# Orbital reconstruction in nonpolar tetravalent transition-metal oxide layers

Nikolay A. Bogdanov  Institute for Theoretical Solid State Physics, IFW
Dresden, Helmholtzstr. 20, 01069 Dresden, Germany  Vamshi M. Katukuri
Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20,
01069 Dresden, Germany  Judit Romhányi  Institute for Theoretical Solid State
Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany  Viktor
Yushankhai  Joint Institute for Nuclear Research, Joliot-Curie 6, 141980
Dubna, Russia  Institute for Theoretical Solid State Physics, IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany  Vladislav Kataev  Institute for
Solid State Research, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
Bernd Büchner  Institute for Solid State Research, IFW Dresden, Helmholtzstr.
20, 01069 Dresden, Germany  Department of Physics, Technical University
Dresden, 01062 Dresden, Germany  Jeroen van den Brink  Institute for
Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany  Department of Physics, Technical University Dresden, 01062 Dresden,
Germany  Liviu Hozoi  Institute for Theoretical Solid State Physics, IFW
Dresden, Helmholtzstr. 20, 01069 Dresden, Germany

(October 27, 2014)

###### 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 $d$-level structure of layered Sr2IrO4 by electron spin resonance.
While canonical ligand-field theory predicts $g_{\parallel}$-factors
$\\!<\\!2$ for positive tetragonal distortions as present in Sr2IrO4, the
experiment indicates $g_{\parallel}\\!>\\!2$. This implies that the iridium
$d$ levels are inverted with respect to their normal ordering. State-of-the-
art electronic-structure calculations confirm the level switching in Sr2IrO4,
whereas we find them in Ba2IrO4 to be instead normally ordered. Given the
nonpolar character of the metal-oxygen layers, our findings highlight the
tetravalent transition-metal 214 oxides as ideal platforms to explore
$d$-orbital reconstruction in the context of oxide electronics.

Their unique diversity of transport and magnetic properties endows transition-
metal (TM) oxides with a long-term 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 Heber (2009); Mannhart and Schlom (2010).
As compared to the bulk material, at interfaces the modification of the nearby
surroundings can significantly affect the valence electronic structure, in
particular, the occupation of the $d$-shell levels Heber (2009); Mannhart and
Schlom (2010); Chakhalian  _et al._ (2007); Benckiser  _et al._ (2011); Hwang
_et al._ (2012). This is often referred to as orbital reconstruction
Chakhalian  _et al._ (2007); Benckiser  _et al._ (2011); Hwang  _et al._
(2012) and brings to the fore the most basic aspect in electronic-structure
theory: how energy levels in quantum matter are formed and populated.

A variety of intrinsically stacked crystalline oxides is presently known. The
high-temperature cuprate superconductors Leggett (2006), for example, fall in
this category but also iridates of the type A2IrO4 (A=Sr2+, Ba2+) which
closely resemble undoped cuprates, both structurally and magnetically Kim  _et
al._ (2008, 2009, 2012a, 2014a, 2014b). Sr2IrO4 has a rather simple
crystalline structure displaying stacked, quasi two-dimensional (2D) IrO2 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
ligand-field theory due to electrostatics that involves both types of metal-
oxygen sheets. In particular, we show that, as compared to the isostructural
cuprate La2CuO4, a different distribution of ionic charges between the TM-O2
and A-O layers modify the sequence of energy levels within the
$t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$ 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 low-symmetry
crystal fields at $d$-metal 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 $5d$-shell electronic structure of crystalline Sr2IrO4, in
particular, the exact order of the Ir $t_{\mathrm{2g}}$ levels. The single
$s\\!=\\!1/2$ hole present in these $t_{\mathrm{2g}}$ orbitals carries an
angular moment $l_{\mathrm{eff}}\\!=\\!1$ and is subject to a large spin-orbit
coupling (SOC), which in first approximation results in an effective Ir4+
moment, or pseudospin,
$j_{\mathrm{eff}}\\!=\\!l_{\mathrm{eff}}\\!-\\!s\\!\approx\\!1/2$ Kim  _et
al._ (2008); Jackeli and Khaliullin (2009); Abragam and Bleaney (1970). We
compare the experimental properties of these pseudospins to the ones we have
calculated by ab initio quantum chemistry methods. This combined approach,
explored here on a strongly spin-orbit 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 $d$-level ordering in Sr2IrO4 is inverted
with respect to the normal ordering in the sister iridate Ba2IrO4 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. Mott-Hubbard physics in $d$-metal
compounds has been traditionally associated with first-series ($3d$) TM
oxides. However, recently, one more ingredient entered the TM-oxide “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 $t_{\mathrm{2g}}$ components Jackeli and
Khaliullin (2009); Abragam and Bleaney (1970), changes the conditions for
localization Kim  _et al._ (2008), the criteria of Mottness and further gives
rise to new types of magnetic ground states and excitations Kim  _et al._
(2012a, 2014b). While various measurements indicate that indeed spin-orbit-
coupled $j_{\mathrm{eff}}\\!\approx\\!1/2$ states form in A2IrO4 Kim  _et al._
(2008, 2012a); Boseggia  _et al._ (2013a), it has been also pointed out that
off-diagonal SOC’s may mix into the ground state (GS) wave function
substantial amounts of $t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$ character
Thornley (1968); Hill (1972); Andlauer  _et al._ (1976); Stebler  _et al._
(1984). Such $t_{\mathrm{2g}}^{5}$–$t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$
many-body interactions were shown to produce remarkable effects in x-ray
absorption and x-ray 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”
$j_{\mathrm{eff}}\\!=\\!1/2$ system Haskel  _et al._ (2012). Additionally,
low-symmetry noncubic fields produce sizable splittings of the $5d$
$t_{\mathrm{2g}}$ levels, in some cases close to or even larger than $\sim$1/2
eV Liu  _et al._ (2012); Bogdanov  _et al._ (2012); Hozoi  _et al._ (2014),
and therefore admix the $j_{\mathrm{eff}}\\!=\\!1/2$ and
$j_{\mathrm{eff}}\\!=\\!3/2$ $t_{\mathrm{2g}}^{5}$ components Abragam and
Bleaney (1970); Thornley (1968). The structure of the spin-orbit GS depends on
both the strength and sign of these splittings. Interestingly, the best fits
of the x-ray absorption and XMCD data are achieved in Sr2IrO4 with a negative
$t_{\mathrm{2g}}$ tetragonal splitting Haskel  _et al._ (2012), although the
oxygen octahedra in this material display a distinct positive tetragonal
distortion – the IrO6 octahedra are substantially elongated Crawford  _et al._
(1994) (a negative tetragonal splitting should occur when the IrO6 octahedra
are compressedBallhausen (1962); Abragam and Bleaney (1970), 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 Sr2IrO4.

The interactions between a pair $\langle ij\rangle$ of nearest-neighbor (NN)
1/2 pseudospins in the presence of an external magnetic field $\mathbf{h}$ is
given by the effective Hamiltonian

$\mathcal{H}^{i,j}_{\mathrm{eff}}=J\tilde{\mathbf{S}}_{i}\cdot\tilde{\mathbf{S}}_{j}+{\mathbf{D}}\cdot{\tilde{\bf
S}}_{i}\times{\tilde{\bf
S}}_{j}+\tilde{\mathbf{S}}_{i}\cdot\bar{\bar{\bm{\Gamma}}}\cdot\tilde{\mathbf{S}}_{j}\,+\mu_{\rm
B}\\!\sum_{k\\!=\\!i,j}\mathbf{h}\cdot\bar{\bar{\mathbf{g}}}_{k}\cdot\tilde{\mathbf{S}}_{k},$
(1)

where $\tilde{\mathbf{S}}_{i}$, $\tilde{\mathbf{S}}_{j}$ are pseudospin
($j_{\mathrm{eff}}\\!\approx\\!1/2$) operators, $J$ is the isotropic
Heisenberg exchange, $\mathbf{D}\\!=\\!(0,0,D)$ defines the antisymmetric
Dzyaloshinskii-Moriya (DM) coupling, $\bar{\bar{\bm{\Gamma}}}$ is a symmetric
traceless second-rank tensor describing the symmetric anisotropy and due to
the staggered rotation of the IrO6 octahedra the $\bar{\bar{\mathbf{g}}}$
tensor splits for each of the two sites into uniform and staggered components
$\bar{\bar{\mathbf{g}}}_{k}^{\mathrm{u}}\pm\bar{\bar{\mathbf{g}}}_{k}^{\mathrm{s}}$
(see, e.g., Refs. [Oshikawa and Affleck, 1997; Hill, 1972]). This effective
spin Hamiltonian is of direct relevance to the interpretation of the ESR data.

