Perpendicular magnetic anisotropy in a single Dy adatom ferrimagnet

Abstract

The electronic structure and magnetism of individual Dy atoms adsorbed on ferromagnetic (Gr)/Ni(111) substrate are investigated using a combination of the density functional theory with the Hubbard-I approximation to the Anderson impurity model (DFT+U(HIA)). The divalent Dy adatom in configuration with [ ] is found. The values of spin =3.4 , orbital =5.2 , and total =8.6 magnetic moments calculated for the Dy f-shell are noticeably different from the atomic second Hund’s rule. There is almost zero moment on (Gr)-atoms. The ferromagnetic Ni substrate moments are anti-aligned to the Dy 4f-shell moment. The X-ray absorption (XAS) and magnetic circular dichroism (XMCD) spectra are calculated and can be compared to the experimental data. The magnetic anisotropy energy (MAE) is calculated from the ground state energy difference for different directions of the magnetization, E[100] - E[001] = 3.5 meV and E[010] - E[001] = 2.2 meV. This large and positive MAE can be important for ultra-high density magnetic recording. The magnetization of Dy@(Gr)/Ni(111) is tilted with respect to the (Gr)/Ni(111) substrate normal by due to a competition between negative first and third order magnetic anisotropies and strong and positive second order magnetic anisotropies. Our studies assist in resolving ambiguities of conventional DFT+U applied to Dy on graphene. They can provide a viable route for further investigation and prediction of the rare-earth based magnetic nanostructures.

Introduction

The unique magnetic properties of RE ions adsorbed on surfaces or embedded into molecular complexes, in virtue of their large spin and orbital magnetic moments and a gigantic magnetic anisotropy energy (MAE), provide a viable pathway for creating atomic-scale magnetic memories¹. These so-called single-atom magnets (SAM), terbium (Tb), dysprosium (Dy), holmium (Ho) adatoms deposited on an insulating MgO surface exhibit a large uniaxial MAE, which acts as a barrier against the spontaneous reversal of the magnetic moments and enhances their magnetic lifetimes². The Dy atoms deposited on the graphene on the top of Ir substrate also display a SAM behaviour³. Experimental research into 4f adatoms on graphene^3,4 and MgO^{5, 6–7} has been carried out for over resent years. Nevertheless, it remains challenging to explain theoretically the influence of the substrate and adsorption geometry on the 4f-shell charge and magnetic configurations and the XAS/XMCD spectra⁸ without prior knowledge of the experimental data.

It is now well established⁹ that density functional theory (DFT) methodologies fall short of describing the basic spin and orbital magnetic properties of the rare-earth (RE), including the RE-transition metal (TM) magnets, correctly. We have recently extended DFT to treat the f-electron materials (RE and actinides), making use of the charge self-consistent implementation of the DFT++ methodology¹⁰ and applied it to Dy adatom on insulating MgO substrate¹¹. In our approach, DFT is combined with an exact diagonalization of a single-impurity many-body quantum impurity model (the so-called Anderson model)¹²(AIM) to account for the full structure of the f-orbital atomic multiplets.

This work aims to explain the spin and orbital characteristics of currently existing magnetic materials and to predict the properties of new materials yet to be synthesized experimentally. For example, we consider the electronic structure and magnetism of individual Dy atoms adsorbed on ferromagnetic graphene (Gr)/Ni(111) substrate. Graphene on Ni(111) is of particular interest because it can be epitaxially grown¹³ and as a building block of spintronic devices¹⁴. Moreover, Dy exhibits a large magnetic anisotropy and has been used for single-atom magnets with a large magnetic anisotropy^3,6.

Our calculations reveal a change of Dy valence from Dy , typical of RE-TM bulk magnets, to Dy at the GR/Ni(111) surface, obtained self-consistently within DFT+U(HIA) calculations. The Dy atom adopts an many-body ground state, , representing a clear deviation from the conventional Hund’s rule configuration. Moreover, we find that the first-order, second-order, and third-order magnetic anisotropy constants have different signs, leading to competing magnetic anisotropy contributions. This competition results in a pronounced tilt of the Dy magnetic moment away from the crystallographic high-symmetry axes, a prediction that can be experimentally verified via the angular dependence of element-specific XMCD spectra. Together, these findings provide new microscopic insight into the electronic configuration and magnetic anisotropy of surface supported RE adatoms.

Results and discussion

Fig. 1 [Images not available. See PDF.]

