Spin-polaron ladder spectrum of the spin-orbit-induced Mott insulator Sr{}_{2}IrO{}_{4} probed by scanning tunneling spectroscopy

Spin-polaron ladder spectrum of the spin-orbit-induced Mott insulator SrIrO probed by scanning tunneling spectroscopy

Jose M. Guevara Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Zhixiang Sun Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Ekaterina M. Pärschke Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA    Steffen Sykora Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Kaustuv Manna Present address: Max-Planck-Institute for Chemical Physics of Solids, 01187 Dresden, Germany Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Johannes Schoop Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Andrey Maljuk Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Sabine Wurmehl Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany Institute for Solid State Physics, TU Dresden, 01069 Dresden, Germany    Jeroen van den Brink Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany    Bernd Büchner Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany Institute for Solid State Physics, TU Dresden, 01069 Dresden, Germany Center for Transport and Devices, TU Dresden, 01069 Dresden, Germany    Christian Hess c.hess@ifw-dresden.de Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01069 Dresden, Germany Center for Transport and Devices, TU Dresden, 01069 Dresden, Germany
July 14, 2019
Abstract

The motion of doped electrons or holes in an antiferromagnetic lattice with strong on-site Coulomb interaction touches one of the most fundamental open problems in contemporary condensed matter physics. The doped charge may strongly couple to elementary spin excitations resulting in a dressed quasiparticle which is subject to confinement. This ’spin-polaron’ possesses internal degrees of freedom with a characteristic ’ladder’ excitation spectrum. Despite its fundamental importance, clear experimental spectroscopic signatures of these internal degrees of freedom are scarce. Here we present scanning tunneling spectroscopy results of the spin-orbit-induced Mott insulator SrIrO. Our spectroscopy data reveal distinct shoulder-like features for occupied and unoccupied states beyond a measured Mott gap of  meV. Using the self-consistent Born approximation we assign the anomalies in the unoccupied states to the spin-polaronic ladder spectrum in this material. These results confirm the strongly correlated electronic structure of this compound and underpin the previously conjectured paradigm of emergent unconventional superconductivity in doped SrIrO.

pacs:
74.25.nd, 74.20.Pq, 74.70.Xa, 75.30.Fv

The concept of confinement in physics is ubiquitous. An interesting example is that of quantum chromodynamics concerning the internal structure of hadrons, such as protons or neutrons. These particles are constituted by fractionally charged quarks held together by gluons, the virtual gauge bosons which mediate the strong force. Unlike other fundamental forces, the interaction potential between the quarks does not decay with distance but increases linearly with distance. As a consequence, quarks practically cannot exist in an isolated form – they are confined to form the hadron. Experimental evidence of quarks thus relies on indirect fingerprints, namely the detection of excited states proving internal degrees of freedom of the hadrons in deep inelastic scattering experiments Greiner ().

A well known and important solid-state analog of a confined particle is the so-called spin polaron, which describes the motion of a single charge (hole or doublon) added to an antiferromagnetic and insulating ground state of an effective correlated background medium. Thereby, the magnetic excitations of the antiferromagnetic (AF) background can be theoretically described by a system of bosons (magnons) which couple to the introduced charge carrier via creating virtual bosonic fluctuations. In this way the charge interacts strongly with its environment of ordered spins and forms a new quasiparticle – the spin polaron. Its excitations have been investigated by, e.g., the self-consistent Born approximation (SCBA) Martinez1991 (); Kane1989 (), quantum wave function methods Reiter1994 () for a single hole within the - model, and exact diagonalization Hamad2008 (). These studies show that the spin polaron is characterized by an environment of misaligned spins (Fig. 1(a-d)) forming an effective confinement potential, where the charge can occupy excited states of different orbital character  Wrobel2008 () (see Fig. 1(e-h)). In the one-particle spectral function these excitations manifest themselves by the occurrence of a rather flat and ladder-like structure (Fig. 1(i)). Due to quantum fluctuations the spin defects can relax and the quasiparticle becomes dispersive. Despite of the proven fundamental importance for rationalizing many open problems in the physics of correlated electron systems, a direct measurement of the internal degrees of freedom of the spin polaron which proves its peculiar confined nature is still lacking. Here we use scanning tunneling spectroscopy (STS) to specifically probe the excited states of the spin polaron in a correlated material. We compare the tunneling spectra with theoretical calculations based on the self-consistent Born approximation for the spin-polaronic ladder spectrum and find excellent agreement.