ESR measurements. For a single crystal of Sr2IrO4 we observe antiferromagnetic
resonance (AFR) modes in the sub-THz frequency domain Bahr  _et al._ (2014) as
displayed in Fig. 2. There are two modes if $\mathbf{h}\|z$: a gapless
Goldstone mode $\nu_{\|1}\\!=\\!0$ and a gapped excitation

$\displaystyle\nu_{\|2}^{2}=\Delta^{2}+{2g^{2}_{\parallel}h^{2}_{\|}\tilde{J}}/({2(J+\Gamma_{\\!zz})+\tilde{J}}),$
(2)

where $\Delta=\sqrt{2\tilde{J}(\tilde{J}-2(J+\Gamma_{\\!zz}))}$,
$\tilde{J}=\sqrt{4D^{2}+(2J-\Gamma_{\\!zz})^{2}}$ and $\Gamma_{\\!zz}$ couples
the ${\tilde{S}}_{i}^{z}$ and ${\tilde{S}}_{j}^{z}$ components (along the $c$
axis, perpendicular to the $ab$-plane IrO2 layers Crawford  _et al._ (1994))
in the third term of Eq. (1) (see Methods for details). Experimental results
are shown for $\nu_{\|2}$ in Fig. 2(a). The data comprise at
$T\\!\ll\\!T_{\mathrm{N}}\\!=\\!240$ K a group of overlapping resonances [Fig.
2(a), inset], possibly due to some distribution of internal fields in the
sample. Though revealing some scatter, the $\nu_{\|2}^{2}$ AFR data follow
approximately a parabolic dependence on $h$ and, most importantly, they lie
substantially above the curve corresponding to the free-electron Landé factor
$g_{\mathrm{e}}\\!=\\!2$ [dashed line in Fig. 2(a)]. The experimental
dependence $\nu_{\|2}^{2}(h^{2})$ can be reasonably well modeled with
$g_{\parallel}$ values of 2.3–2.45. The solid line in Fig. 2(a) is obtained by
using $g_{\parallel}\\!=\\!2.31$, as derived from quantum chemistry
calculations that will be discussed later on.

Figure 1:  TM $\bm{t_{\mathrm{2g}}}$ splittings for tetragonal distortions of
the oxygen octahedron sans SOC. (a) $z$-axis compression of the octahedron
corresponds to a tetragonal splitting $\delta\\!<\\!0$, causes an orbital
singlet to be lowest in energy and the $g$ factors to be ordered as
$g_{\perp}\\!<\\!2\\!<\\!g_{\parallel}$. (b) Elongation of the octahedron
($\delta>0$) causes an orbital doublet to be lowest in energy and the $g$
factors to be ordered as $g_{\parallel}\\!<\\!2\\!<\\!g_{\perp}$. Purple
dashed lines indicate the conventional zero level used to define the sign of
$\delta$.

Sizable deviations to values larger than 2 of $g_{\parallel}$ is clear
indication of the presence of low-symmetry, noncubic crystal fields. In the
simplest approximation, i.e., restricting ourselves to the Ir4+
$t_{\mathrm{2g}}^{5}$ manifold, the anisotropic $g$ factors in axial noncubic
environment can be expressed up to the sign as Abragam and Bleaney (1970)
$g_{\parallel}=g_{c}=(2+2k)\cos^{2}\\!\alpha-2\sin^{2}\\!\alpha$ and
$g_{\perp}=g_{a}=g_{b}=2\sin^{2}\alpha+2\sqrt{2}k\cos\alpha\sin\alpha,$ where
$k$ is a covalency reduction factor,
$\alpha\\!=\\!(1/2)\arctan[2\sqrt{2}\lambda/(\lambda\\!-\\!2\delta)]$
parameterizes the deviation from octahedral symmetry, $\lambda$ is the SOC
constant and $\delta$ the Ir $t_{\mathrm{2g}}$ splitting. A plot for the
dependence of the diagonal $g$ factors on the distortion parameter $\alpha$ is
shown in the inset to Fig. 2(b). For simplicity, $k\\!=\\!1$ is for the moment
assumed but smaller values of $k$ do not bring qualitative changes. In cubic
symmetry $\delta\\!=\\!0$, $\alpha_{\mathrm{cub}}\\!=\\!35.26^{\circ}$ and the
$g$ matrix is isotropic with $g_{\parallel}\\!=\\!g_{\perp}\\!=\\!2$.
According to standard textbooks on ligand-field theoryBallhausen (1962), an
elongation of the out-of-plane Ir-O bond induces a positive tetragonal
splitting of the Ir $t_{\mathrm{2g}}$ levels, with $\delta\\!>\\!0$ and
$\alpha\\!>\\!\alpha_{\mathrm{cub}}$, whereas a bond compression yields
$\delta\\!<\\!0$ and $\alpha\\!<\\!\alpha_{\mathrm{cub}}$ (see Fig. 1). As in
Sr2IrO4 the IrO6 octahedra are substantially elongated in the $z$ direction
Crawford  _et al._ (1994), the $g$ factors are expected to correspond to the
case of positive splitting $\alpha\\!>\\!\alpha_{\mathrm{cub}}$, see the area
to the right of the crossing point shown in the inset to Fig. 2(b). It follows
from the plot that $g_{\parallel}\\!<\\!2$, which obviously contradicts our
AFR data for $\mathbf{h}\\!\parallel\\!z$ [Fig. 2(a)]. The value
$g_{\parallel}\\!=\\!2.31$ used to draw the curve connecting the open circles
in Fig. 2(a) in fact corresponds to
$\alpha\\!=\\!32.05^{\circ}\\!<\\!\alpha_{\mathrm{cub}}$ [see the inset in
Fig. 2(b)] and indicates that, despite the positive $c$-axis tetragonal
distortion, a counterintuitive negative tetragonal splitting of the $5d$
$t_{\mathrm{2g}}$ levels is present in Sr2IrO4.

It should be noted that there is no Goldstone mode for finite in-plane
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

$\nu^{2}_{\bot
1}=m^{2}g^{2}_{\bot}h^{2}_{\bot}+mg_{\bot}h_{\bot}(-4J-\Gamma_{\\!zz}+mJ^{\prime})$
(3)

and

$\displaystyle\nu^{2}_{\bot 2}$
$\displaystyle=m^{2}g^{2}_{\bot}h^{2}_{\bot}+mg_{\bot}h_{\bot}(-4J+5\Gamma_{\\!zz}+3mJ^{\prime})$
(4)
$\displaystyle+(-4J+2\Gamma_{\\!zz}+2mJ^{\prime})(3\Gamma_{\\!zz}+mJ^{\prime})\;,$

where $J^{\prime}\\!=\\!-4D\sqrt{1-m^{2}}+2m(2J-\Gamma_{\\!zz})$ and $m$ is
the in-plane ferromagnetic component of the effective moments. To first order
in magnetic field, $m$ can be expressed as

$\displaystyle
m=-\sqrt{\frac{\tilde{J}-2J+\Gamma_{\\!zz}}{2\tilde{J}}}-\frac{g_{\bot}h_{\bot}}{2\sqrt{6D^{2}+(2J-\Gamma_{\\!zz})^{2}}}\;,$
(5)

which holds for weak fields where $m$ remains small. The first term
corresponds to the zero-field canting, arising from the DM interaction, while
the second term shows how this canting evolves with increasing $h_{\bot}$.
Plots based on Eqs. (3,4) and the quantum chemically derived interaction
parameters (see below) are displayed in Fig. 2(b) together with ESR data.

