Magnetic anisotropy of iridium dimers on two-dimensional materials

Abstract

Magnetic dimers with very large magnetic anisotropy have great potential in information storage applications. By using density functional theory calculations (DFT), we systematically investigated the magnetic anisotropy energy (MAE) of bi-iridium (Ir₂) dimers on four types of graphene-like two-dimensional (2D) material substrates. We considered four possible adsorption sites for Ir₂ on each substrate. The Ir₂ dimer prefers to remain at the single vacancy site with the largest binding energy for all the 2D materials considered. The spin moment and MAE of Ir₂ can be largely affected by the substrate. On the substrate of germanene, the MAE of Ir₂ can be enlarged to approximately 100 meV, even with the higher Ir₂ areal density of 1.804 nm⁻². Moreover, the direction of the easy magnetization axis is determined by the d states in the vicinity of the Fermi level. The present DFT results can be understood with the help of perturbation theory analysis.

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1. Introduction

As one of the oldest branches of physics, magnetism has been utilized for both fundamental science and innovative spintronic devices.¹ From low-dimensional nanostructures, clusters, and even down to single atoms,^2,3 there has been great interest in nano-information storage devices because of their potential widespread applications. The low blocking temperature (BT),⁴ which denotes the threshold of temperature for maintaining thermal fluctuations of spin orientations, is one of the main stumbling blocks for development of information storage devices in practical applications. Scaled with BT, magnetic anisotropy energy (MAE), defined as the energy difference between the easy-axis and hard-axis, is a crucial parameter for magnetic stability. To maintain the spin orientation at room temperature at a long period, it is known that MAE should be at least 30 meV.⁵ Therefore, many efforts in past decades have been undertaken to search and tune candidates with larger MAE values.^6–9

Because of the strong spin–orbital coupling (SOC),^10,11 it is expected that nanostructures containing 4d or 5d transition metal (TM) atoms will possess a possible larger MAE.¹² There have been many attempts to increase the anisotropy barrier by disturbing the local ligand field.^13,14 First, after assembling the magnetic atom into a single-molecule magnet with tens or hundreds of atoms,¹⁵ the MAE can be increased to 40 meV.⁶ Moreover, down to the TM dimer, the MAE can be remarkably enlarged,^16–20 especially for Ni₂²¹ and Pt₂ dimers,¹⁸ and because of the unique molecular orbital, a huge MAE (up to 220 meV per dimer) can be observed. Our recent work also suggested that by attaching a Br atom at one end of the Ir–Ir bond, a giant anisotropic magnet with MAE up to 294 meV can be obtained.⁷ However, a supporting substrate may be necessary for actual device realization. Related reports describe how magnetic atoms can be placed on substrates, e.g., single Co²² or single Co and Fe atoms on Pt(111),¹² Pb(111),²³ or MgO(001)²⁴ substrates, and the MAE of 0.045 meV in bulk Co metal can be increased to 9 meV and 58 meV, respectively.

Since the discovery of graphene,²⁵ there has been great interest in two-dimensional (2D) nanostructures because of their unique properties. Naturally, 2D materials can be utilized as supporting substrates. As a pioneering work, Xiao et al. studied Co–Co and Co–Ir dimers on benzene and graphene,²⁶ and reported a very large MAE of the order of 200 meV per molecule. Hu et al. systematically investigated homo- and hetero-dimers on both perfect and defective 2D materials, such as graphene,⁵ h-BN,²⁷ and 2D covalent organic frameworks.²⁸ They found that on the double vacancy site of h-BN, Ir₂ dimer presented a large MAE (approximately 126 meV) and high structural stability.²⁷ Additionally, Cao et al. found an enhanced MAE of Os atoms on fluorine functionalized graphene²⁹ and Ir atoms on g-C₃N₄,³⁰ the latter of which can be tuned by applying an electric field. Moreover, Zhang et al. found that when the dimers of (Tm, Er, Sm)–Ir are adsorbed onto graphene oxide, the MAE exceeded 200 meV, and Os–(Fe, Ru, Rh) dimer adsorbed on graphene possessed a huge MAE, typically larger than 100 meV.^31,32 Ge et al. used double-vacancy graphene as the substrate and decorated it with Ir atoms, and then manipulated the MAE with an external electric field.³³

All the above works suggest that as the supporting substrate, 2D materials can tailor the MAE of TM atoms/dimers and then have great potential for realize their promise. Therefore, it is worthwhile to continue exploring more alternative 2D substrates. Except for graphene, other atomic 2D materials with flat or buckled honeycomb lattice have also been proposed and fabricated, such as silicene,^34,35 germanene,^35,36 stanene,³⁷ and black phosphorus (BP).³⁸ However, the interaction between TM dimers and these 2D materials is still elusive.