Figure 1: Illustration of the nature of the spin polaron. Its simplest form is captured by the so-called t- model which describes a hole in an antiferromagnetic background of Ising spins with the kinetic energy of the hole and the antiferromagnetic Ising exchange energy between neighboring spins. Panels (a-d) illustrate that a gain in kinetic energy through hopping processes of the hole immediately results in increasing magnetic energy through the destruction of the antiferromagnetic correlation. Thus, the motion of the hole is inhibited and it remains confined to its original position. Nevertheless, the hole in interaction with the antiferromagnetic background possesses well-defined excited states with different orbital character, the wavefuctions of which (Reiter functions) resemble atomic states. These are characterized by the size (in terms of the number of virtual hopping processes) of the spin polaron, the wavefunctions’ symmetry, and a discrete ladder excitation spectrum. (e) -wave like ground state of the spin polaron. (f-h) First excited states corresponding to -wave (f,d) and -wave-like states (h). , , , denote the coefficients of the Reiter wave function. (i) Ladder spectrum of the t- model Martinez1991 () where the exchange interaction lies in the parameter region relevant for SrIrO. See Refs. Kane1989, ; Martinez1991, ; Reiter1994, and the Supplementary Materials (SM) for details.

A hallmark of the spin polaron is its connection to the nature of unconventional superconductivity in the cuprates which is believed to emerge from a quasi two-dimensional correlated Mott-insulating antiferromagnetic parent state upon charge doping Martinez1991 (). The spin polaron and its itinerancy straightforwardly explains the rapid destruction of the antiferromagnetic parent state of the cuprates upon hole doping. Furthermore, it is one key ingredient in many theoretical models which address superconductivity as well as competing phases such as stripe correlations in the underdoped regime of the cuprates’ phase diagram Chernyshev2003 ().

In recent years, it has been increasingly noticed that quasi-2D iridium oxides exhibit correlated physics that is quite similar to that of the cuprates. In particular, SrIrO shares many parallels with the isostructural LaCuO, a prominent Mott-insulating parent compound of the cuprate high-temperature superconductors damascelli2003angle (). The intricate interplay of strong spin-orbit coupling (SOC) and Coulomb repulsion causes the 5 electrons to localize in a state with pseudospins forming the lower Hubbard band (LHB) of the material with a strong AF exchange interaction kim2008novel (); kim2012magnetic (). AF order occurs below 240 K () cao1998weak (), and quasi-2D magnon excitations have been detected kim2012magnetic (); Steckel2016 (). In view of these similarities it is reasonable to expect spin polaron physics to be relevant in SrIrO Paerschke2017 () and it has been argued that a proper doping scheme can drive the material into a high-temperature superconducting phase wang2011twisted ().

Results

We used as-grown single crystals of SrIrO for which a reduced resistivity as compared to the stoichiometric parent compound (see SM Fig. S1) allows for high-resolution scanning tunneling spectroscopy (STS) measurements even at very low temperature ( K). Scanning tunneling microscopy (STM) data obtained at the cold-cleaved (about 10 K) crystals’ surface (Fig. 2(a)) yield atomically resolved SrO-terminated flat terraces with several local defects (about 2% with respect to Ir).

Figure 2: Topography and the tunneling conductance of the clean area. (a) Topography of the sample surface measured at K, with bias voltange V and tunneling current pA. (b) Representative large scale tunneling conductance spectrum taken on a clean place at K, the tip was stabilized at mV and pA. Red arrows indicate fine structure signatures arising from excited states of the spin polaron.

Fig. 2 (b) shows an overview tunneling conductance () spectrum taken at 8.8 K at a place free of defects. The data reveal a sizeable gap  meV where between about  mV and  mV, in agreement with earlier findings dai2014local (); yan2015electron () which we identify as the signature of the Mott gap of the material. At bias voltages below and above this gap, the tunneling conductance reveals a shoulder-like increase which straightforwardly can be associated with the density of states of the LHB and the UHB. At  V, the increases sharply, which we attribute to further high-energy states, in qualitative agreement with optical spectroscopy propper2016optical (), and a possible energy dependence of the tunneling matrix element. We therefore restrict the following discussion to  V.

