Critical photoinduced reflectivity relaxation dynamics in single-layer Bi-based cuprates near the pseudogap end point

TL;DR

This study investigates the ultrafast photoinduced reflectivity relaxation in overdoped Bi-based cuprates near the pseudogap end point, revealing quantum critical behavior through power-law divergence in relaxation times.

Contribution

It provides the first detailed ultrafast spectroscopy analysis of cuprates near the pseudogap end point, identifying quantum critical dynamics in the relaxation processes.

Findings

Power-law divergence of relaxation time near the pseudogap end point.

Pseudogap response in overdoped cuprates resembles that in optimally doped samples.

Quantum critical behavior evidenced by divergence in relaxation dynamics.

Abstract

A comprehensive study of photoinduced transient reflectivity dynamics in heavily overdoped single-layer cuprate (Bi,Pb)$_{2}$Sr$_{2}$CuO$_{6+\delta}$ (Pb-Bi2201), across the end points of the pseudogap and superconducting phases, has been conducted using optical ultrafast time-resolved pump-probe spectroscopy. In the Pb-Bi2201 just before reaching the pseudogap end point, the transient reflectivity dynamics above $T_{{\rm c}}$ resemble the pseudogap response observed in optimally doped La-Bi2201. At low temperatures, however, the relaxation time exhibits a power-law divergence, $\tau\sim10\hbar/k_{\mathrm{B}}T$, signaling quantum critical behavior at the pseudogap end point doping. A similar power-law increase in relaxation time is also observed in Pb-Bi2201 just beyond the pseudogap doping end point, though it is less pronounced.

Full text

I Introduction

The pseudogap (PG) phase in the cuprate high-$T_{{\rm c}}$ superconductors exhibits anomalous metallic properties, distinct from those of conventional Fermi-liquid metals, and has long been investigated as a major factor in elucidating the mechanism of high-$T_{{\rm c}}$ superconductivity. [1, 2, 3] The phase is characterized by presence of an incomplete gap structure above $T_{{\rm c}}$ on an energy scale $\Delta_{{\rm PG}}$ reaching up to $\sim 80$ meV [4] in the underdoped regime. With increasing doping the pseudogap is monotonically suppressed and vanishes at the PG end point doping (PGED) lying in the overdoped superconducting (SC) phase-diagram region.

In recent years, high-quality single crystals of single-layer cuprate superconductors, such as Pb-doped Bi2Sr2CuO6+δ (Pb-Bi2201) [5, 6], La2-xSrxCuO4 (LSCO) [7], and Tl2Bi2CuO6+δ (Tl2201) [8], have become available even in the heavily overdoped regime, where both the SC and the PG phases vanish. This advancement has facilitated the investigation of the intrinsic electronic properties of this regime, offering new insights into the fundamental nature of high-$T_{{\rm c}}$ superconductivity. Taking overdoped Bi2201 as an example, scanning tunneling microscopy (STM) experiments have revealed that charge ordering and electronic nematicity persist in the overdoped regime as a glassy, short-range charge order. [9, 10] Angle-resolved photoemission spectroscopy (ARPES) measurements have identified a Lifshitz transition in the Fermi surface topology and suggested its potential connection to the disappearance of superconductivity. [11, 12, 13] Moreover, high-resolution ARPES has provided compelling evidence of strong electron correlations in the strange-metal state, indicating the presence of many-body interactions beyond electron-phonon coupling [14]. The presence of ferromagnetic fluctuations has been suggested by several transport properties and muon spin relaxation ($\mu$SR) experiments[15, 16], consistent with theoretical predictions. [17] Furthermore, transport experiments have indicated strange metal behavior [18, 19, 20, 8, 21] in the region of the PGED. While in various overdoped cuprates some evidence [22, 18, 19, 23, 20] points towards quantum criticality at the PGED, the transport properties and carrier concentration, as well as the SC properties, evolve rather smoothly across the PGED, suggesting absence of the criticality. [24, 25, 26, 8, 21]

In this study, we investigate the ultrafast photoinduced nonequilibrium relaxation dynamics in strongly overdoped Pb-Bi2201, covering the overdoped end of the SC dome including the PGED and the putative quantum critical point (QCP). While the dynamics has been thoroughly studied in the finite-PG doping region of the phase diagram [27, 28, 29, 30, 31, 32] the relaxation dynamics data in the strongly overdoped region are still lacking.