2. Methods

In this study, by using density functional theory calculations (DFT), we systematically investigated the interaction between TM Ir₂ dimers and 2D materials, i.e., silicene, germanene, stanene, and BP, and compared these with the interaction that occurs with graphene substrate. Different adsorption sites and areal densities were considered, on both perfect and defective 2D materials. Ir₂ exhibited a larger MAE on germanene than that on graphene, and this large MAE (109.4 meV) was maintained even on higher areal density. The physics mechanism can be understood with the assistance of density of states analysis. Our studies showed that germanene may be a good candidate for supporting Ir₂ dimer in high-storage device applications.

The structural, magnetic, and electronic properties of Ir₂ dimer-adsorbed 2D materials were studied by first-principle calculations as implemented in Vienna Ab initio Simulation Package (VASP) code.³⁹ The ion-electron interaction was treated with projector-augmented plane wave (PAW) potentials,⁴⁰ and the exchange–correlation potential was described by generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional.⁴¹ Different supercells, i.e., (7 × 7) for graphene, (5 × 5) for silicene, (4 × 4) for stanene, (4 × 4) for germanene, and (5 × 4) for BP were selected to ensure that the periodic separation in the in-plane direction was larger than 15 Å. A vacuum space of 30 Å in the out-of-plane direction was used to avoid spurious interactions between periodic images. The energy cutoff for the plane-wave basis was set to 500 eV. The atomic structures were fully relaxed without any symmetric constrains, with the total energy and force convergence criteria of 10⁻⁴ eV and 0.01 eV Å⁻¹, respectively. In magnetic and electronic investigations, the convergence criterion for total energy was set to 10⁻⁶ eV, taking into consideration the effect of SOC. The Brillouin zone of the supercells was sampled by Monkhorst–Pack k-point mesh with separation of 0.02 Å⁻¹.

3. Results and discussion

Our investigation of the properties of the Ir₂ dimer, as shown in Table 1, reveal that the calculated Ir–Ir bond length is 2.21 Å, which is similar to the value obtained from previous DFT calculations^7,18,42 and experiments.⁴³ Additionally, the spin moment of 3.427 μ_B is also consistent with previous theoretical results.^7,18,42 For the Ir₂ dimer, the MAE can be defined as the energy difference between the magnetic moment parallel (‖) and perpendicular (⊥) to the Ir–Ir bond when considering SOC: MAE = E(‖) − E(⊥). The MAE of the free Ir₂ dimer is 40.71 meV, which is a value that is very similar to that of a previous calculation (40.6 meV).⁴⁴ It is noteworthy that the MAE is sensitive to the selection of exchange–correlation potential, e.g., the MAE of the Ir₂ dimer is 77 meV from local density approximation (LDA) calculations.⁸ Then, we investigated the Ir₂ adsorbed on 2D materials, i.e., graphene, silicene, germanene, stanene, and black phosphorus. Using graphene as an example, in Fig. 1, we depicted four types of adsorption locations: (1) top (T) site, (2) hollow (H) site, (3) bridge (B) site, and (4) single vacancy (SV) defect site.

Table 1 Ir–Ir bond length (d, Å), Ir–2D bond length (h, Å), spin moment (M_s, μ_B) of Ir₂ adsorbed on 2D materials, spin moment for the Ir atom (Ir¹: near the substrate, Ir²: far from the substrate), binding energy (E_b, eV per Ir₂) and MAE (meV per Ir₂) of the Ir₂ dimer on 2D materials

Adsorption site	2D Material	d	h	M s	M s (Ir^1)	M s (Ir^2)	E b	MAE
Free Ir2 dimer		2.210		3.427	1.713	1.713		40.71
H	Graphene	2.210	1.781	0.382	0.037	0.344	2.136	31.56
	Silicene	2.257	1.247	2.791	0.623	1.980	5.231	21.25
	Germanene	2.240	1.187	2.451	0.626	1.692	4.686	94.85
	BP	2.253	1.427	0.061	−0.054	0.175	4.670	—
T	Graphene	2.206	2.125	3.277	1.265	1.836	1.216	97.93
	BP	2.260	1.801	1.649	0.322	1.195	2.973	51.31
SV	Graphene	2.318	1.060	1.495	0.118	1.425	8.810	18.45
	Germanene	2.286	1.021	1.362	0.130	1.219	7.006	99.63

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Fig. 1 Schematic diagram of the geometry for Ir₂ adsorbed on 2D material from the (a) top view and (b) side view. In (a), different adsorption sites are labeled as T (top site), H (hollow site), B (bridge site), and V (vacancy site), respectively. In (b), d represents the Ir–Ir bond length, and h is the distance between the bottom Ir atom and the 2D material.

