Spin-Orbit Coupled Insulators and Metals on the Verge of Kitaev Spin Liquids in Ilmenite Heterostructures

Spin-Orbit Coupled Insulators and Metals on the Verge of Kitaev Spin Liquids in Ilmenite Heterostructures Yi-Feng Zhao,1, ∗ Seong-Hoon Jang,2 and Yukitoshi Motome1, † 1 Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan Institute for Materials Research, Tohoku University, Aoba, Sendai, 980-8577, Japan arXiv:2403.09112v1 [cond-mat.str-el] 14 Mar 2024 2 Competition and cooperation between electron correlation and relativistic spin-orbit coupling give rise to diverse exotic quantum phenomena in solids. An illustrative example is spin-orbit entangled quantum liquids, which exhibit remarkable features such as topological orders and fractional excitations. The Kitaev honeycomb model realizes such interesting states, called the Kitaev spin liquids, but its experimental feasibility is still challenging. Here we theoretically investigate hexagonal heterostructures including a candidate for the Kitaev magnets, an ilmenite oxide MgIrO3 , to actively manipulate the electronic and magnetic properties toward the realization of the Kitaev spin liquids. For three different structure types of ilmenite bilayers MgIrO3 /ATiO3 with A = Mn, Fe, Co, and Ni, we obtain the optimized lattice structures, the electronic band structures, the stable magnetic orders, and the effective magnetic couplings, by combining ab initio calculations and the effective model approaches. We find that the spin-orbital coupled bands characterized by the pseudospin jeff = 1/2, crucially important for the Kitaev-type interactions, are retained in the MgIrO3 layer for all the heterostructures, but the magnetic state and the band gap depend on the types of heterostructures as well as the A atoms. In particular, one type becomes metallic irrespective of A, while the other two are mostly insulating. We show that the insulating cases provide spin-orbit coupled Mott insulating states with dominant Kitaev-type interactions, accompanied by different combinations of subdominant interactions depending on the heterostructural type and A, while the metallic cases realize spin-orbit coupled metals with various doping rates. Our results indicate that these hexagonal heterostructures are a good platform for engineering electronic and magnetic properties of the spin-orbital coupled correlated materials, including the possibility of Majorana Fermi surfaces and topological superconductivity. I. INTRODUCTION Strong electron correlations, represented as Coulomb repulsion U , play a pivotal role in 3d transition metal compounds and lead to a plethora of intriguing phenomena, such as the Mott transition and high-temperature superconductivity [1, 2]. The other key concept of quantum materials, the spin-orbit coupling (SOC), represented as λ, is a relativistic effect entangling the spin degree of freedom and the orbital motion of electrons, which is an essential ingredient in the topological insulators [3, 4]. Beyond their independent effects, synergy between U and λ has attracted increasing attention recently due to the emergence of new states of matter, such as axion insulators [5, 6] and topological semimetals [7– 9]. In general, it is difficult for the SOC to dramatically influence the electronic properties in 3d transition metal compounds since λ is much smaller than U . However, when proceeding to 4d and 5d systems, the d orbitals are spatially more spread out, which reduces U , and at the same time, the relativistic effect becomes larger for heavier atoms, enhancing λ. Hence, in these systems, the competition and cooperation between U and λ play a decisive role in their electronic states and allow us to access the intriguing regime that yields the exotic correlated states of matter [10]. ∗ † zyf@g.ecc.u-tokyo.ac.jp motome@ap.t.u-tokyo.ac.jp One of the striking examples is the spin-orbit coupled Mott insulator, typically realized in the iridium oxides with Ir4+ valence, e.g., Sr2 IrO4 [11, 12]. In each Ir ion located in the center of the IrO6 octahedron, the crystal field energy, which is significantly larger than U and λ, splits the d orbital manifold into low-energy t2g and highenergy eg ones. For Sr2 IrO4 , five electrons occupy the t2g orbitals and make the system yield the t52g low-spin state. Usually, the partially-filled orbital causes a metallic state according to the conventional band theory, but the insulating state was observed in experiments [13]. Considering the large SOC, the t2g manifold continues to split into high-energy doublet characterized with the pesudospin jeff = 1/2 and low-energy quartet with jeff = 3/2; the latter is fully occupied and the former is half filled. Finally, a Mott gap is opened in the half-filled jeff = 1/2 band by U . This accounts for the insulating nature of the system, and the Mott insulating state realized in the spin-orbital coupled bands is called the spin-orbit coupled Mott insulator. The quantum spin liquid (QSL), one of the most exotic quantum states in the spin-orbit coupled Mott insulators, has received increasing attention due to the emergence of remarkable properties, e.g., fractional excitations [14] and topological orders [15]. In the QSL, long-range magnetic ordering is suppressed down to zero temperature due to strong quantum fluctuations, though the localized magnetic moments are quantum entangled [16–19]. The presence of fractional quasiparticles that obey the nonabelian statistics is not only of great fundamental phys- 2 ical research, but also promising toward quantum computation [20]. In general, one route to realizing the QSL depends on geometrical frustration. Indeed, experiments have evidenced several candidates of QSL in antiferromagnets with lattice structures including triangular unit, where magnetic frustration is the common feature [21– 23]. The other route to the QSL is the so-called exchange frustration caused by the conflicting constraints between anisotrpic exchange interactions [24]. The strong spinorbital entanglement in the spin-orbit coupled Mott insulators, in general, gives rise to spin anisotropy, offering a good playground for the exchange frustration, even on the lattices without geometrical frustration. The Kitaev model is a distinctive quantum spin model realizing the exchange frustration [25]. The model is defined for S = 21 local magnetic moments on the twodimensional (2D) honeycomb lattice with the Ising-type bond-dependent anisotropic interactions, whose Hamiltonian is given by X X H= Kα Siα Sjα , (1) α ⟨i,j⟩α where α = x, y, z denote the three bonds of the honeycomb lattice, and Kα is the exchange coupling constant on the α bond; the sum of ⟨i, j⟩α is taken for nearestneighbor sites of i and j on the α bonds. In this model, the orthogonal anisotropies on the x, y, z bonds provide the exchange frustration. Importantly, the model is exactly solvable, and the ground state is a QSL with fractional excitations, itinerant Majorana fermions and localized Z2 fluxes [25]. Moreover, the anyonic excitations in this model hold promise for applications in quantum computing [25, 26]; especially, non-Abelian anyons, which follow braiding rules similar to those of conformal blocks for the Ising model, appear under an external magnetic field. The exchange frustration in the Kitaev-type interactions can be realized in real materials when two conditions are met [27]. First, at each magnetic ion, the spin-orbit coupled Mott insulating state with