Controlling the magnetic anisotropy of Ru_mIr_n (m + n = 3) clusters using the MgO(001) substrate

Abstract

Large perpendicular magnetic anisotropy energy (MAE) and flexible regulation of the magnitude and direction of MAE have great potential for application in information storage devices. Here, utilizing first-principles calculations, we investigated the magnetic properties of free and MgO(001) supported Ru_mIr_n clusters (Ru_mIr_n@MgO(m + n = 3)). The results indicate that the MAE of mixed clusters increases with the number of Ir atoms due to Ir having a strong coupling between the non-degenerate d_xy and d_x²−y² states. The MAE of free Ir₃ is −8.18 meV with the easy magnetization direction parallel to the x-axis, while the MAE of supported Ir₃ on the MgO substrate increases by a factor of 2.6, and the easy magnetization axis of the structure is shifted to a direction perpendicular to the substrate surface. This change in MAE is due to the significant enhancement in the coupling between the non-degenerate d_yz and d_x²−y² states near the Fermi level of Ir₃ atoms. Moreover, Ir₃@MgO possesses high thermodynamic stability. These results give a new method for manipulating MAE and the direction of easy magnetization, which has great potential for application in magnetic nanodevices.

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Introduction

In conventional electronics, the charge properties of electrons or holes are used to transmit and process information, and the spin properties of the polarized electrons in magnetic materials are used to store the information.^1,2 The most critical point for high-density magnetic storage applications in reality is to explore materials with a large perpendicular magnetic anisotropy (PMA) to resist thermal perturbations so that the information will not be lost.³ Magnetic anisotropy mainly results from the spin–orbit coupling (SOC) interactions, a combination of the orbital magnetic moments and the coordination field. Strong SOC and large spin and orbital magnetic moments will lead to large magnetic anisotropy energy (MAE).^4–8 The mixed clusters of 4d and 5d transition metals (TMs) may have large MAE due to the strong SOC and large spin moments of TMs.^9–14

A supporting substrate may be necessary for the actual device realization of clusters, and large PMA is highly desirable for realizing atomic-scale magnetic data storage.^15,16 Nonmagnetic 2D materials are an active interdisciplinary research topic to support clusters for spintronics and nanoscience applications due to the appropriate ligand field they provide.^14,17–19 Xing et al. studied the MAE of the transition metal dimer supported by benzene (BZ) and found that the MAEs of OsRu@BZ and IrRh@BZ were about 58 meV and 72 meV, respectively.¹⁴ The MAEs of graphene supported mixed clusters of 4d and 5d TM with a single vacancy defect (SV), a single nitrogen-decorated vacancy defect (NSV), and a double nitrogen-decorated vacancy defect (NDV) have been extensively studied.^20–22 Compared with the substrates mentioned above, the MgO(001) surface provides a unique ligand field and degenerate levels induced by the orbital multiplet effects.^23–27 For instance, the orbital multiplicity effect leads to degenerate levels and then produces a huge MAE (60 meV) of Co@MgO(001) due to the special tetragonal ligand field of MgO(001).²⁸ And, thus MgO(001) may be a suitable substrate for supporting clusters to obtain a large MAE. What is more, studies have pointed out that the MAE of Co@MgO(001) is 6 times larger than that of a single Co atom on the surface of Pt(111).²⁹ If Co is replaced by a Ru atom with stronger SOC, the MAE of Ru@MgO(001) reaches 110 meV/Ru.³⁰ Although the magnetic properties of mixed dimers with 4d-TM and 5d-TM have been widely studied, the magnetic properties of TM trimers with 4d and 5d have not been reported. Hence, the research on the MAE of the Ru_mIr_n trimers supported using the MgO(001) substrate has important practical and theoretical significance.

In this work, we investigated the magnetic properties of free Ru_mIr_n (m + n = 3) clusters and manipulated the magnetic anisotropy of Ru_mIr_n using the MgO(001) substrate employing first-principles calculations. The results show that RuIr₂ has a large MAE of 23.40 meV and Ir atoms contribute most to the MAE of mixed Ru_mIr_n. The MgO substrate increases the projected density of states (PDOS) of Ru_mIr_n clusters, leading the MAE of Ir₃@MgO to increase by a factor of 2.6 and regulating the easy magnetized axis from axial to out-of-plane. These changes result from a large enhancement of the mutual coupling between the non-degenerate d-orbits of d_yz and d_x²−y² states. This regulation of the magnitude and direction of the MAE provides new ideas for the future applications of materials in magnetic storage.

