Valence state determines the band magnetocrystalline anisotropy in 2D rare-earth/noble-metal compounds

In intermetallic compounds with zero-orbital momentum ($L=0$) the magnetic anisotropy and the electronic band structure are interconnected. Here, we investigate this connection on divalent Eu and trivalent Gd intermetallic compounds. We find by X-ray magnetic circular dichroism an out-of-plane easy magetization axis in 2D atom-thick EuAu$_2$. Angle-resolved photoemission and density-functional theory prove that this is due to strong $f-d$ band hybridization and Eu$^{2+}$ valence. In contrast, the easy in-plane magnetization of the structurally-equivalent GdAu$_2$ is ruled by spin-orbit-split $d$-bands, notably Weyl nodal lines, occupied in the Gd$^{3+}$ state. Regardless of the $L$ value, we predict a similar itinerant electron contribution to the anisotropy of analogous compounds.


I. INTRODUCTION
The spin-orbit coupling (SOC) is responsible for the splitting of a 4f n orbital into multiplet states |JJ z defined by the total angular momentum J and its projection J z on the magnetization direction. If these levels are accessible by the crystal field energies, quenching of J z can occur. The orientation dependence of this mixing of states is at the origin of the strong magnetocrystalline anisotropy of systems based in rare earths (RE) [1,2]. In recent years, magnets in the zero-dimensional (0D) limit have been successfully realized using REs, either as isolated ad-atoms [3][4][5][6][7] or as centers in organic molecules [8][9][10][11][12][13][14]. Prior to these achievements it was known that the 4f electron shell confers to bulk intermetallic RE compounds uniaxial anisotropy and high Curie temperatures T C (and consequently large coercive fields) [15], where the magnetic coupling follows a RKKY mechanism [16]. In two-dimensional (2D) rare-earth/noblemetal (RE/NM) surface compounds large T C values are retained [17][18][19][20][21][22]. Anisotropy is a requisite for stable longrange magnetic ordering in 2D [23][24][25]. Magneto-optic Kerr effect (MOKE) and X-ray magnetic circular dichroism (XMCD) studies reveal that the magnetocrystalline anisotropy in these compounds does not follow a common trend. For example, the HoAu 2 monolayer (ML) grown on Au(111) is strongly anisotropic with out-ofplane (OOP) easy axis, whereas the GdAu 2 one is more easily magnetized in-plane (IP). The HoAu 2 case can be understood in terms of the 5 H 8 state given by Hund's rule for 4f 10 (Ho 3+ ), since the oblate shape of the orbital and the surface charge distribution favour an OOP easy axis [9,22]. However, explaining the GdAu 2 anisotropy demands an extension of the model. In fact, the 4f shell of Gd 3+ is half-filled, i.e. the total orbital quantum number is L = 0 ( 8 S 7/2 ground state), and the hybridization with the surface bands is negligible, and therefore the anisotropy must have a different origin.
At half-filling 4f 7 , classical dipolar interactions tend to dominate the anisotropy in bulk Gd compounds [26,27], yet the magnetocrystalline contribution is not fully quenched, as observed in metallic hcp Gd [28]. The latter results from the spin polarization of the 5d conduction electrons, which are also subject to SOC [29,30]. This leads to sizable magnetocrystalline anisotropy energy (MAE) values and non-zero orbital momenta [31].
