# Evolution of electronic and magnetic properties of Sr2IrO4 under strain

Ekaterina M. Pärschke1,∗ Wei-Chih Chen2 Rajyavardhan Ray3,4,5 & Cheng-Chien
Chen2

###### Abstract

Motivated by properties-controlling potential of the strain, we investigate
strain dependence of structure, electronic and magnetic properties of Sr2IrO4
using complementary theoretical tools: ab-initio calculations, analytical
approaches (rigid octahedra picture, Slater-Koster integrals), and extended
$t-{\mathcal{J}}$ model. We find that strain affects both Ir-Ir distance and
Ir-O-Ir angle, and the rigid octahedra picture is not relevant. Second, we
find fundamentally different behavior for compressive and tensile strain. One
remarkable feature is the formation of two subsets of bond- and orbital-
dependent carriers, a compass-like model, under compression. This originates
from the strain-induced renormalization of the Ir-O-Ir superexchange and O on-
site energy. We also show that under compressive (tensile) strain, Fermi
surface becomes highly dispersive (relatively flat). Already at a tensile
strain of $1.5\%$, we observe spectral weight redistribution, with the low-
energy band acquiring almost purely singlet character. These results can be
directly compared with future experiments.

Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg,
Austria

Department of Physics, University of Alabama at Birmingham, Birmingham,
Alabama 35294, USA

Leibniz IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany

Dresden Center for Computational Material Science (DCMS), TU Dresden, 01062
Dresden, Germany

Department of Physics, Birla Institute of Technology Mesra, Ranchi, India
835215

Email<EMAIL_ADDRESS>

## Introduction

Exploring the physics of quasi-two-dimensional (2D) spin-orbit Mott insulators
can help to understand high-temperature superconductivity as well as the
general interplay of spin-orbit coupling, Hund’s, and Coulomb interactions. In
particular, a lot of studies have been devoted to the quasi-2D iridates
Sr2IrO4 and Ba2IrO4. [(1), (2), (3)] Iridates show eminent similarities to the
cuprate family of high-temperature superconductors, both in structure and low-
energy physics, and were expected to become superconducting upon doping.
However, so far no superconductivity has been reported in iridates.

In general, Sr2IrO4 behavior often deviates from theoretical predictions. For
example, Mott insulators normally become metallic at high enough pressure as
the unit cell becomes smaller and the bands broaden. This is also true for
spin-orbit coupled Mott insulators, such as ruthenates. [(4), (3)] In Sr2IrO4,
resistance indeed decreases until the pressure of around 25-30 GPa (which
according to Ref. 5, corresponds approximately to a strain of -4%), Haskel2012
or, according to a very recent study 6, 32-38 GPa (-5.1% strain). Then,
however, resistance starts to increase, showing a peculiar U-shaped dependency
and persisting insulating behavior up to at least 185 GPa. ChunhuaChen2019  So
far, no metallization in Sr2IrO4 or other iridates (Sr3Ir2O7 Ding2016 ;
JZhang2018 , BaIrO3, Korneta2010 ; CaoReview2018  etc) has been observed at
pressures up to 40-185 GPa. CaoReview2018 ; ChunhuaChen2019  Moreover, there
is also surprisingly little correlation between the insulating behavior and
magnetism GCao1998  as the latter disappears at around 20 GPa (roughly -2.9 %
strain) in Sr2IrO4 Haskel2012 ; HaskelPRL2021  and 14.4 GPa (roughly -2.1 %
strain) in Sr3Ir2O7, JZhang2018  without the onset of a metal-insulator
transition.

Furthermore, iridates emerge as a good functional playground for manipulation
of the magnetic and electronic properties, which is an exciting goal both
fundamentally and practically. RayanSerraoPRB2013 ; CaoReview2018  Iridium-
based heterostructures and superlattices have therefore emerged as a whole new
field very recently. NicholsNatCom2016 ; DiYiPNAS2016 ; OkamotoNanoLet2017 ;
GruenewaldAdvMat2017 ; JWKimPRB2018 ; WenNJP2019 ; MeyersSciRep2019 ;
DasaJMatChemC2019 ; Mohanta2019  Strain and pressure in particular are
powerful tools on hand to control the magnetic properties of the material. It
has been shown that misfit strain can directly control dispersion of magnetic
excitations in Sr2IrO4, LupascuPRL2014 ; Paris2020 ; SeoPRB2019  as well as
transport properties. SouriJAP2019  A shift of the two-magnon Raman peak to
higher energies was observed under tensile strain, SeoPRB2019  albeit much
weaker than the shift observed in the canonical Mott-Hubbard insulator K2NiF4
and cuprates like Bi1.98Sr2.06Y0.68Cu2O8+δ. CukPRL2008

In Ref. [ 23], the authors used resonant inelastic scattering (RIXS) to show
that magnetic dispersion in Sr2IrO4 is strongly affected by strain. In
particular, the contribution of the second and third nearest-neighbor (NN)
exchange was suppressed (enhanced) upon tensile (compressive) strain. The
tensile strain was shown to drive the system closer to a shorter-range first-
NN only Heisenberg limit, with only little magnon branch softening left at
($\rm{\pi}/2,\rm{\pi}/2$) already upon the tensile strain of $+2\%$. Upon
compressive strain, the energy of ($\mathrm{\pi},0$) magnon was shown to
increase. Paris2020 ; LupascuPRL2014