Figure 2:  ESR data for Sr2IrO4. (a) Out-of-plane magnetic-field geometries;
the insert shows a representative AFR spectrum. (b) In-plane magnetic fields;
the insert demonstrates the $g$-factor anisotropy as function of the
tetragonal distortion parameter $\alpha$ (see text). Symbols denote
experimental data points — solid lines are theoretical curves using Eqs.
(2,3,4) and the quantum chemically computed $g$ factors
$g_{\parallel}\\!=\\!2.31$, $g_{\perp}\\!=\\!1.76$ (see Table 1); dashed lines
are calculated assuming the isotropic $g$ factors.

Quantum chemistry calculations of $\bm{g}$ factors. Results of ab initio
quantum chemistry calculations for the $g$ factors in Sr2IrO4 and in the
structurally related material Ba2IrO4 are listed in Table 1. Our computational
scheme follows the prescription of Bolvin Bolvin (2006) and Vancoillie et al.
Vancoillie  _et al._ (2007). It maps the matrix elements (ME’s) of the ab
initio Zeeman Hamiltonian $\mathcal{H}_{\rm
Z}=-\bm{\mu}\cdot\mathbf{h}=\mu_{\rm
B}(\mathbf{L}+g_{\mathrm{e}}\mathbf{S})\cdot\mathbf{h}$ onto the ME’s of the
effective pseudospin Hamiltonian $\mathcal{H}_{\rm S}=\mu_{\rm
B}\mathbf{h}\cdot\bar{\bar{\mathbf{g}}}\cdot\tilde{\mathbf{S}}$, where
$\bm{\mu}$, $\mathbf{L}$ and $\mathbf{S}$ are magnetic moment, angular-
momentum and spin operators, respectively. The spin-orbit GS wave functions
are computed either at the complete-active-space self-consistent-field
(CASSCF) or multireference configuration-interaction (MRCI) level of theory
Helgaker  _et al._ (2000), as described in Ref. [Berning  _et al._ , 2000] and
using the MOLPRO quantum chemistry package Werner  _et al._ . All necessary
angular-momentum ME’s are calculated as well with MOLPRO (see Methods). In a
first set of calculations, only the three $t_{\mathrm{2g}}$ orbitals at a
given Ir site and five electrons were considered in the active space. The
self-consistent-field optimization was carried out for the corresponding
${}^{2}T_{\mathrm{2g}}$($t_{\mathrm{2g}}^{5}$) state. We use here the more
convenient notations associated to $O_{h}$ symmetry, although the calculations
were performed for the actual experimental geometry, with point-group symmetry
lower than octahedral. Inclusion of SOC yields in this case a set of three
Kramers doublets (KD’s), see Table 1.

Table 1:  $\bm{g}$ factors for Sr2IrO4 and Ba2IrO4. Results of many-body quantum chemistry calculations are shown. The left column displays the electron configurations entering the spin-orbit treatment. Only the high-spin sextet state is considered out of the $t_{\mathrm{2g}}^{3}e_{\mathrm{g}}^{2}$ manifold. Final results are indicated in bold face.  States considered | CASSCF | MRCI
---|---|---
g⟂ | g∥ | g⟂ | g∥
Sr2IrO4 ($\delta\\!=-155$ meV) : |  |  |  |
$t_{\mathrm{2g}}^{5}$ (3 KD’s) | 1.67 | 2.25 | 1.60 | 2.35
$t_{\mathrm{2g}}^{5}$, $t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$, $t_{\mathrm{2g}}^{3}e_{\mathrm{g}}^{2}$ (27 KD’s) | 1.81 | 2.27 | 1.76 | 2.31
Ba2IrO4 ($\delta\\!=\\!65$ meV) : |  |  |  |
$t_{\mathrm{2g}}^{5}$ (3 KD’s) | 2.00 | 1.61 | 2.01 | 1.60
$t_{\mathrm{2g}}^{5}$, $t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$, $t_{\mathrm{2g}}^{3}e_{\mathrm{g}}^{2}$ (27 KD’s) | 2.09 | 1.77 | 2.10 | 1.76

Subsequently we performed calculations with larger active spaces, including
also the Ir $e_{\mathrm{g}}$ orbitals. One $t_{\mathrm{2g}}^{5}$
(${}^{2}T_{\mathrm{2g}}$) plus four $t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$
(${}^{2}\\!A_{\mathrm{2g}}$, ${}^{2}T_{\mathrm{1g}}$,
${}^{2}\\!E_{\mathrm{g}}$, and ${}^{2}T_{\mathrm{2g}}$) spin doublets, two
spin quartets
[${}^{4}T_{\mathrm{1g}}$($t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$) and
${}^{4}T_{\mathrm{2g}}$($t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$)] and one spin
sextet [${}^{6}\\!A_{\mathrm{1g}}$($t_{\mathrm{2g}}^{3}e_{\mathrm{g}}^{2}$)]
entered here the spin-orbit treatment. The orbitals were optimized for an
average of all these terms.

The effect of enlarging the active space to include
$t_{\mathrm{2g}}^{4}e_{\mathrm{g}}^{1}$ terms in the reference wave function
is in the range of $10\%$, in line with earlier semi-empirical estimates for
$4d^{5}$ and $5d^{5}$ systems Thornley (1968); Hill (1972); Andlauer  _et al._
(1976); Stebler  _et al._ (1984). Most importantly, the calculations yield a
negative tetragonal splitting of the Ir $t_{\mathrm{2g}}$ levels in Sr2IrO4,
$\delta\\!=-155$ meV by MRCI, and positive $t_{\mathrm{2g}}$ splitting in
Ba2IrO4 (see Table 1 and Methods). Similar signs, negative in Sr2IrO4 and
positive in Ba2IrO4, but much larger magnitudes ($\approx\\!0.7$ eV) are found
for the computed Ir $e_{\mathrm{g}}$ splittings (not shown in Table 1).

Taken together, the ESR and quantum chemistry results unequivocally point at
an anomalous order of the split Ir 5$d$ levels in Sr2IrO4, related to the
important role of the extended crystalline surroundings in generating low-
symmetry fields that compete with “local” distortions of the ligand cage.
Similar effects were found by ab initio calculations on the 214 layered
rhodate Sr2RhO4 Katukuri  _et al._ (2014a) and the 227 pyrochlore iridates
Hozoi  _et al._ (2014). In contrast, in Ba2IrO4, the stretch of the apical
Ir-O bonds is strong enough Okabe  _et al._ (2011) to overcome the longer-
range electrostatics, turning the tetragonal $t_{\mathrm{2g}}$ positive again,
as discussed in more detail in the following. Consequently, the structure of
the $\bar{\bar{\mathbf{g}}}$ tensor in Ba2IrO4 is qualitatively different,
with $g_{\perp}\\!>\\!2$ and $g_{\parallel}\\!<\\!2$ (see Table 1), the
ordering that one normally expects and encounters for elongated octahedra
Abragam and Bleaney (1970).

Exchange couplings from quantum chemistry.

Figure 3:  Planar IrO2 network in Sr2IrO4. (a) Coordination of the Ir site.
Dashed lines show the boundaries of the crystallographic unit cell within a
given IrO2 layer. (b) The point-group symmetry of the [Ir2O11] block is
$C_{\mathrm{2v}}$; associated symmetry elements are indicated in the figure.