Remarkably, the closer inspection of the of the LHB and the UHB reveals a distinct fine-structure of peak- or shoulder-like features (indicated by arrows in Fig. 2 (b)). These interesting features indicate a direct coupling of the tunneling electrons/holes to specific excitations of the Mott state. Indeed, the theoretical investigation of spin polaron physics of SrIrO, which we discuss in the next section, brings into light that this signature is directly related to the excited states of the confined spin polaron quasiparticles including their inherent ladder spectrum.

Figure 3: Comparison of the theoretical and the experimental differential conductance. The black solid line shows the theoretical spectrum calculated from Eq. (2) using the momentum-integrated Green’s function of the hole. The yellow line reproduces the experimental spectrum shown in Fig. 2(b). At negative bias, the total spectrum is composed of separate contributions due to the multiplet structure of the polaron, i.e. the singlet (blue line) and the triplet (red line) contributions. Since the SCBA inherently provides only the distance between excited levels, the calculated results at positive and negative bias are plotted as to match the respective lowest energy anomaly in the experimental data. Furthermore, the experimental spectrum has been scaled along the axis in order to achieve a rough match with the calculated results. To highlight the typical ladder structure of the spectrum at positive bias we show in the inset the calculated spectrum in a wider energy range. On the negative side equidistant features are not present due to dispersive internal degrees of freedom of the charge excitation.

The clear signatures of the AF Mott physics in the quasi-2D SrIrO  kim2012magnetic (); kim2014excitonic () suggest that the underlying electron system can be modeled by a multi-orbital 2D Hubbard model with spin-orbit coupling which has an AF ground state as it is known from the usual one-band Hubbard model  Kane1989 (). This property simplifies the theoretical treatment by the possibility to apply the well-known “single-hole problem” to describe the relevant excitations of the magnetically ordered pseudospin ground state Schmitt1988 (); Ramsak1993 ().

Quantitive interpretation of the tunneling spectrum

Constructing a polaronic model to calculate spectra, we address separately the positive and negative bias regions since the strong on-site correlations render these two cases very different Paerschke2017 ().

When a negative bias voltage is applied, the electrons are removed from the sample, tunneling towards the tip. This creates an excitation in the configuration, which can be locally described as a configuration with its complicated intrinsic multiplet structure Chaloupka2016 (). In the lowest energy subspace of the Hilbert space this charge excitation would form a singlet and a triplet state. Therefore, to describe a charge excitation on the negative bias side, we introduce a charge excitation creation operator h with an additional degree of freedom . Opposite to this, applying positive bias voltage, results in adding an electron to the local Ir site with configuration. Hence, the charge excitation that one has to consider would resemble the filled-shell configuration and can be described by polaronic excitations shown in Fig. 1.

The motion of the charge excitation on positive(+) and negative(-) sides of the spectra is described by the Hamiltonian: {linenomath*}

(1)

where is the part which includes the low energy excitations of the AF ground state. The hopping part of the Hamiltonian, , describes the kinetic energy of the charge coupled to the magnons, which gives rise to the polaron quasiparticle. The low-energy effective polaron model described here has the same operator structure as the effective polaron model of the t-J model but has more components than the latter due to the multiplet structure of the polaron. Interestingly, this additional degree of freedom also allows for more hopping channels, including free (i. e. not coupled to magnons) hopping between first neighbors, see Materials and Methods for details. The Hamiltonian and its parametrization used here also gives very good quantitative description of the measured ARPES spectra on SrIrO Paerschke2017 (). Specifically, we have evaluated the one-particle Green’s function within the self-consistent Born approximation.

To relate the described modeling to our measurements we exploit the usual proportionality between the tunneling differential conductance and the density of states and calculate the using the relation

(2)

where the time evolution in is determined by the Hamiltonian (1). Fig. 3 shows the results in direct comparison with the experimental data. In the inset of Fig. 3, one clearly sees that the calculated spectra on the positive bias side possess ladder spectral features which are similar to those of the much more simplistic spin polaron in terms of the t- model (see Fig. 1(I)) in a remarkable way. Indeed, the ladder structure on the positive bias side can be clearly identified in the experimental data by the shoulder-like anomalies at about 700 meV and 1100 meV. We point out that the experimental tunneling spectra are subject of significant broadening, presumably by electron-phonon scattering processes and higher-order tunneling processes, which account for the differences between the theoretical and the experimental results.

