Non-local Coulomb correlations in pure and electron-doped ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$: spectral functions, Fermi surface and pseudogap-like spectral weight distributions from oriented cluster dynamical mean field theory
Cyril Martins Benjamin Lenz Luca Perfetti Véronique Brouet François Bertran Silke Biermann
cond-mat.str-el





These two authors contributed equally to this work.

Laboratoire de Chimie et Physique Quantiques, UMR 5626, Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France

These two authors contributed equally to this work.

 Centre de Physique Théorique, Ecole Polytechnique, CNRS UMR 7644, Université Paris-Saclay, 91128 Palaiseau, France

Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA, Université Paris-Saclay, 91128 Palaiseau, France

Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubin-BP 48, 91192 Gif sur Yvette, France

 silke.biermann@polytechnique.edu
 Centre de Physique Théorique, Ecole Polytechnique, CNRS UMR 7644, Université Paris-Saclay, 91128 Palaiseau, France
Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France


                 
71.30.+h, 71.10.Pm, 71.10.Fd, 71.27.+a 






























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    e.g.2g1g
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    δω_1/2δω_c
    εε_v kε_c kw_v kw_c k
    We address the role of non-local Coulomb correlations and short-range
      magnetic fluctuations in the high-temperature phase of
    Sr_2IrO_4 within state-of-the-art spectroscopic and 
    first-principles theoretical methods.
    Introducing a novel cluster dynamical mean field scheme,
    we compute momentum-resolved spectral functions, which we find to be in
    excellent agreement with angle-resolved photoemission spectra.
    We show that while short-range antiferromagnetic fluctuations are crucial
    to account for the electronic properties of the material even in the 
    high-temperature paramagnetic phase, long-range magnetic order is not a
    necessary ingredient of the insulating state.
    Upon doping, an exotic metallic state is generated, exhibiting cuprate-like pseudo-gap spectral properties, for which we propose a surprisingly
    simple theoretical mechanism.
    Non-local Coulomb correlations in pure and electron-doped
      Sr_2IrO_4:
      spectral functions, Fermi surface and pseudogap-like spectral weight
      distributions from oriented cluster dynamical mean field theory
    
        Silke Biermann
        December 30, 2023
    ================================================================================================================================================================================================================
    
    
    
    § INTRODUCTION
    
    
    The iridium oxide Sr_2IrO_4 is not only isostructural to the celebrated
    high-temperature superconducting copper oxides of the La_2CuO_4 family.
    The similarities in the low-energy electronic structure of these compounds,
    with a single orbital forming the low-energy states, are even more intriguing.
    In the case of cuprates it is a single half-filled x^2-y^2 orbital, 
    subject to strong electronic Coulomb correlations, that determines the 
    low-energy properties, presumably including superconductivity <cit.>.
    In the t_2g^5 system Sr_2IrO_4 a complex spin-orbit entangled
    compound orbital carries a single hole <cit.>.
    While in the cuprates the single-orbital nature of the low-energy
    electronic structure results from the single-particle band structure,
    in the iridate it is the result of the joined action of Coulomb interactions
    and spin-orbit interactions that effectively suppress the degeneracy
    <cit.>.
    
    As in the cuprates, stoechiometric samples of Sr_2IrO_4
    are insulating as a result of Coulomb interaction effects beyond the single-particle band picture, and at low temperatures antiferromagnetic order sets in <cit.>. The phase diagram has been explored extensively
    both, using different theoretical methods
    <cit.>
    and
    by various experimental probes:
    optics <cit.>,
    photoemission <cit.>, 
    transport <cit.>,
    scanning tunneling microscopy and spectroscopy experiments  <cit.>, and X-ray spectroscopy <cit.>
    have contributed to establish
    the picture of an insulator with a strongly
    temperature-dependent gap that is however not affected by the onset of
    magnetic order. 
    This is  a fortiori intriguing since even in the paramagnetic phase long-range two-dimensional antiferromagnetic fluctuations exist, which have been found to exceed 100 lattice spacings <cit.>.
    Magnetic exchange interactions seem to 
    be of correspondingly long-range nature <cit.>.
    First principles dynamical mean field theory (DMFT) calculations <cit.>
    can rationalize the insulating nature of the compound, even in its
    paramagnetic phase. This is consistent with the experimentally found
    insensitivity of spectral or transport properties on the presence of
    absence of magnetic order <cit.>.
    Nevertheless the detailed analysis of band dispersions obtained within
    single-site DMFT  
    reveals interesting discrepancies between theory and experiment (see below).
    For the case of the related Ba_2IrO_4<cit.> 
    cluster DMFT leads to much better agreement with angle-resolved 
    photoemission data <cit.>.
    
