Evolution of electronic and magnetic properties of Sr$\mathbf{_2}$IrO$\mathbf{_4}$ under strain

Motivated by properties-controlling potential of the strain, we investigate strain dependence of structure, electronic and magnetic properties of Sr$_2$IrO$_4$ using complementary theoretical tools: {\it 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.


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 Sr 2 IrO 4 and Ba 2 IrO 4 . [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, Sr 2 IrO 4 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. 3,4 In Sr 2 IrO 4 , resistance indeed decreases until the pressure of around 25-30 GPa (which according to Ref. 5, corresponds approximately to a strain of -4%), 5 or, according to a very recent study6, 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. 6 So far, no metallization in Sr 2 IrO 4 or other iridates (Sr 3 Ir 2 O 7 7, 8 , BaIrO 3 , 3, 9 etc) has been observed at pressures up to  GPa. 3,6 Moreover, there is also surprisingly little correlation between the insulating behavior and magnetism 10 as the latter disappears at around 20 GPa (roughly -2.9 % strain) in Sr 2 IrO 4 5, 11 and 14.4 GPa (roughly -2.1 % strain) in Sr 3 Ir 2 O 7 , 8 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. 3,12 Iridium-based heterostructures and superlattices have therefore emerged as a whole new field very recently. [13][14][15][16][17][18][19][20][21] 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 Sr 2 IrO 4 , 2, 22, 23 as well as transport properties. 25 A shift of the two-magnon Raman peak to higher energies was observed under tensile strain, 2 albeit much weaker than the shift observed in the canonical Mott-Hubbard insulator K 2 NiF 4 and cuprates like Bi 1.98 Sr 2.06 Y 0.68 Cu 2 O 8+δ . 26 In Ref. [23], the authors used resonant inelastic scattering (RIXS) to show that magnetic dispersion in Sr 2 IrO 4 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 (π/2, π/2) already upon the tensile strain of +2%. Upon compressive strain, the energy of (π, 0) magnon was shown to increase. 22,23 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. 3,12 Unveiling the details of the interplay of lattice, magnetic, and other degrees of freedom in Sr 2 IrO 4 is needed to understand the recently observed electrical control of octahedra rotation 27,28 and the much-debated strong magnetoelastic coupling. [29][30][31][32] Currently, a clear understanding of neither how exactly nor by which mechanism do superexchange and hopping parameters in Sr 2 IrO 4 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 θ of the oxygen octahedra (see Fig. 1). 33 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, 1 so one needs to resort to theoretical methods where correlations are treated nonperturbatively, 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, 4, 35-38 so ab-initio methods are demanded.
Another difficulty is that as one eventually approaches a possible metal-insulator transition at high 3 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 scalarrelativistic DFT bandstructure obtained within the generalized gradient approximation (GGA). 39 Subsequently, we solve an extended t − J model within the self-consistent Bohr approximation (SCBA) to obtain the angle-resolved photoemission spectra (ARPES) and study the straincontrolled 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
Evolution of hopping parameters under strain. Sr 2 IrO 4 shows an in-plane staggered octahedra rotation characterized by a single parameter: O − Ir − Ir angle denoted by θ in Fig. 1. Under ambient conditions, the octahedral rotation is found to be θ ≈ 13.6 • for the relaxed structure, which is close to the reported experimentally value of 11.8 • . 40 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 θ, 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.
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), 5 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 or b axes is denoted as t 1 , between xz(yz) along a (b ) axis as t 2 , and between xz(yz) along b (a ) 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 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 (b ), 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 , and yz carriers which can only propagate along b ) and suppression of the bond-independent and thus truly twodimensional xy carriers. This compass-model-like 41 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 ∼ 20 GPa, not too far from the value of resistivity minimum under pressure. 5,6 To disentangle the contribution of inter-atomic distance d and the octahedral rotation θ to the hopping parameter trends with strain, we employ the analytical approaches of Glazer and Slater-Koster. The Glazer picture 1 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 θ. However, as  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.
We then proceed with a more specific orbital-resolved Slater-Koster-integrals-based approach. 3, 44 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 ddσ ,V ddπ , V ddδ for a known bond direction l, m, n as tabulated in Slater-Koster tables. 3,44 In Sr 2 IrO 4 , we also need to account for the rotation of the d orbitals within the t 2g sector due to the in-plane octahedral rotation. 45 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 elementẼ 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:Ẽ where θ is the in-plane rotation of the IrO 6 octahedra (see Fig.1). Similarly, for the overlap between the rotated xz(yz) orbitals along the a direction, we get and for interatomic interorbital overlap along x: (3) Figure 3 shows the resulting hybridization matrix elementsẼ as a function of the in-plane octahedral rotation θ, 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 elementsẼ on strain ( Fig. 3c) then accounting for the change of bond angle θ. This is also consistent with the quantum chemistry study. 22 However, not all trends obtained from the DFT calculations are well reproduced: the hopping parameter E § †, § † 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.
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 α = p x , p y orbitals of the oxygen,Ẽ xy,O,xy = α=px,py E xy,α,xy . The hopping integral is calculated as of t 1 (xy-xy hopping) under compressive strain suggests an electronic state crossover as the role of xy orbitals in the composite J eff = 1/2 is decreasing. Notably, a pressure-induced phase transition was also suggested in a recent X-ray powder diffraction study 46 at pressures around 20 GPa, which should correspond to approximately −3% strain and is in good agreement with our findings.
Overlap of the spin-orbit coupled J eff states. We now estimate the overlap between the J eff = 1/2 states for NN, 2NN, and 3NN (denoted τ , τ and τ , correspondingly), which can be calculated from t 2g orbitals overlap using the Clebsh-Gordon coefficients. 2,47 In Fig Table   1 in Ref. 23), which in ambient conditions are assumed to scale with τ 2 . 2 However, in a recent RIXS study on strained Sr 2 IrO 4 , 23 , the authors suggest that the simple J ∝ τ 2 /U relationship fails for strained Sr 2 IrO 4 due to the polaronic renormalization of the charge excitations. In particular, the first-neighbor exchange interaction J 1 was shown to decrease slightly upon the tensile strain, while J 2 and J 3 decreased much faster, based on a fit of the Heisenberg model to the measured magnon dispersion. 23 An earlier RIXS study also suggested that magnetic exchange interaction J increases upon the compressive strain. 22 Consistent with both RIXS studies, 22,23 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 τ 's is significantly slower than that observed for superexchange interaction J , indicating that J ∝ τ 2 /U relationship indeed fails for strained Sr 2 IrO 4 . Our DFT strain trends are also consistent with the modest increase of magnetic exchange interaction J 1 under compressive strain reported in the two-magnon Raman study. 2 It is interesting to compare the trends observed in Sr 2 IrO 4 to those in 3d transition metal oxides -cuprates. In Ref. [ 48] authors used XAS at Cu 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-meanfield level.
To study the evolution of the Fermi surface under strain we calculate photoemission spectral functions of strained Sr 2 IrO 4 , using extended t − J model formalism developed in Ref. [53]. The extended t − J model used in the calculation (see Methods) depends on two sets of parameters: the magnetic exchange parameters J 1 , J 2 , J 3 , Ising anisotropy coefficient ∆, 54 , and the hopping parameters t i describing overlap of the t 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 − J model. 53 To properly account for the changes in the electronic structure, we need to account for the a) -0.52% b) +1.53%  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. 23 Second, upon compressive strain, the photoemission spectra of Sr 2 IrO 4 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 Sr 2 IrO 4 becomes 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  is shifted to (π/2, π/2) at the tensile strain and to (π, 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 We also calculated the photoemission spectra of Sr 2 IrO 4 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 lowenergy 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 (π/2, π/2) point. These features can be readily observed in the future ARPES measurementsa smoking gun test of our findings.