Not surprisingly however, such a ladder spectrum is not present on the negative bias side – the polaron motion on the negative voltage is additionally greatly complicated by the internal degree of freedom of the charge excitation, which not only creates additional interacting channels but also provides a possibility for a nearest-neighbor free hopping of the polaronic quasiparticle. Therefore, the polaron quasiparticle becomes more dispersive and a considerable amount of spectral weight is transferred to the incoherent part of the one-particle spectrum. Altogether these two effects lead in the momentum summation in Eq. (2) to a more complex on the negative bias side (black line in Fig. 3). Nevertheless, one can study the different contributions to the total Green’s function which are carried by spin polarons of the two different values of the total quantum momentum, and . The calculated contributions are shown in Fig. 3 in blue and red, correspondingly. Unlike on the positive bias side, the two large peaks observed on the negative side of the tunneling conductance spectra separately correspond to singlet and triplet polarons. Accordingly, we assign the two shoulders/peaks at about  meV and  meV in the experimental data to be primarily of singlet and triplet character, respectively.

After having assigned these most salient aspects of the tunneling spectrum of SrIrO to essential spin polaron physics, we finally mention that a careful analysis of the experimental and theoretical spectra shown on Fig. 3 also allows to extract the value of Coulomb repulsion . We estimate in the range between 2.05 eV and 2.18 eV, where is the binding energy of the polaron for negative/positive bias (see Supplementary Materials) and the Mott gap accurate to the lowest quasiparticle peak bandwidth (both on positive and negative bias sides).

Discussion

Our findings experimentally and theoretically confirm the important role of the spin-polaronic quasiparticle for the physics of SrIrO. More specifically, our data for positive bias voltage, which correspond to electron doping of the AF, reveal clear-cut signatures for the prototypical spin polaron ladder spectrum, i.e., the fingerprints of the internal degrees of freedom of the electron being confined within the AF background. This suggests that the electron-doped regime of SrIrO – in analogy to the hole-doped cuprates wang2011twisted () – is particularly promising to exhibit similar phenomena as the hole doped cuprates. Indeed, the reported signatures of a -wave gap kim2016observation (); yan2015electron () and of stripe-like correlations Battisti2017 () indicate a phenomenology that can be traced back to the spin polaron physics Chernyshev2003 (); kim2014excitonic (). Thus, the current experimental and theoretical efforts  wang2011twisted (); kim2012magnetic (); de2015collapse (); yan2015electron (); kim2016observation (); Battisti2017 () to find the route to unconventional superconductivity in this material are strongly supported by our study. The situation is, however, less clean for the hole-doped regime where the physics is more complicated. Since here the spectral features are dominated by both singlet and triplet polarons, with singlet states being of lower energy with respect to the Fermi level, the analogy to the cuprates is not present. Nevertheless, if chemically achievable, an intricate and fascinating doping evolution governed by the interplay of the singlet and triplet polarons may be expected in the hole-doped regime, too.

Methods

Sample preparation

SrCO and IrO powders (with 4N purity) were taken in stoichiometric ratio and mixed with SrCl flux with a 1:5 sample-to-flux weight ratio. The mixture was heated to 1210 C for 12 h and then slowly cooled to 1000 C with a cooling rate of 4 C/h, followed by a rapid cool-down to room temperature at 150 C/h. Crystals of SrIrO with diameter up to 5 mm were filtered from the residue after dissolving the flux in hot water. The crystals were grown in a Pt crucible (50 ccm) with a lid to reduce a flux evaporation. The crystals were characterized regarding microstructure (scanning electron microscopy), composition (energy dispersive x-ray analysis), crystallographic structure (single crystal diffraction), magnetic properties (magnetometry), and resistivity (see SM).

Scanning tunneling microscopy/spectroscopy measurements

STM and STS measurements were carried out using a home-built variable temperature STM schlegel2014design (). Scanning tunneling microscopy singles crsytal samples were cold-cleaved in cryogenic vacuum at about 10 K.

Theory

The motion of the charge excitation on positive(+) and negative(-) sides of the spectra is described by the Hamiltonian: {linenomath*}

(3)

where describes low energy excitations of the AF ground state. It is given by {linenomath*}

(4)

where is the dispersion of the (iso)magnons represented by the quasiparticle states and . The hopping part of the Hamiltonian, , describes the transfer of the charge excitation in the bulk coupled to the magnons, which we will also address as polaron quasiparticle. It is given by: {linenomath*}

(5)
(6)

where are the two AF sublattices. The dispersions , ( ) describe the nearest, next nearest, and third neighbor free hopping. The terms ( and ) are vertices describing the polaronic hopping of the charge excitation on the positive (negative) side and are given explicitly in the supplementary materials (also see Ref. Paerschke2017 ()). All the vertices were obtained analytically in a limit of strong on-site Coulomb repulsion and depend on the five hopping parameters of the minimal tight-binding model obtained as the best fit of the latter to the LDA calculations. The model used here is based on the polaronic model we used to calculate ARPES spectra on SrIrO, see Ref. Paerschke2017 () for details.

