Research Article

Reaching the magnetic anisotropy limit of a 3d metal atom

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Science  30 May 2014:
Vol. 344, Issue 6187, pp. 988-992
DOI: 10.1126/science.1252841

Maximizing atomic magnetic memory

A study of the magnetic response of cobalt atoms adsorbed on oxide surfaces may lead to much denser storage of data. In hard drives, data are stored as magnetic bits; the magnetic field pointing up or down corresponds to storing a zero or a one. The smallest bit possible would be a single atom, but the magnetism of a single atom —its spin—has to be stabilized by interactions with heavy elements or surfaces through an effect called spin-orbit coupling. Rau et al. (see the Perspective by Khajetoorians and Wiebe) built a model system in pursuit of single-atom bits—cobalt atoms adsorbed on magnesium oxide. At temperatures approaching absolute zero, the stabilization of the spin's magnetic direction reached the maximum that is theoretically possible.

Science, this issue p. 988; see also p. 976


Designing systems with large magnetic anisotropy is critical to realize nanoscopic magnets. Thus far, the magnetic anisotropy energy per atom in single-molecule magnets and ferromagnetic films remains typically one to two orders of magnitude below the theoretical limit imposed by the atomic spin-orbit interaction. We realized the maximum magnetic anisotropy for a 3d transition metal atom by coordinating a single Co atom to the O site of an MgO(100) surface. Scanning tunneling spectroscopy reveals a record-high zero-field splitting of 58 millielectron volts as well as slow relaxation of the Co atom’s magnetization. This striking behavior originates from the dominating axial ligand field at the O adsorption site, which leads to out-of-plane uniaxial anisotropy while preserving the gas-phase orbital moment of Co, as observed with x-ray magnetic circular dichroism.

Magnetic anisotropy (MA) provides directionality and stability to magnetization. Strategies to scale up the MA of ferromagnetic 3d metals have relied on introducing heavy elements within or next to the ferromagnet in order to enhance the spin-orbit coupling energy. Rare-earth transition-metal alloys, such as TbCoFe (1), and binary multilayers, such as Co/Pt and Co/Pd (2), are used as magnetic recording materials because of their large perpendicular MA (3). Recent experiments, however, have shown that Co and Fe thin films deposited on metallic oxides such as AlOx and MgO present MA energies on the order of 1 meV/atom (4, 5), which is similar to that of Co/Pt interfaces but driven by the electronic hybridization between the metal 3d and O 2p orbitals (6, 7). Perpendicular magnetic tunnel junctions, including CoFeB/MgO layers, are being intensively investigated for nonvolatile MRAM (magnetic random access memory) applications (5, 8, 9), in which the lateral dimensions of a magnetic bit approach 20 nm (10).

A fundamental constraint to the downscaling of magnetic devices is the total amount of MA energy that can be induced in the storage layer, which limits its thermal stability factor and influences the rate of magnetization switching (11). As the dimensions of a magnetic bit shrink to the atomic scale, quantum-mechanical excitation and relaxation effects, which greatly affect the magnetization can come into play. We explore the limit of how much MA can be stored in an atom and for how long it can retain a given spin state in a model system of a single Co atom bound to an MgO layer. We show that this “bit” achieves the maximum possible MA energy for a 3d metal. This MA limit is ∼60 meV, set by the atomic spin-orbit coupling strength times the unquenched orbital angular momentum. We measured spin relaxation times on the order of 200 μs at 0.6 K and show that the rate-limiting relaxation step for a Co atom is determined by the mixing of excited spin states into the ground state induced by nonaxial ligand field components.

