Long-range ferrimagnetic order in a two-dimensional supramolecular Kondo lattice

Realization of long-range magnetic order in surface-supported two-dimensional systems has been challenging, mainly due to the competition between fundamental magnetic interactions as the short-range Kondo effect and spin-stabilizing magnetic exchange interactions. Spin-bearing molecules on conducting substrates represent a rich platform to investigate the interplay of these fundamental magnetic interactions. Here we demonstrate the direct observation of long-range ferrimagnetic order emerging in a two-dimensional supramolecular Kondo lattice. The lattice consists of paramagnetic hexadeca-fluorinated iron phthalocyanine (FeFPc) and manganese phthalocyanine (MnPc) molecules co-assembled into a checkerboard pattern on single-crystalline Au(111) substrates. Remarkably, the remanent magnetic moments are oriented in the out-of-plane direction with significant contribution from orbital moments. First-principles calculations reveal that the FeFPc-MnPc antiferromagnetic nearest-neighbour coupling is mediated by the Ruderman–Kittel–Kasuya–Yosida exchange interaction via the Au substrate electronic states. Our findings suggest the use of molecular frameworks to engineer novel low-dimensional magnetically ordered materials and their application in molecular quantum devices.

. The external magnetic field B is parallel to the k-vector of the incoming X-rays. (b,c) XAS and XMCD measured for FeFPc and MnPc at the L 3,2 edges in zero field and (d,e) in an applied field of 6.8 T, at T = 2.5 K. The XMCD spectra acquired on MnPc molecules in zero applied external magnetic field reveal no dichroic signal in c, which suggests that the remanent magnetic moments of the MnPc molecules are fully aligned along the out-of-plane direction. The XMCD spectrum acquired at the Fe L 3,2 edge of FeFPc molecules in zero field resolves a small remanent magnetisation in b. (f) Element-specific XMCD peak height vs B curves measured on the Fe and Mn ions reveal no remanent magnetisation. The curves have been recorded according to the protocol described in the Methods section.

Supplementary Note 1.
To investigate the magnetic coercivity of our 2D spin array, we performed XAS/XMCD measurements at T = 2.5 K with the magnetic field continuously ramped down from 6.8 T to -0.1 T (see orange and violet arrows in Supplementary Figure 1a). As shown in Supplementary Figure 1b,c the sign of the XMCD of the FeFPc molecules did not change, whilst that of the MnPc molecules did, in comparison to XMCD spectra acquired at 6.8 T (cf. Figure 1c). The observed behaviour of the XMCD signals suggests that the coercivity of the spin array is smaller than 0.1 T. In case the system would exhibit a coercivity higher than 0.1 T, one would observe an opposite behaviour, i.e. the FeFPc molecules changing the XMCD sign, whilst the MnPc species do not. Note that due to the spiky behaviour of the total electron yield around zero field and the continuous field ramping with the need for binning into certain B-field bins and interpolation, there is an increased noise in XMCD peak height vs B curves around zero field.
Applying the mean field approximation to the bipartite Ising model and assuming only a nearest neighbour interaction between MnPc and FeFPc molecules, we can extract the strength of the indirect exchange coupling from the XMCD peak height vs B curves. At B = 2 T the magnetic moments of FeFPc molecules are equal to zero, which means that the Zeeman energy and exchange coupling are of equal strength. The FeFPc molecules are surrounded with 4 MnPc species, therefore the effective magnetic field induced by the 4 MnPc molecules is equal to 4 〈 〉, where 〈 〉 = 1 µ B (i.e. estimated at B = 2 T). Hence, we extract the exchange coupling strength to be about 0.12 meV, which is significantly lower than the Kondo temperature of about 10 K (see below). Note that in our calculation we neglected anisotropy terms. Knowing the strength of the indirect exchange coupling we estimate an ordering temperature of T C ~ 3.7 K. This value is in line with our XMCD measurements performed at B = 0 T and T = 5 K, shown in Supplementary Figure   1f,g, in which we cannot resolve a detectable XMCD signal.
