Design of the experiment

We have fabricated a supramolecular lattice consisting of two different metallo-phthalocyanines, hexadeca-fluorinated iron phthalocyanine (FeFPc, Fig. 1a) and manganese phthalocyanine (MnPc, Fig. 1b) by co-deposition on an inert, non-magnetic Au(111) substrate. The Au(111) substrate facilitates molecular self-assembly and, in addition, hosts Shockley-type spin-split surface states that are ideal for promoting RKKY coupling between adsorbed spin moments27. The presence of the peripheral fluorine atoms in the FeFPc molecules and hydrogen atoms on the MnPc species directs the self-assembly into a checkerboard arrangement as resolved by scanning tunnelling microscopy (STM) (see Fig. 1c) through the C–H···F–C interactions28,29. On this heterogeneous molecular monolayer, we applied a combination of molecule-selective probing (that is, XMCD and STS) of the magnetic moments, which allows us to address the element-specific magnetic moments residing on the long-range ordered Fe and Mn ions, as well as the presence of the Kondo effect individually.

Figure 1: Assessing competing fundamental magnetic interactions. (a) Scheme of the FeFPc and (b) scheme of the MnPc molecules. The former has all hydrogen atoms on the periphery of the molecule replaced with fluorine. The pictograms shown next to the molecular sketches are used to distinguish the molecules. (c) STM image (Bias=−2.2 V, I t =5 pA) acquired on an extended domain of FeFPc and MnPc molecules on a Au(111) substrate co-assembled in a checkerboard pattern. Scale bar, 10 nm. The inset shows a zoom of the STM image providing details of the checkerboard pattern with one MnPc species surrounded by four FeFPc molecules and vice versa (scale bar, 1 nm). (d) Sketch of the two occurring magnetic interactions in the remanent state, the short-range many-body Kondo screening and the long-range RKKY exchange interaction of the magnetic molecular centres mediated by the conduction electrons of the Au(111) substrate. (e) The magnetic moments of the two molecular species are antiferromagnetically coupled and align their moments in the out-of-plane direction. Full size image

Topographic images of molecules

The molecules are easily identified from the inset of Fig. 1c, with the small bright species attributed to MnPc molecules, while the FeFPc molecules appear bigger due to the fluorine ligands. Each molecule has four nearest neighbours of the opposite kind, whereas the next-nearest neighbours are of the same type. The periodic zig-zag-like pattern resolved by STM depicts a characteristic fingerprint of atomically clean Au(111) surfaces, the so-called herringbone reconstruction of the top-most layer. In Fig. 1d, we illustrate the fundamental magnetic interactions involved: the spin-bearing molecules act as magnetic centres, to which the spins of substrate electrons couple locally through the Kondo effect, whereas two spin centres can couple via the RKKY interaction mediated by the Au(111) substrate electronic states.

