Abstract
A major challenge in understanding the cuprate superconductors is to clarify the nature of the fundamental electronic correlations that lead to the pseudogap phenomenon. Here we use ultrashort light pulses to prepare a nonthermal distribution of excitations and capture novel properties that are hidden at equilibrium. Using a broadband (0.5–2 eV) probe, we are able to track the dynamics of the dielectric function and unveil an anomalous decrease in the scattering rate of the charge carriers in a pseudogaplike region of the temperature (T) and holedoping (p) phase diagram. In this region, delimited by a welldefined T*_{neq}(p) line, the photoexcitation process triggers the evolution of antinodal excitations from gapped (localized) to delocalized quasiparticles characterized by a longer lifetime. The novel concept of photoenhanced antinodal conductivity is naturally explained within the singleband Hubbard model, in which the shortrange Coulomb repulsion leads to a kspace differentiation between nodal quasiparticles and antinodal excitations.
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Introduction
Superconductivity in the cuprates takes place when charge carriers are injected into a chargetransfer insulator^{1} in which the carriers are localized by the strong electron–electron interactions. This surprising phenomenon has motivated a remarkable effort to understand to what extent the electron–electron interactions determine the physical properties of cuprates when tuning these materials away from the insulating state by increasing the hole concentration p and the temperature T. The physics of superconducting cuprates is further complicated by an anisotropic gap (pseudogap) in the electronic density of states (DOS), which opens well above the superconducting critical temperature (T_{c}) at moderate doping and takes on maximal values close to the k=(±π,0), (0,±π) regions (antinodes) of the Brillouin zone (BZ) (ref. 2). In the same region of the p–T phase diagram, different ordered phases have been observed. Such phases can be favoured by the reduced kinetic energy of the carriers and might be associated with a quantum critical point underneath the superconducting dome T_{c}(p)^{3,4}. Indeed, a wealth of different broken symmetries, such as unusual q=0 magnetism^{5,6,7}, chargedensity waves (CDW)^{8,9}, stripes^{10}, nematic and smectic phases^{11,12,13}, have been reported. The universal mechanism underlying the formation of the pseudogap continues to be the subject of intense research^{14}, and the relation between the electronic interactions and the various ordering tendencies remains one of the major open questions.
A possible scenario that reconciles this phenomenology is that the pseudogap emerges as an inherent effect of the strong shortrange Coulomb repulsion, U, between two electrons occupying the same lattice site^{15}. Considering the case of an isotropic Mott insulator, the Udriven suppression of charge fluctuations is expected to reduce the electron kinetic energy and render the electronic excitations increasingly localized in real space. Cluster generalization of dynamical meanfield theory (DMFT) calculations^{16,17,18} suggests that the cuprates exhibit a similar Udriven reduction of kinetic energy that is, however, not uniform in momentum space. In the underdoped region, the antinodal excitations are indeed quasilocalized, with a T=0 divergent scattering rate that is reminiscent of the Mott insulator (see Fig. 1a), whose selfenergy (SE) diverges at low frequency and decreases at finite temperature. In contrast, the nodal (N) excitations in the vicinity of k=(±π/2,±π/2) still retain the essential nature of the quasiparticles (QPs) at large hole concentrations, that is, a scattering rate that increases when external energy is provided (see Fig. 1b). The possible kspace differentiation between nodal QPs and antinodal excitations in the pseudogap phase has been hitherto elusive both to kspaceintegrated equilibrium techniques, such as optical spectroscopy and resistivity measurements, that are mostly sensitive to the properties of nodal QPs and to conventional angleresolved photoemission spectroscopy (ARPES), which has limits in capturing small temperature and kdependent variations of the electronic scattering rate.
Here we adopt a nonequilibrium approach^{19} based on ultrashort light pulses (≈100 fs) used to artificially prepare the system in a nonthermal state with an excess of antinodal (AN) excitations^{20}. The key element of our experiment, which goes beyond singlecolour pumpprobe techniques^{21,22,23}, is that the dynamics of the optical properties are simultaneously probed over an unprecedentedly broad energy range (0.5–2 eV). This technique allows us to probe the damping of the infrared reflectivity plasma edge and to directly relate the transient reflectivity variation, δR(ω,t), to the instantaneous value of the optical scattering rate. From the outcome of timeresolved broadband spectroscopy, we infer that the transient nonthermal state created in the pseudogap phase is characterized by a scattering rate smaller than that at equilibrium. The emerging picture is that, on excitation, the localized antinodal states transiently evolve into more mobile states with a reduced scattering rate. This scenario is corroborated by DMFT calculations within the singleband Hubbard model and by timeresolved and ARPES (TRARPES) data in the pseudogap state. Finally, the generality of the concept of photoenhanced antinodal conductivity in the pseudogap regime is demonstrated by combining the results obtained on different families of copper oxides (Bi and Hgbased) in a single and universal phase diagram.