Magnetic Anisotropy in Quantum Systems

The microscopic origin of MA is the combined effect of the anisotropy in the atom’s orbital angular momentum (L), together with the interaction between L and the atom’s spin angular momentum (S). This interaction is given by HSOC = λL·S, where λ is the atomic spin-orbit coupling parameter. In solids and molecules, L tends to align along specific symmetry directions, set by the spatial dependence of the ligand field. The strength of the MA is defined here by the so-called zero-field splitting (ZFS) (12), which is the energy difference between the electronic ground state and the first excited state that has its spin pointing in a different direction with respect to the ground state, in the absence of an external field. For spin-flip transitions that leave L unchanged, the ZFS is thus proportional to λL, where λ is ~−22 meV for Co (13). However, in most magnetic compounds the orbital moment magnitude L is either quenched or strongly diminished by ligand field (14) and hybridization (15) effects, leading to MA energies on the order of 0.01 meV/atom in bulk magnets and up to ~1 meV/atom in thin films (16) and nanostructures (17). Achieving large ZFS in transition metals requires somehow breaking the spatial symmetry of the atomic wavefunctions without quenching the orbital magnetization. The most promising strategy to preserve the large L of a free atom and induce uniaxial anisotropy is to use low-coordination geometries, as shown for atoms deposited on the threefold coordinated sites of a (111) surface (18, 19) and molecular complexes and crystals with two-coordinate metal species (2024). This strategy, if specific conditions are met, can be brought to its limit by coordinating one magnetic atom to a single substrate atom.

We achieved this extreme by using cobalt, which has L = 3, the highest in the transition metal series, and a thin film of MgO as a substrate with a onefold (“atop”) coordinated site for adsorbed transition metal atoms (25). Co atoms were deposited on a single MgO layer grown on Ag(100) (2628). They appear as protrusions that are 0.15 ± 0.02 nm high when imaged with scanning tunneling microscopy (STM) (Fig. 1A). The preferred binding site, determined with density functional theory (DFT), is on top of oxygen (Fig. 1B) (26) with four Mg atoms as neighbors, resulting in C4v symmetry. Despite the presence of these four Mg atoms, the spin density of the valence electrons of the Co is rotationally symmetric around z (effectively Cv) (Fig. 1D). This axial coordination can preserve the orbital moment of the free atom along the vertical axis but quench it in-plane (Fig. 1C). The DFT density of states of the Co d-levels (Fig. 2) shows that the interaction with the Mg atoms is weak and the Co dx2y2, dxy orbitals remain largely degenerate. The dominant bond is between the out-of-plane d orbitals of Co and p orbitals of O, resulting in an uniaxial ligand field along z. DFT calculations further indicate that the Co atom is charge-neutral and has spin magnitude S = 1.39 ± 0.05.

Fig. 1 Co on MgO films.

(A) Constant current STM image of seven Co atoms on 1 ML of MgO on Ag(100) at T = 1.2 K (10 pA, 50 mV, 7.5 nm × 7.5 nm). Shown are schematic diagrams of STM (left) and XAS (right). (B) DFT-calculated structure and valence electron charge density of one Co atom atop an O atom in 1 ML MgO on Ag(100). Charge density color scale is in atomic units. (C) Schematic model of the orbital occupancy of Co in a free atom (left), in its 4F term (L = 3, S = 3/2), and Co on the MgO surface (right). The orbital moment is preserved along the easy-axis of the Co in this cylindrical ligand field (LZ = 3, SZ = 3/2). (D) Top view of ball model of the atomic structure (top) and DFT calculation of the Co atom spin density of the valence electrons (bottom). Oblique view shows contours of constant positive (red) and negative (blue) spin polarization.

Fig. 2 DFT spin-resolved partial density of states of the Co, O, and Mg levels.

The figure shows the large ligand field splitting induced by O ligation for the out-of-plane orbitals and also shows the degeneracy between the Co (dxz,dyz) and (dxy,dx2y2) orbitals. Large overlap between the Co dz2 and O pz as well as dxz, dyz and px, py orbitals indicates hybridization between Co and O, whereas no Co-Mg overlap is visible.

Measurement of the ZFS with STM

We used inelastic electron tunneling spectroscopy (IETS) (19, 2932) to probe the quantum spin states of the Co atoms (26). In such a measurement, electrons tunneling from the STM tip may transfer energy and angular momentum to a magnetic atom and induce spin-flip excitations above a threshold voltage. The IETS spectrum of a Co atom on 1 monolayer (ML) MgO at 0.6 K is shown in Fig. 3A. We observed a sudden stepwise increase in conductance at ±57.7 mV, symmetric around zero bias, as expected for an inelastic excitation. The dI/dV step is magnetic in origin and splits into two in an applied magnetic field (Fig. 3, B and C). For ease of discussion, we begin by approximating the magnetic state of the Co atom as an S = 3/2 system with uniaxial anisotropy (later in the paper, we will include the effects of configuration mixing and the presence of large orbital moment). We assign these excitations to transitions between the ground (Sz = ±3/2, labeled as states 0 and 1 in Fig. 3D) and excited states (Sz = ±1/2, states 2 and 3). At zero field, the states 0 and 1, as well as 2 and 3, are degenerate and yield identical excitation voltages (V02 = V13). The two steps shift in accord with Zeeman energies, with the 0 → 2 step shifting up and the 1 → 3 shifting down in energy with increasing magnetic field, to yield a well-resolved splitting of 1.8 ± 0.2 meV at 6 T.