    Following the analogy with the cuprates, one of the most intriguing
    questions is the evolution of the electronic structure when
    additional carriers are introduced.
    Recently, several groups have succeeded to dope Sr_2IrO_4,
    both on the electron- and hole-doped side. Most interestingly, however,
    despite all the analogies with the cuprates, to date no superconducting
    phase could be observed.
    
    A metallic state can be realized via cationic substitution both in the case of hole doping, e.g. with K <cit.>, Rh <cit.> or Ru <cit.>, and in the case of electron doping with La <cit.>.
    For La doping x≥0.04 the antiferromagnetic Néel temperature vanishes and the system (Sr_1-xLa_x)_2IrO_4 remains a paramagnetic metal down to lowest temperatures <cit.>.
    Other possibilities of electron doping are depositing a K surface layer <cit.> and via oxygen depletion <cit.>.
    
    Angle-resolved photoemission spectroscopy (ARPES) has been able to identify
    the band dispersions and Fermi surfaces in electron- <cit.>
    and hole-doped <cit.> samples.
    On the electron-doped side, the dominant Fermi surface feature is a
    lens-shaped electron pocket centred around the M-point of the
    crystallographic Brillouin zone. A detailed analysis of the onset of
    spectral weight suggests the existence of depletion regions <cit.>
    reminiscent of the pseudogap behavior observed in the cuprates <cit.>.
    
    In this paper, we establish a first principles description of the
    spectral properties of pure and electron-doped Sr_2IrO_4 beyond
    the DMFT approximation of a purely local
    many-body self-energy. We demonstrate that including short-range
    fluctuations is crucial to reliably assess spectral properties,
    which we find in excellent agreement with experiments.
    In the electron-doped case we find an exotic metallic state, whose properties
    we relate to recent angle-resolved photoemission spectra. 
    In particular, our calculations offer a surprisingly simple picture
    for the putative antinodal pseudogap found in experiments.
    It is in fact a direct consequence of strong inter-site Ir-Ir
    fluctuations.
    Our findings suggest that while the similarities in the electronic
    structure between iridates and cuprates cover various quite different
    aspects, these common features may not be considered as proxy for
    superconductivity.
    
    
    
    
    § ASSESSING ELECTRONIC PROPERTIES OF SR_2IRO_4
    
    
    The 5d transition metal oxide Sr_2IrO_4 crystallizes
    in a tetragonal crystal structure 
    derived from the K_2NiF_4 structure,
    well-known in  or ,  by lowering the
    symmetry by a 11^∘ rotation of its IrO_6 octahedra
    around the c-axis <cit.>[Further reduction of the symmetry by staggered tetragonal distortion of the IrO_6 octahedra <cit.> has not been considered here.].
    Below 240 K, canted antiferromagnetic (AF) order sets in<cit.>.
    Here, we focus on the paramagnetic (PM) insulating phase of Sr_2IrO_4 above 240 K.
    The rotations of the IrO_6 octahedra result in a doubling of the
    unit cell along the c-direction, and a primitive cell in the paramagnetic
    phase that is as large as in the antiferromagnetic one.
    Its four crystallographically equivalent Ir atoms are pairwise related by
    a screw-symmetry, with the orientation of pairs being rotated by 90 degrees
    between neighboring planes, see Fig. <ref>A and B.
    