pseudospin jeff = 1/2 should be realized. Second, the pseudospins need to interact with each other through the ligands shared by neighboring octahedra which forms edgesharing network. Over the past decade, enormous efforts have been devoted to exploring the candidate materials for the Kitaev QSL that meet these conditions [28–31]. Fortunately, dominant Kitaev-type interactions were indeed discovered in several materials, such as A2 IrO3 with A = Li and Na [32–38] and α-RuCl3 [39–46]. Not only the 5d and 4d candidates, 3d transition metal compounds like Co oxides have also been investigated [47, 48]. In addition to these examples with ferromagnetic (FM) Kitaev interactions, the antiferromagnetic (AFM) Kitaev QSL candidates were also predicted by ab initio calculations, e.g., f -electron based magnets [49, 50] and polar spinorbit coupled Mott insulators α-RuH3/2 X3/2 with X = Cl and Br in the Janus structure [51]. Although a plethora of Kitaev QSL candidates have been investigated, those realizing the Kitaev QSL in the ground state are still missing, since a long-range magnetic order due to parasitic interactions such as the Heisenberg interaction hinders the Kitaev QSL. Considerable efforts have been dedicated to suppressing the parasitic interactions and/or enhancing the Kitaev-type interaction. One way is to utilize heterostructures that incorporate the Kitaev candidates. For example, the Kitaev interaction is promoted more than 50% for the heterostructure composed of 2D monolayers of α-RuCl3 and graphene compared to the pristine α-RuCl3 , predicted by ab initio calculations [52]. The heterostructures between a 2D αRuCl3 and three-dimensional (3D) topological insulator BiSbTe1.5 Se1.5 evidenced the charge transfer phenomena, albeit the magnetic properties were not reported [53]. Within the realm of Kitaev heterostructures, remarkably few studies have been designed for the composite 3D/3D superlattices due to the fabrication challenge [54]. To date, few attempts [55] have been made to investigate the development of the electronic band structure and the magnetic properties, particularly whether the Kitaev interaction is still dominant when constructing the 3D/3D heterostructures using Kitaev QSL candidates. In this paper, we theoretically study the electronic and magnetic properties in bilayer heterostructures as an interface of 3D/3D superlattices using a recentlysynthesized iridium ilmenite MgIrO3 [56] and other ilmenite magnets ATiO3 with A = Mn, Fe, Co, and Ni as the substrate. This material choice is motivated by two key considerations: (i) All of these materials have been successfully synthesized in experiments, which is helpful for the epitaxial growth of multilayer heterostructures, and (ii) the ilmenite MgIrO3 is identified as a good candidate for Kitaev magnets [57, 58]. We consider three configurations of heterostructures, classified by type-I, II, and III, which are all chemically allowed due to the characteristics of the alternative layer stacking in ilmenites, as shown in Fig. 1. The electronic band structures, magnetic ground states, and the effective exchange interactions are systematically investigated by employing the combinatorial of ab initio calculations, construction of the effective tight-binding model, and perturbation expansions. We find that (i) the spin-orbit coupled bands characterized by the effective pseudospin jeff = 1/2, a key demand for Kitaev-type interactions, are still preserved in the MgIrO3 layer for all types of the heterostructures, (ii) type-I and III heterostructures realize spin-orbit coupled Mott insulators excluding Mn typeIII, whereas type-II ones are spin-orbit coupled metals with doped jeff = 1/2 bands with various carrier concentrations, and (iii) in almost all of the insulating cases, the Kitaev-type interactions are predominant, whereas the forms and magnitudes of the other parasitic interactions depend on the specific types of the heterostructures and the A atoms. The structure of the remaining article is as follows. In Sec. II, we provide a detailed description of the optimized lattice structures of MgIrO3 /ATiO3 heterostructures with A = Mn, Fe, Co, and Ni. In Sec. III, we 3 introduce the methods employed in this work, including the means for structural optimization and the ab initio calculations with LDA+SOC+U scheme (Sec. III A), the estimates of the effective transfer integral and construction of the multiorbital Hubbard model (Sec. III B), and the second-order perturbation that is used in the estimation of exchange interactions (Sec. III C). In Sec. IV, we systematically display the results of the electronic band structures for three types of heterostructure MgIrO3 /ATiO3 . In Sec. IV A, we present the electronic band structures for the paramagnetic state obtained by LDA+SOC, together with the projected density of states (PDOS) derived by the maximally-localized Wannier function (MLWF). In Sec. IV B, we discuss the stable magnetic states within LDA+SOC+U and show their band structures and PDOS. In Sec. V, we derive the effective exchange coupling constants for the heterostructures for which the LDA+SOC+U calculations suggest spin-orbit coupled Mott insulating nature, and show their location in the phase diagram for the K-J-Γ model. In Sec. VI, we discuss the possibility of the realization of Majorana Fermi surfaces (Sec. VI A) and exotic superconducting phases (Sec. VI B) in the heterostructures, and the feasibility of these heterostructures in experiments (Sec. VI C). Section VII is devoted to the summary and prospects. In Appendix A, we present the details of the energy difference of the magnetic orders and the effective exchange couplings between the A ions. We present additional information on orbital projected band structures for different specific types of heterostructures of A atoms in Appendix B and the band structures of monolayer MgIrO3 in Appendix C. II. HETEROSTRUCTURES MgIrO3 and ATiO3 both belong to ilmenite oxides ABO3 with R3̄ space group. The lattice structure consists of alternative stacking of honeycomb layers with edge-sharing AO6 octahedra and those with BO6 octahedra. The common stacking layer structures reduce the lattice mismatch to form the heterostructures and also make them feasible to fabricate in experiments. In this study, we consider heterostructures composed of monolayer of MgIrO3 and ATiO3 with the balance chemical formula, to clarify the interface effect on the electronic properties of 3D/3D superlattices. Specifically, we construct three types of heterostructures, distinguished by the intersurface atoms and pertinent octahedra in the middle layer, labeled as type-I, II, and III and shown in Fig. 1. For type-I, the top and bottom layers are made of honeycomb networks of IrO6 and TiO6 octahedra, respectively, whereas the sandwiching honeycomb layer is formed of alternating MgO6 and AO6 octahedra. In the type-II, the bottom layer is replaced by AO6 honeycomb layer, resulting in a mixture of MgO6 and TiO6 in the middle layer. The type-III has a similar constitution of top and bottom layers to type-I, while the middle (a) type-I Ir Ir Mg A Mg/A Ti c b (b) type-II a Ti c b a Ir Mg/Ti A (c) type-III Ir A Ti FIG. 1. Schematic pictures of crystal structures for three types of the heterostructures MgIrO3 /ATiO3 : (a) type-I, (b) type-II, and (c) type-III with A = Mn, Fe, Co, or Ni. The left and right panels show the side views and the bird’seye views, respectively. The type-I is composed