⇦Chinese Physics Letters, 2023, Vol. 40, No. 3, Article code 037101 Tuning the Mottness in Sr$_{3}$Ir$_{2}$O$_{7}$ via Bridging Oxygen Vacancies Miao Xu (徐妙)1†, Changwei Zou (邹昌炜)1†, Benchao Gong (龚本超)2†, Ke Jia (贾可)3,4, Shusen Ye (叶树森)1, Zhenqi Hao (郝镇齐)1, Kai Liu (刘凯)2, Youguo Shi (石友国)3,4, Zhong-Yi Lu (卢仲毅)2*, Peng Cai (蔡鹏)2*, and Yayu Wang (王亚愚)1,5 Affiliations 1State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 2Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 4Songshan Lake Materials Laboratory, Dongguan 523808, China 5Frontier Science Center for Quantum Information, Beijing 100084, China Received 12 December 2022; accepted manuscript online 21 February 2023; published online 28 February 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: zlu@ruc.edu.cn; pcai@ruc.edu.cn Citation Text: Xu M, Zou C W, Gong B C et al. 2023 Chin. Phys. Lett. 40 037101    Abstract The electronic evolution of Mott insulators into exotic correlated phases remains puzzling, because of electron interaction and inhomogeneity. Introduction of individual imperfections in Mott insulators could help capture the main mechanism and serve as a basis to understand the evolution. Here we utilize scanning tunneling microscopy to probe the atomic scale electronic structure of the spin-orbit-coupling assisted Mott insulator Sr$_{3}$Ir$_{2}$O$_{7}$. It is found that the tunneling spectra exhibit a homogeneous Mott gap in defect-free regions, but near the oxygen vacancy in the rotated IrO$_{2}$ plane the local Mott gap size is significantly enhanced. We attribute the enhanced gap to the locally reduced hopping integral between the 5$d$ electrons of neighboring Ir sites via the bridging planar oxygen $p$ orbitals. Such bridging defects have a dramatic influence on local bandwidth, thus provide a new way to manipulate the strength of Mottness in a Mott insulator.
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 DOI:10.1088/0256-307X/40/3/037101 © 2023 Chinese Physics Society Article Text Since the discovery of high-temperature superconductivity in copper oxides, which are widely believed to be doped Mott insulators, there has been a considerable revival in the field of Mott physics. A Mott insulator is characterized by strong onsite Coulomb repulsion that outweighs the bandwidth, leading to an insulating ground state even for a half-filled band. The simplest scenario can be captured by the one-band Hubbard model,[1,2] which involves two parameters: the nearest-neighbor hopping integral $t$ and the onsite Coulomb repulsion $U$. The competition between them can drive a transition between insulating and metallic states, which can be achieved by tuning the Coulomb interaction strength, hopping integral, dimensionality, and so on.[3] In practice, there are several means to metallize a Mott insulator. For example, applying external pressure on Mott insulators[4] reduces the lattice constant and increases the overlap of atomic wave functions, which in turn increases the bandwidth. Or like in the cuprates, introducing electrons/holes into parent Mott insulators via chemical doping can create the metallic and even superconducting states.[5-7] However, it is much less common to realize the opposite process in solids, i.e., making a Mott insulator more insulating by reducing $t$ or enhancing $U$. The strong spin orbit coupling (SOC) assisted Mott insulating phase in 5$d$ iridates[8-11] provides new opportunities for understanding and manipulating the Mottness because the comparable $t$ and $U$ scales make the system highly tunable. The Ruddlesden–Popper (RP) series Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$ ($n = 1$, 2, $\infty$) members have been widely studied because their structural,[10] electronic,[12-14] and magnetic[15] properties bear striking similarities to the parent compound of cuprates. However, the much-expected superconductivity remains elusive in these systems although a metallic state can be realized by chemical doping.[16] It is generally believed that the Mottness is essential for superconductivity in cuprates because it is responsible for the local spins of unpaired Cu 3$d$ electrons and the antiferromagnetic interaction between them.