We considered the initial configurations with the Ir₂ dimer perpendicular and parallel to the 2D sheets. The structural properties of Ir₂-adsorbed 2D materials are listed in Table 1, where it has been noted that in some configurations, Ir–Ir bonds are not perpendicular but are parallel to or merged into the 2D sheets. These configurations (silicene: T and SV; germanene: T; BP: SV, and all configurations for stanene) are, therefore, not listed in Table 1. Compared with that for the Ir₂ dimer, there is a less than 0.1 Å increase for most Ir–Ir bond lengths (d) of Ir₂ when adsorbed on 2D materials, suggesting the negligible effect of the substrate. Additionally, the distance (h) from the bottom Ir atom (Ir¹) to the substrate (see Fig. 1) ranged from 1.021 to 2.125 Å, depending on the detailed adsorption site. The structural stability can be measured by the binding energy defined as: E_b (in eV per Ir₂) = E_Ir2 + E_2D − E_Ir2–2D, where E_Ir2 is the total energy of isolated Ir₂ dimer, E_2D is the total energy of 2D material, and E_Ir2–2D is the total energy of the system with Ir₂ adsorbed on a 2D material. The binding energy of Ir₂ on the SV site of graphene (germanene) is 8.81 eV per Ir₂ (7.006 eV per Ir₂), which is much larger than that of the other sites, but slightly lower than that of the defective h-BN (9.13 eV per Ir₂).²⁷ This suggests that Ir₂ can be quickly attracted to the single vacancy site on either the graphene or germanene substrate.

Table 1 shows that all considered systems are spin polarized, and the spin moments (M_s) are mainly localized on the Ir₂. Compared with the free Ir₂ dimer, M_s of Ir₂ is lower and very sensitive to the substrate. Consistent with previous results of TM dimer on defective graphene,⁵ Ir¹ has a smaller M_s (−0.054–1.265 μ_B), and the Ir atom far away the substrate (Ir²) is less affected by 2D material and has a larger M_s (0.175–1.980 μ_B). It is noteworthy that the in-plane easy-axis can be observed when Ir₂ is adsorbed on the H site of BP, in which M_s will decrease to −0.054 μ_B for Ir¹ and 0.175 μ_B for Ir², respectively. When an Ir₂ dimer is adsorbed on germanene, the system is unstable, and one Ir atom flies away from the T site. For both the H and SV sites of germanene, Ir¹ has a smaller M_s than Ir². In order to understand the magnetic moment of the Ir₂ dimer as a function of the distance h, using Ir₂ on the T site of graphene (ρ_gra = 0.451 nm⁻²) as an example, the magnetic moment for either Ir₂ dimer or Ir atoms linearly changes with the increase in h. For Ir¹ (near the substrate), the magnetic moment increases from 1.219 μ_B (h = 2.029 Å) to 1.543 μ_B (h = 2.429 Å), due to the weakened interaction between Ir¹ and the C atom at the T site. Additionally, the magnetic moment of the other Ir atom decreases from 1.88 μ_B to 1.841 μ_B, and the magnetic moment for the Ir₂ dimer slightly increases with the increase in h.

We next investigated the MAE of these configurations. By definition, the MAE is the energy difference between the magnetic moment parallel (‖) and perpendicular (⊥) to the atomic plane:

MAE = E(‖) − E(⊥) (1)

Both x and y directions have been considered for the magnetic moment parallel and perpendicular to the atomic plane. A positive MAE indicates a preference for out-of-plane magnetization, namely, an easy-axis along the z-axis, which is essentially required for magnetic information storage. As can be found in Table 1, the MAE highly relies on the selection of supporting substrates. Note that the MAE for the SV of graphene from the present calculations is only 18.45 meV per Ir₂ (see Table 1), which is quite smaller than 51.4 meV per Ir₂⁴³ or approximately 30 meV per Ir₂.⁵ The main reason for the underestimated MAE is the larger vacuum space in the out-of-plane direction, which is 30 Å in the present calculations, as mentioned above. When the vacuum space is decreased to 15 Å, as in the previous calculations in ref. 5 and 43, the MAE will increase to 42.7 meV per Ir₂, which is consistent with previous calculations. Fig. 2 shows E_b and MAE and a comparison with free Ir₂ dimer. Interestingly, when Ir₂ is adsorbed on the T site of graphene and the BP, H, and SV sites of germanene, the MAE is 40.71 meV larger than that of the free Ir₂ dimer. Among them, germanene not only enlarged the MAE of Ir₂, but it also stabilized the Ir₂ dimer with a large E_b (4.686 for H and 7.006 eV per Ir₂ for SV). This result suggests that germanene may be an ideal candidate as a supporting substrate for Ir₂ in real applications.

