Abstract
In the underdoped copperoxides, hightemperature superconductivity condenses from a nonconventional metallic ”pseudogap” phase that exhibits a variety of nonFermi 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 T_{c}, 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, τ_{qp}, as a function of temperature and magnetic field in underdoped HgBa_{2}CuO_{4+δ} (Hg1201) and YBa_{2}Cu_{3}O_{6+x} (YBCO) single crystals by ultrafast timeresolved reflectivity. We find that τ_{qp}(T ) exhibits a local maximum in a small temperature window near T_{c} that is prominent in underdoped samples with coexisting charge order and vanishes with application of a small magnetic field. We explain this unusual, nonBCS behavior by positing that T_{c} marks a transition from phasefluctuating SC/CDW composite order above to a SC/CDW condensate below. Our results suggest that the superfluid in underdoped cuprates is a condensate of coherentlymixed particleparticle and particlehole pairs.
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Introduction
First observed as stripelike order in the “214” cuprates^{1}, as checkerboard order in vortex cores^{2}, and subsequently at the surface of Bi and Cl based compounds^{3,4,5}, the universality of CDW order in the cuprate phase diagram has been established, through NMR^{6} and Xray scattering^{7,8,9,10,11,12} probes. In YBa_{2}Cu_{3}O_{6+x} (YBCO) near hole concentration 1/8 application of large magnetic fields stabilizes longrange CDW order at a temperature that approaches T_{c}^{6,13}. The neardegeneracy 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 shortrange antiferromagnetic fluctuations drive the formation of CDW and SC states^{14,15,16,17,18}. Moreover, the temperature dependence of the CDW amplitude in YBCO, as determined from Xray scattering, can be reproduced by a model of fluctuating, multicomponent order, of which CDW and SC states are two projections^{17}.
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 spin^{19}. Subsequent to BCS, Kohn and Sherrington^{20} 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 phasecoherent 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 timeresolved measurements of their lifetime against recombination, whereby two quasiparticles of opposite spin repair and scatter into the condensate. As we discuss below, this scattering rate is sensitive to the phasecoherence 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 T_{c}. 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 experiments^{21,22} 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.
Results
Correlation with CDW order
In Fig. 1 we show ΔR(t)/R at several temperatures for underdoped samples of Hg1201 with T_{c} = 55, 71, and 91 K. At high temperature we observe a shortlived 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 nearoptimal doping this positive, longlived component appears close to T_{c}^{24} 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 T_{c} (data at T_{c} 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 Hg1201 samples with T_{c}’s ranging from 55 to 91 K. The temperature (T_{onset}) 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 T_{onset} and T_{c} 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 T_{c}.
Based on a correlations with other probes, we believe that the appearance of the slow, positive component of ΔR at the temperatures T_{onset}(p) shown in Fig. 2g corresponds to the onset of local CDW order. There is a clear correspondence between the dome of T_{onset} as determined by ΔR(T) in Hg1201 and the region of the phase diagram where a CDW is detected in YBCO^{25,26}. Although the phase space region of CDW in Hg1201 is yet to be mapped in as much detail as in YBCO, a CDW has been detected in Hg1201 samples with T_{c} = 71 K^{12} at a temperature (indicated by the green circle in Fig. 2g) coincident with T_{onset}. Another correlation linking ΔR to the CDW in underdoped cuprates is that the positive component of ΔR in YBCO OrthoVIII (to be discussed further below) has the same temperature dependence as a zero wavevector vibrational mode that arises from CDWinduced zonefolding^{27}.
Recombination lifetime
We turn now to measurements of the T dependence of the recombination lifetime of quasiparticles for T < T_{onset}. 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 longlived 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 τ_{qp}(T) with hole concentration in the Hg1201 system. At each hole concentration we observe structure in the Tdependence of the quasiparticle recombination time at T_{c}. In underdoped samples there is a peak in τ_{qp}(T) at T_{c} that is most prominent in the T_{c} = 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 Hg1201 (T_{c} = 71 K) and YBCO OrthoVIII (T_{c} = 67 K), illustrating that generality of the phenomena described above. Figure 3b,c show the temperature dependence of the amplitude ΔR_{qp} ≡ A and τ_{qp} respectively for the two underdoped samples. In both we observe the onset of positive ΔR(T) well above T_{c} and a smooth variation through the SC transition. As shown in Fig. 3c, peak in τ_{qp}(T) centered on T_{c} is strikingly similar in the two representative samples.