[17-20] One possible reason for the absence of superconductivity in doped iridates is the much smaller Mott gap compared to the cuprates due to the more extended 5$d$ orbitals. Therefore, it is highly desirable to find a way to manipulate the local electronic structure of the iridates and to enhance the strength of Mottness. With excellent spatial and energy resolutions, scanning tunneling microscopy (STM) is highly suitable for investigating the local electronic structure of Mott insulators. Notably, the relatively small Mott–Hubbard gap in iridates is more friendly for STM studies, and the special crystal structure with rotated octahedra[16] makes it possible to identify different types of defects in the IrO$_{2}$ plane.[21,22] In this Letter, we use STM to investigate the atomic scale electronic structure of intrinsic oxygen vacancy (O$_{\rm v}$) in the rotated IrO$_{2}$ plane of bilayer parent iridate Sr$_{3}$Ir$_{2}$O$_{7}$. It is found that such defects locally induce lattice distortions in the SrO surface with a four-fold degeneracy. Furthermore, the planar O$_{\rm v}$ strongly suppresses the high energy density of state (DOS), and enhances the local Mott gap size. We attribute this to the locally reduced hopping integral between neighboring Ir 5$d$ electrons via bridging planar oxygen $p$ orbitals, which provides a new knob for tuning the Mottness in strongly correlated electron systems. High-quality Sr$_{3}$Ir$_{2}$O$_{7}$ single crystals were grown by the flux method and the details have been reported elsewhere.[23,24] The crystals are cleaved in the ultrahigh vacuum preparation chamber (with base pressure below $1 \times 10^{-10}$ mbar) either at 77 K or at room temperature, and then directly transferred into the STM head cooled by liquid nitrogen. Because pristine Sr$_{3}$Ir$_{2}$O$_{7}$ is highly insulating, the STM experiments here are all performed at 77 K to avoid possible charging effect.[25] The exposed surface retains intact during the measurement over several days, indicating extremely low absorption rate at low temperature. Moreover, the exposed surfaces cleaved at room temperature show no distinguishable features as those cleaved at 77 K. Before the measurements, an electrochemically etched tungsten tip is treated and calibrated on an Au(111) surface.[26] The differential conductance $dI/dV$, which is approximately proportional to the local DOS, is obtained by the standard ac lock-in method with modulation frequency $f = 587$ Hz.
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Fig. 1. Crystallographic and topographic images of bilayer iridate Sr$_{3}$Ir$_{2}$O$_{7}$. (a) Schematic crystal structure of Sr$_{3}$Ir$_{2}$O$_{7}$. (b) The top view illustrating the rotated IrO$_{6}$ octahedra with perfect stoichiometry. (c) The topographic image taken with tunneling current $I = 30$ pA and bias voltage $V = -700$ mV. (d) The topographic images of four O vacancies located in the four red squares in (c) with the surface Sr sites indicated by red dots. (e) The STM line-cut profile along the defect-free area and defect area. (f) The side view of RP perovskite bilayer with $\sqrt 2 \times \sqrt 2$ 2D unit cell with planar O$_{\rm v}$ by using structural optimization calculation.
Figure 1(a) displays the crystal structure of the bilayer Sr$_{3}$Ir$_{2}$O$_{7}$ Mott insulator, in which the corner-sharing double-IrO$_{6}$-octahedra are sandwiched by SrO layers. The crystal is easy to cleave between two weakly bonded neighboring SrO layers, as indicated by the gray planes. The IrO$_{6}$ octahedra are rotated about the $c$-axis by 11$^{\circ}$[16] and the rotations of neighboring octahedra alternate in sign within a layer, as schematically shown in Fig. 1(b). The topographic image in Fig. 1(c) shows an exposed SrO surface, where the Sr atoms with lattice constant $\sim$ 3.9 Å are clearly visible. The surface is not perfectly flat, similar to that reported previously in Sr$_{2}$IrO$_{4}$ and Sr$_{3}$Ir$_{2}$O$_{7}$,[21,22,27-30] either due to the imperfect surface cleaving or intrinsic defects in the IrO$_{2}$ plane. The latter is particularly important because it acts as a local perturbation to the strongly correlated state, which is at the heart of Mott physics. The new feature revealed here is one type of defect in the IrO$_{2}$ plane accompanied by an off-centered Sr atom [red squares in Fig. 1(c)] that can be easily identified by STM topography. The shifted Sr atoms are slightly away from the center of surrounding dip regions. To further analyze this defect, we only focus on those individual defects away from other ones, such as bigger blackholes (small cluster of defects) and rarely seen dumbbell-like bright spots nearby/inside clusters. Interestingly, the shifted Sr atoms have four different orientations that are energetically