In this work, we show that the magnetocrystalline anisotropy of RE-NM 2 MLs is not defined by the 4f single-orbital anisotropy only, but there exists an additional term originated at the itinerant electrons. Since all the compounds of the RE-NM 2 family display similar band dispersion features, the calculations presented here for L = 0 systems, namely EuAu 2 and GdAu 2 , predict that the itinerant electrons contribute to the MAE with ≈ 1 meV in general and that the RE valence state determines whether this contribution favours an OOP or IP easy axis of magnetization. This means that, for RE materials with a large 4f single orbital anisotropy, the total anisotropy will not depend on the band dispersion. For other RE metals, however, the band dispersion may define the magnetic anisotropy. We show that Eu in the EuAu 2 ML on Au(111) behaves as a divalent species and thus it is nominally in a 8 S 7/2 ground state, as Gd 3+ in the GdAu 2 ML. However, an EuAu 2 ML presents an OOP easy axis of magnetization and MAE = 1.6 meV, in contrast to the IP easy axis observed in GdAu 2 . The IP easy axis of GdAu 2 is explained in terms of the SOC lifting degeneracies in dispersive valence bands with Gd(d) character. On the other hand, in divalent EuAu 2 the Eu(d) states are unoccupied and do not contribute. Instead, the anisotropy is caused by the strong f − d band hybridization.

A. Experimental Methods
Sample preparation of EuAu 2 has been carried out by thermal deposition of Eu onto a clean Au(111) single crystalline surface. The formation of the monolayer is achieved when a complete layer with moiré is observed in Scanning Tunneling Microscopy (STM), or the surface state emission of the Au(111) Shockley state in angleresolved photoemission spectroscopy (ARPES) [32,33] has completely vanished. The substrate temperature is around 675 K for GdAu 2 , whereas for EuAu 2 a precise temperature of 575 K is required. The prepared EuAu 2 and GdAu 2 ML systems reveal a √ 3 × √ 3R30 • atomic arrangement and a long range order moiré lattice with respect to the underlaying Au(111) substrate. This superstructure is easily distinguishable in low-energy electron diffraction (LEED) images and serves as a quality indicator in XMCD synchrotron preparations.
The EuAu 2 XMCD measurements were performed at Boreas beamline of the Spanish synchrotron radiation facility ALBA using a 90% circularly polarized light from a helical undulator. The measurements were undertaken at 2-20 K with a variable magnetic field up to ±6 T. The applied magnetic field H was aligned with the photon propagation vector. The XMCD spectrum is the difference between the two X-ray absorption spectroscopy (XAS) spectra recorded with opposite orientation of the magnetic field and/or the circular helicity of the light, which we call µ + and µ − for simplicity. The XMCD signal is proportional to the projection of the magnetization in the direction of the applied magnetic field. At normal light incidence, the field is also normal to the sample surface (out-of-plane geometry) while at grazing incidence (here 70 • ), the magnetic field is nearly parallel to the surface (in-plane geometry). The magnetization curves are taken by varying continuously the applied field with the sample kept in one of the mentioned geometries for both circular helicities. For normalization issues, at each field value two XAS absorption values were taken, the first one corresponding to the photon energy of the maximum of the XMCD signal, the second one at a slightly lower photon energy prior to the XAS absorption edge. In order to determine the Curie temperature T C of the material such magnetization curves were taken for several temperatures below and above T C .
ARPES experiments were taken at CASSIOPEE beamline of SOLEIL synchrotron, France, and at our home laboratory in San Sebastián (Spain) together with the STM and LEED analysis. Photoemission data in San Sebastian were acquired using Helium Iα (hν = 21.2 eV) light. In San Sebastián and at SOLEIL a channelplate-based display type hemispherical analyzer was used (Specs 150 and Scienta R4000 electron analyzers) with angular and energy resolution set to 0.1 • and 40 meV, respectively. At the synchrotron, p-polarized light was used and the sample temperature during measurements was 70 K, while sample temperature during Helium Iα ARPES measurements were 120 K. Resonant photoemission spectroscopy (ResPES) is achieved at the Eu 4d →4f absorption edge. In such measurements the photoemission signal is resonantly enhanced due to a superposition/interference of the direct photoemission process and an Auger decay and leads to a broad resonant maximum above the 4d absorption threshold accompanied by a number of narrow peaks caused by several decay processes that was initially explained for the 4f 7 configuration of Gd [34] and later for the same configuration of Eu [35]. In mixed-valent Eu compounds slightly different resonant energies [36] can be used to differentiate di-and tri-valent contributions. Here, we used hν = 141 eV and 146 eV corresponding to two close photon energies in order to minimize photoemission cross-section changes but still resonantly enhance di-or tri-valent signals. The offresonant energy hν = 130 eV corresponds to a simple (4f 7 ) 8 S 7/2 → (4f 7 ) 7 F J photoemission transition. The individual J components cannot be resolved easily at such high photon energies due to worse energy resolution, but have been observed for pure Eu metal at low photon energies [37].

