Quantum Criticality in a Layered Iridate

Iridates provide a fertile ground to investigate correlated electrons in the presence of strong spin-orbit coupling. Bringing these systems to the proximity of a metal-insulator quantum phase transition is a challenge that must be met to access quantum critical fluctuations with charge and spin-orbital degrees of freedom. Here, electrical transport and Raman scattering measurements provide evidence that a metal-insulator quantum critical point is effectively reached in 5 % Co-doped Sr2IrO4 with high structural quality. The dc-electrical conductivity shows a linear temperature dependence that is successfully captured by a model involving a Co acceptor level at the Fermi energy that becomes gradually populated at finite temperatures, creating thermallyactivated holes in the Jeff = 1/2 lower Hubbard band. The so-formed quantum critical fluctuations are exceptionally heavy and the resulting electronic continuum couples with an optical phonon at all temperatures. The magnetic order and pseudospin-phonon coupling are preserved under the Co doping. This work brings quantum phase transitions, iridates and heavy-fermion physics to the same arena. 1 ar X iv :2 10 4. 01 09 8v 1 [ co nd -m at .s tr -e l] 2 A pr 2 02 1


INTRODUCTION
A rare combination of strong spin-orbit coupling and substantial electron correlations in 5d 5 iridates leads to exotic quantum states and remarkable physical phenomena such as spin liquid phases [1], Kitaev physics [2][3][4], Fermi arcs in the electronic structure with a possible connection to high-T c superconductivity [5][6][7][8][9][10], and control of the crystal structure and magnetic properties by an electric current [11,12]. An intensively investigated iridate is Sr 2 IrO 4 (SIO), crystallizing in a tetragonal structure with an ABCD stacking of layers with tilted IrO 6 octahedra, as shown in Figs. 1(a) and 1(b) [13,14]. As for the electronic structure, the octahedral crystal field splits the Ir 5d level into t 2g and e g levels. The strong spin-orbit coupling breaks the 6-fold degeneracy of the t 2g levels, giving rise to lower J eff = 3/2 and upper J eff = 1/2 sublevels. Finally, the J eff = 1/2-derived band is broken into lower and upper Hubbard bands, rendering SIO insulating [15][16][17]. The Ir pseudospins order in a non-collinear magnetic structure below T N ∼ 240 K [see Fig. 1 [16,18,19].
Injection of charge carriers through electron or hole doping tends to reduce T N , although it is seems possible to destroy the insulating state without necessarily destroying the magnetic order [20] and vice-versa [21]. The most obvious approach to access the possible quantum critical fluctuations (QCF) [22,23] associated with a metal-insulator transition would be to force the closure of the bandgap by application of external pressures. Nonetheless, the nonmetallic state persists up to at least 185 GPa despite the identification of exotic states at intermediate pressures [24][25][26][27][28][29]. The alternative approach employed here involves the quest for a specific dilute cationic substitution that induces acceptor or donor impurity levels within the bandgap without actually charge doping the Ir-derived bands at T = 0 K, as shown in Fig. 1(c). A possible contender might be the Sr 2 Ir 1−x Rh x O 4 system, as Rh and Ir are in the same group of the periodic table. However, it has been demonstrated that dilute Rh-substitution induces Rh 3+ /Ir 5+ charge partitioning, in practice doping the system with holes and rapidly suppressing the magnetic order [30][31][32][33][34][35][36][37][38][39][40][41]. Thus, the energy of the Rh 3+ acceptor level is inferred to lay below the Fermi energy, E Rh 3+ < E F , at least for x < 0.24.
Another candidate is the Sr 2 Ir 1−x Co x O 4 system. Based on first principle calculations for Sr 2 Ir 0.5 Co 0.5 O 4 , it was inferred that Co 3+ /Ir 5+ charge partitioning is energetically stable [42], which was confirmed by subsequent experiments on this material [43][44][45]. On the other hand, more dilute Co substitutions have not been investigated in much detail yet, although it is already known that even moderate Co substitution levels (x ≤ 0.1) are not sufficient to reduce T N appreciably [46]. In this work, we investigate the charge transport, magnetic, structural and vibrational properties of Sr 2 Ir 0.95 Co 0.05 O 4 (SICO). We demonstrate that this material shows charge transport quantum critical behavior that is consistent with a simple picture where the Co 3+ acceptor level coincides with the top of the lower Hubbard band at Fig. 1(c)], triggering QCF with remarkably large renormalized masses. SICO. An additional feature is observed at ∼ 340 cm −1 in SIO, indicative of a stoichiometric sample [48,49], whereas in SICO this feature is washed out and broad additional signals are observed at ∼ 260 and ∼ 420 cm −1 . The 260 cm −1 peak was previously observed and becomes stronger in more disordered samples [48,49]. The weak intensity of the additional Raman features associated with disorder and the relatively sharp M 1 − M 4 modes in SICO indicate that the individual layer structures are preserved to a large extend through the 5 % Co substitution, in contrast to the Sr 2 Ir 1−x Ru x O 4 series that shows much larger modifications in the phonon Raman spectra even at low doping levels [49].