degenerate, as indicated by the yellow arrows. Figure 1(d) is a closer look at the four defects in Fig. 1(c) superposed with the ideal square Sr lattice (red dots), which clearly reveal that each bright atom is shifted between two neighboring Sr sites. Figure 1(e) displays the STM line-cut profiles across defect-free area and defect area, which reflects an abrupt atomic elevation and collapse simultaneously within a few lattice distances around defect area. The typical corrugation of the defect is smaller than 15 pm, indicating it originates from small lattice distortion or electronic structure effect rather than absorbed molecule. This new type of defect can be attributed to the O$_{\rm v}$ residing on the underlying rotated IrO$_{2}$ plane, which is the most favorable O$_{\rm v}$ site in parent Sr$_{3}$Ir$_{2}$O$_{7}$.[31] The observed shift of the Sr atom and its four-fold degeneracy is a natural consequence of the unbalanced lattice environment caused by such O$_{\rm v}$. To get a better insight on this lattice distortion, we performed theoretical simulation using the Vienna ab initio simulation package (VASP)[32,33] and carry out density functional theory (DFT) calculations (see theoretical methods for computational details). Furthermore, we have considered a $\sqrt 2 \times \sqrt 2$ supercell consisting of eight formula units in our simulations, which correspond to about 1.8% of O$_{\rm v}$ concentration, very close to the intrinsic O$_{\rm v}$ concentration of about 1.0% (roughly estimated from the area with a single O$_{\rm v}$). After fully structural relaxation, we find that Sr atoms around O$_{\rm v}$ undergoes an apparent lattice reconstruction, as vividly shown by side view in Fig. 1(f). Once O$_{\rm v}$ appears in the rotated IrO$_{2}$ plane, one of the nearest surface Sr atoms would climb and is easily visible while the other would sink from their initial atomic positions and behaves as a dark hole. In addition, the surface Sr atoms around O$_{\rm v}$ move slightly along the same Sr-O$_{\rm v}$-Sr direction as marked by blue arrows. Notably, due to the complicated environment of O$_{\rm v}$s in the underlying layers, together with finite size effect and complex tunneling matrix element on oxide surface, the calculated simulation cannot be compared quantitatively with STM results. Overall, the simulation captures the main STM features that the shifted and raised Sr atoms are slightly away from the center of surrounding dip regions. The surface Sr atoms is indeed susceptible to the intrinsic O$_{\rm v}$s in the rotated IrO$_{2}$ plane, which is consistent with our experimental STM results.
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Fig. 2. The typical $dI/dV$ spectra of Sr$_{3}$Ir$_{2}$O$_{7}$. (a) Spatially averaged $dI/dV$ spectrum taken at a defect-free area. The associated topography is shown in the inset. (b) The four colored curves are $dI/dV$ spectra taken on the four O vacancies shown in Fig. 1(d). Inset shows a representative topography of the defect (left panel) and the corresponding $dI/dV$ map (right panel) at $V = 450$ mV.
Figure 2(a) displays the spatially averaged $dI/dV$ spectrum obtained on an area far from any defect (denoted as “defect-free” hereafter). Due to the relatively high measurement temperature, the SOC-assisted Mott gap is significantly broadened compared to the low temperature spectrum reported previously.[21] To extract the size of the Mott gap, we perform linear extrapolations (red dashed lines) to estimate the band edge positions for both the upper Hubbard band (UHB) and lower Hubbard band (LHB). The gap size is estimated to be $\sim$ 140 meV, similar to that reported before in pristine Sr$_{3}$Ir$_{2}$O$_{7}$.[21,34,35] To investigate the effect of local impurities on the Mott insulating phase, the colored curves in Fig. 2(b) display $dI/dV$ spectra taken directly on the four O$_{\rm v}$s shown in Fig. 1(d). The four spectra are nearly identical to each other, but all show apparent differences from the defect-free spectrum (black curve). The main effect of the defects is a strong suppression of DOS for energy range between 100 and 500 meV, namely the lower part of UHB, while the spectral lineshape of the LHB is nearly intact. Consequently, the bottom of UHB is lifted and the effective Mott gap on the four defects is increased to 600 meV as extracted by the linear extrapolation (green dashed lines) of the LHB and UHB spectral lines. The $dI/dV$ map in the inset of Fig. 2(b) is taken over