In the underdoped copper-oxides, high-temperature superconductivity condenses from a nonconventional metallic ”pseudogap” phase that exhibits a variety of non-Fermi liquid properties. Recently, it has become clear that a charge density wave (CDW) phase exists within the pseudogap regime. This CDW coexists and competes with superconductivity (SC) below the transition temperature Tc, suggesting that these two orders are intimately related. Here we show that the condensation of the superfluid from this unconventional precursor is reflected in deviations from the predictions of BSC theory regarding the recombination rate of quasiparticles. We report a detailed investigation of the quasiparticle (QP) recombination lifetime, τqp, as a function of temperature and magnetic field in underdoped HgBa2CuO4+δ (Hg-1201) and YBa2Cu3O6+x (YBCO) single crystals by ultrafast time-resolved reflectivity. We find that τqp(T ) exhibits a local maximum in a small temperature window near Tc that is prominent in underdoped samples with coexisting charge order and vanishes with application of a small magnetic field. We explain this unusual, non-BCS behavior by positing that Tc marks a transition from phase-fluctuating SC/CDW composite order above to a SC/CDW condensate below. Our results suggest that the superfluid in underdoped cuprates is a condensate of coherently-mixed particle-particle and particle-hole pairs.
First observed as stripe-like order in the “214” cuprates1, as checkerboard order in vortex cores2, and subsequently at the surface of Bi and Cl based compounds3,4,5, the universality of CDW order in the cuprate phase diagram has been established, through NMR6 and X-ray scattering7,8,9,10,11,12 probes. In YBa2Cu3O6+x (YBCO) near hole concentration 1/8 application of large magnetic fields stabilizes long-range CDW order at a temperature that approaches Tc6,13. The near-degeneracy of the characteristic temperatures of CDW and SC phases suggests that these two order parameters are related, as opposed to simply coexisting and competing. Several theoretical works have suggested that the same short-range antiferromagnetic fluctuations drive the formation of CDW and SC states14,15,16,17,18. Moreover, the temperature dependence of the CDW amplitude in YBCO, as determined from X-ray scattering, can be reproduced by a model of fluctuating, multi-component order, of which CDW and SC states are two projections17.
The basic premise of BCS theory is that the SC condensate is made up of Cooper pairs, which are bound states of two electrons with opposite momenta and spin19. Subsequent to BCS, Kohn and Sherrington20 showed that a CDW state is likewise a pair condensate, but of electron and holes, whose net momentum determines the wavelength of charge order. Quasiparticles (or broken pairs) are the fundamental excitations of paired condensates such as the SC and CDW states. It is important for what follows to note that, although quasiparticles in SC and CDW states are fermions, they have properties distinct from the quasiparticles that constitute the normal state. SC quasiparticles are phase-coherent linear superpositions of normal state electrons and holes, while CDW quasiparticles are superpositions of electrons (or holes).
Our experiments probe quasiparticle states in underdoped cuprates through time-resolved measurements of their lifetime against recombination, whereby two quasiparticles of opposite spin re-pair and scatter into the condensate. As we discuss below, this scattering rate is sensitive to the phase-coherence of quasiparticle superposition states. To measure the recombination lifetime, we first generate a nonequilibrium quasiparticle population by photoexcitation with an ultrashort optical pulse. We use a low pump fluence so as to probe the linear regime in which the photogenerated quasiparticle population is small compared to its thermal value near Tc. The rate of return to equilibrium is measured by resolving the photoinduced change in optical reflectivity, ΔR(t), as a function of time, t, after absorption of the pump pulse. A wealth of experiments21,22 have demonstrated that the appearance of a ΔR signal reflects the opening of a gap (or gaps) at the Fermi level and that its amplitude is proportional to the nonequilibrium quasiparticle population.
