Abstract
A promising route to tailoring the electronic properties of quantum materials and devices rests on the idea of orbital engineering in multilayered oxide heterostructures. Here we show that the interplay of interlayer charge imbalance and ligand distortions provides a knob for tuning the sequence of electronic levels even in intrinsically stacked oxides. We resolve in this regard the dlevel structure of layered Sr_{2}IrO_{4} by electron spin resonance. While canonical ligandfield theory predicts g_{}factors less than 2 for positive tetragonal distortions as present in Sr_{2}IrO_{4}, the experiment indicates g_{} is greater than 2. This implies that the iridium d levels are inverted with respect to their normal ordering. Stateoftheart electronicstructure calculations confirm the level switching in Sr_{2}IrO_{4}, whereas we find them in Ba_{2}IrO_{4} to be instead normally ordered. Given the nonpolar character of the metaloxygen layers, our findings highlight the tetravalent transitionmetal 214 oxides as ideal platforms to explore dorbital reconstruction in the context of oxide electronics.
Introduction
Their unique diversity of transport and magnetic properties endows transitionmetal (TM) oxides with a longterm potential for applications in microelectronics and electrical engineering. Nowadays the search for new or superior properties goes beyond known bulk phases and includes oxide interfaces and stacked superlattices^{1,2}. As compared with the bulk material, at interfaces the modification of the nearby surroundings can significantly affect the valence electronic structure, in particular, the occupation of the dshell levels^{1,2,3,4,5}. This is often referred to as orbital reconstruction^{3,4,5} and brings to the fore the most basic aspect in electronicstructure theory: how energy levels in quantum matter are formed and populated.
A variety of intrinsically stacked crystalline oxides is presently known. The hightemperature cuprate superconductors^{6}, for example, fall in this category but also iridates of the type A_{2}IrO_{4} (A=Sr^{2+}, Ba^{2+}) that closely resemble undoped cuprates, both structurally and magnetically^{7,8,9,10,11}. Sr_{2}IrO_{4} has a rather simple crystalline structure displaying stacked, quasi twodimensional (2D) IrO_{2} and double SrO layers. We shall demonstrate that in this system the occupation of the valence d electronic levels differs from what is expected in textbook ligandfield theory due to electrostatics that involves both types of metaloxygen sheets. In particular, we show that, as compared with the isostructural cuprate La_{2}CuO_{4}, a different distribution of ionic charges between the TMO_{2} and AO layers modifies the sequence of energy levels within the t_{2g} and e_{g} manifolds and consequently very fundamental physical properties such as the magnetic g factors, which determine the relation between the magnetic moment and quantum number of a magnetic particle. Our findings are of direct relevance to the field of stacked oxide heterostructures and provide a guideline on how lowsymmetry crystal fields at dmetal sites can be altered and potentially engineered through the appropriate design of successive ionic layers.
To show this we first use electron spin resonance (ESR) measurements to untangle the 5dshell electronic structure of crystalline Sr_{2}IrO_{4}, in particular, the exact order of the Ir t_{2g} levels. The single s=1/2 hole present in these t_{2g} orbitals carries an angular moment l_{eff}=1 and is subject to a large spinorbit coupling (SOC), which in first approximation results in an effective Ir^{4+} moment, or pseudospin, j_{eff}=l_{eff}−s≈1/2 (refs 7, 12, 13). We compare the experimental properties of these pseudospins with the ones we have calculated by ab initio quantum chemistry methods. This combined approach, explored here on a strongly spinorbit coupled material for the first time, provides direct access to the spatial anisotropies of the g factors and further to the detailed microscopic superexchange interactions. The ESR measurements and theory are found to agree on a quantitative level and moreover undoubtedly show that the dlevel ordering in Sr_{2}IrO_{4} is inverted with respect to the normal ordering in the sister iridate Ba_{2}IrO_{4} or the isostructural 214 cuprate superconductors. The good agreement between the ESR data and the outcome of the computational methodology we describe and employ here establishes the latter as a reliable tool for the investigation of nontrivial electronic structures and magnetic couplings.