an isolated defect, which reveals an elliptical pattern with suppressed DOS due to the O$_{\rm v}$ in the IrO$_{2}$ plane. The enlarged gap observed here is in stark contrast to other phenomena associated with defect in a Mott insulator,[21,22,26,36] which usually exhibits an in-gap electronic state. To visualize the spatial extension of Mott gap and the anomalously enhanced gap size due to planar O$_{\rm v}$ in Sr$_{3}$Ir$_{2}$O$_{7}$, we acquire spectral line cuts along clean surface and through an O$_{\rm v}$, as indicated by different colored arrows in Fig. 3(a). The Mott gap is homogeneous at defect-free locations, as demonstrated by Figs. 3(b) and 3(c). Near the O$_{\rm v}$ defect, on the other hand, there is a pronounced spatial distribution of the spectra with suppressed UHB spectral weight and enhanced gap size [Figs. 3(d)–3(f)]. Notably, the suppressed state associated with the defect is strongly localized within about three lattice distances, as revealed by the $dI/dV$ map in the insets of Fig. 2(b). In Fig. 3(h), we display the theoretical simulation of TDOS in the pristine Sr$_{3}$Ir$_{2}$O$_{7}$ with perfect stoichiometry, which exhibits expected insulator characteristics. When single planar bridging O$_{\rm v}$ is introduced into the supercell, as shown in Fig. 3(i), the calculated TDOS is still an insulating state but displays emergence of electronic state (marked by red arrow) within the initial Mott gap, consistent with previous calculation results,[31] but not reproducing the experimental STM results. The filling of the Mott gap in the calculation can be attributed to charge doping effect, which has been observed in Iridates doped with either electrons or holes.[21,28,30] Therefore, the unconventional enhancement of Mott gap observed here is not lead by charge doping effect. Noticeably, the planar O$_{\rm v}$s observed in our STM experiment shows the strong localization of electronic state. Then we simulate the site dependence of local DOS (LDOS) in Fig. 3(i). It shows remarkable suppression at UHB above the planar O$_{\rm v}$ [marked by blue arrow in Fig. 3(i)], while the LDOS above next-nearest-neighbor planar oxygen (around 4 Å away from the planar O$_{\rm v}$, marked by orange arrow) are nearly similar to that obtained in pristine case. The strong localization of electronic structure and the LDOS suppression at lower part of UHB are captured by the LDOS calculation. There are several relevant energy scales in SOC-assisted Mott insulating materials: the crystal field $\Delta$, the SOC strength $\lambda$, the on-site Coulomb repulsion energy Hubbard $U$, and the electron kinetic energy described by hopping integral $t$. As a common motif of iridates, each Ir$^{4+}$ ion is surrounded by an octahedral cage of O$^{2-}$ anions [Figs. 1(a) and 4(a)], which give rise to crystal field splitting of the $d$ orbitals as large as 3–4 eV. The two-fold degenerate $e_{\rm g}$ levels are raised in energy and the three-fold degenerate $t_{\rm 2g}$ levels are lowered,[37] as shown in the left panel of Fig. 4(b), leading to a partially filled $t_{\rm 2g}$ band.[38] As SOC is taken into account, the $J_{\rm eff }=3/2$ band can accommodate four 5$d$ electrons, leaving the remaining one occupying the $J_{\rm eff }=1/2$ state. This narrow band generated by SOC is more susceptible to Mott localization even by a moderate $U$, making it a $J_{\rm eff }=1/2$ Mott insulator [middle panel of Fig. 4(b)]. Therefore, for the perovskite iridates Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$ ($n = 1$, 2), the interaction of $J=1/2$ electrons are determined by the atomic interaction of the Ir$^{4+}$ ion and the hopping of the $t_{\rm 2g}$ electrons. As the number of IrO$_{2}$ planes within a unit cell increases, the paths for hopping between neighboring IrO$_{2}$ planes also proliferate, which in turn enhances the electron itinerancy. Recent optical spectroscopy and first principles calculations[39] reveal the insulator to metal transition (IMT) with increasing $n$, leaving Sr$_{2}$IrO$_{4}$ ($n = 1$) and Sr$_{3}$Ir$_{2}$O$_{7}$ ($n = 2$) on the insulating side and SrIrO$_{3}$ ($n=\infty$) on the metallic side.[40] The enhanced hopping and hence larger bandwidth make Sr$_{3}$Ir$_{2}$O$_{7}$ a substantially weaker Mott insulator than Sr$_{2}$IrO$_{4}$,[41-44] as shown in the right panel of Fig. 4(b). It lies near the boundary of IMT with a small Mott gap, which provides a unique opportunity for manipulating the subtle balance between $t$ and $U$.