of the top honeycomb layer with edge-sharing IrO6 octahedra and the bottom honeycomb layer of TiO6 , sandwiching a honeycomb layer of alternating MgO6 and AO6 octahedra. In the typeII, the bottom is replaced by the AO6 honeycomb layer, leaving a mixture of MgO6 and TiO6 in the middle, and in the type-III, the middle is replaced by the AO6 honeycomb layer. In type-I and II, the chemical formula is commonly given by Mg2 Ir2 O6 /A2 Ti2 O6 , but that for type-III is MgAIr2 O6 /A2 Ti2 O6 . The crystal structures are embodied by VESTA [59]. layer is fully composed of AO6 octahedra. We intentionally design these structures to balance their chemical valences and prevent the presence of redundant charges. This can be derived from the chemical formula for each type of the heterostructure, such as Mg2 Ir2 O6 /A2 Ti2 O6 for type-I and II, and MgAIr2 O6 /A2 Ti2 O6 for type-III, respectively. We optimize the lattice structures of the heterostructures by the optimization scheme in Sec. III A. The information of the stable lattice structures, including the in-plane lattice constant, the bond distance between adjacent Ir atoms and O atoms, and the angle between the neighboring Ir, O, and Ir atoms, are listed in Table I for three types of the heterostructures with different A atoms. For comparison, the experimental structures of the bulk MgIrO3 are also listed. We find all the in-plane constants are close to the bulk value of 5.158 Å, in which 4 the maximum and minimum lattice mismatch is 1.2% and 0.1% respectively of type-II for the Fe atom and type-III for the Mn atom. See also the discussion in Sec. VI C. Meanwhile, not only the in-plane constants but also the bond distances of Ir atoms are both enlarged as the increase of ionic radii of A atoms. The heterostructural type can significantly influence the bond distance and angle as well. For example, the angle between neighboring Ir atoms and the intermediate O atom θIr−O−Ir = 96.69◦ of type-II for the Ni case largely increases from that of 94.03◦ of the bulk case. In terms of the Ir-Ir bond length (dIr−Ir ), the length of 2.986 Å for the bulk [56] is substantially decreased to 2.930 Å of type-II for the Fe case. Meanwhile, the Ir-O bond length dIr−O of all cases are enlarged compared with that of the bulk system of 1.942 Å, in which type-III with Co atoms is maximally influenced. TABLE I. Structural information of optimized heterostructures for MgIrO3 /ATiO3 (A = Mn, Fe, Co, and Ni): a denotes the in-plane lattice constant, and d and θ represent the bond distance and the angle between neighboring ions, respectively. The experimental information on the bulk MgIrO3 is also shown for comparison. A Mn Fe Co Ni type I II III I II III I II III I II III bulk [56] a(Å) 5.167 5.127 5.152 5.104 5.068 5.083 5.116 5.113 5.115 5.148 5.181 5.173 5.158 III. A. dIr−Ir (Å) 2.986 2.962 2.977 2.951 2.930 2.940 2.961 2.955 2.960 2.977 2.994 2.994 2.986 dIr−O (Å) 1.997 2.009 1.985 2.012 2.004 2.008 1.999 1.990 2.019 2.006 1.997 1.995 1.942 θIr−O−Ir (◦ ) 95.92 95.57 95.96 94.32 94.00 93.98 94.76 95.26 94.31 95.03 96.69 94.18 94.03 METHODS Ab initio calculations In the ab initio calculations, we use the QUANTUM ESPRESSO [60] based on the density functional theory [61]. The exchange-correlation potential is treated as Perdew-Zunger functional by using the projectoraugmented-wave method [62, 63]. Under the consideration of the SOC effect, the fully relativistic functional is utilized for all the atoms except oxygens [64]. To obtain stable structures for heterostructures, we initially construct a bilayer MgIrO3 structure using the experimental structure for the bulk material. Subsequently, we replace the lower half with ATiO3 layer to create three different types of heterostructures in Fig. 1. Then, we perform full optimization for both lattice parameters and the position of each ion until the residual force becomes less than 0.0001 Ry/Bohr. During the optimization procedure, the structural symmetry is retained as R3̄ space group. The 20 Å thick vacuum is adopted to eliminate the interaction between adjacent layers. The 6×6×1 and 12×12×1 Monkhorst-Pack k-points meshes are utilized for the structural optimization and self-consistent calculations, respectively [65]. The self-consistent convergence is set to 10−8 Ry and the kinetic energy is chosen to 80 Ry for all the structural configurations, which are respectively small and large enough to guarantee accurate results. To simulate the electron correlation effects for 3d electrons of A atoms and 5d electrons of Ir atom, we adopt the LDA+SOC+U calculations [66] with the Coulomb repulsions UA = 5.0 eV, 5.3 eV, 4.5 eV, and 6.45 eV with A = Mn, Fe, Co, and Ni atoms, respectively, and UIr = 3.0 eV, accompanying with the Hund’srule coupling with JH /U = 0.1 according to previous works [67, 68]. Based on the ab initio results, we also obtain the MLWFs by using the k points increased to 18×18×1 within the Momkhorst-Pack scheme [65]. We select the t2g , 2p, and 3d orbitals respectively of Ir, O, and A atoms to construct the MLWFs by employing the Wannier90 [69]. Herein, we include O 2p and A 3d orbitals due to their significant contribution near the Fermi level, as detailed in Figs. 2-5. By utilizing the MLWFs, we construct the tight-binding models and calculate their band structures for comparison. We also calculate the PDOS of each atomic orbital, including the effective angular momentum of Ir atoms jeff , from the tight-binding models. We consider the non-relativistic ab initio calculations and relative MLWFs for the estimation of transfer integrals (see Sec. III B). B. Multiorbital Hubbard model To estimate the effective exchange interactions between the magnetic Ir ions, we need the effective transfer integrals between neighboring Ir t2g orbitals with the association of O 2p orbitals by constructing MLWFs with LDA calculation in the paramagnetic state. It is noticeable that the effects of relativistic SOC and electron correlation are not taken into account in this calculation to circumvent the doublecounting in constructing the effective spin models. Specifically, the effective transfer integral t is estimated as [57] tiu,jv + X tiu,p t∗jv,p p ∆p−uv . (2) The first term denotes the direct hopping between two adjacent Ir atoms, where tiu,jv represents the transfer integral between orbital u at site i and orbital v at site j. The second term denotes the indirect hopping between the two Ir atoms via the shared O 2p orbitals, where tiu,p represents the transfer integral between Ir atom u orbital at site i and ligand atom p orbital, and ∆p−uv 5 is the harmonic mean of the energy of u and v orbitals measured from that of p orbitals. Herein, we consider only hopping processes between the nearest-neighbor Ir atoms. Using the effective transfer integrals, we construct a multiorbital Hubbard model with one hole occupying the t2g orbitals, whose Hamiltonian is given by C. Second-order perturbation (3) For Ir5+ ions, the t2g manifold splits into a doublet and a quartet under the SOC, which are respectively characterized by the pseudospin jeff = 1/2 and 3/2. In the ground state, the latter is fully occupied and the former is half filled, which is described by the Kramers doublet |jeff = 1/2, +⟩ and |jeff = 1/2, −⟩ [27, 74]: The first term denotes the kinetic energy of the t2g electrons as X † γ Hhop = − ci (T̂ij ⊗ σ0 )cj , (4) 1 |jeff = 1/2, +⟩ = √ (|dyz ↓⟩ + i|dzx ↓⟩ + |dxy ↑⟩) , (9) 3 1 |jeff = 1/2, −⟩ = √ (|dyz ↑⟩ − i|dzx ↑⟩ − |dxy ↓⟩) . (10) 3 H = Hhop + Htri + Hsoc + HU . i,j where the matrix T̂ijγ includes the effective transfer integrals estimated by Eq. (2), γ is the x, y, and z bond connected by neighboring sites i and j which belong to different honeycomb sublattices, and σ0 denotes the 2×2 identity matrix; c†i = (c†i,yz,↑ , c†i,yz,↓ , c†i,zx,↑ , c†i,zx,↓ , c†i,xy,↑ , c†i,xy,↓ ) denote the creation of one hole in the t2g orbitals (yz, zx, and xy) carrying spin up (↑) or down (↓) at site i. The second term in Eq. (3) denotes the trigonal crystal splitting as X † ci (T̂tri ⊗ σ0 )ci , (5) Htri = − i  0 ∆tri ∆tri T̂tri = ∆tri 0 ∆tri  . ∆tri ∆tri 0 The third term denotes the SOC as  iσz λ X † 0 0 ci −iσz Hsoc = − 2 i iσ −iσ y x  −iσy iσx  ci , 0 (6) (7) where σ {x,y,z} are Pauli matrices, and λ is the SOC coefficient; for instance, λ of Ir atom is estimated at about 0.4 eV [70, 71]. The last term denotes the onsite Coulomb interactions as [72, 73] X U niu↑ niu↓ HU = i X i,u<v,σ + X (2) Eσ′ ,σ′ ;σi ,σj = i with T̂tri in the form of  + When the system is in the spin-orbit coupled Mott insulating state with the low-spin d5 configuration, the low-energy physics can be described by the pseudospin jeff = 1/2 degree of freedom. In this case, the effective exchange interactions between the pseudospins can be estimated by using the second-order perturbation theory in the atomic limit, where the three terms in Eq. (3), Htri + Hsoc + HU , are regarded as unperturbed Hamiltonian, and Hhop is treated as perturbation. The energy correction for a neighboring pseudospin pair in the second-order perturbation is given by [U ′ niuσ nivσ̄ + (U ′ − JH )niuσ nivσ ] JH (c†iu↑ c†iv↓ ciu↓ civ↑ + c†iu↑ c†iu↓ civ↓ civ↑ ), (8) i,u̸=v with niuσ = c†iuσ ciuσ ; σ̄ =↓ (↑) for σ =↑ (↓). In Eq. (8), the first, second, and third summations represent the intraorbital Coulomb interaction in the same orbital with opposite spins, the interorbital Coulomb interactions between orbital u and orbital v, and the spin-flip and pairhopping processes, respectively. j X ⟨σi′ σj′ |Hhop |n⟩⟨n|Hhop |σi σj ⟩ E 0 − En n , (11) where σi and σi′ denote the pseudospin + or − at site i, |σi σj ⟩ and ⟨σi′ σj′ | is the initial and final states during the perturbation process, respectively, and |n⟩ is the intermediate state with 5d4 -5d6 or 5d6 -5d4 electron configuration; E0 is the ground state energy for the 5d5 5d5 electron configuration, and En is the energy eigenvalue for the intermediate state |n⟩. Here, |n⟩ and En are obtained by diagonalizing the unperturbed Hamiltonian Htri + Hsoc + HU . The effective pseudospin Hamiltonian is written in the form of X X γ H= SiT Jij Sj , (12) γ=x,y,z ⟨i,j⟩ where i, j denote the neighboring sites, and γ denotes the three types of Ir-Ir bonds on the MgIrO6 honeycomb layer that are related by C3 rotation. The coupling constant γ Jij is explicitly given, e.g., for the z bond as   J Γ Γ′ z Jij =  Γ J Γ′  , Γ′ Γ′ K (13) where J, K, Γ, and Γ′ represent the coupling constants for the isotropic Heisenberg interaction, the bonddependent Ising-like Kitaev interaction, and two types of the symmetric off-diagonal interactions. Using the per(2) turbation energy Eσ′ ,σ′ ,σi ,σj obtained by Eq. (11), the i j 6 coupling constants are calculated as (2) J = 2E+,−;−,+ ,   (2) (2) K = 2 E+,+;+,+ − E+,−;+,− , o n (2) Γ = 2Im E−,−;+,+ , o n (2) Γ′ = 4Re E+,+;+,− . IV. A. (14) (15) (16) (17) ELECTRONIC BAND STRUCTURE LDA+SOC results for paramagnetic states Let us begin with the electronic band structures from LDA+SOC calculations. The results for MgIrO3 /ATiO3 with A = Mn, Fe, Co, and Ni are shown in Figs. 2(a), 3(a), 4(a), and 5(a), respectively. Here we display the band structures in the paramagnetic state obtained by the ab initio calculations (black solid lines) and the MLWF analysis (blue dashed lines), together with the atomic orbitals PDOS including 5d of Ir atoms, 3d of A atoms, and 2p of O atoms related to IrO6 and AO6 octahedra. It is obvious that the systems are metallic for all the types of heterostructures, regardless of the choice of A atoms. The strong SOC splits the t2g bands of Ir atoms into jeff = 1/2 and jeff = 3/2 bands, as depicted in the PDOS. Specifically, the jeff = 1/2 bands are predominated to form the metallic bands in the proximity of the Fermi level, covering the energy region almost from −1.0 eV to 0.2 (0.4) eV for A = Mn with type-II (type-I and III), from −0.8 eV to 0.2 eV for A = Fe with all types, from −0.6 (−1.0) eV to 0.2 eV for A = Co with type-II (type-I and III), and from −0.4 eV to 0.4 eV for A = Ni with type-I and II, and −0.8 eV to 0.2 eV with type-III. On the other hand, the jeff = 3/2 bands primarily occupy the energy region below the jeff = 1/2 bands. From the PDOS in right panels of Figs. 2(a), 3(a), 4(a), and 5(a), the 3d bands of A atoms and the O 2p bands of AO6 octahedra simultaneously across the Fermi level, with hybridization with the Ir 5d bands. Notably, the energy range of the A 3d bands closely overlaps with that of the Ir 5d jeff = 1/2 bands for A = Mn and Fe, but it overlaps with both jeff = 1/2 and jeff = 3/2 manifold for A = Co and Ni. As to the O 2p bands, the energy range of the PDOS overlaps with that for corresponding Ir 5d or A 3d encapsulated in the octahedra, suggesting Ir-O and A-O hybridization. B. LDA+SOC+U results for magnetic states 1. Magnetic ground states The bulk counterpart of each constituent of the heterostructures exhibits some magnetic long-range orders in the ground state. In the bulk MgIrO3 , Ir ions show a zigzag-type AFM order with the magnetic moments lying almost within the honeycomb plane [58]. In the bulk ATiO3 , the A = Mn and Fe ions show Néel-type AFM orders with out-of-plane magnetic moments [75], while the A = Co and Ni ions support Néel orders with inplane magnetic moments [75, 76]. It is intriguing to examine how these magnetic orders in the bulk are affected by making heterostructures. We determine the stable magnetic ground states for each heterostructure through ab initio calculations by including the effect of electron correlations based on the LDA+SOC+U method. To determine the potential magnetic ground state for each heterostructure, we compare the energy across a total of 16 magnetic configurations among all combinations of following types of the magnetic orders: FM and Néel orders with in-plane and out-of-plane magnetic moments for A layer, and Néel and zigzag orders with in-plane magnetic moments, as well as FM with in-plane and out-of-plane magnetic moments for Ir layer, within a 2×2×1 supercell setup. TABLE II. Stable magnetic orders obtained by the LDA+SOC+U calculations: FM, Néel, and zigzag denotes the ferromagnetic, Néel-type antiferromagnetic, and zigzagtype antiferromagnetic orders, respectively. While the directions of the magnetic moments are all in-plane for the Ir layers, those for A can be in-plane (“in”) or out-of-plane (“out”) depending on A and type of the heterostructure. A Mn Fe Co Ni layer Ir Mn Ir Fe Ir Co Ir Ni type-I in-zigzag in-Néel in-Néel out-Néel out-FM in-Néel in-Néel out-Néel II in-zigzag out-FM in-Néel in-FM in-FM in-Néel in-zigzag in-FM III in-zigzag in-Néel in-FM in-FM in-Néel in-Néel in-zigzag in-Néel The results of the most stable magnetic state are listed in Table II. The details of the energy comparison are shown in Appendix A. In most cases, the A ions show Néel orders as in the bulk cases, but the direction of magnetic moments is changed from the bulk in some cases. For instance, type-I and III with A = Mn and type-III with A = Fe switch the moment direction from out-ofplane to in-plane. While all the types with A = Co retain the in-plane Néel states, type-I with A = Ni is changed into the out-of-plane Néel state. The results indicate that the direction of magnetic moments are sensitively altered by making the heterostructures with MgIrO3 . Meanwhile, the other cases, type-II with A = Mn, Fe, and Ni as well as type-III with A = Fe, are stabilized in the FM state. These results are in good accordance with the effective magnetic couplings between the A ions estimated by a similar perturbation theory in Sec. III C [77, 78], attesting to the reliability of magnetic properties in heterostructures (see Appendix A). 