Results
Optical properties of cuprates
Nonequilibrium optical spectroscopy is emerging as a very effective tool to unravel the different degrees of freedom coupled to electrons in correlated materials^{19,24}. After the photoexcitation process, small variations of the equilibrium optical properties can be measured on a timescale faster than the recovery of the equilibrium QPs’ distribution, the complete restoration of the (longrange) orders^{25} and the heating of the phonons^{26}, thus unveiling an intriguing physics that cannot be accessed under equilibrium conditions. Singlecolour timeresolved reflectivity measurements^{21,22,23} have been applied in the past to study the pseudogap state, evidencing a characteristic dynamics proportional to the pseudogap amplitude (Δ_{pg}) and a change of sign of the photoinduced reflectivity variation at T≈T*. Nonetheless, the inherent lack of spectral information of singlecolour techniques precluded the understanding of the origin of this pseudogaprelated reflectivity signal. Recent developments in ultrafast techniques now permit to overcome this limitation by probing the dynamics of the dielectric function over a broad frequency range. This paves the way to a quantitative modelling of the ultrafast optical response of the pseudogap phase.
To properly model the dynamics of the optical properties measured in nonequilibrium conditions, we start by introducing the basic elements that characterize the equilibrium reflectivity, R_{eq}(ω), of optimally doped Bi_{2}Sr_{2}Y_{0.08}Ca_{0.92}Cu_{2}O_{8+δ} (OPYBi2212, holedoping p=0.16, T_{c}=96 K), shown in Fig. 2a. In the infrared region, the normal incidence R_{eq}(ω) is dominated by a metalliclike response, which is characterized by a broad edge at the dressed plasma frequency ω_{p}≈1.25 eV. The infrared reflectivity of optimally and overdoped cuprates is well reproduced by the extended Drude model (EDM)^{27,28} in which a frequencydependent optical scattering time, τ(ω,T), accounts for all the processes that affect the QP lifetime, such as the electronic scattering with phonons, spin fluctuations or other bosons of electronic origin. Within the EDM, the optical scattering rate is connected to the singleparticle SE by:
where f is the Fermi–Dirac distribution and Σ(ζ,T) and Σ*(ζ,T) are the electron and hole kspaceaveraged SEs. In Fig. 2a, we report the best fit of the EDM to the experimental R_{eq}(ω) of OPYBi2212 at T=100 K. In the calculation of the SE in equation (1), we use a recently developed model^{29,30,31} that takes into account a nonconstant QPs’ DOS, Ñ(ω,T), characterized by a pseudogap width Δ_{pg}=40 meV (see Methods section). The kspaceintegrated electronic DOS is recovered completely between Δ_{pg} and 2Δ_{pg}. The Ñ(ω,T) extracted by optical spectroscopy (see Methods; Supplementary Notes 1 and 2; Supplementary Figs 1 and 2) is in agreement with the outcome of tunnelling experiments^{32}. The quality of the fit to the optical properties (Fig. 2a) demonstrates that the EDM with a nonconstant DOS is a very effective tool to extract the scattering properties of the charge carriers, at least for moderate holedoping concentrations. The possible failure of the EDM model is expected for holedoping concentrations smaller than those of the samples studied in this work. The interband transitions at ħω>1.5 eV are accounted for by additional Lorentz oscillators at visible/ultraviolet frequencies. Focusing on the energy range that will be probed by the timeresolved experiment (0.5–2 eV), we note that τ(ω) has almost approached the asymptotic value τ_{∞}≈2 fs (see inset of Fig. 2a). Therefore, we can safely assume that, in the 0.5–2 eV range, the damping of the reflectivity edge depends on the asymptotic value of the scattering rate, γ_{∞}=ħ/τ_{∞}.
Nonequilibrium optical spectroscopy
In the nonequilibrium optical spectroscopy, snapshots of the reflectivity edge of OPYBi2212 are taken with 100 fs time resolution, as a function of the delay from the excitation with the 1.5 eV pump pulses. Considering that the temporal width of the probe pulses is much wider than the inherent scattering time of the charge carriers (τ_{∞}≈2 fs), we can assume that the electronic excitations have completely lost any coherence on a timescale faster than the observation time. Therefore, the transient optical properties can be rationalized by changing some effective parameters in the equilibrium reflectivity.
Notably, R_{eq}(ω,γ_{∞}) exhibits an isosbestic point (see Fig. 2a) at the frequency . At this frequency, the reflectivity is independent of γ_{∞} and, for small δγ_{∞} changes, it can be expanded as δR(ω,γ_{∞})=[∂R(ω)/∂γ_{∞}]δγ_{∞}, where [∂R(ω)/∂γ_{∞}]>(<)0 for (see Supplementary Note 3; Supplementary Fig. 3). Therefore, the reflectivity variation, measured over a broad frequency range across , can provide a direct information about the instantaneous value of the total scattering rate during the thermalization process of the photoinduced nonequilibrium QP population. In the bottom panel of Fig. 2a, we report the normalized reflectivity variation δR/R(ω), calculated from the equilibrium EDM model by assuming a positive (red line) or negative (blue line) variation of the total scattering rate, γ_{∞}. The reflectivity variation associated with a change in γ_{∞} extends over a broad frequency range and it changes sign when crossing the isosbestic point .