Correlation with CDW order
In Fig. 1 we show ΔR(t)/R at several temperatures for underdoped samples of Hg-1201 with Tc = 55, 71, and 91 K. At high temperature we observe a short-lived negative component of ΔR(t) that is associated with the pseudogap (PG)23. With decreasing T a larger amplitude positive component with a much longer lifetime appears and quickly dominates the signal. In cuprates with near-optimal doping this positive, long-lived component appears close to Tc24 and was therefore associated with the onset of superconductivity. However, this association breaks down in underdoped samples in which the positive component is already large at Tc (data at Tc are highlighted in red).
In Fig. 2a–f we plot the maximum value of ΔR/R(t) (which occurs at ≈1 ps after photoexcitation) as a function of T in underdoped Hg-1201 samples with Tc’s ranging from 55 to 91 K. The temperature (Tonset) at which the positive component of ΔR appears is indicated by a blue down arrow in each panel. Note that ΔR continues to increase continuously with further decrease of T, without a clear feature at the critical temperature for superconductivity (indicated by red arrows). Both Tonset and Tc are plotted as a function of hole concentration, p, in Fig. 2g. The onset temperatures of positive ΔR outline a dome that peaks on the underdoped side of the phase diagram and extends to temperatures 130 K above Tc.
Based on a correlations with other probes, we believe that the appearance of the slow, positive component of ΔR at the temperatures Tonset(p) shown in Fig. 2g corresponds to the onset of local CDW order. There is a clear correspondence between the dome of Tonset as determined by ΔR(T) in Hg-1201 and the region of the phase diagram where a CDW is detected in YBCO25,26. Although the phase space region of CDW in Hg-1201 is yet to be mapped in as much detail as in YBCO, a CDW has been detected in Hg-1201 samples with Tc = 71 K12 at a temperature (indicated by the green circle in Fig. 2g) coincident with Tonset. Another correlation linking ΔR to the CDW in underdoped cuprates is that the positive component of ΔR in YBCO Ortho-VIII (to be discussed further below) has the same temperature dependence as a zero wavevector vibrational mode that arises from CDW-induced zone-folding27.
We turn now to measurements of the T dependence of the recombination lifetime of quasiparticles for T < Tonset. The decay curves in Fig. 1 were fit using a function of the form, , where the first term describes QP recombination, the second term accounts for finite risetime and the presence of a negative PG component, and the constant offset C captures a long-lived contribution that we attribute to local heating by the pump pulse (see Supplement for details on the fitting procedure). Figure 3a displays the evolution of τqp(T) with hole concentration in the Hg-1201 system. At each hole concentration we observe structure in the T-dependence of the quasiparticle recombination time at Tc. In underdoped samples there is a peak in τqp(T) at Tc that is most prominent in the Tc = 71 K sample and decreases in amplitude at lower and slightly higher hole concentration in a manner that appears to be correlated with the strength of the CDW.
Figure 3b,c compare ΔR(t, T) in Hg-1201 (Tc = 71 K) and YBCO Ortho-VIII (Tc = 67 K), illustrating that generality of the phenomena described above. Figure 3b,c show the temperature dependence of the amplitude ΔRqp ≡ A and τqp respectively for the two underdoped samples. In both we observe the onset of positive ΔR(T) well above Tc and a smooth variation through the SC transition. As shown in Fig. 3c, peak in τqp(T) centered on Tc is strikingly similar in the two representative samples.
