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The RKKY Mechanism for Magnetic Ordering of Sparse Fe Adatoms on Graphene Yan Zhu, Yanfei Pan, Zhongqin Yang, Xinyuan Wei, Jun Hu, Yuan Ping Feng, Hongbin Zhang, and Ruqian Wu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11803 • Publication Date (Web): 31 Jan 2019 Downloaded from http://pubs.acs.org on February 2, 2019

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The Journal of Physical Chemistry

The RKKY Mechanism for Magnetic Ordering of Sparse Fe Adatoms on Graphene Y. Zhua,1,*, Y. F. Pana,1, Z. Q. Yangb, X. Y. Weib, J. Hud, Y. P. Fenge, H. Zhangf,* and R. Q. Wu b,c,* a

College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b State Key Laboratory of Surface Physics and Key Laboratory for Computational Physical Sciences (MOE) & Department of Physics, Fudan University, Shanghai 200433, China c Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA d College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China e Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore. f Institute of Materials Science, Technical University of Darmstadt, Darmstadt 64287, Germany

Abstract: First-principles calculations are carried out to elucidate the mechanism of the exchange interaction among magnetic adatoms on graphene, using Fe/graphene as an example. By fitting the calculated spin wave spectra of different sparse adsorption configurations to the Heisenberg Hamiltonian, the exchange parameters between different neighbors (Ji) were found to follow a single RKKY-type function, indicating that the RKKY mechanism plays the dominant role in governing the magnetic order in these systems. Furthermore, calculated formation energies suggest that the 3×3 structure can be a metastable state for the growth of Fe on graphene, which otherwise may easily form clusters.

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Transition metal adatoms, clusters and ultrathin films on graphene (TM/Gra) may produce new magnetic properties such as the formation of Skyrmions and the generation of topological quantum states that are promising for the design of spintronics devices1,2,3,4,5,6,7,8. As a prototypical magnetic element, Fe is often used to bring exchange splitting and spin orbit coupling (SOC) in bands of graphene or other emergent two-dimensional materials for the manipulation of their transport properties. Although Fe adatoms prefer a three-dimensional growth mode on graphene9,10, single-atom-thick Fe layers on graphene have been experimentally synthesized and observed by using in-situ low-voltage aberration-corrected transmission electron microscopy11. This makes the functionalization of graphene with magnetic metals feasible and interesting. For the exploitation in graphene-based microelectronics devices, the structural stability and long-range magnetic interaction of transition metal atoms are critical issues for fundamental studies. Nevertheless, most studies for TM/Gra systems have focused mainly on the effect of the TM-graphene interaction on the band features, and very few of them deals with the exchange coupling between TM atoms and clusters1,2,4,5,6,8,9,10,12,13. To control the magnetic state and critical temperature of TM/Gra systems for practical applications, it is hence important to unravel the mechanism that governs their magnetic orders. Although it was suggested that the dominant contribution for the long range magnetic ordering among TM atoms originates from indirect exchange interactions mediated through the delocalized Dirac electrons in graphene (or RKKY type)12,13, the validation of this proposal has been absent12.

In this work, we study the exchange interaction between Fe adatoms on graphene in a variety of configurations to shed light for the exchange coupling in Fe/Gra. Using the spin spiral algorithm, we calculate directly the spin wave spectra of these structures at the level of density functional theory (DFT). A rather complete data set of the exchange, Ji, is obtained according to the Heisenberg model, and these data surprisingly fall on a single J(r) curve described by the RKKY function. Our work provides support that the RKKY mechanism plays the leading role in magnetic 3



ordering of TM/Gra at low TM concentrations, and this finding should be useful for the understanding of magnetic and transport properties of TM/Gra systems.