To obtain ab initio quantum chemistry values for the inter-site effective
magnetic couplings in Sr2IrO4 [see Eq. (1)], we carried out additional
calculations on larger clusters that incorporate two $5d^{5}$ sites. The two-
octahedra cluster has $C_{\mathrm{2v}}$ symmetry (see Fig. 3), which implies a
diagonal form for $\bar{\bar{\bf\Gamma}}$ and
$\bar{\bar{\mathbf{g}}}_{i}^{\mathrm{u}}$ in equation (1) (see Methods). By
one-to-one correspondence between the ME’s of the ab initio Hamiltonian

$\mathcal{H}^{i,j}=\mathcal{H}_{\rm SR}^{i,j}+\mathcal{H}_{\rm
SO}^{i,j}+\mathcal{H}_{\rm Z}^{i,j}$ (6)

and the ME’s of the effective spin Hamiltonian (1) in the basis of the lowest
four spin-orbit 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), $\mathcal{H}_{\rm SR}^{i,j}$
is the scalar-relativistic Born-Oppenheimer Hamiltonian, $\mathcal{H}_{\rm
SO}^{i,j}$ describes spin-orbit interactions Berning  _et al._ (2000) and
$\mathcal{H}_{\rm Z}^{i,j}$ is the two-site Zeeman Hamiltonian.

Diagonalization of the spin Hamiltonian (1) provides the expected singlet
$|\tilde{s}\rangle$ and three (split) triplet components $|t_{x}\rangle$,
$|t_{y}\rangle$ and $|\tilde{t}_{z}\rangle$. Due to the DM interaction,
$|\tilde{s}\rangle$ and $|\tilde{t}_{z}\rangle$ are admixtures of “pure”
$|0,0\rangle$ and $|1,0\rangle$ spin functions. Our mapping procedure yields
$J\\!\approx\\!48$ meV, somewhat lower than $J$ values of 55–60 meV derived
from experiment Kim  _et al._ (2012a); Subramanian  _et al._ (1994), and a
ratio between the DM and Heisenberg couplings $D/J\\!=\\!0.25$, in agreement
with estimates based on effective superexchange models Jackeli and Khaliullin
(2009); Kim  _et al._ (2012b); Perkins  _et al._ (2014) and large enough to
explain the nearly rigid rotation of magnetic moments that is observed when
the IrO6 octahedra revolve Boseggia  _et al._ (2013b); Ye  _et al._ (2013). Ab
initio results for the NN anisotropic couplings $\bar{\bar{\bf\Gamma}}$, also
relevant for a detailed understanding of the magnetic properties of Sr2IrO4,
are shown as well in Table 2. In our convention the $x$ axis is taken along
the Ir-Ir link, i.e., it coincides with the $\langle 110\rangle$
crystallographic direction Crawford  _et al._ (1994), and $z\\!\parallel\\!c$.
We obtain $\Gamma_{\\!xx}\\!\approx\\!\Gamma_{\\!zz}$, which then allows to
recast the Heisenberg and symmetric anisotropic terms in (1) as
$(J-\Gamma_{\\!yy}/2)\,\tilde{\mathbf{S}}_{i}\cdot\tilde{\mathbf{S}}_{j}+(3\Gamma_{\\!yy}/2)\,\tilde{S}_{i}^{y}\,\tilde{S}_{j}^{y}$,
with $\Gamma_{\\!xx}\\!=\\!\Gamma_{\\!zz}\\!=-\Gamma_{\\!yy}/2$. Equally
interesting, for no rotation of the IrO6 octahedra and straight Ir-O-Ir bonds
in Ba2IrO4, it is $\Gamma_{\\!yy}$ and $\Gamma_{\\!zz}$ which are
approximately the same, providing a realization of the compass-Heisenberg
model Katukuri  _et al._ (2014b); Perkins  _et al._ (2014) since the DM
coupling is by symmetry 0 in that case.

Table 2:  Nearest-neighbonr magnetic couplings in Sr2IrO4. Results of spin-orbit MRCI calculations on two-octahedra clusters are displayed (meV). $\Gamma_{\\!xx}\\!+\\!\Gamma_{\\!yy}\\!+\\!\Gamma_{\\!zz}\\!=\\!0$ since $\bar{\bar{\bf\Gamma}}$ is traceless.  $J$ | $D$ | $\Gamma_{\\!xx}$ | $\Gamma_{\\!yy}$ | $\Gamma_{\\!zz}$
---|---|---|---|---
47.8 |  $\pm$11.9 | 0.42 | –0.84 | 0.42

The two-site magnetic Hamiltonian (1) features in-plane symmetric-anisotropy
couplings $\Gamma_{\\!xx}$ and $\Gamma_{\\!yy}$ which were not considered in
previous studies Jackeli and Khaliullin (2009); Bahr  _et al._ (2014). In the
presence of two-sublattice order, terms containing these couplings cancel each
other in the mean-field energy but they are in general relevant for pseudospin
fluctuations and excitations. Using spin-wave 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 zero-field gap, in particular, we used $J$, $D$ and $g$-factor
values as listed in Table 2 and a somewhat larger $\Gamma_{\\!zz}$ parameter
of 0.98 meV. To leading order, the dependence of $\nu^{2}_{\bot 1}$ and
$\nu^{2}_{\bot 2}$ on $h$ is linear, see Eqs. (3,4), and the slope is
proportional to $m$. At low fields ($\leq\\!1$ T), $m$ can be actually
replaced with its field independent value Bahr  _et al._ (2014). Using the
MRCI coupling constants, the first term in equation (5) then yields a moment
$m\\!\approx\\!0.12\mu_{\mathrm{B}}$, in good agreement with the experiment
Kim  _et al._ (2009); Cao  _et al._ (1998).

Discussion
The exact $d$-level order is of fundamental importance in TM oxides, dictating
for instance the symmetry of the quasiparticle states in photoemission Kim
_et al._ (2014a, 2006); Katukuri  _et al._ (2014a), the sign of the single-ion
anisotropy Bogdanov  _et al._ (2013) and the nature of the magnetic ordering
Bogdanov  _et al._ (2013); Fang  _et al._ (2004). In 214 iridates
specifically, it determines the various isotropic as well as anisotropic
contributions to the magnetic exchange couplings Jackeli and Khaliullin
(2009); Kim  _et al._ (2012b); Perkins  _et al._ (2014); Katukuri  _et al._
(2014b), the evolution of those magnetic interactions with strain Lupascu  _et
al._ (2014) and/or pressure Haskel  _et al._ (2012) and most likely the nature
of the intriguing transition to a nonmagnetic phase in Sr2IrO4 under high
pressure Haskel  _et al._ (2012). Having established that in Sr2IrO4 the $d$
levels are inverted and that in the closely related Ba2IrO4 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 IrO6 octahedron. As a simple numerical experiment that
preserves charge neutrality of the A2IrO4 system, we assigned the 4 NN iridium
sites (in-plane, see Fig. 4) the charge $Q_{\mathrm{TM}}\\!-\\!2\Delta q$ and
the 8 closest A-site cations (out of plane) the valence
$Q_{\mathrm{A}}\\!+\\!\Delta q$. In a fully ionic picture, $Q_{\mathrm{TM}}$
and $Q_{\mathrm{A}}$ are $4+$ and $2+$, respectively. However, since in our
calculations the NN TM and A sites are not modeled as just formal point
charges (see Methods) the actual valence states depart from their formal
values, with larger “deviations” for $Q_{\mathrm{TM}}$. The way we introduce
$\Delta q$ in the computations is therefore by appropriately modifying the
nuclear charge at the respective site. For variable $\Delta q$, this
interpolates linearly between nearby surroundings corresponding to $5d$ 214
layered perovskites (with $\Delta q\\!=\\!0$ and TM4+, A2+ formal valence
states) and their cuprate 214 equivalents (with $\Delta q\\!=\\!1$, TM2+/A3+
formal ionic charges and “normal” order of the TM $t_{\mathrm{2g}}$ and
$e_{\mathrm{g}}$ levels Hozoi  _et al._ (2011)).