In Supplementary Figure 2 we present XAS/XMCD data acquired on the chessboard FeFPc+MnPc array on Au(111) in grazing incidence geometry, where the XMCD probes predominantly the in-plane components of the magnetic moments, and partly the out-of-plane components contributing to the overall signal 1,2 . The XMCD spectrum acquired at the Fe L 3,2 edge of FeFPc molecules resolves a small remanent magnetisation (Supplementary Fig. 2b), which is attributed to the out-of-plane component. Applying an external magnetic field of B = 6.8 T leads to alignment of both magnetic moments along the field. A sum-rule analysis applied on the XMCD spectra of both molecules (see Supplementary Table 1) resolves that in the field-aligned state (B = 6.8 T), the total magnetic moments are larger in grazing incidence than in the normal incidence geometry. This finding is in line with a previous report on thin films of FePc, which were found to exhibit easy-plane magnetic anisotropy 3 . A salient difference between these two cases, FePc films and the here-studied supramolecular layer, is discussed below.  (4) for the orbital and spin effective moments, respectively, to obtain the isotropic spin moments. The dashed lines depict the extracted respective isotropic spin moments.

Supplementary Note 2.
In order to extract absolute values of the spin and orbital magnetic moments of the central metal ions of the MnPc and FeFPc molecules, the acquired XMCD data have been processed with the sumrule analysis 4,5 . The orbital magnetic moment has been extracted using the following equation, (1) and the effective spin magnetic moment has been obtained from where the integration of the circularly polarised XA spectra (C ! ,C ) is over the energy range defined by the respective L 3 and L 2 edges, <N h > stands for average number of holes at a metal ion (<N h > = 4 for FeFPc and <N h > = 5 for MnPc). The term m S stands for the spin magnetic moment and m T is the intra-atomic dipolar moment resulting from the spin-quadrupolar anisotropy 6 .
The square planar symmetry of the phthalocyanine ligand at the metal ion causes both the effective spin and orbital magnetic moments to be angle dependent (see Supplementary Figure 3) with the following dependency 9 , -= ./ 012 * -+ 3/ 245 * - where ./ and 3/ are the out-of-plane and in-plane components of the orbital magnetic moment, respectively, andis the angle between the k-vector of the incoming X-rays and the surface normal. The term ( 3/ in supplementary equation (4) 10 .
In Supplementary Table 1 we present the thus-obtained values of the magnetic moments of the FeFPc and MnPc molecules both in normal and grazing incidence.
Supplementary Table 1. Values of the measured magnetic moments of the FeFPc and MnPc molecules both in normal and grazing incidence and with and without applied external magnetic field, extracted from a sum-rule analysis. The negative sign of the magnetic moments on the FeFPc molecules means an opposite direction of the XMCD signal as compared to that of the MnPc species.
Since the sum-rule analysis tends to overestimate the effective spin moment of Mn 2+ ions by 33% 7,8 . The values of the Mn magnetic moments have therefore been corrected by multiplying them by a factor 0.7.  Figure 3 reveals an unusual angular dependence of the magnetic moments. In the field aligned state the size of the orbital moment of FeFPc decreases from a higher value observed in normal incidence to a smaller one measured in grazing incidence geometry. On the other hand, the angle dependence of the effective spin moment shows an opposite behaviour with a higher value in grazing incidence and smaller one in normal incidence. A somewhat similar behaviour is observed for MnPc, however with the change in the orbital moment being comparably smaller. Notably, the behaviour found here for FeFPc on Au(111) is different from that observed previously for thin films of FePc 3 and for the FePc/graphene/Ir system 11 , where both effective spin moment and orbital moment increased with the incident angle resolving an easy-plane anisotropy. Considering that the orbital moment is pivotal to the magnetic anisotropy 12 our data reveal an out-of-plane easy axis for FeFPc and MnPc in the supramolecular layer.
In a similar, but more pronounced way the effective spin and orbital moments behave in remanence (see Supplementary Table 1). The moments measured for MnPc are Ising-like, i.e. only non-zero in the out-of-plane direction. For FeFPc the orbital moment is again considerably larger in the out-ofplane direction, consistent with an out-of-plane easy axis in the remanent magnetic state. Interestingly, our measurements evidence a different behaviour of the magnetic anisotropy for the two-dimensional supramolecular layer as compared to thick films of FePc 3 .

Supplementary Note 3.
In this Supplementary Note we report XAS/XMCD spectra acquired on the supramolecular FeFPc+MnPc array assembled on Ag(111) measured at T = 2.5 K in normal incidence geometry.
The supramolecular array was deposited on a Ag(111) substrate to investigate the role of the substrate on the inter-molecular magnetic coupling. In Supplementary Figure 4 we present XAS/XMCD data acquired on this system in normal incidence geometry. This is the same geometry where the supramolecular array assembled on the Au(111) substrate shows ferrimagnetic order in remanence. However, the dichroic spectra acquired on both molecules at zero magnetic field show zero magnetization, i.e. no remanence. Next, we apply an external magnetic field (B = 6.8 T) leading to alignment of the molecular magnetic moments along the field direction. We thereby confirm the pivotal role of the Au(111) substrate in mediating the indirect magnetic exchange coupling between MnPc and FeFPc molecules.