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Fig. 3. Evolution of tunneling spectra across bridging O vacancy. (a) Topographic image including defect-free region and typical defects where the linecut shown in (b), (d), (e) are obtained. (b) Tunneling spectra along the black arrowed line in (a) and the corresponding spectral intensity map are shown in (c). (d) Tunneling spectra along the red arrowed line in (a). (e) Tunneling spectra along the blue arrowed line in (a) and the corresponding spectral intensity map is shown in (f). All the spectra are taken with $V = -700$ mV and $I = 50$ pA. (g) Schematic crystal structure for theoretical calculation. (h) TDOS (up) and LDOS of empty atoms P1 (middle) and P2 (down) of pristine structure. (i) TDOS (up) and LDOS of empty atoms P1 (middle) and P2 (down) of Sr$_{3}$Ir$_{2}$O$_{7}$ with about 1.8% O$_{\rm v}$ introduced. Here, empty atoms P1 (near O$_{\rm v}$) and P2 (away from O$_{\rm v}$) locate at the coordinates of (0.25, 0.5, 0.7) and (0.25, 0.5, 0.7), respectively.
For bilayer Sr$_{3}$Ir$_{2}$O$_{7,}$ we examine the Hubbard model on the square lattice with intralayer nearest-neighbor hopping $t$, interlayer hopping $t_{\bot}$, and onsite Coulomb repulsion $U$. The planar O$_{\rm v}$ [left panel of Fig. 4(c)] breaks the hybridization between the 5$d$ states and the $p$-states originally at the O$_{\rm v}$ site, which controls the intralayer hopping $t$ of the 5$d$ electrons and hence the bandwidth $W$. As revealed in Fig. 3(i), the DOS at UHB outside Mott gap is remarkably suppressed above planar O$_{\rm v}$, suggesting an enlarger Mott gap would in turn appears. Recent theoretical simulation[45] indicates that the markedly increased Mott gap appears as the width of lower and upper Hubbard bands is reduced within the half-filled Hubbard model. In addition, we studied the apical O$_{\rm v}$ that bridges adjacent IrO$_{2}$ planes along the $c$-axis, as shown in Fig. 4(d) as well as the blue square in Fig. 1(c), whose topography is characterized by the four sinking Sr atoms on the surface. According to previous quantum Monte Carlo analysis of this bilayer Hubbard model,[46] a continuous increase of single particle gap is expected for decreasing interlayer hopping $t_{\bot}$. As revealed by the $dI/dV$ spectrum in the down panel of Fig. 4(f), this defect indeed suppresses the DOS of UHB and induces an enlarged Mott gap, just like the planar O$_{\rm v}$. In addition, poor Coulomb screening resulting from the reduced electron itinerancy around O$_{\rm v}$ may in turn enlarge the effective electron interaction, which is essential to reconcile theoretical calculation[31] and experiments.
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Fig. 4. Planar and apical bridging O vacancy in bilayer iridate. (a) Side view of RP perovskite bilayer with $\sqrt 2 \times \sqrt 2$ 2D unit cell. (b) Schematic band diagram of Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$ ($n = 1$, 2) showing the Mott–Hubbard gap with a total angular moment $J_{\rm eff }= 1/2$ due to strong SOC. (c) Left: schematic diagram showing the missing planar O on the rotated IrO$_{2}$ plane in Sr$_{3}$Ir$_{2}$O$_{7}$. Middle: the corresponding defect topography. Right: $dI/dV$ spectrum taken on the defect (red curve) and defect-free area (black curve). (d) Schematic diagram showing the missing apical O between IrO$_{2}$ planes along with the defect topography [middle, zoomed in from the blue square in Fig. 1(c)] and $dI/dV$ spectrum (right).