A clear understanding of the electronic and magnetic properties of iridates
and their evolution with strain is, therefore, of interest not only from a
fundamental point of view but also for applications. RayanSerraoPRB2013 ;
CaoReview2018  Unveiling the details of the interplay of lattice, magnetic,
and other degrees of freedom in Sr2IrO4 is needed to understand the recently
observed electrical control of octahedra rotation CaoPRL2018 ; YePRB2020  and
the much-debated strong magnetoelastic coupling. SonNPJQM2019 ;
DashwoodPRB2019 ; Liu2019 ; SamanthaAPL2019  Currently, a clear understanding
of neither how exactly nor by which mechanism do superexchange and hopping
parameters in Sr2IrO4 change with strain is available, not even on a
phenomenological level. One of the interesting questions is whether the change
in electronic and magnetic properties upon the strain is mostly associated
with bond length change, as argued in e.g. Ref. [ 22], or the change of the
in-plane rotation angles $\theta$ of the oxygen octahedra (see Fig.
1).NicholsAPL2013

Studying the behavior of iridates under strain and pressure is a demanding
task not only experimentally, but also theoretically. On one hand, iridates
are strongly correlated Mott insulators, Kim2008  so one needs to resort to
theoretical methods where correlations are treated non-perturbatively,
employing effective descriptions like Hubbard or Heisenberg models. On the
other hand, microscopic changes of orbitals, their overlap, and structural
changes are essential to understand the behavior of a crystal under strain,
BhandariNJP2019 ; ZaitsevJPCM2020 ; ChoiPRB2020 ; EngstromPRB2021 ;
NicholsonComNat2021  so ab-initio methods are demanded. Another difficulty is
that as one eventually approaches a possible metal-insulator transition at
high pressure and/or strain, effective models, such as the Heisenberg
superexchange model, fail.

In this paper, we focus on the effect of strain and combine various
complementary theoretical tools to provide a comprehensive analysis of how the
magnetic properties are affected by strain. For different (compressive and
tensile) strain values, we use density functional theory (DFT) based ab-initio
calculations to access microscopic changes in the crystal structure, and study
the corresponding changes in the electronic properties through Wannierization
of the scalar-relativistic DFT bandstructure obtained within the generalized
gradient approximation (GGA).PerdewPRL1996  Subsequently, we solve an extended
$t-{\mathcal{J}}$ model within the self-consistent Bohr approximation (SCBA)
to obtain the angle-resolved photoemission spectra (ARPES) and study the
strain-controlled evolution of the Fermi surface. Realistic values of the
input parameters for these calculations were used: the hopping parameters were
obtained from the DFT calculations, while the extended-range exchange
couplings were obtained by direct comparison to the magnon dispersion measured
with RIXS. In this way, the presented analysis contains no free parameters
apart from an overall constant energy shift (chemical potential) in the SCBA
calculations.

## Results and Discussion

###  0.1 Evolution of hopping parameters under strain.

Sr2IrO4 shows an in-plane staggered octahedra rotation characterized by a
single parameter: $\measuredangle{\rm O}-{\rm Ir}-{\rm Ir}$ angle denoted by
$\theta$ in Fig. 1. Under ambient conditions, the octahedral rotation is found
to be $\theta\approx 13.6^{\circ}$ for the relaxed structure, which is close
to the reported experimentally value of $11.8^{\circ}$. Boseggia2013  The
epitaxial strain on iridates then affects not only the distance between the Ir
atoms but also the Ir-O-Ir bond angle, as can be seen in Fig. 2a. The in-plane
octahedra rotation angle $\theta$, obtained using DFT (see Methods for
details), monotonically increases (decreases) upon compressive (tensile)
strain in the studied range of -7.5$\%$ to 7.5$\%$, where negative strains
correspond to compression.

Figure 1: In-plane structure of Sr2IrO4 and the notation for orbital overlap
parameters. For nearest neighbors (NN), interorbital overlap parameters are
denoted $t_{1}$, $t_{2}$, $t_{3}$ and intraorbital $t_{4}$, and for second
nearest neighbors (2NN) we consider interorbital overlap only:
$t_{1}^{\prime}$, $t_{2}^{\prime}$, $t_{3}^{\prime}$. Orbital overlap between
third neighbors is defined in the same way, not shown. Due to the symmetry
considerations, the overlap of out-of-plane $t_{\mathrm{2g}}$ orbitals is
anisotropic: $t_{2}$ describes hopping between $xz$($yz$) along $a^{\prime}$
($b^{\prime}$) axis. The O-Ir-Ir angle $\theta$ characterizes the in-plane
octahedra rotation as shown.

To ascertain the influence of structural changes on the electronic properties,
we study the evolution of Wannier tight-binding model hoppings derived from
DFT (see Methods for details), as a function of strain (Fig. 2b). The
notations for the hoppings are shown in Fig 1: the intraorbital hoppings
between $xy$ orbitals along $a^{\prime}$ or $b^{\prime}$ axes is denoted as
$t_{1}$, between $xz$($yz$) along $a^{\prime}$ ($b^{\prime}$) axis as $t_{2}$,
and between $xz$($yz$) along $b^{\prime}$ ($a^{\prime}$) as $t_{3}$. The
interorbital hopping between $yz$ and $xz$ orbitals is denoted as $t_{4}$, all
other interorbital hoppings are negligible. Further neighbor interorbital
hoppings are denoted as $t_{i}^{\prime}$ and are shown in Fig. 1.

Upon compression, direction-dependent hopping parameter $t_{2}$ is increasing,
but surprisingly, $t_{1}$ is decreasing (Fig. 2b). This emerging anisotropy in
hopping parameters is interesting, as $t_{2}$ hopping describes the
propagation of an electron with $xz$ ($yz$) orbital character along only one
axis, $a^{\prime}$ ($b^{\prime}$), whereas $t_{1}$ allows an $xy$ electron to
hop in both directions. We thus see that upon compressive strain, the system
favors the separation of the entire Fermi sea into two Fermi seas with bond-
dependent propagation ($xz$ carriers which can only propagate along
$a^{\prime}$, and $yz$ carriers which can only propagate along $b^{\prime}$)
and suppression of the bond-independent and thus truly two-dimensional $xy$
carriers. This compass-model-likeNussinov2015  propagation is quite unusual
and could cause the formation of charge density wave.