    
    
    DMFT-based first principles calculations
    which introduce effective Hubbard interactions and assess the
    resulting quantum fluctuations locally on each Ir atom could
    indeed identify the insulating state of Sr_2IrO_4<cit.>,
    even in the absence of long-range magnetic order.
    The resulting description of spectral properties (see Fig. 3c of
    Ref. <cit.>, which is replotted as Fig. 3
    in the Supplemental Material[ See Supplemental Material at [URL will be inserted by publisher] ]) is however not in agreement with experiment.
    
    Here, we introduce
    an extension of  ab initio
    cluster DMFT that extends the concept to
     oriented clusters as
    representative entities of parts of the solid. As in dynamical
    mean field theory, a self-consistency condition restores the original
    symmetries of the lattice, which – in the case of an oriented cluster
    – concern however not only the usual translational symmetries, but
    also the point group of the solid. Details of the method and its
    practical implementation are discussed in the Supplemental Material [footnote].
    
    Our starting Hamiltonian is the three-orbital Hubbard-type
    Hamiltonian of Ref. <cit.>, where a
    single-particle Hamiltonian for the t_2g-manifold is derived from
    Density Functional Theory (DFT) calculations and augmented by Hubbard
    and Hund's interaction terms.
    As shown in Ref. <cit.>, however, the effect of the interactions onto
    the ȷ32 states is essentially a global shift that makes these orbitals
    completely filled, leaving only the 12 states around the Fermi level.
    Therefore, we take the result for the ȷ32 states directly from
    Ref. <cit.>, and include only the 12 states into our new
    cluster theory treatment. This is done via a tight-binding parametrization
    of the corresponding bands.
    
    The oriented cluster DMFT (OC-DMFT) treatment of this Hamiltonian focuses onto the local Green's function
    
        G_loc (ω) = ∑_𝐤,α G_α(𝐤, ω)
    
    with the momentum 𝐤- and orientation α-resolved Green's function 
    
        G_α(𝐤, ω) = [ω + μ - H(𝐤) - Σ^α_dimer(ω)]^-1.
    
    Here, Σ^α_dimer is the self-energy of a dimer impurity
    problem, augmented by an orientation α=a± b, where a and b are the unit-cell vectors.
    The 𝐤 sum runs over the first Brillouin zone, restoring the
    translational invariance of the solid after calculating its self-energy
    from the quantum dimer problem, and the sum over α restores its
    point group. More details on the general philosophy and practical
    implementation can be found in the Supplemental Material [footnote].
    
    In a metal, electronic screening is drastically enhanced as compared to an
    insulator with profound consequences for the spectra <cit.>.
    This is even more true in 5d compounds, where the relatively extended
    nature of the 5d orbitals induces inter-atomic interactions that are relatively
    large as compared to the local ones [H. Jiang et al., in preparation].
    For this reason, the effective Hubbard interactions are expected to be
    smaller in the La-doped compound than in the pure sample.
    We mimic this effect here by using a smaller on-site Hubbard interaction
    for the doped case (U_eff=0.6 eV) than for the undoped one (U_eff=1.1 eV).
    
    
    
