Introduction

Finding universal features in interacting electronic systems is a major theme in modern condensed matter research. By universality we refer to the low-energy properties that do not depend on the details of the interactions. For materials with relatively weak electron correlations, the low-energy excitations are well-described by the Fermi liquid theory. For doped Mott insulators where correlations are strong, pinpointing the hallmarks common to all Mott-metal crossovers1,2,3 has proven a formidable task. This could largely be attributed to the long candidate list of competing electronic orders, including long-range magnetic order and Fermi surface instabilities, among others, which yield a complex global doping-temperature phase diagram. For example, while the past three decades have witnessed tremendous progress in characterizing the phenomenology of the high-Tc cuprates, the interpretation of most of these experimental observations remains highly controversial. It has become essentially inseparable whether such exotic phenomena as pseudogaps and marginal Fermi liquid (MFL) scattering rates2,3,4,5 arise from the metal-insulator transition, certain density-wave instabilities6 or are fluctuations of the superconductivity7. Moreover, the charge insulation in most known Mott insulators arises solely from the Coulomb repulsion. It would be highly desired to study new doped Mott insulators, especially those with a cleaner phase diagram (thus fewer competing orders) and a different mechanism that forbids electron double-occupancy.

Sr2IrO4 has attracted much interest recently as a new family of Mott insulator8,9, with the potential of realizing novel phases of matter10 and achieving higher superconducting TC11. The Ir-O planes are similar to the Cu-O planes in cuprates, with the Ir atoms sitting at the centre of the Ir-O octahedra, except for the 22° Ir-O-Ir bond angle (Fig. 1a). There are five t2g electrons per Ir. The Ir t2g level splits into the J3/2 doublet (filled with four electrons) and the J1/2 singlet as a result of the strong spin-orbit coupling. The half-filled J1/2 further splits into lower (filled with 1 electron) and upper (empty) bands, with this splitting general believed to be due to the Coulomb repulsion, which is why these bands are considered upper and lower Hubbard bands (UHB and LHB, see Fig. 1c)9. The insulating behaviour in Sr2IrO4 derives from the coupling of strong spin-orbit interaction with Coulomb repulsions, which is drastically different from that in cuprates. However, the Sr2IrO4 parent compound, just like undoped cuprates, is antiferromagnetically ordered12. Driving Sr2IrO4 towards metallicity thus provides a unique opportunity to investigate the universal features of the Mott-metal crossover.

Figure 1: Sr2IrO4 as a Mott insulator on the square lattice.
figure 1

(a) The real-space unit cells of Sr2IrO4 and La2CuO4. (b) The k-space unit cells of the same, with matching colour scaling and with near-EF ARPES spectral weight. Ignoring the 22° Ir-O twists gives the blue cells in real and k-space, and corresponds to the regular unit cell of La2CuO4. Including these twists in Sr2IrO4 (black, (a)) back-folds the k-space cell in k-space (white), similar to the AF order in the parent cuprates. (c) The formation of the Mott gap in Sr2IrO4 as a result of the spin-orbit coupling and Coulomb interaction. (d,e) Possible schematics of chemical potential evolution with Rh doping.

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Different approaches of doping Sr2IrO4 and related compounds have been found13,14,15,16,17, and of special interest is the Sr2Ir1−xRhxO4 series. With as little as 4% Rh substitution, the normalized resistivity drops by 104. The long-range magnetic order decays more slowly and still survives with a TN∼17 K for 15% Rh (see Supplementary Fig. 1). Rh is directly above Ir in the periodic table, so is expected to be isovalent. It has been proposed that the metallicity in the Rh-doped iridates comes from the reduced spin-orbit coupling of Rh (due to the smaller atomic number) which then leads to the reduced splitting of the J3/2 and J1/2 bands15,16, as well as the formation of in-gap states16 (Fig. 1e). As we will show below, the Rh atoms in fact act as effective hole dopants to Sr2IrO4 (Fig. 1d). The chemical potential moves to the top of the LHB, without major band renormalizations.