Figure 4:  Effect of interlayer charge imbalance in A2IrO4 iridates. (a) The
nearby surroundings of TM sites in A2TMO4 layered perovskites. In test
calculations one can assign the adjacent (in-plane) TM ions the formal charge
$Q_{\mathrm{TM}}\\!-\\!2\Delta q$, which is compensated by assigning the NN A
sites the charge $Q_{\mathrm{A}}\\!+\\!\Delta q$. (b) Tetragonal crystal-field
energy splitting between $t_{\mathrm{2g}}$ orbitals ($\delta$) as a function
of the charge redistribution $\Delta q$ for Sr2IrO4 and Ba2IrO4.

As is illustrated in Fig. 4(a), increasing $\Delta q$ amounts to moving
positive charge from the IrO2 plane to the adjacent A-O layers. The
calculations show that upon moving charge in such a manner, the Ir
$t_{\mathrm{2g}}$ splitting $\delta$ increases, see Fig. 4(b). In other words,
this redistribution of charge counteracts the level inversion in Sr2IrO4 and
further increases the already positive $\delta$ in Ba2IrO4. In Sr2IrO4 the
cubic-like $j_{\mathrm{eff}}\\!=\\!1/2$ limit occurs for $\Delta
q\\!=\\!0.22$. This effect can easily be understood: placing more positive
charge out of the IrO2 plane stabilizes the out-of-plane $t_{\mathrm{2g}}$
orbitals, corresponding to the ($yz$,$zx$) orbital doublet, and thus enhances
$\delta$. One can also do the opposite and drive $\Delta q$ negative. In this
case more positive charge piles up in the IrO2 plane, which one expects to
lower the energy of the $xy$ orbital singlet, thus enhancing the level
inversion in Sr2IrO4. This is indeed what happens, see Fig. 4b. What is more,
driving $\Delta q$ negative even causes a level inversion in Ba2IrO4, when
$\Delta q\\!\lesssim\\!-0.25$. It is interesting to note that the slope of the
$\delta$ versus $\Delta q$ lines in Sr2IrO4 is much steeper than in Ba2IrO4,
which is caused by the significantly smaller Ir–Ir distances in Sr2IrO4.

From these test calculations it is clear that low-symmetry crystal fields
associated to neighbors beyond the first ligand coordination shell – in
particular, the highly charged Ir4+ NN’s – counteract the local tetragonal
crystal field that is caused by the elongation of the IrO6 octahedra in both
Sr2IrO4 and Ba2IrO4. In the case of Ba2IrO4 the local distortion is still
strong enough to overcome these longer-range effects but in Sr2IrO4, with a
slightly smaller tetragonal distortion, the longer-range electrostatics wins,
causing the observed level inversion.

While the role of the high ionic charge of in-plane ions has been earlier
invoked in the tetravalent Ru oxide compound Ca2RuO4 Fang  _et al._ (2004) and
in mixed-valence manganites Wu  _et al._ (2011), we here explicitly prove it
by combined ESR measurements and many-body ab initio calculations on
structurally and chemically simpler systems in which additional complications
arising from further distortions Fang  _et al._ (2004); Braden  _et al._
(1998) or the presence of multiple TM valence states Wu  _et al._ (2011) are
excluded. A reversed order of the Ir $t_{\mathrm{2g}}$ levels in Sr2IrO4 has
been also indirectly implied by fits of x-ray absorption Haskel  _et al._
(2012) and x-ray magnetic scattering Fujiyama  _et al._ (2014); Boseggia  _et
al._ (2013b) 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 many-body 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. Haskel  _et al._ , 2012, and not the
$t_{\mathrm{2g}}$-$e_{\mathrm{g}}$ orbital hybridization invoked in Ref.
Fujiyama  _et al._ , 2014, but basic interlayer electrostatics.

We have, in sum, provided an integrated picture on the $d$-level structure and
magnetic anisotropies in Sr2IrO4, a prototype spin-orbit driven magnetic
insulator. Both the single-site $\bar{\bar{\mathbf{g}}}$ tensor and intersite
effective exchange interactions are analyzed in detail. To access the latter,
we build on an earlier computational scheme for deriving intersite matrix
elements in mixed-valence spin-orbit coupled systems Bolvin (2007). While the
ratio $D$/$J$ of the antisymmetric Dzyaloshinskii-Moriya and isotropic
Heisenberg couplings is remarkably large in Sr2IrO4 and concurs with an in-
plane rotation pattern of the Ir magnetic moments that follows nearly rigidly
the staggered rotation of the IrO6 octahedra Boseggia  _et al._ (2013b); Ye
_et al._ (2013), the most prominent symmetric anisotropic terms are according
to the quantum chemistry data in-plane, perpendicular to the Ir-Ir links. The
structure of the $\bar{\bar{\mathbf{g}}}$ tensor, as measured by ESR and
computed with first-principles electronic-structure methods, is such that
$g_{\parallel}\\!=\\!g_{c}\\!>\\!g_{\perp}\\!=\\!g_{a}\\!=\\!g_{b}$ and
distinctly indicates a negative tetragonal-like splitting of the Ir
$t_{\mathrm{2g}}$ levels, in spite of sizable positive tetragonal distortions
in Sr2IrO4. We further observe that a much stronger tetragonal distortion in
Ba2IrO4 renders the tetragonal $d$-level splitting positive and
$g_{\parallel}\\!<\\!g_{\perp}$. The interesting situation arises that
nevertheless the magnitude of the Ir $t_{\mathrm{2g}}$ splitting is largest in
Sr2IrO4. The $d$-level inversion in Sr2IrO4 and the surprisingly small
splitting in Ba2IrO4 have to do with the way the positive ionic charge is
distributed between adjacent Ir4+O2 and A2+O layers, having in contrast to the
214 cuprate superconductors, for example, more positive charge in the TM-O
planes. This almost compensates the “local” tetragonal field arising from the
$z$-axis elongation of the IrO6 octahedra in Ba2IrO4 and overcompensate it in
Sr2IrO4.

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 Lupascu  _et al._ (2014) and
pressure Haskel  _et al._ (2012) experiments in square-lattice iridates, for
example, in connection to the spin-flop transition earlier predicted in
Sr2IrO4 Jackeli and Khaliullin (2009); Perkins  _et al._ (2014). It also opens
up the perspective of manipulating this way the $d$-level ordering in oxide
heterostructures with highly charged, trivalent and tetravalent, species.
Compounds with tetravalent species within the TM-O2 layers, in particular,
given the nonpolar character of the quasi 2D sheets, provide ideal playgrounds
to explore the mechanism of $d$-level ordering pointed out here since that
will not be hindered by “interface” charge redistribution and structural
reconstruction occuring in polar heterostructures from polar discontinuities
Ohtomo and Hwang (2004); Koster  _et al._ (2008). A recent experimental
realization of such mixed, tetravalent/divalent TM-oxide interfaces is for
example the SrRuO3/NiO interface Ning  _et al._ (2013), one system that
requires in this respect closer theoretical examination.