Charge transport at zero magnetic field
The electrical resistivity curves ρ(T ) of SIO and SICO are shown in Fig. S4(a). The values of ρ(T ) are significantly reduced for SICO with respect to the parent compound.
In addition, the temperature dependence of the former is remarkably well captured by a simple power-law behavior, ρ(T ) = A −1 · T −x with x = 1.075(1), over the entire investigated temperature interval 3 < T < 300 K [dashed line in Fig. S4(a)]. Such power-law behavior is not found in SIO (see Supplementary Figure S3). Figure S4(b) displays the same data in terms of electrical conductivity σ(T ) in a double-log scale for SICO only. The fit to the lowtemperature data (T < 6 K) is optimized by a slightly different exponent x = 1.003(6) that is even closer to unity. Note that the positive exponent implies that σ → 0 as T → 0, i.e., the material is non-metallic. On the other hand, an exponential behavior is expected for insulating materials, with 1/4 ≤ α ≤ 1 for either a bandgap insulator (α = 1) or for a variable range hopping mechanism where the specific α is defined by the system dimensionality and the specific energy-dependence of the density of states n(E) near E F [50].
Neither of such exponential behaviors for σ(T ) are observed for SICO (see Supplementary Figure S4), so this material is not classified either as a Mott, Slater or an Anderson insulator.
Actually, according to very general scaling considerations, a power-law behavior σ(T ) ∝ T x signals a metal-insulator quantum phase transition [22]. The positive exponent contrasts with the negative values found in metallic materials where the quantum criticality is not due to the proximity of a metal-insulator transition but interfaces two different conducting states, such as in a magnetic/heavy fermion quantum phase transition [51,52]. The specific value x ∼ +1 for SICO must be captured by an appropriate microscopic model (see below).

Magnetic properties and magnetoresistance
Magnetization M (T ) curves of SICO and SIO are shown in Fig. 4(a), taken on warming after zero-field cooling and warming after field cooling, with H = 5 kOe. Both compounds order at T N ∼ 240 K, in line with Ref. [46]. The shapes of the M (T ) curves of SIO and SICO are distinct and the magnetization values are substantially reduced by the Co substitution. The latter result either indicates that the Co moments partially compensate the Ir net moments or the magnetic canting angle of the Ir moments [see Fig. 1 reduced for SICO. Also, both materials feature a separation of the zero-field cooling and warming after zero-field cooling curves that becomes more prominent below T ∼ 100 K [53]. This is understood in terms of a field-induced transition of the ↓↑↑↓ magnetic structure and symmetry-related domains at zero-field to ↑↑↑↑ under an applied field along the ab-plane H ab ∼ 2 kOe [16,48,54]. In a polycrystal, the H a projection is different for each crystallite, and with an external field H = 5 kOe a substantial fraction of the crystallites will still have H a < 2 kOe, leading to a coexistence of different phases at low temperatures in proportions that are arguably sensitive to the thermomagnetic sample history. This is manifested not only in the different zero-field cooling and warming after zero-field cooling M (T ) curves, but also in the hysteresis of the isothermal M (H) curves [see Fig. 4  SIO shows a substantial negative magnetoresistance of ∼ 9 % for H = 35 kOe. A fieldhysteretic behavior is also observed. This phenomenon is again originated in the fieldinduced transition between the ↓↑↑↓ and ↑↑↑↑ structures, where the former is substantially more resistive [20]. Remarkably, the initial high-resistance state is not recovered by cycling the field. This is likely due to a powder distribution of internal fields that are created after the ↑↑↑↑ state is first activated, which may prevent the homogeneous high-resistance ↓↑↑↓ state to be simultaneously recovered over the entire sample volume. The magnetoresistance of SICO is largely suppressed with respect to SIO [see Fig. 4(c)], remaining below 1 % up to H = 35 kOe. This indicates that, for SICO, the additional electronic scattering channel introduced by the Co impurities reduces the mean-free path of the thermally-activated charge carriers, overwhelming the magnetic scattering channel that is responsible for the large magnetoresistance of SIO. We therefore estimate that the electronic mean free path of SICO is D ∼ 8Å, which is the average separation between the Co scattering centers for 5 % substitution. We should note that in SIO the conductivity in the ab plane is higher than along c [20,55]. Thus, in polycrystals the resistivity and magnetoresistance curves are presumably dominated by the charge transport in the ab plane. This is also likely valid for SICO, although this assumption is not essential for the conclusions of this work.