The normal-state transient reflectivity in the finite-PG hole-doping region is characterized with a relaxation component that shows a characteristic $T$ dependence, set by the PG energy scale $\Delta_{\mathrm{PG}}$ [33], and relaxes on a sub-picosecond time scale [27, 28, 29, 30, 31, 32] down to the lowest $T$. While we find that such behavior is reproduced in optimally doped La-Bi2201 the normal-state dynamics in overdoped Pb-Bi2201 shows quite different behavior with a divergent-like relaxation-time slowing down with decreasing $T$, $\tau\propto T^{-z}$, near the PGED, supporting the presence of criticality at the PGED. As a similar normal-state transient reflectivity relaxation slowing-down behavior was observed in the electron-doped cuprates [34, 35, 36, 37, 38] the present data suggest commonality of underlying physics in heavily hole-doped and electron-doped cuprates.

II Experimental

High-quality single-crystal Bi2201 samples were prepared using the floating zone method. The optimally doped reference sample (La-Bi2201; OPD34) was obtained by partially substituting Sr with La. Highly overdoped samples were prepared through partial substitution of Bi with Pb, followed by annealing under various conditions. The Pb-Bi2201 sample with $T_{{\rm c}}=10$ K is labeled as VOD10, the one with $T_{{\rm c}}=7$ K as VOD7, and the non-SC Pb-Bi2201 as VOD0. As depicted in Fig. 1, the doping levels of these three samples correspond to the pre-PGED, the beyond-PGED, and the beyond-SC end point doping, respectively.

Ultrafast optical time-resolved spectroscopy was performed using cavity-dumped Ti:Al2O3 laser (pulse width 120 fs, repetition rate 270 kHz, $\lambda_{{\rm pr}}=800$ nm) for the probe pulse and its second harmonic ($\lambda_{{\rm P}}=400$ nm) for the pump pulse. The sample was mounted in a copper holder within a helium-flow cryostat, and the coaxially aligned pump and probe beams were focused onto the sample using a lens. The beam diameter was set to 15 $\mu$m. The optical permittivity changes induced by the pump pulse carrier excitation were detected as the probe pulse reflectivity changes with a time delay $t_{{\rm Ppr}}$, which was controlled by the optical path delay. To enhance the detection sensitivity, the weak reflectivity changes of the probe were amplified using a lock-in detection technique, where the pump pulse intensity is modulated with an optical chopper.

III Results

The $T$-dependence of the transient reflectivity in all samples is summarized in Fig. 2 as color density plots (upper) and the transient reflectivity at representative temperatures (lower). Figure 2 (a) and (e) show the results for the optimally doped sample (OPD34), while (b)–(d) and (f)–(h) correspond to the overdoped samples (VOD$x$). The excitation fluence for each sample is adjusted according to the transient-response saturation threshold, to the fluence-linear amplitude region, as detailed in Supplement. The OPD34 sample exhibits a typical transient reflectivity response that reflects the characteristics of the phase diagram (Fig. 1), as shown in Fig. 2(e), where the carrier dynamics in the metallic (top), pseudogap (middle), and superconducting (bottom) states can be identified.[39, 32] A notable feature of the Bi-based cuprate superconductors is the negative sign [39, 30, 32] of the near-infrared transient reflectivity relaxation component associated with the PG. Similar behavior is observed in the VOD10 sample while the VOD7 and VOD0 samples exhibit the transient reflectivity with the positive sign over the entire temperature range. In all VOD$x$ samples the low-$T$ non-SC-state relaxation dynamics appears significantly slower in comparison to the OPD34 samples. Moreover, there is no notable transient-reflectivity change when entering the SC state in the VOD7 and VOD10 samples.