Computational details

Density functional theory (DFT) computations were carried out in the VASP³¹ code using the projector augmented-plane wave (PAW)³² method. Perdew, Burke, and Ernzerhof (PBE) exchange–correlation functionals in the generalized gradient approximation (GGA)³³ was used to model the interaction between ions and electrons. 500 eV was chosen as the plane wave cut-off energy. A 4 × 4 × 1 Brillouin zone k-point grid was used for structural optimization, and 6 × 6 × 1 for static and MAE calculations. The MAE was calculated taking the SOC into consideration. The convergence criteria for force and total energy were 1 × 10⁻² eV Å⁻¹ and 1 × 10⁻⁷ eV, respectively. We constructed a 4 × 4 rectangular supercell of MgO(001) with a lattice constant of 8.4918 Å. To prevent effects between periodic lattices, we added a vacuum layer of 17 Å in the direction perpendicular to the substrate. To consider the Coulomb interaction, we used a method of GGA+U with U–J values of 2.3–2.5 eV for 4d and 5d states to describe the d electrons.^34,35 To verify the rationality of the selected U_eff (U–J) value, we tested the U_eff value as shown in Tables S1 and S2 (ESI†). The results suggest that the MAEs of these systems have a minor change in the U_eff values of 1.92–2.92 eV for Ru and 1.84–2.84 eV for Ir. In addition, the M_S value remains constant with a U_eff of 1.92–2.92 for Ru. Therefore, the used U_eff values for Ru and Ir of 2.42 and 2.34 eV, respectively, in our calculations are reasonable. The MAE is defined by this formula: MAE = E_x − E_z, where E_x and E_z represent the total energies of self-consistent calculations in the x (in-plane) and z (vertical) magnetization directions, respectively.

Results and discussion

Geometry and stability of Ru_mIr_n and Ru_mIr_n@MgO (m + n = 3)

The lowest energy structures of Ru_mIr_n are shown in Fig. 1. The lowest energy structures of Ru₃ and Ru₂Ir are near the isosceles triangle. The lengths of isosceles sides of Ru₃ (2.235 Å) are shorter than those of the Ru–Ir bond of Ru₂Ir (2.392 Å), while the Ru–Ru bond of Ru₃ is longer than the based side of Ru₂Ir (2.178 Å). The lowest energy structure of RuIr₂ is still an isosceles triangle, while the Ru–Ir bond length of RuIr₂ is slightly smaller than that of Ru₂Ir. When all Ru atoms are replaced, the lowest energy structure of Ir₃ is linear, with the smallest bond length (2.180 Å) in the clusters of Ru_mIr_n.

Fig. 1 The lowest energy structures of Ru_mIr_n clusters and Ru_mIr_n@MgO systems (m + n = 3). The red and green balls represent Mg atoms and O atoms, respectively. The bond length is in angstrom (Å).

To further explore the influence of different components on the stability of clusters, we calculated the binding energy (E_b) of a cluster with the following formula:

E_b = E_RumIrn − mE_Ru − nE_Ir (3-1)

where E_RumIrn, E_Ru, and E_Ir are the energies of the lowest energy Ru_mIr_n, Ru atom, and Ir atom, respectively. As listed in Table 1, the binding energies of Ru_mIr_n steadily increase from −11.75 eV to −10.05 eV as the number of Ir atoms increases, and these large absolute binding energies suggest that the clusters are quite stable.

Table 1 The total spin moment (M_S), binding energy (E_b), and MAE of Ru_mIr_n (m + n = 3), respectively

Cluster	Ru3	Ru2Ir	RuIr2	Ir3
M S (μB)	6.00	5.00	4.00	1.00
E b (eV)	−11.75	−11.32	−10.56	−10.05
MAE (meV)	−4.39	4.07	23.40	−8.18

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To elucidate the charge transfer of Ru_mIr_n, Table 2 demonstrates the Bader charge (Q) of each atom in Ru_mIr_n, respectively. For Ru₂Ir, the charge of Ir is 0.296 e, indicating that 0.296 electrons transfer from Ir to Ru. The charge of the Ru atom in RuIr₂ is −0.274 e. The transfer charge of atoms in Ru₃ is quite small. While for Ir₃, the charge of Ir2 reached −0.323 e, and thus the Ir–Ir bond is the smallest with 2.181 Å among all the studied Ru_mIr_n clusters. The Bader charge analysis demonstrates that the capacity to obtain electrons of Ru is substantially greater than that of Ir.