The Fe/Gra is simulated using a slab model in which Fe atoms taking the hollow sites over the hexagons of graphene. Two representative structures are shown in Fig. 1. For convenience of discussions below, we mark the central Fe atom by “0” and its ith nearest neighbors by “i”. DFT calculations are carried out by using the pseudopotential plane-wave method as implemented in the Vienna ab-initio simulation package (VASP)14,15. The projector-augmented wave (PAW) approach is used for electron-ion interaction while the Perdew-Burke-Ernzerhof (PBE) functional is adopted to describe the exchange-correlation interactions among valence electrons16,17,18. A cutoff energy of 500 eV is chosen for the plane wave basis expansion. K-point grids of 13×13×1, 13×11×1, 13×9×1, 11×11×1, 11×9×1, 9×9×1 and 7×7×1 are used to sample the two-dimensional Brillouin zones (BZs) of the 2×2, 2×3, 2×4, 3×3, 3×4, 4×4 and 5×5 supercells, respectively. With one Fe atom in these supercells, the corresponding Fe coverages are 12.5%, 8.3%, 6.3%, 5.6%, 4.2%, 3.1% and 2.0%, respectively. All structures are optimized until the force on each atom becomes less than 10-2 eV/Å. The optimized lattice constant of graphene is 2.47 Å, which is in good agreement with results of previous DFT calculations and experimental data2,19.

A convenient way of determining the exchange parameters through DFT calculations is to calculate the energy-momentum dependence of the spin spiral structures, as done for many systems in the literature, such as the bulk γ-Fe20, freestanding TM layers21, monoatomic chains22,23, and TM monolayers on substrates24,25. Here, we adopt the spin spiral approach to calculate the spin wave spectra of different Fe/Gra configurations and extract the exchange parameters by fitting these spectra with the Heisenberg model. For the spin spiral calculations, a full vector-field description of the magnetization density that allows continuous variation of the direction of local magnetic moments is required. We perform fully unconstrained noncollinear magnetic 4


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calculations, as implemented in the VASP code by Hobbs et al.26 In addition, we apply the generalized Bloch equations to analyze the incommensurate spin spiral states13,27,28. The spin orbit coupling is included in the present calculations. The convergence criterion for self-consistent total-energy calculations is 1.0×10−6 eV/supercell, and test calculations show that changes in the spin spiral excitation energy are negligible if a stricter convergence criterion is applied.

Fig. 1 (Color online) Top view of atomic structures of Fe/Gra with one Fe atom in (a) the 2×2 structure; (b) the 2×4 structure. The large grey and small black balls represent Fe and C atoms respectively. The ith nearest neighbor of the Fe atom adsorbed at site “0” is marked by “i”. The heights, magnetic moments, and adsorption energies of Fe adatoms with respect to the geometries are shown in (c), (d), and (e), respectively.

Similar to previous findings, Fe atoms prefer to take the hollow sites on graphene2,20. Figure 1 shows two configurations of Fe/Gra, forming 2×2 and 2×4 patterns with different symmetries, respectively. In the 2×2 structure, each Fe atom (Fe0) has six nearest Fe neighbors and six second nearest Fe neighbors. The 2×4 structure has a lower symmetry, and there are only two first nearest neighboring Fe atoms and two 5



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second nearest neighbor Fe atoms. The optimized heights of Fe adatoms over the graphene plane (h) are displayed in Fig. 1(c). Overall, one may see that h increases along with the coverage. As shown in Fig. 1(d), the total magnetic moment converges to 2.0 μB when the supercell is larger than 3×3, which is consistent with the value reported in the literature2,19,20,29. In contrast, the magnetic moment increases in small supercells and approaches toward that of a free-standing Fe monolayer, 3.1 μB.

To characterize the strength of the Fe-graphene interaction, adsorption energies are calculated as Eads_x = (EG + EFe_x) – EFe/G

(1),

where EFe/G and EG are the total energies of graphene with and without Fe adatoms, and EFe_x is the energy of an Fe atom in vacuum (x=a), bulk (x=b), or film (x=f, i.e. in the same supercell but without graphene), respectively30. As shown in Fig. 1(e), Eads_a of all structures are above zero but Eads_b are below zero, which means that Fe atoms tend to form clusters on graphene. Beyond the 3×3 structure, the adsorption energies Eads_a and Eads_f converge to 1.15 eV which is comparable to 1.04 eV and 1.21 eV reported in previous studies2, 29. One important finding here is that the largest Eads_a is reached in the 3×3 structure due to the competition between Fe-Fe and Fe-C interactions. It appears that an ordered 3×3 structure gives a local energy minimum and can be fabricated for Fe/Gra, which deserves experimental effort.