B. Theoretical Methods
Density-functional theory (DFT) calculations were performed in the full-potential linearized augmented plane wave (FLAPW) formalism [38][39][40] at the GGA+U level to describe strong correlation [41,42] in the fulllocalized limit approximation, since the Eu(4f ) orbital is half filled [43]. The Perdew-Burke-Ernzerhof (PBE) exchange and correlation functional [44] was used. The parameter U = 5.5 eV is found to match the 4f bands binding energies observed in ARPES. In the case of the calculation parameters for GdAu 2 , we refer the reader to previous work in Ref. [21], where U = 7.5 eV was employed. The lattice constant of the model geometries was fixed to the EuAu 2 experimental value. For the supported models a √ 3 × √ 3R30 • supercell geometry was used with f cc stacking of atomic planes and an interlayer distance of 2.25Å between the flat EuAu 2 monolayer and the Au(111) substrate. For the local FLAPW basis, as in previous works with REAu 2 , RE-6s, 4f, 5d and Au-6s, 5d electrons were included as valence electrons, and RE-5s, 5p and Au-5p as linear orbitals. Partial wave expansions up to l max = 8 and 10 were set inside the Eu and Au muffin-tin spheres of radii 1.43 and 1.48Å, respectively.
SOC was included in the calculation both selfconsistently and in the force theorem perturbative approximation [45][46][47][48], i.e. without carrying out further self-consistent optimization of the charge density, for spins oriented in-plane (X) and out-of-plane (Z) [see Fig. 2(a)]. Using the force theorem, the MAE is computed as the energy diference between the band energies, i.e. it is a rigid-band approximation. The working principles of the method are described in Section III E. The MAE accuracy is highly dependent on the fine details of the band structure, specially the gap openings at band crossings and the Fermi level. Therefore, a fine sampling of the first Brillouin zone (BZ) and the sharpest possible Fermi-Dirac function are required. In this work, the MAE was converged with a tolerance of ∼ 0.1 meV using a smearing width σ = 5 meV for the Fermi level (firstorder Methfessel-Paxton method [49]) and a mesh of at least 25 × 25 k-points, with plane wave expansion cutoffs of 4 and 12 bohr −1 for the wavefunctions and potential, respectively.

A. Structure Characterization
The EuAu 2 ML is characterized by a hexagonal moiré superstructure visible in the scanning tunneling microscopy (STM) micrograph and the LEED pattern of Fig. 1(a). Similar moiré superstructures are found in other RE/NM metal surfaces [18,21,22,52]. The STM analysis of EuAu 2 yields a moiré superlattice constant of a m = (3.3 ± 0.1) nm with a coincidence lattice of (11.5 ± 0.3) × (11.5 ± 0.3) with respect to the Au(111) surface (a Au = 0.289 nm), in agreement with LEED. The • reconstruction on top of the Au(111) surface and a lattice parameter of 0.55 nm. The a m of EuAu 2 is 10% smaller than the one found in GdAu 2 , HoAu 2 , or YbAu 2 , which show a m ≈ 3.6 nm [18,22].