Upon tensile strain, $t_{1}$ is nearly independent of the strain value and is
the dominant hopping, while $t_{2}$ decreases steadily (Fig. 2b). Different
behavior of $t_{1}$ upon compressive and tensile strain reflects the change of
Fermi surface topology between compressive and tensile strain.

It is also interesting to note that the smallest hopping parameter $t_{3}$,
describing the hopping between almost parallel $d$-orbitals with very small
overlap goes to zero around -3$\%$, which corresponds to compression of $\sim
20\,$GPa, not too far from the value of resistivity minimum under pressure.
Haskel2012 ; ChunhuaChen2019

To disentangle the contribution of inter-atomic distance $d$ and the
octahedral rotation $\theta$ to the hopping parameter trends with strain, we
employ the analytical approaches of Glazer and Slater-Koster. The Glazer
picture Glazer1972  is often used in rigid octahedra approximation whereby the
main effect of the modest strain is assumed to be the change of the in-plane
rotation angle $\theta$. However, as detailed in Supplementary Figure 2, the
trends obtained within the Glazer picture disagree with the DFT results in
Fig.2b, and even contradict them in rigid octahedra approximation. Therefore,
the Glazer picture has limited applicability for iridates, and rigid octahedra
approximation is improper.

a
b

Figure 2: Dependency of the octahedra rotation angle $\theta$, Ir-Ir distance
$d$ and NN hopping parameters on strain. a The in-plane IrO6 octahedral
rotation angle $\theta$ and Ir-Ir distance $d$ as functions of strain,
obtained with full structure relaxation in DFT calculations (see Methods). b
NN hopping parameters for different values of strain, see Fig.1 for notation.

We then proceed with a more specific orbital-resolved Slater-Koster-integrals-
based approach. HarrisonBook2004 ; Slater1954  Slater-Koster integrals are
hybridization matrix elements $E$ between atomic $d$-states on neighboring
atoms obtained via integrating over relevant spherical harmonics. The
resulting interatomic matrix elements $E$ are proportional to the $d$-wave
functions overlap and can be expressed via cubic harmonic matrix elements
$V_{\mathrm{dd\sigma}}$,$V_{\mathrm{d}\mathrm{d}\mathrm{\pi}}$,
$V_{\mathrm{dd\delta}}$ for a known bond direction ${l,m,n}$ as tabulated in
Slater-Koster tables.HarrisonBook2004 ; Slater1954  In Sr2IrO4, we also need
to account for the rotation of the $d$ orbitals within the $t_{\mathrm{2g}}$
sector due to the in-plane octahedral rotation.WangSenthil2011  Therefore, we
decompose the rotated $d$ orbital in the basis of non-rotated $d$ orbitals
before evaluating the Slater-Koster matrix elements. For example, the
hybridization matrix element $\tilde{E}$ between the two rotated NN $xy$
orbitals can be obtained as a superposition of hybridization matrix elements
$E$ of non-rotated $xy$ and $x^{2}-y^{2}$ orbitals obtained as:

$\displaystyle\tilde{E}_{\mathrm{xy,xy}}=\cos^{2}(2\theta)E_{\mathrm{xy,xy}}-\sin^{2}(2\theta)E_{\mathrm{x^{2}-y^{2},x^{2}-y^{2}}},$
(1)

where $\theta$ is the in-plane rotation of the IrO6 octahedra (see Fig.1).
Similarly, for the overlap between the rotated $xz$($yz$) orbitals along the
$a^{\prime}$ direction, we get

$\displaystyle\tilde{E}_{\mathrm{xz,xz(yz,yz)}}=\cos^{2}(\theta)E_{\mathrm{xz,xz(yz,yz)}}-\sin^{2}(\theta)E_{\mathrm{yz,yz(xz,xz)}},$
(2)

and for interatomic interorbital overlap along $x$:

$\displaystyle\tilde{E}_{\mathrm{yz,xz}}=E_{\mathrm{yz,xz}}+\sin\theta\cos\theta(E_{\mathrm{xz,xz}}+E_{\mathrm{yz,yz}}).$
(3)

Figure 3 shows the resulting hybridization matrix elements $\tilde{E}$ as a
function of the in-plane octahedral rotation $\theta$, the Ir-Ir distance $d$,
as well as both the parameters (see Fig. 2a). We find that at least in the
Slater-Koster approximation, accounting for the change of the distance $d$
alone (Fig. 3b) can provide a better approximation to a full dependency of
matrix elements $\tilde{E}$ on strain (Fig. 3c) then accounting for the change
of bond angle $\theta$. This is also consistent with the quantum chemistry
study. LupascuPRL2014

However, not all trends obtained from the DFT calculations are well
reproduced: the hopping parameter $E_{\mathcal{xy,xy}}$ is increasing under
compressive strain (Fig. 3c), unlike the $t_{1}$ hopping extracted from DFT
(Fig. 2b). To address this, we also consider the O-mediated indirect
Ir$-$O$-$Ir hoppings.

a
b
c

Figure 3: Dependency of Slater-Koster Ir-Ir integrals on strain. Slater-Koster
integrals for rotated NN orbitals $\tilde{E}$ as a function of a octahedra in-
plane rotation angle $\theta$, b Ir-Ir distance $d$, c both $\theta$ and $d$
(calculated using values of $\theta$ and $d$ extracted from DFT shown in Fig.
2a).

