron occupation numbers of the d orbitals on the captured Co adatom in a K 1 /Au(111) composite, along with their contributions to the local spin moment S z . In distinct contrast to the Co ion in the CoPc mold which has a negligible S z , the captured Co adatom has an appreciable value of S z ≈ 0.5, which originates mainly from the d π (d xz and d yz ) orbitals. Thus, the Co adatom is considered to be in a local S = 1 2 state.
Note that a halfly occupied orbital often refers to the scenario that the orbital is occupied by a spin-up electron with no spin-down electron occupancy, and so the overall spin moment is S z = 1 2 . In contrast, for the captured Co adatom in a K 1 , each of the d π orbitals is occupied by roughly one spin-up electron but also by half a spin-down electron; see Supplementary  Table 1 as well as the inset of Fig. 3(b) in the main text. Because of the fractional (nonzero) occupation of spin-down electrons, each d π orbital contributes only about s z = 0.21 to the local spin moment, and it is the sum of s z of all the five Co d orbitals that amounts to the spin moment (S z = 1 2 ) of a purely halfly occupied orbital. For the K 2 /Au(111) composite in which two Co adatoms locate at the para positions of the Pc ligand, the local spins on the two distantly separated Co adatoms can be parallel or anti-parallel to each other, corresponding to a ferromagnetic (FM) or antiferromagnetic (AFM) spin-state of the composite, respectively.
Because of the long-range SE interaction between the two local spins mediated by the delocalized Kohn-Sham orbitals on the CoPc mold (see Supplementary Figure 4), there is a finite energy gap between the FM and AFM spin-states, Table 1: Electron occupation numbers for the d orbitals on the Co ion in the CoPc mold and on the captured Co adatom in a K 1 /Au(111) composite. Here, n occ = n ↑ + n ↓ and S z = 1 2 (n ↑ − n ↓ ), with n ↑ and n ↓ being the majority-spin and minority-spin occupation numbers, respectively.  may suffer from static correlation error, 9 calculation of J eff by more accurate ab initio quantum chemistry methods is appealing. This is however rather challenging because of the large size of the K 2 /Au(111) composite.  Figure 8(A) depicts the PDOS of d π (d xz and d yz ) orbitals of a bare Co adatom adsorbed at a hollow site and a bridge site on the Au(111) surface, and there is an appreciable distinction in the line shape of PDOS between these two sites. In contrast, for a captured Co adatom in a K 1 , the PDOS of d π orbitals exhibit a minor change upon the variation of adsorption sites; see Supplementary Figure 8(B). This indicates that the strong bonding between the Co d π orbitals and the conjugated π-orbitals of the isoindole unit has a predominant influence on the local electronic structure of the Co adatom. Consequently, regularized by the d π -π bonding interaction, the local electronic structure of the Co adatom is hardly affected by the chemical environment beyond the CoPc molecular mold. It should be mentioned that the top-site adsorption is also explicitly considered in the calculation, but the captured Co adatom always relaxes to a hollow site through geometric optimization. The influence of spin-orbit coupling (SOC) on the electronic structures of K n /Au(111) composites is examined by turning on non-collinear spin in the DFT calculation. It is found that aligning the local spin in different magnetization directions results in a rather small variation in the total energy (less than 0.3 meV per Co adatom), indicating that the SOC has a negligible effect on the local electronic structures of the K n complexes. This is also consistent with the designation of a local S = 1 2 state for each captured Co adatom. In a K n /Au(111) composite, each Co adatom interacts with its surrounding environment, which consists of an isoindole ligand of the CoPc molecular mold and the gold substrate. In the following, we will elaborate on how the environment affects the local electronic structure of the Co adatom. In particular, we scrutinize the evolution of the PDOS of d xz orbital, one of the d π orbitals which contribute predominantly to the formation of Kondo states. For an isolated Co atom, the PDOS of its d xz orbital exhibits a sharp peak; see Supplementary Figure 9(A). This affirms that the d xz orbital is an eigen energy level of the Co atom. In an isolated K 1 complex, the Co adatom binds to an isoindole ligand of the CoPc mold through the d π -π bonding interaction. The isoindole ligand field results in the rearrangement of the Co d orbitals and regularization of the local spin distribution (analogous to that depicted in Fig. 3(a) of the main text). On the other hand, it is evident from Supplementary Figure 9(B) that the PDOS of the d xz orbital retains a single-peak structure -the main peak amounts to 90% of the total area. This suggests that the d xz orbital of the Co atom largely preserves its atomic feature even under the influence of the ligand field.