Methods
Single-site magnetic properties. The $g$ factors were obtained by computations
on clusters which contain one central IrO6 octahedron, the four NN IrO6
octahedra and the nearby ten Sr ions. The solid-state surroundings were
modeled as a large array of point charges fitted to reproduce the crystal
Madelung field in the cluster region. To obtain a clear picture on crystal-
field effects and spin-orbit interactions at the central Ir site, we cut off
the magnetic couplings with the adjacent Ir ions by replacing the tetravalent
open-shell $d^{5}$ NN’s with tetravalent closed-shell Pt4+
$t_{\mathrm{2g}}^{6}$ species. This is a usual procedure in quantum chemistry
studies on TM systems, see, e.g., Refs. de Graaf  _et al._ , 1999; Hozoi  _et
al._ , 2002, 2011; Domingo  _et al._ , 2012; Bogdanov  _et al._ , 2013;
Maurice  _et al._ , 2013. We used energy-consistent relativistic
pseudopotentials and valence basis sets of quadruple-zeta quality supplemented
with $f$ polarization functions for the central Ir ion Figgen  _et al._ (2009)
and all-electron triple-zeta basis sets for the six adjacent ligands Dunning
(1989). For the TM NN’s, we applied energy-consistent relativistic
pseudopotentials and triple-zeta basis functions Figgen  _et al._ (2009) along
with minimal atomic-natural-orbital basis sets Pierloot  _et al._ (1995) for
the O’s coordinating those TM sites but not shared with the central
octahedron. The Sr and Ba species were modeled by divalent total-ion effective
potentials supplemented with a single $s$ function Fuentealba  _et al._
(1985). All O $2p$ and metal $t_{\mathrm{2g}}$ 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. Werner and Knowles, 1988; Knowles and Werner, 1992),
which is referred to as MRCISD. To separate the metal $5d$ and O $2p$ valence
orbitals into different groups, i.e., central-octahedron and adjacent-
octahedra orbitals, we used the Pipek-Mezey localization module Pipek and
Mezey (1989) available in MOLPRO. The computations with hypothetical
$(Q_{\mathrm{TM}}\\!-\\!2\Delta q)$ and $(Q_{\mathrm{A}}\\!+\\!\Delta q)$
ionic charges at the TM and Sr/Ba sites next to the reference Ir ion were
carried out as frozen-orbital multideterminant calculations (also referred to
as CASCI) with three Ir $t_{\mathrm{2g}}$ and five electrons in the active
space and orbitals optimized for $\Delta q\\!=\\!0$.

Table 3:  Ir $\bm{t_{\mathrm{2g}}}$ splittings and $\bm{g}$ factors for Sr2IrO4 and Ba2IrO4. Results as obtained with the ORCA program Neese (2012) are shown. Only the ${}^{2}T_{\mathrm{2g}}$($t_{\mathrm{2g}}^{5}$) states were included in the CASSCF optimization and in the spin-orbit treatment.  |  $\delta_{t_{\mathrm{2g}}}$ (meV) | $g_{\perp}$ | $g_{\parallel}$
---|---|---|---
Sr2IrO4 : |  |  |
CASSCF | –127 | 1.66 | 2.23
NEVPT2 | –199 | 1.55 | 2.41
Ba2IrO4 : |  |  |
CASSCF | 30 | 1.93 | 1.74
NEVPT2 | 70 | 2.01 | 1.58
Table 4:  Matrix elements of the ab initio spin-orbit Hamiltonian. The latter is described by expression (6). Results of spin-orbit MRCI calculations are shown (meV). The two-site singlet and (split) triplet states are labeled $|\tilde{s}\rangle$ and {$|t_{x}\rangle$, $|t_{y}\rangle$, $|\tilde{t}_{z}\rangle$}, respectively. Due to the antisymmetric exchange, $|\tilde{s}\rangle$ and $|\tilde{t}_{z}\rangle$ are admixtures of “pure” $|0,0\rangle$ and $|1,0\rangle$ spin functions.  $\mathcal{H}^{i,j}$ | $|\tilde{s}\rangle$ | $|t_{x}\rangle$ | $|t_{y}\rangle$ | $|\tilde{t}_{z}\rangle$
---|---|---|---|---
$\langle\tilde{s}|$ | $0$ | $0.2308i\,\mu_{\mathrm{B}}h_{y}$ | $-0.1768i\,\mu_{\mathrm{B}}h_{x}$ | $0$
$\langle t_{x}|$ | $-0.2308i\,\mu_{\mathrm{B}}h_{y}$ | $48.3328$ | $2.3083i\,\mu_{\mathrm{B}}h_{z}$ | $-1.6854i\,\mu_{\mathrm{B}}h_{y}$
$\langle t_{y}|$ | $0.1768i\,\mu_{\mathrm{B}}h_{x}$ | $-2.3083i\,\mu_{\mathrm{B}}h_{z}$ | $48.9626$ | $1.6266i\,\mu_{\mathrm{B}}h_{x}$
$\langle\tilde{t}_{z}|$ | $0$ | $1.6854i\,\mu_{\mathrm{B}}h_{y}$ | $-1.6266i\,\mu_{\mathrm{B}}h_{x}$ | $49.0630$
Table 5:  Matrix elements of the effective spin Hamiltonian. The explicit form of the latter is given by (1); for additional notations we use here, see Table 6.  $\mathcal{H}^{i,j}_{\mathrm{eff}}$ | $|\tilde{s}\rangle$ | $|t_{x}\rangle$ | $|t_{y}\rangle$ | $|\tilde{t}_{z}\rangle$
---|---|---|---|---
$\langle\tilde{s}|$ | 0 | $ih_{y}\frac{g_{yy}D-g_{xy}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}}$ | $-ih_{x}\frac{g_{xx}D+g_{xy}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}}$ | 0
$\langle t_{x}|$ | $-ih_{y}\frac{(g_{yy}D-g_{xy}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}}$ | $\frac{1}{2}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}-\Gamma_{\\!x\\!-\\!y}+\Gamma_{z}\right)$ | $-ig_{zz}h_{z}$ | $-ih_{y}\frac{g_{yy}D+g_{xy}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}}$
$\langle t_{y}|$ | $ih_{x}\frac{g_{xx}D+g_{xy}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}\right)}}$ | $ig_{zz}h_{z}$ | $\frac{1}{2}\left({}^{\tilde{J}}\\!/_{2}+J_{\\!x\\!+\\!y}+\Gamma_{x\\!-\\!y}+\Gamma_{z}\right)$ | $ih_{x}\frac{g_{xx}D-g_{xy}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}}$
$\langle\tilde{t}_{z}|$ | 0 | $ih_{y}\frac{g_{yy}D+g_{xy}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}}$ | $-ih_{x}\frac{g_{xx}D-g_{xy}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}{\sqrt{\tilde{J}\left({}^{\tilde{J}}\\!/_{2}-J_{\\!x\\!+\\!y}\right)}}$ | ${}^{\tilde{J}}\\!/_{2}$

The spin-orbit treatment was performed according to the procedure described in
Ref. Berning  _et al._ , 2000. In a first step, the scalar relativistic
Hamiltonian is used to calculate correlated wave functions for a finite number
of low-lying states, either at the CASSCF or MRCI level. In a second step, the
spin-orbit 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 spin-
orbit coupled wave functions.

The $g$ factors were computed following the scheme proposed by Bolvin Bolvin
(2006) and Vancoillie et al. Vancoillie  _et al._ (2007) (for alternative
formulations, see, e.g., Ref. Chibotaru and Ungur, 2012). For the KD GS
$\left\\{\psi,\bar{\psi}\right\\}$, the Abragam-Bleaney tensor Gerloch and
McMeeking (1975); Abragam and Bleaney (1970) ${\bf G}={\bf g}{\bf g}^{T}$ can
be written in matrix form as