Supercell model for rare-earth impurity on graphene/Ni(111) (Dy@GR/Ni). Dy atoms are shown in blue, C atoms are in gold, and Ni atoms in magenta (A); The spin-resolved f-projected (fDOS) DOS , The graphene and Dy-5d projected DOS (B); The edge XAS and XMCD spectra (normal incidence) (C) for Dy@GR/Ni.

To discern the spin and orbital magnetism of the rare-earth Dy adatom on GR/Cu(111) we make use of the supercell model with 36 Ni-atoms (4 layers) for the substrate, and the graphene layer with eighteen C atoms on the top of the Ni(111) surface. The graphene overlayer is placed in the so-called “HCP” (or “1-3”) position, where one of the C atoms sits on the top of the Ni surface, and another is over the second Ni subsurface layer. The experimental Ni lattice constant of 4.705 a.u. is used. The rare-earth Dy atom is placed in the hollow position atop the graphene/Ni(111) surface. The top view of the supercell is shown in Fig. 1A.

The structure relaxation is performed using the Vienna ab initio simulation package (VASP¹⁵), employing the DFT exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE¹⁶) in combination with the projector augmented-wave method (PAW¹⁷). Assuming that localized 4f electrons have a negligible effect on the geometry, we used a rare-earth Lu adatom as a substitute for Dy, treating the 14 closed 4f-shell electrons of Lu as valence states. The atomic positions were relaxed until the forces on the Lu adatom, the carbon atoms of graphene, and the topmost two monolayers (ML) of Ni were reduced below 0.001 eV/Å. The use of Lu instead of Dy is further justified by comparison with previous total energy minimization calculations for Dy on graphene/Cu(111)²⁰, where the optimal distance of 4.23 a.u. is reasonably close to the optimized distance of 4.15 a.u. obtained for Lu on graphene/Ni(111).

The relaxed structural data obtained from the VASP simulations was used as input for further DFT+U(HIA) electronic structure calculations that employ the relativistic version of the full-potential linearized augmented plane-wave method (FP-LAPW)^18,19. Further details of DFT+U(HIA) calculations are described in the “Theoretical Method” section. This two-step approach combines the speed and efficiency of the highly optimized VASP package with the state-of-the-art accuracy of the FP-LAPW method.

The results of DFT+U(HIA) calculations are shown in Table 1. The optimal distance a.u. between the Dy atom and the graphene surface is slightly reduced compared with the previously reported Dy@GR/Cu(111) case²⁰. The spin , orbital , and the total ( + ) moments shown in Table 1 are noticeably different from what would be expected from the atomic second Hund rule. The f-shell moments are aligned with the 6s (0.05 ) and 5d (0.09 ) spin magnetic moments in the Dy “muffin-tin” . The spin polarisation of the C-atoms of graphene is very small, less than 0.04 , and directed along the Dy atom 6s5d polarisation. Remaining Ni and interstitial spin moments are anti-aligned to the 4f moment of Dy. The total non-f spin of -18.03 and orbital -1.80 are mainly the 3d-moments of the Ni substrate.

The spin-resolved f-projected (fDOS) DOS are shown in Fig. 1 (B) together with the d-projected (dDOS) of Dy and indicate the semi-atomic character of the f-states of Dy acquiring some d-occupation. Also, the partial DOS for the graphene C-atoms is shown in Fig. 1 (B). It indicates a significant n-doping due to the charge transfer to graphene and the metallic character of Dy@GR/Ni(111) similar to Dy@GR/Cu(111) case²⁰.

The Dy valence state is not assumed a priori but instead emerges naturally from the self-consistent DFT+U(HIA) calculations. It corresponds to the many-body ground-state (GS) solution of Eq. (4). The GS is characterized by (number of localized f electrons), quantum numbers , , (total, orbital, and spin moments, respectively), and a Lande factor of . These values correspond to the ionic configuration and define the Dy adatom valence as Dy . The fact that Dy is obtained self-consistently, rather than imposed as an input, underscores the reliability of the method in capturing the electronic configuration.

The f-electron occupation calculated with the aid of Eq. (5) is consistent with the configuration obtained from Eq. (4). The many-body GS of Eq. (4) is with the total magnetic moment = 8.72 . This value is consistent with the total + magnetic moment (cf. Table 1) calculated with the aid of Eq. (5).

Recently, the magnetic properties of Dy on a free-standing graphene were investigated with the aid of DFT+U²¹. Two different self-consistent solutions were found, the state with , , , and another one with partially quenched . They were interpreted to correspond and f-electron configurations. From the total energy difference between these two magnetic solutions, the solution was found as the lowest energy state. The present DFT+U(HIA) calculations assist in resolving this ambiguity and allow us to identify this state as .