Figure 2b shows the frequencyresolved measurements on OPYBi2212 at T=100 K, as a function of the delay t from the pump pulse. The relative reflectivity variation δR/R(ω,t)=[R_{neq}(ω,t)−R_{eq}(ω)]/R_{eq}(ω), where R_{neq} is the nonequilibrium reflectivity, is reported as a twodimensional (2D) plot. The colour scale represents the magnitude of δR/R(ω,t) as a function of t and of the probe photon energy ħω. Soon after the excitation, δR/R(ω,t) is positive below , whereas a negative signal is detected above . When compared with the δR/R(ω) calculated for δγ_{∞}<0 (bottom panel of Fig. 2a), the measured δR/R(ω,t) suggests a transient decrease in the QP scattering rate soon after the excitation with the pump pulse.
To substantiate the possible transient variation of γ_{∞}, a differential model is used to quantitatively analyse the δR/R(ω,t) signal. This approach consists in finding out the minimal set of parameters that should be changed in the equilibrium dielectric function to satisfactorily reproduce the measured δR/R(ω,t) at a given time t. The main goal of this procedure is to disentangle the contributions of the genuine variation of γ_{∞}, from that of possible photoinduced bandstructure modifications, such as the transient filling of the pseudogapped electronic states^{31,33}. In Fig. 3, we report a slice of δR/R(ω,t) (red dots) at fixed delay time, that is, t=100 fs. δR/R(ω, t=100 fs) is qualitatively reproduced, over the whole probed frequency range, by modifying only two parameters in the equilibrium EDM, that is, Ñ(ω,T) and γ_{∞}. In contrast, no change in the interband transitions and in the plasma frequency is required. This demonstrates that, in the probed spectral range, the δR/R(ω) signal is not significantly affected by the transient change in the electronic occupation at the pump energy scale or by other excitoniclike processes and that the density of the charge carriers is not significantly modified by the excitation process. Similarly, Fig. 3 also displays the contributions to the δR/R(ω, t=100 fs) signal, as arising from the variation of Ñ(ω,T) (green line) and γ_{∞} (blue line), separately. The central result is that a negative γ_{∞} variation is necessary to reproduce δR/R(ω,t=100 fs). δγ_{∞}<0 corresponds to a narrowing of the reflectivity plasma edge around the isosbestic point at (see Fig. 2a), which is detected as a broad reflectivity variation changing from positive to negative when moving to high frequencies. Quantitatively, the relative variation δγ_{∞}/γ_{∞}=−(1.2±0.4) × 10^{−2} is extracted from the fitting procedure. For sake of clarity, we show in Fig. 3 the total δR/R(ω) calculated (black dashed line) by constraining a positive δγ_{∞}/γ_{∞}=+1.2 × 10^{−2} variation (red dashed line). In this case, the main features of the measured δR/R(ω) cannot be, even qualitatively, reproduced. This demonstrates that, while the experimental uncertainties can affect the absolute value of δγ_{∞}/γ_{∞}, the transient photoinduced decrease in the scattering rate is a robust experimental fact that is unaffected by possible uncertainties, such as fluence fluctuations or changes in the pumpprobe overlap.
The characteristic relaxation time ŧ of the measured δγ_{∞} is obtained by fitting the function δR/R(t)=δR/R(0)exp(−t/ŧ) to the timeresolved traces at fixed frequencies. In particular, we focus on the time traces at ħω=0.68 and 1.55 eV, shown in Fig. 3b,c, for which the contribution from the Ñ(ω,T) variation is negligible. The resulting value, ŧ=600±50 fs (see Supplementary Note 4; Supplementary Fig. 4), is of the same order as that of the time required for the complete heating of the lattice and for the recovery of the equilibrium QPs’ population, as will be shown by the timeresolved photoemission measurements. Interestingly, on the picosecond timescale, the δR/R(ω) signal is qualitatively opposite to the signal at t≈100 fs, as shown by the time traces in Fig. 3b,c. In particular, it changes from negative to positive as ω increases and crosses . The δR/R(ω, t>1 ps) signal can be reproduced by a broadening of the reflectivity edge, equivalent to an increase in the scattering rate (δγ_{∞}/γ_{∞}=10^{−4}), with a negligible contribution from the variation of Ñ(ω,T). This γ_{∞} increase is compatible with a local effective heating (δT=0.6 K) of the lattice, which can be estimated considering the pump fluence (10 μJ cm^{−2}) and the specific heat of the sample (see Supplementary Note 5; Supplementary Fig. 5).
The transient decrease in the total electronic scattering rate, measured on the subpicosecond timescale at T=100 K, is a novel finding that contrasts both with the results^{26} obtained well above the pseudogap temperature and with the behaviour expected for a metallic system. In the latter case, the scattering rate should increase as energy is delivered, either adiabatically or impulsively. Furthermore, as inferred from the δR/R(ω,t) measured in local quasiequilibrium conditions (t>1 ps), this anomalous δγ_{∞}<0 cannot originate from the partial quench of an order parameter or a temperatureinduced decrease in the electron–boson coupling. In fact, the expected positive δγ_{∞}>0 is recovered at t>1 ps, although on this timescale an increase in the effective electronic and bosonic temperatures and the related amplitude decrease in the possible order parameter are attained.