B. Valency Analysis by ResPES
The ResPES spectrum [see Fig. 1(b)] displays unequivocally the presence of a single Eu 2+ multiplet peak and no further peaks related to Eu 3+ revealing the presence of only divalent Eu atoms exclusively located at the EuAu 2 ML. Eu atoms below the surface (either di-or trivalent) would give rise to an additional shifted multiplet [36] that does not appear in the spectra. Therefore, the existence of a single EuAu 2 ML with divalent character is probed.  Fig. 1(d) are taken at different geometries and at the photon energy of the maximum M 5 XMCD peak while changing µ 0 H. The out-of-plane magnetization curves with the field applied perpendicular to the surface saturates at approx. 2 T, while the in-plane magnetization curve reaches saturation at much higher applied fields, close to 6 T. This clearly reveals that the easy axis of magnetization is perpendicular to the plane. Magnetization loops were recorded at various temperatures, allowing an Arrot plot estimation [21,54] of the Curie temperature of the surface compound of T C = 13 K [see inset of Fig. 1(d)]. Further details are found in Fig. S5.
The OOP easy axis of magnetization in EuAu 2 MLs differs from the IP one of GdAu 2 [20,21], despite of the identical 4f electronic configuration ( 8 S 7/2 ) and atomic structure. In the following, we explain this behavior in terms of their respective band structure measured by ARPES, which exhibits a clear dependence on the valence character of the REs, namely, divalent for Eu and trivalent for Gd. Further analysis by DFT provides insight in the spin-orbit effects on the individual bands.

D. Electronic Structure by ARPES and DFT
Results of ARPES measurements performed on EuAu 2 MLs show the characteristic surface dispersive localized bands with Eu(d)-Au(s, p) character that are commonly found in all REAu 2 surface compounds [19,21,22], see Fig. 2(d,e) and SM Figs. S1 and S2. Following the notation of Ref. [22], these valence bands are labelled A, B, C, and C ′ . As already seen in other REAu 2 materials, some of these bands are better detected in the first BZ while other bands gain intensity in higher order BZ's. The 4f emissions of Eu are seen in the binding energy range between 1.1 and 0.4 eV, in agreement with ResPES of Fig. 1(b). To better visualize the valence band structure, a low photon energy hν = 21.2 eV was choosen, where the 4f cross-section is relatively low. Signs of hybridization between the localized Eu 4f level and the EuAu 2 valence bands are observed as kinks in the linear dispersion of the C bands in the second BZ and are marked with arrows in Fig. 2(e) (see also Fig. S2). Such 4f -valence band hybridizations have been observed for other Eu-based bulk [55] and in YbAu 2 surface compounds [22]. The C band in EuAu 2 is characterized by a nearly conical shape atΓ, with the apex above the Fermi level E F . DFT band structure calculations for isolated and supported EuAu 2 MLs disclose the intersection above E F of the C band, of d xy character, with bands of d x 2 −y 2 and d xz,yz character, as shown in Fig. 2(c). These band crossings occur below (above) the Fermi energy for trivalent (divalent) RE ions [21,22]. Adding the 3 ML Au(111) substrate slab provide additional dispersive bands [21,22], which cause an upward shift of ≈ 0.1 eV in the RE(d)-Au(s) hybrid band manifold, and a strong renormalization of 4f levels. Indeed, in the calculated free-standing EuAu 2 band structure the 4f states lie ≈ 0.7 eV lower. The 4f band of EuAu 2 is half-filled but, as the individual bands are heavily split by hybridization with the A and C bands, it shows some degree of dispersion across the BZ. In contrast, Gd(4f ) orbital in GdAu 2 barely interacts with the surrounding metal, showing a small crystal field splitting and no significant dispersion (see SM Fig. S3).

E. Force Theorem Analysis
SOC at the crossings of the two-dimensional RE(d)-Au(s) bands are sources of anisotropy that we characterize by DFT in the following. The SOC matrix element for two electrons belonging to orbitals of the same shell is where l and s are the one-electron orbital and spin momentum operators; l, m and σ indices label the eigenstates of l, l z , and s, respectively; and ξ l is the SOC strength, which is approximately constant for all the electrons in each l-shell. Intershell SOC is negligible. For hydrogenic atomic orbitals, these matrix elements are analytical and have been tabulated [56,57]. Selected combinations of m values and spin orientations give non-zero matrix elements. A second-order perturbative treatment of the SOC hamiltonian term shows that this behaviour is inherited by hybrid bands of an extended system [58][59][60][61][62][63][64]. In such case, degeneracy lifting at band crossings obeys the rules imposed by the crystallographic symmetry of the system and the orbital symmetry of the bands for a given orientation of the spins.
