Two component dynamics of the superconducting order parameter revealed by time-resolved Raman scattering
R. P. Saichu, I. Mahns, A. Goos, S. Binder, P. May, S. G. Singer, B. Schulz, A. Rusydi, J. Unterhinninghofen, D. Manske, P. Guptasarma, M.S. Williamsen, M. Rübhausen
TTwo component dynamics of the superconducting order parameter revealed bytime-resolved Raman scattering
R. P. Saichu , I. Mahns , A. Goos , S. Binder , P. May , S. G. Singer , B. Schulz , A. Rusydi , ,J. Unterhinninghofen , D. Manske , P. Guptasarma , M.S. Williamsen , and M. R¨ubhausen ∗ Institut f¨ur Angewandte Physik, Universit¨at Hamburg, Jungiusstrasse 11, D-20355 Hamburg,Germany. Center for Free Electron Laser Science (CFEL), Notkestrasse 85, D-22607 Hamburg, Germany Nanocore, Department of Physics, Faculty of Science, NUS, Singapore 117542, Singapore Institut f¨ur Theoretische Physik, Universit¨at Bremen, Postfach 330440, D-28334 Bremen, Germany Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53211, USA (Dated: December 8, 2018)We study the dynamics of the superconducting order parameter in the high- T c cuprateBi Sr CaCu O − δ by employing a novel time-resolved pump-probe Raman experiment. We findtwo different coupling mechanisms that contribute equally to the pair breaking peak. One couplingsets in very fast at 2 ps and relaxes slow, while the other one is delayed and sets in roughly at 5 psand relaxes fast. A model that couples holes through phonons is able to reproduce one part of thecondensate dynamics, thus, we argue that hole-spin interactions are of importance as well. PACS numbers: 74.25.Gz, 74.40.+k, 74.72.Hs, 78.47.-p, 78.30.-j, 78.47.jc
The nature of the interaction between holes leading tosuperconductivity is encoded in the properties of the su-perconducting order parameter [1, 2]. These propertiesare reflected by the energy, the momentum dependence,and the time scales on which the order parameter reactsto an external perturbation [1, 2, 3, 4]. In a material withcompeting interactions, there is a potential for the devel-opment of competing ordering phenomena [2, 3]. Un-doped high temperature superconductors are antiferro-magnetic insulators that become superconducting upondoping. In the superconducting state, the suppressed an-tiferromagnetic order might prevail on short length andtime scales and, hence, affects thermodynamic proper-ties [1, 2, 3]. Therefore, it is crucial to understand thetransient physics that is directly connected to a specificphase transition [4]. Such an approach allows the study ofcompeting order parameters, their individual relaxationchannels and elucidates the potential interplay betweenthem.Major progress in the development of pulsed laser, syn-chrotron, and free electron laser sources has led to in-novative time-resolved techniques that can directly ad-dress the transient physics of, e.g. a correlated mate-rial [5, 6, 7, 8, 9, 10]. However, several pump-probetechniques that have been applied to understand the su-perconducting condensate in high-temperature supercon-ductors, fail to probe the superconducting order param-eter ∆( k ) directly [6]. In this context an intense debateexists on the question of whether or not one deals withone or more coupling processes for the pairing mecha-nism [9]. The two mechanisms which attracted the great-est attention invoke the coupling between holes throughphonons or spin fluctuations [1, 2, 3, 11]. However, un- ∗ corresponding author: [email protected] til today there is no clear understanding to which degreethese mechanisms jointly contribute to the superconduct-ing state. Different coupling mechanisms can be expectedin different time scales in the response to an external per-turbation.In this letter, we present a unique time-resolvedtwo-color inelastic light (Raman) scattering experiment,which allows us to probe directly the superconductingorder parameter. We report measurements of the dy-namics of ∆( k ). The samples are slightly overdopedBi Sr CaCu O − δ (Bi-2212) high-temperature supercon-ductors with a T c of 82 K and are