For a prototypical Mott–Hubbard insulator, low energy excitations are given by the superexchange coupling $J$ between local moments as $J = 4t^{2}/U$.[47] Previous experimental work implies that the charge transfer gap (CTG) size is a crucial energy scale that determines the superconductivity of cuprates,[17,18,48] and theoretical calculation proposes that CTG can be broadly tuned through chemical substitution and strain.[49] Mn-doped ruthenate[50] even induces quantum phase transition from a metal to an antiferromagnetic Mott insulator with enhanced Coulomb correlation between localized 3$d$ orbitals. Notably, an appropriate Mott gap size is necessary for realizing superconductivity because too large a gap will lower $J$ and weaken pairing strength. On the other side, too small a gap will impair the strength of correlation between unpaired electrons. Compared with 3$d$ transition-metal oxides such as cuprates, iridates possess relatively small Mott gap hence a possible strategy for realizing superconductivity along this direction is to enhance the Mottness gap size. The observed planar oxygen vacancy here markedly tunes the hopping integral of unpaired electrons and locally induces larger Mott gap, which may be beneficial for achieving superconductivity in iridates when charge carriers are introduced. In summary, we have used STM to investigate the bridging oxygen vacancy in the rotated IrO$_{2}$ plane of 5$d$ transition-metal oxide Sr$_{3}$Ir$_{2}$O$_{7}$ and found suppressed DOS outside the Mott gap. We attribute this phenomenon to the reduced hopping between the 5$d$ electrons on the Ir$^{4+}$ ion via the bridging O sites. Our results show that the bridging defects significantly affect electronic structure in Mott insulators and promote the electron correlation effect, indicating a crucial new role of oxygen vacancies in oxide functional materials.[51] These results provide a new knob for manipulating the Mottness of the iridates, which may have important implications to the emergence of superconductivity. Acknowledgements. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0302900) and the Basic Science Center Project of National Natural Science Foundation of China (Grant No. 51788104). It was supported in part by the Beijing Advanced Innovation Center for Future Chip (ICFC) and Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics. Peng Cai was supported by the National Natural Science Foundation of China (Grant No. 12074424), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China. Computational resources were provided by the Physical Laboratory of High Performance Computing at Renmin University of China and Beijing Super Cloud Computing Center. Theoretical Methods. The first-principles electronic structure calculations were performed by using the projector augmented wave method[52] as implemented in the VASP package.[32,33] For the exchange-correlation functional, the generalized Perdew–Burke–Ernzerhof gradient approximation[53] was employed. The kinetic energy cutoff of the planewave basis was set to be 520 eV. A $\sqrt 2 \times \sqrt 2$ supercell of Sr$_{3}$Ir$_{2}$O$_{7}$ containing eight chemical formulae was used. It corresponds to about 1.8% of O$_{\rm v}$ concentration, very close to the intrinsic O$_{\rm v}$ concentration of about 1.0% (roughly estimated from the area with a single O$_{\rm v}$). In structural optimization, both the lattice parameters and the internal atomic positions were optimized until the forces on all atoms were smaller than 0.01 eV/Å. For the Brillouin zone sampling, $8 \times 8 \times 4$ and $6 \times 6 \times 3$ $k$-point meshes were adopted for structural optimization and the DOS calculations, respectively. To include the strong correlation effect among Ir 5$d$ electrons, the on-site Coulomb repulsion with an effective Hubbard $U$ of 1.6 eV was adopted in accordance with recent Sr$_{3}$Ir$_{2}$O$_{7}$ calculations.