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Fig. 2 (Color online) (a) The spin spiral excitation energy E(q) as a function of the spiral wave vector q for different structures of Fe/Gra. The scattered symbols are the calculated results and the lines are fitted ones. The special q-points shown in (b) are labeled; the reference energy (E=0) is set to the value at Γ point corresponding to a ferromagnetic state. The unfilled symbols are the non SOC calculation results for the 2 × 2 structure. (b) The spin spiral state at Γ (FM state), M, and K (TAFM state) points based on the Generalized Bloch equation in Eq. (S1). The magnetic moment is in the plane and the color circle is guide for eyes. (c) The contour plot of the energy 7



density of spin spiral excitation energy at all q-points in the first BZ. The contour values are in meV and labeled in the figures. The special k-points are also labeled.

The SOC is important for transition metal atoms on graphene4,31 and hence is included in the present DFT calculations. One of the important consequences of SOC is the magnetic anisotropy. Defined as MAE=Eperpendicular-Ein-plane, the uniaxial magnetic anisotropy energy of Fe/Gra is about -1.0 meV per Fe atom in all calculated structures – indicating that the easy axis of Fe/Gra systems is in-plane. Hence, the spin spiral is arranged in the plane in the following discussions. We wish to point out that the spin spiral excitation energies of Fe/Gra, E(q), rarely change in the calculations with and without the inclusion of SOC, as illustrated by the closeness of filled and opened black squares in Fig. 2(a) for the 2×2 cell. This is understandable since the strength of SOC of Fe is much weaker than that of the exchange interaction.

As shown in Fig. 2(a), E(q) curves show different dispersions as from one structure to another. It is noted that most E(q) curves are positive, suggesting FM ground states. For the low symmetric 2×4 and 3×4 structures, E(q) at the N point are lower than those in the 3×3 and 4×4 structures, because Fe atoms form rows in these low symmetry structures, as shown in Fig. 1(b). The exchange interaction across rows is weaker (shown in E(q) along ΓN path) than that within the rows (shown in E(q) along ΓM path). Interestingly, the E(q) curves of the 4×4 and 5×5 structure become negative territory around the K point, indicating that their magnetic ground state has a triangular antiferromagnetic (TAFM) pattern. Since frustrated spin spiral states provide a test ground for various theories regarding magnetic ordering32, Fe/Gra can be a useful model system for further studies of exchange interactions in low-dimensional systems.

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Fig. 3 (Color online) (a) Exchange interaction parameters between Fe0 and the ith neighbors in different Fe/Gra structures. The meaning of J3V and J6V in the 2×4 structure is marked in the insert. (b) The fitting of Ji of the 4×4 and 5×5 structures to the RKKY function (black line), a0=2.47Å is the lattice constant of graphene.

The extended Heisenberg model includes the exchange, Dzyaloshinskii-Moriya interaction (DMI)33,34. 1 E   [ J i (1  s 0  s i )  d i  (s 0  s i )] i 2

(2)

where si is the normalized spin vector. For thin films, we use di=di(z×ui), with ui being 9



the unit vector between sites 0 and i, with z being the unit vector normal to the graphene plane. For the convenience of discussions, the energy of the ferromagnetic (FM) ground state is set as zero. Ji is the exchange interaction energy and di is the DMI energy for the ith-neighbor. As the DMI is much weaker than the exchange interaction, di can be neglected in model analysis. Now, we may reliably obtain effective Ji for different configurations by fitting their DFT spin wave spectra to Eq. (S2)-Eq.(S5). In turn, the contour plots of spin spiral energies in the entire first BZ can be reconstructed for each system with these J parameters and the results are displayed in Fig. 2(c). As expected, the lowest E(q) occurs at Γ for structures that have the ferromagnetic ground state. The 4×4 and 5×5 configurations are the exceptions and their E(q) minimizes at the K point. In these two structures, the 5×5 configuration has much more stable TAFM than that of 4×4. This reminds us that complex magnetic structures may still occur even in sparse configurations. For low-symmetry structures, contours of E(q) also suggest anisotropic behavior as energy increases faster along the direction of atomic row away from Γ.

