Graphyne as a promising substrate for high density magnetic storage bits

Abstract

Applying magnetic nanostructures in high density magnetic data storage is hindered by a lack of suitable substrate. Using density functional theory, we explored the potentiality of graphyne as a template for nanomagnetic bits. Due to the unique porous structure of graphyne, Os atom tightly binds to the graphyne at the hollow site with an in-plane MAE of 18 meV. Through a rigid band model, we introduced the strategy to manipulate the MAE by rearrangement of d-orbitals of Os atom. A large MAE of about 48 meV was obtained for F functionalized Os@graphyne. To obtain the out-of-plane easy axis, we have considered the MAE of several transition-metal combinations. We finally identified Os–Os@graphyne as an excellent candidate for room temperature applications due to the high MAE and the structural stability.

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Stimulated by the fast-growing demand and need of high density magnetic data storage, research efforts in the field of nanomagnetism are extremely vigorous in recent years.^1,2 The key issue in this realm is how to increase the magnetic anisotropy energy (MAE), which denotes the threshold energy barrier for inhibiting magnetization reversal against thermal fluctuations.^3–6 For practical applications at room temperature, it is crucial to find magnetic nanostructures with MAE up to 30–50 meV. An important direction of research has focused on very small transition-metal (TM) clusters, which might satisfy this criterion.^7–11 Free TM clusters were predicted to show large value of MAE according to tight-binding model and density functional calculations.^4,7 More recently, much of the interest has concentrated on the magnetic properties of TM dimer, which is the smallest chemical objects that possess a magnetic anisotropy.^2,4,6,9,12 It was reported that several freestanding TM dimers might provide large MAE with the order of 40–70 meV.^10,12 It is impossible, however, to exploit this unusual property unless they are hosted by a suitable supporting substrate, and such a system should fulfill some conditions, i.e., high MAE, high stability, etc. Many efforts have been devoted to seek a suitable host for TM dimers. It is found that TM dimers on benzene or defective graphene are extremely stable and possess high MAE.^6,13,14 This offers great opportunity for technological developments of robust magnetic nanostructures. However, despite considerable efforts, creating well-ordered and uniformly distributed vacancies in graphene in a controlled way is always experimentally challenging. Thus, it is necessary to find new alternative substrate materials that could trap TM dimers strongly, evenly distributed, and not deteriorate the dimers MAE.

In comparison with graphene, graphyne possesses the natural “holes”. Graphyne, a 2D carbon allotrope with the same symmetry as graphene, is made up of hexagonal carbon rings and acetylene linkages.^15,16 The unique porous structure with well-ordered vacancies makes graphyne a promising substrate for stable and uniformly dispersed adatoms due to the fact that each pore can only host one TM atom.^17–19 This raises the hope of designing a highly stable system of small clusters deposited on the graphyne.²⁰ In the present work we will demonstrate that graphyne is a suitable substrate material for TM dimer with fascinating MAE value by systematic first-principles calculations. As a typical example, the 5d metal absorbed on graphyne is selected as our prototype. We chose Os element adsorbed on graphyne (Os@graphyne) as a typical example. According to first-principle calculations, we found the single Os atom is tightly bound to the porous site of graphyne. The calculated MAE is about 18 meV with the easy axis in the plane of the graphyne. To enhance the MAE of Os@graphyne, nonmetal elements are used to manipulate the d-orbitals of Os. Our results show that huge MAE about 48 meV can be obtained for the F functionalized Os@graphyne with the easy axis in the plane of the graphyne. Moreover, we also investigated the MAE of TM combinations. We filtered Os–Os@graphyne as good candidate with huge MAE of 34.5 meV as well as out-of-plane easy axis.

As shown in Fig. 1(a), the structure of Os atom embedded in graphyne was mimicked as a supercell composed of 2 × 2 graphyne unit cells with one Os atom at porous site, which can safely avoid the interactions between Os atom within neighboring images. The nonmetal atoms (H, O, F) or TM atoms (Fe, Co, Ru, Rh, Os) are put right on the top of Os atom, as depicted in Fig. 1(b). The vacuum thickness along the z axis was set to 15 Å, which is large enough to avoid the interaction between adjacent layers. First-principles calculations were performed based on the density-functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP).^21,22 The exchange–correlation potential is treated with the Perdew–Burke–Eznerh of generalized-gradient approximation (PBE-GGA).²³ We used the projector augmented wave (PAW) method for the description of the electron–ion interaction. A 15 × 15 Γ-centered k-point mesh was used to sample the Brillouin zone of the supercell. For geometry optimization, all the internal coordinates are fully relaxed until the Hellmann–Feynman forces are less than 0.01 eV Å⁻¹. The climbing-image nudged elastic band (cNEB) method was employed to determine the energy barrier of the single Os atom dissociation and the Os dimer migration on the surface of graphyne.²⁴ The MAE was obtained by applying the torque approach which has been proved to be an effective method for the reliable determination of MAE.^25,26

Fig. 1 Top view (a) and side view (b) of the 2 × 2 supercell for functionalized Os@graphyne. The gray, blue and the green spheres stand for C atom, and Os atom and the functionalization atom, respectively.

