Dynamical Screening of Local Spin Moments at Metal–Molecule Interfaces

Transition-metal phthalocyanine molecules have attracted considerable interest in the context of spintronics device development due to their amenability to diverse bonding regimes and their intrinsic magnetism. The latter is highly influenced by the quantum fluctuations that arise at the inevitable metal–molecule interface in a device architecture. In this study, we have systematically investigated the dynamical screening effects in phthalocyanine molecules hosting a series of transition-metal ions (Ti, V, Cr, Mn, Fe, Co, and Ni) in contact with the Cu(111) surface. Using comprehensive density functional theory plus Anderson’s Impurity Model calculations, we show that the orbital-dependent hybridization and electron correlation together result in strong charge and spin fluctuations. While the instantaneous spin moments of the transition-metal ions are near atomic-like, we find that screening gives rise to considerable lowering or even quenching of these. Our results highlight the importance of quantum fluctuations in metal-contacted molecular devices, which may influence the results obtained from theoretical or experimental probes, depending on their possibly material-dependent characteristic sampling time-scales.

We performed our first-principles calculations using density functional theory (DFT) as implemented within the Vienna ab-initio Simulation Package (VASP). 1 To obtain parameters for the many-body calculations, that take into account dynamical correlations effects, specifically, we performed non-spin polarised DFT calculations using the DFT-relaxed structures.
In the following, we discuss how the ab-initio parameters are extracted to describe a correlated sub-space within the Anderson Impurity Model as well as the details of many-body simulations.

Many-body effects: DFT++
To obtain a realistic description of screening of the local magnetic moments of the molecules at metal contact, we employed a combined approach: density functional theory (DFT) plus many-body technique within multi-orbital Anderson impurity model (AIM). The method is often referred to as the DFT++ method. 2 The first realistic description of the molecule-surface hybrids is obtained within DFT.
This allows us to extract ab-initio parameters to describe a correlated sub-space, i.e., the impurity, corresponding to the TM-3d multiplet, which is coupled to an electron bath, corresponding to the rest of the system: phthalocyanine ring plus the Cu(111) surface, through a retarded hybridization function. In the AIM, the impurity Hamiltonian is supplemented with a Coulomb interaction, and the concomitant many-body problem is solved numerically.
The complete Hamiltonian of the AIM can be written as where d iσ (d † iσ ) are the annihilation (creation) operators of an electron in impurity (TM-S-2 3d) orbital i with spin σ, and c mσ (c † mσ ) denote the annihilation (creation) operators of an electron in bath orbital m with spin σ and energy b m . V im denotes the coupling between the TM-3d and the bath orbitals, d ij describes the crystal field matrix and U ijkl represents the full Coulomb tensor within the TM-3d multiplet. The rotationally-invariant Coulomb interaction is parametrized via the Slater radial integrals 3,4 F 0 , F 2 , and F 4 , such that U = F 0 and J = 1 14 (F 2 + F 4 ), with the ratio F 4 /F 2 = 0.625, yielding a spherically symmetric tensor. 5,6 In all of our calculations, we have used U = 4.0 eV and J=1.0 eV. We note that when the spherical symmetry is lifted in presence of a crystal field, the effect can be taken into account, e.g., within the constrained random phase approximation. 7 In n our analysis, we do not expect this neglected effect to change our main conclusions.
For the numerical solution of the many-body problem, we have used a continuous-time hybridization-expansion quantum Monte Carlo (QMC) solver as implemented in w2dynamics program package. 8 The dynamical nature of the hybridization function is fully taken into consideration within the QMC algorithm. We emphasize that discretised Hamiltonian approaches, such as exact diagonalisation, are inadequate in the case of molecules adsorbed on metallic surfaces due to a broad nature of the hybridization function. The double counting correction term is implemented by fixing the chemical potential corresponding to the occupation of the impurity orbitals, n = iσ n iσ . In this work, the total impurity occupations considered as, n, are 2.57e (TiPc), 3.47e (VPc), 4.67e (CrPc), 5.81e (MnPc), 6.62e (FePc), 7.66e (CoPc), 8.13e (NiPc), and 9.32e (CuPc). We note that the off-diagonal elements of the Hamiltonian and the hybridization functions are significantly small and have been neglected to avoid technical complications, and this approximation provides an orbital-diagonal QMC self-energy. S-3