[31] The SOC effect was also included in the DOS calculations. References Electron correlations in narrow energy bandsExchange splitting in ferromagnetic nickelSpringer Tracts in Modern PhysicsMott Transition in Cr-Doped V 2 O 3 A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2-xNaxCuO2Cl2Ubiquitous Interplay Between Charge Ordering and High-Temperature Superconductivity in CupratesVisualizing the evolution from the Mott insulator to a charge-ordered insulator in lightly doped cupratesStructure and magnetic properties of Sr 2 − x A x IrO 4 ( A =Ca and Ba)Quasi-Two-Dimensional Mott Transition System Ca 2 − x Sr x RuO 4 Structural and magnetic studies of Sr 2 IrO 4 Spin ordering and electronic texture in the bilayer iridate Sr 3 Ir 2 O 7 Novel J eff = 1 / 2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4 Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr2 IrO4Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnetMagnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate SuperconductorsThe challenge of spin–orbit-tuned ground states in iridates: a key issues reviewRelationship between the parent charge transfer gap and maximum transition temperature in cupratesScaling of the transition temperature of hole-doped cuprate superconductors with the charge-transfer energyDoping a Mott insulator: Physics of high-temperature superconductivityImaging the evolution of metallic states in a correlated iridateEvidence for a percolative Mott insulator-metal transition in doped Sr 2 IrO 4 Strong pseudospin-lattice coupling in Sr 3 Ir 2 O 7 : Coherent phonon anomaly and negative thermal expansionFlux growth of Sr+1Ir O3+1 (n = 1, 2, ∞) crystalsPoor electronic screening in lightly doped Mott insulators observed with scanning tunneling microscopyVisualizing the atomic-scale electronic structure of the Ca2CuO2Cl2 Mott insulatorLocal density of states study of a spin-orbit-coupling induced Mott insulator Sr 2 Ir O 4 Universality of pseudogap and emergent order in lightly doped Mott insulatorsAtomic-scale fragmentation and collapse of antiferromagnetic order in a doped Mott insulatorDoping induced Mott collapse and possible density wave instabilities in (Sr1−xLax)3Ir2O7Magnetic properties of bilayer Sr 3 Ir 2 O 7 : Role of epitaxial strain and oxygen vacanciesAb initio molecular dynamics for liquid metalsEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setPhonon-assisted optical excitation in the narrow bandgap Mott insulator Sr 3 Ir 2 O 7 Dimensionality-controlled Mott transition and correlation effects in single-layer and bilayer perovskite iridatesImaging the atomic-scale electronic states induced by a pair of hole dopants in Ca2CuO2Cl2 Mott insulatorOrbital Physics in Transition-Metal OxidesCrystal field splitting in Sr n + 1 Ir n O 3 n + 1 ( n = 1 , 2 ) iridates probed by x-ray Raman spectroscopyDimensionality-Controlled Insulator-Metal Transition and Correlated Metallic State in 5 d Transition Metal Oxides Sr n + 1 Ir n O 3 n + 1 ( n = 1 , 2, and ∞ )Interplay between spin-orbit coupling and Hubbard interaction in SrIrO 3 and related Pbnm perovskite oxidesLarge Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator TransitionWeak antiferromagnetism of J eff = 1 2 band in bilayer iridate Sr 3 Ir 2 O 7 Spectroscopic indications of polaronic behavior of the strong spin-orbit insulator Sr 3 Ir 2 O 7 Canted antiferromagnetic ground state in Sr3 Ir2 O7Charge dynamics of the antiferromagnetically ordered Mott insulatorGround-state phase diagram of the half-filled bilayer Hubbard modelEffective Hamiltonian for the superconducting Cu oxidesStrength of correlations in electron- and hole-doped cupratesTuning the charge-transfer energy in hole-doped cupratesEmergent charge order near the doping-induced Mott-insulating quantum phase transition in Sr3Ru2O7Oxygen vacancies: The (in)visible friend of oxide electronicsProjector augmented-wave methodGeneralized Gradient Approximation Made Simple
[1] Hubbard J 1963 Proc. R. Soc. A 276 238
[2] Hubbard J 1964 Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 84 455
[3] Gebhard F 1997 Metal—Insulator Transitions: Models and Methods. Springer Tracts in Modern Physics vol 137 pp 1–48
[4] McWhan D B, Rice T M, and Remeika J P 1969 Phys. Rev. Lett. 23 1384