For the convenience of discussion, we remind that the contributions to MAE can be subdivided into E_uu, E_dd, and E_ud,du, where uu and dd denote the contributions from the same spin channel and ud and du denote the contributions from the cross-spin channel. According to the second-order perturbation,²⁷ the contributions from the same spin channel can be expressed as:

where o and u denote occupied and unoccupied electronic states. E_o and E_u are their band energies, respectively. L_z and L_x are the angular momentum operators. ξ is the strength of SOC. Similarly, the corresponding contributions to the MAE from cross spin channel can be expressed as

For the contributions from the d states, the nonzero matrix elements of the L_z and L_x operators are 〈xz|L_z|yz〉 = 1, 〈x² − y²|L_z|xy〉 = 2, 〈xy|L_x|xz〉 = 1, 〈x² − y²|L_x|yz〉 = 1 and 〈z²|L_x|yz〉 = . It would be easy to find that the positive contributions to the MAE originates from two matrix elements 〈xz|L_z|yz〉 and 〈x² − y²|L_z|xy〉, while 〈xy|L_x|xz〉, 〈x² − y²|L_x|yz〉 and 〈z²|L_x|xz, yz〉 donate negative contributions, according to eqn (1). However, contributions to MAE from the nonzero matrix elements of the L_z and L_x operators are opposite in sign for different spin coupling, described as in eqn (2).

The calculated lattice constant of the unit cell for the graphyne monolayer is 7.00 Å, consistent with previous studies.^18,28 The Os atom was initially placed at the porous site of graphyne, as shown in Fig. 1(a). After full relaxation, we found that the Os atom moves outward by a height of h₁ 1.12 Å. This is reasonable because the strong interaction between carbon atoms at the edge of vacancy and Os atom, as was also found in previous studies.¹⁷ However, we previously found that the TM atoms almost remain in the plane of the g-C₃N₄ sheet due to the large vacancy.²⁹ On the other hand, the carbon atoms near the Os atom also shift outward of the graphyne plane by a height of h₂ 0.4 Å. The stability of the Os atom on graphyne was examined through the binding energy, which is defined as E_b = E (graphyne) + E (Os) − E (Os@graphyne). Here, E (graphyne), E (Os), and E (Os@graphyne) refer to the total energies of graphyne, Os atom, and the whole system consisting of Os atom and graphyne, respectively. We found that the Os atom strongly binds to graphyne with a large binding energy of 5.85 eV. The other possible initial adsorption sites for Os adatom were also taken into our consideration and we found that the most favorable adsorption site is porous site.

The magnetic moment for Os@graphyne is 1.42 μ_B and the calculated MAE is −18.0 meV, with the easy axis in the plane of the graphyne sheet. Nevertheless, the in-plane MAE of Os@graphyne (the MAE between x- and y-directions is only 0.4 meV) is too small to withstand thermal fluctuation. In order to understand the origin of the MAE in electronic structures, we present the projected density of states (PDOS) of Os atom as shown in Fig. 2(a). Under the influence of carbon atoms in the graphyne, the five d-orbitals of Os split into three groups, specifically, the d_z², d_xy/x²−y² and d_xz/yz as shown in Fig. 2(a). According to eqn (1) and (2), we found that the d_z² states in both the majority- and minority-spin channels play key roles in MAE. The positive contributions to MAE is from the cross-spin coupling 〈z²|L_z|yz〉 as indicated in eqn (2). It is found that the E_dd is about −44 meV which is mainly contributed from d_z² states and d_xy or d_yz in minority-spin channel near the Fermi level through the 〈z²|L_x|yz〉 couplings as in eqn (1). The large negative contributions to MAE result in the easy-axis in the plane of the graphyne sheet. Obviously, there is an effective way to manipulate the MAE by moving d_z² orbitals in the minority-spin channel above the Fermi level.

Fig. 2 (a) The calculated projected density of states (PDOS) of the d-orbitals and (b) Fermi level dependent total and decomposed MAEs from rigid band model of Os@graphyne. The vertical dashed lines mark the Fermi level.