Nonequilibrium broadband optical spectroscopy crucially broadens the information that could be reached in the past by singlecolour techniques. The quantitative modelling of the dynamics of the dielectric function in the 0.5–2 eV energy range demonstrates that, after the impulsive photoexcitation, the optimally doped YBi2212 sample at 100 K is driven into a nonequilibrium state, characterized by a scattering rate smaller than that at equilibrium (δγ_{∞}<0). In terms of the a.c. conductivity, this corresponds to a transient increase in the conductivity, once assumed a constant density of the charge carriers. The relaxation time of this photoenhanced conductivity is ŧ=600±50 fs. On the picosecond timescale, the equilibrium distribution, dominated by nodal QPs, is recovered and a more conventional behaviour of the scattering rate (δγ_{∞}>0) is observed, in agreement with the outcomes of conventional equilibrium techniques, such as resistivity and optics^{34,35}.
The kspacedependent nonequilibrium QP population
Since the energy scale of the pump photons (1.5 eV) is much larger than the thermal (k_{B}T≈10 meV) and the pseudogap (Δ_{pg}≈40 meV) energy scales, the photoinduced QPs’ distribution is expected to have no relation with the equilibrium one. The photoexcitation process can be roughly reduced to two main steps. In the first step, the pump pulse is absorbed creating electron–hole excitations extending from −1.5 to +1.5 eV across the Fermi energy (E_{F}) and with a distribution that is regulated by the joint DOS of the photoexcitation process. In consequence of the extremely short scattering time of highenergy excitations (τ_{∞}=ħ/γ_{∞}≈2 fs), this photoexcited population undergoes a fast energy relaxation related to multiple scattering processes that lead to the creation of excitations at energies closer to E_{F}. Considering the kspaceintegrated DOS in the pseudogap state, a high density of excitations is expected to accumulate, within the pumppulse duration (0–100 fs), in the Δ_{pg}−2Δ_{pg} energy range from E_{F}. In the second and slower step, the subset of scattering processes that allow large momentum exchange, while preserving the energy conservation, leads to the recovery of a quasiequilibrium distribution dominated by nodal QPs. Considering the phasespace constraints for these processes, the redistribution of excitations in the kspace is expected to be effective on the picosecond timescale.
To directly investigate the kspace electron distribution after the photoexcitation process with 1.5 eV ultrashort pulses, we apply a momentumresolved technique. TRARPES (see Methods section) is applied to measure the transient occupation at different kvectors^{36,37} along the Fermi arcs (see Fig. 4a) of a Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} single crystal at T=100 K. When considering the TRARPES spectra at fixed angles Φ from the antinodal direction (see Fig. 4a), the pump excitation results in a depletion of the filled states below E_{F} and a filling of the states above E_{F}^{36}. In the following, we will focus on the integral (I) of the TRARPES spectra for E>E_{F}, which is proportional to the total excess of excitations in the empty states. Figure 4b reports the variation of I normalized to the intensity before the arrival of the pump pulse (ΔI_{Φ}(t)/I_{Φ}) as a function of Φ. For each value of Φ, ΔI_{Φ}(t)/I_{Φ} is fitted by a single exponential decay of the form ΔI_{Φ0}exp(−t/ŧ_{Φ}), convoluted with a Gaussian function to account for the finite temporal resolution (~100 fs). Figure 4c,d shows the kdependent values of ΔI_{Φ0} and ŧ_{Φ} extracted from the fitting procedure. Within the error bars, ΔI_{Φ0} increases by a factor 3 when moving from the node (Φ=45°) towards the antinodal region of the BZ, indicating an effective photoinjection of AN excitations. This situation is dramatically different from that expected in equilibrium conditions, in which the number of excitations is governed solely by the Fermi–Dirac distribution at temperature T. In this case, the excitations’ density should significantly decrease when approaching the AN region of the BZ, as a consequence of the gap Δ_{pg}>>k_{B}T in the density of electronic states. Another notable result is that the relaxation time ŧ_{Φ} is kdependent, increasing from 300 to 800 fs when moving from the N to the AN region. These values are much larger than the total optical scattering time (τ_{∞}=ħ/γ_{∞}≈2 fs), which demonstrates that the recovery of a quasiequilibrium QPs’ distribution is severely constrained, either by the phase space available for the scattering processes simultaneously conserving energy and momentum, or by some bottleneck effect related to the emission of gapenergy bosons during the relaxation of AN excitations.
TRARPES demonstrates that, soon after the excitation with 1.5 eV pump pulses, an excess number of lowenergy electron excitations is accumulated in the antinodal region. Furthermore, the relaxation time of this nonequilibrium distribution is significantly kdependent. Taken together, these results prove that this nonequilibrium electron distribution in the kspace cannot be described by a Fermi–Dirac distribution with a single effective electronic temperature that is evolving in time. At the simplest level, we can assume that the effective temperature increase in the excitations in the antinodal region is larger than that of the nodal QP population. Notably, the kspaceaveraged relaxation time of the nonequilibrium distribution, 〈ŧ_{Φ}〉=550±200 fs, is the same, within the error bars, as of the relaxation time of the transient δγ_{∞}<0 measured by nonequilibrium optical spectroscopy. This result demonstrates the direct relation between the creation of a nonthermal distribution with an excess of antinodal excitations and the transient decrease in the scattering rate measured by nonequilibrium optical spectroscopy.