The $T$-dependent transients are analyzed by fitting them with a single exponential decay function ${\displaystyle\Delta R/R=A\exp{\left(-t/\tau\right)}+C}$. The amplitude ($A$) and relaxation time ($\tau$) are shown as functions of $T$ in Fig. 3. We focus on the data below $T=150$ K for the OPD34 sample in Fig. 3 (a, e) and below $T=100$ K for the VOD$x$ samples in Fig. 3 (b)–(d) and (f)–(h).

For OPD34, the PG carrier dynamics dominate the high-temperature side above $T_{{\rm c}}$. In Fig. 3(a), the amplitude of OPD34 increases gradually with decreasing $T$ and saturates below $T=70$ K. The decrease observed below $T=45$ K can be attributed to the contributions from the SC response and its fluctuating component, which exhibits an opposite sign compared to the PG response. For temperatures above $T\sim 45$ K, the $T$-dependence of the OPD34 transient-reflectivity amplitude is well described by a $T$-independent gap bottleneck model [33], previously used to describe the PG component in various cuprate superconductors. The solid line in Fig. 3(a) represents the best-fit result, yielding $\Delta_{{\rm PG}}=52\pm 6$ meV. It should be noted that, during the fitting procedure, the positive metallic response, observed at high-$T$, is assumed to remain constant throughout the entire $T$ range under consideration.

In Fig. 3 (b), the $T$-dependence of the amplitude in the VOD10 sample is qualitatively similar to that in the OPD34 sample, showing a gradual increase followed by a plateau as the temperature decreases. Notably, the fitting curve based on the $T$-independent gap bottleneck model fairly matches the experimental data (solid line in Fig. 3(b)). Here, the value of $\Delta_{{\rm PG}}=9.8\pm 1.6$ meV is obtained.

On the other hand, when looking at the relaxation time, the VOD10 sample data (Fig. 3(f)) show a significant difference from those of the OPD34 sample (Fig. 3(e)). In the OPD34 sample the relaxation time, $\tau$, remains below $\sim 1$ ps in the temperature range of $T=30-140$ K, despite some slowing-down with decreasing $T$. In contrast, VOD10 displays a pronounced increase of $\tau$ with decreasing $T$, following a power-law dependence of the form $\tau\propto T^{-z}$. From the fit, depicted by the solid line in Fig. 3(f), we determine the exponent to be $z=0.9$.

In Figs. 3(c) and (d), we present the transient reflectivity analyses for VOD7 and VOD0. The amplitude shows only a weak dependence on temperature in both samples. However, the relaxation time $\tau$ of VOD7 sample exhibits a power-law increase (Fig. 3(g)), resembling the behavior observed in the VOD10 sample(Fig. 3(f)). The magnitude of $\tau$ is quantitatively comparable between the two samples, while a power-law fit (solid line in Fig. 3(g)) yields a smaller exponent of $z=0.55$.

Finally, the analysis of the VOD0 data reveals an almost $T$-independent amplitude (Fig. 3(d)) and relaxation time (Fig. 3(h)).

IV Discussion

The comparison of the OPD34 and VOD10 samples in Fig. 2 suggests that the negative transient reflectivity in the VOD10 sample below $T=$100 K originates from the PG response. Indeed, the $T$-dependence of the transient reflectivity amplitude of VOD10 can be well described by the phonon-bottleneck model with a $T$-independent PG (Fig. 3(b)) [33], and its fluence dependence exhibits a saturation behavior [Fig. S3(b)], which can be attributed to the photoinduced suppression of the PG phase. [40]

On the other hand, the relaxation time of the VOD10 sample increases and appears to diverge as the temperature decreases (Fig. 3(f) and 4), differently than in the OPD34 sample. A power-law transient-reflectivity relaxation-time $T$-dependence with $z=1.18$ was previously reported in the normal state of electron doped Nd2-xCexCuO4+δ single crystal [37] at $x=0.156$ doping. Moreover, a temporal scaling of the transient reflectivity when comparing different $T$ was also observed so the normal-state transient reflectivity was attributed to the dynamics of an order parameter showing critical slowing-down with decreasing $T$. Later, doping dependent transient reflectivity study [38] in La2-xCexCuO4+δ thin films showed that at lower dopings the power-law relaxation time $T$-dependence is present only at higher $T$, where the antiferromagnetic correlations are absent. The temporal scaling was not found at lower dopings and the power-law exponent showed increase with decreasing doping to $z\sim 2$ at $x=0.08$.