The Au(111) substrate hosts spin-split surface states with the following Fermi-wavevectors: 1.67 and 1.92 nm -1 , which are much larger than the Fermi-wavevector of Ag(111) surface states (k F = 0.8 nm -1 ) (supplementary ref. 13). Simulating the oscillatory behaviour of the RKKY coupling using the formula ∝ sin 2= > / 2= > * (supplementary ref. 14), with the aforementioned Fermi wavevectors of the surface states and d being the distance from the scattering centre, we find that at a distance 1.4 nm (i.e., the observed distance between nearest neighbouring FeFPc and MnPc species on the Au(111) substrate), the molecular spins are expected to couple antiferromagnetically. Furthermore, nextnearest neighbours at a distance of 1.98 nm are expected to couple ferromagnetically. If we, however, assume the same distances between magnetic centres of a FeFPc+MnPc checkerboard array, but assembled on a Ag(111) substrate, the RKKY interaction would lead to a ferromagnetic coupling between nearest neighbours and weak antiferromagnetic coupling for next-nearest neighbours. Thus, the above given argument and the ab initio calculations presented in the main text support that the long-range ferrimagnetic order is mediated by the RKKY coupling via the Au substrate.
Recently, for tetracyano-p-quinodimethane (TCNQ) molecules assembled on graphene/Ru(0001) the existence of a ferromagnetic ordered ground state with an in-plane moment orientation was claimed on the basis of spin polarised STM/STS measurements performed on different monomolecular domains 15 . However, the magnetic contrast which was resolved only after averaging across small areas of differently oriented domains on rather noisy data, exhibits a contrast between parts of the sample that does not discriminate long-range ferromagnetism from paramagnetism with disordered patches of TCNQ molecules having magnetisation either up or down, and is hence of limited evidence for establishing long-range order in the system. Furthermore, the proposed direct Heisenberg exchange mechanism 15 via the intermolecular hybridisation between frontier orbitals of the TCNQ molecules would not be able to lead to long-range ferromagnetic order in a truly 2D system at finite temperatures, as this would contradict the Mermin-Wagner theorem.

Supplementary Note 4.
In this Supplementary Note we report the measurement of the Kondo temperatures of MnPc and FeFPc molecules in the supramolecular array as well as scanning tunnelling spectroscopy on a MnPc vacancy.
Scanning tunnelling spectroscopy recorded on a missing MnPc molecule of the supramolecular monolayer displays the so-called "cloning" of the zero-bias features of the neighbouring FeFPc molecules (see Supplementary Figure 5a). Such a "cloning" of the Kondo state present on neighbouring atoms adsorbed on a metallic substrate has been previously demonstrated for Co atoms adsorbed on Cu(111) by placing the atom in one of the focal points in the artificially constructed quantum corrals 16,17 . In our case, the Kondo signature measured in the centre of a MnPc vacancy is replicated without the need of an additionally constructed confined state. The observed effect readily demonstrates that the Au(111) surface states are weakly affected by the adsorbed molecules and consequently that these surface states play an important role in mediating the RKKY exchange coupling between MnPc and FeFPc molecules. This observation is in good agreement with a previous report, where a "cloning" was observed for a missing FePc molecule in a 2-dimensional FePc lattice adsorbed on Au(111) 18 . The here-measured Kondo temperature of the MnPc molecules is slightly higher T K = 9.4±1.8 K than the one of the FeFPc species T K = 9.2±2 K (see the fits in Supplementary Fig. 5b,c).
The Kondo temperatures have been obtained by measuring scanning tunnelling spectra at the centre of the magnetic molecules at different temperatures. The STS spectra have subsequently been fitted with the Frota function 19 in the form given by Prüser et al. 20 : @A @B~ D * Im H4I JK L JM NB N O !JM P + Q * R + 0 where S is phase factor defining the shape of the Kondo zero bias feature, T stands for a resonance width, I U is centre of the Kondo resonance and a, b, c are multiplication factors and a constant, respectively. The temperature dependent width of the Kondo resonance, extracted from the fit to function in supplementary equation (5) has been fitted with the Fermi-liquid model [Supplementary Eq. (6)]. The Kondo temperature was extracted from the fit that models the resonance width as a function of the temperature (see Supplementary Fig. 5b,c) employing the expression 21,22 where V rms is the root mean square of modulation voltage, X Z the Kondo temperature, = is the Boltzmann constant and W is a multiplication factor defining the slope of the curve at X ≫ X Z .