So far, no superconductivity has yet been reported in these Rh-doped compounds, which is different from doped cuprates. The absence of superconductivity reduces possible competing orders and makes the Rh-doped iridates a cleaner system to study—the long-range canted antiferromagnetism (AF) is the only confirmed order in the system. In the following, we show in the effectively hole-doped Sr2IrO4 the emergence of pseudogaps, Fermi arcs and marginal-Fermi-liquid-like electronic scattering rates, features first reported and/or most famous in doped cuprate Mott insulators. It appears these iridates resemble cuprates not only structurally but also in terms of electronic structures.

Results

Effective hole doping via Rh substitution

Angle-resolved photoemission spectroscopy (ARPES) proves an invaluable tool for directly observing both electronic structure and low-energy electron dynamics in doped Mott insulators3 and bears implication for electronic order, including density-wave instabilities. We performed ARPES on single crystals of Rh-doped Sr2IrO4. Details of the sample growth are listed in the Methods section and Supplementary Note 1; and the ARPES experiment setups are presented in the Methods section and Supplementary Note 2. Due to the lattice reconstruction, the Brillion zone (BZ) of Sr2IrO4 and its Rh-doped compounds is reduced in half, similar to the formation of the antiferromagnetic (AF) BZ in the parent cuprates. We show the constant energy surfaces of Sr2IrO4 (Fig. 1b, left panel) and as a comparison, of Pb-doped Bi2Sr2CaCu2O8+δ (Fig. 1b, right panel). The folded and ‘original’ BZs for the iridates are marked in white solid and blue dashed lines, respectively. The ‘original’ BZ corresponds to one Ir-O plaquette in real space, as shown by the blue dashed line in the left panel in Fig. 1a. As we will show later in the paper, the close-to-EF features of the doped iridates are best captured not by the folded BZ, but instead by the ‘original’ blue BZ. To avoid confusion, we use (π, 0) to mark the X point in the BZ as defined in ref. 9, and use Γ’ to denote (π, π), which is the Γ point in the second folded BZ. In Fig. 2a,b, we show the constant energy surfaces as a function of binding energy for Sr2IrO4 and Sr2Ir1−xRhxO4 with x=15% at T=50 K. While there is no Fermi surface for the parent compound, there are states at EF in the x=15% iridate, corresponding to enhanced conductivity in the Ir-O plane. The constant energy surface of the x=15% compound is quite similar to that of the parent, except that it is shifted in binding energy by ∼200 meV. To identify the Fermi surface topology, we plot the ARPES spectrum along Γ’−(π, 0) for both samples (Fig. 2c; along the yellow lines in Fig. 2a,b). There is a hole-pocket centred at (π, 0), which comes from the J1/2 LHB9,18. The top of the valence band is ∼170 meV below EF in the parent compound, and is above EF for the x=15% sample. Indeed, both the J3/2 band (white dashed line in Fig. 2c) and J1/2 LHB (green dashed line)19,20 are shifted by ∼200 meV. It appears that rather than a reduced splitting of the J3/2 and J1/2 bands, Sr2IrO4 is hole doped with Rh substitution. Note, that while the Rh-doped compound displays strong spectral weight extending towards (π/2, π/2), the M point near EF, the band dispersion at the M point lies below EF (see Supplementary Fig. 2 and Supplementary Note 3). Therefore, the ‘Fermi surface’ is made up only of the states encircling X or (π, 0), that is, it encompasses holes.

Figure 2: Constant energy surfaces and high-symmetry cuts of the parent and Rh-doped Sr2IrO4.
figure 2

(a,b) The constant energy surfaces of the parent (a) and x=15% Rh-substituted (b) Sr2IrO4. The numbers are binding energies relative to EF. The solid white/dashed blue lines are the folded/original BZs. (c) ARPES energy-momentum intensity plots along Γ’−(π, 0)−Γ’ (yellow lines in panels (a,b)). The dashed green and white lines through the data guide the eye for the J1/2 and J3/2 bands, respectively.