7 (a) type-I 1 type-II Ir OIr Mn OMn jeff =1/2 jeff =3/2 1 Ir 3d type-III OIr Mn OMn jeff =1/2 jeff =3/2 1 Ir OIr Mn OMn jeff =1/2 jeff =3/2 3d 2p -2 M K -1 -2 ! M K ! Energy (eV) -1 -2 1! ! 0 0 M K 0 -1 -2 ! M K ! 2p 2p 0 -1 -2 1! ! Energy (eV) -1 (b) 1! Energy (eV) Energy (eV) 0 2p 3d 2p Energy (eV) Energy (eV) 2p M K ! M K ! 0 -1 -2 ! FIG. 2. The band structures of MgIrO3 /MnTiO3 for type-I, II, and III with (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+U calculations for the stable magnetic orders (see Sec. IV B 2). The black lines represent the electronic structure obtained by the ab initio calculations, and the light-blue dashed curves represent the electronic dispersions obtained by tight-binding parameters using the MLWFs. The right panels in each figure denote the PDOS for different orbitals on specific atoms: The red and blue lines represent the jeff manifolds of Ir atoms, the cyan and orange lines represent the 2p orbitals of O atoms in IrO6 octahedra (OIr ) and MnO6 octahedra (OMn ), respectively, and the green line represents the 3d orbitals of Mn atoms. The Fermi energy is set to zero. The magnetic states in the Ir layer are more complex due to the possibility of the zigzag state. For A = Mn, the magnetic ground states of the Ir layer in all three types prefer the in-plane zigzag state as in the bulk of MgIrO3 . In contrast, for A = Fe, type-I and II are stable in the in-plane Néel state, but type-III prefers the in-plane FM state. For A = Co, only type-III stabilizes the in-plane Néel state, while others exhibit the out-of-plane FM state for type-I and the in-plane FM state for type-II. Lastly, for A = Ni, both type-II and type-III prefer the in-plane zigzag state, while it changes into the in-plane Néel state in type-I. These results indicate that the magnetic state in the Ir honeycomb layer is susceptible to both A ions and the heterostructure type. We will discuss this point from the viewpoint of the effective magnetic couplings in Sec. V. 2. Band structures We present the band structures obtained by the LDA+SOC+U calculations in Figs. 2(b), 3(b), 4(b), and 5(b) for A = Mn, Fe, Co, and Ni, respectively. In these calculations, we adopt the stable magnetic states in Table II, except for the cases with in-plane zigzag order in the Ir layer. For the zigzag cases, for simplicity, we replace them by the in-plane Néel solutions, keeping the A layer the same as the stable one. This reduces significantly the computational cost of the MLWF analysis for the zigzag state with a larger supercell. We confirm that the band structures for the Néel state are similar to those for the zigzag state, and the energy differences between the two states are not large as shown in Appendix A. When we turn on Coulomb repulsions for both Ir and A atoms, most of the type-I and III heterostructures become insulating, except for Mn type-III. The band gaps, obtained by Eg = Ec − Ev , are shown in Fig. 6, where Ec denotes the energy of conduction band minimum and Ev is that of valence band maximum. In all cases, except for type-I with A = Mn and Ni and type-III with A = Co, the gap is defined by the jeff = 1/2 bands of Ir ions, that is, both conduction and valence bands are jeff = 1/2, and the jeff = 1/2 bands is half filled. It is worth highlighting that there are four jeff = 1/2 bands, which originate from different sites of Ir atoms with opposite magnetic moments; in the bulk and monolayer cases they are degenerate in pair [57], but the degeneracy is lifted in the heterostructures and two out of four are occupied in the half-filled insulating state. Meanwhile, in the cases of type-I with A = Mn and Ni and type-III with A = Co, 8 (a) type-I 1 Ir type-II OIr Fe OFe jeff =1/2 jeff =3/2 1 Ir type-III OIr Fe OFe jeff =1/2 jeff =3/2 1 Ir OIr Fe OFe jeff =1/2 jeff =3/2 2p 2p 3d 2p 2p 3d 2p -2 M K -1 M K M K -1 M K 0 -1 -2 1! ! 0 -2 ! ! Energy (eV) -1 -2 1! ! 0 -2 ! 0 Energy (eV) -1 (b) 1! Energy (eV) Energy (eV) 0 Energy (eV) Energy (eV) 2p 3d K ! M K ! 0 -1 -2 ! ! M FIG. 3. The band structures of MgIrO3 /FeTiO3 for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+U calculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2. (a) type-I 1 Ir type-II OIr Co OCo jeff =1/2 jeff =3/2 1 Ir type-III OIr Co OCo 1 Ir jeff =3/2 M K 0 -1 -2 ! M K ! Energy (eV) -1 -2 1! ! 2p 3d 2p 0 M K 0 -1 -2 ! M K ! 0 -1 -2 1! ! Energy (eV) -2 (b) 1! Energy (eV) Energy (eV) -1 Energy (eV) Energy (eV) 2p 3d 2p 0 OIr Co OCo jeff =1/2 jeff =1/2 jeff =3/2 2p 3d 2p M K ! M K ! 0 -1 -2 ! FIG. 4. The band structures of MgIrO3 /CoTiO3 for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+U calculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2. 9 (a) type-I 1 Ir type-II OIr Ni ONi 1 Ir jeff =1/2 jeff =3/2 2p 3d 2p type-III OIr Ni ONi 1 jeff =1/2 jeff =3/2 Ir jeff =3/2 3d 2p 2p 3d 2p M K -1 -2 ! M K ! Energy (eV) -1 -2 1! ! 0 0 M K 0 -1 -2 ! M K ! 0 -1 -2 1! ! Energy (eV) -2 (b) 1! Energy (eV) Energy (eV) -1 Energy (eV) Energy (eV) 2p 0 OIr Ni ONi jeff =1/2 M K ! M K ! 0 -1 -2 ! FIG. 5. The band structures of MgIrO3 /NiTiO3 for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+U calculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2. the 3d bands of A ions hybridized with O 2p orbitals intervene near the Fermi level, and the gap is defined between the jeff = 1/2 and 3d bands. In these cases, however, a larger gap is well preserved in the jeff = 1/2 bands, as shown in Figs. 2(b), 4(b), and 5(b). We note that the Co type-III is a further exception since the gap opens between the highest-energy jeff = 1/2 band and the Co 3d band; the jeff = 1/2 bands are not half filled but 3/4 filled (see Appendix B). We plot the band gap defined by the jeff = 1/2 bands by red asterisks in Fig. 6, including the 3/4-filled case for the Co type-III. These results clearly indicate that the inclusion of both SOC and U effects results in the opening of a band gap in the jeff = 1/2 bands at half filling in type-I and III heterostructures excluding Mn type-III and Co type-III. This suggests the formation of spin-orbit coupled Mott insulators in the Ir honeycomb layers, which are cornerstone of the Kitaev candidate materials [27], motivating us to further investigate the effective exchange interaction in Sec. V. The Co type-III is in an interesting state with 3/4 filling of jeff = 1/2 bands, but we exclude it from the following analysis of the effective exchange interactions in Sec. V. Distinct from the emergence of spin-orbit coupled Mott insulator, the LDA+SOC+U band structures show metallic states for type-II heterostructures. The jeff = 1/2 bands do not show a clear gap and cross the Fermi level, resulting in the spin-orbit coupled metals. Notably, in all cases, the upper jeff = 1/2 bands are partially TABLE III. Electronic states of each heterostructure obtained by the LDA+SOC+U calculations. SOCI and SOCM denote spin-orbit coupled insulator and metal, respectively. e and h in the parentheses represent the carriers in the SOCM doped to the mother SOCI. The asterisk for the Co type-III indicates that the system is in the 3/4-filled insulating state of the jeff = 1/2 bands. A Mn Fe Co Ni type-I SOCI SOCI SOCI SOCI II SOCM(e) SOCM(e) SOCM(e) SOCM(e) III SOCM(h) SOCI SOCI* SOCI doped, realizing electron-doped Mott insulators. The doping rate varies with A atoms. We note that the typeIII heterostructure of Mn also exhibits a metallic state, but in this case, holes are doped to the lower jeff = 1/2 band. See Appendix B for the orbital projected band structures. We summarize the electronic states in Table III. The type-I and III heterosuructures are all spin-orbit coupled Mott insulators (SOCI) except for the type-III Mn case, while the type-II are all spin-orbit coupled metals (SOCM). For the SOCM, we also indicate the nature of carriers, electrons or holes; the type-II heterostructures are all electron doped, while the type-III Mn is hole doped. 10 0.7 type-I type-III type-I: jeff=1/2 Band gap (eV) 0.6 type-III: jeff=1/2 0.5 0.4 0.3 0.2 0.1 0 Mn Fe Atoms Co Ni FIG. 6. The band gap in the insulating states for type-I and III obtained by the LDA+SOC+U calculations. The open circles represent the gaps opening in the jeff = 1/2 bands for Mn and Ni of type-I and Co of type-III. Note that the Co type-III is exceptional since the jeff = 1/2 bands are at 3/4 filling, rather than half filling in the other cases; see the text for details. V. EXCHANGE INTERACTIONS The electronic band structure analysis reveals the emergence of the spin-orbit coupled Mott insulating state in the MgIrO6 layer of type-I and III heterostructures, except for the Mn and Co type-III. In these cases, the low-energy physics is expected to be described by effective pseudospin models with dominant Kitaev-type interactions [27]. The effective exchange interactions can be derived by means of the second-order perturbation for the multiorbital Hubbard model (Secs. III B and III C). We set UIr = 3.0 eV, JH /UIr = 0.1, and λ = 0.4 eV in the perturbation calculations. The results are summarized in Fig. 7. For comparison, we also plot the estimates for monolayer and bulk MgIrO3 . For the bulk case, its potential for hosting Kitaev spin liquids was demonstrated in the previous study [57]. Regarding the monolayer, we obtain the results from the band structures shown in Appendix C, which illustrate the preservation of the jeff = 1/2 manifold and spin-orbit coupled insulating nature. In type-I heterostructures, the dominant interaction is the FM Kitaev interaction K < 0 for almost all A atoms, except for Mn. Particularly for A = Ni, the absolute value of K is significantly larger than the others, even considerable when compared with the monolayer and bulk MgIrO3 . The subdominant interaction is the off-diagonal symmetric interaction Γ > 0. The other offdiagonal symmetric interaction Γ′ as well as the Heisenberg interaction J is weaker than them. In the Mn case, all the interactions are exceptionally weak, presumably because of the intervening Mn 3d band and its hybridiza- tion with the Ir jeff = 1/2 bands. Meanwhile, for the type-III heterostructures, since the Mn and Co cases exhibit a metallic state and 3/4 occupation of jeff = 1/2 bands, respectively, we only calculate the effective magnetic constants for Fe and Ni. In these cases also, the dominant interaction is the FM K, accompanied by the subdominant Γ interaction, as shown in Fig. 7(a). Thus, in all cases except the Mn type-I heterostructure, the dominant magnetic interaction in the spin-orbit coupled Mott insulating state in the Ir honeycomb layer is effectively described by the FM Kitaev interaction. Since Γ′ is smaller than the other exchange constants, the lowenergy magnetic properties can be well described by the generic K-J-Γ model [79, 80], which has been widely and successfully applied to study the Kitaev QSLs. We summarize the obtained effective exchange interactions of K, J, and Γ by using the parametrization (K, J, Γ) = N (sinθsinφ, sinθcosφ, cosθ), (18) where N = (K 2 + J 2 + Γ2 )−1/2 is the normalization factor. Figure 7(b) presents the results except for Mn type-I. Our heterostructures distribute in the region near the FM K only case (θ = π/2 and φ = 3π/2). We find a general trend that larger A atoms make the systems closer to the FM K only case; the best is found for Ni type-I and III. In the previous studies for the K-J-Γ model [79, 80], a keen competition between different magnetic phases was found in this region, which does not allow one to conclude the stable ground state in the thermodynamic limit. Given that this region appears to be connected to the solvable point for the FM Kitaev QSL, our heterostructures provide a promising platform for investigating the Kitaev QSL physics and related phase competition by finely tuning the magnetic interactions via the proximity effect in the heterostructures. VI. DISCUSSION Our systematic study of ilmenite heterostructures MgIrO3 /ATiO3 with A = Mn, Fe, Co, and Ni reveals their fascinating electronic and magnetic properties. The heterostructures in the paramagnetic state are metallic in terms of band structures obtained by LDA+SOC, regardless of types and A atoms. When incorporating the effect of electron correlation by the LDA+SOC+U calculations, type-II heterostructures remain metallic across entire A atoms, whereas type-I and III heterostrcutures turn into insulating states, except for Mn type-III. As a consequence, the electronic states of heterostructures are classified into the spin-orbit coupled insulators and metals, each holding unique properties. The insulating cases possess the jeff = 1/2 pseudospin degree of freedom, and furnish a fertile playground to investigate the Kitaev QSL. In these cases, however, due to the magnetic proximity effects from the A layer, we may expect interesting modification of the QSL state, as discussed in Sec. VI A below. Meanwhile, the metallic cases open avenues for 11 Coupling const. (meV) (a) 50 Γ J 0 Γ′ -50 K -100 -150 Mn Fe Co Ni monolayer bulk Atom (b) AFM K only 𝜙 = 𝜋/2 𝜃 = = 2 𝜋/ 𝜃 = /8 3𝜋 𝜃 = 4 𝜋/ 𝜃 𝜙=𝜋 8 𝜋/ FM J only Γ only 𝜙=0 AFM J only Co-I ★ ◆ monolayer ♡ Fe-III ◆ Fe-I Ni-III ▲♡ ▲ Ni-I bulk 𝜙 = 3𝜋/2 FM K only FIG. 7. The effective magnetic constants of heterostructures for different A atoms (A = Mn, Fe, Co, and Ni) of (a) type-I (solid line with pentagram) and type-III (dashed line with hollow pentagram). For comparison, we also show the results for monolayer and bulk. In (b), we summarize the results of K, J, and Γ in (a) except for Mn type-I by using the parametrization in Eq. (18). The parameters of the intraorbital Coulomb interaction, Hund’s coupling, and spin-orbit coupling are set to UIr = 3.0 eV, JH /UIr = 0.1, and λ = 0.4 eV, respectively, in the perturbation calculations. exploring spin-orbit coupled metals, relatively scarce in strongly correlated systems [81–85]. In Sec. VI B, we discuss the possibility of exotic superconductivity in our selfdoped heterostructures. In addition, we discuss the feasibility of fabrication of these heterostructures and identification of the Kitaev QSL nature in experiments in Sec. VI C. A. Majorana Fermi surface by magnetic proximity effect In the pure Kitaev model, the spins are