$\begin{split}G_{kl}&=2\displaystyle\sum\limits_{u,v=\psi,\bar{\psi}}\langle
u|\hat{L}_{k}+g_{\mathrm{e}}\hat{S}_{k}|v\rangle\langle
v|\hat{L}_{l}+g_{\mathrm{e}}\hat{S}_{l}|u\rangle\\\
&=\displaystyle\sum\limits_{m=x,y,z}(\Lambda_{km}+g_{\mathrm{e}}\Sigma_{km})(\Lambda_{lm}+g_{\mathrm{e}}\Sigma_{lm})\,,\end{split}$
(7)

where

$\displaystyle\Lambda_{kx}\\!=\\!2Re\left[\langle\bar{\psi}|\hat{L}_{k}|\psi\rangle\right]\,,$
$\displaystyle\Sigma_{kx}\\!=$
$\displaystyle\,2Re\left[\langle\bar{\psi}|\hat{S}_{k}|\psi\rangle\right]\,,$
(8)
$\displaystyle\Lambda_{ky}\\!=\\!2Im\left[\langle\bar{\psi}|\hat{L}_{k}|\psi\rangle\right]\,,$
$\displaystyle\Sigma_{ky}\\!=$
$\displaystyle\,2Im\left[\langle\bar{\psi}|\hat{S}_{k}|\psi\rangle\right]\,,$
$\displaystyle\Lambda_{kz}\\!=\\!2\langle\psi|\hat{L}_{k}|\psi\rangle\,,$
$\displaystyle\Sigma_{kz}\\!=$
$\displaystyle\,2\langle\psi|\hat{S}_{k}|\psi\rangle\,.$

The ME’s of $\hat{L}$ are here provided by MOLPRO while those of $\hat{S}$ are
derived using the conventional expressions for the generalized Pauli matrices
:

$\begin{split}(\hat{S}_{z})_{MM^{\prime}}&=M\delta_{MM^{\prime}}\;,\\\
(\hat{S}_{x})_{MM^{\prime}}&=\frac{1}{2}\sqrt{(S+M)(S-M+1)}\delta_{M-1,M^{\prime}}\\\
&+\frac{1}{2}\sqrt{(S-M)(S+M+1)}\delta_{M+1,M^{\prime}}\;,\\\
(\hat{S}_{y})_{MM^{\prime}}&=-\frac{i}{2}\sqrt{(S+M)(S-M+1)}\delta_{M-1,M^{\prime}}\\\
&+\frac{i}{2}\sqrt{(S-M)(S+M+1)}\delta_{M+1,M^{\prime}}.\end{split}$ (9)

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 cross-check the $g$-factor values computed with our subroutine, we further
performed $g$-factor calculations using the module available within the ORCA
quantum chemistry package Neese (2012). We applied all-electron DKH (Douglas-
Kroll-Hess) basis sets of triple-zeta quality for the TM ions Pantazis  _et
al._ (2008), triple-zeta basis functions for the ligands of the central
octahedron Dunning (1989) and double-zeta basis functions for additional O’s
at the NN octahedra Dunning (1989). Dynamical correlation effects were
accounted for by $N$-electron valence-state second-order perturbation theory
(NEVPT2) Angeli  _et al._ (2001); Schapiro  _et al._ (2013). CASSCF and NEVPT2
results are listed in Table 3, for both Sr2IrO4 and Ba2IrO4. It is seen that
the data in Tables 1 and 3 compare very well and indicate the same overall
trends.

Superexchange interactions in Sr2IrO4. NN magnetic coupling constants were
obtained for Sr2IrO4 by calculations on an embedded cluster that includes two
IrO6 octahedra as magnetically active units. As for the calculation of single-
site magnetic properties, to accurately describe the charge distribution in
the immediate neighborhood, the adjacent IrO6 octahedra (six) and the closest
Sr2+ ions (16) were also incorporated in the actual cluster. We used energy-
consistent relativistic pseudopotentials along with quadruple-zeta basis sets
for the valence shells of the two magnetically active Ir ions Figgen  _et al._
(2009), all-electron quintuple-zeta basis sets for the bridging ligand Dunning
(1989) and triple-zeta basis functions for the other O’s associated with the
two reference octahedra Dunning (1989). 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 Figgen  _et al._ (2009); Dunning (1989). Additional
ions defining the NN octahedra, the nearby Sr2+ species and the farther
crystalline surroundings were modeled as in the single-site study, see above.

Table 6:  Notations used for anisotropic exchange coupling parameters in Sr2IrO4. The definitions on the left-hand side are applied in Table 5. Other conventions presently employed in the literature are also shown for comparison.  | $J,\Gamma_{xx}\\!=\\!-\Gamma_{yy}\\!-\\!\Gamma_{zz}$ | $J_{x},J_{y},J_{z}$ |  $J,\delta\\!J_{xy},\delta\\!J_{z}$ ($x$ bond)
---|---|---|---
| (present study) |  (Ref. Perkins  _et al._ , 2014) |  (Refs. Jackeli and Khaliullin, 2009; Perkins  _et al._ , 2014; Kim  _et al._ , 2012b)
$J_{\\!x\\!+\\!y}$ | $J-{}^{\Gamma_{\\!zz}}\\!/_{2}$ | ${}^{\left(J_{x}+J_{y}\right)}\\!/_{2}$ | $J+{}^{\delta\\!J_{xy}}\\!/_{2}$
$\tilde{J}$ |  2$\sqrt{D^{2}+J^{2}_{\\!x\\!+\\!y}}$ |  2$\sqrt{D^{2}+J^{2}_{\\!x\\!+\\!y}}$ |  2$\sqrt{D^{2}+J^{2}_{\\!x\\!+\\!y}}$
$\Gamma_{x\\!-\\!y}$ | ${}^{\left(\Gamma_{xx}-\Gamma_{yy}\right)}\\!/_{2}$ | ${}^{\left(J_{x}-J_{y}\right)}\\!/_{2}$ | $\left\\{\begin{matrix}\delta\\!J_{xy}&{\mathrm{for}}~{}\langle t_{x}|\mathcal{H}|t_{x}\rangle\\\ 0&{\mathrm{for}}~{}\langle t_{y}|\mathcal{H}|t_{y}\rangle\end{matrix}\right.$
$\Gamma_{z}$ | ${}^{3}\\!/_{2}\Gamma_{zz}$ | $J_{z}-J_{\\!x\\!+\\!y}$ | $\delta\\!J_{z}$

For two adjacent magnetic sites, the
${{}^{2}T_{\mathrm{2g}}}(t_{\mathrm{2g}}^{5})\\!\otimes\\!{{}^{2}T_{\mathrm{2g}}}(t_{\mathrm{2g}}^{5})$
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 $\mathcal{H}_{\rm
SR}^{i,j}$. In the subsequent MRCI treatment, only the Ir $t_{\mathrm{2g}}$
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 5$d^{5}$ iridates Bogdanov  _et al._
(2012); Katukuri  _et al._ (2014c).

Diagonalization of the Hamiltonian $\mathcal{H}_{\rm
SR}^{i,j}+\mathcal{H}_{\rm SO}^{i,j}$ in the basis of the lowest nine singlet
and nine triplet states provides a total of 36 spin-orbit-coupled $|j_{\rm
eff}^{i},j_{\rm eff}^{j}\rangle$ eigenfunctions, namely, four
$|{}^{1}\\!/_{2},{}^{1}\\!/_{2}\rangle$, eight
$|{}^{1}\\!/_{2},{}^{3}\\!/_{2}\rangle$, eight
$|{}^{3}\\!/_{2},{}^{1}\\!/_{2}\rangle$ and sixteen
$|{}^{3}\\!/_{2},{}^{3}\\!/_{2}\rangle$ states. In the simplest picture, the
lowest four $|{}^{1}\\!/_{2},{}^{1}\\!/_{2}\rangle$ roots imply either singlet
or triplet coupling of the spin-orbit $j_{\rm eff}\\!=\\!1/2$ (or, more
generally, pseudospin $\tilde{S}\\!=\\!1/2$) on-site objects and are separated
from higher-lying states by a gap of $\gtrsim\\!0.5$ eV, much larger than the
strength of the intersite exchange. It is this set of lowest four spin-orbit
MRCI roots that we map onto the eigenstates of the effective two-site
(pseudo)spin Hamiltonian (1). The Zeeman interaction shows up on the quantum
chemistry side as $\mathcal{H}_{\rm Z}^{i,j}\\!=\\!\mu_{\rm
B}(\mathbf{\hat{L}}^{i,j}+g_{e}\mathbf{\hat{S}}^{i,j})\cdot\mathbf{h}$, where
$\mathbf{\hat{L}}^{i,j}$ and $\mathbf{\hat{S}}^{i,j}$ are transformed to the
spin-orbit-coupled basis using the spin-orbit wave functions as unitary
transformation matrix. ME’s of the ab initio model Hamiltonian
$\mathcal{H}_{\rm SR}^{i,j}+\mathcal{H}_{\rm SO}^{i,j}+\mathcal{H}_{\rm
Z}^{i,j}$ are shown in Table 4. Diagonal components show the energies of the
zero-field states, while the off-diagonal ME’s describe the coupling to
magnetic field.