Electron-phonon and pseudospin-phonon couplings
We now return to a more detailed investigation of the Raman spectra In opposition to SICO, the M 1 lineshape for SIO is symmetric at low temperatures, in agreement with previous studies [57,58]. Figure 5(b) shows the temperature dependence of |1/q| for mode M 1 of SICO. A substantial asymmetry parameter |1/q| = 0.15 is observed at the base temperature. This reveals a continuum of low-energy excitations at T → 0 that is not present in SIO. Considering that a hypothetical metallic state in SICO is dismissed by resistivity data, this result indicates that the insulating gap in SICO must be below ∼ 0.02 eV, placing this material very close to a quantum critical point. The |1/q| parameter tends to increase on warming, with anomalies at T ∼ 100 K and ∼ T N . In particular, the anomaly at T ∼ 100 K suggests there is another physically meaningful temperature below T N for SICO, as previously inferred for SIO [12,20,53,59,60]. The reduced magnetization below

DISCUSSION
We propose the schematic energy diagram displayed in Fig. 1(c) where E ≡ |E − E F | and m * is the lower Hubbard band effective mass [65]. The number of thermally-activated holes per unit volume at low temperatures is where Γ(u) is the gamma function. These carriers have mean energyĒ(T ) = 3k B T /2. We now treat these thermally-activated holes semiclassically with a renormalized mass m ren that is not necessarily equal to m * . The time between collisions is where D is the carrier mean free path. From the Drude relation σ(T ) = q 2 N (T )τ /m ren , where q is the electron charge, we obtain Thus, this simple model is able to reproduce the observed power-law behavior for the elec- This is reasonable once a non-zero density of charge carriers is presumably necessary to destroy the magnetic ordering at T = 0 K. The thermally-activated holes possess charge and spin-orbital degrees of freedom, and this system offers an opportunity to access the resulting QCF in the magnetically ordered regime.

CONCLUSIONS
In summary, a detailed analysis of our electrical resistivity and Raman scattering data uncovers the presence of QCF in SICO. The inferred quantum phase transition is related to the crossing of the Co 3+ acceptor level with the top of the lower J eff = 1/2 band at E F .

The linear behavior observed in σ(T ) is captured by a Drude-like semiclassical model that
indicates a large renormalized electron mass for the thermally-activated charge carriers.

Sample synthesis
The Sr 2 IrO 4 (SIO) and Sr 2 Ir 0.95 Co 0.05 O 4 (SICO) polycrystalline samples were synthesized by a standard solid-state reaction method employing high-purity SrCO 3 , IrO 2 and CoO as described in Ref. [26]. Since the reagents are mixed in stoichiometric proportions, this method leaves minimum margin for compositional Sr/Ir/Co variations, which is critical for the conclusions of this work. Also, the large absolute resistivity values at low temperatures of SIO, and especially the large low-T /high-T ratio ρ(50 K)/ρ(300 K) ∼ 10 4 [see Fig. S4(a)] compares to the highest reported values in the literature [30], unfavoring the possibility of any relevant doping due to off-stoichiometric oxygen content in this sample. This conclusion is likely extensible to SICO, considering that both samples were synthesized by the same procedure.

X-ray diffraction and energy-dispersive X-ray spectroscopy
Laboratory X-ray diffraction data were taken in a Bruker D2 diffractometer equipped with a linear detector, employing Cu Kα radiation (λ = 1.542Å). Rietveld refinements were performed with the GSAS-II suite [66]. The refined Co occupancy at the Ir site is 4.2(7) % for SICO, close to the 5 % nominal value. The other refined crystallographic parameters are given in the Supplementary Table S1. Energy-dispersive X-ray spectroscopy measurements were performed on two different spots of SICO using a scanning electron microscope FEI beamtime. The authors also acknowledge the Center for Semiconducting Compounds and Nanotechnologies (CCS) at UNICAMP for providing the equipment and technical support for the energy-dispersive X-ray spectroscopy measurements.

DATA AVAILABILITY
The authors declare that the main data supporting the findings of this study are available within the article and its Supplementary Information file. Extra data are available from the corresponding author upon reasonable request.
Optical signature of a crossover from mott-to slater-type gap in  (c) Unpolarized Raman scattering spectra of the two investigated samples at T = 20 K. The vertical arrows highlight the presence of a peak at ∼ 340 cm −1 for SIO that is indicative of a stoichiometric sample [48,49], and extra features at ∼ 260 and ∼ 420 cm −1 for SICO that are associated with weak disorder.