    § RESULTS
    
    
     §.§ Undoped Sr_2IrO_4
    
    
    We have measured ARPES spectra of Sr_2IrO_4,
    under the experimental conditions described in the Supplemental Material [67].
    Figure <ref>C displays the resulting spectra along the Γ-M-X-Γ
    path in the first and second Brillouin zone. Due to matrix element effects,
    these results display characteristic differences, with spectra in the first
    Brillouin zone amplifying the 12 contribution, while in the second
    Brillouin zone the ȷ32 contribution is dominant. 
    This strong matrix element effect expresses the fact that the corresponding Fourier
    component of the potential (lowering the symmetry from I_4 / mmm to I_4 / acd) is weak.
    In agreement with ARPES spectra in the literature <cit.>, we find the
    first removal state at Γ to be of ȷ32 character, while the
    12 form a strongly dispersive feature that displays a maximum at the
    X point.
    In our theoretical calculations, we do not address matrix element effects,
    which would differentiate results between different Brillouin zones.
    For this reason, the theoretical spectral function is compared to the
    sum of the experimental spectra.
    Most intriguingly, in single-site DMFT calculations (see Fig. 3 in the Supplemental Material [67]), 
    the 12 states
    form a very weakly dispersive feature with an onset of spectral weight at -0.2 eV below the Fermi level, in strong disagreement with the experimental spectra in Fig. <ref>C.
    
    Figure <ref>D displays the result of our present calculations that
    include non-local many-body correlations within our new OC-DMFT scheme.
    The comparison to the experimental spectra 
    yields impressive agreement, demonstrating that non-local many-body
    effects were indeed the missing ingredient for assessing spectral
    properties of this compound. The OC-DMFT treatment effectively
    includes inter-iridium site fluctuations in the half-filled 12
    manifold, and in particular includes the inter-site magnetic exchange
    of energy scale 4t^2/U into the description. With the present
    parameters we obtain a value of ∼108 meV (78 meV) for states of inter-layer (anti)bonding nature, 
    which coincides with the experimental estimate of the magnetic exchange coupling J=4t^2/U∼100 meV <cit.>.
    As a consequence of the antiferromagnetic fluctuations, the 12 band is much more dispersive than the single-site DMFT calculation and each of the Hubbard bands [From the dimer cluster used within our OC-DMFT treatment four spectral branches emerge: A bonding and antibonding band and two corresponding satellites, see also Ref. <cit.>. Therefore one should rather refer to the branches with large spectral weight visible in the spectral plot of Fig. <ref>D as (anti)bonding bands. Actually the correlation satellites or Hubbard bands are the outer excitations and carry less spectral weight. However, to be consistent with the notation in standard literature on Sr_2IrO_4 we use the notion of Hubbard bands when discussing the undoped compound.] has a width of ∼0.8 eV.
    
    
    
    Our analysis further allows for a refinement of the identification of
    the character of the spectral features in the first and second Brillouin zone.
    In Fig. <ref>C1, the signal comes from the first Brillouin zone and is dominated by the 12 manifold, even though along the MX direction some
    spectral weight around -1.1 eV originates from the ȷ32 manifold. 
    In Fig. <ref>C2, the signal comes from the second Brillouin zone and is dominated by the ȷ32 bands. 
    Our calculations show that even within the ȷ32 manifold the experimental signal is selective with respect to the m_j quantum number.
    The main contribution to the spectral weight seen in the second Brillouin zone comes from the m_j=1/2 band whereas the m_j=3/2 contributes to the first Brillouin zone.
    
    The left panel of Fig. <ref>A shows an ARPES spectrum acquired at
    the X point at -0.25 eV at room temperature.
    According to ARPES measurements of the magnetic phase, the lowest energy excitations disperse up to the X point and never cross the Fermi level. 
    This spectrum is qualitatively similar to the one measured below T_N in previous works <cit.>, demonstrating that spectral properties are
    largely insensitive to the presence or absence of long-range magnetic order.
    The lack of a Fermi level crossing excludes the presence of metallic quasiparticles even in the paramagnetic phase, in agreement between theory and experiment.
    By comparing the measured spectrum with the calculated spectral function, the overall agreement allows for a clear identification of the peak at -0.25 eV with the 12 lower Hubbard band. 
    We show in the left panel of Fig. <ref>B the photoelectron intensity map collected at -0.5 eV. 
    The intense blobs observed at Γ originate from the top level of the ȷ32 band whereas the spectral features around M and X arise from the 12 band which form circles centered around Γ with a radius of 0.5 Å^-1. 
    