The indirect oxygen-mediated overlap between the two rotated NN $xy$ orbitals
can be calculated as a sum of the hopping integrals between two Ir atoms via
$\alpha=p_{\mathrm{x}}$, $p_{\mathrm{y}}$ orbitals of the oxygen,
$\tilde{E}_{\mathrm{xy,O,xy}}=\sum_{\alpha=p_{\mathrm{x}},p_{\mathrm{y}}}{E_{\mathrm{xy},\alpha,\mathrm{xy}}}$.
The hopping integral is calculated as

$\displaystyle\tilde{E}_{\mathrm{xy},\alpha,\mathrm{xy}}$
$\displaystyle=(\cos(2\theta)E_{\alpha,\mathrm{xy}}^{-l,m,n}+\sin(2\theta)E_{\alpha,\mathrm{x^{2}-y^{2}}}^{-l,m,n})$
(4)
$\displaystyle\times(\cos(2\theta)E_{\alpha,\mathrm{xy}}^{l,m,n}-\sin(2\theta)E_{\alpha,\mathrm{x^{2}-y^{2}}}^{l,m,n})/\Delta_{\mathrm{pd}},$

where $l=\cos{\theta}$, $m=\sin{\theta}$, $n=0$ are the directional cosines of
the vector from the oxygen O to the Ir atom HarrisonBook2004  along the
$a^{\prime}$-axis in the units of Ir-O distance $d_{\rm Ir-O}=0.5d_{0}^{\rm
Ir-Ir}/\cos{\theta}$ ($d_{0}^{\rm Ir-Ir}$ is the distance between the NN Ir
ions under ambient conditions), and $E_{\alpha,\mathrm{xy}}$,
$E_{\alpha,\mathrm{xy}}$ are the $p$-$d$ Slater-Koster integrals.
HarrisonBook2004  We note that $l,m,n$ indexes were omitted for Eqs.(1-3),
because for the $d-d$ overlap, Slater-Koster integrals are quadratic in
directional cosines, HarrisonBook2004  and simply $\\{l,m,n\\}=\\{1,0,0\\}$
for a pair of Ir atoms along the $a^{\prime}$ bond. For the indirect oxygen-
mediated overlap, however, one has to account for the sign of the directional
cosines. Moreover, for the indirect hopping, the Ir-O-Ir hopping has to be
renormalized by the charge transfer energy
$\Delta_{\mathrm{pd}}=E^{\mathrm{on-site}}_{\mathrm{xy}}-E^{\mathrm{on-
site}}_{\alpha}$, the energy difference between corresponding Ir-$d$ and O-$p$
orbitals (see Supplementary Figure 3a). Surprisingly, the charge transfer
energy $\Delta_{\mathrm{pd}}$ has a strongly non-linear dependency on the
strain (see Supplementary Figure 3a).

We plot the resulting indirect superexchange matrix elements $\tilde{E}$ as a
function of the in-plane octahedral rotation $\theta$, Ir-Ir distance $d$ in
Fig. 4. Indeed, the indirect Ir-Ir overlap decreases drastically under
compressive strain, unlike the direct $d-d$ overlap. Interestingly, this
behavior is directly linked to both the change in the distance between the
atoms as well as the angle $\theta$ describing octahedra rotation (see
Figs.4a,b). We also note that taking into account the strain dependence of the
iridium and oxygen on-site energies is crucial to obtain correct trends. In
fact, non-linear behavior of charge transfer energy $\Delta_{\mathrm{pd}}$
seems to be directly responsible for the non-linear strain dependency of the
indirect oxygen-mediated hopping $\tilde{E}$. This suggests that the role of
oxygens in the low-energy physics of strained iridates and other transitional-
metal oxides might be underestimated and requires further investigation.

a
b
c

Figure 4: Dependency of Slater-Koster Ir-O-Ir integrals on the strain. Slater-
Koster integrals for indirect hopping between rotated Ir NN orbitals
$\tilde{E}$ via oxygen $p$ orbitals as a function of a octahedra in-plane
rotation angle $\theta$, b Ir-Ir distance $d$, c both $\theta$ and $d$
(calculated using values of $\theta$ and $d$ extracted from DFT shown in Fig.
2a).

Accordingly, the contribution of the indirect orbital overlap should be small
for $xz$ and $yz$ orbitals. Indeed, the indirect $yz$-$yz$ orbitals overlap
along $a^{\prime}$-axis is zero. The $xz$ orbitals hybridize with
O-$p_{\mathrm{z}}$ orbital, however, this hybridization decreases with strain
much slower than for $xy$ orbitals, explaining the different behavior of $xy$
and $xz$ orbitals under compressive strain.

The fact that relative Ir-O hybridization is directly responsible for the
resulting suppression of $t_{\mathrm{1}}$ ($xy$-$xy$ hopping) under
compressive strain suggests an electronic state crossover as the role of $xy$
orbitals in the composite $J_{\mathrm{eff}}=1/2$ is decreasing. Notably, a
pressure-induced phase transition was also suggested in a recent X-ray powder
diffraction study Samanta2020  at pressures around 20 GPa, which should
correspond to approximately $-3\%$ strain and is in good agreement with our
findings.

###  0.2 Overlap of the spin-orbit coupled $J_{\mathrm{eff}}$ states.

We now estimate the overlap between the $J_{\mathrm{eff}}=1/2$ states for NN,
2NN, and 3NN (denoted $\tau$, $\tau^{\prime}$ and $\tau^{\prime\prime}$,
correspondingly), which can be calculated from $t_{\mathrm{2g}}$ orbitals
overlap using the Clebsh-Gordon coefficients. Jackeli2009 ; Plotnikova2016  In
Fig. 5, we show the change of the overlap of NN $J_{\mathrm{eff}}=1/2$ states
calculated from the DFT values obtained here. As experimental values of
$J_{\mathrm{eff}}=1/2$ states overlap $\tau$ are hard to measure, one can try
to compare hopping parameters $\tau$ with available experimental estimates of
magnetic exchange interactions $\mathcal{J}$ (see Table 1 in Ref. 23), which
in ambient conditions are assumed to scale with $\tau^{2}$. Jackeli2009
However, in a recent RIXS study on strained Sr2IrO4, Paris2020 , the authors
suggest that the simple $\mathcal{J}\propto\tau^{2}/U$ relationship fails for
strained Sr2IrO4 due to the polaronic renormalization of the charge
excitations. In particular, the first-neighbor exchange interaction
$\mathcal{J}_{\mathrm{1}}$ was shown to decrease slightly upon the tensile
strain, while $\mathcal{J}_{\mathrm{2}}$ and $\mathcal{J}_{\mathrm{3}}$
decreased much faster, based on a fit of the Heisenberg model to the measured
magnon dispersion. Paris2020  An earlier RIXS study also suggested that
magnetic exchange interaction $\mathcal{J}$ increases upon the compressive
strain. LupascuPRL2014

