Introduction

The continuous miniaturization of spintronics devices for modern technologies such as magnetic data storage will eventually reach the ultimate length scale (i.e., one to a few atoms)1,2,3,4,5,6. Recently, reading and writing quantum magnetic states in magnetic nanostructures with only a few transition-metal (TM) or rare-earth (RE) atoms were achieved by several experimental groups1,2,3,4,5,6,7. These investigations demonstrate the fascinating possibility to utilize magnetic nanostructures and even single atoms in nanometer-scale spintronics devices. In this realm, the magnetic anisotropy of a magnetic nanostructure is a critical factor because it prevents the random spin reorientation induced by thermal fluctuations. Therefore, large magnetic anisotropic energy (MAE) is desired in magnetic nanostructures that serve as the building blocks of spintronics devices. Except for Co adatoms on Ag(001)8, most magnetic nanostructures based on 3d TM atoms have MAEs of only a few meV which corresponds to a blocking temperature (TB) under 50 K, implying that their magnetic states are stable only at very low temperature2,9. For practical applications of magnetic nanostructures at room temperature, large MAEs of 30–50 meV are necessary.

The magnetic anisotropy of a magnetic nanostructure originates from the spin-orbit coupling (SOC) effect. By analysis of the SOC Hamiltonian with the second-order perturbation theory, Wang et al. expressed the MAE as a competition between the angular momentum components along z-axis (Lz) and x-axis (Lx)10,11:

$${\mathrm{MAE}} \approx \xi ^2\mathop {\sum }\nolimits_{u\alpha ,o\beta } (2\delta _{\alpha \beta } - 1)\left[ {\frac{{\langle u\alpha {\mathrm{|}}L_z{\mathrm{|}}o\beta \rangle ^2}}{{E_{u\alpha } - E_{o\beta }}} - \frac{{\langle u\alpha {\mathrm{|}}L_x{\mathrm{|}}o\beta \rangle ^2}}{{E_{u\alpha } - E_{o\beta }}}} \right]$$
(1)

Here ξ is the SOC constant; Euα and Eoβ are the energy levels of an unoccupied state with spin α (|uα〉) and an occupied state with spin β (|oβ〉), respectively. Therefore, there are two keys to achieve a large MAE: (i) large SOC constant ξ which exists in heavy atoms such as 5d TM atoms; (ii) specific energy diagram to reduce the denominator in Eq. (1) that can be realized by appropriate ligand field. For example, a large MAE of 9 meV in a single Co atom was induced by placing it on Pt(111) substrate12, while the MAE in bulk Co metal is only 0.045 meV per Co atom13. On the other hand, a giant zero field splitting (ZFS) of 58 meV for a Co adatom on a MgO(001) surface was observed in a recent experiment4, and such a giant ZFS was attributed to the special ligand field of the substrate. Roughly speaking, a large ZFS value usually implies a large MAE14. However, the ZFS cannot be obtained directly by regular density functional theory (DFT) calculations due to the single-electron approximation15. Alternatively, DFT calculations were performed to determine MAE of a TM adatom under various coordination environments8,15,16,17,18. Interestingly, combining the effects of the ligand field, orbital multiplet, and large SOC constants, a larger MAE of 208 meV for an Os adatom on a MgO(001) surface were predicted by DFT calculations19. Recently, attempts to investigate the magnetic anisotropy of a single RE adatom such as Dy, Ho, and Er on different substrates were made20,21,22. Despite the large SOC constants in these RE atoms, the resulting ZFSs are relatively small, i.e., 21.4 meV for a Dy adatom on graphene, 5.3 meV and 3.9 meV for Er and Ho adatoms on Pt(111), respectively. Meanwhile an appreciable MAE of about 10 meV/atom for a Er adatom on Cu(111) was reported by Singha et al.23. Nevertheless, the RE adatoms have longer magnetic lifetime than the 3d TM adatoms such as Co.

In addition to single TM and RE adatoms on certain substrates, TM and RE dimers are of particular interest23,24,25,26,27,28 due to their special symmetry and molecular orbitals. A homo-nuclear TM dimer is rotationally invariant around the molecular axis. Consequently, its magnetic anisotropy will arise at the first-order perturbation treatment of SOC interaction and therefore can be anomalously large compared to most other magnetic nanostructures24. In fact, appreciable MAEs of 30–70 meV were predicted theoretically for some free-standing TM dimers27,29,30 (positive MAE means easy axis parallel to the dimer axis).