Co ion in CoPc
Supplementary Figure 9(C) plots the PDOS of the d xz orbital of the Co adatom in a K 1 /Au(111) composite, whose line shape is distinctly different from those displayed in Supplementary Figure 9(A) and (B). The PDOS of the d xz orbital is no longer a single peak. Instead, it has a rather scattered distribution over a wide energy range. Moreover, there is an apparent correspondence between the peak positions of the PDOS of Co d xz orbital and those of s orbitals of the nearest gold atoms. Such a correspondence highlights a strong hybridization between the Co d xz orbital and the Au s orbitals. It is this hybridization that leads to the formation of Kondo states at the K 1 /Au(111) interface, since the latter is known to originate from the screening of local spin moment (contributed predominantly by the Co d π orbitals) by the spins of itinerant electrons in the metal substrate (contributed by the Au s orbitals).
In general, the broadening (or splitting) of an atomic d orbital reflects its interaction with the surrounding environment. As shown in Supplementary Figure 6, the d xy , d x 2 −y 2 , and d z 2 orbitals of the Co adatom have much smaller broadening under the influence of the environment. Considering also the fact that these orbitals only present a minor contribution to the local spin moment (see Supplementary Table 1), we conclude that it is the d π (d xz and d yz ) orbitals that contribute predominantly to the formation of Kondo states.
As shown in Supplementary Figure 9, the splitting of the d xz orbital is caused solely by the hybridization with the s orbitals of surrounding gold atoms. Therefore, the energy span of the split d π orbitals characterizes the s-d hybridization strength (∆ s ), 10 which is a decisive factor of the strength of Kondo screening. 11 Conventionally, the magnitude of ∆ s is extracted as the HWHM for the broadened or split d orbital. 10,12 However, for the K n /Au(111) composites in our study, the PDOS of the split d π orbitals has a complicated line shape, and thus we propose to extract ∆ s as half of the full width of the split peaks; see The values of the Anderson impurity model (AIM) parameters, including d , U , ∆ α , Ω α and W α , are extracted from the results of DFT calculation. 10, 12,13 • The energy of the impurity orbital is estimated as d =´ρ d (ω)ωdω/´ρ d (ω)dω, where ρ d (ω) is the PDOS of the impurity orbital. Here, each impurity orbital represents the sum of d π (d xz and d yz ) orbitals on a Co adatom. The electron occupation number on the impurity orbital is calculated by • W α is estimated as the width of the energy bands of the STM tip (α = t) or the gold substrate (α = s) which have nonzero couplings to the d π orbitals.
• Ω α is taken to be the same as the chemical potential of the α-reservoir.
• The on-site electron-electron Coulomb interaction energy U is extracted as the energy difference between the centers of the spin-up and spin-down d π orbitals.
• ∆ s is the hybridization strength between the impurity and substrate orbitals, as it represents the amplitude of the hybridization function Λ s (ω) ≡ π k |t kd | 2 δ(ω − ks ) = ∆s 2 In our experiments, the STM tip is in the tunneling regime, and we thus assume ∆ t has a small constant value (∆ t ∆ s ).
In the wide-band limit (∆ α W α ), the spectral function of the impurity orbital is where G r d (ω) is the retarded Green's function of the impurity orbital, and ∆ = ∆ s + ∆ t ≈ ∆ s .
Note that ρ d (ω) is just the PDOS of the d π orbitals of the captured Co adatoms, so ∆ s could be extracted as the HWHM of the broadened (or split) peak of ρ d (ω), provided that ρ d (ω) retains a single-peak structure. However, as shown in Supplementary Figure 9(C), ρ d (ω) has a rather complicated line shape because of the strong s-d hybridization. Thus, ∆ s is extracted as the half width of the split peaks which constitute the major contribution to the local spin moment; see Fig. 3(b) in the main text. Numerically, the value of ∆ s is determined by cutting the split peaks with a sufficiently small threshold. For instance, the thresholds of 0.05 and 0.1 eV −1 result in ∆ s = 0.261 and 0.258 eV for the K 1 /Au(111) composite, respectively. It is estimated that the uncertainty of the cutting threshold could cause an uncertainty of ∼ 5 K in the predicted Kondo temperature.
• The SE interaction between the two spatially separated Co adatoms in the K 2 /Au(111) complex, ∆ SE , is taken as a tunable parameter. Its value is finally settled by comparing the Kondo temperature T K determined by the HEOM calculation carried out on a two-orbital AIM to the experimental measurement; see Fig. 4 in the main text and Supplementary Note 10 below.