The Tdependence of the quasiparticle lifetime in the sample of Hg1201 at nearoptimal doping (topmost data set of Fig. 3a) is qualitatively different from what is seen in underdoped samples. In the nearoptimal sample τ_{qp} grows monotonically as T → T_{c}, suggesting a tendency to diverge as ΔR_{qp}(T) goes to zero, while in underdoped samples the peak in τ_{qp} at T_{c} is a small feature on a smooth background. The behavior of τ_{qp} in nearoptimal Hg1201, which is observed in other optimally doped cuprates as well^{24,28}, can be understood within the meanfield theory of superconductivity. The theoretical description of τ_{qp}(T) in the context of BCS began in the 1960’s and its subsequent history is reviewed in ref. 29. In the meanfield picture, the relaxation of a nonequilibrium quasiparticle population is described by a pair of coupled equations: a LandauKhalatnikov equation for the energy gap and a Boltzmannlike equation that governs the quasiparticle distribution^{29}. According this analysis, (where τ is the electron inelastic scattering time) and diverges as the superconducting gap Δ(T) vanishes as T approaches T_{c} from below. In more recent work on cuprate superconductors, effects associated with the phonon bottleneck^{28,30,31,32} are included, which leads to replacing τ by the lifetime of phonons with energy greater than 2Δ.
Phase coherence and recombination
The behavior of τ_{qp}(T) in underdoped samples is at odds with the meanfield picture described above. Two observations–continuous variation of ΔR_{qp} for T < T_{onset} and the smoothly varying background in τ_{qp}(T) that underlies the small peak–suggest that a quasiparticle gap has opened well above T_{c}. Given a preexisting gap, it is very difficult to explain the modulation of the quasiparticle lifetime near T_{c}. In view of the difficulties with the meanfield picture, we are led to consider the onset of phase coherence at T_{c}, rather than gapopening, as the origin of the structure in τ_{qp}(T).
Coherence effects have been observed previously elastic QP scattering, as detected by QP interference in scanning tunneling microscopy experiments^{2,33,34}. 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 phaseincoherent state the recombination rate is proportional to the square modulus of the interaction matrix element between electrons and holes. In a phasecoherent 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 interactions^{19}.
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 timereversal 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 energies^{35}. In the limit that E and E′ approach the gap energy, these factors reduce to F_{SC} = 1 and F_{CDW} = 0, while in the normal state F = 1/2. We note that the result F_{CDW} = 0 is related to the π phase shift between occupied and unoccupied states, which has recently been observed in CDW states in Bi2212^{36}.
The simplest scenario, in which a phasefluctuating superconductor becomes fully phase coherent at T_{c}, is inconsistent with a peak in τ_{qp}(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 τ_{qp}(T) that agrees well with experiment.
Fluctuation of composite SCCDW 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 particlehole 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 angleresolved photoemission measurements observe rapid thermalization of quasiparticles to gap edge after photoexcitation^{37,38}, 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 dwave 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 amplitude^{34}, 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 τ_{c} is the phasecorrelation time and τ_{0} 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 multicomponent order parameter^{17} whose amplitude is constant and whose fluctuations are described by a single phase ϕ. With these assumptions the coherence factor can be written,
where .
Figure 4d shows a fit to τ_{qp}(T) for the Hg1201 T_{c} = 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 Tdependence of τ_{0} is determined by a polynomial fit to τ_{qp}(T) that ignores the peak at T_{c}. The quasiparticle decoherence rate is constrained to be constant below T_{c} and to vary ∝(T − T_{c}) above the transition, as suggested by angleresolved photoemission spectra^{39}. The slope of vs. T − T_{c} extracted from our fit (≈0.15 THz/K) is consistent with estimates of coherence times obtained from optical conductivity^{40}. The peak in τ_{qp}(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 T_{c}, the order parameter is CDWlike with a short coherence time. As the temperature is lowered towards T_{c} 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 SClike near T_{c}, 〈F〉 increases, giving rise to a peak τ_{qp}.