The T-dependence of the quasiparticle lifetime in the sample of Hg-1201 at near-optimal doping (topmost data set of Fig. 3a) is qualitatively different from what is seen in underdoped samples. In the near-optimal sample τqp grows monotonically as T → Tc, suggesting a tendency to diverge as ΔRqp(T) goes to zero, while in underdoped samples the peak in τqp at Tc is a small feature on a smooth background. The behavior of τqp in near-optimal Hg-1201, which is observed in other optimally doped cuprates as well24,28, can be understood within the mean-field theory of superconductivity. The theoretical description of τqp(T) in the context of BCS began in the 1960’s and its subsequent history is reviewed in ref. 29. In the mean-field picture, the relaxation of a nonequilibrium quasiparticle population is described by a pair of coupled equations: a Landau-Khalatnikov equation for the energy gap and a Boltzmann-like equation that governs the quasiparticle distribution29. According this analysis, (where τ is the electron inelastic scattering time) and diverges as the superconducting gap Δ(T) vanishes as T approaches Tc from below. In more recent work on cuprate superconductors, effects associated with the phonon bottleneck28,30,31,32 are included, which leads to replacing τ by the lifetime of phonons with energy greater than 2Δ.
Phase coherence and recombination
The behavior of τqp(T) in underdoped samples is at odds with the mean-field picture described above. Two observations–continuous variation of ΔRqp for T < Tonset and the smoothly varying background in τqp(T) that underlies the small peak–suggest that a quasiparticle gap has opened well above Tc. Given a pre-existing gap, it is very difficult to explain the modulation of the quasiparticle lifetime near Tc. In view of the difficulties with the mean-field picture, we are led to consider the onset of phase coherence at Tc, rather than gap-opening, as the origin of the structure in τqp(T).
Coherence effects have been observed previously elastic QP scattering, as detected by QP interference in scanning tunneling microscopy experiments2,33,34. However, the implications of coherence for a process in which two quasiparticles scatter into the condensate have not previously been considered in the context of the cuprates. In a phase-incoherent state the recombination rate is proportional to the square modulus of the interaction matrix element between electrons and holes. In a phase-coherent paired state, recombination is more complex because, as mentioned previously, QPs are linear superpositions of electrons and holes. As a result, the matrix element for recombination reflects multiple channels that can interfere constructively or destructively, depending on the nature of the pairing state and QP interactions19.
The rate of QP recombination in a paired condensate is the product of the normal state rate and a “coherence factor” (F) that is a function of the Bogoliubov coefficients u and v. For the case of time-reversal invariant QP interactions the coherence factors for SC and CDW condensates are given by
where Δ and Δ′ are the amplitudes of the SC gap at the two QP momenta, Φ and Φ′ are the analogous CDW gaps, and E and E′ are the QP energies35. In the limit that E and E′ approach the gap energy, these factors reduce to FSC = 1 and FCDW = 0, while in the normal state F = 1/2. We note that the result FCDW = 0 is related to the π phase shift between occupied and unoccupied states, which has recently been observed in CDW states in Bi221236.
The simplest scenario, in which a phase-fluctuating superconductor becomes fully phase coherent at Tc, is inconsistent with a peak in τqp(T) (or local minimum in recombination rate). Instead, SC coherence yields a doubling of the recombination rate, corresponding to the factor of two jump in F from 1/2 to 1 upon crossing from a normal to SC state. However, we have found that a model that takes into account the dual presence of fluctuating CDW and SC order leads to a singular feature in τqp(T) that agrees well with experiment.
Fluctuation of composite SC-CDW order
To formulate this model quantitatively, we derived the recombination coherence factor for a state with coexisting SC and CDW order. The composite SC/CDW condensate is made of particle-hole quadruplets, pairing electrons at ±k and holes at ±(k + Q) separated by the CDW wavevector Q. The structure of the quasiparticle eigenstates of the composite SC/CDW condensate is shown schematically in Fig. 4a. Based on these eigenstates, we determined the dependence of the mixed state coherence factors on the quasiparticle energy (see Methods). As time and angle-resolved photoemission measurements observe rapid thermalization of quasiparticles to gap edge after photo-excitation37,38, we focus on the coherence factor in the limit that E, E′ → ∆, ∆′ which simplifies the expression for F considerably. In this limit, the QP recombination coherence factor becomes,
where and are the relative phases of the QP pair undergoing recombination. In the case of d-wave pairing, these phases will be 0 or π, depending on which k points the QPs occupy. STM QP interference demonstrates that the strongest scattering channels are between states with the same sign of gap amplitude34, so we restrict our attention to these recombination processes.