    
    
     §.§ La-doped Sr_2IrO_4 and the putative pseudogap
    
    
    In the following, we investigate spectral properties of the electron-doped system and compare our results to experimental findings on (Sr_1-xLa_x)_2IrO_4 with x=0.05<cit.>. In this doping regime the system is a paramagnetic metal down to lowest temperatures.
    
    
    Figure <ref> displays the spectral function A(𝐤,ω) along the Γ-M-X-Γ path of the Brillouin zone resulting from our calculations for an electron doping of 10%. 
    The features of the upper Hubbard band (UHB) of the undoped compound can be continuously connected to features of the quasiparticle states at the Fermi level, which are visible in Fig. <ref>.
    In our dimer picture, this reminiscent feature of the UHB arises as bonding and antibonding states of the cluster, which is why we will refer in the following to the lower (upper) branch of the 12 band as (anti)bonding band [footnote].
    
    
    The antibonding band (ABB) crosses the Fermi level and causes a Fermi pocket around the M point.
    The positions of the band maxima at the high symmetry points have changed slightly as compared to the undoped case.
    At the M point the bonding band (BB) of the 12 appears around -0.37 eV and at the X point it is in close proximity to the Fermi energy.
    There, depending on resolution 
    finite weight of the tail of the broad peak can be picked up at the Fermi energy.
    At the Γ point the signal of the ȷ32 manifold can be seen at -0.4 eV.
    
    Overall, the spectral function displays excellent agreement with available ARPES data <cit.>.
    In agreement with Ref. <cit.> we find a nearly linear dispersion close to the Fermi level, which can be traced back to the ABB of the 12 band.
    Compared to <cit.> our spectrum differs slightly at the M point in that in <cit.> the dispersion close to the Fermi energy 
    
    was interpreted to extrapolate to a Dirac point around -0.1 eV and a quasilinear continuation for even lower energies, possibly with a small gap at the high symmetry line.
    Here, we see a distinct gap between bonding and antibonding 12 band at the M point and can clearly distinguish between states of the ABB forming the electron pocket and those of the BB.
    Although the lower part of the experimentally measured dispersion matches our calculated spectrum, the states right below the putative Dirac point are not found here.
    Our calculation rather suggests a scenario similar to the one of Ref. <cit.>, where a similar quasilinear dispersion was found for a doping of x=0.04.
    There, the vertical dispersion was discussed as stemming from spectral weight of the large tail of the 12, which is picked up by the momentum distribution curve <cit.>.
    
    
    
    We now turn to the Fermi surface.
    The ABB crosses the Fermi energy close to the M point resulting in a lens-shaped electron pocket, 
    which is centered around the M point and elongated along the Brillouin zone boundary (see Fig. <ref>A).
    Our theoretical findings agree well with recent ARPES measurements on the electron-doped system (Sr_1-xLa_x)_2IrO_4, both considering the shape and the size of the pockets <cit.>.
    
    The ARPES measurements also revealed unusual metallic behavior <cit.> with an antinodal pseudogap at a La concentration of x=0.05<cit.>. 
    Figure <ref>B shows
    particle-hole averaged spectral functions (“symmetrized spectra”) as they
    are often plotted from experimental data. We move
    along a semicircular k-path, which is a natural extension of the Fermi
    surface (see Fig. <ref>A).
    At small angles 4^∘≤α the symmetrized spectral function shows a pseudogap, which closes again at α≈15^∘.
    For angles 21^∘≤α≤26^∘ an additional feature emerges at sufficiently large temperatures, which could be misinterpreted as a pseudogap, too.
    However, since the hole-part of the spectrum is still close to the Fermi energy, this feature can be traced back to contributions of hole-parts of the spectrum due to large enough temperatures, which then resemble a pseudogap in the symmetrized spectral function.
    Finally, for angles α≲ 45^∘, i.e. close to the antinodal point, spectral features closely resemble the pseudogap features emerging at smaller angles close to the tips of the electron pockets.
    In experiment, based on symmetrized energy distribution curves, the regions outside the pockets were identified as showing a pseudogap, which reaches up to the antinodal point <cit.>.
    Our calculations reveal that the antinodal regions do not show the same pseudogap, which stems from the upper Hubbard band that forms the pockets, but rather relicts of the lower 12 Hubbard band (Fig. <ref>). 
    To better understand the origin of pseudogap-like features in the symmetrized spectral function, we focus in Fig. <ref>A on Fermi surface cuts of the spectral function. 
    Within OC-DMFT we have access to the precise position of the quasiparticle excitations. 
    However, in plots of the spectral function A(𝐤,ω) we add a finite broadening η to mimic the experimental 𝐤 resolution.
    The top left panel shows the spectral density at the Fermi level with a realistic broadening η=0.10 eV. 
    It shows the lens-shaped Fermi surface pocket, which is in good agreement with ARPES spectra <cit.> (top right panel of Fig. <ref>A).
    Plotting the extracted pseudogap from the symmetrized spectral function in Fig. <ref>B along the semicircular 𝐤-path resembles the pseudogap found in ARPES (bottom left panel).
    