To obtain more hints to manipulate the MAE, it is useful to calculate the total and spin decomposition MAE as a function of the position of the Fermi level according to the rigid band model. As shown in Fig. 2(b), when the Fermi level shifts downward, the positive contributions from ud/du, which is given in eqn (2), increase fastly. However, the negative contributions from dd, through 〈z²|L_x|yz〉 couplings, continuously decrease. And simultaneously, the negative contributions from uu slightly change with the variation of Fermi level. The competition between the contributions to MAE from different spin parts will lead to produce a positive total MAE when the Fermi level further shifts downward, which corresponds to less electrons occupancy into the d_z² orbital of Os. Positive MAE about 5 meV can be obtained for Os@graphyne, when the Fermi level shifts downward by 0.1 eV which corresponds one electron taken out of the system as seen in insert of Fig. 2(b). In particular, MAE can be extremely enhanced to be 47 meV by removing two electrons. Accordingly, the MAE can change sign and be enhanced to a huge value through rearrangement of d-orbitals occupancy of Os atom.

To modify the redistribution of electrons within d-orbitals of Os atom, we introduce nonmetal elements on the top of Os@graphyne. It has been recently shown that the introduction of light atoms or small molecules to metal-molecules can substantially control the magnetic properties, such as spin state, easy magnetization axis, and MAE.^30–34 The effects of H, O and F adatoms on top of the Os atom are investigated. Considering the symmetries of orbitals, the z-oriented orbitals d_z² is strongly hybridized with s or p_z states of adatoms, leading to the occupied (unoccupied) d_z² states of Os and shifting energy level of d_z² orbitals at Fermi level. After fully relaxation, we found that all our considered atoms strongly bind to Os with bond lengths in a range of 1.6–1.9 Å. The strong chemical effect of these adatoms significantly changes the arrangements of Os d-orbitals. For H absorbed on the top of Os@graphyne, the d_z² orbitals are almost occupied and the energy level d_z² deeply move away from the Fermi level due to the hybridization between H s states and Os d_z² states as shown in Fig. 3(a). Hence, the negative contributions to the MAE from the 〈z²|L_x|yz〉 coupling are greatly reduced. However, the positive contributions induced by cross-spin channel coupling 〈z²|L_z|yz 〉 are simultaneously eliminated due to the shift of the energy leveld_z². Eventually, the MAE of H functionalized Os@graphyne is only 4.7 meV as listed in Table 1. The positive MAE is mainly produced by the coupling between occupied and unoccupied d_xz/yz through 〈xz|L_z|yz〉 for both minority- and majority-spin channel. F functionalized Os@graphyne is different from H functionalized case. As shown in Fig. 3(c), the unoccupied Os d_z² orbitals shift towards the Fermi level and the occupied Os d_z² orbitals move far below the Fermi level due to the stronger hybridization between F p_z and Os d_z² orbitals. The coupling between majority-spin occupied and minority-spin unoccupied d_xz/yz through 〈xz|L_z|yz〉 and coupling between majority-spin occupied d_xz/yz and unoccupied d_z² through 〈z²|L_z|yz〉 results in a huge negative MAE of 47.7 meV. The d-orbitals arrangement of Os atom for O functionalized Os@graphyne is completely different from the H and F cases. As shown in Fig. 3(b), we found that all electrons are paired, which gives rise to the quenchment of magnetism for O functionalized Os@graphyne. Our results clearly indicate an effective approach to rearrange the d-orbitals for modulation the MAE through element functionalization.

Fig. 3 (a)–(c) The calculated PDOS of d-orbitals Os and (d)–(f) s and p orbitals of the functionalization atoms for H, O and F. The vertical dashed lines refer to the Fermi level.

Table 1 Total and local spin magnetic moments (μ_B) and total MAEs of element functionalized Os@graphyne

System	Ms			MAE
	Total	Os	Element
Os@graphyne	1.42	1.0	—	−18.0
H–Os@graphyne	1.08	0.6	0.0	4.7
O–Os@graphyne	0.00	0.0	0.0	—
F–Os@graphyne	1.56	0.9	0.1	−47.7
Fe–Os@graphyne	0.00	0.0	0.0	—
Co–Os@graphyne	2.69	0.2	1.9	14.7
Ru–Os@graphyne	1.36	0.1	1.1	8.2
Rh–Os@graphyne	0.00	0.0	0.0	—
Os–Os@graphyne	1.07	0.05	0.8	34.5