In the presence of phase fluctuations, the cos ϕ factors are replaced by their ensemble average , where τc is the phase-correlation time and τ0 is the QP lifetime in the fully incoherent regime. The coherence factor obtained by this substitution would apply to systems with coexisting, but independent, CDW and SC order. However, in the light of evidence that these orders are strongly coupled in underdoped cuprates, we consider a description in terms of a multi-component order parameter17 whose amplitude is constant and whose fluctuations are described by a single phase ϕ. With these assumptions the coherence factor can be written,
Figure 4d shows a fit to τqp(T) for the Hg-1201 Tc = 71 K sample using Eq. 4. Figure 4b,c show the temperature dependence of the three parameters from which the fits were generated. The mixing angle θ(T) is plotted in Fig. 4b. The reciprocal of the coherence time, , and reciprocal of the incoherent recombination time, , are plotted as solid and dashed lines, respectively in Fig. 4c. The T-dependence of τ0 is determined by a polynomial fit to τqp(T) that ignores the peak at Tc. The quasiparticle decoherence rate is constrained to be constant below Tc and to vary ∝(T − Tc) above the transition, as suggested by angle-resolved photoemission spectra39. The slope of vs. T − Tc extracted from our fit (≈0.15 THz/K) is consistent with estimates of coherence times obtained from optical conductivity40. The peak in τqp(T) shown in Fig. 4d arises from the interplay between the mixing angle θ(T) and the onset of quasiparticle coherence. Starting at temperatures well above Tc, the order parameter is CDW-like with a short coherence time. As the temperature is lowered towards Tc and the CDW becomes more coherent, 〈F〉 dips below its normal state value of 1/2 and the QP lifetime is enhanced. However, as the order parameter crosses over from CDW to SC-like near Tc, 〈F〉 increases, giving rise to a peak τqp.
Magnetic field effect
To further examine the relationship between phase coherence and quasiparticle recombination, we investigated the effect of magnetic field, B, on τqp(T). An overview of τqp(B,T) in fields applied perpendicular to the Cu-O planes is shown in Fig. 5. Figure 5a compares the quasiparticle lifetime in zero field and 6 Tesla for a sample of Hg-1201 with Tc = 71 K. The peak in τqp(T) is entirely washed out by the field and replaced by a smooth background that is described by the model parameter τ0(T) discussed previously. In Fig. 5c, the change in recombination rate caused by the field, , is plotted vs. B at three temperatures: well below, above, and at Tc. It is clear that strong dependence of τqp on B is observed only near Tc. The field dependence of the quasiparticle lifetime is qualitatively different in the near optimal Hg-1201 sample (shown in Fig. 5b), where we find that the maximum τqp shifts to lower T but is not reduced, consistent with what is expected for a mean-field gap opening transition.
The B dependence of τqp can be understood to be a consequence of dephasing in the vortex liquid induced by the field. Assuming a total dephasing rate, , where Γ(B) is the dephasing rate associated with vortex diffusion, yields
The dashed line in Fig. 5c is a fit to this functional form, with Γ(B) = (0.08 THz/T)B. This result can be compared with the theory of phase fluctuations in the vortex liquid41, where it is shown that , where ξ and lB are the coherence and magnetic length, respectively. Equating this estimate with the measured Γ(B) yields a reasonable value for the coherence length of 2.4 nm, strongly suggesting that B-induced dephasing accounts for wipeout of the τqp peak.