Embedding the transition metal(TM)-atom: Hybridization function
In this section, we explain how the TM atoms embed in the phthalocyanine plus Cu-surface environment, we calculate the ab-initio hybridization function from the Kohn-Sham Green's function, G KS , in the Lehmann representation: where ψ nk and nk are the Kohn-Sham eigenstates and eigenvalues for band n and reciprocalspace point k, while δ is an infinitesimal broadening, meaning that G KS is the retarded propagator. The G KS is then projected down to form the local (or the impurity) propagator G 0 , This projection is defined using atom-centered, localized orbitals χ i . In this basis, the local Green's function is written as where P i nk = χ i |ψ nk are projection matrices, which are ortho-normalized according tõ using the effective overlap operator Finally, the hybridization function is calculated from the local impurity Green's function from the expression S-4 At this point, one can obtain the impurity-bath coupling matrix elements by fitting the hybridization function with the following: We note that, if the projected TM-3d orbitals are not effectively orthonormal (i.e., if d i =j = 0), then the hybridization function includes also off-diagonal terms. In the present scenario, ∆ i =j ∆ ii for any pair (i, j). We show the diagonal elements of the hybridization Fig. 2 in the main text, as well as in Fig. S1. In Fig. S1, we depict the hybridization function of the out-of-plane orbitals i. e., the d z 2 , d xz and d yz orbitals for all of the TMPc molecules on the Cu surface. We note that the hybridization of the To show how the hybridization changes in TMPc molecules, we obtained the coupling strengths V im (or simply V i ) corresponding to the axial bonds between the d x 2 −y 2 orbital and predominantly the N-2p orbitals of the phthalocyanine ring. We also obtained the bath energy, b m , corresponding to the strongest peak in the ∆ d x 2 −y 2 (as seen in Fig. 2 in the main text). The values are presented in Table 1.  Energy -0.62 -0.43 -0.22 -0.26 -0.12 -0.37 -0.08 -0.14 The absorption energies reported in Table 2 without the D2 vdW energy corrections present a different picture from the one reported in Table 1 in the main text. The adsorption values are equal to or less than 0.4 eV (except for the Ti case). Such an energy range is typical of the physisorption regime. It has to be noted, however, that the PBE functional tends to underestimate the absorption energy at metallic surfaces 9-11 while the PBE-D2 approach tends to overestimate them. 12 We thus expect the TMPc to be in a hybrid absorption regime between chemisorption and physisorption. The structural and atomic charge analysis in the main text and in the paragraph below support such hypothesis. In fact, while the TMPc molecules are distorted when put in contact with the Cu(111) surface, and a significant hybridization of the out-of-plane 3d orbitals with the metallic states of Cu(111) surface is observed (Figure 2c, main text), no visible bond breaking or rearrangement is found in our analysis. In addition, while we observed a noticeable charge transfer between the Cu(111) surface and the TMPc (Figure S2), the electronic charge redistributed is always less than 1 e − in total. These charge transfers are redistributed thanks to delocalization throughout the whole organic framework of the TMPc, causing distortion but not bond breaking.
In the following we discuss the changes in the TMPc molecules due to adsorption in comparison with the free TMPc molecules. S-7

Structural changes
Below, we compare the change in the TM-N bond lengths of free and adsorbed molecules, as those are crucial for the ligand field and hence the spin-state. The structural distortions in connection with the non-planarity are discussed in the main text (Fig. 1). In general, adsorption breaks the D 4h symmetry of the isolated molecules due to local structural mismatch. Crucial to the ligand field, the TM-N bond lengths also change due to the adsorption. Molecules embedding early TM ions show a larger expansion of the TM-N core, while as the surface molecule distance is increased in the later part of the TM series, the TM-N bond distances appear similar to the free molecule counterparts.