Table 2 Bader charge (Q) of Ru or Ir atoms for the Ru_mIr_n clusters and Ru_mIr_n@MgO systems (m + n = 3), respectively

Q (e)	Ru1	Ru2	Ru3	Ir1	Ir2	Ir3	RumIrn
Ru3	−0.053	0.022	0.030	—	—	—	0
Ru2Ir	−0.142	−0.154	—	0.296	—	—	0
RuIr2	−0.274	—	—	0.149	0.125	—	0
Ir3	—	—	—	0.145	−0.323	0.178	0
Ru3@MgO	0.086	0.091	0.097	—	—	—	0.274
Ru2Ir@MgO	−0.009	−0.076	—	0.442	—	—	0.358
RuIr2@MgO	−0.262	—	—	0.411	0.264	—	0.413
Ir3@MgO	—	—	—	0.343	−0.013	0.412	0.742

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The lowest energy structures and the sub-stable structure of Ru_mIr_n@MgO are also shown in Fig. 1 and Fig. S1 of the ESI,† respectively. The lowest energy structures of Ru_mIr_n@MgO are the Ru or Ir atoms situated on top of the O atom. The effect of the substrate on the structure of clusters is minimal, and the relative changes in the bond lengths of Ru_mIr_n do not surpass 4.6% except for Ir₃. Ir atoms at both ends of Ir₃ are attracted to O atoms, resulting in an increase in the length of the Ir–Ir bond. The minimum distance (d) between Ru/Ir and the nearest O of the substrate ranges from 2.07 Å to 2.18 Å (see Table 3), showing a strong binding between the MgO substrate and clusters.

Table 3 The minimum distance (d) between the Ru/Ir and O atoms, the absorption energy (E_A) of the cluster, and the MAE of Ru_mIr_n@MgO (m + n = 3) with GGA+U, respectively

Structure	Ru3@MgO	Ru2Ir@MgO	RuIr2@MgO	Ir3@MgO
d (Å)	2.18	2.11	2.07	2.12
E A (eV)	−1.91	−2.27	−2.80	−2.31
MAE (meV)	−2.86	3.93	4.18	20.56

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To evaluate the stability of Ru_mIr_n supported by the MgO substrate, the adsorption energy E_A (see Table 3) is calculated using the following equation:

E_A = E_RumIrn@MgO − E_RumIrn − E_MgO (3-2)

where E_RumIrn@MgO, E_RumIrn, and E_MgO refer to the energies of Ru_mIr_n@MgO, free Ru_mIr_n, and pristine MgO, respectively. It shows that E_A is proportional to the distance d. Ru₃@MgO has a maximum E_A with −1.91 eV. The absorption energies of Ru₂Ir@MgO and RuIr₂@MgO reach to −2.27 eV and −2.80 eV, respectively. Ir₃@MgO has a large absolute adsorption energy of 2.31 eV and a short distance d of 2.12 Å, indicating that Ir₃ binds tightly with the substrate.

To clarify the physical essence of Ru_mIr_n firmly anchored on the MgO substrate, we performed the Bader charge analysis of Ru_mIr_n@MgO (see Table 2). As displayed in Table 2, Ru₃ supported by MgO loses the least number of electrons (0.274 e) among the Ru_mIr_n@MgO systems, and the number of lost electrons for Ru_mIr_n gradually increases with the increasing Ir atom. The total charge reaches 0.742 e in Ir₃@MgO, suggesting that Ir₃ interacts strongly with the substrate. Compared with the charge of free clusters, the MgO support induces clusters to transfer more electrons to the substrate.

To further investigate the interaction between Ru_mIr_n and the MgO substrate, we show the differential charge density of Ru_mIr_n@MgO in Fig. 2. The light yellow and blue areas represent charge accumulation and dissipation, respectively. It is clear that oxygen atoms mainly obtain electrons for the MgO substrate. All the differential charge densities show that the O atoms near the Ir atoms lose more charge than the O atoms near the Ru atoms, indicating that the Ir atoms have a stronger interaction with the MgO substrate.