Results
Pseudospins and effective Hamiltonian
MottHubbard physics in dmetal compounds has been traditionally associated with firstseries (3d) TM oxides. However, recently, one more ingredient entered the TMoxide ‘Mottness’ paradigm—large SOC’s in 5d systems. SOC in 5d and to some extent 4d anisotropic oxides modifies the very nature of the correlation hole of an electron, by admixing the different t_{2g} components^{12,13}, changes the conditions for localization^{7}, the criteria of Mottness and further gives rise to new types of magnetic ground states and excitations^{9,11}. While various measurements indicate that indeed spinorbitcoupled j_{eff}≈1/2 states form in A_{2}IrO_{4} (refs 7, 9, 14), it has been also pointed out that offdiagonal SOC’s may mix into the ground state (GS) wavefunction substantial amounts of character^{15,16,17,18}. Such manybody interactions were shown to produce remarkable effects in Xray absorption and Xray magnetic circular dichroism (XMCD): the branching ratio between the L_{3} and L_{2} Ir 2p absorption edges reaches values as large as 4, nearly 50% higher than the 2.75 value for a ‘pure’ j_{eff}=1/2 system^{19}. In addition, lowsymmetry noncubic fields produce sizeable splittings of the 5d t_{2g} levels, in some cases close to or even larger than ∼1/2 eV (refs 20, 21, 22), and therefore admix the j_{eff}=1/2 and components^{13,15}. The structure of the spinorbit GS depends on both the strength and sign of these splittings. Interestingly, the best fits of the Xray absorption and XMCD data are achieved in Sr_{2}IrO_{4} with a negative t_{2g} tetragonal splitting^{19}, although the oxygen octahedra in this material display a distinct positive tetragonal distortion—the IrO_{6} octahedra are substantially elongated^{23} (a negative tetragonal splitting should occur when the IrO_{6} octahedra are compressed^{13,24}, see Fig. 1). This is already a first indication of the level inversion that our ESR measurements and quantum chemistry calculations show to take place in Sr_{2}IrO_{4}.
The interactions between a pair 〈ij〉 of nearestneighbour (NN) 1/2 pseudospins in the presence of an external magnetic field h is given by the effective Hamiltonian
where , are pseudospin (j_{eff}≈1/2) operators, J is the isotropic Heisenberg exchange, D=(0,0,D) defines the antisymmetric DzyaloshinskiiMoriya (DM) coupling, is a symmetric traceless secondrank tensor describing the symmetric anisotropy and due to the staggered rotation of the IrO_{6} octahedra the tensor splits for each of the two sites into uniform and staggered components (see for example, refs 16, 25). This effective spin Hamiltonian is of direct relevance to the interpretation of the ESR data.
ESR measurements
For a single crystal of Sr_{2}IrO_{4} we observe antiferromagnetic resonance (AFR) modes in the subTHz frequency domain^{26} as displayed in Fig. 2. There are two modes if hz: a gapless Goldstone mode v_{1}=0 and a gapped excitation
where , and Γ_{zz} couples the and components (along the c axis, perpendicular to the abplane IrO_{2} layers^{23}) in the third term of equation (1) (see Methods for details). Experimental results are shown for v_{2} in Fig. 2a. The data comprise at T≪T_{N}=240 K a group of overlapping resonances (Fig. 2a, inset), possibly due to some distribution of internal fields in the sample. Though revealing some scatter, the AFR data follow approximately a parabolic dependence on h and, most importantly, they lie substantially above the curve corresponding to the freeelectron Landé factor g_{e}=2 (dashed line in Fig. 2a). The experimental dependence can be reasonably well modelled with g_{} values of 2.3–2.45. The solid line in Fig. 2a is obtained by using g_{}=2.31, as derived from quantum chemistry calculations that will be discussed later on.
Sizeable deviations to values >2 of g_{} is clear indication of the presence of lowsymmetry, noncubic crystal fields. In the simplest approximation, that is, restricting ourselves to the Ir^{4+} manifold, the anisotropic g factors in axial noncubic environment can be expressed up to the sign as^{13} g_{}=g_{c}=(2+2k)cos^{2}α−2sin^{2}α and where k is a covalency reduction factor, α=(1/2) arctan parameterizes the deviation from octahedral symmetry, λ is the SOC constant and δ the Ir t_{2g} splitting. A plot for the dependence of the diagonal g factors on the distortion parameter α is shown in the inset to Fig. 2b. For simplicity, k=1 is for the moment assumed but smaller values of k do not bring qualitative changes. In cubic symmetry δ=0, α_{cub}=35.26° and the g matrix is isotropic with . According to standard textbooks on ligandfield theory^{24}, an elongation of the outofplane IrO bond induces a positive tetragonal splitting of the Ir t_{2g} levels, with δ>0 and α>α_{cub}, whereas a bond compression yields δ<0 and α<α_{cub} (see Fig. 1). As in Sr_{2}IrO_{4} the IrO_{6} octahedra are substantially elongated in the z direction^{23}, the g factors are expected to correspond to the case of positive splitting α>α_{cub}, see the area to the right of the crossing point shown in the inset to Fig. 2b. It follows from the plot that g_{}<2, which obviously contradicts our AFR data for hz [Fig. 2a]. The value g_{}=2.31 used to draw the curve connecting the open circles in Fig. 2a in fact corresponds to α=32.05°<α_{cub} (see the inset in Fig. 2b) and indicates that, despite the positive caxis tetragonal distortion, a counterintuitive negative tetragonal splitting of the 5d t_{2g} levels is present in Sr_{2}IrO_{4}.