Fig. 3 Magnetic excitations measured in STM at T = 0.6 K.

(A) Differential conductance spectrum (dI/dV). The tip is positioned above a Co atom on 1 ML MgO on Ag(100) (red) and bare MgO (brown). (B) Expanded view of the steps near 58 mV for 0 T (red) and 6 T (blue) (tip height setpoint 5 nA, 100 mV). (C) The step position as a function of magnetic field, showing the field-induced splitting. Only one step is resolvable at 0 T (red point). Dashed lines are linear fits to the data points. (D) Schematic energy level diagram. The states are labeled in order of increasing energy from 0 to 3. The arrows V02 and V13 indicate the transitions measured in IETS.

The IETS measurements reveal a remarkable ZFS of 57.7 meV between ground and excited states. The ZFS is much larger than the typical values of several millielectron volts reported before for single atoms on surfaces (19, 3033), which indicates an exceptionally high MA for Co on MgO. Moreover, the presence of the V13 step, in addition to the V02 step, at finite magnetic field is surprising because at low temperature (kBT << eV01, where kB is the Boltzmann constant and T is temperature) and low applied voltage (Vbias < V02) one would expect only state 0 (the ground state) to be occupied for an appreciable fraction of the time. The observation of the 1 → 3 transition for Co on MgO is an indication that the excited state 1 has a lifetime above 1 ns (the mean tunneling time between electrons at the measured currents).

Electronic Structure Probed with X-ray Absorption Spectroscopy

To understand the large energy and time scales revealed by the IETS measurements, we performed x-ray absorption spectroscopy (XAS) of isolated Co atoms deposited on 2 to 4 MLs of MgO on Ag(100). By measuring the excitation cross-section for 2p to 3d transitions, L-edge x-ray absorption spectra provide a probe of the bonding and the magnetic properties of transition metal ions (34) that is highly complementary to IETS. Spectra acquired at the L3 Co edge with circularly polarized light are shown in Fig. 4A (26). The XAS lineshape differs from that of Co atoms adsorbed on metal substrates (18, 35) as well as from typical CoO phases (36), showing that the bonding of Co is specific to the MgO surface. The x-ray magnetic circular dichroism (XMCD) intensity measured at normal incidence is larger than at grazing incidence (Fig. 4B), which implies that the Co magnetic moment has an out-of-plane easy axis. The XMCD spectra reveal a large orbital-to-effective spin moment ratio, in the range of 0.9 to 1.2, which indicates that L is very large on this surface. A discussion of the XMCD sum rule analysis (37, 38) and technical challenges related to x-ray–induced desorption on thin insulating films (39) is reported in (26).

Fig. 4 XMCD measurements and multiplet calculations.

(A) Experimental and simulated x-ray absorption spectra of Co/MgO/Ag(100) at normal (θ = 0°) and grazing (θ = 60°) incidence recorded over the L3 Co edge at T = 3.5 K and B = 6.8 T. The Co coverage is 0.03 ML. The spectra are the sum of positive and negative circular polarization, (I+ + I). (B) XMCD spectra (II+). The XMCD intensity is given as percentage of the total absorption signal shown in (A). (C) Out-of-plane magnetization versus field at 3.5 K measured by XMCD after saturating the sample at 6.8 T (black, red, and green squares) and −6.8 T (blue) at each point. Different colors refer to different samples. The solid line represents the expectation value of 〈Lz〉 + 〈2Sz〉 ≈ 6μB at 3.5 K. The inset (top left quadrant) compares fits between 3 (red curve) and 3.5 K (black). (D) Lowest energy levels obtained with the multiplet calculations as a function of ligand field, spin-orbit coupling, and applied magnetic field. The color code of the energy levels highlights the different orbital symmetry of the states: blue for E and red for B2. The two transitions seen in IETS are indicated by arrows.