Bader charge analysis
Bader charge analysis was carried out on the basis of the total charge density, accounting for both the electronic and ionic core charges. We performed the charge analysis on both the full systems (TMPc/Cu(111)) and the TMPc molecules in the gas phase in order to understand the charge transfer behaviour at the interface. Upon adsorption on the Cu-substrate, all the TMPc molecules acquire electronic charge from the surface as shown in Figure S2   We see significantly enhanced charge fluctuations and weakened spin fluctuations in the d R subspace in TiPc, MnPc, and CoPc. As we discussed in the main paper, both of these S-10 effects significantly screen the local spins in the same subspace, which is responsible for the local magnetic moment in these molecules. We, therefore, see a large deviation in the longtime moments from the corresponding instantaneous moments, as presented in Fig. 3 in the main paper. The scenario is different in CuPc. There the d R subspace is completely filled, yielding non-existent charge and spin fluctuations (i. e. C ij and S ij ≈ 0). In Fig. S3, we observe a relatively strong ferromagnetic spin fluctuation in the d x 2 −y 2 orbital, which is the origin of the local moment in CuPc.
In Fig. S4, we show the orbital-diagonal terms of spin susceptibility, χ ii (τ ), of all the TMPc molecules, studied in this paper. As we discussed in the main paper, the gaps between Figure S4: Orbital resolved spin susceptibility (χ ii ) in imaginary time τ for all TM ions in TMPc/Cu(111) systems. the χ ii (τ = 0) and χ ii (τ = β/2) mark the screening of the spins in those orbitals. In Fig. S4, we show the different degree of such screening in different 3d orbitals within the same TMPc molecule, as well as the variation in different TMPc molecules adsorbed on Cu(111) surface. However, we note that in CuPc the scenario is different from the rest of the molecules. Unlike in other molecules, there both χ(0) and χ(β/2) are comprised solely of the d x 2 −y 2 orbital contribution, signifying that the instantaneous and the screened local moments have a d x 2 −y 2 origin.

Temperature dependence of dynamical screening
We discuss the temperature dependence of the dynamical processes that leads to screening of the local moments in TMPc molecules. We considered FePc/Cu(111) for this specific study.
In Fig. S5(a), we plot the instantaneous (S inst ) and screened (S scr ) spin moments of the FePc molecule (adsorbed on Cu(111)) at temperatures ranging from 30K to 290K. We note that here we performed the AIM calculations in a reduced orbital space comprising of the d xz , d yz , and d z 2 orbitals of the Fe atom. Such an approximation is justified because in FePc, the d xy orbital is completely filled and, hence, does not contribute to the local moment. As discussed in the main paper, the spin on the d x 2 −y 2 orbital is screened completely; therefore, as far as the screened moment is concerned, d x 2 −y 2 does not have any contribution. Nonetheless, the instantaneous moment has a significant contribution from the d x 2 −y 2 orbital. To elucidate this contrast, in Fig. S5(a) we compare the S inst and S scr as calculated from the full 5 3d-orbital AIM calculations, against those from the reduced-space 3 3d-orbital AIM calculations, all at 290K. One can see that the S inst from a 5 orbital calculation differs from the same for 3 orbital calculation, while the screened moments do not substantially differ.
From these results, we also see that the S inst remain constant in the temperature range, while the screened moments show a clear temperature dependence. We note that even at 290K, the long-time moment is strongly suppressed due to the dynamical screening, as also discussed in the main paper. Upon lowering the temperature, the S scr (almost) linearly S-12 reaches approximately 0 µ B . This is consistent with what is expected in a metallic system, namely that at a long enough time, a complete screening takes place at low-T due to strong fluctuations of the local moment induced by the high electron mobility.
We finally elucidate the role of hybridization, hence the electron mobility, in the temperaturedependent screening, in Fig. S5(b). Here, we plot the temperature dependence of the S scr upon an artificial lowering of the hybridization, ∆. At ∆/2, the behaviour is linear-like although screening is significantly less, particularly at the higher temperature regime. If further decreased, at ∆/3, the linear behavior is lifted and local spin is not quenched at the lowest temperature we can reach (30K), while at ∆/5, the local moment is only partially S-13 screened, retaining large S scr even at that low temperature.