It should be noted that there is no Goldstone mode for finite inplane magnetic fields. The canting angle depends in this case both on the strength of the DM interaction and the applied field. The two modes are
and
where and m is the inplane ferromagnetic component of the effective moments. To first order in magnetic field, m can be expressed as
which holds for weak fields where m remains small. The first term corresponds to the zerofield canting, arising from the DM interaction, while the second term shows how this canting evolves with increasing h_{⊥}. Plots based on equations (3, 4) and the quantum chemically derived interaction parameters (see below) are displayed in Fig. 2b together with ESR data.
Quantum chemistry calculations of g factors
Results of ab initio quantum chemistry calculations for the g factors in Sr_{2}IrO_{4} and in the structurally related material Ba_{2}IrO_{4} are listed in Table 1. Our computational scheme follows the prescription of Bolvin^{27} and Vancoillie et al.^{28}. It maps the matrix elements (MEs) of the ab initio Zeeman Hamiltonian onto the MEs of the effective pseudospin Hamiltonian , where μ, L and S are magnetic moment, angularmomentum and spin operators, respectively. The spinorbit GS wave functions are computed either at the completeactivespace selfconsistentfield (CASSCF) or multireference configurationinteraction (MRCI) level of theory^{29}, as described in ref. 30 and using the MOLPRO quantum chemistry package^{31}. All necessary angularmomentum MEs are calculated as well with MOLPRO (see Methods). In a first set of calculations, only the three t_{2g} orbitals at a given Ir site and five electrons were considered in the active space. The selfconsistentfield optimization was carried out for the corresponding state. We use here the more convenient notations associated to O_{h} symmetry, although the calculations were performed for the actual experimental geometry, with pointgroup symmetry lower than octahedral. Inclusion of SOC yields in this case a set of three Kramers doublets (KDs), see Table 1.
Subsequently we performed calculations with larger active spaces, including also the Ir e_{g} orbitals. One plus four (^{2}A_{2g}, ^{2}T_{1g}, ^{2}E_{g} and ^{2}T_{2g}) spin doublets, two spin quartets [ and ] and one spin sextet entered here the spinorbit treatment. The orbitals were optimized for an average of all these terms.
The effect of enlarging the active space to include terms in the reference wavefunction is in the range of 10%, in line with earlier semiempirical estimates for 4d^{5} and 5d^{5} systems^{15,16,17,18}. Most importantly, the calculations yield a negative tetragonal splitting of the Ir t_{2g} levels in Sr_{2}IrO_{4}, δ=−155 meV by MRCI, and positive t_{2g} splitting in Ba_{2}IrO_{4} (see Table 1 and Methods). Similar signs, negative in Sr_{2}IrO_{4} and positive in Ba_{2}IrO_{4}, but much larger magnitudes (≈0.7 eV) are found for the computed Ir e_{g} splittings (not shown in Table 1).
Taken together, the ESR and quantum chemistry results unequivocally point at an anomalous order of the split Ir 5d levels in Sr_{2}IrO_{4}, related to the important role of the extended crystalline surroundings in generating lowsymmetry fields that compete with ‘local’ distortions of the ligand cage. Similar effects were found by ab initio calculations on the 214 layered rhodate Sr_{2}RhO_{4} (ref. 32) and the 227 pyrochlore iridates^{22}. In contrast, in Ba_{2}IrO_{4}, the stretch of the apical IrO bonds is strong enough^{33} to overcome the longerrange electrostatics, turning the tetragonal t_{2g} positive again, as discussed in more detail in the following. Consequently, the structure of the tensor in Ba_{2}IrO_{4} is qualitatively different, with g_{⊥}>2 and g_{}<2 (see Table 1), the ordering that one normally expects and encounters for elongated octahedra^{13}.