XAS/XMCD measurements

We have investigated the magnetic properties of the supramolecular array using element-selective X-ray absorption spectroscopy (XAS) and XMCD, which allows us to probe the magnetic moments of Mn and Fe individually (see Methods). The measurement set-up and our major finding are illustrated in Fig. 2. The applied magnetic field B, k-vector of incoming X-rays and surface normal are parallel to each other in normal incidence geometry (Fig. 2a). Figure 2b–e shows the XAS/XMCD spectra acquired at normal incidence, at T=2.5 K, and in static magnetic fields of various strengths of 0 and 6.8 T, respectively. In absence of an external magnetic field, that is, for B=0 T (after being magnetized at 6.8 T at 2.5 K), the XMCD spectra demonstrate remanent magnetic moments on both species with out-of-plane orientation (Fig. 2b,c), substantiating the direct observation of long-range magnetic order in a 2D spin-bearing molecular layer. Interestingly, the magnetic moment on the Fe ion of FeFPc molecules is aligned antiparallel to that on the Mn ion of the MnPc molecules, as seen by the opposite signs of the XMCD signals. The antiparallel alignment of Fe and Mn magnetic moments hints at an antiferromagnetic coupling between Mn and Fe ions in the nearest-neighbour positions. Applying a magnetic field of B=6.8 T, both magnetic moments of MnPc and FeFPc are found to align parallel with the field (Fig. 2d,e). The remanent magnetization of FeFPc molecules can also be readily recognized from the XMCD peak height versus B curve as a discontinuity at B∼0 T (Fig. 2f). The XMCD peak height versus B curve of the MnPc molecules on the other hand, does not display an easily recognizable discontinuity at B=0 T. It is noteworthy that data points in the curve taken at small fields are more susceptible to noise due to the continuous measurement protocol that causes spiky behaviour of the total electron yield (TEY) around zero field (see Methods) and due to excitations induced by the X-ray beam radiation. Although some points close to zero field might suggest a ferromagnetic coupling (open down triangles in Fig. 2f), the antiferromagnetic coupling and remanence (at B=0 T) of MnPc moments is confirmed by the XMCD spectra shown in Fig. 2c. The antiparallel alignment of the magnetizations on the FeFPc and MnPc sublattices persists for magnetic fields smaller than ∼2 T. At B=2 T, the corresponding Zeeman energy on FeFPc molecules and the long-range magnetic coupling energy becomes comparably large and the net magnetization of the Fe ions goes to zero (Fig. 2f). Applying the mean field approximation to the bipartite Ising model with nearest-neighbour interaction, we estimate the strength of the magnetic coupling from the XMCD peak height versus B curves to be about J Fe−Mn =0.12 meV, which leads to an ordering temperature of T C ∼3.7 K. The full lines shown in Fig. 2f depict the fits of the XMCD curves to the mean field approximation model without anisotropy terms; the latter could play a role for small magnetic fields. XMCD spectra measured at T=5 K do not show remanence, which confirms the estimated critical temperature (cf. Supplementary Fig. 1f,g). To elucidate the role of the Au(111) substrate on the magnetic ordering, we prepared an FeFPc+MnPc supramolecular array assembled on Ag(111) substrates, which, similar to the Au(111) substrates, host Shockley-like surface states; however, with very different Fermi wave vectors k F and Fermi density of states27, crucial for the long-range RKKY coupling, which depends as on the distance d between magnetic centres30. Our XMCD spectra acquired on the FeFPc+MnPc/Ag(111) system (see Supplementary Note 3 and Supplementary Fig. 4) show no remanent magnetization on either molecules, which suggests the pivotal role of the Au(111) substrate on the emergent long-range order.

Figure 2: Observation of long-range ferrimagnetic order in a 2D supramolecular layer. (a) Sketch of the experimental set-up, in which the XMCD and XAS measurements are performed in normal incidence geometry, with the external magnetic field B and the k-vector of the X-rays parallel to the surface normal, that is, [111] direction. Illustration of the ferrimagnetically ordered molecular spins in the ground state (B=0 T) and the ferromagnetically aligned spins at B=6.8 T. (b,c) XAS/XMCD spectra measured at the Fe L 3,2 edge and B=0 T demonstrate remanent magnetic moments of FeFPc molecules, which are aligned antiparallel to the remanent magnetic moments observed for the MnPc molecules. (d,e) In the applied external magnetic field of B=6.8 T, both molecular spins are aligned parallel to the applied field. (f) The measured individual XMCD peak height versus B curves of FeFPc and MnPc molecules show long-range order with antiferromagnetic coupling between the two sublattices. For the FeFPc molecules, the magnetic moment becomes aligned with the applied field for B≥2 T, as is evidenced by the zero crossing, which appears when the Zeeman energy wins over the FeFPc-MnPc antiferromagnetic exchange coupling. Open down triangles depict the data points at B∼0 T that possess higher noise level due to the measurement protocol (see Methods). All measurements were performed at T=2.5 K. Full lines show the fit of the XMCD data to a Brillouin function, adopting a mean field approximation without anisotropy terms (m(Mn)=2.3 μ B , m(Fe)=−1.2 μ B and J Fe−Mn =0.12 meV). Full size image