Fig. 2 Top and side views of the differential charge density of (a) Ru₃@MgO, (b) Ru₂Ir@MgO, (c) RuIr₂@MgO, and (d) Ir₃@MgO, respectively.

We further conducted the stability analysis of support systems using an ab initio molecular dynamics simulation (MDS) at 300 K, taking RuIr₂@MgO and Ir₃@MgO as examples. The thermodynamic stability of RuIr₂@MgO and Ir₃@MgO was evaluated by 10 ps using a Nose–Hoover thermostat with a step of 1 fs, and the canonical ensemble was employed. As shown in Fig. 3, the structures of the two systems do not change significantly after huge temperature fluctuations, and the energy fluctuations are minor, indicating their high thermodynamic stability.

Fig. 3 The fluctuations of the temperature and total energy of (a) RuIr₂@MgO and (b) Ir₃@MgO during the process of MDS. The illustrations represent the top and side views of RuIr₂@MgO and Ir₃@MgO after the MDS.

Magnetic properties of Ru_mIr_n and Ru_mIr_n@MgO (m + n = 3)

The total spin moments M_S (the total spin moment of a unit cell) of Ru_mIr_n are given in Table 1. The M_S of Ru_mIr_n decreases with the number of Ru atoms, where Ru₃ has the largest (6 μ_B), and Ir₃ has the smallest (1 μ_B) spin magnetic moment among all the studied Ru_mIr_n clusters. Furthermore, we listed the MAEs of clusters in Table 1. Ru₃ exhibits an MAE of −4.39 meV with an easy magnetization axis parallel to the triangle plane. Ru₂Ir has a small MAE (4.07 meV) with an easy magnetization axis perpendicular to the triangle plane. RuIr₂ presents a maximum MAE (23.40 meV) with the same easy magnetization direction as that of Ru₂Ir. For Ir₃, the easy magnetization axis parallels the x-axis with an MAE of −8.18 meV. Single component clusters exhibit in-plane or axial magnetic anisotropy, while mixed clusters with different Ru and Ir elements produce the perpendicular magnetic anisotropy.

To better understand the physical picture of the effect of different components on the MAE of clusters, the projected density of states (PDOS) of the d-orbital of free Ru_mIr_n is illustrated in Fig. 4. Following the second-order perturbation formula,³⁶ the distribution of the d-orbits near the Fermi level affects the magnitude of MAE. The second-order perturbation formula is as follows:

where ξ is the SOC constant, σ(σ′) = ↑(↓), namely spin up(spin down), o and u represent the occupied and unoccupied states, and L_x(z) represents the x(z) component of the orbital angular momentum operator, respectively. For the d electrons, the non-zero matrix elements of the L_z and L_x operators are 〈xz|L_z|yz〉 = 1, 〈x² − y²|L_z|xz〉 = 2, 〈xy|L_x|xz〉 = 1, 〈x²− y²|L_x|yz〉 = 1 and . As eqn (3-3), the main contribution to MAE is derived from the states near the Fermi level. In the following analysis, the matrix element difference |〈o↑/↓|u↑/↓〉|² corresponds to the orbital–momentum matrix element squared difference in eqn (3-3).

Fig. 4 Projected density of states of the d-orbital of each atom in (a) Ru₃, (b) Ru₂Ir, (c) RuIr₂, and (d) Ir₃ without SOC.