fractionalized into itinerant Majorana fermions and localized Z2 gauge fluxes [25]. The former has gapless excitations at the nodal points of the Dirac-like dispersions at the K and K’ points on the Brillouin zone edges, while the latter is gapped with no dispersion. When an external magnetic field is applied, the Dirac-like nodes of Majorana fermions are gapped out, resulting in the emergence of quasiparticles obeying non-Abelian statistics [25]. Beyond the uniform magnetic field, the Majorana dispersions are further modulated by an electric field and a staggered magnetic field [86, 87]. For instance, with the existence of the staggered magnetic field, the Dirac-like nodes at the K and K’ points are shifted in the opposite directions in energy to each other, leading to the formation of the Majorana Fermi surfaces. Moreover, the introduction of both uniform and staggered magnetic fields can lead to further distinct modulations of the Majorana Fermi surfaces around the K and K’ points, which are manifested by nonreciprocal thermal transport carried by the Majorana fermions [87]. In our heterostructures of type-I and III, the A layer supports a Néel order in most cases (Table II). It can generate an internal staggered magnetic field applied to the Ir layer through the magnetic proximity effect. This mimics the situations discussed above, and hence, it may result in the Majorana Fermi surfaces in the possible Kitaev QSL in the Ir layer. The combination of the uniform and staggered magnetic fields could also be realized by applying an external magnetic field to these heterostructures. Thus, the ilmenite heterostructures in proximity to the Kitaev QSL in the Ir layer hold promise for the formation of Majorana Fermi surfaces and resultant exotic thermal transport phenomena, providing a unique platform for identifying the fractional excitations in the Kitaev QSL. B. Exotic superconductivity by carrier doping QSLs have long been discussed as mother states of exotic superconductivity [19, 88, 89]. There, the introduction of mobile carriers to insulating QSLs possibly induces superconductivity in which the Cooper pairs are mediated by strong spin entanglement in the QSLs. A representative example discussed for a long time is highTc cuprates; here, the d-wave superconductivity is induced by carrier doping to the undoped antiferromagnetic state that is close to a QSL of so-called resonating valence bond (RVB) type [89–91]. A similar exotic superconducting state was also discussed for an iridium oxide Sr2 IrO4 with spin-orbital entangled jeff = 1/2 bands [92, 93]. Carrier doped Kitaev QSLs have also garnered extensive attention due to its potential accessibility to unconventional superconductivity that may possess more intricate paring from the unique QSL properties. It was reported that doping into the Kitaev model with additional Heisenberg interactions (K-J model) led to a spin-triplet topological superconducting state [94], where the pairing nature is contingent upon the doping concentration. Furthermore, the competition between K and 12 J also significantly impacts the superconducting state; for example, K prefers a p-wave superconducting state, whereas J tends to favor a d-wave one [95, 96]. Even topological superconductivity is observed in an extended K-J-Γ model [97]. In the present work, we found metallic states in the spin-orbital coupled jeff = 1/2 bands in the Ir layer for all type-II heterostructures (electron doping) and the typeIII Mn heterostructure (hole doping) (see Table III). Besides, in the type-III Co heterostructure, electron doping occurs in the Ir layer, resulting in the 3/4-filled insulating state in the jeff = 1/2 bands. These appealing results suggest that our ilmenite heterostructures offer a platform for studying exotic metallic and superconducting (even topological) properties with great flexibility by various choices of materials combination, which have been scarcely realized in the bulk systems. C. and experiments on, for instance, dynamical spin structure factors [44, 114–119] and the thermal Hall effect and its half quantization [45, 120–122], have been developed to identify the fractional excitations in Kitaev QSL, directly applying them on the heterostructures is still a great challenge. A promising experimental tool would be the Raman spectroscopy, given its successful application to not only bulk [41, 123, 124] but also atomically thin layers [125, 126]. The signals might be enhanced by piling up the heterostructures. Besides, many proposals for probing the Kitaev QSL in thin films and heterostructures have been recently made, such as local probes like scanning tunneling microscopy (STM) and atomic force microscopy (AFM) [127–131] as well as the spin Seebeck effect [132]. Additionally, as mentioned in Sec. VI A, the observation of the Majorana Fermi surfaces by thermal transport measurements in some particular heterostructures is also interesting. Experimental feasibility VII. The bulk compounds of ilmenite ATiO3 with A = Mn, Fe, Co, and Ni have been successfully synthesized and investigated for over half a century due to its fruitful magnetic and novel electronic properties [75, 76, 98–100]. Technically, the Fe case, however, is more challenging compared to the others, as its synthesis needs very high pressure and high temperature conditions [101, 102]. Besides, the iridium ilmenite MgIrO3 has also been synthesized as a power sample, where a magnetic phase transition was observed at 31.8 K [56]. The experimental lattice parameters are 5.14 Å for ATiO3 with A = Mn [75, 103–105], 5.09 Å for A = Fe [101, 106, 107], 5.06 Å for A = Co [76, 108, 109], and 5.03 Å for A = Ni [75, 107, 110, 111], respectively, as well as that is 5.16 Å for MgIrO3 . The relatively small lattice mismatch between these materials also ensure the possibility of combining them to create heterostructure with different compounds. Indeed, we demonstrated this in Sec. II; see Table I. More excitingly, the IrO6 honeycomb lattice has been successfully incorporated into the ilmenite MnTiO3 with the formation of several Mn-Ir-O layers [54]. This development lightens the fabrication of a supercell between MgIrO3 and ATiO3 . The verification of Kitaev QSL poses a significant challenge even though the successful synthesis of aforementioned heterostructures. First of all, it is crucial to identify the spin-orbital entangled electronic states with the formation of the jeff = 1/2 bands in these heterostructures, as they are essential for the Kitaev interactions between the pseudospins. Several detectable spectroscopic techniques are useful for this purpose, applicable to both bulk and heterostructures [11, 34, 35, 39, 42, 43, 46, 112, 113]. Even the Kitaev exchange interaction can be directly uncovered in experiment [36, 37]. However, the key challenge lies in probing the intrinsic properties of Kitaev QSL, such as fractional spin excitations. Thus far, despite cooperative studies between theories SUMMARY To summarize, we have conducted a systematic investigation of the electronic and magnetic properties of the bilayer structures composed by the ilmenites ATiO3 with A = Mn, Fe, Co, and Ni, in combination with the candidate for Kitaev magnets MgIrO3 . We have designed and labeled three types of heterostructure, denoted as type-I, II, and III, distinguished by the atomic configurations at the interface. Our analysis of the electronic band structures based on the ab initio calculations