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Acknowledgements

General: We thank K. Wohlfeld, C. Renner, B.J. Kim and M. Allan for helpful discussions. Furthermore we thank D. Baumann and U. Nitzsche for technical assistance.

Funding: The project is supported by the Deutsche Forschungsgemeinschaft through SFB 1143 (projects C05, C07, A03, and B01) and by the Emmy Noether programme (S.W. project WU595/3-3). Furthermore, this project has received funding from the European Research Council (ERC) under the European Unions’ Horizon 2020 research and innovation programme (grant agreement No 647276 – MARS – ERC-2014-CoG).

Author Contributions: JMG. and ZS performed STM experiments. EMP and SS performed theoretical calculations. JMG., ZS, EMP, SS, JvdB, and CH analyzed and interpreted the data. Samples were grown and characterized by KM, AM, and SW. JS performed the resistance measurements. JMG, ZS, EMP, SS, JvdB and CH wrote the manuscript. All authors discussed the data and contributed to the manuscript. JvdB, BB, and CH designed the theoretical and experimental work.

Competing Interests: The authors declare that they have no competing financial interests.

Data and materials availability: Additional data and materials are available online and/or may be requested from CH.

Supplementary Materials

Resistivity data of the SrIrO samples

The in-plane resistance of the as grown SrIrO single crystals was measured using a standard 4-probe technique (5K-300K). Here, accounts for a possible oxygen deficiency. In the as-grown single crystals, is not controlled which is known to lead to a variation of the resistivity korneta2010electron (). Fig. S1 shows representative resistivity data for samples with very different temperature dependencies of the resistivity, where the more insulating characterisitics (labelled ’Insulating Sample’ in Fig. S1) can be attributed to an almost stoichiometric oxygen content () of the corresponding sample de2015collapse (). On the other hand, the only weakly insulating character of the other sample implies a small amount of oxygen vacancies korneta2010electron (). In order to be able to perform high-resolution tunneling spectroscopy at low temperature, we took advantage of the reduced resistivity of this sample (labelled ’STM Sample’ in Fig. S1) and used it for the tunneling experiments in our study. Note that the observed impurity amount of about 2% with respect to Ir in the topographic data shown in Fig. 2(A) is consistent with an oxygen deficiency of the same order korneta2010electron ().

Figure S1: Normalized in-plane resistivity of selected SrIrO samples. The strong reduction of the low-temperature upturn of the resistivity of the sample labelled ’STM Sample’ evidences a significant amount of oxygen vacancies.

Ladder spectrum in the t- model

Assuming that the ground state at half-filling can be described by a classical Néel state with spin excitations, one can describe the motion of the hole in the AF background by an effective Hamiltonian which naturally follows from an anisotropic t-J model Kane1989 () by assuming a finite ratio of the exchange parameters, ,

(7)

where a charge excitation is represented by a spinless fermion with creation operator and spin excitations are represented by the boson operators . The spin-wave dispersion where is the spin and is the coordination number of the underlying square lattice and the Bogoliubov factors are given by

(8)

The coupling of the hole to magnons is described by . Substituting to Eq. (7) we obtain the polaron representation of the t-J model,

(9)

where the coefficients become q-independent: and .

To show that indeed different excitations in the ladder spectrum of the t-J model can be directly observed in the tunneling spectroscopy experiment, we map the polaronic Hamiltonian (7) onto an effective system of free fermions and bosons

(10)

where the new effective Hamiltonian is related to the original Hamiltonian via a unitary transformation,

(11)

Such a method enables to study the polaron excitations as projected on the effective free particle, which can directly couple to the tunneling electrons in an STS experiment.

The unitary transformation (11) in general renormalizes the spin excitations of the background (first term of Eq. (10)) and generates the second term of Eq. (10). Since the Eq. (10) has the quadratic diagonal form and the transformation (11) is unitary, the energy quantities and can be seen as eigenenergies of the original model. The transformation (11) can be constructed and numerically carried out by using the projective renormalization method (PRM) (see Ref. Cho2016 ()). Within this method the polaronic term of the Hamiltonian is integrated out in steps (1500 in the actual calculation), leading to the renormalization of the fermion and boson energy parameters.