Exchange couplings from quantum chemistry
To obtain ab initio quantum chemistry values for the intersite effective magnetic couplings in Sr_{2}IrO_{4} (see equation (1)), we carried out additional calculations on larger clusters that incorporate two 5d^{5} sites. The twooctahedra cluster has C_{2v} symmetry (see Fig. 3), which implies a diagonal form for and in equation (1) (see Methods). By onetoone correspondence between the MEs of the ab initio Hamiltonian
and the MEs of the effective spin Hamiltonian (1) in the basis of the lowest four spinorbit states defining the magnetic spectrum of two NN octahedra, we can derive in addition to the g factors the strengths of the Heisenberg and anisotropic intersite couplings. In equation (6), is the scalarrelativistic BornOppenheimer Hamiltonian, describes spinorbit interactions^{30} and is the twosite Zeeman Hamiltonian.
Diagonalization of the spin Hamiltonian (1) provides the expected singlet and three (split) triplet components t_{x}〉, t_{y}〉 and . Owing to the DM interaction, and are admixtures of ‘pure’ 0,0〉 and 1,0〉 spin functions. Our mapping procedure yields J≈48 meV, somewhat lower than J values of 55–60 meV derived from experiment^{9,34}, and a ratio between the DM and Heisenberg couplings D/J=0.25, in agreement with estimates based on effective superexchange models^{12,35,36} and large enough to explain the nearly rigid rotation of magnetic moments that is observed when the IrO_{6} octahedra revolve^{37,38}. Ab initio results for the NN anisotropic couplings , also relevant for a detailed understanding of the magnetic properties of Sr_{2}IrO_{4}, are shown as well in Table 2. In our convention the x axis is taken along the IrIr link, that is, it coincides with the 〈110〉 crystallographic direction^{23}, and zc. We obtain Γ_{xx}≈Γ_{zz}, which then allows to recast the Heisenberg and symmetric anisotropic terms in (1) as , with Γ_{xx}=Γ_{zz}=−Γ_{yy}/2. Equally interesting, for no rotation of the IrO_{6} octahedra and straight IrOIr bonds in Ba_{2}IrO_{4}, it is Γ_{yy} and Γ_{zz} which are approximately the same, providing a realization of the compassHeisenberg model^{36,39} since the DM coupling is by symmetry 0 in that case.
The twosite magnetic Hamiltonian (1) features inplane symmetricanisotropy couplings Γ_{xx} and Γ_{yy}, which were not considered in previous studies^{12,26}. In the presence of twosublattice order, terms containing these couplings cancel each other in the meanfield energy but they are in general relevant for pseudospin fluctuations and excitations. Using spinwave theory and effective parameters derived from the quantum chemistry calculations, we nicely reproduce the correct GS and character of the modes, as shown in Fig. 2. To reproduce the experimental zerofield gap, in particular, we used J, D and gfactor values as listed in Table 2 and a somewhat larger Γ_{zz} parameter of 0.98 meV. To leading order, the dependence of and on h is linear, see equations (3, 4), and the slope is proportional to m. At low fields (≤1 T), m can be actually replaced with its field independent value^{26}. Using the MRCI coupling constants, the first term in equation (5) then yields a moment m≈0.12μ_{B}, in good agreement with the experiment^{8,40}.
Discussion
The exact dlevel order is of fundamental importance in TM oxides, dictating for instance the symmetry of the quasiparticle states in photoemission^{10,32,41} and the nature of the magnetic ordering^{42,43}. In 214 iridates specifically, it determines the various isotropic as well as anisotropic contributions to the magnetic exchange couplings^{12,35,36,39}, the evolution of those magnetic interactions with strain^{44} and/or pressure^{19} and most likely the nature of the intriguing transition to a nonmagnetic phase in Sr_{2}IrO_{4} under high pressure^{19}. Having established that in Sr_{2}IrO_{4} the d levels are inverted and that in the closely related Ba_{2}IrO_{4} they are not raises the question what actually drives the inversion. To address this, we performed an additional set of calculations, in which we change the charges around the reference IrO_{6} octahedron. As a simple numerical experiment that preserves charge neutrality of the A_{2}IrO_{4} system, we assigned the 4 NN iridium sites (inplane, see Fig. 4) the charge Q_{TM}−2Δq and the 8 closest Asite cations (out of plane) the valence Q_{A}+Δq. In a fully ionic picture, Q_{TM} and Q_{A} are 4+and 2+, respectively. However, since in our calculations the NN TM and A sites are not modelled as just formal point charges (see Methods), the actual valence states depart from their formal values, with larger ‘deviations’ for Q_{TM}. The way we introduce Δq in the computations is therefore by appropriately modifying the nuclear charge at the respective site. For variable Δq, this interpolates linearly between nearby surroundings corresponding to 5d 214 layered perovskites (with Δq=0 and TM^{4+}, A^{2+} formal valence states) and their cuprate 214 equivalents (with Δq=1, TM^{2+}/A^{3+} formal ionic charges and ‘normal’ order of the TM t_{2g} and e_{g} levels^{45}).