T^{*}_{neq}(p) line emerging from nonequilibrium spectroscopy
The use of ultrashort light pulses to manipulate the equilibrium QP distribution is crucial to investigate the cuprate phase diagram from a perspective that was hitherto inaccessible. In this section, we report on the T^{*}_{neq}(p) temperature below which the transient reduction of γ_{∞} is observed via nonequilibrium optical spectroscopy. As shown in Fig. 3a, for ω significantly above ω_{p}, the contribution to δR/R(ω,t) due to the modification of Ñ(ω) is negligible. Therefore, singlecolour pumpprobe measurements at 1.55 eV probe photon energy contain the direct signature of the transient δγ_{∞}<0, in the form of a negative component^{21,22} with relaxation time ŧ=600 fs^{23}. Figure 5a reports some of the singlecolour time traces measured as a function of the temperature (from 300 to 20 K) for different hole concentrations. The negative component (blue colour) appears in the δR/R(t) signal at T^{*}_{neq}=240±20 K for p=0.13 and at T^{*}_{neq}=165±20 K for p=0.16. Above p=0.18, this negative δR/R(t) signal is never detected on cooling the sample down to T_{c} (see Supplementary Notes 6 and 7; Supplementary Figs 6, 7 and 8). A similar behaviour is also found for underdoped HgBa_{2}CuO_{4+δ} (Hg1201; see Supplementary Note 8; Supplementary Figs 9,10 and 11). Hg1201 is considered a nearly ideal singlelayer cuprate and exhibits a maximal critical temperature close to that of doublelayer YBi2212^{38,39}. Collecting the results for the two materials on the same plot, we obtain a phase diagram that shows a universal behaviour for the properties of different copperoxidebased superconductors having the same maximal critical temperature. Figure 5b shows that the p–T phase diagram of cuprates is dominated by an ubiquitous and sharp T^{*}_{neq}(p) boundary. Below this line, the pumpinduced nonthermal distribution of the charge carriers exhibits a scattering rate smaller than that at equilibrium. This pseudogaplike T^{*}_{neq}(p) boundary meets the superconducting dome slightly above P=0.18. Below T_{c}, the continuation of the T^{*}_{neq}(p) line delimits two regions of the superconducting dome that exhibit opposite variation of the optical spectral weight of intra and interband transitions^{24,40,41,42}, related to the crossing from a kinetic energy gain to a potential energy gaindriven superconducting transition, consistent with predictions for the 2D Hubbard model^{43}.
Further insight into the T^{*}_{neq}(p) line unveiled by nonequilibrium optical spectroscopy is provided by the comparison with the pseudogap temperature T*(p) estimated from complementary equilibrium techniques^{7,14,44,45} (elastic neutron scattering, resistivity and resonant ultrasound spectroscopy) and with the onset temperature of different ordered states^{46,47,48} (CDW, timereversal symmetrybreaking states and fluctuating stripes). The picture sketched in Fig. 5c compares the main phenomenology on the most common materials with similar critical temperatures (Bi2212, Hg1201, YBCO). Remarkably, the T^{*}_{neq}(p) line almost exactly coincides with the pseudogap line estimated by resistivity measurements^{45} and ultrasound spectroscopy^{14} and with the onset of the newlydiscovered q=0 exotic magnetic order^{7,44}. The appearance of ordered states at T<T* is likely the consequence of the instability of the correlated pseudogap ground state on further cooling.
The antinodal scattering rate
The results reported in the previous sections suggest that the T^{*}_{neq}(p) line delimits a region in which the AN states evolve into more metallic ones (δγ_{∞}<0) on photoexcitation with the pump pulses. The generality of the results obtained calls for a general model that accounts for the phase diagram unveiled by the nonequilibrium optical spectroscopy. Considering that the measured transient decrease in the carrier scattering rate is faster than the complete heating of the lattice, we focus on the minimal model that neglects electron–phonon coupling and retains the genuine physics of correlations, that is, the 2D Hubbard Hamiltonian^{49,50}. To compute the temperaturedependent SE in different positions of the BZ, we use the dynamical cluster approximation, a cluster extension of DMFT that captures the kspace differentiation of the electronic properties^{17} between different regions of the BZ (see Methods section). Furthermore, longrange correlations are neglected to focus on the intrinsic effect of shortrange correlations inside the chosen foursite cluster (see Methods section). At this stage, the increase in energy related to the pump excitation is mimicked by selectively increasing the effective temperature of the nodal and antinodal SEs.