In the so-called force theorem approach, where the SOC correction enters non-self-consistently in the DFT calculation, the MAE is obtained from the spinorientation-dependent band energy contribution to the total energy of the system [45,47]: (2) where the sum runs over the eigenenergies ǫŝ a kn (k and n are the k-point and band indices, respectively), f F D is the Fermi-Dirac function, and Eŝ a F are the Fermi energies for the spin orientationsŝ a = Z (out-of-plane) and X [the in-plane direction along nearest RE atoms, shown in Fig. 2(a)]. Azimuthal dependence of the MAE is small. According to this equation, non-zero contributions to the MAE are generated at band crossings that become gapped for one spin orientation, but not for the other. The final easy-axis direction results from the integration of all individual gap contributions, of positive and negative small values, over occupied states. At crossings of fully occupied bands, the energy dispersion around the degeneracy point has to be asymmetrical in order to have a sizable contribution to MAE, otherwise band energies around the gapped feature will cancel out each other. If the degeneracy lifting occurs exactly at the Fermi level, the contribution to the MAE will be large [47,65].
A spectral analysis of Eq. 2 is not possible, as the Fermi energy depends on the spin orientation. Instead, the common practice is to plot MAE vs. N e , i.e. MAE(N e ), where N e is the number of electrons with respect to the neutral case (N e = 0) calculated as an integral over the density of states of the system for each spin orientation, i.e. each N e value provides E X,Z F (N e ) values [64][65][66][67][68]. Therefore, the MAE(N e ) curve allows to asign a positive or a negative MAE contribution to the region of the energy range close to E X,Z F (N e ). As the Fermi energy fluctuation with the spin direction is small (a few meV), we use the E F (N e ) value calculated without SOC to guide the eye in the graphs. The value obtained at neutrality, MAE(N e = 0), corresponds to the expected anisotropy of the system. This type of MAE analysis relies on the validity of the force theorem approximation, namely, it assumes that there are only minor changes in the electron density (essentially, loss of collinearity) caused by the effect of SOC [46,47]. Despite the two REAu 2 compounds under study being heavy-atom systems, by comparing to fully self-consistent calculations of the band structures with SOC, we find that the force theorem properly accounts for the spin-orbit induced gaps in an energy range of several eV around the Fermi energy (see SM Figs. S6 and S7 for GdAu 2 and EuAu 2 , respectively).

GdAu2
We apply this analysis to the band structure of freestanding and supported GdAu 2 MLs, shown in Fig. 3(bc). The band crossings labelled α and α ′ in the figures are degenerate for in-plane magnetization, but split by OOP magnetization. Fig. 3(a) panels shows, as a guide for the eye, the orbital characters and spin polarities in the absence of SOC for free-standing GdAu 2 . The α and α ′ features are crossings between d xy and d x 2 −y 2 (m = ±2) with equal spin polarization, a situation where the matrix element Eq. 1 foresees splitting by OOP magnetization. The Weyl nodal line β [69], which is a ring-shaped crossing around the Γ point between d xy and d x 2 −y 2 bands with opposite spin polarization [see Fig. 3(a)], is split by in-plane magnetization according to Eq. 1. Fig. 3(d) shows the obtained MAE(N e ). The curves E F (N e ) − E F (0) without SOC, shown in the SM Fig. S9 establish an approximate mapping between band fillings and binding energies [a few discrete values are given also in the right-hand side axes of Fig. 3(b,c)]. For GdAu 2 close to neutrality (N e = 0) the essential contributions to the MAE are those of α, α ′ and β features. In the free-standing case, the β nodal ring results in the net sharp MAE peak of negative values in the dashed red line at N e = −0.9 (i.e. contributing to IP easy axis of magnetization), whereas features α and α ′ contribute to the OOP easy magnetization axis with peaks of positive MAE at N e = −1 and just above N e = 0 in the red dashed curve of Fig. 3(d). At neutrality we obtain an OOP easy axis with MAE(0) = 0.17 meV.