Using the unitary transformation (11), the one-particle spectral function can be calculated immediately,

(12)

where and are calculated in the renormalization process described above and represent the spectral weight of the particular polaron excitation.

The internal excitations of the polaron can also be visualized by its wave function. Following Reiter1994 (), we write it in the form

(13)

Here, is the product of the hole vacuum and the spin-wave vacuum, and , are so-called Reiter coefficients. We see that the wave function of the doped hole can in principle be approximated by a superposition of the wave function of a free hole and wave functions of the hole dressed with magnons.

Fig. S3 shows the calculated effective dispersion of the spin polaron for different values of the ratio . For small values of , i.e. close to the Ising limit, one clearly sees that the energy of the polaron increases as a function of its momentum k in stair-step fashion: as soon as the momentum k is sufficiently large to produce a magnon, the polaron is raised to the next excited level. The ground state of the polaron is characterized by momentum states around where only a finite range of momentum values is occupied. Since the values of examined in Fig. S3 are small (), the magnon dispersion is almost momentum-independent and approximately equal to the magnon dispersion in the t-J model (9). For smaller values of (shown in Fig. S3 with red circles), the polaron dispersion within each rung is quite flat, whereas for larger values of (green circles), the polaron becomes more dispersive. Overall, the polaron becomes less localized with the ratio decreasing.

The possibility to map the spin polaron model to an effective model of free charge carriers (dressed with characteristic ladder-like quasiparticle dispersion) indicates that it must indeed be possible to detect internal excitations of spin polarons in STM experiment.

To get better understanding of the nature of the polaron states shown on the Fig. 1, we calculate the first two Reiter coefficients and from Eq. (13) using perturbation theory with respect to the parameter assuming (strong coupling limit):

(14)

In this approximation, the spectral function of the hole has the form (similar to Eq.( 12))

(15)

This equation includes two different types of internal excitations. As one can see from the momentum dependence of the Reiter coefficients, the lowest excitation has -wave character and represents a rather localized state of the hole. The second excitation is spatially more extended due to its proportionality to -functions, and the sign of the coefficient changes as a function of momentum, which means that it is orthogonal to the first term.

.1 Relevance of the t- ladder physics to SrIrO

To show the relevance of the above discussed theory to the case of SrIrO we have calculated the scaling of the energy spacing between first and second excited states on the positive side of the tunneling conductance as a function of ratio for the material specific model (3). It is known Bulaevskii1968 () that for the t- model this energy spacing scales as , see Supplemental figure S2(a). As one can see on the Supplemental figure S2(b), energy gap calculated for model (3) follows the same law in the region of parameters relevant to the real material (shown in light gray).

Figure S2: (a,b) Comparison of energy spacing between first and second excitation state of polaron scaling as a function of ratio: (A) t- model, (b) material-specific t-J model defined by in Eq. (3). In light gray the region of the values relevant for the SrIrO is shown: eV, first neighbor hoppings takes values from eV to eV depending on the orbital character.
Figure S3: Polaronic quasiparticle dispersion for the effective Hamiltonian of the anisotropic t-J model given by Eq. (7) calculated for three different values of the ratio . The value of is fixed throughout the calculation. The spectrum becomes more ladder-like as approaches the Ising limit . For the lowest value of (red circles) the value of lies in the relevant for SrIrO parameter region (as indicated in Fig. S2(b)) and the energy spacing between the first and second excitation is of the order of .

Estimating the Coulomb repulsion

By careful analysis of the spectra shown in Fig. 2(b) one can also extract the value of the Coulomb repulsion . It is connected to the Mott gap value as

(16)

where () is the binding energy of the polaron formed when a hole (electron) is added to the ground state of the system. We estimate polaron binding energies by performing SCBA calculations setting the hopping part of Hamiltonian Eq. (3) to zero separately for positive and negative bias cases. In this way the polaron is artificially fully localized and its spectral function is simply a delta function. The binding energies are then given by a relative shift between these delta function peaks and the quasiparticle peaks of the full calculation (Fig. 3). From such a consideration the particular energy values are estimated to be

(17)

Then the Coulomb repulsion takes a value between and since the Mott gap correct to the lowest quasiparticle peak bandwidth (both on positive and negative bias sides).

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