The XMCD signals of FeFPc and MnPc measured at normal and grazing incidence (shown in Supplementary Fig. 2) were analyzed with the sum-rule method. In the magnetic field oriented state (B=6.8 T), we resolve sizable induced spin magnetic moments, which are larger when the magnetic field is applied in-plane as compared with out-of-plane (cf. Supplementary Table 1). In contrast, the XMCD spectra acquired at remanence in normal incidence geometry are dominated by the orbital magnetic moments (m L =−0.39 μ B for FeFPc and m L =0.28 μ B for MnPc). The unequal sizes of the moments together with the observed antiparallel coupling establish a ferrimagnetic ground state in the 2D supramolecular lattice. The dominating contribution of the orbital moment can be recognized already from the fact that the L 3 and L 2 XMCD peaks (Fig. 2b,c and Supplementary Table 1) have the same sign (cf. ref. 6). The shape of the XMCD spectra is consistent with earlier XMCD measurements on FePc films31,32 and on an MnPc thin film33. Previously, thick films of FePc molecules were reported to exhibit magnetic anisotropy with in-plane easy axis31. Remarkably, here we have observed the first remanent magnetization in the out-of-plane direction for MnPc and FeFPc molecules. Such re-orientation of the predominant magnetization direction is likely to be related to a change of the molecular symmetry upon adsorption onto the Au(111) surface; specifically, here we find a symmetry change from D 4h to C 4v and C 2v for MnPc and FeFPc, respectively (see Supplementary Note 5). A similar change in the magnetic easy axis from in-plane to out-of-plane direction was observed within STS data taken on the FePc species adsorbed on oxygen-reconstructed Cu surfaces34. Furthermore, using the XMCD technique a reduction of magnetic moment anisotropy in FePc molecules was observed for the FePc/graphene/Ir system35. Our observations suggest that a change in magnetic anisotropy is most likely to be caused by the symmetry reduction; however, other contributions, as for example the hybridization of d-orbitals or the balance of in-plane and out-of-plane orbitals of the d-electrons should be taken into account.

STS measurements

The local Kondo interactions on both molecules have been investigated by the STS technique with the tip positioned above the centre of the molecules. Pronounced Kondo resonances have previously been reported for magnetic molecules on non-magnetic substrates15,18,21,36, caused by spin flips of the conduction electrons in vicinity of the magnetic impurity. This many-body renormalization of the energy levels leads to a sharp Kondo resonance, which is observed at the Fermi level for S=1/2 spins17. In our case, both molecular species comprise magnetic centres; however, of higher spins. Figure 3 shows differential conductance spectra (dI/dV) acquired above both FeFPc and/or MnPc molecules around the Fermi energy, that is, around zero bias, in the temperature range of 2.6–9.0 K. The spectra of both molecules exhibit zero bias anomalies, that is, a dip-like feature in the spectra of FeFPc molecules and a step-like shape for the MnPc species, well in line with data reported previously21,37. Measurements of differential conductance spectra in the temperature range 2.6–9.0 K result in a smearing and gradual suppression of these features towards higher temperatures confirming their Kondo character18. The strength of the Kondo coupling is often expressed in terms of the Kondo temperature that is extracted from a fit of the temperature-dependent spectra. The fit to the STS data using a modified Frota function (for details, see Supplementary Note 4) is shown as red curves in Fig. 3. The fitted line shapes resemble the measured data well. In our case, the Kondo temperature of MnPc and FeFPc molecules is T K =9.4±1.8 K and T K =9.2±2.0 K, respectively (see Supplementary Fig. 5). These values are somewhat lower/higher compared with those reported for single MnPc (T K =36 K) and FePc molecules (T K =2.6 K) on Au(111)21,37. Importantly, the main outcome of our STS and XMCD measurements is the emergence of long-range ferrimagnetic order in the supramolecular Kondo lattice at temperatures well below the Kondo temperature of ∼10 K.

Figure 3: STS measurements of FeFPc and MnPc. (a) The temperature-dependent differential conductance dI/dV spectra acquired above the centre of the FeFPc molecules show Kondo features around zero bias voltage. The dip-like feature measured on the centre of the FeFPc molecules broadens and becomes shallower with increasing temperature. (b) Spectra acquired above the centre of the MnPc species show a step-like shape, which is a signature of the Kondo resonance that broadens and vanishes with increasing temperature. Red full curves are fits to the temperature dependent dI/dV spectra with a modified Frota function17 to determine the Kondo temperatures. The inset shows the area where the spectra where acquired, with a scale bar of 2 nm. Full size image