Intuitively, the energy diagram of a given TM dimer can be modified effectively by chemical functionalization, which can in turn affect the magnetic anisotropy as clearly expressed in Eq. (1). Therefore, it is tempting to explore feasible ways to enhance the magnetic anisotropy of TM dimers. Besides placing a TM dimer on substrates, one possible tactic is to attach a light nonmetal atom which can form a strong chemical bond(s) with the TM atom(s) (thus affect the energy diagram) yet still retain the spin moment of the entire cluster.

Here, we show the possibility to engineer the magnetic anisotropy of a TM dimer by halogen functionalization. We choose the iridium dimer (Ir2) as a prototype since Ir2 possesses the largest MAE (70 meV) among free-standing homonuclear TM dimers27. Our first-principles calculations demonstrate that a MAE up to 294 meV can be achieved in Ir2 functionalized with a halogen atom (F, Cl, Br, I) and is of great potential for application in molecular spintronics devices.

Results

Magnetic anisotropy of Ir2

First, we calculate the structural and magnetic properties of Ir2 as given in Table 1. The equilibrium Ir–Ir bond length is 2.24 Å, slightly shorter than the experimental value (2.27 Å)31 but close to a previously computed value (2.22 Å) based on the DFT/PW91 method27. The calculated binding energy is 5.4 eV, significantly larger than the experimentally estimated upper limit (4.66 ± 0.21 eV)31,32 but in good agreement with previous calculation using the LDA method and LanL2DZ-BSSE basis set correction (5.5 eV)33. It is known that LDA tends to overbind metal clusters33,34,35 and the difference between the computational and experimental values originates from the approximation of LDA itself, as well as the zero-temperature nature of DFT static calculations (while there are temperature effects in experiments). Qualitatively, the large binding energy reflects a strong interaction between the two Ir atoms. The spin moment (MS) is 4 μB without considering the SOC effect. After including the SOC effect, MS reduces slightly to 3.86 μB while a large orbital moment (ML) of 2.06 μB is induced, in line with previous calculation27. The MAE is 77 meV with easy axis parallel to the molecular axis, and it is slightly larger than that reported previously (70 meV)27, probably due to different choice of DFT methods (e.g., exchange-correlation functional and basis set).

Table 1 Summary of basic properties of Ir2 and Ir2X
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This table lists geometry, interatomic distance (d), binding energy (Eb), spin moment (MS), orbital moment (ML), and magnetic anisotropy energy (MAE). The interatomic distances are shown as Ir–Ir (Ir-X) bond lengths for Ir2X. A positive MAE indicates the easy axis parallel to the molecular axis.