A power-law normal-state transient-reflectivity relaxation-time temperature dependence was observed also in hole doped La2-xSrxCuO4+δ [28] where the relaxation time is much faster and $z\sim 0.5$ can be inferred in the normal state at and slightly below the optimal doping, $x\sim 0.15$ (see Fig. 4). The slightly overdoped, $x=0.2$ sample showed a crossover to a steeper, $z\sim 1,$ behavior below $T\sim 150$ K. The origin of the $T$-dependent relaxation time was tentatively attributed [41] to the $T$-dependent acoustic phonon escape length in the framework of the phonon-bottleneck model.

The normal-state transient-reflectivity relaxation-time in other higher-$T_{\mathrm{c}}$ hole doped cuprates has not been studied in such detail, in particular in the overdoped region. In Fig. 4 we compile some of the results from the present authors previous works. There appears to be no systematic power-law $T$-dependence.

As the $T$-dependent normal-state transient reflectivity amplitude behaves consistently with the PG phonon-bottleneck model [33] in a large number of hole doped cuprates [27, 29, 44, 28] we plot in Fig. 4 also the theoretical prediction for the $T$ dependence of the energy relaxation time, assuming, that the dominant relaxation process is anharmonic high-energy phonon decay and taking the into account a realistic phonon density of states [45]. As expected from the model, the energy relaxation time shows an exponential divergence at low-$T$, where in the bottleneck regime most of the deposited energy remains in the nonequilibrium quasiparticle population and the anharmonic phonon relaxation channel becomes ineffective. At higher-$T$, $k_{\mathrm{B}}T\gtrsim\Delta_{\mathrm{PG}}$, where a significant amount of the deposited energy is transferred into the high-energy phonon subsystem, the $T$-dependence flattens out becoming similar to the $z\sim 0.5$ behavior.

Taking into account the $\Delta_{\mathrm{PG}}$ values obtained from the amplitude fits [27, 29, 44, 28], we find that the $T$-dependence of the relaxation time in the near-optimally holed-doped cuprates is qualitatively consistent with the anharmonic relaxation mechanism. La-214 with the smaller $\Delta_{\mathrm{PG}}\sim 20$ meV [28] shows a flatter while Hg-1223 with $\Delta_{\mathrm{PG}}\sim 65$ meV shows a steeper $T$-dependence. We note, however, that the phonon cutoff energy in the hole doped cuprates is $\sim 70-90$ meV [46, 47] so the $\Delta_{\mathrm{PG}}$ magnitudes in excess of $\sim 40$ meV obtained from the bottleneck-model [33] fits are inconsistent with the assumptions of the model.

In the case of VOD10 sample the anharmonic bottleneck-model energy $T$-dependent relaxation time is clearly inconsistent with the data suggesting that either (i) another energy relaxation channel exists with the relaxation rate proportional to $T$, $1/\tau\propto T$, ($z\sim 1$), or (ii) the transient reflectivity (in the strongly overdoped region) does not correspond to the nonequilibrium quasiparticle density, but to another (presumably collective) degree of freedom, as in proposed [37] in the case of Nd2-xCexCuO4+δ.

Possibility (i) of another relaxation channel is plausible as the PG state is not fully gaped and inelastic quasiparticle scattering to the states within the PG is not completely suppressed. It is not clear, however, how the $\tau\sim 1/T$ behavior would be reproduced down to $T\sim 10$ K. The low energy phonons ($\hbar\omega\lesssim 1$ meV) with $n\sim k_{\mathrm{B}}T/\hbar\omega$ are unlikely to provide a rate-limiting effect for the out-of-anti-node quasiparticle scattering as their momenta are too small. The quasiparticle-quasiparticle scattering, which is expected to be less phase-space restricted as in the case of a $d$-wave superconductor [48], could easily scatter quasiparticles out of the anti-node, but the amount of the low-energy thermally excited quasiparticles ($n_{\mathrm{qp}}^{\mathrm{th}}\sim T$) should not play a major role in such process.