very well characterized[12, 13, 14, 15]. In our pump-probe Raman experimentwe employ the UT-3 Raman spectrometer, which is afully reflective achromatic spectrometer for the frequencyrange from deep ultraviolet to near infrared [16]. In orderto obtain well defined time resolution we use two pulsedlaser beams. The pump beam at 3.44 eV photon energy[full width at half maximum (FWHM) is 0.8 ps] initiatesthe experiment and drives the system out of its equilib-rium state. With the Raman probe beam at 1.72 eV pho-ton energy (FWHM = 0.9 ps) we observe the energy andheight of the pair breaking peak ∆( k ) as a function of de-lay, i.e. time difference between the pump and the probebeam. The large energy difference between the pump andthe probe beam avoids any spurious signal in the Ramanprobe. We have employed B g -polarization ∝ (x -y ) re-sembling the d -wave symmetry of the superconductingorder parameter. B g -symmetry can be studied by usingcrossed polarization between incident and scattered lightwith respect to the a and b axes in the CuO planes. Thepump beam heats the sample roughly 40-65 K above itsequilibrium temperature, whereas the probe beam heatsthe sample by about 3 K. These values are estimatedby comparing the non-equilibrium Raman spectra to thecorresponding temperature corrected equilibrium Ramanspectra from a continuous wave (CW) source, and numer- a r X i v : . [ c ond - m a t . s up r- c on ] F e b (a) (b) FIG. 1: (color online) Steady state and time-resolved Ramanspectra of superconductors and metals depending on temper-ature in B g geometry. (a) shows the spontaneous Ramanscattering intensity at 10 K and at 300 K. A gap opens be-low 250 cm − (blue area) and a pair breaking peak appearsaround 420 cm − (red area). (b) Temperature evolution of theRaman difference spectrum at a delay between the pump andthe probe beam of 3 ps. Positive values indicate an increaseof intensity in the pumped state compared to the equilib-rium state, negative values indicate the opposite. The insetshows the integrated intensity of the difference spectra be-tween 300 cm − and 600 cm − . ical estimates [17, 18]. In the superconducting state thetemperature rise is large enough to break Cooper pairs.Thus, changes in the energy and the height of the pair-breaking peak, i.e. the superconducting order parameter,can be probed.In Fig. 1(a) we show Bose-corrected steady statespectra for a transferred energy (Raman shift) between100 cm − and 600 cm − in the normal and superconduct-ing state of Bi-2212. We employ a probe energy of 1.72 eVin order to minimize resonance effects that are known tooccur for higher incident photon energies [12, 19]. In thenormal state one can observe a flat background (blackcurve) that results from the scattering of charges withina marginal Fermi liquid [19, 20, 21, 22]. On the otherhand, in the superconducting state a gap opens (blueshaded area) and a pair breaking peak forms at roughlytwice the maximum value of ∆( k ) (red shaded area) at ≈
420 cm − ∼ = 52 meV [19]. The height of the peak is pro-portional to the number of Cooper pairs that are brokenaround ( ± π ,0) and (0, ± π ) in the Brillouin zone.If we pump the superconducting state, we raise thesample temperature. For different equilibrium tempera-tures and at a fixed delay of 3 ps Raman difference spec-tra normalized to the integrated scattering intensity I int are shown in Fig. 1(b). In the pumped state, we clearlyfind a difference spectrum with positive values at lowenergies and negative values at higher energies. Thisobservation reflects directly a loss of Cooper pairs andcorrespondingly a decreased intensity in the pair break-ing peak. Due to charge conservation, this decrease ofintensity is accompanied by an increase of quasi-particlespectral weight within the gap. Finally, the Raman dif-ference spectrum vanishes at T c . The integral over themagnitude of the Raman difference spectrum is shown (a) (b) Time (ps) R a m an s h i ft ( c m ) - B1g