[5] Hanaguri T, Lupien C, Kohsaka Y, Lee D H, Azuma M, Takano M, Takagi H, and Davis J 2004 Nature 430 1001
[6] da Silva Neto E H, Aynajian P, Frano A et al. 2014 Science 343 393
[7] Cai P, Ruan W, Peng Y, Ye C, Li X, Hao Z, Zhou X, Lee D H, and Wang Y 2016 Nat. Phys. 12 1047
[8] Shimura T, Inaguma Y, Nakamura T, Itoh M, and Morii Y 1995 Phys. Rev. B 52 9143
[9] Nakatsuji S and Maeno Y 2000 Phys. Rev. Lett. 84 2666
[10] Crawford M K, Subramanian M A, Harlow R L, Fernandez-Baca J A, Wang Z R, and Johnston D C 1994 Phys. Rev. B 49 9198
[11] Dhital C, Khadka S, Yamani Z et al. 2012 Phys. Rev. B 86 100401
[12] Kim B J, Jin H, Moon S J et al. 2008 Phys. Rev. Lett. 101 076402
[13] Kim B J, Ohsumi H, Komesu T, Sakai S, Morita T, Takagi H, and Arima T 2009 Science 323 1329
[14] Kim Y K, Krupin O, Denlinger J, Bostwick A, Rotenberg E, Zhao Q, Mitchell J, Allen J, and Kim B 2014 Science 345 187
[15] Kim J, Casa D, Upton M et al. 2012 Phys. Rev. Lett. 108 177003
[16] Cao G and Schlottmann P 2018 Rep. Prog. Phys. 81 042502
[17] Ruan W, Hu C, Zhao J et al. 2016 Sci. Bull. 61 1826
[18] Weber C, Yee C, Haule K, and Kotliar G 2012 Europhys. Lett. 100 37001
[19] Lee P A, Nagaosa N, and Wen X G 2006 Rev. Mod. Phys. 78 17
[20]Roy S B 2019 Mott Insulators (Bristol: IOP Publishing)
[21] Okada Y, Walkup D, Lin H et al. 2013 Nat. Mater. 12 707
[22] Sun Z X, Guevara J M, Sykora S, Pärschke E M, Manna K, Maljuk A, Wurmehl S, van den Brink J, Büchner B, and Hess C 2021 Phys. Rev. Res. 3 023075
[23] Hu L L, Yang M, Wu Y Let al. 2019 Phys. Rev. B 99 094307
[24] Manna K, Aslan-Cansever G, Maljuk A, Wurmehl S, Seiro S, and Büchner B 2020 J. Cryst. Growth 540 125657
[25] Battisti I, Fedoseev V, Bastiaans K M, De La T A, Perry R S, Baumberger F, and Allan M P 2017 Phys. Rev. B 95 235141
[26] Ye C, Cai P, Yu R, Zhou X, Ruan W, Liu Q, Jin C, and Wang Y 2013 Nat. Commun. 4 1365
[27] Dai J X, Calleja E, Cao G, and McElroy K 2014 Phys. Rev. B 90 041102
[28] Battisti I, Bastiaans K M, Fedoseev V et al. 2017 Nat. Phys. 13 21
[29] Zhao H, Manna S, Porter Z, Chen X, Uzdejczyk A, Moodera J, Wang Z, Wilson S D, and Zeljkovic I 2019 Nat. Phys. 15 1267
[30] Wang Z, Walkup D, Maximenko Y, Zhou W, Hogan T, Wang Z, Wilson S D, and Madhavan V 2019 npj Quantum Mater. 4 43
[31] Kim B, Liu P, and Franchini C 2017 Phys. Rev. B 95 024406
[32] Kresse G and Hafner J 1993 Phys. Rev. B 47 558
[33] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[34] Park H J, Sohn C H, Jeong D, Cao G, Kim K, Moon S, Jin H, Cho D Y, and Noh T 2014 Phys. Rev. B 89 155115
[35] Wang Q, Cao Y, Waugh J, Park S, Qi T, Korneta O, Cao G, and Dessau D 2013 Phys. Rev. B 87 245109
[36] Li H W, Ye S S, Zhao J F, Jin C Q, and Wang Y Y 2021 Sci. Bull. 66 1395
[37] Tokura Y and Nagaosa N 2000 Science 288 462
[38] Moretti Sala M, Rossi M, Al-Zein A et al. 2014 Phys. Rev. B 90 085126
[39] Moon S J, Jin H, Kim K W et al. 2008 Phys. Rev. Lett. 101 226402
[40] Zeb M A and Kee H Y 2012 Phys. Rev. B 86 085149
[41] Kim J, Said A, Casa D, Upton M, Gog T, Daghofer M, Jackeli G, Van Den Brink J, Khaliullin G, and Kim B 2012 Phys. Rev. Lett. 109 157402
[42] Fujiyama S, Ohashi K, Ohsumi H, Sugimoto K, Takayama T, Komesu T, Takata M, Arima T, and Takagi H 2012 Phys. Rev. B 86 174414
[43] King P D C, Takayama T, Tamai A et al. 2013 Phys. Rev. B 87 241106
[44] Nagai I, Yoshida Y, Ikeda S, Matsuhata H, Kito H, and Kosaka M 2007 J. Phys.: Condens. Matter 19 136214
[45] Han X J, Liu Y, Liu Z Y, Li X, Chen J, Liao H J, Xie Z Y, Normand B, and Xiang T 2016 New J. Phys. 18 103004
[46] Golor M, Reckling T, Classen L, Scherer M M, and Wessel S 2014 Phys. Rev. B 90 195131
[47] Zhang F C and Rice T M 1988 Phys. Rev. B 37 3759
[48] Weber C, Haule K, and Kotliar G 2010 Nat. Phys. 6 574
[49] Yee C H and Kotliar G 2014 Phys. Rev. B 89 094517
[50] Leshen J, Kavai M, Giannakis I, Kaneko Y, Tokura Y, Mukherjee S, Lee W C, and Aynajian P 2019 Commun. Phys. 2 36
[51] Gunkel F, Christensen D V, Chen Y Z, and Pryds N 2020 Appl. Phys. Lett. 116 120505
[52] Blöchl P E 1994 Phys. Rev. B 50 17953
[53] Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865