The self-consistently determined crystal field potential matrix can be expressed in terms of Wybourne’s spherical tensor operators ²², and a set of Stevens crystal field parameters (CF) [ ] can be extracted²³. The non-zero CF parameters [ ], and the exchange field are shown in Table 1. We used Eq. (4), with these parameters as an input for the Quanty code^24,25 to evaluate the -edge XAS and XMCD spectra. The computed spectra are shown in Fig. 1C. They can be used to make a comparison of our predictions with the experimental data.Table 1

Average distance (in a.u.) from the graphene overlayer on Ni(111) (GR/Ni), and between the graphene overlayer and the Ni surface, f-electron occupation , spin , orbital , and the total + moments (in ) for the Dy-atom. The non-f spin and the orbital magnetic moments, together with the total non-f magnetic moment m in the Dy@GR/Ni unit cell. The non-zero CF parameters [ ], and the exchange field for Dy adatom on GR/Ni(111) (in meV). The magnetic anisotropy constants K, and the corresponding magnetic anisotropy energy (MAE, meV) for the Dy adatom on GR/Ni(111) (in meV).

Dy@GR/Ni					+
Dy	4.145	9.94	3.38	5.26	8.64
GR/Ni	4.136		-18.03	-1.80	-19.83
CEF
(meV)	14.20	-4.19	-0.36	-8.50	11.0
MAE					[010]- [001]
(meV)	-8.39	11.80	-0.60	0.66	2.15
					[100]- [010]
					1.32

The magnetic anisotropy of Dy@Gr/Ni(111) is evaluated making use of the “two sublattices” model of 3d-4f intermetallics^26,27. In this model, which is commonly used for the bulk rare-earth-transition metal ferrimagnets²⁷, the hamiltonian is divided in two parts. First, there is the 4f local Hamiltonian for the Dy,

1

where are the Stevens operator equivalents, are the Stevens factors for a given ground state multiplet J, the exchange and the external magnetic fields are acting on the 4f shell. The magnitude of is equal to in Eq. (4).

The second part, the 3d (Gr)/Ni(111) sublattice, is described by

2

where and are the magnetization and the magnetic anisotropy of the Gr/Ni(111) substrate. The polar angles define the direction of the magnetization .

Fig. 2 [Images not available. See PDF.]

(A) and for Dy adatom on graphene/Ni(111) (Dy@GR/Ni).

The Dy adatom contribution to the magnetic anisotropy ,

3

Note, that the direction of is anti-parallel to with and are expressed using the polar coordinates which define the direction of .

The angular dependence of with is shown in Fig. 2A. The angular dependence of with is shown in Fig. 2B reflecting the in-plane sixfold symmetry. Fitting angular dependence by Eq.(3), we obtain = -8.39 meV, = 11.80 meV, = -0.60 meV and = 0.66 meV. The and constants are negative, and the magnetic anisotropy is dominated by the term. The positive sign of leads to a nonzero basal angles for the minimal energy. The uniaxial MAE = 2.2 meV and = 3.5 meV are evaluated. The positive MAE sign indicates the perpendicular magnetic anisotropy for Dy@Gr/Ni(111).

As follows from the ( ) angular dependence shown in Fig. 2, the magnetic anisotropy energy Eq.(3) for the Dy adatom on (Gr)/Ni(111) has minima at . The tilting angle is determined²⁸ by where .

To evaluate the magnetic anisotropy of the (Gr)/Ni(111) substrate, we calculated the MAE for a GR/Ni /GR slab model (see Supplemental Material^{29, 30, 31, 32, 33, 34, 35-36}). The slab MAE was found to be 0.18 meV, which is further reduced to 0.11 meV after appying the shape anisotropy correction. Thus, the magnetic anisotropy of the (Gr)/Ni(111) substrate is small and positive, and will be neglected in the following discussion. Furthermore, since is dominated by the spin contribution (see Table 1), we also neglect the small anisotropy of the orbital moments of the Ni substrate atoms.

The magnetic structure of the whole system is determined by the minimum of the total energy, a sum of Eq.(2) and Eq.(3). Since , and in a zero external field, it is determined by the minima of the magnetic anisotropy energy Eq.(3) at . For the tilting angle and , the 4f-magnetic moment is aligned at and . Corresponding is aligned in the plane with and . Thus, the Dy f-magnetic moment is essentially anti-parallel to the . The total magnetization ( plus ) of Dy@(Gr)/Ni(111) is tilted in the plane (Fig. 1A) to the surface normal direction by and Dy and Ni moments remain almost anti-aligned. This prediction of magnetization tilt can be experimentally verified by measuring the angular dependence of element-specific XMCD signal intensities.