To determine the electronic ground state and the structure of the lowest lying magnetic states, we simulated the x-ray experimental results using multiplet ligand field theory (34). The multiplet calculations include charge transfer (σ-donation) via the dz2 orbital and take into account the mixing between d7 and d8l configurations, where l describes a ligand hole on the O site. As shown in Fig. 4, A and B, there is excellent agreement between the simulated and experimental XAS and XMCD. The resulting d-shell occupancy is 7.44 electrons, which is in good agreement with the DFT results [7.27 electrons in a Löwdin analysis (26, 40)]. The evolution of the calculated Co states as a function of ligand field splitting and spin-orbit interaction is shown in Fig. 4D [the complete energy diagram is provided in fig. S3 (26)]. The lowest energy level (Fig. 4D, left edge) is an octuplet (blue) with Lz = ±3 ⊗ Sz = ±1.25, Sz = ±0.42, where the spin moment is slightly less than the free atom value of S = 3/2 because of mixing of the ground state 4F and 3F terms of the d7 and d8 configurations, respectively.

Origin of the ZFS

The electronic states of Co after including all interactions—namely, ligand field, spin-obit coupling, and external magnetic field—are shown on the right side of Fig. 4D. What is most unusual about the resulting spin doublet ground state is that it is composed of a mixture of states dominated by Lz = ±3 and thus has an orbital moment near the free atom limit. Unlike previous reports, such a large L for a surface-adsorbed transition metal atom was observed here because the ligand field is essentially uniaxial [it does not lift the degeneracy between the (dxz, dyz) or between the (dx2y2, dxy) orbitals], and both d7 and d8 configurations have the same orbital multiplicity so that configuration mixing—which takes place here, as it does on most substrates—does not reduce the magnitude of L.

The substantial orbital contribution can also be seen in the magnetization measured by XMCD as a function of applied field, which indicates a local moment of ∼6μB per atom (Fig. 4C) (where μB is the Bohr magneton). This result is in agreement with the magnetization μz = 〈Lz〉 + 〈2Sz〉 calculated by using the wave functions and energy levels obtained from the multiplet simulations (Fig. 4C, solid black line). Both experimental and theoretical curves saturate very fast, as expected for strong MA. At low magnetic fields, the measured values remain above the calculated values (Fig. 4C, inset), which could be the result of slow relaxation effects or induced magnetic moment contributions from the substrate atoms.

The multiplet energy diagram in Fig. 4D, derived from the model fit to the XAS data, provides a detailed interpretation of the IETS spectra. The calculated energy separation between the ground-state spin doublet (states 0 and 1) and the first excited spin doublet (states 2 and 3) at zero field is 55 meV, which closely matches the energy of the conductance step (V02 = V13 = 57.7 mV) measured with IETS (26). This level of agreement between XAS and IETS is remarkable considering that these are independent experiments that take place at radically different energy scales (hundreds of electron volts for the x-ray measurements as compared with millielectron volts for IETS).

The multiplet results establish that the separation of the first two spin doublets at 0 T is the ZFS seen in IETS spectra and explain its magnitude. The key is the nearly unquenched orbital moment of the lowest energy levels, which allows the Sz = ±3/2 states to be split maximally from the Sz = ±1/2 states by the spin-orbit interaction. In this case, the ZFS is equal to λL ΔSz, which for Co (with L = 3 and ΔSz = 1) gives λL ≈ 60 meV, reaching up to the full magnitude of the spin-orbit coupling energy intrinsic to a Co atom. This value is much higher than usually observed for transition metal systems, in which L arises as a perturbative effect because of spin-orbit coupling, and the ZFS has a second-order dependence on λ2 (41, 42).

Spin Lifetime Measurements

The ZFS defines the energy for the lowest-order process required to surmount the barrier that separates 0 and 1, the states with large and opposite magnetic moments. Our experiments are at low temperature (kBT << ZFS), which effectively suppresses thermal excitations of the magnetic moment over the MA energy barrier. These conditions offer the possibility to probe in detail nonthermal magnetization reversal mechanisms that become important when a magnet is scaled to atomic dimensions. In the case of magnetic atoms placed near electrodes (here, the Ag substrate and STM tip), spin relaxation can occur through ΔSz = ±1 transitions induced by electrons from these electrodes that scatter off the magnetic atom and either tunnel across the junction or return to the original electrode (43). These scattering processes result in quantum tunneling of the magnetization (44, 45). These mechanisms are extremely sensitive to the local environment, such as the electronic density of states of the substrate and distortions of the ligand field surrounding the magnetic adsorbates (45).