As is illustrated in Fig. 4a, increasing Δq amounts to moving positive charge from the IrO_{2} plane to the adjacent AO layers. The calculations show that upon moving charge in such a manner, the Ir t_{2g} splitting δ increases, see Fig. 4b. In other words, this redistribution of charge counteracts the level inversion in Sr_{2}IrO_{4} and further increases the already positive δ in Ba_{2}IrO_{4}. In Sr_{2}IrO_{4} the cubiclike j_{eff}=1/2 limit occurs for Δq=0.22. This effect can easily be understood: placing more positive charge out of the IrO_{2} plane stabilizes the outofplane t_{2g} orbitals, corresponding to the (yz,zx) orbital doublet, and thus enhances δ. One can also do the opposite and drive Δq negative. In this case more positive charge piles up in the IrO_{2} plane, which one expects to lower the energy of the xy orbital singlet, thus enhancing the level inversion in Sr_{2}IrO_{4}. This is indeed what happens, see Fig. 4b. What is more, driving Δq negative even causes a level inversion in Ba_{2}IrO_{4}, when Δq−0.25. It is interesting to note that the slope of the δ versus Δq lines in Sr_{2}IrO_{4} is much steeper than in Ba_{2}IrO_{4}, which is caused by the significantly smaller Ir–Ir distances in Sr_{2}IrO_{4}.
From these test calculations it is clear that lowsymmetry crystal fields associated to neighbours beyond the first ligand coordination shell, in particular, the highly charged Ir^{4+} NNs, counteract the local tetragonal crystal field that is caused by the elongation of the IrO_{6} octahedra in both Sr_{2}IrO_{4} and Ba_{2}IrO_{4}. In the case of Ba_{2}IrO_{4} the local distortion is still strong enough to overcome these longerrange effects but in Sr_{2}IrO_{4}, with a slightly smaller tetragonal distortion, the longerrange electrostatics wins, causing the observed level inversion.
While the role of the high ionic charge of inplane ions has been earlier invoked in the tetravalent Ru oxide compound Ca_{2}RuO_{4} (ref. 43) and in mixedvalence manganites^{46}, we here explicitly prove it by combined ESR measurements and manybody ab initio calculations on structurally and chemically simpler systems in which additional complications arising from further distortions^{43,47} or the presence of multiple TM valence states^{46} are excluded. A reversed order of the Ir t_{2g} levels in Sr_{2}IrO_{4} has been also indirectly implied by fits of Xray absorption^{19} and Xray magnetic scattering^{37,48} spectra. As a more direct and more sensitive experimental technique to such details of the valence electronic structure and with back up from truly ab initio manybody calculations, ESR now provides irrefutable evidence for such physics. The numerical ‘experiment’ outlined in Fig. 4 further shows that at the heart of this effect is not the intersite exchange, as assumed in ref. 19, and not the t_{2g}e_{g} orbital hybridization invoked in ref. 48, but basic interlayer electrostatics.
We have, in sum, provided an integrated picture on the dlevel structure and magnetic anisotropies in Sr_{2}IrO_{4}, a prototype spinorbit driven magnetic insulator. Both the singlesite tensor and intersite effective exchange interactions are analysed in detail. To access the latter, we build on an earlier computational scheme for deriving intersite matrix elements in mixedvalence spinorbit coupled systems^{49}. While the ratio D/J of the antisymmetric DzyaloshinskiiMoriya and isotropic Heisenberg couplings is remarkably large in Sr_{2}IrO_{4} and concurs with an inplane rotation pattern of the Ir magnetic moments that follows nearly rigidly the staggered rotation of the IrO_{6} octahedra^{37,38}, the most prominent symmetric anisotropic terms are according to the quantum chemistry data inplane, perpendicular to the IrIr links. The structure of the tensor, as measured by ESR and computed with firstprinciples electronicstructure methods, is such that and distinctly indicates a negative tetragonallike splitting of the Ir t_{2g} levels, in spite of sizable positive tetragonal distortions in Sr_{2}IrO_{4}. We further observe that a much stronger tetragonal distortion in Ba_{2}IrO_{4} renders the tetragonal dlevel splitting positive and g_{}<g_{⊥}. The interesting situation arises that nevertheless the magnitude of the Ir t_{2g} splitting is largest in Sr_{2}IrO_{4}. The dlevel inversion in Sr_{2}IrO_{4} and the surprisingly small splitting in Ba_{2}IrO_{4} have to do with the way the positive ionic charge is distributed between adjacent Ir^{4+}O_{2} and A^{2+}O layers, having in contrast to the 214 cuprate superconductors, for example, more positive charge in the TMO planes. This almost compensates the ‘local’ tetragonal field arising from the z axis elongation of the IrO_{6} octahedra in Ba_{2}IrO_{4} and overcompensates it in Sr_{2}IrO_{4}.