Fig. 2 Binding energy (E_b) and MAE of the Ir₂ dimer adsorbed on the H, T, and SV sites of different 2D materials. The dashed lines mark the MAE of the Ir₂ dimer (40.71 meV) as a visual guide.

We then studied germanene with different Ir₂ densities (ρ), and compared them to the graphene case. The Ir₂ density ranged from 0.389 to 4.77 nm⁻² and from 0.451 to 1.804 nm⁻² for graphene (ρ_gra) and germanene (ρ_ger), respectively. With the increase in ρ_gra, the variation of d is less than 0.6% for all adsorption sites, but h can be enlarged by 4.6% for H and 11.7% for SV, respectively, with respect to that in the (7 × 7) graphene supercell (see Table 1). With the increase in ρ_ger, the variation of d was less than 0.1%, which is consistent with that on graphene substrate. Being contrary with graphene, h for germanene slightly decreased (less than 0.75%) with the increase in ρ_ger. We found that MAE is linearly functionalized with h, and only a 1.1% decrease in MAE occurred with a 6.7% increase in h. Therefore, we can safely neglect its effects. Because of the repulsive interaction of Ir₂ from the periodic images, for the higher areal density, Ir₂ can either fly away from the T site of graphene, or merge into the 2D sheet for the H or T site of germanene.

Fig. 3 shows E_b, M_s, and the MAE of Ir₂ on graphene and germanene with different adsorption locations and different densities. The SV is always the most preferred site, with a larger E_b for both graphene and germanene. The E_b difference (approximately 2 eV) between the H and SV sites is quite small when Ir₂ is absorbed on the germanene substrate, but it is quite large (approximately 7 eV) for Ir₂ at different sites on the graphene substrate. A similar trend can be found for M_s, which is not sensitive to either ρ_gra or ρ_ger, but is sensitive to the adsorption locations. The changes in MAE with different ρ values are shown in Fig. 3. For the most stable adsorption site (SV) of graphene, the MAE generally increases with the increase in ρ_gra, but most of the MAE values are smaller than that of free Ir₂ dimer (40.71 meV). For the T site on graphene, the MAE is generally larger than 40 meV, but it is reduced with the increase in ρ_gra. We note that for ρ_gra = 4.77 nm⁻², the MAE is abnormally large again, which probably results from the interactions between the periodic images of Ir₂ in the (2 × 2) supercell. Furthermore, the E_b value is one of the most essential for Ir₂ absorption. Therefore, low absorption energy (1.216 eV per Ir₂ from Table 1) determines that Ir₂ attached at the T site of graphene is not a practical device, even with a high MAE value. On germanene, for both the H and SV sites, the MAE generally increases with the increase in ρ_ger, and the values are much larger than that on the graphene substrate or that of the free Ir₂ dimer. All the above results suggest that because germane exhibits a larger E_b and MAE than graphene, germanene may be a more suitable substrate for use in information storage devices.

Fig. 3 E _b, M_s, and MAE of the Ir₂ dimer on (a) graphene and (b) germanene with different densities. The dashed lines mark the MAE of the Ir₂ dimer (40.71 meV).

To understand the physical picture behind the varying MAE values of the Ir₂ dimer, we discuss the difference between Ir₂ on graphene and on germanene with the assistance of the second order perturbation approach proposed by Wang.¹⁰ Accordingly, MAE can be determined as:

in which ξ denotes the SOC strength, O(U) denotes the occupied (unoccupied) state, l_z(l_x) denotes the angular momentum operator, and E_U(E_O) denotes the energy level of the unoccupied (occupied) state. Then, MAE can be approximately determined by the angular momentum operators l_z and l_x across the states around the Fermi level. The projected density of states (PDOS) diagram derived from the major contributions close to the Fermi level can be found in Fig. 4. According to the PDOS of Fig. 4, the d-related molecular orbitals in the spin-up channel are almost completely occupied, and the matrix elements of the l_z and l_x operators are zero, which result in the interaction between the spin-up bands (u) being negligible. The empty states mainly belong to spin-down bands (d). Thus, MAE can be approximately obtained by summarizing the contributions from the d–d interaction and the u–d interaction between the U and O states:Due to the rotational symmetry of Ir₂, the 5d orbitals split into three groups: d_xy/x²−y², d_xz/yz, and d_z². For contributions from the d states, the nonzero matrix elements of the l_z and l_x operators are:^10,45 〈xz|l_z|yz〉 = 〈xy|l_x|xz〉 = 〈x² − y²|l_x|yz〉 = 1, 〈x² − y²|l_z|xy〉 = 2 and . Then, MAE can be obtained by analyzing the PDOS in a fixed energy range near the Fermi energy.