We now focus on measurements of the spin lifetime as a probe of the nonthermal decay mechanisms. The relaxation time T1 of excited spin states can be measured with spin-polarized STM with a pump-probe scheme (33). The current in a spin-polarized tunnel junction sensitively depends on the relative alignment of tip and sample spins (46). Thus, the tunnel current with the atom in the ground state is generally different from the current in an excited state. Sufficiently large pump pulses put the atom into excited spin states, from which it eventually decays back to the ground state. This decay was monitored with a probe pulse. Such a pump-probe measurement is shown in Fig. 5A with an exponentially changing current, yielding a lifetime T1 = 232 ± 17 μs at 1 T. To determine which state is giving the long lifetime signals observed here, we measured the amplitude of the pump-probe signal as a function of pump voltage (Fig. 5, B and C), which shows an onset of the signal at 59 ± 2 meV (Fig. 5B) and another sharp onset at 1.9 ± 0.1 meV (Fig. 5C). The first threshold is in good agreement with V02 and indicates when state 1 can be reached via state 2. The 1.9-meV threshold corresponds to the direct excitation 0 → 1, which is in agreement with V01 = 2(LZ + 2SZBB calculated from the multiplet model at 3 T, demonstrating that we are measuring the lifetime of state 1. This Zeeman splitting yields a total magnetic moment of 5.5 ± 0.3μB, which matches the magnetic moment determined from the XMCD measurements (Fig. 4C) (47), including the large orbital moment. The relaxation time remains independent of pump voltage, and we conclude that the measured T1 is always that of state 1; the other states decay too quickly to be observed.

Fig. 5 Relaxation time and excitation threshold of Co at T = 0.6 K.

(A) Pump-probe measurement of the excited state relaxation time at B = 1 T showing tunnel current as a function of delay time. The exponential fit (black line) yields T1 = 232 ± 17 μs. The data are taken with the tip height setpoint at Iset = 10 pA and Vset = 100 mV. The pulse sequence parameters are Vpump = 90 mV, Vprobe = 20 mV. (B and C) Pump-probe signal amplitude at B = 3 T as a function of pump voltage. For signal-to-noise reasons, the setpoint is Iset = 500 pA and Vset = 100 mV, which corresponds to the tip 0.2 nm closer to the atom than in (A). This gives T1 = 7.6 ± 0.1 μs (26). The vertical line at −59 mV in (B) shows the transition seen in dI/dV spectra. (C) The pump-probe amplitude for a smaller range of pump voltages. Linear fits (black lines) extrapolate to −2 ± 0.1 mV and +1.8 ± 0.2 mV at zero amplitude. Error bars are comparable with symbol size.

It is surprising that the pump signal is detectable for pump voltages below V02 because the ZFS is large, and quantum tunneling of the magnetization is forbidden in odd half-integer spin systems in the absence of a transverse magnetic field. However, the multiplet analysis shows that the weak distortion of the symmetry caused by the interaction of the Co with the Mg atoms mixes states from higher multiplets, mostly with |0,±1/2〉 character (Fig. 4D, in red), with the lowest states |±3,±3/2〉. Although this mixing is small (on the order of 4%) and does not change the total moments substantially, it allows the coupling of states 0 and 1 by a ΔSz = ±1 spin-flip transition between their |0,−1/2〉 and |0,+1/2〉 components, which can explain the observed quantum tunneling induced via substrate electrons. In addition, Co has a nuclear spin I = 7/2, which may facilitate otherwise prohibited electron spin relaxation. Tunneling of the magnetization because of hyperfine coupling could explain the lack of remanence in the magnetization curve measured with XMCD. However, the hyperfine coupling is usually effective at low fields (48) and is unlikely to be the cause of the spin relaxation observed for pump-probe experiments at B ≥ 1 T.