It should be noted that the present calculations are carried out at K. Nevertheless, the uniaxial MAE of 2.15 meV can be used to qualitatively estimate the thermal stability of Dy@GR/Ni(111) at low temperatures within the framework of the Arrhenius-Neel law for thermally activated switching, where is the magnetic lifetime, is the characteristic attempt time between successive reversal events, and is the energy barrier for thermally assisted magnetization reversal. Using meV (the MAE) and s, taken from experimental data on Dy@GR/Ir(111)², we obtain for K an estimated lifetime of s.

Summary

In summary, we address the electronic and magnetic structure and the magnetic anisotropy of Dy@Gr/Ni(111), making use of the DFT+U(HIA) methodology and a supercell model. The divalent Dy adatom in configuration with [ ] and the Lande factor =1.25 is found with many-body ground state of Eq. (4). The values of spin =3.4 , orbital =5.2 , and total =8.6 calculated for the Dy f-shell are noticeably different from the atomic second Hund’s rule. There is almost zero moment on (Gr)-atoms. The ferromagnetic Ni substrate moments are anti-aligned to the Dy 4f-shell moment. The magnetic anisotropy energy (MAE) is calculated from the energy difference for different directions of the magnetization = 2.2 meV and = 3.5 meV. The magnetization of Dy@(Gr)/Ni(111) is tilted to the (Gr)/Ni(111) substrate normal by due to a competition between negative first and third order magnetic anisotropies and strong and positive second order magnetic anisotropies. Our studies can provide a viable route for further investigation and prediction of the rare-earth single-atom magnets. The large and positive MAE in Dy@(Gr)/Ni(111) can be utilized for ultra-high-density magnetic recording.

Theoretical method

We employ recently proposed DFT+U(HIA) extension^11,20 of the DFT+U method in a rotationally invariant, full potential implementation^37,38. In the DFT+U(HIA) approach, DFT is combined with the exact diagonalization of the Anderson impurity model³⁹. Assuming the weakness of the hybridization between the localized 4f-electrons and the itinerant s, p, and d-states, the Anderson impurity model is reduced to the atomic limit and corresponds to the Hubbard-I approximation (HIA). The corresponding Anderson impurity model Hamiltonian can be written as,

4

where creates a 4f electron. The parameter ( , the chemical potential) defines the number of f-electrons. The parameter specifies the 4f-SOC strength, and is taken from DFT calculations. is the crystal-field potential. These parameters are determined in the charge self-consistent DFT+U(HIA) calculations as described in Ref. ^20,39. is the strength of the exchange field acting on the 4f shell of Dy (see Supplemental material^{29, 30, 31, 32, 33, 34, 35-36}). The last term in Eq.(4) describes the Coulomb interaction in the 4f-shell. The Slater integrals  eV, and eV, eV, and 4.83 eV were chosen to parametrize the Coulomb interaction term. They correspond to the values for Coulomb   eV and exchange  eV.

The band Lanczos method⁴⁰ is employed to find the lowest-lying eigenstates of the many-body Hamiltonian and to calculate the one-particle Green’s function in the subspace of the f orbitals at low temperature ( eV). The selfenergy is obtained from the inverse of the Green’s function matrix , and used to calculate the local Green’s function G(z) for the f-electrons,

5

in a single-site approximation. With the aid of the local Greens function G(z), we evaluate the occupation matrix . The matrix is used to construct an effective DFT+U potential , which is used in the Kohn–Sham equations

6

These equations are iteratively solved until self-consistency over the charge density is reached. In each iteration, a new value of the 4f-shell occupation is obtained from the solution of Eq. (6). Subsequently, a new self-energy corresponding to the updated 4f-shell occupation is constructed. Finally, the next iteration is started by evaluating the new local Green’s function, Eq. (5). Further details of our implementation and calculations can be found in the Supplemental material^{29, 30, 31, 32, 33, 34, 35-36}.

Acknowledgements

We acknowledge stimulating discussions with Dr. A. Barla. This work was co-funded by the European Union and the Czech Ministry of Education, Youth and Sports (Project TERAFIT - CZ.02.01.01/00/22 008/0004594). We acknowledge partial support provided by the Czech Science Foundation (GACR) Grant No. 24-11992S. DL acknowledges the project e-INFRA CZ (ID:90254) by the Czech Ministry of Education, Youth and Sports.

Author contributions

AS, IH, and DL conceived and supervised the project. AS, FM, and DL performed the computations. All authors contributed to the interpretation of the data and the writing of the manuscript.

Data availability

The data used during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

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