The same features α, α ′ and β appear for GdAu 2 /3 ML Au(111), albeit at binding energies higher by ∼ 0.1 eV [see Figs. 3(b,c)], which also result in positive and negative contributions to the MAE around the neutrality point. The β line causes the in-plane peak at N e = −0.5 (solid blue line), and α and α ′ contribute to the OOP magnetization easy axis with positive-valued MAE peaks at N e = −1.3 and 0.8, respectively. At the neutrality point we find MAE(0) = −0.9 meV, i.e. IP easy axis of magnetization, same as observed experimentally. This is due to the contribution of the β gapped line, which dominates over α and α ′ single-point gaps at Γ, aided by the fact that the β gapped line lies closer to E F in the supported case. The contribution of the GdAu 2 4f 7 electrons to the anisotropy lies within the force theorem error, since the departure from sphericity of this orbital is negligible when embedded in the alloy (in other words, the Gd(4f ) spin density cloud would be almost insen- sitive to the applied field orientation, as shown in SM Fig. S3). Therefore, we attribute the experimental easy magnetization plane essentially to the MAE contribution of the dispersive Gd(d)-Au(s) bands.

EuAu2
Next, we apply the force theorem methodology to the The force theorem Eq. 2 applied to the U = 5.5 eV band structures yields an OOP easy magnetization axis both for free-standing and supported EuAu 2 , with MAE(0) = 1.0 and 1.54 meV, respectively [these are, respectively, the values taken by the thick red dashed and blue solid curves, at neutrality in Fig. 4(a)]. The MAE peaks due to α ′ , β and α band crossing features are visible in the free-standing EuAu 2 bands above the charge neutrality level (see also Fig. S8). For EuAu 2 /3 ML Au(111), Fig. 4(b) shows two large broad peaks at each MAE(N e ) curve, which take values ≃ ±0.1 eV at the electron filling ranges that correspond to the binding energy ranges where the hybridized 4f bands lie. The peak pairs account for the filling of the SOC-split band manifolds 4f 7/2 and 4f 5/2 . In the calculation with U = 5.5 eV, these peaks occur at N e = −10 and −5, respectively, corre-sponding to binding energies close to -1 eV [see also SM Fig. S9(b,c)]. These bands yield a non-negligible positive residual contribution to MAE(0) by virtue of the f − d hybridization. In a single-ion picture, this would be interpreted as a loss of sphericity of the half-filled 4f shell, which acquires a net non-zero L [9]. The SM Fig. S3 shows the magnetization distribution projected onŝ a [mŝ a (r)] of the Eu(4f ) shell embedded in the freestanding alloy monolayer, obtained by integration of the Kohn-Sham states in the binding energy range between -2.25 and -1.25 eV. The magnetization anisotropy distribution m X (r) − m Z (r) takes positive and negative values in-plane and out-of-plane, respectively, which can be interpreted as the Eu(f ) orbital spin density tendency to become deformed along the applied field direction. Note that the same quantity integrated for Gd(f ) between -10 and -8.5 eV is an order of magnitude smaller.
The 4f contribution to the MAE of EuAu 2 can be further probed by modifying the U correlation parameter. We have carried out force theorem analyses for supported EuAu 2 with U = 3.5 and 7.5 eV. When U is decreased, the MAE energy is increased by ≃ 1 meV and when U is increased, the easy axis of magnetization remains out-ofplane, but the MAE(0) value is reduced (see Fig. 4). The densities of states [SM Fig. S9(d)] show that a larger U value implies less hybridization between the 4f electrons and the hybrid bands of d character crossing the Fermi level. Indeed, for U = 7.5 eV the 4f band lies below E F − 1.5 eV, i.e. close to the Au substrate states and below the d xy band. For U = 3.5 eV, however, the 4f band is centred at E F − 0.5 eV, producing hybrid f − d states close to the Fermi level. This shows that the degree of f − d hybridization drives the MAE behaviour of EuAu 2 .