has revealed that the spin-orbital coupled bands characterized by the pseudospin jeff = 1/2, one of the fundamental component for the Kitaev interactions, is retained in the MgIrO3 layer for all the types of heterostructures. We found that the MgIrO3 /ATiO3 heterostructures of typeI and III are mostly spin-orbit coupled insulators, while those of type-II are spin-orbit coupled metals, irrespective of the A atoms. In the insulating heterostructures of type-I and III, based on the construction of the multiorbital Hubbard models and the second-order perturbation theory, we further found that the low-energy magnetic properties can be described by the jeff = 1/2 pseudospin models in which the estimated exchange interactions are dominated by the Kitaev-type interaction. We showed that the parasitic subdominant interactions depend on the type of the heterostructure as well as the A atoms, offering the playground for systematic studies of the Kitaev spin liquid behaviors. Moreover, the stable Néel order in the ATiO3 layer acts as a staggered magnetic field through the magnetic proximity effect, leading to the potential realization of Majorana Fermi surfaces in the MgIrO3 layer. Meanwhile, in the metallic heterostructures of type-II as well as type-I Mn, we found that the nature of carriers and the doping rates vary depending on the heterostructures. This provides the possibility of systematically studying the spin-orbit coupled metals, including exploration of unconventional superconductivity 13 due to the unique spin-orbital entanglement. In recent decades, significant progress has been made in the study of QSLs, primarily focusing on the discovery and expansion of new members in bulk materials. However, there has been limited exploration of creating and manipulating the QSLs in heterostructures despite the importance for device applications. Our study has demonstrated that the Kitaev-type QSL could be surveyed in ilmenite oxide heterostructures, displaying remarkable properties distinct from the bulk counterpart, such as flexible tuning of the Kitaev-type interactions and other parasitic interactions, and carrier doping to the Kitaev QSL. Besides the van der Waals heterostructures such as the combination of α-RuCl3 and graphene, our finding would enlighten an additional route to explore the Kitaev QSL physics including the utilization of Majorana and anyonic excitations for future topological computing devices. TABLE IV. Energy differences between different magnetic ordered states obtained by the LDA+SOC+U calculations for MgIrO3 /MnTiO3 : FM, Néel, and zigzag denotes the ferromagnetic, Néel-type antiferromagnetic, and zigzag-type antiferromagnetic orders, respectively. While the directions of the magnetic moments are all in-plane for the Ir layers, those for A can be in-plane (“in”) or out-of-plane (“out”). The bold numbers denote the low-energy states used for calculating the band structures in Sec. IV B 1. Ir magnetic state Mn in in-FM out in out-FM out in ACKNOWLEDGMENTS We thank Y. Kato, M. Negishi, S. Okumura, A. Tsukazaki, and L. Zh. Zhang, for fruitful discussions. This work was supported by JST CREST Grant (No. JPMJCR18T2). Parts of the numerical calculations were performed in the supercomputing systems of the Institute for Solid State Physics, the University of Tokyo. Appendix A: Detailed ab initio data for energy and magnetic coupling In this Appendix, we present the details of ab initio results for various types of heterostructures. Tables IVVII list the energy differences between different magnetic states for MgIrO3 /ATiO3 heterostructures with A = Mn, Fe, Co, and Ni. The bold elements in these tables are the lowest-energy state in each type, utilized for the calculations of band structures in Sec. IV B 1. We also show in Table VIII the effective magnetic coupling constants between A atoms, in which negative and positive value indicates the FM and AFM coupling, respectively. Note that the A atoms comprise a triangular lattice at the interface in type I, a honeycomb lattice at the ATiO3 layer, and a honeycomb lattice at the interface, as depicted in Fig. 1. Néel out in zigzag out FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel I 323.5 1.492 18.13 3.695 4.266 0.773 35.71 76.55 27.18 26.98 20.45 2.953 0.307 0.000 20.05 4.893 energy/Ir (meV) II III 4.868 975.4 46.52 1002 4.268 1065 44.554 978.2 7.343 990.8 43.68 993.2 1.584 1014 40.34 879.6 8.127 78.52 42.83 48.51 4.474 121.0 39.39 22.45 2.923 78.80 21.55 0.000 0.000 120.49 39.62 22.95 TABLE V. Energy differences between different magnetic ordered states for MgIrO3 /FeTiO3 . The notations are common to Table IV. Ir magnetic state Fe in in-FM out in out-FM out in Néel out in zigzag out FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel I 9.090 17.00 15.71 1.170 10.68 12.12 55.58 8.115 24.64 11.80 14.48 0.000 15.58 4.376 14.35 4.743 energy/Ir (meV) II III 1.437 0.000 15.75 13.14 21.80 2.631 27.02 14.25 0.973 0.092 0.948 13.01 21.13 1.739 26.29 8.833 0.000 0.056 0.721 12.97 20.21 1.244 25.37 8.596 1.233 0.056 13.60 4.222 21.42 2.244 26.59 9.608 Appendix B: Orbital projected band structures In this Appendix, we show the projection of the band structure to the Ir 5d orbitals for type-II heterostructures in Fig. 8 and type-III of Mn and Co in Fig. 9. The green shaded bands include high-energy four jeff = 1/2 bands and low-energy eight jeff = 3/2 bands. The results in Fig. 8 indicate that electrons are doped to the half-filled jeff = 1/2 bands, realizing the spin-orbit coupled metallic states for all A atoms. The doping rates are large (small) for A = Mn and Co (Fe and Ni). Meanwhile, Fig. 9(a) shows that the jeff = 1/2 bands are slightly hole doped in the type-III with A = Mn. Figure 9(b) indicates that the type-III with A = Co achieves an insulating state with 3/4-filled jeff = 1/2 bands. 14 magnetic state Co in in-FM out in out-FM out in Néel out in zigzag out FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel I 188.2 191.3 40.23 1.201 0.174 0.000 295.3 47.55 189.5 184.3 17.53 0.617 191.8 189.6 40.81 1.476 energy/Ir (meV) II III 20.02 0.795 0.000 0.253 25.63 72.39 235.7 87.28 20.06 0.800 0.019 0.071 25.85 260.4 0.044 49.64 19.56 0.749 19.36 0.000 24.82 33.16 234.9 31.16 19.88 0.748 19.39 1.969 25.45 34.02 130.0 32.03 TABLE VII. Energy differences between different magnetic ordered states for MgIrO3 /NiTiO3 . The notations are common to Table IV. Ir magnetic state Ni in in-FM out in out-FM out in Néel out in zigzag out FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel FM Néel I 2.990 4.998 3.536 0.156 6.901 7.311 20.94 6.220 14.06 13.00 4.941 0.000 4.971 6.275 3.451 3.876 energy/Ir (meV) II III 0.097 37.80 70.93 35.99 1.255 26.76 69.45 35.74 2.039 37.48 70.74 35.98 2.385 24.79 69.61 61.76 1.678 48.98 18.68 54.35 0.294 26.51 18.631 37.36 0.000 36.97 33.08 0.000 0.042 26.20 68.36 35.45 TABLE VIII. Effective magnetic coupling constants between the A atoms for three types of heterostructures. The unit is in meV. A Mn Fe Co Ni type-I 0.667 0.017 0.065 0.088 II -0.712 -0.607 0.262 -0.126 1.0 (b) Fe 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 -2.0 1.0 Γ M K Γ -2.0 1.0 Γ (c) Co 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 Γ M K Γ M K Γ (d) Ni 0.5 -2.0 III 2.417 -0.149 6.458 0.405 1.0 (a) Mn Energy (eV) Ir lations. In the LDA+SOC result, the system behaves as an insulating state with a tiny band gap of approximately ∼0.096 eV. 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