As shown in Fig. 4, the 5-fold degenerate d states of all atoms in Ru_mIr_n are split into three non-degenerate states (d_xy, d_z², and d_x²−y²) and two 2-fold degenerate states (d_yz and d_xz). The MAEs of clusters are mainly from the spin-down occupied and unoccupied states except for Ru₃. For Ru₃, the non-zero matrix elements |〈d_x²−y²↑|d_xy↓〉|² of the three Ru atoms have the main contribution, leading to an in-plane MAE (as shown in Fig. 4(a)). The non-zero matrix element |〈d_xy↓|d_x²−y²↓〉|² of the Ir1 atom dominates the out-of-plane MAE of Ru₂Ir (as shown in Fig. 4(b)). The MAE of RuIr₂ is still largely donated by the non-zero matrix element |〈d_xy↓|d_x²−y²↓〉|² of Ir1 (see Fig. 4(c)), but Ir2 also contributes positively to MAE, resulting in a large out-of-plane MAE with 23.40 meV. For the MAE of Ir₃, it is mainly contributed by the non-zero matrix elements |〈d_z²↓|d_yz↓〉|² of Ir1 and Ir3, leading to the axial MAE of −8.18 meV (see Fig. 4(d)). The MAE of Ir₃ is larger than that of Ru₃. Meanwhile, Ir contributes most to the MAE of mixed clusters, indicating that Ir may bring large MAE, and the MAE can be increased by changing the composition of Ru_mIr_n.

To reveal the potential mechanism of how composition change impacts the MAE, the d-orbital resolved MAE of Ru_mIr_n with the SOC effect is analyzed according to the method of second-order perturbation theory.³⁷Fig. 5 shows the d-orbital resolved MAE of each atom in Ru_mIr_n. As shown in Fig. 5(a), the matrix elements (d_xy, d_x²−y²) of three Ru atoms play a dominant role in the MAE of Ru₃, while their contributions are extremely small, resulting in a minute in-plane MAE (−4.39 meV). For Ru₂Ir, the MAE is largely contributed by the matrix element (d_xy, d_x²−y²) of Ir1, while the coupling between the d_xy and d_x²−y² states of Ru2 gives a negative contribution, which leads to a small out-of-plane MAE (4.07 meV). For RuIr₂, as shown in Fig. 5(c), the positive contribution is mainly from the matrix elements (d_xy, d_x²−y²) of Ir1 and Ir2, which leads to a large out-of-plane MAE (23.40 meV). The MAE of Ir₃ is mainly contributed by the matrix elements (d_z², d_yz) of Ir1 and Ir3, resulting in an MAE of −8.18 meV with the easy axis parallel to the Ir–Ir bond.

Fig. 5 The d-orbital resolved MAE of each atom in the (a) Ru₃, (b) Ru₂Ir, (c) RuIr₂, and (d) Ir₃ clusters, respectively.

For the Ru_mIr_n@MgO systems, we investigated the total spin moment M_S as shown in Fig. 6(a). Ir₃@MgO possesses a large spin moment (3.00 μ_B). Compared with Ru_mIr_n, all the M_S values of Ru_mIr_n@MgO increased by 2 μ_B, respectively. According to the Bader charge analysis, the oxygen atoms in the substrate obtain electrons from Ru_mIr_n, which increases the number of unpaired electrons in Ru_mIr_n and leads to the increment in M_S. To investigate the origin of the magnetic moment, we presented the spin densities in the inserted figure of Fig. 6(a). It can see that the spin densities of Ru_mIr_n@MgO are mainly located on the Ru_mIr_n clusters, suggesting that the Ru_mIr_n clusters contribute to the major magnetic moment.

Fig. 6 (a) The spin moments (M_S) and orbital moment differences ΔM_L of Ru_mIr_n@MgO, respectively. The illustrations are the spin densities of Ru_mIr_n@MgO. (b) The MAEs of Ru_mIr_n and Ru_mIr_n@MgO, respectively.

To reveal the regulatory effect of the support on the MAE, we listed the MAEs of free and supported Ru_mIr_n as shown in Fig. 6(b). Compared with free clusters, the MAEs of Ru₃@MgO and Ru₂Ir@MgO change little. However, the MAE of RuIr₂@MgO changes a lot and decreases to 4.18 meV, and the easy magnetization axis remains unchanged. In comparison, the MAE of Ir₃@MgO (21.40 meV) increases to more than twice the original and the easy magnetization axis changes from in-plane to out-of-plane. Such changes provide a guidance for the regulation of MAE. To investigate the origin of MAE, we give the orbital magnetic moment difference (ΔM_L = M_L‖ − M_L⊥) as shown in Fig. 6(a). As prescribed by Bruno's relationship,³⁸ ΔM_L and MAE follow a linear relationship according to the formula , where is the effective coefficient to represent the interactions between spins and orbitals. The MAE and ΔM_L of Ru_mIr_n@MgO largely follow Bruno's relationship, and Ir₃@MgO has a large ΔM_L (0.088 μ_B) and the largest MAE of 21.40 meV.