In the context (i) an inhomogeneous scenario similar to the one proposed in Ref. [41] should also be considered. In such scenario coexistence of gaped and ungapped regions is assumed and the excess energy relaxation in the gaped region is attributed to the energy transport to the ungapped regions. In [41] it was proposed that the energy is removed by ballistic acoustic phonons, however, as discussed above, the anharmonic relaxation channel slows down exponentially at low-$T$ so the $\tau\sim 1/T$ behavior cannot be reproduced.

Alternatively, the energy could be carried away by the high energy degrees of freedom, either the nonequilibrium quasiparticles or the phonons with $\hbar\omega>\Delta_{\mathrm{PG}}$. In the case of ballistic transport the characteristic relaxation time would be proportional to the characteristic inhomogeneity length scale, $l_{\mathrm{inh}}$, implying a critical inhomogeneity-scale behavior, $l_{\mathrm{inh}}\propto 1/T$.

In the case of diffusive transport, the characteristic relaxation time is set by the diffusion time, \tau_{\mathrm{D}}=$$l_{\mathrm{inh}}{}^{2}/D, where $D$ corresponds to an appropriate111High-energy quasiparticle or high-energy phonon diffusion constant. $T$-dependent diffusion constant. As $D$ usually increases with decreasing $T$ the $\tau\sim 1/T$ behavior implies increasing, $l_{\mathrm{inh}}\sim\sqrt{D/T}$, with decreasing $T$. The $\tau\sim 1/T$ behavior therefore cannot be simply a consequence of a static $T$-independent (chemical) sample inhomogeneity with constant $l_{\mathrm{inh}}$. Any critical behavior, $l_{\mathrm{inh}}\propto 1/T^{\nu}$, would imply also critical behavior of $D\sim 1/T^{2\nu-1}$.

We therefore conclude that possibility (i) can consistently reproduce the observed transient-reflectivity amplitude and relaxation time behavior only when assuming the inhomogeneous scenario with the $T$-dependent characteristic length that diverges with decreasing $T$, suggesting a critical behavior.

Assuming possibility (ii) the transient reflectivity with the $\tau\sim 1/T$ relaxation time divergence in the VOD10 sample could be interpreted in terms of dynamics of an order parameter that couples to the dielectric function. The $\tau\sim 1/T$ relaxation time behavior would correspond to the critical slowing down due to a low-$T$ second order phase transition or possibly a $T=0$ quantum phase transition at the PGED. As $\tau$ does not diverge at $T_{\mathrm{c}}$ and the SC response is undetectable even below $T_{\mathrm{c}}$ we can rule out the SC order parameter. Moreover, the observed timescale follows a power-law divergence, $\tau\sim 10\hbar/k_{\mathrm{B}}T$, indicating a connection to the Planckian time scale, which is predicted [50] in the quantum critical region of the phase diagram and often discussed in the context of the T-linear resistivity in cuprates [51].

Differently from the case of Nd2-xCexCuO4+δ [37], the temporal scaling of the transient reflectivity is absent for the present data. This, however, does not rule out the order parameter scenario as the scaling, $\Delta R(t)\propto te^{-t/\tau}$ [37], is expected only in the case of near-critically damped order parameter dynamics, while in the case of strongly overdamped order parameter dynamics a two exponent, $\Delta R(t)\propto e^{-t/\tau}-e^{-t/\tau_{\mathrm{r}}}$, which does not possess any intrinsic scaling, is expected.

Due to the requirement of the coupling to the dielectric function and the experimental probe polarization the symmetry of the order parameter is confined to zero-wavevector and Ag point symmetry [43]. Assuming that the order parameter is related to the PG and disorder is not strong, this rules out the possibility of any density-wave as well as magnetic orders as the origin of the PG.

Similar low-$T$ power-law relaxation time behavior is observed also in the VOD7 sample, but with a different $z=0.56$. As the doping of this sample is also in the vicinity of the PGED, it is plausible to assume that the low-$T$ relaxation time divergence has the same origin as in the VOD10 sample. In the VOD7 sample, however the pump fluence dependence (see Supplemental) is quite different from the VOD10 sample, showing only a partial saturation. This, together with the weak amplitude $T$-dependence, suggests that the contribution of the metallic response to the transient reflectivity in the VOD7 sample is significant also at low-$T$. The single exponential relaxation fit procedure therefore does not allow us to extract the intrinsic relaxation time of the saturable component that is associated to the order parameter dynamics. As the metallic component relaxation time is only weakly $T$ dependent the effective relaxation time obtained from the single exponential fit is also expected to show a weaker $T$ dependence with different $z$.