Effective local spins
We discuss the magnetic behavior of the free TMPc molecules in the absence of the surface underneath. We note that the structures of the free molecules are relaxed following the criteria mentioned in the main text (sec. Methods). The results provide 1) a direct comparison with the existing experimental results, and 2) insight into the role of hybridization in the dynamical screening process, in particular via the out-of-plane orbitals.
In Fig. S6, similarly as performed for the adsorbed molecules in the main text, we plot the total spins of the free TMPc molecules obtained with DFT+AIM and DFT+U calculations. The purple and the cyan bars depict the instantaneous (S inst ) and screened (S scr ) effective spin moments, respectively, while the green bars represent spin moments obtained in DFT+U.
One can immediately notice that, in general, the S inst are atomic-like, mostly driven by inter-orbital Hund's coupling, and have similar values as the adsorbed TMPc molecules on Cu. It is noteworthy that, as discussed earlier, surface adsorption causes a structural change, and hence a change in the ligand field in the molecule, while simultaneously allowing an effective charge transfer. The effect of surface hybridization in the instantaneous local moments is minimal. However, in comparison with the adsorbed molecules, the long-time spin moments are significantly less screened. We obtained, S scr ≈ 0.5µ B (TiPc), 1.3µ B (VPc),  17,18,26 In these calculations, the double counting correction term is implemented by fixing the chemical potential corresponding to the occupation of the impurity S-14 orbitals, n = iσ n iσ . The total impurity occupations considered as, n, are 2.38e (TiPc), 3.51e (VPc), 4.61e (CrPc), 5.73e (MnPc), 6.52e (FePc), 7.57e (CoPc), 8.50e (NiPc), and 9.26e (CuPc). Figure S6: Dynamical screening of the effective moments in isolated TMPc molecules. Unscreened (purple), screened (cyan), and DFT+U effective spin moments (green) for TMPc molecules. ∆ s is the difference between the instantaneous and screened spin moments.
It is to be noted that, unlike in the case of adsorbed molecules, the DFT+U moments are very similar to the screened moments obtained in AIM. In absence of the surface induced valence fluctuation, the charge fluctuation is rather small. As discussed in the following subsection, one sees a flattening of χ(τ ) which reflects poor screening, and which is a signature of correlated insulators. In such a scenario, treating of static exchange and correlation effects at the level of DFT+U works reasonably well, albeit without providing insights on transient behaviour. S-15

Spin susceptibility
In Fig. S7, we present the orbital resolved local spin-susceptibility, χ ii (τ ), for the free TMPc molecules. Distinguishable from the adsorbed molecules (particularly the out-of-plane orbitals), one observes a flattening in χ ii (τ ), which stems from poor screening in absence of the hybridization with the surface. In CrPc, all d R orbitals contribute (almost) equally to Figure S7: Orbital resolved spin susceptibility (χ ii ) in imaginary time τ for all TM ions in isolated TMPc molecules. the long-time moment, signifying strong spin fluctuations dictated by Hund's physics. As expected, the ∆ s is the lowest among all TMPc molecules. Generally, in the absence of hybridization through the out-of-plane orbitals, the lack of screening results in significant S-16 χ ii (β/2) values, hence large persisting moments. Interestingly, in free CuPc, both S inst and S scr are comparable with the same in adsorbed CuPc. The magnetic moment in CuPc is carried out by the in-plane d x 2 −y 2 orbital, which is barely impacted by the surface induced screening, as also observed in experiments. 26,27