The values of all the AIM parameters for the K n /Au(111) composites are listed in Supplementary Table 2. We calculate the differential conductance spectra (dI/dV versus V ) of the K n /Au(111) composites by applying the hierarchical equations of motion (HEOM) method and adopting a single-orbital AIM. The results are compared with the experimental measurements in Supplementary Figure 10(A). The agreement between the theoretically obtained line shape and the experimental counterpart is remarkable. It has been demonstrated both theoretically 14 and experimentally 15,16 that the zero-bias conductance G ≡ ( dI dV ) V =0 of a Kondo impurity obeys the following empirical scaling relation:

Supplementary
Here, T is the environmental temperature, G 0 is the conductance at T → 0, g b is the background conductance due to the electron transport through the non-Kondo 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 on a spin-1 2 impurity. [14][15][16] The Kondo temperature T K can thus be determined by fitting the calculated G versus T to Supplementary Equation (3). As an example, Supplementary Figure 10  . It is remarkable that the theoretically determined T K by a single-orbital AIM almost coincide with the experimental values for K n (n = 1, 2, 3, 4) except for K 2 . If the K 2 /Au(111) composite is represented by a single-orbital AIM (such a reference system is denoted as K * 2 ), its T K is predicted to be 157 K, which is only slightly higher than the T K of a K 2 but considerably lower than the experimental value of 191 K. This indicates a single-orbital AIM is inadequate to address the Kondo effect in the K 2 /Au(111) composite, and the long-range interaction between the two remote Co adatoms has a nontrivial influence on the Kondo states at the K 2 /Au(111) interface.
We then calculate G(T ) for the K 2 /Au(111) composite by adopting a two-orbital AIM with different values of ∆ SE ; see Supplementary Figure 11(B). The resulting T K are given in Fig. 4 of the main text and in the top-right corner of Supplementary Figure 11 In addition to the empirical scaling relation of the zero-bias conductance versus temperature, Kondo temperature has also been determined by using the Fermi liquid theory, i.e., the relation of Supplementary Equation (1). In fact, this is how the experimental value of T K are  (1) is also carried out, which is computationally rather expensive, because it requires the calculation of dI/dV for the whole low bias region, rather than just the zero-bias conductance. Supplementary Figure 12(A) and (B) depict the calculated dI/dV spectra for the K 1 /Au(111) and K 2 /Au(111) composites at different temperatures, respectively. Apparently, the conductance peak gets broadened as the environmental temperature increases, and the calculated data fit well to a single Fano function; see Eq. (1) of the main text. The Kondo temperature T K is extracted by fitting the width of the peak, Γ(T ), to the relation of Supplementary Equation (1) while taking α to be a tunable parameter; see the bottom-right corner of Supplementary Figure 12(C) and (D). Such determined T K agree consistently with those obtained via the empirical scaling relation; see the direct comparison displayed in Fig. 2(d) of the main text.
The remarkable agreement between the two sets of theoretical data, as well as between the theoretical and experimental results, justifies the AIMs adopted to represent the K n /Au(111) composites, and also affirms the accuracy of the DFT+HEOM calculations. The average binding energy between a captured Co adatom and its surrounding environment in a K n /Au(111) composite is defined by E b ≡ (E Kn/Au(111) −E CoPc/Au(111) −nE atom )/n. Here, n is the number of captured Co adatoms, E atom is the energy of an isolated Co atom, and E Kn/Au(111) and E CoPc/Au(111) are the total energies of the K n /Au(111) and CoPc/Au(111) composites, respectively.
Supplementary Table 3 presents the evolution of E b with the increasing n. The average binding energy for the captured Co adatoms in a K n /Au(111) composite relative to that in a K 1 /Au(111) composite is defined as Obviously, the magnitude of ∆E b increases with the enlarging n, indicating a stronger coupling with the environment; see also Supplementary Figure 13. It is consistent with the result of ∆ s versus n for the K n /Au(111) composites. For a K 2 complex, the average binding energy E b for the captured Co adatoms is somewhat larger than its counterpart in a K 2 .