The subtle interplay between local distortions of the O ligand cage and additional uniaxial fields associated with the anisotropic extended surroundings opens new perspectives on strain^{44} and pressure^{19} experiments in squarelattice iridates, for example, in connection to the spinflop transition earlier predicted in Sr_{2}IrO_{4} (refs 12, 36). It also opens up the perspective of manipulating this way the dlevel ordering in oxide heterostructures with highly charged, trivalent and tetravalent species. Compounds with tetravalent species within the TMO_{2} layers, in particular, given the nonpolar character of the quasi 2D sheets, provide ideal playgrounds to explore the mechanism of dlevel ordering pointed out here since that will not be hindered by ‘interface’ charge redistribution and structural reconstruction occuring in polar heterostructures from polar discontinuities^{50,51}. A recent experimental realization of such mixed, tetravalent/divalent TMoxide interfaces is for example the SrRuO_{3}/NiO interface^{52}, one system that requires in this respect closer theoretical examination.
Methods
Singlesite magnetic properties
The g factors were obtained by computations on clusters which contain one central IrO_{6} octahedron, the four NN IrO_{6} octahedra and the nearby ten Sr/Ba ions. The solidstate surroundings were modelled as a large array of point charges fitted to reproduce the crystal Madelung field in the cluster region. To obtain a clear picture on crystalfield effects and spinorbit interactions at the central Ir site, we cutoff the magnetic couplings with the adjacent Ir ions by replacing the tetravalent openshell d^{5} NNs with tetravalent closedshell Pt^{4+} species. This is a usual procedure in quantum chemistry studies on TM systems, see for example, refs 42, 45, 53, 54, 55, 56. We used energyconsistent relativistic pseudopotentials and valence basis sets of quadruplezeta quality supplemented with f polarization functions for the central Ir ion^{57} and allelectron triplezeta basis sets for the six adjacent ligands^{58}. For the TM NNs, we applied energyconsistent relativistic pseudopotentials and triplezeta basis functions^{57} along with minimal atomicnaturalorbital basis sets^{59} for the Os coordinating those TM sites but not shared with the central octahedron. The Sr and Ba species were modelled by divalent totalion effective potentials supplemented with a single s function^{60}. All O 2p and metal t_{2g} electrons at the central octahedron were correlated in the MRCI calculations. The latter are performed with single and double substitutions with respect to the CASSCF reference (for technicalities, see refs 61, 62), which is referred to as MRCISD. To separate the metal 5d and O 2p valence orbitals into different groups, that is, centraloctahedron and adjacentoctahedra orbitals, we used the PipekMezey localization module^{63} available in MOLPRO. The computations with hypothetical (Q_{TM}−2Δq) and (Q_{A}+Δq) ionic charges at the TM and Sr/Ba sites next to the reference Ir ion were carried out as frozenorbital multideterminant calculations (also referred to as CASCI) with three Ir t_{2g} and five electrons in the active space and orbitals optimized for Δq=0.
The spinorbit treatment was performed according to the procedure described in ref. 30. In a first step, the scalar relativistic Hamiltonian is used to calculate correlated wavefunctions for a finite number of lowlying states, either at the CASSCF level or at the MRCI level. In a second step, the spinorbit part is added to the initial scalar relativistic Hamiltonian, matrix elements of the aforementioned states are evaluated for this extended Hamiltonian and the resulting matrix is finally diagonalized to yield SO wavefunctions.