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We determined the chemical potential shift quantitatively from the valence bands, as shown in Fig. 3a. With the increase of Rh concentration, the chemical potential is pushed deeper into the J1/2 LHB, confirming that Rh acts as an effective hole dopant. We extrapolate the chemical potential shift at finite Rh densities and derive an ∼170 meV intercept in the zero doping limit. Note, the top of the valence band in the parent compound locates at (π, 0) and is around 170 meV below EF, from both ARPES18 and scanning-tunneling21 experiments. Thus on effective hole doping the chemical potential immediately jumps to the edge of the LHB (Fig. 1d), as opposed to competing models (Fig. 1e)16,22 where new quasiparticle-like states emerge in the middle of the Hubbard gap. The doping schematic in Fig. 1d also agrees with the recent optical conductivity measurements16, while the interpretation in ref. 16 (Fig. 1e) could not explain the ARPES measured band structure (see Supplementary Note 4).

Figure 3: Rh atoms act as hole dopants in Sr2IrO4.
figure 3

(a) Chemical potential shift versus Rh concentration. The chemical potential shift is measured from the shift of both J1/2 LHB and J3/2 bands. The error bar for each Rh concentration is the half-width of the positive peaks of the second-derivatives of EDCs, averaged over a few high-symmetry k points. (b) Simple atomic picture of hole doping, ignoring band effects and Mott splitting. With a roughly similar average energy for both Ir and Rh sites, the smaller on-site spin-orbit splitting on the Rh sites lowers the J1/2 energy relative to that of the host Ir sites. This causes an electron transfer to Rh, that is, hole doping of the Ir lattice.

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The effective hole doping is quite plausible when considering the simple atomic model depicted in Fig. 3b. Rh atoms have smaller spin-orbit coupling than Ir, leading to the smaller splitting between the J3/2 and J1/2 states. Assuming the average energy of all six t2g states is similar for both Rh and Ir, the empty J1/2 state of Rh would then have a lower energy than that of Ir. Thus a J1/2 electron from a neighbouring Ir atom will fill the J1/2 state on the Rh site, leaving behind a hole on the Ir site, as well as a filled and immobile Rh3+ site. Of course this is a simplistic model that neither takes into account the finite bandwidth of the effective J states nor the Coulomb repulsion U. Recent X-ray absorption experiments at the Rh L3-edge have confirmed Rh indeed has a valence of 3+ in these compounds23.

Fermi arcs and pseudogaps

Hereafter, we focus on the low-energy electronic dynamics of these hole-doped compounds in search for ‘universal’ features during the Mott-metal crossover. In Fig. 4a we show the Fermi surface topology for the x=15% sample at 50 K. The segments of the Fermi surface centred at Γ (Γ’) and equivalent k locations are highlighted with solid yellow (blue) lines, and labelled FS1 and FS2, respectively. Energy-distribution curves (EDCs) from many different k points on the Fermi surface are plotted in Fig. 4c. There is a marked difference between the EDCs from FS1 and FS2—those from FS2 are generally pushed away from EF. We refer to this spectral weight suppression as a ‘pseudogap’ and use the standard ‘midpoint of leading edge’ method24,25 to quantify it, by fitting the EDCs to a shifted, broadened leading edge (see Supplementary Note 5), with the amount of shift (defined as the gap size) giving the 50% point of the leading edge. Note that in contrast to the case of a BCS-like gap that works well for superconductivity or a standard charge or spin density-wave gap, this shifted edge does not have a pile-up of spectral weight beyond the gap edge, like many other pseudogaps4,25, that is, it does not enforce spectral weight conservation on the opening of the gap. This absence of quasiparticle peaks in the doped iridates is one of the signatures of their non-Fermi liquid nature. We assign the gap size Δ1 and Δ2 from the leading edge fitting to FS1 and FS2, respectively. For x=15%, Δ1 as defined by this leading edge method vanishes, and FS1 can be considered a ‘regular’ piece of Fermi surface (albeit without quasiparticle peaks), while EDCs from FS2 show a partial depletion of near-EF spectral weight. As FS1 is only topologically connected to the ‘gapped’ FS2, we describe the FS1 as a Fermi surface ‘arc’ and FS2 as ‘pseudogapped’.