Supplementary Table 3: Average binding energy for the captured Co adatoms and the average distances between a captured Co adatom and its neighboring Au (R Co−3Au ) and C (R Co−6C ) atoms in the K n /Au(111) composites. All energies are in units of eV, and all distances are in units ofÅ. Note that spin-orbit coupling is explicitly included.  We have replaced the CoPc by CuPc and H 2 Pc molecules to construct similar K n (CuPc) and K n (H 2 Pc) complexes on the Au(111) surface. Supplementary Figure 14 shows the topographic images of those complexes and their dI/dV spectra measured by placing the STM tip on top of the captured Co adatoms. The lines shapes of the dI/dV spectra much resemble those of K n (CoPc), and the Kondo peak at zero bias gets broader as n increase from 1 to 2 (or 2 ); see Supplementary Figure 14(E) and (J). These results are generally similar to the ones from CoPc except for K 2 (H 2 Pc) and K 2 (CuPc). The measured T K of K 2 (CuPc) and K 2 (H 2 Pc) are only slightly higher than their isomers K 2 (CuPc) and K 2 (H 2 Pc) by 4∼5 K.
On the other hand, the construction of the K n (CuPc) and K n (H 2 Pc) complexes with n ≥ 3 was unsuccessful.  Figure 15 depicts the PDOS of d orbitals that are supposedly relevant to the superexchange interaction between the two captured Co adatoms in a K 2 (CuPc) complex. The dashed line at −2 eV clearly indicates that the PDOS of d orbitals on the captured Co adatoms are energetically separated from that of the d orbitals on the central Cu ion. This confirms that the central Cu ion cannot bridge the long-range superexchange interaction between two captured Co adatoms in a K 2 (CuPc) complex. Similarly, the coupling channel for superexchange interaction also breaks down due to the lack of the central hub in a K 2 (H 2 Pc) complex. Supplementary Figure 16 displays the Kohn-Sham orbitals relevant to the SE interaction between the spatially separated Co adatoms in K 2 /Au(111) and K 3 /Au(111). As shown in the insets of Supplementary Figure 16(A) and (B), the delocalized orbitals on the CoPc mold appear to be much less connected than their counterparts in K 1 /Au(111); see Supplementary  Figure 4. Consequently, the long-range SE interaction is supposed to be much weaker in K 2 /Au(111) and K 3 /Au(111) composites.
Supplementary Figure 17 displays the delocalized Kohn-Sham orbitals in the K 2 /Au(111) and K 4 /Au(111) composites. Because of the high structural symmetry of the atom-mold complex, the connectivity of the delocalized orbitals is well preserved in a K 2 or a K 4 . These orbitals play the role as the bridging channels for the long-range SE interaction. The channel in a K 4 complex is less effective than in a K 2 , because in the former the d π orbitals on different Co adatoms are only partially connected by the channel; see remarks in Supplementary Note 15. As described in Supplementary Note 6, each of the captured Co adatoms in a K n /Au(111) composite is in a local S = 1 2 state. Since it is the d π orbitals that contribute predominantly to the Kondo screening, the local spin associated with the Kondo states is S z 0.42.
For describing the Kondo states in the K 2 /Au(111) composite, we adopt a two-orbital AIM which involves an off-diagonal hybridization function, Λ 12 (ω) = ∆ SE 2 W 2 s (ω−Ωs) 2 +W 2 s , to characterize the long-range SE interaction between the two Co adatoms mediated by the CoPc mold. To verify that the SE interaction indeed results in an effective FM coupling between the two Co adatoms, we compute S 12 = Ŝ 1 ·Ŝ 2 and Ŝ 2 as functions of ∆ SE by using the HEOM method. Here,Ŝ i = 1 2 ss ĉ † is σ ss ĉ is is the spin operator for the d π orbitals of the ith Co adatom (i = 1, 2), with σ representing the set of Pauli matrices; and S =Ŝ 1 +Ŝ 2 . The calculated results are shown in Supplementary Figure 18.  Figure 18: Calculated S 12 and Ŝ 2 versus ∆ SE for a K 2 /Au(111) composite. A two-orbital AIM which explicitly involves ∆ SE is adopted. Figure 18, the spin-spin correlation S 12 increases monotonically from zero to a positive value with the increasing ∆ SE . This indicates that the two remotely separated local spins are independent of each other at ∆ SE = 0, while they are aligned toward a mutually parallel direction at a nonzero ∆ SE . It is noted that two lines almost overlap each other in Supplementary Figure 18. This is because the local spin moments on the Co adatoms (S 1 and S 2 ) remain almost constant upon the variation of ∆ SE . By using the relation Ŝ 2 = S(S + 1), we have Ŝ 2 2 ≈ Ŝ 2 1 = S 1 (S 1 + 1) = Ŝ 2 − 2S 12 /2, and thus S 1 ≈ S 2 = 0.4. Therefore, the results yielded by the HEOM calculation on the two-orbital AIM agree closely with those of the DFT calculation, cf. Supplementary Table 1.