The g factors were computed following the scheme proposed by Bolvin^{27} and Vancoillie et al.^{28} (for alternative formulations, see, for example, ref. 64). For the KD GS , the AbragamBleaney tensor^{13,65}, G=gg^{T} can be written in matrix form as
where
The MEs of are here provided by MOLPRO while those of are derived using the conventional expressions for the generalized Pauli matrices :
G is next diagonalized and the g factors are obtained as the possitive square roots of the three eigenvalues. The corresponding eigenvectors specify the rotation matrix to the main magnetic axes. In our case, the magnetic z axis is along the crystallographic c coordinate, while x and y are ‘degenerate’ and can be any two perpendicular directions in the ab plane.
To crosscheck the gfactor values computed with our subroutine, we further performed gfactor calculations using the module available within the ORCA quantum chemistry package^{66}. We applied allelectron DKH (DouglasKrollHess) basis sets of triplezeta quality for the TM ions^{67}, triplezeta basis functions for the ligands of the central octahedron^{58} and doublezeta basis functions for additional Os at the NN octahedra^{58}. Dynamical correlation effects were accounted for by Nelectron valencestate secondorder perturbation theory (NEVPT2)^{68,69}. CASSCF and NEVPT2 results are listed in Table 3, for both Sr_{2}IrO_{4} and Ba_{2}IrO_{4}. It is seen that the data in Tables 1 and 3 compare very well and indicate the same overall trends.
Superexchange interactions in Sr_{2}IrO_{4}
NN magnetic coupling constants were obtained for Sr_{2}IrO_{4} by calculations on an embedded cluster that includes two IrO_{6} octahedra as magnetically active units. As for the calculation of singlesite magnetic properties, to accurately describe the charge distribution in the immediate neighbourhood, the adjacent IrO_{6} octahedra (six) and the closest Sr^{2+} ions (16) were also incorporated in the actual cluster. We used energyconsistent relativistic pseudopotentials along with quadruplezeta basis sets for the valence shells of the two magnetically active Ir ions^{57}, allelectron quintuplezeta basis sets for the bridging ligand^{58} and triplezeta basis functions for the other Os associated with the two reference octahedra^{58}. We further employed polarization functions at the two central Ir sites and for the bridging anion, namely 2 Ir f and 4 O d functions^{57,58}. Additional ions defining the NN octahedra, the nearby Sr^{2+} species and the farther crystalline surroundings were modelled as in the singlesite study, see above.
For two adjacent magnetic sites, the manifold entails nine singlet and nine triplet states. The CASSCF optimization was carried out for an average of these nine singlet and nine triplet eigenfunctions of the scalar relativistic Hamiltonian . In the subsequent MRCI treatment, only the Ir t_{2g} and the O 2p electrons at the bridging ligand site were correlated. Results in good agreement with the experimental data were recently obtained with this computational approach for related 5d^{5} iridates^{21,70}.
Diagonalization of the Hamiltonian in the basis of the lowest nine singlet and nine triplet states provides a total of 36 spinorbitcoupled eigenfunctions, namely, four , eight , eight and sixteen states. In the simplest picture, the lowest four roots imply either singlet or triplet coupling of the spinorbit j_{eff}=1/2 (or, more generally, pseudospin =1/2) onsite objects and are separated from higherlying states by a gap of 0.5 eV, much larger than the strength of the intersite exchange. It is this set of lowest four spinorbit MRCI roots that we map onto the eigenstates of the effective twosite (pseudo)spin Hamiltonian (1). The Zeeman interaction shows up on the quantum chemistry side as , where and are transformed to the spinorbitcoupled basis using the spinorbit wavefunctions as unitary transformation matrix. MEs of the ab initio model Hamiltonian are shown in Table 4. Diagonal components show the energies of the zerofield states, while the offdiagonal MEs describe the coupling to magnetic field.
For the experimentally determined crystal structure of Sr_{2}IrO_{4} (ref. 23), the twooctahedra [Ir_{2}O_{11}] cluster displays C_{2v} symmetry. Having the x axis along the 〈110〉 crystallographic direction^{23} and zc, the effective anisotropic couplings read D=(0,0,D),
for IrIr links along x and
for IrIr links along y, with Γ_{xx}+Γ_{yy}+Γ_{zz}=0. The uniform and staggered components of the tensor take for individual Ir sites the following form :
g_{zz}=g_{} while g_{xx} and g_{yy} are directly related to g_{⊥} but not restricted to be equal due to the lower symmetry of the twooctahedra cluster as compared with the IrO_{6} unit.