a
b

Figure 5: Dependency of $J_{\mathrm{eff}}=1/2$ overlap on the strain. a NN,
2NN and 3NN hopping parameters ($\tau$, $\tau^{\prime}$ and
$\tau^{\prime\prime}$, correspondingly) between $J_{\mathrm{eff}}=1/2$ states
calculated as a superposition of obtained from DFT calculations $xy$, $yz$ and
$xz$ hopping parameters. b Relative change of NN, 2NN and 3NN
$J_{\mathrm{eff}}=1/2$ overlap normalized by their values in pristine
condition ($\tau_{0}$, $\tau_{0}^{\prime}$ and $\tau_{0}^{\prime\prime}$,
correspondingly. 2NN and 3NN hoppings for $t_{\mathrm{2g}}$ orbitals used to
calculate $\tau$’s are shown in SI.)

Consistent with both RIXS studies, LupascuPRL2014 ; Paris2020  calculated here
values of NN, 2NN, and 3NN hopping parameters are all decreasing upon tensile
strain (Fig. 5). As discussed in Ref. [ 23], this trend for $\tau$’s is
significantly slower than that observed for superexchange interaction
$\mathcal{J}$, indicating that $\mathcal{J}\propto\tau^{2}/U$ relationship
indeed fails for strained Sr2IrO4. Our DFT strain trends are also consistent
with the modest increase of magnetic exchange interaction
$\mathcal{J}_{\mathrm{1}}$ under compressive strain reported in the two-magnon
Raman study. SeoPRB2019

It is interesting to compare the trends observed in Sr2IrO4 to those in $3d$
transition metal oxides - cuprates. In Ref. [ 48] authors used XAS at Cu
$L_{3}$-edge of La2CuO4 together with analytical and DFT theoretical
approaches to show that both bandwidth and electron-electron correlations were
increasing upon the compressive strain. As a result, magnetic exchange
$\mathcal{J}$ was shown to increase (decrease) almost linearly upon the
compressive (tensile) strain. In comparison, while the orbital-dependent
hoppings in Sr2IrO4 behave very differently from this (Fig. 2b), showing the
surprising decrease in $t_{1}$ with compressive strain, effective
$J_{\mathrm{eff}}=1/2$ orbitals in Sr2IrO4 have strain dependence (Fig. 5)
somewhat similar to those of cuprates, echoing the famous parallel between
Sr2IrO4 and La2CuO4.

###  0.3 The evolution of the Fermi surface under strain.

Structural response to strain may also be accompanied by changes in the Fermi
surface. Thus, strong changes of Fermi surface upon uniaxial pressure have
recently been reported in Ru-based compound Sr2RuO4, SunkoNPJQuantMat2019 ;
BarberPRB2019  along with the more than double increase of the superconducting
transition temperature. TaniguchiJPhysSocJap2015  A recent work on Sr2IrO4
employing a tight-binding model has pointed out that out-of-plane tilting of
the oxygen octahedra can induce shrinking of the Fermi surface and suppress
nesting and the $d$-wave superconductivity. LindquistPRB2019  There is no out-
of-plane tilting in Sr2IrO4 under pristine conditions or modest strain - it
was very recently shown to appear only under the pressure of as much as 40
GPa. ChunhuaChen2019  However, in-plane octahedra rotation is strongly
affected by the modest strain already, and it is important to understand if
and how the Fermi surface is affected, particularly on the beyond-mean-field
level.

a) -0.52%
b) +1.53%

Figure 6: Dependency of the spectral function of Sr2IrO4 on the strain.
Spectral function calculated for various strain values: a LSAT, -0.52%
(compressive strain), b GSO, +1.53% (tensile strain) corresponding to ARPES
(negative energies) and inversed photoemission spectra (positive energy). The
horizontal axis is the 2D crystal momentum. The vertical axis is the energy
(eV), where zero energy represents the Fermi level.

To study the evolution of the Fermi surface under strain we calculate
photoemission spectral functions of strained Sr2IrO4, using extended
$t-{\mathcal{J}}$ model formalism developed in Ref. [ 53]. The extended
$t-{\mathcal{J}}$ model used in the calculation (see Methods) depends on two
sets of parameters: the magnetic exchange parameters
$\mathcal{J}_{\mathrm{1}}$, $\mathcal{J}_{\mathrm{2}}$,
$\mathcal{J}_{\mathrm{3}}$, Ising anisotropy coefficient $\Delta$,ValePRB2015
, and the hopping parameters $t_{i}$ describing overlap of the
$t_{\mathrm{2g}}$ orbitals. We obtain the set of $t_{i}$’s for each strain
value from DFT calculations as discussed in detail above. Using this Wannier
Hamiltonian as a starting point describing single-particle hopping processes,
we consider all possible many-body hopping processes to derive the hopping
part of the extended $t-{\mathcal{J}}$ model. Paerschke2017

To properly account for the changes in the electronic structure, we need to
account for the evolution of the magnetic exchange parameters with the strain.
It has been obtained from the publishedParis2020  fits to the RIXS spectra on
strained samples. As experimental data is available for small strain values
range only, we restrict calculations of photoemission spectra to that range
and show the photoemission (and inverse photoemission) spectra in Fig. 6 for
two substrates: (LaAlO3)${{}_{0}.3}$(Sr2AlTaO6)${{}_{0}.7}$ (100) (LSAT) and
GdScO3 (110) (GSO), providing a strain of -0.52$\%$ and +1.53$\%$,
correspondingly.

a -0.52%, $J=0$
b +1.53%, $J=0$
c -0.52%, $J=1$
d +1.53%, $J=1$

Figure 7: Contribution of singlet $J=0$ and triplet $J=1$ states to the
spectral function of Sr2IrO4 for various strain values. a-b: singlet $J=0$
contribution to the ARPES spectral function: a LSAT, -0.52% (compressive
strain), b GSO, +1.53% (tensile strain); c-d: triplet $J=1$ contribution to
the ARPES spectral function: c LSAT, -0.52% (compressive strain), d GSO,
+1.53% (tensile strain). The horizontal axis is the 2D crystal momentum. The
vertical axis is the energy (eV), where zero energy represents the Fermi
level.