The FS of a lightly doped system is reported in Fig. 6a and exhibits a progressive smearing when moving from the nodes to the antinodes. This result is qualitatively in agreement with the FS experimentally measured by conventional ARPES in prototypical cuprates (compared with the ARPES data on optimally doped Bi_{2}Sr_{2}Y_{0.08}Ca_{0.92}Cu_{2}O_{8+δ} at 100 K reported in Fig. 4a). The smearing of the FS at antinodes reflects a strong dichotomy between the scattering rate of nodal and antinodal excitations that can be captured by plotting the imaginary parts of the calculated SE, that is, the inverse QP lifetime, as a function of the effective temperature (see Fig. 6c). In contrast to nodal QPs, whose scattering rate increases with temperature, the scattering rate of antinodal excitations exhibits a completely different evolution, decreasing as the effective temperature rises. This anomalous behaviour is related to the localized and gapped character of the antinodal fundamental excitations that experience very strong electronic interactions at low temperatures. The concept of delocalized QPs, characterized by a smaller scattering rate, is progressively recovered when the temperature increases. This striking dichotomy of the nature of the elementary excitations in the kspace is the consequence of a momentumspace selective opening of a correlationdriven gap, which eventually evolves into the full Mott gap at p=0. The same physics has been previously identified in calculations based on similar approaches (also using larger clusters to achieve a better momentum resolution)^{16,17,18,51}. When the hole doping is increased (see Fig. 6b,d), the kspace differentiation of N–AN fundamental excitations is washed out and a more conventional metallic behaviour is recovered. In this case, the delocalized QPs exhibit a gapless energy spectrum and a scattering rate that is proportional to a power function of the temperature over the entire BZ.
DMFT calculations thus confirm an intrinsic Udriven momentumspace differentiation of the electronic properties of cuprates at finite hole concentrations and temperatures. The nature of the AN states is similar to that of a Mott insulator in the sense that the scattering rate of AN states decreases when the internal energy of the system is increased.
Discussion
Wrapping up the experimental outcomes collected in this work, we can gain a novel insight into the pseudogap physics of highT_{c} cuprates. Timeresolved optical spectroscopy demonstrates that, on excitation with 1.5 eV pump pulses, the optical properties of cuprates transiently evolve into those of a more conductive system (δγ_{∞}<0). TRARPES shows that the kspace distribution of the electron excitations created by the pump pulse is characterized by an excess of AN electrons. Finally, the reported observation of a transient enhancement of the conductivity unveils a universal and sharp pseudogaplike T^{*}_{neq}(p) boundary in the p–T phase diagram. All these results cannot be rationalized in terms of the simple consequence of an anisotropic scattering of QPs with bosonic fluctuations, such as antiferromagnetic spin fluctuations. Although antiferromagnetic correlations may strongly influence the dynamics of AN excitations, the energy provided by the pump pulse should necessarily result in an increase in the boson density, leading to an increase in the scattering rate for all the timescales. Furthermore, the transient decrease in the scattering rate on the subpicosecond timescale cannot be related to the presence of incipient charge orders, such as CDW, which are quenched by the pump pulse. The picosecond dynamics of conventional CDW has been widely studied in weakly correlated (U≈0) materials^{25,52}. After the impulsive excitation, the characteristic timescale for the CDW recovery is on the order of several picoseconds, that is, much longer than the transient δγ_{∞}<0 measured in our work. Even assuming a purely electronic (and faster) density wave mechanism in cuprates, the photoinduced decrease in the scattering rate should monotonically decrease eventually approaching a zero value. This is in contrast with the experimental observation of a transition from δγ_{∞}<0 to δγ_{∞}>0 on the picosecond timescale.
On the other hand, the Hubbard model provides a minimal framework for the interpretation of the pseudogap region of cuprates as unveiled by timeresolved spectroscopies. The pump pulse provides energy to the system in a nonthermal way, which can be schematized as a larger increase in the effective temperature of AN excitations as compared with that of nodal QPs. The transient decrease in the scattering rate (δγ_{∞}<0) is thus related to the evolution of AN excitations from Mottlike gapped excitations to delocalized QPs with a longer lifetime. On the picosecond timescale, the equilibrium electronic distribution is recovered and the expected δγ_{∞}>0 is measured. Although this picture does not exhaust all the properties of the pseudogap, it captures a key element of the universal and fundamental nature of the antinodal states, providing a backbone for more realistic multiband descriptions^{3} that could give rise to a brokensymmetry state originating from a quantum phase transition^{4} at T=0. Furthermore, the shortrange Coulomb repulsion induces a suppression of the charge fluctuations below the T^{*}_{neq}(p) line that is opposite to the effect of temperature. Therefore, the region of the cuprate phase diagram delimited by T^{*}_{neq}(p) is intrinsically prone to bulk^{53,54} and surface^{55,56} phaseseparated instabilities, whose nature depends on the details of the Fermi surface (FS) of the particular system considered. Interestingly, the region of the cuprate phase diagram in which the optical photoexcitation creates a nonequilibrium state with longer lifetime closely corresponds to that in which the possibility of creating a transient superconductive state by tetrahertz excitation has been recently discussed^{57,58}. These findings suggest a general tendency of copper oxides to develop, when photoexcited, a transient nonequilibrium state that is more conductive than the equilibrium phase.
Methods
Samples
The Ysubstituted Bi2212 single crystals were grown in an image furnace by the travelling solvent floatingzone technique with a nonzero Y content to maximize T_{c}^{38}. The underdoped samples were annealed at 550 °C for 12 days in a vacuumsealed glass ampoule with copper metal inside. The overdoped samples were annealed in a quartz test tube under pure oxygen flow at 500 °C for 7 days. To avoid damage of the surfaces, the crystals were embedded in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} powder during the annealing procedure. In both cases, the quartz tube was quenched into icewater bath after annealing to preserve the oxygen content at annealing temperature.