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Although huge MAE can be obtained for F functionalized Os@graphyne, the magnetic easy axis is in the plane of graphyne. The in-plane easy axis is disadvantage for magnetic record in conventional technology. Previous investigations have been pointed out that TM dimers are an alternative route to achieve a huge MAE.^4,6,12,14,35 We considered a series of TM atoms such as Fe, Co, Ru, Rh, and Os on the top of Os@graphyne as TM dimer combinations. After structural relaxation, the vertical geometry for all combinations is obtained right at the middle porous site as shown in Fig. 1. The bond lengths for the metal combinations are in the range 2.0–2.2 Å. Our calculation shows that all systems except Fe–Os@graphyne and Rh–Os@graphyne are magnetic as listed in Table 1. The largest total spin moment of 2.69 μ_B is found in Co–Os@graphyne whereas the smallest is 1.07 μ_B for Os–Os@graphyne. The magnetic moments of Os atoms near graphyne are significantly suppressed, compared to that of Os in Os@graphyne. Interestingly, the MAE of Os–Os@graphyne is about 34.5 meV with magnetic easy axis perpendicular to the graphyne plane. The huge MAE of Os–Os@graphyne is promising for the practical applications at room temperature. To better understand the driving factor for the development of huge MAE, we plotted the PDOS of the d orbitals for the two Os atoms in Os–Os@graphyne in Fig. 4(a) and (b). We identified that the couplings between minority-spin occupied and unoccupiedd_xy/x²−y² of upper Os, through 〈x² − y²|L_z|xy〉, play the dominant role for the giant positive MAE of Os–Os@graphyne. Particularly, the d_xy and d_x²−y² orbitals show a large energy separation and the Fermi level sits between them, which corresponds to a plateau of MAE near Fermi level, as indicated in Fig. 4(c).

Fig. 4 PDOS of the d-orbitals of the upper Os (a) and the lower Os atom (b) in Os–Os@graphyne. (c) Fermi level dependent total MAE. The vertical dashed lines mark the Fermi level.

The structural stability is another important factor for the practical application in magnetic storage. Firstly, we investigate the probability of the dissociation of Os atom away from Os–Os@graphyne and the dissociation of Os dimer away from Os–Os@graphyne. The probability is scaled with the dissociation energy,¹⁴ which is defined as:

E_d = E_dp + E_re − E_w, (3)

where E_dp, E_re, and E_w denote the total energy of the dissociated part, the remainder, and the whole system before dissociation, respectively. The dissociation energy E_d is 4.50 eV for single Os atom away from Os–Os@graphyne while 5.21 eV is needed for moving the Os dimer away from Os–Os@graphyne. It is shown that dissociation of one Os atom is relatively easier than Os dimer from Os–Os@graphyne. To get further insight into the migration of the upper Os atom on the graphyne, we estimate the diffusion barrier using the cNEB approach. As sketched in Fig. 5(a), two possible diffusion paths are considered in our calculations. The calculated energy barriers shown in Fig. 5(b) and (c) were found to be about 2.7 eV and 2.9 eV along for path H₁ and path H₂, respectively. Moreover, we found that the energy of the final state is larger than the initial state. Our results indicate that Os–Os@graphyne should be stable for application in magnetic record device.

Fig. 5 Os diffusion pathways (a) and energy profiles (b) and (c) for Os–Os@graphyne. The total energy of the initial geometry is set to zero.

To summarize, we report graphyne as an excellent template for nanomagnetic bits in application through a systematic DFT study. Due to the unique porous structure of graphyne, Os atom tightly binds to the graphyne at the hollow site. The MAE is about 18 meV with in-plane easy axis. Using rigid band model analysis, we introduced the strategy to manipulate the MAE by rearrangement of d-orbitals occupancy of Os atom through nonmetal functionalization. We found that the MAE is enhanced to be 47.7 meV for F functionalized Os@graphyne. To obtain the out-of-plane easy axis, we have investigated the MAE of the TM combinations. In view of MAE = 34.5 meV and high structural stability, we finally identified Os–Os@graphyne as an excellent candidate for room temperature applications. Our studies may pave an effective way towards robust nanomagnetic units for high density magnetic record.

Acknowledgements

This work is supported by National Natural Science Foundation of China (No. 11074212, 11474245, 11204259, 11402221), the Program for New Century Excellent Talents in University (No. NCET-12-0722), Natural Science Foundation of Hunan Province (No. 2015JJ6106) and Changjiang Scholars and Innovative Research Team in University (No. IRT13093). Computational support is provided by National Supercomputing Center in Tianjin and Research Center of Supercomputing Application in National University of Defense Technology.

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