Density functional theory +U calculations

To examine the origin of the observed long-range magnetic order, we have performed electronic structure calculations using the density functional theory with on-site Coulomb U correlations (DFT+U) added to capture the strong d-electron interactions at the Fe and Mn centres (see Methods). The supramolecular checkerboard pattern is described by a periodic simulation-cell containing 2 × 2 molecules, whereas the Au(111) substrate is modelled with three atomic Au layers each consisting of 120 Au atoms. The atomic positions have been optimized by complete self-consistent relaxation of all forces in the 588-atom simulation cell. We find an optimized Fe–Mn distance of 14.35 Å, in agreement with the 13.9±0.3 Å, which we measured with STM. Importantly, the calculations predict a ground state with antiparallel coupling between the spin moments on nearest-neighbour FeFPc and MnPc molecules, and a ferromagnetic coupling of each of the species to their next-nearest neighbour. To confirm the lowest total energy of this predicted ferrimagnetic arrangement we have performed total energy calculations assuming an entirely ferromagnetic state and found it to have a higher total energy by 9 K (0.78 meV) for the structural unit cell (that is, one FeFPc+MnPc pair), in accordance with the XMCD measurements. This energy corresponds to a nearest-neighbour Fe–Mn exchange interaction of J Fe−Mn ≈0.10 meV (see Supplementary Table 2), consistent with the estimated experimental value. The ab initio computed spin density, shown in top view in Fig. 4a, attests the antiparallel coupling between the FeFPc and MnPc spin moments. The computed 3d spin-only moments of −2.1 and 3.7 μ B , for FeFPc and MnPc (see Supplementary Note 5 and Supplementary Fig. 6 for 3d-orbital density of state plots), respectively, are in good agreement with those determined by XMCD (of 1.3 and 3.8 μ B , in the field-aligned state, see Supplementary Note 2 and Supplementary Fig. 3). The measured spin magnetic moment of FeFPc is reduced from the free molecule value S=1 (computed as spin moment m S =2.1 μ B , DFT+U) to S∼1/2 (m S =1.3 μ B , obtained from XMCD), which indicates an underscreened Kondo effect in the limit of strong coupling20. Our experimental measurements further provide the practically unperturbed magnetic moment of the MnPc (S∼2) molecule, m S =3.8 μ B , which is in good agreement with the value obtained from our DFT+U calculations, m S =3.7 μ B .

Figure 4: Results of density-functional theory-based calculations. (a) Top-view of the DFT+U computed spin densities of the 2D supramolecular layer on the Au(111) substrate. The red iso-surfaces depict the positive spin density on the Mn atoms of the MnPc molecules and the green iso-surfaces the negative spin density on the Fe atoms of the FeFPc molecules. It is noteworthy that the spin densities on the ligand atoms are opposite to those on the metal centres. (b) Side view of the spin density plot at an enlarged iso-density value (5 × 10−3 eÅ−3), which shows the interaction of the spin magnetization on the metallo-phthalocyanines with the Au substrate atoms. At this iso-density value, the dominant spin polarization is clearly visible. The direction of the side view is given by the arrow in a. Colour code: yellow atoms depict the Au(111) substrate atoms, brown the carbon atoms, blue the nitrogen atoms, white the hydrogen atoms and orange the fluorine atoms. Full size image

To determine the origin of the ferrimagnetic coupling we have performed additional calculations with the Au substrate removed. Without the Au substrate, no notable magnetic interaction between the neighbouring molecules was obtained, establishing that the molecule–molecule spin coupling is not due to a weak exchange interaction mediated by the overlap of phthalocyanine ligands. Rather, the electrically conductive Au(111) substrate is pivotal for mediating the magnetic interaction between FeFPc and MnPc, which implies that the RKKY interaction is responsible for stabilizing the long-range magnetic order. In Fig. 4b, we show a side view of the calculated magnetization density revealing that the electron density underneath the metal-ion centres exhibits a small spin polarization. This spin-polarized lobe is opposite to the dominant spin polarization resolved on the adjacent molecular centre, evidencing an antiferromagnetic coupling between mobile substrate electrons, predominantly the surface electrons, and the molecular spin.

The magnetic dipole–dipole interaction has previously been proposed as a possible source of coupling between spins that could stabilize long-range magnetic order in pure 2D systems7,8. For the here studied supramolecular layer, the dipole–dipole interaction between FeFPc and MnPc moments is however extremely small; for the remanent moments on two nearest neighbours at a distance of 14 Å, the corresponding dipolar energy is <2 × 10−9 eV. The dipolar interaction could thus support a long-range ordering temperature of the order of 10−5 K. This is obviously much less than our observed temperature of a few Kelvin and hence the dipolar interaction can be excluded as source of the observed long-range ferrimagnetic order. Lastly, we mention that a different form of magnetic order in a pure 2D spin system could result from Kosterlitz–Thouless topological order, caused by chiral magnetic vortices38. Such topological vortex phase would however have a zero mean magnetization, in contrast to the non-zero XMCD signals that we measured at T=2.5 K.