    In our calculation, we can trace back the origin of the pseudogap features by choosing an artificially reduced resolution 
    (see bottom right panel of Fig. <ref>A), which reveals the fine structure of the Fermi pocket.
    What seemed to be a sole pocket is actually composed of two separate lens-shaped pockets, which are tilted off-axis in opposite directions by roughly 15^∘ each.
    Given the experimental resolution it is clear that ARPES cannot resolve them separately.
    The dimer cluster used in our OC-DMFT calculation is the minimal setup to account for the two different IrO_6 orientations of adjacent Ir sites and leads to two configurations (related by a 90 degree rotation), which are schematically shown in Fig. <ref>C. 
    It is the breaking of orientation symmetry within each configuration, which gives rise to the stretched, tilted Fermi pocket at the Fermi energy as shown in the Fermi surface cuts of the spectral function in Fig. <ref>C. 
    
    Since both orientations alternate between the different Ir layers along the c-direction, Figure <ref>A shows the Fermi surface of the orientation average of both configurations shown in Fig. <ref>C.
    A realistic broadening, however, renders it impossible to identify two separate pockets, but rather suggests an interpretation as a single pocket structure (see Fig. <ref>A for η=0.10 eV).
    As a consequence, the path along which one extracts the energy distribution curves of this putative single pocket resembles the circular shape of the DFT result that was used in Ref. <cit.> and does not coincide with the Fermi surface of the (two-pocket) Fermi surface visible at higher resolution.
    
    Since the putative sole pocket visible at larger broadening is a superposition of the two canted pockets, its tip corresponds to the region between the two pockets. 
    When measuring there, the emerging symmetrized spectral function shown in Fig. <ref>B features a pseudogap. 
    In this sense, the appearance of the pseudogap close to the pocket's tip is a direct consequence of the non-local fluctuations. 
    
    
    
    
    § SUMMARY AND OUTLOOK
    
    In conclusion, we have performed  ab initio calculations within a novel
    oriented cluster dynamical mean field theory scheme, which includes non-local
    quantum fluctuations based on an oriented dimer as a reference system for a dynamical mean field theory scheme. We find excellent agreement between the resulting spectral functions for both, pure and electron-doped , with experimental photoemission data, emphasizing the role of non-local quantum correlations in these 5d compounds. 
    These findings provide new evidence for the electronic analogies between iridates and cuprates, making the absence of superconductivity an even more intriguing feature of the iridate compounds.
    In particular, the single-orbital nature of a compound, the emergence of a metallic state from a doped insulating parent compound, and the pseudogap-like features discussed in  do not 
    seem to be proxies for superconductivity.
    
    
    
    
    This work was supported by a Consolidator Grant of the European Research
    Council (Project CorrelMat-617196), the French Agence Nationale de la
    Recherche under project SOCRATE and IDRIS/GENCI Orsay
    (Project No. t2017091393).