Figure 4: Fermi surface segments and pseudogaps in hole-doped Sr2IrO4.
figure 4

All the data were taken at 50 K unless otherwise noted. (a) Fermi surface spectral weight of the x=15% sample, with a hole-like Fermi pocket centred around the (π, 0) point of the unfolded (blue dashed) BZ. The FS pocket is separated into segments FS1 (yellow) and FS2 (blue), with FS1 facing Γ and FS2 facing Γ’. Q vectors (white arrows) are possible density-wave nesting vectors. (b,c) EDCs from multiple locations along the FS1 and FS2 segments (yellow and blue, respectively) taken from the x=4% and x=15% samples. Locations of the individual EDCs are marked by the open coloured circles in a. The leading edges of most EDCs do not reach EF, that is, they are gapped or pseudogapped. Gap sizes extracted using the ‘midpoint of leading edge’ method, are shown in b,c and compiled in d, with Δ1 labelling the gaps from FS1 and Δ2 the gaps from FS2. The uncertainties of Δ1 and Δ2 are defined as the standard deviation of the fitted gap sizes of individual EDCs from the averaged value. (e) EDCs from FS1 (dashed) and FS2 (solid) showing minimal temperature dependence across the magnetic phase transition of the x=11% sample.

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We track how Δ1 and Δ2 evolve with reduced Rh concentration, as the material gets closer to the Mott insulator. At x=4% (Fig. 4b), both Δ1 and Δ2 are finite, indicating that both FS1 and FS2 are pseudogapped, with Δ1∼3 meV and an increased Δ2∼38 meV. In Fig. 4d we plot the doping dependence of Δ1 and Δ2. For x=4%∼<11%, the entire Fermi surface is pseudogapped, which resembles the deeply underdoped cuprates26,27. We mark the presence of both pseudogaps as suppressed spectral weight near EF (the ‘notch’) in Fig. 1d. The pseudogaps observed here are not to be mistaken for the matrix element effect as these gaps happen over a very narrow-binding energy range that is essentially identical for different photon energies (see Supplementary Note 6).

The pseudogap phase in correlated electron systems is often considered a symmetry-broken phase of matter. Thus the origin of the pseudogap could be reflected in its k-space symmetry, as well as its thermal evolution. For Rh concentration 4–15%, Δ1 (Δ2) is roughly independent of k along the Fermi surface segment FS1 (FS2). In this sense, the pseudogaps in the non-superconducting Rh-doped iridates are clearly different from the near-nodal prepairing pseudogap in the near-optimal cuprates28, where pseudogaps follow the superconducting paring symmetry.

Figure 4e shows the temperature dependence of EDCs at kF from FS1 and FS2 for the x=11% sample, with the temperature range straddling the AF ordering temperature TN=57 K. Within the error bar no obvious changes with temperature are observed, indicating that the pseudogap is not directly related to the long-range canted AF order. This is further confirmed in the x=15% sample (Fig. 4c), where Δ2 is finite at 50 K and persists to at least 200 K (see Supplementary Fig. 3 and Supplementary Note 5), above TN=17 K. This observation suggests the pseudogap is not tied to the long-range magnetic order, and it is likely the pseudogap phase persists down to the zero-temperature quantum ground state in these hole-doped iridates.

Another commonly considered origin of pseudogaps is the density-wave instabilities in the form of Fermi surface nesting, as has been discussed in the manganites25 and cuprates6,29. In the case of iridates, it is temping to draw nesting vectors such as the white arrows (Fig. 4a) between FS2’s with the same gap size Δ2. However, the same ordering vector Q also connects FS1’s, yet with a much smaller gap Δ1. The Fermi surface nesting scenario does not explain the preference for a larger gap along FS2 than along FS1. It appears the pseudogap in iridates is inconsistent with many influential explanations for the antinodal pseudogap25,29 in manganites and near optimally doped cuprates though it may be more connected with the more-recently observed nodal pseudogap in heavily underdoped cuprates26,27.