For free Ru_mIr_n, the d_x²−y² and d_xy states of Ir move to a high and low energy level with the number of Ru atoms decreasing, respectively, as shown in Fig. 4. When clusters are placed on the surface of MgO, the O atoms mainly obtain electrons and lead to redistribution of electrons for Ru_mIr_n clusters. One can find that the lost electrons of Ir increase and the bond of Ir–O changes from a covalent bond to an ionic bond with the decreasing number of Ru atoms, which cause the d_x²−y² and the occupied d_xz(yz) states of Ir shift to a low energy level and a high energy level, respectively, as displayed in Table 2 and Fig. 2. The above analyses indicate that Ru atoms and the MgO substrate can cause the redistribution of states near the Fermi level, which leads to the change of MAE.

Next, we studied the PDOS of the d-orbital of Ru_mIr_n@MgO in Fig. 7 to analyze the origin of MAE. Similar to the PDOS of Ru_mIr_n, the 5-fold degenerate d-states of Ru_mIr_n atoms in Ru_mIr_n@MgO are split into three non-degenerate states (d_xy, d_z², and d_x²−y²) and two 2-fold degenerate states (d_yz and d_xz). As shown in Fig. 7(a), for Ru₃@MgO, the non-zero matrix elements |〈d_xy↑|d_x²−y²↓〉|² and |〈d_xy↓|d_x²−y²↓〉|² of Ru1 have the main and opposite contributions to the MAE, resulting in the small in-plane MAE (−2.86 meV). For Ru₂Ir@MgO, the non-zero matrix element |〈d_yz↑|d_z²↓〉|² of Ir1 contributes mainly to the out-of-plane MAE of 3.93 meV (see Fig. 7(b)). The non-zero matrix element |〈d_xy↑|d_x²−y²↓〉|² of Ir1 has a negative contribution to the MAE of RuIr₂@MgO, while the non-zero matrix elements |〈d_xy↓|d_x²−y²↓〉|² and |〈d_z²↓|d_yz↓〉|² of Ir2 positively contribute to the MAE of RuIr₂@MgO, resulting in an out-of-plane MAE of 4.18 meV (see Fig. 7(c)). For Ir₃@MgO (see Fig. 7(d)), its MAE is mainly contributed by the non-zero matrix element |〈d_yz↑|d_x²−y²↓〉|² of Ir1 and Ir2, resulting in a large out-of-plane MAE (20.56 meV).

Fig. 7 Spin-polarized PDOS of the d-orbital of each atom in Ru_mIr_n (m + n = 3), leaving out the SOC effect of (a) Ru₃@MgO, (b) Ru₂Ir@MgO, (c) RuIr₂@MgO, and (d) Ir₃@MgO, respectively.

To elucidate the physical picture behind the impact of the MgO substrate, the d-orbital resolved MAE of Ru_mIr_n@MgO with the SOC effect is shown in Fig. 8. For the Ru₃@MgO system, as shown in Fig. 8(a), the contribution to MAE is mainly from the matrix element (d_xy, d_x²−y²) of Ru1, resulting in a small in-plane MAE (−2.86 meV). As shown in Fig. 8(b), for Ru₂Ir@MgO, the matrix element (d_z², d_yz) of Ir1 has a relatively large positive contribution, resulting in an out-of-plane MAE of 3.93 meV. For RuIr₂@MgO, MAE is mainly contributed by the matrix elements (d_xy, d_x²−y²) and (d_z², d_yz) of Ir2, but the negative contribution of the matrix element (d_xy, d_x²−y²) of Ir1 is large, resulting in a small out-of-plane MAE (4.18 meV). In Fig. 8(d), the matrix elements (d_yz, d_x²−y²) of all Ir atoms in Ir₃@MgO give the main positive contribution, resulting in a large out-of-plane MAE (20.56 meV).

Fig. 8 The d-orbital resolved MAE of (a) Ru₃@MgO, (b) Ru₂Ir@MgO, (c) RuIr₂@MgO, and (d) Ir₃@MgO systems, respectively.