Calculated photoemission spectra show one conductance band at positive
energies, and two valence bands at negative energies: a sharp singlet band
around -0.25 eV and a more incoherent triplet band at -0.5 eV. (see Fig. 7 for
a detailed discussion on the band character). We see that the strain-induced
changes of the photoemission spectra are quite prominent for samples with a
strain difference of $2\%$. First, for tensile strain, as compared to
compressive strain, the Mott gap increases (Fig. 6b), suggesting stronger
polaron binding of the photoinduced hole to the magnetic background. Paris2020
Second, upon compressive strain, the photoemission spectra of Sr2IrO4 show a
highly dispersive singlet band (Fig. 6a), while upon tensile strain, both
singlet and triplet bands are much less dispersive, and the Fermi surface of
Sr2IrO4becomes relatively flat (Fig. 6b). It is important to note that the
relative flattening of the Fermi sheet upon tensile strain is a many-body
effect distinct from the anisotropic compass-like hoppings under compressive
strain. One should be able to observe such significant renormalization of the
spectral weight in the ARPES data even for small, realistic values of strain.

The conduction band is only weakly affected by strain. Fig. 6 shows a marginal
flattening of the conduction band upon the tensile strain. We, therefore,
expect a minimal effect of epitaxial strain on possible superconductivity.

To explore the effect of strain on the ARPES spectra in more detail, we plot
in Fig. 7 separate contributions of the singlet ($J=0$) and triplet ($J=1$)
charge excitations to the full spectra. As one can see, under tensile strain,
$J=0$ contributes the most at ($\mathrm{\pi}$,0), whereas under compressive
strain, its contribution is slightly more widespread. On the contrary, the
$J=1$ spectral weight is shifted to $(\mathrm{\pi}/2,\mathrm{\pi}/2)$ at the
tensile strain and to $(\mathrm{\pi},0)$ at the compressive strain. In
particular, $J=1$ contribution to the ”lower energy” band of the photoemission
spectra is strongly reduced upon the tensile strain. We thus observe strain-
controlled spectral weight redistribution between the charge carriers of
singlet and triplet characters. Already moderate tensile strain is sufficient
to make the lower energy band of almost purely singlet character.

In summary, we predict a dramatic strain dependence of the electronic
properties of Sr2IrO4 for compressive v/s tensile strain. The most remarkable
feature is the appearance of the compass-model-like contribution of electron
propagation due to the separation of the Fermi sea in Sr2IrO4 into two subsets
of bond- and orbital-dependent carriers under compressive strain. This enables
the formation of charge density wave due to nesting and could be connected to
the puzzling metalization avoidance in Sr2IrO4 upon pressure. The Fermi sea
separation originates from strain dependency of relative Ir-O hybridization,
as well as on-site O energy, suggesting an important role of oxygens in a low-
energy physics of strained iridates and other transitional-metal oxides.

Despite the suppression of the bond-independent $xy$ hopping $t_{1}$ under
compressive strain, the hopping amplitude of the composite
$J_{\mathrm{eff}}=1/2$ state still increases under compressive strain owing to
the contribution from hoppings between $xz/yz$ orbitals ($t_{2}$) — the
largest of two in-plane nearest-neighbor direction-dependent hoppings. The
obtained trends for $J_{\mathrm{eff}}=1/2$ hopping, $\tau$, are in good
agreement with available experimental data.

We also calculated the photoemission spectra of Sr2IrO4 upon compressive and
tensile strain (for samples grown on LSAT and GSO substrates, respectively).
We find that under compressive (tensile) strain, the singlet band becomes
significantly more (less) dispersive, and both the singlet and triplet bands
shift up (down) in energy. We also show that the electronic properties of the
low-energy model can be controlled by strain, since the already moderate
tensile strain is sufficient to make the lower energy band of almost purely
singlet character, and shift the triplet spectral weight to
$(\mathrm{\pi}/2,\mathrm{\pi}/2)$ point. These features can be readily
observed in the future ARPES measurements — a smoking gun test of our
findings.

###  0.4 DFT calculations.

The DFT results are calculated by the Quantum Espresso package Giannozzi_2009
using Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional
PerdewPRL1996  and projector-augmented-wave pseudopotentials BloechPRB1994
with the unit cell containing 16 Sr, 8 Ir, and 32 O atoms. In the calculations
of structure relaxation and electronic structure, spin-orbit interaction is
not included. The unstrained structure (a = 5.5098 Å, c = 26.1522 Å) is
acquired through relaxing both the lattice geometry and the atomic positions.
For biaxial-strained structures, the $a$\- and $b$-axis are fixed as per the
strain value while the $c$-axis and the atomic positions are simultaneously
relaxed. The plane wave cutoff energy is set to be 40 Ry, and a
$\Gamma$-centered $10\times 10\times 2$ $k$-points in the full Brillouin zone
are sampled. The structural and electronic convergence criteria were set to be
$10^{-4}$ Ry/$a_{0}$ and $10^{-6}$ Ry, respectively.