To understand the origin of the novel magnetic characteristic of Ir2, the energy diagram of its molecular orbitals is plotted in Fig. 1a. Due to the rotational symmetry of Ir2, the Ir-5d orbitals split into three groups:\(d_{xy/x^2 - y^2},d_{xz/yz}\) and \(d_{z^2}\). The basal plane of \(d_{xy/x^2 - y^2}\) is perpendicular to the Ir–Ir bond, and those of dxz/yz and \(d_{z^2}\) cross the Ir–Ir bond. Consequently, the interaction between the two Ir atoms results in three types of hybridizations: \(d_{xy/x^2 - y^2} - d_{xy/x^2 - y^2}\), \(d_{xz/yz} - d_{xz/yz}\), and \(d_{z^2} - d_{z^2}\). The corresponding bonding and antibonding states (i.e., six molecular orbitals in each spin channel) can be thus notated as \(\delta _d/\delta _d^ \ast\), \(\pi _d/\pi _d^ \ast\) and \(\sigma _d/\sigma _d^ \ast\). Based on the radial wave functions in Fig. 1b and the spin-polarized projected density of states (PDOS) in Fig. 2a of these molecular orbitals, we identified the orbital characters of all energy levels and marked them in Fig. 1a. Clearly, the hybridization \(d_{z^2} - d_{z^2}\) is much stronger than the others, which leads to large energy separation between \(\sigma _d\) and \(\sigma _d^ \ast\). The interaction between the two \(d_{xy/x^2 - y^2}\) orbitals is weakest, thus the corresponding energy separation is smallest. It can be seen that all these molecular orbitals are occupied in the majority spin channel. While in the minority spin channel, the doubly degenerate \(\delta _d^ \ast\) is half occupied, and \(\pi _d^ \ast\) and \(\sigma _d^ \ast\) are unoccupied. Therefore, the electronic configuration of these molecular orbitals is (σd)2(πd)4(δd)4(\(\delta _d^ \ast\))3(\(\pi _d^ \ast\))2(\(\sigma _d^ \ast\))1. On the other hand, the interaction between the Ir-6s orbitals is also strong, and the antibonding-state molecular orbitals (\(\sigma _s^ \ast\)) in both spin channels are unoccupied, resulting in an electronic configuration (σs)2(\(\sigma _s^ \ast\))0. In addition, there is moderate hybridization between \(d_{z^2}\) and s orbitals, as shown in Fig. 2a. Note that the electronic configuration of an isolated Ir atom is (5d)7(6s)2. Accordingly, we can conclude that each Ir-6s orbital donates one electron to the Ir-5d orbitals. As a consequence, the total Ms of Ir2 is 4 μB, contributed by \(\delta _d^ \ast\) (1 μB), \(\pi _d^ \ast\) (2 μB), and \(\sigma _d^ \ast\) (1 μB), respectively. With the SOC effect, the energy diagram does not change much if the magnetization is along the hard axis. On the contrary, most orbitals split significantly when the magnetization is along the easy axis (i.e., Ir–Ir bond). In particular, the splitting of the energy level of \(\delta _d^ \ast\) is as large as ~0.9 eV, so that the half occupancy in the minority spin channel is eliminated. Such different SOC effect according to the magnetization directions leads to large anisotropies for both spin and orbital moments. As seen in Supplementary Table 1, the spin moments along the easy and hard axes are 3.86 and 3.28 μB, while the corresponding orbital moments are 2.06 and 1.16 μB, respectively. Intuitively, the SOC splitting of the energy levels and the anisotropies of the spin and orbital moments are responsible for the large MAE of Ir224.

Fig. 1
figure 1

Molecular orbitals of an iridium dimer (Ir2). a Energy diagram of the molecular orbitals of Ir2 without and with the spin-orbit coupling (SOC) effect (separated by a vertical dashed line). Black and red colors refer to energy levels in the majority (up) and minority (down) channels, respectively. Both magnetizations perpendicular (⊥, hard axis) and parallel (||, easy axis) to the Ir–Ir bond are presented. The dashed lines connect the energy levels of each molecular orbital between the two magnetizations. b Radial wave functions of the molecular orbitals without the SOC effect. The paratactic orbitals are energetically degenerate. Yellow and blue colors stand for positive and negative wave functions, respectively, with a cutoff radius of ±0.02 a.u

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Fig. 2
figure 2

Projected density of states (PDOS) and orbital resolved magnetic anisotropy energy (MAE) of an iridium dimer (Ir2). a PDOS of the s and d orbitals of Ir2. The vertical dashed line marks the Fermi level (EF). b Nonzero contribution from each pair of molecular orbitals in the minority spin channel (Edd) and in the opposite spin channels (Eud). The horizontal dashed line shows the position of EF. The thickness of each vertical line scales with the magnitude of the corresponding contribution to MAE distinguished by different colors

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To further elucidate the underlying mechanism of the large MAE of Ir2, we applied Eq. (1) to distinguish respective contribution from each pair of orbitals to the MAE. For convenience, we divide total MAE into three parts: MAE = MAEuu + MAEdd + MAEud, respectively from coupling between majority spin states (uu), minority spin states (dd), and cross-spin states (ud). Since the s orbital is not involved in SOC, the MAE of Ir2 is determined by the Ir-5d related molecular orbitals. From the energy diagram in Fig. 1a and the PDOS in Fig. 2a, the Ir-5d related molecular orbitals in the majority spin channel are occupied completely, thus MAEuu is negligible. For MAEdd and MAEud, the nonzero contributions from the angular moment components in Eq. (1) are plotted in Fig. 2b. Clearly, for MAEdd, the contribution from Lz is much larger than that from Lx, resulting in a positive MAEdd (~10ξ2). For cross-spin coupling, Lz still contributes more to MAEud than Lx, but this leads to a negative MAEud (~ −3ξ2) due to the negative sign in Eq. (1). Nevertheless, the absolute magnitude of MAEdd is larger than that of MAEud, resulting in a positive total MAE (about 7ξ2) for Ir2. Furthermore, it can be seen from Fig. 2b that both the sign and magnitude of MAEdd and MAEud are dominated by Lz that are related to the δd and \(\delta _d^ \ast\) orbitals: \(\delta _d, \downarrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow\) (~ 17ξ2), \(\delta _d, \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow\) (~ 8ξ2) and \(\delta _d^ \ast , \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow\) (~ 16ξ2). Therefore, the MAE of Ir2 is mainly attributed to the SOC effect associated with the \(d_{xy/x^2 - y^2}\) orbitals through the angular moment component Lz.