Marginal-Fermi-liquid like single-electron scattering rate

The non-Fermi liquid nature of these doped iridates is not only reflected in the absence of quasiparticle peaks along the EDC (Fig. 4b,c), but also in the single-electron scattering rate. In the Fermi liquid theory, the quasiparticle scattering rate grows linearly with the binding energy (and temperature) squared. For up to 15% Rh substitution, as shown in Fig. 5b (with raw data shown in Fig. 5a) the scattering rates increase roughly linearly with binding energy—a signature of the MFL5. We could rule out this linear scattering rate as from the Rh-Ir substitution, and further details are shown in the Supplementary Fig. 4 and Supplementary Note 7. Moreover, as shown in Fig. 5c, there is a linear relation between the resistivity and the sample temperature, as highlighted by the black dashed line. Here we have ignored the upturn of the resistivity at low temperatures that is likely due to a localization effect, as has also been observed in most of the underdoped cuprates30. The linear MFL scattering rate is one of the most iconic features of the cuprates and other correlated materials such as the ruthenates31 and has been attributed to a wide range of ideas including quantum critical fluctuations5 and the break-up of quasparticles32.

Figure 5: Marginal-Fermi-liquid-like single-electron scattering rates.
figure 5

(a) The energy-momentum intensity plot along (−π, 0)−Γ−(π, 0) for an x=11% sample at T=50 K, passing through four pieces of FS1 Fermi surface. The peak centroids obtained from double-Lorentzian fittings to the momentum-distribution curves (MDCs) are marked with blue lines. The inset shows where the cut in the main figure is taken from the BZ. (b) The full-width-half-maximum (FWHM) from the double-Lorentzian fitting is plotted versus the binding energy, showing a linear ‘MFL’ scattering rate. (c) Resistivity versus temperature for the same sample showing a linear dependence at intermediate temperatures. The black dashed line is a guide to the eye.

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Discussion

The simultaneous appearance of Fermi arcs, pseudogaps and the MFL state in both iridates and cuprates is unlikely to be coincidental. We note recent measurements of electron-doped iridates using a range of spectroscopies, including ARPES33,34,35, scanning-tunneling microscope36 and resonant inelastic X-ray scattering37, also with features similar to those in cuprates. Our study and these new findings suggest there is a wide resemblance between doped iridates and cuprates in terms of band structures, electron dynamics and collective excitations, and on both the hole and electron-doped sides of the phase diagram. As cuprate is a canonical doped Mott insulator, and the most thoroughly studied one so far, our knowledge of a doped Mott insulator is to a great extend bounded by the cuprate literature. In this sense, the aforementioned hallmarks are most likely intrinsic and universal, rather than material specific, during the Mott-metal crossover.

We have ruled out prepairing and Fermi surface nesting as accountable for the pseudogaps in the iridates. Also there is no known quantum critical point in the doped iridates. We could also rule out long-range AF and structural distortions: both the pseudogap and the MFL survive above TN in the iridates, and static structural distortions are less likely to be related to the electron dynamics.

The precise definition of a pseudogap is in itself debated. From an experimental perspective, a widely accepted working criterion is well-articulated by Timusk and Statt4 in the mid-1990s in the study of cuprates: a ‘pseudogap’ is a partial gap, or more general, an incomplete suppression of spectral weight, that occurs only on some regions of the Fermi surface. Experimental features observed in, for example, ARPES, scanning-tunneling microscope, optical conductivity, all stem out of this seemingly naive (and more general) working definition. Admittedly, this definition does not point to the nature of pseudogaps. There are some cases where the origin of pseudogaps are understood theoretically to various degrees. For example, pseudogaps from preformed Cooper pairs usually have a momentum dependence characteristic of the pairing symmetry3,28,38, with relatively well-defined Bogoliubov quasiparticles. On the other hand, pseudogaps due to Fermi surface nesting features ‘hotspots’ on the Fermi surface connected by the charge/orbital/magnetic ordering vector25,29, which could be detected in other techniques.

Notably, even in these cases, not all facets of the pseudogap of interest, for example, its evolution against temperature, follows known theories. For example, the prepairing pseudogap28 in the optimally doped cuprates does evolve continuously into a ‘real’ hardened gap with decreasing temperature into the superconducting regime. However, as temperature rises, the prepairing pseudogap has been reported38,39 to be killed due to density of states filling-in rather than a reduction in gap size with increasing temperature, unlike standard Bogoliubov quasiparticles, reflecting their non-Fermi liquid nature. In other cases, a ‘real’ hardened gap and a defined phase transition may never develop when lowering the temperature towards absolute zero from the pseudogap phase. For example, in the p-type cuprates with a hole doping of 5–6%, the pseudogap temperature is generally above 200 K, and the ground state is considered by many a spin-glass without any defined phase transition3,40. There is no known clear boundary between the pseudogap phase and the spin-glass in this regime.