FIG. 2: (color online) Temporal evolution of the time-resolvedRaman difference spectra at an equilibrium temperature of10 K in B g geometry. In (a) three difference spectra of threedifferent delay times are shown. The contour plot shown in(b) presents 12 Raman difference spectra for different delaytimes. The dashed line separates two energy regions of thepair-breaking peak that reveal different characteristic behav-ior. The intensity changes are color coded demonstrating thetransfer of spectral weight from high to low energies after 1 ps,respectively. in the inset of Fig. 1(b) clearly demonstrating that weobserve effects that are related exclusively to the super-conducting state and are not related to the pseudogap[3, 4].In order to analyze the dynamics of Cooper pairs inthe pumped spectrum, we show of example in Fig. 2(a)three Raman difference spectra at 10 K as a function ofdelay time between pump and probe pulse. Already at1.65 ps after the pump beam, the Raman probe detects apronounced loss of spectral weight at high energies whichcorresponds to the high-energy tail of the pair-breakingpeak shown in Fig. 1(a), and only a marginal gain of spec-tral weight within the gap. At 6.6 ps the loss of spectralweight is shifted to energies around the maximum inten-sity in the pair-breaking peak and a clear gain of spectralweight within the gap [blue shaded area in Fig. 1(a)] isobserved. Furthermore, the integrated change at 6.6 ps issignificantly larger as compared to the change at 1.65 ps.This clearly demonstrates that the superconducting con-densate reacts on two different time scales to the pumpbeam. Interestingly, these time scales are roughly con-nected to two sides of the pair-breaking peak, i.e. aboveand below 420 cm − . After 16.5 ps the difference spec-trum is overall strongly diminished indicating a completerelaxation back into the equilibrium state.In Fig. 2(b) we show the temporal evolution in moredetail by employing a density plot consisting of 12 Ra-man difference spectra (green color corresponds to nointensity change, i.e. the equilibrium state). The Ramanshift is set on the y-axis, whereas the time delay betweenpump and probe beams is set on the x-axis. From thisdata set we can derive the two different time scales inmore detail: First, a fast response at around 2 ps thatstarts with a suppression of a pair breaking peak above420 cm − . This time scale relates directly to the resultsof novel time-resolved ARPES experiments by Perfetti etal. showing unambiguously that the hot electrons andhot phonons created by a pump pulse thermalize within50 fs and 2 ps, respectively [23]. From this we can con-clude that our observed dynamics after 2 ps is drivenby a homogeneously heated sample. After the initialloss of spectral weight in the pair-breaking peak around2 ps, an increase of low-energy spectral weight is observedroughly 1 ps later, reflecting the transformation of holesfrom the superconducting condensate into quasiparticlesdue to charge conservation. Then, a second, delayed re-sponse after 5 ps is observed in the pair breaking peak,this time below 420 cm − . This spectral weight suppres-sion in the pair-breaking peak also yields an additionalgain in spectral weight within the gap roughly after 1 ps.Thus, summarizing our spectra, we can clearly identifytwo contributions to the suppression of the pair-breakingpeak that have their typical energy and time scales. Weobserve a fast onset of suppression of the pair-breakingpeak above 420 cm − and a delayed suppression below420 cm − . Both regions of the pair-breaking peak areindicated by the differently red shaded areas in Fig. 1(a).As expected from charge conservation, both responsesyield their respective gain of spectral weight within thegap indicating a clear redistribution of the superconduct-ing condensate [blue shaded area in Fig. 1(a)].Having identified two different time scales of the super-conducting condensate, what is their corresponding time(decay) constant? To study this, we have integrated thespectral weight along the energy axis above and below420 cm − as indicated by the dashed line in Fig. 