Fig. 4 Projected density of states (PDOS) of (a) Ir₂-graphene (upper: H site, lower: SV site), (b) Ir₂-germanene (upper: H site, lower: SV site) and (c) free Ir₂ dimer. The dashed lines label the Fermi level (E_F). Black, red, and blue colors denote the contributions from the d_z², d_xz/yz, and d_xy/x²−y² orbital, respectively.

The PDOS in Fig. 4 shows the different contribution of d orbitals of free Ir₂ dimer, and Ir₂ on graphene (7 × 7) and germanene (4 × 4). Clearly, compared with the PDOS of Ir₂ dimer, the band shifting in the Ir₂-substrate system enables the MAE to be tunable. Three differences in the PDOS can be found between graphene- and germanene-supporting substrates. When adsorbed on the H site of graphene, the spin-up occupied state d_xy/x²−y² moved closer with E_F, the spin-up occupied state d_xz/yz changed to an unoccupied state, and the spin-down d_z² states moved down with the creation of two spin-up d_z² states. On the contrary, when Ir₂ is adsorbed on the H site of germanene, the spin-up occupied state d_xy/x²−y² moved away from E_F, and the spin-up occupied state d_xy/x²−y² and d_xz/yz moved closer to E_F. Additionally, the new created spin-up d_z² state is closer to E_F than that of the Ir₂ free dimer.

All these differences suggested that the smaller MAE can be assigned to graphene and the larger one assigned to germanene. A similar orbital shift can be found for SV configurations, as shown in the lower panel of Fig. 4(a) and (b). Considering the contributions from the d_z², d_xz/yz, and d_xy/x²−y² orbitals, it is possible to calculate MAE using eqn (2). The MAE from eqn (2) (DFT calculations) in the energy range from −2 to 2 eV consists of free Ir₂ dimer: 19.3ξ² (40.71 meV), Ir₂ on the H site of graphene: 15.2ξ² (31.54 meV per Ir₂), and Ir₂ on the H site of germanene: 130.3ξ² (94.84 meV per Ir₂). Note that the MAE from eqn (2) depends on the selection of the energy range, which is due to the small fluctuation from the second order perturbation approach. However, the DFT results can generally be reproduced, and the different substrate effect on MAE is understandable.

As shown in Fig. 3(b), the MAE of Ir₂ on the H or V site of germanene nonlinearly changes as the areal density increases from 0.451 nm⁻² to 1.804 nm⁻². To explore this phenomenon, in Fig. 5(a) and (b), we plotted the PDOS of Ir₂ on the germanene H and SV sites, respectively. A nonlinear shift occurs for both d_xy/x²−y² and d_xz/yz states with the increase in areal density. For example, in Fig. 5(a), the spin-up occupied state d_xy/x²−y² shifts away from the Fermi level with the areal density increase from 0.451 nm⁻² to 0.802 nm⁻². However, this state shifts closer to the Fermi level as the areal density increases from 0.802 nm⁻² to 1.804 nm⁻². As shown in Fig. 5(c) and (d), this type of nonlinear behavior can be reproduced from eqn (2), with the summarization within the energy range of (−2, 2 eV).

Fig. 5 The PDOS and MAE of Ir₂ adsorbed at the (a and c) H site and (b and d) SV site of germanene. In (a) and (b), the three parts from the top to bottom correspond to the Ir₂ density of 0.451, 0.802, and 1.804 nm⁻², respectively. The dashed line labels the Fermi level. In (c) and (d), the MAE from DFT and eqn (2) are labeled by black squares and blue triangles, respectively.

4. Conclusions

In summary, the stability and magnetic properties of Ir₂ dimers adsorbed on different 2D materials were investigated using first-principles calculations. When the Ir₂ dimer remained at the single vacancy site, the binding energy was approximately two times larger compared with that of other adsorption sites. We also found that germanene is the ideal supporting substrate for the Ir₂ dimer, because of its larger MAE of 121.01 meV. Moreover, the binding energy and MAE for Ir₂ on germanene remain less affected with the higher areal density of 1.804 nm⁻². The DFT results can be understood by the PDOS analysis with the assistance of the second order perturbation approach.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (201802070) and the National Natural Science Foundation of China (11904251).

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