Magnetic anisotropy of functionalized Ir2

According to the discussions above and Eq. (1), if there is an effective way to modify the energy diagram of Ir2, the contribution from each pair of the molecular orbitals will be revised, which alters the total MAE. One possible tactic is to attach an additional nonmetal atom to Ir2 to tailor the interaction between the two Ir atoms and thus the magnetic property of Ir2. To this end, we examined a series of nonmetal atoms (X), including C, Si, N, P, O, S, and halogen atoms, to construct Ir2X trimers. As shown in Supplementary Tables 2 and 3, there are three types of possible equilibrium structures: (i) linear chain with an X atom at one end of the Ir–Ir bond; (ii) isosceles triangle with an X atom over the middle point of the Ir–Ir bond; (iii) linear chain with an X atom at the middle point. We obtained ground-state structures and MAEs for all Ir2X clusters and found that only the halogen atoms can result in huge MAEs. Hereafter, we only take the halogen as a prototype to discuss the strategy to engineer the magnetic anisotropy of TM dimers.

For all halogen elements X (X = F, Cl, Br, I), the Ir2X trimers prefer type-I configuration (see Supplementary Table 3). As seen from Table 1, the Ir–Ir bond length in Ir2X changes slightly, while the Ir-X bond length increases monotonically with X from F to I due to the increasing atomic radius. Moreover, the binding energies of Ir2X are about twice of that of Ir2, indicating strong binding between the Ir and X atoms and high structural stability of Ir2X. Consequently, the Ir–Ir bond is significantly weakened, which leads to remarkable change of the energy levels of Ir-5d orbitals as manifested by the PDOS in Fig. 3a. Clearly, the F-2p orbitals hybridize strongly with the dxz/yz and \(d_{z^2}\) orbitals of Ir1 (the Ir atom bonding with X). Compared to Ir2, the energy levels of the bonding states of both dxz/yz and \(d_{z^2}\) orbitals of Ir1 shift downward by about 0.3 eV in the majority spin channel and 0.6 eV in the minority spin channel, while those of the corresponding antibonding states shift upward by about 0.7 eV and 0.3 eV, respectively. As a result, the hybridization between the two Ir atoms through the dxz/yz and \(d_{z^2}\) orbitals is markedly weakened. For Ir2 (the Ir atom at the other end of Ir2X), the energy levels of the dxz/yz and \(d_{z^2}\) orbitals shift upwards, and the energy separations between the corresponding bonding and antibonding states become narrower. Meanwhile, some electrons are transferred from the dxz/yz and \(d_{z^2}\) orbitals in Ir1, as shown in Supplementary Fig. 1. On the other hand, the \(d_{xy/x^2 - y^2}\) orbitals do not change much because these orbitals do not hybridize with the F-2p orbitals and the Ir–Ir bond length changes little. Therefore, the δd and \(\delta _d^ \ast\) are similar to those of Ir2, but their energy separation is narrowed by about 0.24 eV and the exchange splitting increases by about 0.52 eV [see Figs. 2a and 3a].