As we pointed out at the forefront of this paper, here we define pseudogaps following the definition from Timusk and Statt4. We have shown the pseudogaps and Fermi arcs in these iridates are distinctive from the prepairing and charge/orbital/spin ordered ones. The survival of pseudogaps to these seemingly high temperatures, compared with the Neel temperature, is a reflection of the Mott nature of iridates—something that is very relevant considering the recent debates over whether Sr2IrO4 is a Mott insulator or a Slater insulator41,42. The main disagreement between the two types of insulators is the role of short-range AF correlations. The Slater insulator is a mean-field concept built on the formation of long-range order—in the present case the long-range AF order. The short-range AF correlations that underlie the long-range AF order is ignored, or at best considered as a higher-order fluctuation. It is expected that the gap in a Slater insulator will diminish with decreasing long-range magnetic order, trending to zero as the phase transition is approached. Experimentally, there is no clear change in the band structure for both the parent compound18,43 and the doped iridates (this work) across the onset of long-range magnetism. This indicates the long-range magnetic order is not necessary for the formation of the gap and Sr2IrO4 is a Mott insulator.

Our observation of hallmarks during the Mott-metal crossover due to their proximity to the Mott state is in contrast with many of the prevailing views in the literature, but similar to those suggested in a few recent theoretical works44,45. More specifically, the smooth crossover from the Mott insulator Sr2IrO4 to the hole-doped ‘bad metal’, suggests the essential role of short-range correlations in the low-energy electron dynamics. It is interesting to point out the short-range AF correlations may be responsible for many of the hallmarks during the Mott-metal crossover, and in regions of the phase diagram where long-range AF order is long gone. For example, short-range AF correlations could indicate a widespread distribution of AF interaction strengths and quantum incoherence, giving rise to the lack of quasiparticle peaks and less well-defined gapping near the Fermi level—a possibility for iridates and even cuprates, among other doped Mott insulators.

The relation between the AF energy scale and the pseudogap temperatures is far from clear, with the strengths of the AF interactions converted to >1,000 K in iridates (and cuprates). In the latter the pseudogap temperature T* neither tracks the AF phase nor the superconducting dome. As for iridates, at this stage, we are not yet sure whether the pseudogap closes or could be filled due to thermal effects, at a higher temperature T*. Our data here would suggest most likely that there is a gradual filling-in of the density of states with rising temperature, while the actual T* (if it can be well-defined) is yet to be determined (see Supplementary Note 5 for details). Interestingly the community does not have a consensus whether the pseudogap actually closes or is filled in the underdoped cuprates. Regardless, this type of study, in which we closely consider the similarities and differences in the behaviour of the iridates and cuprates as they evolve away from the Mott insulator is a powerful tool to delve deeply into the nature of the doped Mott insulating state.

Methods

Sample preparation

All the ARPES, transport and magnetization data were taken from bulk Sr2Ir1−xRhxO4 samples. Single crystals were grown from off-stoichiometric quantities of SrCl2, SrCO3, IrO2 and RhO2 using self-flux techniques.

ARPES measurements

The ARPES experiments were performed at the PGM-A end station at the Synchrotron Radiation Center of University of Wisconsin-Madison, the Beamline 4.0.3 and 7.0.1 ARPES end stations at the Advanced Light Source, Lawrence Berkeley National Laboratory and the Surface and Interface Science beamline of the Swiss Light Source at the Paul Scherrer Institut. The samples were cleaved in situ with vacuum better than 5 × 10−11 Torr. The band structure and low-energy spectra near the Fermi level were taken with hν=77, 80, and 90 eV, with an energy resolution ∼25 meV, which is sharp on the scale of the principle spectral features.

Additional information

How to cite this article: Cao, Y. et al. Hallmarks of the Mott-metal crossover in the hole-doped pseudospin-1/2 Mott insulator Sr2IrO4. Nat. Commun. 7:11367 doi: 10.1038/ncomms11367 (2016).