Methods
The hopping H t part is derived by projecting multiorbital Hubbard model employing orbitaldependent hopping parameters t i onto spin-orbit coupled basis. 53,58 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. 53,58 We evaluate the Green function G(k, ω) using the selfconsistent Born approximation (SCBA). 60 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. 60 The spectral functions are calculated numerically for a 16×16 cluster.
Data availability All the data that support the findings of this study are available from the corresponding author (E.M.P.) upon reasonable request.
Code availability The codes used in this study are available from the corresponding author (E.M.P.) on reasonable request.

Supplemental Information
We start by presenting hopping parameters for further neighbors extracted from DFT. In Supplemental Figure 1 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 inplane rotation angle θ 1 . Using strain-dependent angle values extracted from our DFT calculations, we get that the overall hopping t ∝ cos 4.5 θ/(0.5d) 3.5 should decrease under compressive strain (Supplemental Figure 2(a)), as indeed expected for metal oxides with perovskite structure 2 . 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 S1 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. 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 E xz/yz,z,xz/yz = (cos(θ)E −l,m,n z,xz/yz ∓ sin(θ)E −l,m,n z,yz/xz ) × (cos(θ)E l,m,n z,xz/yz ± sin(θ)E l,m,n z,yz/xz )/∆ pd , We obtain that indirect overlap for yz orbitals along a 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 ∆ 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 ∆ pd smaller and thus the overall contribution of the indirect hopping bigger for xy orbital. Moreover, upon compressive strain, the octahedra get further elongated 4 , and, therefore, the denominator would further increase (decrease) for xz-(xy-) indirect hopping contribution. and neighboring oxygen p orbitals along a -axis. We note that due to the lattice symmetry, for an oxygen neighboring iridium atom along b -axis, the energies of p x and p y orbitals are interchanged.
We note that when averaged over the unit cell, the energies of p x and p y orbitals are the same, as expected from symmetry considerations.(b) Slater-Koster integrals for indirect hopping between rotated Ir NN orbitalsẼ via oxygen p orbitals (along a -axis) as a function of octahedra in-plane rotation angle θ, Ir-Ir distance d.