The spin lifetime of Co/MgO is much lower than that reported for electrons bound to shallow donors in Si (49) as well as that reported for Ho atoms on Pt (19), both exceeding a few minutes at cryogenic temperatures. However, it is very large for a transition metal atom, for which typical T1 times are on the order of 100 ns on insulating substrates (33) and 100 fs on metals (31). This difference can be attributed to the MgO layer serving two separate purposes. First, because the binding site symmetry preserves the orbital moment, state 0 and 1 are decoupled from each other not only by the large change in spin Sz, but also by the large change in Lz. Second, even a single MgO layer is very efficient in reducing the decay of the excited state by scattering with substrate electrons. This scattering rate could be tuned by increasing the number of MgO monolayers, while still being able to electrically probe the magnetic states. Furthermore, the presence of the STM tip imposes a limit on the lifetime, and the measured 200 μs value sets a lower bound on the intrinsic T1 of Co atoms on this surface.


This work elucidates the interplay between the MA, spin, and orbital degrees of freedom in systems at the border of free atoms and the solid state and highlights the atomistic limits on the miniaturization of magnetic systems. Additionally, this system realizes the single-atom analog of magnetic tunnel junctions based on perpendicular CoFeB/MgO layers. As such, it provides microscopic understanding of materials with strong perpendicular MA, which are required for further downscaling of spintronic devices (9, 10). Our measurements of ZFS and spin relaxation time demonstrate the advantages and impediments intrinsic to size reduction in such materials. Despite the very large MA, the strong coupling of d-electrons to the environment makes the spin lifetime of transition metal atoms very sensitive to perturbations caused by the ligand field and scattering from conduction electrons. Nonetheless, the large energy and time scales measured in this experiment indicate that relatively long-lived quantum states are possible for single Co atoms on MgO surfaces. Judging from the knowledge accumulated on magnetic tunnel junctions and this work, Co/MgO and possibly Fe/MgO represent a very favorable combination for the miniaturization of magnetic devices beyond the present technological limits.

On a more fundamental note, our results show that the combination of IETS and XAS is extremely powerful to describe the many-body interactions that determine the spin and the orbital degrees of freedom of magnetic atoms on surfaces, going beyond the spin Hamiltonian description successfully used in previous STM studies of nanosized magnetic structures (19, 30, 32, 33). Aside from a consistent description of the electronic and magnetic ground state, the role of nonthermal spin relaxation mechanisms can be determined based on independent input obtained through the multiplet analysis of the x-ray spectra and pump-probe measurements.

Supplementary Materials

Materials and Methods

Figs. S1 to S7

Table S1

References (5058)

References and Notes

  1. Magnetic anisotropy is often defined as a classical energy barrier, or as the energy needed to orient the magnetization perpendicular to the easy-axis. Our temperature was too low to observe Arrhenius behavior because nonthermal processes dominate the relaxation in the range of temperatures accessed. Additionally, studies of quantum magnets sometimes infer a barrier using DS2 or DJ2, but these are applicable only for pure-spin or pure-J systems. Consequently, we use the ZFS as definition of the MA.
  2. Materials and methods are available as supplementary materials on Science Online.
  3. The quantum mechanical description of the energy levels derived from the multiplet calculation indicates that J is not a good quantum number. Therefore, we use the Zeeman energy (LZ + 2SZBB instead of a description based on the Landé g-factor.
  4. Acknowledgments: S.S. and P.G. acknowledge support from the Swiss Competence Centre for Materials Science and Technology (CCMX). J.D. acknowledges funding by an Ambizione grant of the Swiss National Science Foundation. I.G.R., S.B., C.P.L., and A.J.H. thank B. Melior for expert technical assistance. C.P.L and A.J.H. thank the Office of Naval Research for financial support. S.G., O.R.A., and B.A.J. thank the National Energy Research Scientific Computing Center (NERSC) for computational resources. O.R.A. was supported by the National Science Foundation under grant DMR-1006605. B.A.J. thanks the Aspen Center for Physics and National Science Foundation grant 1066293 for hospitality while doing the calculations which appear in this paper. We thank A. Cavallin for helping in developing the Igor code used to analyze the XAS spectra. I.G.R., S.B., R.M.M., C.P.L., and A.J.H. performed the STM experiments and data analysis. S.G., O.R.A., and B.A.J. carried out the DFT calculations. S.R., F.D., L.G., S.B., J.D., C.P., and F.N. carried out the XMCD experiments. F.D., S.R., S.S., and P.G. analyzed the XMCD data. S.S. wrote the multiplet calculation code and performed the simulations. All authors discussed the results and participated in writing the manuscript. A.J.H., P.G., and H.B. initiated and directed this research. The authors declare that they have no competing financial interests.

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