Fig. 3
figure 3

Projected density of states (PDOS) and orbital resolved magnetic anisotropy energy (MAE) of Ir2F. a PDOS of the F atom and two Ir atoms. Ir1 and Ir2 denote the Ir atom close to the F atom and at the other end of the trimer, respectively. The vertical dashed line indicates the Fermi level (EF). b, c Nonzero contribution from each pair of molecular orbitals in the minority spin channel (Edd) and in the opposite spin channels (Eud). The horizontal dashed line shows the position of EF for each case. The width of each vertical line scales with the magnitude of the corresponding contribution to MAE distinguished by different colors

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The magnetic moments are also significantly modified by attaching an X atom to Ir2. Without considering the SOC effect, the total spin moments of these Ir2X trimers are all 5 μB as listed in Supplementary Table 3, increasing by 1 μB compared to Ir2. According to the Milliken population analysis, Ir2 donates the electron on \(\delta _d^ \ast ( \downarrow )\)to the halogen atom X, so that the Fermi level (EF) is pinned between the energy levels of \(\pi _d^ \ast ( \uparrow )\) and \(\delta _d^ \ast ( \downarrow )\). Thus, the spin moment of Ir2X is contributed by \(\delta _d^ \ast\) (2 μB), \(\pi _d^ \ast\) (2 μB), and \(\sigma _d^ \ast\) (1 μB). However, the SOC effect notably reduces Ms of Ir2X to 3 μB (see Table 1), which is even smaller than that of Ir2 by 1 μB. On the contrary, MS of Ir2 is only altered slightly by the SOC effect. From the PDOS without SOC (Fig. 3a), we can see that the Fermi level (EF) is pinned within the small gap (61 meV) between \(\pi _d^ \ast ( \uparrow )\) and \(\delta _d^ \ast ( \downarrow )\). In other words, \(\pi _d^ \ast ( \uparrow )\) is fully occupied with two electrons and \(\delta _d^ \ast ( \downarrow )\)is empty. After including the SOC effect with easy-axis magnetization, the degenerate energy levels of both \(\pi _d^ \ast ( \uparrow )\) and \(\delta _d^ \ast ( \downarrow )\) split by 498 meV and 896 meV, respectively (see Supplementary Fig. 2). Consequently, the higher energy level stemmed from \(\pi _d^ \ast ( \uparrow )\) shifts upward and locates above EF, while the lower energy level stemmed from \(\delta _d^ \ast ( \downarrow )\) shifts downward and locates below EF. Therefore, both \(\pi _d^ \ast ( \uparrow )\) and \(\delta _d^ \ast ( \downarrow )\) are occupied by one electron. Obviously, one electron on \(\pi _d^ \ast ( \uparrow )\) transfers to \(\delta _d^ \ast ( \downarrow )\), resulting in the reduction of MS by 1 μB. Accordingly, the spin configuration becomes \(\delta _d^ \ast\) (1 μB), \(\pi _d^ \ast\) (1 μB), and \(\sigma _d^ \ast\) (1 μB). In addition, the orbital moments of Ir2X are about 1 μB, which is about half of that of Ir2. As shown in Supplementary Table 1, both the local spin moment and orbital moment on Ir1 are smaller than those on Ir2, which is caused by the hybridization between Ir1 and X. It is noteworthy that the orbital moment of Ir2X show little anisotropy between the easy and hard axes, despite of the enhanced anisotropy of the spin moment with respect to Ir2.

Remarkably, the MAEs of all Ir2X trimers are largely enlarged (see Table 1). Among them, Ir2Cl possesses the least MAE yet it is still as large as 223 meV; Ir2Br has the largest MAE of 294 meV; and the MAEs of the other two trimers are about 230 meV. All these MAE values are comparable to the highest theoretical values reported in literature, i.e., the lower and upper estimates of the MAE for the heteronuclear dimer CoIr decorated by benzene are 248 meV and 289 meV, respectively36.