IV. CONCLUSIONS
To summarize, our XMCD experiments have determined that a EuAu 2 ML on Au(111) shows out-of-plane easy magnetization axis. Using photoemission experiments combined with DFT calculations, we show that this is a consequence of the Eu 2+ valence state and the f − d band hybridization. In contrast, the easy magnetization plane of GdAu 2 ML is explained by Gd 3+ valence and spin-orbit splitting of the point and line degeneracies of bands with d xy,x 2 −y 2 character. The role of the RE valence is to determine whether these specific states are occupied or empty. This model of the contribution of the itinerant electrons to the magnetocrystalline anisotropy is a general result for the REAu 2 family of monolayer compounds. For RE=Eu,Gd this is the only mechanisms at work, since the 4f orbital is half-filled, i.e. L = 0. For RE atoms with non-zero L quantum numbers, there is a single-ion anisotropy that results from the interplay between spin-orbit and crystal-field splittings of the 4f multiplet. For example, this multiplet mechanism dominates the OOP easy magnetization axis behaviour in HoAu 2 , where the valence is Ho 3+ [22]. We predict the itinerant electrons to yield MAE values of ≈ 1 meV in REAu 2 monolayers, irrespective of the L value. Therefore, the band and multiplet mechanisms may eventually compete. for Eu(4f ) (h). The m Gd (r) sensitivity to the magnetization axis is smaller than that of m Eu (r) by an order of magnitude, which means that Eu(4f ) is more prone to be deformed by a magnetic field, i.e. it can provide non-negligible contributions to the MAE by departure of its spherical shape, as shown in the main paper Fig. 4  (a),(c) X-ray absorption spectra taken with left and right polarized light at T = 3K and with an applied field of µ0H = 6T with the field applied normal to the surface for EuAu2 and GdAu2, respectively. The resulting difference, the XMCD signals, are shown in the bottom part for fields normal to the surface θ = 0 • , at θ = 70 • and 60 • , respectively. (b),(d) The magnetization curves are obtained by acquisition of the XAS signal at the maximum of the XMCD signal (and a pre-edge value for normalization) varying continually the applied field from +6T to -6T and then from -6T to +6T for both circular light polarizations. The Curie temperature TC can be extracted by the Arrott plot analysis [6,7] of magnetization curves taken at several temperatures around TC . As a result we obtain TC = 13K and 19K for the EuAu2 and GdAu2 surface compounds, respectively. Most important here is the change in the anisotropy, resulting in an out-of-plane easy axis of magnetuzation for Eu atoms in EuAu2 and an in-plane easy axis for Gd in GdAu2. FIG. S6. Self-consistent and force-theorem bands for free-standing and supported GdAu2 structures. Band structures of free-standing and supported GdAu2 are shwon for spins aligned to in-plane (panels labelled as SOC X) and out-of-plane (SOC Z) directions. The eigenenergies obtained for SOC included in the force theorem approximation (labelled FT, red) are compared with the self-consistent ones (SCF, blue). Each individual calculation is referred to its own Fermi level (note that the Fermi energy changes if the spin orientation changes). Agreement between FT and SCF calculations in the case of a 3 ML Au substrate is reduced with respect to that of free-standing GdAu2 due to the intense SOC at the substrate bands, which introduces small overall shifts in the band structure. Despite this, the overall band structure, and in particular the gap openings, for the two spin directions are satisfactorily reproduced by the force theorem method.  Note that the gap openings that were relevant in the anisotropy of trivalent GdAu2 (labelled α, α ′ and β) lie above the Fermi level in the EuAu2 case and therefore do not contribute to the observed MAE.