As shown in Supplementary
As shown in Fig. 4 of the main text, the Kondo temperature of a K 2 /Au(111) composite exhibits a non-monotonic dependence on the strength of SE interaction, i.e., T K first de-creases slowly with the increasing ∆ SE to a minimal value, and then rises quickly as ∆ SE increases further. Such a phenomenon can be understood by referring to a serially coupled two-orbital AIM that has been investigated previously. 17 In a serially coupled two-orbital AIM, one impurity orbital is coupled to only one electron reservoir. The whole impurity is described by . Here, the last term represents the inter-orbital hopping (with t being the strength), which gives rise to an effective AFM coupling between the two orbitals, with the coupling strength being J t = 4t 2 U . 18 Numerical calculations have revealed that the Kondo effect exhibits a non-monotonic trend with the increase of t. Specifically, the Kondo effect is first enhanced as t increases from zero to a certain value, and then it gets suppressed as t increases further. 17 The mechanism of such a non-monotonic behavior is as follows. When t increases from zero, the local spin on one impurity orbital starts to feel the AFM interaction from the other impurity orbital, and thus it is screened more strongly by its surrounding environment. This results in enhanced Kondo correlation. On the other hand, when t reaches a certain value, the two local orbitals start to form delocalized molecular orbitals. Because of the AFM nature of the inter-orbital coupling, this results in an S = 0 ground state within the impurity, leading to the quenching of local spins. Consequently, the Kondo correlation is more and more suppressed with further increasing of t.
The situation is exactly opposite for the K 2 /Au(111) composite. Instead of the AFM tcoupling, the environment-induced SE interaction is of FM nature. Thus, when ∆ SE increases from zero, the local spin on one impurity orbital starts to feel the FM interaction from the other impurity orbital, and thus it is screened less strongly by its surrounding environment. This results in weakened Kondo correlation. Nevertheless, when ∆ SE reaches a certain value, the whole impurity favors an S = 1 ground state because of the FM coupling, resulting in a substantially enlarged spin moment within the total impurity. Consequently, the Kondo correlation is strengthened with further increasing of ∆ SE . This thus explains the nonmonotonic variation of T K versus ∆ SE depicted in Fig. 4 of the main text.
As mentioned in Supplementary Note 6, the long-range SE interaction leads to a finite energy gap J eff between the FM and AFM states of the K 2 /Au(111) composite. A crude estimation gives J eff = E AFM − E FM ≈ γ ∆ 2 SE U +∆ = 14 meV, but the prefactor γ does not have a simple form and thus its value is subject to a large uncertainty.
To have a more accurate assessment of J eff , we explicitly include the effective AFM tcoupling into the two-orbital AIM, and finite-tune the value of t until the resulting S 12 = 0. The latter indicates that the FM and AFM spin-states become degenerate. From the HEOM calculation the zero spin correlation is reached at t 0.027 eV. We thus have J eff − J t = E AFM −E FM = 0, and so J eff = J t = 4t 2 U 3.3 meV. Such a value agrees consistently with the value of 4 meV calculated by using the DFT method. Since J eff < k B T K (the latter is about 16.5 meV), the triplet-to-singlet spin excitation is overwhelmed by the Kondo resonance and is thus invisible in the dI/dV spectra.
As shown in Supplementary Figure 19, the peak height of the PDOS of the d xz orbitals on the captured Co adatoms and on the Co ion in a K 4 /Au(111) composite (at −0.44 eV) is somewhat lower than that of a K 2 Au(111) (at −0.37 eV). This suggests the bridging channel is less effective for the two para-Co adatoms in a K 4 /Au(111), and thus the SE interaction in a K 4 /Au(111) is weaker than that in a K 2 /Au(111).
The delocalized Kohn-Sham orbital that mediates the SE interaction can be expressed as ψ SE = j c j χ j , where j labels the basis functions including the atomic orbitals on the captured Co adatoms. Thus, ψ SE = c 1 χ

8.
Since ∆ SE is determined to be 123 meV for a K 2 /Au(111), its value is estimated to be around 15 meV for a K 4 /Au(111). Therefore, by referring again to the inset of Fig. 4 in the main text, we estimate that the SE interaction in a K 4 /Au(111) could lead to a minor change in T K by about 4 K.