Now we collect Ji from different geometries and inspect the trend of J-distance dependence in Fig. 3(a). Interestingly, they have oscillatory behavior as a group. Especially, Ji of both 4×4 and 5×5 configuration can be guided by a RKKY-type function for two-dimensional magnetic structures.35,36 J 2 (r )  a

cos 2k F a 0 x cos 2k F r a 2 ( 2k F r ) ( 2k F a 0 x ) 2

(3)

where kF is the Fermi wavevector in the conducting host; a is a constant; r is magnetic distance from Fe0; a0 is lattice constant of graphene and x is relative distance as shown the horizontal ordinate in Fig. 3. This trend is particularly clear in Fig. 3(b), where we only use data from symmetric configurations to avoid the complexity of anisotropic exchange interactions. We believe that the RKKY exchange interaction between Fe atoms is mediated by the π electrons of graphene. The periods of oscillations depend kF ( k F  2 J , where σJ is the effective electron number 10


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density around the Fermi energy37). By fitting the J-distance curve in Fig. 3(b) to the function in Eq. (3), we get kFa0=1.01 and a=217.2 meV. Hence, kF=0.409 Å-1 and we have σJ=0.0266 e/Å-2, indicating that about 0.07 e per carbon atom mediate the RKKY interaction. As shown in Fig.4a, Bader charge analysis indicates that each Fe adatom donates 0.83-0.85 electrons to graphene in the sparse structures. Most of these electrons are however rather localized in the region around the Fe cores as shown in Fig. 4b, and only a small portion is responsible for mediating the magnetic interaction.

Fig. 4 (Color online) (a) Charge (electron) transfer from Fe atom to graphene in various

Fe/Gra

structures,

and

(b)

charge

density

difference

contours

(CFe/Gra-CGra-CFe) in a plane mid-way between the Fe atom and graphene, indicated by the red line in the lower-left inset, for the 4×4 structure. Contour values are ρn=0.001×n (e/Å3). 11



To summarize, we have systematically investigated the magnetic ordering of different configurations of Fe/Gra to establish its mechanism. It was found that the 3×3 structure gives a local minimum of formation energy and thus can be fabricated as a metastable structure. The effective exchange parameters (Ji) are extracted by fitting the ab initio magnon spectra to the classical Heisenberg model. Interestingly, Ji values from different low concentration configurations fall on a single line and show the typical RKKY character. We also found interesting frustrated magnetic configurations in the 4×4 and 5×5 structures, which would be good modeling systems for fundamental studies of complex magnetic structures.

■ ASSOCIATED CONTENT s ○

Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI:. The detailed E(q) formulas for four typical structures: 2×2, 2×3, 2×4 and 3×4 (PDF)

■ AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] (Y. Zhu); [email protected] (H. Zhang); [email protected] (R.Q. Wu) Co-first authors 1)

Y. Zhu and Y. F. Pan contributed equally to this work.

Website Y. P. Feng: http://www.physics.nus.edu.sg/phyfyp/ H. B. Zhang: https://www.mawi.tu-darmstadt.de/tmm/theorie_magnetischer_materialen/staff_1/mit arbeiter_tmm_details_61888.en.jsp R. Q. Wu: https://www.physics.uci.edu/wugroup/ 12


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ORCID Y. Zhu: 0000-0001-9620-7505 Z. Q. Yang: 0000-0002-4949-1555 J. Hu: 0000-0002-7575-2165 Notes The authors declare no competing financial interest.

■ACKNOWLEDGMENTS Work at NUAA and Fudan was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos: 11204131, 11574051, 11474056). Work at UCI was supported by DOE-BES (Grant No. DE-FG02-05ER46237). Computer simulations were performed at the U.S. Department of Energy Supercomputer Facility (NERSC) and Fudan University High-end Computing Centre.

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