The Hg1201 single crystals were grown using a flux method, characterized and heat treated to the desired doping level^{39}. The crystal surface is oriented along the abplane with a dimension of about 1 mm^{2}. Hg1201 samples are hygroscopic. Therefore, the last stage of the preparation of the sample surface is done under a continuous flow of nitrogen, on which the sample is transferred to the highvacuum chamber (10^{−7} mbar) of the cryostat within a few minutes. Before each measurement, the surface is carefully checked for any evidence of oxidation.
Optical spectroscopy
The abplane dielectric function at equilibrium of the YBi2212 samples has been measured using conventional spectroscopic ellipsometry^{59}. The dielectric function has been obtained by applying the Kramers–Kronig relations to the reflectivity for 50<ω/2πc<6,000 cm^{−1} and directly from ellipsometry for 1,500<ω/2πc<36,000 cm^{−1}.
The δR/R(ω,t) data presented in Fig. 2 have been acquired combining two complementary techniques^{60}: (i) a pump supercontinuumprobe setup^{61}, based on the white light generated in a photonic crystal fibre seeded by a Ti:sapphire cavitydumped oscillator, to explore the visible–near infrared range of the spectrum (1.1–2 eV); (ii) a pump tunableprobe setup, based on an optical parametric amplifier seeded by a regenerative amplifier, to extend measurements in the infrared spectral region (0.5–1.1 eV). In both cases, the laser systems operate at 250 kHz repetition rate. The pump fluence is set to 10±2 μJ cm^{−2} for spectroscopic measurements. Singlecolour measurements (ħω=1.55 eV), presented in Fig. 5, have been performed directly using the output of a cavitydumped Ti:sapphire oscillator. The highfrequency modulation of the pump beam, combined with a fast scan of the pumpprobe delay and lockin acquisition, ensures a high signaltonoise ratio (~10^{6}) and fast acquisition times, necessary to study the evolution of the timeresolved optical properties as a function of the temperature. Singlecolour measurements have been performed with a pump fluence ranging from 3 to 30 μJ cm^{−2}. In all experiments, the pump photon energy is 1.55 eV. Samples are mounted on the cold finger of a closedcycle cryostat. The temperature of the sample is stabilized within ±0.5 K.
The EDM with a nonconstant DOS
In the EDM, the scattering processes are effectively accounted for by a temperature and frequencydependent scattering rate γ(ω,T), which is often expressed through the socalled memory function, M(ω,T). The dielectric function resulting from the EDM is:
In the conventional formulation of the EDM (for more details, see refs 28, 31), the calculation of the SE Σ(ω,T) is based on the assumption of a constant DOS at the Fermi level. This approximation is valid at T=300 K in optimally and overdoped systems, but fails as the temperature and the doping decrease and a pseudogap opens in the electronic DOS. A further evolution of the EDM, accounting for a nonconstant electronic DOS, has been recently developed^{29}, and has been used to analyse spectroscopic data at equilibrium^{30}. Within this model, the imaginary part of the electronic SE is given by:
where n and f are the Bose–Einstein and Fermi–Dirac distribution functions, respectively; π(Ω) is the Bosonic function; and Ñ(ω,T) is the normalized DOS. The real part of the SE, Σ_{1}(ω,T), can be calculated by using the Kramers–Kronig relations.
The normalized DOS Ñ(ω,T) is modelled by^{30}:
Where Δ_{pg} represents the (pseudo)gap width, while the normalized DOS at E_{F}, that is, Ñ(0,T), represents the gap filling. The π(Ω) function is extracted by fitting the EDM to the normal state optical properties^{26}. π(Ω) is characterized by a lowenergy part (up to 40 meV), a peak centred at ~60 meV and a broad continuum extending up to 350 meV.
The analysis of the time and frequencyresolved data is performed by modelling the nonequilibrium dielectric function (ε_{neq}(ω)) and calculating the reflectivity variation through the expression: δR/R(ω,t)=[R_{neq}(ω,t)−R_{eq}(ω)]/R_{eq}(ω), where the normal incidence reflectivities are calculated as: R_{eq}(ω)=[1−√ε_{eq}(ω)]/[1+√ε_{eq}(ω)]^{2} and R_{neq}(ω)=[1−√ε_{neq}(ω)]/[1+√ε_{neq}(ω)]^{2}.
The role of the finite penetration depth of the pump pulse (d_{pu}=160 nm @ 1.55 eV) is accounted for by numerically calculating δR/R(ω) through a transfer matrix method, when a graded index of the variation of the refractive index n with exponential profile along the direction z perpendicular to the surface, that is, δn=δn_{0}exp(−z/d_{pu}), is assumed.
Photoemission spectroscopy
The FS of YBi2212 reported in Fig. 4a has been measured by ARPES in equilibrium conditions. ARPES has been performed with 21.2 eV linearly polarized photons (Heα line from a SPECS UVS300 monochromatized lamp) and a SPECS Phoibos 150 hemispherical analyzer. Energy and angular resolutions were set to 30 meV and 0.2°. The Bi2212 samples studied by TRARPES are nearly optimally doped single crystals with a transition temperature T_{c}=88 K. The samples have been excited by 55 fs laser pulses with a photon energy of 1.55 eV at 300 kHz repetition rate, at an absorbed fluence of 35 μJ cm^{−2}. The transient electron distribution was probed by timedelayed 80 fs, 6 eV laser pulses, photoemitting electrons, which were detected by a timeofflight spectrometer. The energy resolution was 50 meV, the momentum resolution 0.05 Å^{−1} and the time resolution <100 fs.