To solve now the actual problem, we need to transform the effective spin Hamiltonian (1) to the same form as the ab initio Hamiltonian shown in Table 4, that is, diagonal in zero magnetic field. The result of such a transformation is shown in Table 5. Direct correspondence between homologous MEs in the two arrays yields a set of eight independent equations that finally allow to derive hard values for all effective coupling constants that enter expression (1). While the results for the intersite exchange interactions in (1) and (10,11) are shown in Table 2, the tensor data obtained from the twooctahedra calculations are g_{xx}=1.64, g_{yy}=1.70, g_{zz}=2.31 and g_{xy}=0.02. The way the additional parameters introduced in Table 5 are defined is explained in Table 6. The intermediate steps followed to arrive to the matrix form provided in Table 5 are outlined in Supplementary Tables I–III and the Supplementary Methods.
Spinwave calculations
For understanding all details of the ESR spectrum, we carried out a spinwave analysis using the Hamiltonian (1) and a sitefactorized variational GS wave function
where for each sublattice index L∈{A,B},
For the 2D unit cell displayed in Fig. 3a, D_{2d} symmetry is considered.
The spin components of the GS configuration depend on the α_{L} and variational parameters as
For magnetic fields parallel to the c axis, the GS energy only depends on the parameters α_{A}=α_{B}=α and . is the angle between neighbouring inplane spins and α describes how much the spins are tilted away from the c axis. For α=π/2 the spins are lying within the ab plane, while α=0 corresponds to the fully polarized highmagneticfield case. The GS energy
is minimized when
For zero field we find that α=0, the spins are confined to the ab plane (that is, to the IrO_{2} layer) and the angle is controlled by the strength of D. is not affected by fields along the c axis while α changes smoothly from π/2 to 0 with increasing the field strength.
The two magnons characteristic for spin1/2 antiferromagnets are related to states orthogonal to and . The 4 × 4 Hamiltonian defining these magnons can be derived by using the wellknown HolsteinPrimakoff approach and diagonalized through a Bogoliubov transformation. For hc, the two spinwave modes are the gapless v_{1}=0 Goldstone mode, corresponding to U(1) symmetry breaking, and the gapped mode given by (2). When the magnetic field lies in the ab plane, the z component of Ir spins remains zero (with α=π/2) and the GS energy only depends on the angle :
For simplicity, we select x for the direction of the magnetic field. The result is, however, independent of how this choice is made as there is no anisotropy within the ab plane.
An infinitesimally small inplane field fixes the direction of uniform magnetization as . At finite field both modes thus become gapped. To determine the uniform magnetization m one needs to minimize equation (21), which to leading order in magnetic field leads to equation (5). The first term of equation (5) corresponds to the zerofield moment, which arises due to the canting induced by the DM interaction. This ferromagnetic orderparameter is further enhanced in finite h_{⊥} field. As long as h_{⊥} is small equation (5) remains valid.
As discussed in the main text, in low fields, m can be approximated by its field independent value^{26}. Using the quantum chemically derived interaction parameters (see Table 2), we then find m≈0.12μ_{B}, in good agreement with recent experiments^{8,40}. Yet the zerofield gap comes out too large as compared to experiment. A good fit can nevertheless be reached with J, D and g values as obtained in the MRCI treatment and by increasing Γ_{zz} from 0.42 to 0.98 meV.
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How to cite this article: Bogdanov, N. A. et al. Orbital reconstruction in nonpolar tetravalent transitionmetal oxide layers. Nat. Commun. 6:7306 doi: 10.1038/ncomms8306 (2015).
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Acknowledgements
We thank G. Jackeli, G. Khaliullin, P. Fulde, and H. Takagi for fruitful discussions. N.A.B. and L.H. acknowledge financial support from the Erasmus Mundus Programme of the European Union and the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), respectively. Experimental work was in part supported through the DFG project KA 1694/81.
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V.K. conducted the ESR experiments. N.A.B. and V.M.K. carried out the ab initio quantum chemistry calculations and subsequent mapping of the ab initio data onto the effective spin Hamiltonian, with assistance from L.H., V.Y., J.R. and V.K. J.R. performed the spinwave analysis. L.H., V.K., J.v.d.B. and B.B. designed the project. L.H., J.v.d.B., V.K., N.A.B., V.M.K. and J.R. wrote the paper, with contributions from all other coauthors.
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Bogdanov, N., Katukuri, V., Romhányi, J. et al. Orbital reconstruction in nonpolar tetravalent transitionmetal oxide layers. Nat Commun 6, 7306 (2015). https://doi.org/10.1038/ncomms8306
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