In the fitting process we obtained the following parameters: W The importance of the Coulomb correlations has been shown previously for a number of metallo-porphyrins and metallo-phthalocyanines; in particular, they lead to molecular magnetic moments and spin-crossover temperatures that are significantly improved 24-26 over those given by standard density functional theory (DFT) calculations. In the present calculations U and J were taken to be 4 eV and 1 eV, respectively. These values were shown to provide the correct spin state for free as well as substrate absorbed metallo-phthalocyanines and metallo-porphyrins 27,28 . For the DFT exchange-correlation functional we used the generalised-gradient approximation (GGA) in the parameterisation of Perdew et al. 29 . The electronic structure calculations were performed with the VASP full-potential plane-wave code 30 with a kinetic energy cut-off of 400 eV.
As we find that the molecules are relatively weakly adsorbed on the Au(111) surface with an adsorption energy of 0.88 eV per molecule (see Methods) there could be an influence of the Van der Waals interaction. To investigate this we performed calculations with the Van der Waals-Density Functional (vdW-DF) method 31 . We find however that the vdW-DF calculations predict a significantly larger lattice constant for Au (4.245 Å, as compared to the experimental value of 4.08 Å). As this larger Au atomic distance would have an influence on the adsorbed molecules within the selfconsistent relaxation process of the atomic distances, we decided to use only the DFT+U framework with the GGA exchange-correlation description.
We further mention that our STM images show the so-called herringbone reconstruction of the Au surface (see Fig. 1c). This reconstruction occurs on a length scale of about 20 nm, which is much larger than the sizes of our simulation cell (laterally 28.85 x 29.98 Å 2 ). Hence, the herringbone reconstruction has not been taken into account in our simulations. However, the important length scale for the indirect exchange interaction is the distance between the FeFPc and MnPc molecules, which is much smaller, about 14 Å. Therefore the herringbone reconstruction is expected not to influence this exchange interaction noticeably.
The computed 3d spin moments were -2. the population of the > f g orbital is responsible for a reduction of the total spin S.
Upon adsorption of FeFPc and MnPc on Au(111) we find that the molecular geometry slightly changes. In the gas phase, the FeFPc and MnPc molecules have D 4h symmetry, but adsorbed on the Au(111) surface MnPc adopts C 4v and FeFPc C 2v . In case of FeFPc two of its benzene rings moved slightly towards the substrate, while in case of MnPc the phthalocyanine plane came closer to the substrate yet with the Mn atom positioned 0.17 Å above the macrocyclic ring.
To investigate whether the selfconsistently computed antiparallel alignment of moments on the FeFPc and MnPc molecules represents the lowest energy state, we have carried out additional total energy calculations for two other magnetic phases. Specifically, we calculated the ferromagnetic (FM) phase (all Fe and Mn moments are parallel) and a different ferrimagnetic (AFM1) phase, where one of the Fe moments was reversed, enforcing thus an antiferromagnetic alignment on the FeFPc sublattice. The selfconsistently calculated total energies are given in Supplementary Table 2. We note that in these different magnetic states the same molecular structure has been used, in order to extract purely the magnetic interaction. The total energy calculations confirm that the AFM0 state with antiparallel orientations of the Fe and Mn moments is the ground state, in agreement with our XMCD measurements. In addition we observe that the ferromagnetic state has a total energy that is higher by 1.5588 meV per 2x2 unit cell, or 0.78 meV per structural unit cell (containing one FeFPc and one MnPc molecule), which corresponds to 9 K. The AFM1 state with one flipped Fe magnetic moment has a total energy that is higher than that of AFM0 by 0.876 meV per 2x2 unit cell, or 5 K per structural unit cell. These calculated exchange interactions are expectedly very small, which is consistent with the weak mediation of the long-range exchange RKKY interaction via the Au substrate. The calculated exchange energies are in good agreement with the long-range magnetic ordering temperature of about 5 K estimated from our temperature-dependent XMCD measurements. Assuming a nearestneighbour-only exchange interaction Hamiltonian, m = 8 , provides a calculated value = 0.10 meV, which agrees well with the strength of the exchange interaction estimated from the XMCD versus magnetic field measurements.