To confirm our view of the underlying mechanism for the extraordinary enhancement of MAE due to halogen functionalization, we extracted the energy levels from the PDOS in Fig. 3a and estimated the orbital resolved MAEs of Ir2F using Eq. (1), as plotted in Fig. 3b,c. It can be seen that the main contributions to the total MAE stem from the matrix elements of LZ (\(\langle \delta _d, \downarrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\), \(\langle \delta _d, \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\), \(\langle \delta _d^ \ast , \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\)) and LX \((\langle \pi _d^ \ast , \uparrow {\mathrm{|}}L_x{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle )\) of both Ir atoms. Compared to Ir2 (see Supplementary Table 4), the positive contribution from \(\langle \delta _d, \downarrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\) increases lightly and the negative contributions from \(\langle \delta _d, \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\) and \(\langle \delta _d^ \ast , \uparrow {\mathrm{|}}L_z{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\) are slightly reduced. Interestingly, the contribution from \(\langle \pi _d^ \ast , \uparrow {\mathrm{|}}L_x{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\) is minor in Ir2 due to the relatively large energy separation between the corresponding orbitals. However, the energy levels of \(\pi _d^ \ast ( \uparrow )\) and \(\delta _d^ \ast ( \downarrow )\) become very close (e.g., ~61 meV for Ir2F as shown in Fig. 3a), which results in huge contribution to MAE from \(\langle \pi _d^ \ast , \uparrow {\mathrm{|}}L_x{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\). The final estimated MAEdd and MAEud are 13ξ2 and 44ξ2 respectively, both being larger than the corresponding values in Ir2 (10ξ2 and −3ξ2). Consequently, the total MAE of Ir2F increases dramatically to 232 meV, about three times as in Ir2 (77 meV). In fact, for all Ir2X trimers considered here, the term \(\langle \pi _d^ \ast , \uparrow {\mathrm{|}}L_x{\mathrm{|}}\delta _d^ \ast , \downarrow \rangle\) dominates the MAE (see Supplementary Table 4) because they all have similar energy diagrams (see Supplementary Fig. 3).

Discussion

Generally speaking, the halogen functionalization causes charge transfer and energy shifts in the Ir2 dimer delicately, thus resulting in enormous MAEs. This strategy to tune the MAE of a TM dimer should be universal. Our preliminary results indicate that the MAE of an Os2 dimer functionalized by O (F) atom can reach 290 (240) meV, with easy axis perpendicular to the Os–Os bond. It should also be noted that only maximizing the MAE is not enough to obtain stable nanomagnets. Other physical factors including the quantum number of the ground state and crystal field symmetry also play important roles in the dynamics of magnetization, especially the magnetic lifetime. For practical device applications, the magnetic molecule should be placed on certain substrates which may destroy the magnetic anisotropy. Our preliminary calculations reveal that an Ir2Br molecule supported by fully fluorinated graphene (GF) can retain its linear configuration, as well as large MAE of 238 meV. However, the binding energy between the Ir2Br trimer and the GF substrate is only about 0.1 eV, which is insufficient to withstand thermal perturbation and migration for applications in spintronics devices at room temperature. Therefore, more efforts are needed to search for suitable substrates that can not only hold Ir2X trimers but also retain their large MAEs. This is a challenging issue that deserves further exploration and the future results will be published elsewhere.

Our first-principles calculations have demonstrated that the MAE of an Ir2 dimer can be significantly boosted up to 294 meV by attaching a Br atom at one end of the Ir–Ir bond. Analyses of the energy diagrams and the matrix elements of the SOC Hamiltonian show that the \(d_{xy/x^2 - y^2}\) and dxz/yz orbitals are responsible for the colossal MAE. More specifically, the halogen atoms which have strong electronegativity lead to stable linear configuration for the Ir2X trimers and modify the energy levels of the 5d orbitals of the Ir atoms. The strategy of chemical functionalization introduces a new synthetic approach to chemically engineering the magnetic anisotropy of small magnetic nanostructures towards future-generation magnetic information storages using one to a few atoms per bit.

Methods

The structural relaxations and electronic structure calculations were carried out with the OpenMX package37 based on the density functional theory (DFT)38,39 (see details in Supplementary Note 1 including Supplementary Table 5 and Supplementary Fig. 4). We employed the spin-polarized local density approximation (LDA) in the Ceperley-Alder scheme40,41, norm-conserving pseudopotentials42,43,44,45,46, and pseudo-atomic localized basis functions47,48. The cutoff radii of the radial wave function were 9.0, 7.0, 9.0, 9.0, and 11.0 a.u. and the valence orbitals were s2p2d2f1, s3p3d2, s3p3d2, s4p4d3f1, and s3p3d3f2 for Ir, F, Cl, Br and I, respectively. The fully relativistic pseudopotentials46 were used, and the cutoff energy was set to 300 Ry. The criteria for energy and force convergence were 10−7 Hartree and 10−4 Hartree per Bohr, respectively. To determine MAE and orbital magnetic moments, self-consistent calculations including the SOC effect were performed with fully unconstrained non-collinear DFT method49,50,51,52. The MAEs of the linear molecules Ir2 dimer and Ir2X trimers are defined as the difference between the total energies with magnetizations parallel [E(||)] and perpendicular [E⊥] to the Ir–Ir bond: MAE = E(||) − E(⊥).