ClusterDMFT and the Hubbard model
The Hubbard Hamiltonian^{49,50} is given by:
where creates (annihilates) an electron with spin σ on the i (j) site, is the number operator, t_{ij} the hopping amplitude to nearest and nextnearest neighbours, U the Coulomb repulsion between two electrons occupying the same lattice site and μ is the chemical potential that controls the total number of electrons in the N sites.
The Hubbard model has been studied by means of a ClusterDMFT that maps the full lattice model onto a finite small cluster (here a foursite cluster) embedded in an effective medium that is selfconsistently determined as in standard meanfield theory. The method therefore fully accounts for the shortrange quantum correlations inside the cluster. It has been shown by various authors that different implementations of this approach provide qualitatively similar results and reproduce the main features of the phase diagram of the cuprates, including the dwave superconducting state and the pseudogap region that we discuss in the present paper. The calculations of this paper use the dynamical cluster approximation^{62} prescription and the 4* patching of the BZ introduced in^{17} and have been performed using finitetemperature exact diagonalization^{63} to solve the selfconsistent cluster problem using eight energy levels in the bath as in several previous calculations. The finitetemperature version of the exact diagonalization has been implemented as discussed in ref. 17 including typically 40 states in the lowtemperature expansion of the observables.
Additional information
How to cite this article: Cilento, F. et al. Photoenhanced antinodal conductivity in the pseudogap state of highTc cuprates. Nat. Commun. 5:4353 doi: 10.1038/ncomms5353 (2014).
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Acknowledgements
We thank D. Fausti, G. Zgrablic, G. Sordi, E. Gull, P. Werner, A. Avella, M. Fabrizio, A. Cavalleri, D. Manske, G. Ghiringhelli, B. Keimer, D. Mihailovic for the useful discussions and comments and for their support. The research activities of C.G., S.D.C., N.M., S.M., F.B., G.F., M.C., F.P. and U.B. have received funding from the European Union, Seventh Framework Programme (FP7 20072013), under Grant No. 280555 (GO FAST). F.C., G.C., and F.P. acknowledge the support of the Italian Ministry of University and Research under Grant No. FIRBRBAP045JF2 and FIRBRBAP06AWK3. M.C. is financed by European Research Council through FP7/ERC Starting Grant SUPERBAD, Grant Agreement 240524. L. R. and U. B. acknowledge support by the Mercator Research Center Ruhr (MERCUR). The YBi2212 crystal growth work was performed in M.G.'s prior laboratory at Stanford University, Stanford, CA 94305, USA, and supported by DOEBES. The Hg1201 crystal growth work at the University of Minnesota was supported by DOEBES Award DE SC0006858. The work at UBC was supported by the Killam, Alfred P. Sloan, Alexander von Humboldt, and NSERC's Steacie Memorial Fellowships (A.D.), the Canada Research Chairs Program (A.D.), NSERC, CFI, and CIFAR Quantum Materials. D.v.d.M. acknowledges the support of the Swiss National Science Foundation under Grant No. 200020140761 and MaNEP. The Open Access has been funded by Università Cattolica del Sacro Cuore and ElettraSincrotrone Trieste S.C.p.A.
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C.G., F.C. and F.P. conceived the project and the timeresolved experiments. C.G. coordinated the research activities with input from all the coauthors, particularly F.C., S.D.C., G.C., U.B., A.D., M.C. The pumpsupercontinuumprobe technique at the ILAMP Labs (Brescia) was designed and developed by C.G., F.B. and G.F. The pumpsupercontinuum probe technique at the TREX Labs (Elettra, Trieste) was designed and developed by F.C. and F.P. The timeresolved optical measurements and the data analysis were performed by F.C., S.D.C., G.C., S.P., S.M., N.N. and C.G. The Bi2212 crystals were grown and characterized by H.E., M.G., R.C. and A.D. The Hg1201 crystals were grown and characterized by M.K.C, C.J.D., M.J.V. and M.G. Equilibrium ARPES measurements were perfomed by R.C. and A.D. Nonequilibrium ARPES measurements were performed by L.R and U.B. The equilibrium spectroscopic ellipsometry measurements were performed by D.vdM. The DMFT calculations were carried out by M.C., who provided input for the theoretical interpretation of the experimental data. The text was drafted by C.G., F.C., M.C. and F.P with major input from S.D.C., G.C., R.C., F.B., G.F., U.B., A.D., D.vdM. and M.G. All authors extensively discussed the results and the interpretation, and revised the manuscript.
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Cilento, F., Dal Conte, S., Coslovich, G. et al. Photoenhanced antinodal conductivity in the pseudogap state of highT_{c} cuprates. Nat Commun 5, 4353 (2014). https://doi.org/10.1038/ncomms5353
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DOI: https://doi.org/10.1038/ncomms5353
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