With the DFT eigenvalues and eigenvectors, we then use Wannier90 package
MOSTOFI2008685  and implement a disentanglement procedure to obtain the
Wannier functions and the hopping parameters. During the disentanglement, 120
bands were considered, which accounted for contributions from 96 (32$\times$3)
O-$2p$ orbitals and 24 (8$\times$3) Ir-$t_{\mathrm{2g}}$ orbitals. Four
iridium layers (i.e. 8 Ir atoms) are involved to construct 24 wannierized
$t_{\mathrm{2g}}$ orbitals in one unit cell (6 Wannier functions per Ir
layer).

###  0.5 Calculation of the photoemission spectra.

The ARPES spectral function is obtained as imaginary part of the Green
function $G({\bf k},\omega)$ describing propagation of the photohole in the
ground state, dressed in the low-energy magnetic excitations (magnons):
$G({\bf k},\omega)={\rm Tr}{\langle{\rm AF}|{\bf h}_{{\bf
k}}\frac{1}{\omega-\mathcal{H}_{\rm t-{\mathcal{J}}}+i\delta}{\bf h}_{{\bf
k}}^{\dagger}|{\rm AF}\rangle}$. Here, photohole ${\bf h}_{{\bf k}}$ is a
vector in a full spin-orbital Hilbert space, obtained by projecting three
$t_{\mathrm{2g}}$ orbitals of Ir atom onto the spin-orbit coupled basis
Paerschke2017 ; PaerschkeRay2018 . The motion of the hole is governed by
$t-{\mathcal{J}}$ Hamiltonian $\mathcal{H}_{\rm
t-{\mathcal{J}}}=\mathcal{H}_{\rm t}+\mathcal{H}_{\mathcal{J}}$. The magnetic
$H_{\mathcal{J}}$ part includes NN, 2NN and 3NN Heisenberg interactions
$\mathcal{J}_{\mathrm{1}}$ , $\mathcal{J}_{\mathrm{2}}$,
$\mathcal{J}_{\mathrm{3}}$ and an anisotropic $\Delta$ term, arising from non-
negligible Hund’s coupling ValePRB2015 ; PinciniPRB2017 :

$\displaystyle\mathcal{H}_{\mathcal{J}}=\mathcal{J}_{\mathrm{1}}\sum_{\langle
i,j\rangle}{\left(\mathbf{\tilde{S}}_{i}^{x}\mathbf{\tilde{S}}_{j}^{x}+\mathbf{\tilde{S}}_{i}^{y}\mathbf{\tilde{S}}_{j}^{y}+(1-\Delta)\mathbf{\tilde{S}}_{i}^{z}\mathbf{\tilde{S}}_{j}^{z}\right)}$
(5) $\displaystyle+\mathcal{J}_{\mathrm{2}}\sum_{\langle\langle
i,j\rangle\rangle}{\mathbf{\tilde{S}}_{i}\cdot\mathbf{\tilde{S}}_{j}}+\mathcal{J}_{\mathrm{3}}\sum_{\langle\langle\langle
i,j\rangle\rangle\rangle}{\mathbf{\tilde{S}}_{i}\cdot\mathbf{\tilde{S}}_{j}}$

The hopping $\mathcal{H}_{\rm t}$ part is derived by projecting multiorbital
Hubbard model employing orbital-dependent hopping parameters $t_{i}$ onto
spin-orbit coupled basis. Paerschke2017 ; PaerschkeRay2018  The motion of a
charge excitation in the new spin-orbit coupled basis is then expressed
analytically in terms of these $t_{i}$ which are obtained directly from DFT.
Paerschke2017 ; PaerschkeRay2018  We evaluate the Green function $G({\bf
k},\omega)$ using the self-consistent Born approximation (SCBA).Martinez1991
SCBA is a diagrammatic approach that evaluates Green function of a
quasiparticle dressed with bosons (here, photohole dressed with magnons) that
form diagrams of rainbow type. Martinez1991  The spectral functions are
calculated numerically for a $16\times 16$ cluster.

All the data that support the findings of this study are available from the
corresponding author (E.M.P.) upon reasonable request.

The codes used in this study are available from the corresponding author
(E.M.P.) on reasonable request.

E.M.P. thanks Eugenio Paris, Thorsten Schmitt, Krzysztof Wohlfeld and other
coauthors for an inspiring previous collaboration, Paris2020  and is grateful
to Gang Cao, Ambrose Seo and Jungho Kim for insightful discussions. R.R.
acknowledges helpful discussion with Sanjeev Kumar and Manuel Richter. This
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No
754411. C.C.C. acknowledges support from the U.S. National Science Foundation
Award No. DMR-2142801.

The authors declare that there are no competing interests.

E.M.P. designed the project and carried out the photoemission spectra
calculations. E.M.P. and R.R. performed the Slater-Koster analysis. W.C.C.
carried out the DFT and Wannier function analysis. E.M.P. and C.C.C.
supervised the project. All the authors analyzed the results. E.M.P. wrote the
manuscript with the help of R.R. and C.C.C..

Correspondence and requests for materials should be addressed to E. M.
Pärschke (email: ekaterina.paerschke@ist.ac.at).

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## Supplemental Information

### 0.6

We start by presenting hopping parameters for further neighbors extracted from
DFT. In Supplemental Figure 1, we show the hopping parameters for further
neighbors: second nearest (2NN) and third nearest (3NN) neighbors. These
values were extracted from DFT calculations using Wannier functions fitting,
as discussed in detail in Methods.

a
b

Supplementary Figure 1:  (a) 2NN hopping parameters, (b) 3NN hopping
parameters as a function of strain. Hopping parameters were extracted from the
Wannier function fit of the DFT band structure. Dashed lines are used to show
Wannier function fitting results, while solid lines are used to show results
obtained using Slater-Koster integrals.