To better comprehend the influence of the substrate on the MAE, the PDOS diagrams of the d-orbital of Ru_mIr_n and Ru_mIr_n@MgO (n = 2, 3) are shown in Fig. 9 due to the MAEs of RuIr₂ and Ir₃ greatly change after being supported on the MgO substrate. As shown in Fig. 9(a), the nonzero matrix elements |〈d_x²−y²↑|d_xy↓〉|² and |〈d_xy↑|d_x²−y²↓〉|² of free RuIr₂ are the main contributors to the MAE. After being supported, the occupied states of d_yz and d_xz increase significantly, and then the main contribution to the MAE is from the nonzero matrix element |〈d_yz↑|d_z²↓〉|². Hence, the MAE of RuIr₂@MgO drastically reduce after being supported, and the easy magnetization axis direction remains unchanged (see Fig. 9(b)). For free Ir₃, the occupied spin-down d_z² states are closer to the Fermi level than the occupied spin-up d_z² states, so the non-zero matrix elements |〈d_z²↓|d_yz↓〉|² give major contributions to the MAE, resulting in an in-plane MAE of −8.18 meV (see Fig. 9(c)). After being supported, the occupied spin-up d_xz state increases greatly, which leads to the contribution to the MAE (20.56 meV) mainly from the non-zero matrix element |〈d_yz↑|d_x²−y²↓〉|² of Ir₃ (as shown in Fig. 9(d)), and the change of the easily magnetized axis from in-plane to out-of-plane.

Fig. 9 The PDOS of the d-orbital of Ru_mIr_n (m + n = 3) irrespective of the SOC effect of the (a) RuIr₂, (b) RuIr₂@MgO, (c) Ir₃, and (d) Ir₃@MgO systems.

To better understand the underlying mechanism of the effect of the substrate, the total d-orbital resolved MAE images of Ru_mIr_n and Ru_mIr_n@MgO (n = 2, 3) are shown in Fig. 10. For RuIr₂, the matrix element (d_xy, d_x²−y²) gives a large positive contribution, resulting in a large out-of-plane MAE (23.40 meV) (see Fig. 10(a)). As shown in Fig. 10(b), after being supported, the contribution of the matrix element (d_xy, d_x²−y²) of RuIr₂ decreases resulting in a small out-of-plane MAE (4.18 meV). As shown in Fig. 10(c) and (d), free Ir₃ is mainly negatively contributed by the matrix element (d_z², d_yz), resulting in a small in-plane MAE (−8.18 meV). After being supported, the matrix element (d_z², d_yz) of Ir₃ has a positive contribution to the MAE, while the positive contribution from the matrix element (d_yz, d_x²−y²) increases significantly; hence, the magnitude of the MAE (20.56 meV) remarkably increases meanwhile the easily magnetized axis turns from in-plane to out-of-plane.

Fig. 10 The total d-orbital resolved MAE of the TM atoms in (a) RuIr₂, (b) RuIr₂@MgO, (c) Ir₃, and (d) Ir₃@MgO.

Conclusions

This paper investigated the magnetic anisotropy of Ru_mIr_n trimers and Ru_mIr_n@MgO systems (m + n = 3) and the modulatory effect of the MgO support on the MAE using first-principles calculations. Only Ru₂Ir and RuIr₂ produce a perpendicular magnetic anisotropy among trimers. The Ir atoms contribute most to the MAE of mixed Ru_mIr_n owing to the strong coupling between the non-degenerate d_xy and d_x²−y² states of Ir. The magnitude of MAE and the easy magnetization direction of structures can be manipulated using the MgO support, which is attributed to the increase of the densities of states around the Fermi level of Ru_mIr_n. The free Ir₃ exhibits a small in-plane MAE of −8.18 meV, while Ir₃@MgO possesses high stability and a large perpendicular magnetic anisotropy with an MAE of 21.40 meV due to the strong coupling between the non-degenerate d_yz and d_x²−y² states around the Fermi level of Ir₃. This work provides an effective method for regulating the magnitude and direction of magnetic anisotropy to facilitate practical applications.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11764034 and 12064036), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_15R46), the Yangtze River Scholar Research Project of Shihezi University (No. CJXZ201601), the Shihezi University Young Innovative Talents Cultivation Program (No. CXPY202114 and CXPY202019), and the High-level Talent Scientific Research Start-up Project (No. RCZK202008).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp03760b

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