To summarize, we have described measurements and analysis of the photoinduced transient reflectivity, ΔR(t,T) in representative YBCO and Hg-1201 samples of underdoped cuprates. The onset of ΔR with decreasing temperature is correlated with the appearance of the incommensurate CDW detected previously by NMR, STM, and X-ray scattering measurements. This correlation leads us to conclude that, although the density of states depression known as the pseudogap is formed at a higher temperature, the CDW either enhances it, or leads to a new gap of different origin. A correlation between CDW and gap formation is suggested as well by recent ARPES measurements42,43. We focused attention on two aspects of ΔR(t) as T was lowered through Tc. In underdoped samples, ΔR increases continuously through Tc, suggesting a smooth variation of the gap at the Fermi surface hot spots (or Fermi-arc tips). Second, while the gap varies continuously, the quasiparticle recombination time exhibits a narrow local maximum at Tc. We proposed that this peak in τqp indicates a crossover from fluctuating CDW to SC/CDW order, occurring as the condensate coherence time slides through the time window set by the background recombination time ~2–3 ps. The link between condensate coherence and the peak in τqp(T) was further supported by the observation that magnetic field causes the peak to disappear into the background. Our results show that quasiparticle recombination provides a new method for probing the onset of coherence in systems characterized by a fluctuating multi-component order parameter.
Materials and Methods
Hg-1201 synthesis and preparation
We examined a series of high quality single crystals of Hg-1201 ranging from deeply underdoped to optimal doping with Tc’s of 55 K, 65 K, 71 K, 79 K, 81 K, 91 K, and 94 K. These samples were grown by the self-flux method44 and annealed to achieve different oxygen concentrations45, with Tc determined within ~1 K by magnetic susceptibility measurements. Hole concentration was determined using methods described in ref. 46. Samples were polished under nitrogen flow using 0.3 μm grit films to prevent surface oxidation.
The measurements reported herein were performed using 100 fs pulses from a mode-locked Ti:Sapphire laser at 800 nm center wavelength and ≃1 μJ/cm2 fluence. Pump pulses induced a small (~10−4) fractional change in reflectivity that was monitored by time-delayed probe pulses. At the low laser fluence used in this study, samples are weakly perturbed by the pump pulse, in the sense that the density of photogenerated quasiparticles is much less than the thermal equilibrium value. In this regime the decay rate is a measure of the quasiparticle recombination rate in thermal equilibrium. Measurements in zero magnetic field at Berkeley were performed under vacuum in an Oxford continuous-flow liquid He cryostat. Magneto-optic measurements at Berkeley were performed in a 6T Oxford Spectramag cryostat, and those at MIT in a 7T Janis cryostat.
Calculation of mixed-state coherence factors
The mean-field Hamiltonian is given by 47, with
Diagonalizing the CDW and SC subsystems via successive Bogoliubov transformations yields energy eigenvalues . The eigenvectors give the composite QP operators , where
In the above, u and s are assumed to be real, and the phases of v and t are determined by the phases of Δ and Φ, respectively. We calculate the coherence factors following ref. 48, starting with a quasiparticle interaction of the form . We assume that the interaction is spin-independent. The four four electronic transitions that involve the same QP operators,
must be summed coherently before calculating the squared modulus of the matrix element that determines the quasiparticle scattering rate. Expressing the four bilinear operators in terms of the quasiparticle (γ) operators yields,
and . Summing over recombination channels and then squaring yields a coherence factor for quasiparticle recombination given by .
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We acknowledge J. C. Davis for enlightening discussions, as well as Lina Ji for assistance with crystal growth. Synthesis and characterization of Hg-1201 samples performed at the University of Minnesota was supported by the Department of Energy, Office of Basic Energy Sciences, under Award No. DE-SC0006858. N.B. acknowledges the support of FWF project P2798. Optical measurements and modeling performed at Lawrence Berkeley National Lab was supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
The authors declare no competing financial interests.
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Hinton, J., Thewalt, E., Alpichshev, Z. et al. The rate of quasiparticle recombination probes the onset of coherence in cuprate superconductors. Sci Rep 6, 23610 (2016). https://doi.org/10.1038/srep23610
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