Magnetic field effect
To further examine the relationship between phase coherence and quasiparticle recombination, we investigated the effect of magnetic field, B, on τ_{qp}(T). An overview of τ_{qp}(B,T) in fields applied perpendicular to the CuO planes is shown in Fig. 5. Figure 5a compares the quasiparticle lifetime in zero field and 6 Tesla for a sample of Hg1201 with T_{c} = 71 K. The peak in τ_{qp}(T) is entirely washed out by the field and replaced by a smooth background that is described by the model parameter τ_{0}(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 T_{c}. It is clear that strong dependence of τ_{qp} on B is observed only near T_{c}. The field dependence of the quasiparticle lifetime is qualitatively different in the near optimal Hg1201 sample (shown in Fig. 5b), where we find that the maximum τ_{qp} shifts to lower T but is not reduced, consistent with what is expected for a meanfield gap opening transition.
The B dependence of τ_{qp} 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 liquid^{41}, where it is shown that , where ξ and l_{B} 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 Binduced dephasing accounts for wipeout of the τ_{qp} peak.
Summary
To summarize, we have described measurements and analysis of the photoinduced transient reflectivity, ΔR(t,T) in representative YBCO and Hg1201 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 Xray 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 measurements^{42,43}. We focused attention on two aspects of ΔR(t) as T was lowered through T_{c}. In underdoped samples, ΔR increases continuously through T_{c}, suggesting a smooth variation of the gap at the Fermi surface hot spots (or Fermiarc tips). Second, while the gap varies continuously, the quasiparticle recombination time exhibits a narrow local maximum at T_{c}. We proposed that this peak in τ_{qp} 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 τ_{qp}(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 multicomponent order parameter.
Materials and Methods
Hg1201 synthesis and preparation
We examined a series of high quality single crystals of Hg1201 ranging from deeply underdoped to optimal doping with T_{c}’s of 55 K, 65 K, 71 K, 79 K, 81 K, 91 K, and 94 K. These samples were grown by the selfflux method^{44} and annealed to achieve different oxygen concentrations^{45}, with T_{c} 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.
Ultrafast Measurements
The measurements reported herein were performed using 100 fs pulses from a modelocked Ti:Sapphire laser at 800 nm center wavelength and ≃1 μJ/cm^{2} fluence. Pump pulses induced a small (~10^{−4}) fractional change in reflectivity that was monitored by timedelayed 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 continuousflow liquid He cryostat. Magnetooptic measurements at Berkeley were performed in a 6T Oxford Spectramag cryostat, and those at MIT in a 7T Janis cryostat.
Calculation of mixedstate coherence factors
The meanfield 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 spinindependent. 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,
where,
and . Summing over recombination channels and then squaring yields a coherence factor for quasiparticle recombination given by .
Additional Information
How to cite this article: Hinton, J. P. et al. The rate of quasiparticle recombination probes the onset of coherence in cuprate superconductors. Sci. Rep. 6, 23610; doi: 10.1038/srep23610 (2016).
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Acknowledgements
We acknowledge J. C. Davis for enlightening discussions, as well as Lina Ji for assistance with crystal growth. Synthesis and characterization of Hg1201 samples performed at the University of Minnesota was supported by the Department of Energy, Office of Basic Energy Sciences, under Award No. DESC0006858. 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. DEAC0205CH11231.
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J.H. and J.O. planned the experiment and analyzed the results. J.H., E.T., J.K., Z.A. and F.M. performed the optical experiments in J.O. and N.G.’s laboratories. J.H. and A.K. performed coherence factor calculations. Hg1201 samples were grown by M.C., M.V., C.D. and N.B. in M.G.’s laboratory. D.B., W.H. and R.L. grew the YBCO samples. J.H. and J.O. wrote the manuscript, with contributions from M.G., N.G. and A.L. All authors reviewed the manuscript.
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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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DOI: https://doi.org/10.1038/srep23610
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