### 0.7

We proceed with Glazer’s picture of the rigid octahedral rotation that is
often used for perovskite metal oxides. In this approach, the main effect of
the strain is assumed to be the change of the in-plane rotation angle $\theta$
Glazer1972 . Using strain-dependent angle values extracted from our DFT
calculations, we get that the overall hopping
$t\propto\cos^{4.5}\theta/(0.5d)^{3.5}$ should decrease under compressive
strain (Supplemental Figure 2(a)), as indeed expected for metal oxides with
perovskite structureSeoPRB2019 . This trend, however, clearly contradicts
those obtained in DFT calculations in the main text. Even when changes in the
bond length $d$ are taken into account (Supplemental Fig 2(b)-(c)), this
simple phenomenological approach could only explain the evolution of
direction-dependent $t_{2}$ and $t_{3}$ hoping parameters, but not the $xy-xy$
hopping parameter $t_{1}$. It is, moreover, not consistent with the two-magnon
Raman scattering data, as pointed out by [ 2]. We thus conclude that simple
Glazer’s rigid octahedra picture is not applicable for iridates.

a
b
c

Supplementary Figure 2:  (a)-(c) Value of phenomenological hopping parameter
$t_{\rm{Glazer}}$ as a function of (a) octahedra in-plane rotation angle
$\theta$, (b) Ir-Ir distance $d$, (c) both $\theta$ and $d$ (calculated using
values of $\theta$ and $d$ extracted from DFT).

### 0.8

In the following, we present Slater-Koster analysis of the oxygen-mediated
overlap of $xz$ and $yz$ orbitals. The indirect overlap between the two
rotated NN $xz$/$yz$ orbitals is mediated via the oxygen $p_{z}$ orbital and
calculated as

$\displaystyle\tilde{E}_{\mathrm{xz/yz,z,xz/yz}}$
$\displaystyle=(\cos(\theta)E_{\mathrm{z,xz/yz}}^{-l,m,n}\mp\sin(\theta)E_{\mathrm{z,yz/xz}}^{-l,m,n})$
$\displaystyle\times(\cos(\theta)E_{\mathrm{z,xz/yz}}^{l,m,n}\pm\sin(\theta)E_{\mathrm{z,yz/xz}}^{l,m,n})/\Delta_{\mathrm{pd}},$
(6)

where $l=\cos{\theta}$, $m=\sin{\theta}$, $n=0$ are the directional cosines of
the vector connecting the Ir atom to neighboring (along the $a^{\prime}$-axis)
oxygen O in the units of Ir-O distance $d_{\rm Ir-O}=0.5d_{0}^{\rm Ir-
Ir}/\cos{\theta}$. $E_{\mathrm{z,xz/yz}}$ are the $p-d$ Slater-Koster
integrals HarrisonBook2004 , and $\Delta_{\mathrm{pd}}=E^{\mathrm{on-
site}}_{\mathrm{xz/yz}}-E^{\mathrm{{on-site}}}_{\mathrm{z}}$ is the difference
between on-site energies of the corresponding $d$ and $p$ orbitals, which are
obtained from the projected density of states and shown in Supplemental Figure
3(a).

We obtain that indirect overlap for $yz$ orbitals along $a^{\prime}$ axis is
zero (not shown), and indirect $xz$-$xz$ overlap decreases slightly under
compressive strain (Supplemental Figure 3(b)). The contribution of the
indirect $xz$-$xz$ overlap to the overall $xz$-$xz$ hopping is significantly
reduced compared to the case of the $xy$-$xy$ Ir-Ir overlap by the denominator
$\Delta_{\mathrm{pd}}$, the energy difference between corresponding Ir and O
orbitals (Supplemental Figure 3(a)). The $xy$ orbital is lower in energy than
$xz/yz$ orbitals due to the octahedra elongation, and thus closer to the
oxygen orbitals, making denominator $\Delta_{\mathrm{pd}}$ smaller and thus
the overall contribution of the indirect hopping bigger for $xy$ orbital.
Moreover, upon compressive strain, the octahedra get further elongated
BhandariNJP2019 , and, therefore, the denominator would further increase
(decrease) for $xz$\- ($xy$-) indirect hopping contribution.

a
b

Supplementary Figure 3:  (a) The strain dependency of the on-site energy of
iridium $d$ orbitals and neighboring oxygen $p$ orbitals along
$a^{\prime}$-axis. We note that due to the lattice symmetry, for an oxygen
neighboring iridium atom along $b^{\prime}$-axis, the energies of
$p_{\mathrm{x}}$ and $p_{\mathrm{y}}$ orbitals are interchanged. We note that
when averaged over the unit cell, the energies of $p_{\mathrm{x}}$ and
$p_{\mathrm{y}}$ orbitals are the same, as expected from symmetry
considerations.(b) Slater-Koster integrals for indirect hopping between
rotated Ir NN orbitals $\tilde{E}$ via oxygen $p$ orbitals (along
$a^{\prime}$-axis) as a function of octahedra in-plane rotation angle
$\theta$, Ir-Ir distance $d$.

## References

  * (1) Glazer, A. M.  The classification of tilted octahedra in perovskites.  _Acta Crystallographica Section B_ 28, 3384–3392 (1972).  URL https://doi.org/10.1107/S0567740872007976.
  * (2) Seo, A. _et al._ Compressive strain induced enhancement of exchange interaction and short-range magnetic order in $\mathrm{S}{\mathrm{r}}_{2}\mathrm{Ir}{\mathrm{O}}_{4}$ investigated by Raman spectroscopy.  _Phys. Rev. B_ 100, 165106 (2019).  URL https://link.aps.org/doi/10.1103/PhysRevB.100.165106.
  * (3) Harrison, W. A.  _Elementary Electronic Structure_ (World Scientific Press, Singapore, 2004).
  * (4) Bhandari, C., Popović, Z. S. & Satpathy, S.  Electronic structure and optical properties of ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$ under epitaxial strain.  _New Journal of Physics_ 21, 013036 (2019).  URL https://doi.org/10.1088/1367-2630/aaff1b.