The fluence-dependent slowdown of the relaxation time, observed in both VOD10 and VOD7 samples (see Fig. S3 for details), becomes more pronounced at lower temperatures. This behavior can be understood as a transient increase of the effective electronic (and high-energy-phonon) temperature upon photoexcitation, which temporarily drives the system away from the QCP. In the VOD7 sample, the slowdown is apparent only for $F\lesssim 10~\mu$J/cm2, since at higher fluences the response is dominated by the unsaturated metallic component.

In VOD0, the transient reflectivity is predominantly governed by the metallic carrier dynamics as the saturation is virtually absent in the full studied fluence range and the relaxation time is only weakly $T$-dependent, consistent with theoretical analysis [52].

A number of reports highlighted the presence of critical slowing down associated with a quantum QCP in different cuprate superconductors. $\mu$SR measurements on the electron-doped cuprate Pr1-xLaCexCuO4±δ (PLCCO) have revealed a $T$-dependent relaxation rate that follows a single power-law behavior within the PG phase at optimal doping. [53] This power-law dependence is indicative of critical phenomena near a ferromagnetic or antiferromagnetic QCP [54, 55, 56], suggesting that the optimal doping level in PLCCO lies in proximity to a QCP. Resonant inelastic X-ray scattering (RIXS) measurements have further revealed critical slowing down attributed to charge fluctuations. In YBa2Cu3O7-δ (YBCO) and Bi2Sr2CaCu2O8+δ (Bi2212), this phenomenon is most pronounced near the optimal doping level, reinforcing the notion that a QCP is located in this region. [57] Additionally, the Fano resonance observed in optimally doped Bi2212 suggests that quantum fluctuations contribute to the melting of charge order via interactions with phonons. [58]

Optical time-resolved studies generally reflect the combined effects of charge, spin, and lattice degrees of freedom, making it challenging to isolate specific fluctuation channels. However, the small wave vector of optical photons and their symmetry strongly constrain which fluctuation channels can directly couple to the dielectric tensor. The finite wave vector fluctuations as well as magnetic fluctuations can therefore affect the optical transient dynamics only indirectly by modulating the relaxation channels, as discussed above for the case of the phonon-bottleneck model. Either way, the transient reflectivity data uncover singular-like relaxation-time $T$-dependence in the vicinity of the PGED. Although the observed relaxation timescales exceed the Planckian bound by an order of magnitude, the distinct dynamics near the PGED can be naturally interpreted within the framework of quantum criticality.

V Summary and Conclusions

We investigated the temperature-dependent photoinduced transient reflectivity dynamics in Pb-Bi2201 single-layer cuprate superconductors, focusing on the strongly overdoped region of the phase diagram that spans the pseudogap and superconducting-dome end-point dopings. In the vicinity of the pseudogap end point, below $T\sim 100$ K, we observe a Planckian-like normal-state relaxation time, $\tau\sim 10\hbar/k_{\mathrm{B}}T$. We show that such behavior is incompatible with the anharmonic phonon relaxation described by the phonon-bottleneck model, which consistently accounts for the normal-state relaxation dynamics in underdoped and optimally hole-doped cuprates. Considering possible relaxation scenarios, we suggest that the observed behavior most naturally originates from quantum criticality at the pseudogap end point.

Comparing the present behavior to the behavior in an electron doped cuprate [37] suggests universal behavior in the vicinity of a quantum critical point, however, more data in the strongly hole-overdoped cuprates would be needed to confirm this.

Acknowledgements.

V.V. K., D. M. and T. M. acknowledge the financial support of Slovenian Research and Innovation Agency (research core funding No. P1-0040). Y. T. acknowledges the financial support of Japan Society for Promotion and Science (JSPS, 19H05826).