For the experimentally determined crystal structure of Sr2IrO4 Crawford  _et
al._ (1994), the two-octahedra [Ir2O11] cluster displays $C_{2v}$ symmetry.
Having the $x$ axis along the $\langle 110\rangle$ crystallographic direction
Crawford  _et al._ (1994) and $z\\!\parallel\\!c$, the effective anisotropic
couplings read $\mathbf{D}=\left(0,0,D\right)$,

$\displaystyle\bar{\bar{\bm{\Gamma}}}^{(x)}\\!=\\!\left(\begin{array}[]{ccc}\Gamma_{xx}&0&0\\\
0&\Gamma_{yy}&0\\\ 0&0&\Gamma_{zz}\end{array}\right)$ (10)

for Ir-Ir links along $x$ and

$\displaystyle\bar{\bar{\bm{\Gamma}}}^{(y)}\\!=\\!\left(\begin{array}[]{ccc}\Gamma_{yy}&0&0\\\
0&\Gamma_{xx}&0\\\ 0&0&\Gamma_{zz}\end{array}\right)$ (11)

for Ir-Ir links along $y$, with $\Gamma_{xx}+\Gamma_{yy}+\Gamma_{zz}=0$. The
uniform and staggered components of the $\bar{\bar{\mathbf{g}}}$ tensor take
for individual Ir sites the following form :

$\displaystyle\bar{\bar{\mathbf{g}}}_{i}\\!=$
$\displaystyle\,\bar{\bar{\mathbf{g}}}_{i}^{\mathrm{u}}\pm\bar{\bar{\mathbf{g}}}_{i}^{\mathrm{s}}\\!=\\!\\!\left(\begin{array}[]{ccc}g_{xx}&0&0\\\
0&g_{yy}&0\\\
0&0&g_{zz}\end{array}\right)\\!\pm\\!\left(\begin{array}[]{ccc}0&g_{xy}&0\\\
g_{xy}&0&0\\\ 0&0&0\end{array}\right).$ (12)

$g_{zz}\\!=\\!g_{\parallel}$ while $g_{xx}$ and $g_{yy}$ are directly related
to $g_{\perp}$ but not restricted to be equal due to the lower symmetry of the
two-octahedra cluster as compared to the IrO6 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, i.e., diagonal in zero magnetic field. The result of such a transformation
is shown in Table 5. Direct correspondence between homologous ME’s 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 $\bar{\bar{\mathbf{g}}}$-tensor data
obtained from the two-octahedra 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.

Spin-wave calculations. For understanding all details of the ESR spectrum, we
carried out a spin-wave analysis using the Hamiltonian (1) and a site-
factorized variational GS wave function

$|\Psi\rangle=\prod_{\rm u.c.}|\psi_{A}\rangle|\psi_{B}\rangle\;,$ (13)

where for each sublattice index $L\\!\in\\!\left\\{A,B\right\\}$,

$|\psi_{L}\rangle=\cos\frac{\alpha_{L}}{2}|\uparrow\rangle+e^{i\varphi_{L}}\sin\frac{\alpha_{L}}{2}|\downarrow\rangle\;.$
(14)

For the 2D unit cell displayed in Fig. 3(a), $D_{2d}$ symmetry is considered.

The spin components of the GS configuration depend on the $\alpha_{L}$ and
$\varphi_{L}$ variational parameters as

$S^{x}_{L}=\frac{1}{2}\sin\alpha_{L}\cos\varphi_{L}\;,$ (15)
$S^{y}_{L}=\frac{1}{2}\sin\alpha_{L}\sin\varphi_{L}\;,$ (16)
$S^{z}_{L}=\frac{1}{2}\cos\alpha_{L}\;.$ (17)

For magnetic fields parallel to the $c$ axis, the GS energy only depends on
the parameters $\alpha_{A}=\alpha_{B}=\alpha$ and
$\varphi_{A}-\varphi_{B}=\delta\varphi$. $\delta\varphi$ is the angle between
neighboring in-plane spins and $\alpha$ describes how much the spins are
tilted away from the $c$ axis. For $\alpha=\pi/2$ the spins are lying within
the $ab$ plane, while $\alpha=0$ corresponds to the fully polarized high-
magnetic-field case. The GS energy

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle(J+\Gamma_{\\!zz})\cos\alpha^{2}-g_{\|}h_{\|}\cos\alpha$ (18)
$\displaystyle+$
$\displaystyle((J-\frac{1}{2}\Gamma_{\\!zz})\cos\delta\varphi-D\sin\delta\varphi)\sin\alpha^{2}$

is minimized when

$\delta\varphi=-\arccos\frac{2J-\Gamma_{\\!zz}}{\tilde{J}}\;,$ (19)
$\alpha=\arccos\frac{g_{\|}h_{\|}}{2J+2\Gamma_{\\!zz}+\tilde{J}}\;.$ (20)

For zero field we find that $\alpha=0$, the spins are confined to the $ab$
plane (i.e., to the IrO2 layer) and the angle $\delta\varphi$ is controlled by
the strength of $D$. $\delta\varphi$ is not affected by fields along the $c$
axis while $\alpha$ changes smoothly from $\pi/2$ to $0$ with increasing the
field strength.

The two magnons characteristic for spin-1/2 antiferromagnets are related to
states orthogonal to $|\psi_{A}\rangle$ and $|\psi_{B}\rangle$. The $4\times
4$ Hamiltonian defining these magnons can be derived by using the well known
Holstein-Primakoff approach and diagonalized through a Bogoliubov
transformation. For $h\|c$, the two spin-wave modes are the gapless
$\nu_{\|1}=0$ 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 $\alpha=\pi/2$) and the GS energy
only depends on the $\delta\varphi$ angle :

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle(J+\frac{\Gamma_{\\!zz}}{2})\cos\delta\varphi-
g_{x}h_{x}\cos\frac{\delta\varphi}{2}-D\sin\delta\varphi\,.$

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 in-plane field fixes the direction of uniform
magnetization as $m=-\cos\frac{\delta\varphi}{2}$. At finite field both modes
thus become gapped. To determine the uniform magnetization $m$ one needs to
minimize Eq. (LABEL:eq:e0_hx), which to leading order in magnetic field leads
to Eq. (5). The first term of Eq. (5) corresponds to the zero-field moment
which arises due to the canting induced by the DM interaction. This
ferromagnetic order-parameter is further enhanced in finite $h_{\bot}$ field.
As long as $h_{\bot}$ is small Eq. (5) remains valid.

As discussed in the main text, in low fields, $m$ can be approximated by its
field independent value Bahr  _et al._ (2014). Using the quantum chemically
derived interaction parameters (see Table 2), we then find
$m\\!\approx\\!0.12\mu_{\mathrm{B}}$, in good agreement with recent
experiments Kim  _et al._ (2009); Cao  _et al._ (1998). Yet the zero-field 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 $\Gamma_{\\!zz}$ from 0.42 to 0.98 meV.

Acknowledgments
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 1694/8-1.

Correspondence should be addressed to N.A. B. (email: n.bogdanov@ifw-
dresden.de) or L. H. (email: l.hozoi@ifw-dresden.de).

Authors Contribution
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 spin-wave 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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