2(b).The temporal evolution of these spectral-weight changesare displayed in Fig. 3(a) and (b), respectively. As we willargue below, it is remarkable that the behavior of the fasthigh energy response is quite consistent with a relaxationprocess through in-plane phonons. Our calculation willshow that this characteristic time scale of the relaxationprocess of about 7.4 ps is consistent with previous esti-mates from other experiments [6, 9]. However, the sec-ond, delayed response below 420 cm − is unexpected andhas also an untypical behavior, since rise and decay timesare roughly equal and of only 1.4 ps [see Fig. 3(b)]. Fromneutron scattering experiments we know that for energiesbelow 420 cm − the coherence peak in the spin suscepti-bility yields to a strong coupling between holes even closeto T c [24]. Thus, it is reasonable to assume that remnantspin-mediated coupling between holes yields an enhancedstiffness of the paired holes against an external pertur-bation resulting in this delayed reduction of the Cooperpairs after 5 ps. Such a scenario could be supported by apicture in which charge rich domains are surrounded byfluctuating charge poor domains. Indeed, static domainshave been observed by static probes such as ScanningTunneling Microscopy (STM) experiments [25].Finally, we show in Fig. 3(c) a comparison between amodel and our Raman difference spectra in the pumpedstate. For our calculations, we have employed the methodof density-matrix theory [26, 27] which has been recently (a)(b) (c) FIG. 3: (color online) Integrated intensity for the two differ-ent energy regions and comparison between theoretical andexperimental response. (a) Energy region from 420 cm − to580 cm − . (b) Energy region from 300 cm − to 410 cm − .The green characteristic decay time of (a) τ = 7.4 ps and (b) τ = 1.4 ps are determined by an exponential and Gaussianfit, respectively. (c) Calculated time-resolved Raman differ-ence spectra for two different delay times. The inset showsthe experimental difference spectra. generalized for time-resolved spectroscopy on supercon-ductors. We have used the Hamiltonian of Ref. [18]that invokes the coupling to the most important phononmodes of the copper-oxygen planes, i.e. the breathingand buckling modes. The corresponding electron-phononmatrix elements are treated within LDA. Coupling ofholes to spin fluctuations is not considered. Then, inthe superconducting state, the non-equilibrium Ramanintensity can be calculated from the imaginary part ofthe response function χ ( q = 0 , ω ) ∼ (cid:88) k γ k (cid:16) u k (cid:10) α † k α k (cid:11) + v k (cid:0) − (cid:10) β † k β k (cid:11)(cid:1) + u k v k (cid:0)(cid:10) α † k β † k (cid:11) + (cid:10) β k α k (cid:11)(cid:1)(cid:17) . (1) q and ω denote the transferred momentum and energy,respectively. The summation runs over all (electronic)wavevector k in the first Brillouin zone. γ k ∝ ( k x − k y )is the non-resonant B g Raman vertex. Due to the em-ployed Bogoliubov transformation the Raman suscepti-bility χ ( q = 0 , ω ) includes the mixing amplitudes u k , v k and the creation and annihilation operators α k , α † k , β k and β † k of the Bogoliubov quasiparticles (linear combina-tion of holes and electrons), respectively [28]. The useof the low photon energies in the probe beam of about1.7 eV makes this non-resonant vertex more applicableas compared to visible and UV-photon energies wherestrong resonance effects need to be considered in Bi-2212[12]. We chose a one band tight-binding fit to the mea-sured band structure [19, 29] and a d -wave order param-eter ∆ k = ∆ (cos k x - cos k y )/2, yielding a quasiparti-cle dispersion E k = (cid:112) ( (cid:15) k − µ ) + ∆ k . Then, we calcu-late the time-dependent expectation values (cid:10) α † k α k (cid:11) ( t ), (cid:10) β † k β k (cid:11) ( t ) and (cid:10) β k α k (cid:11) ( t ) = (cid:10) α † k β † k (cid:11) ∗ ( t ) with the helpof coupled Boltzmann-equations for the non-equilibriumsituation in the pumped state. An additional damping δ = 5 meV is used to account for other scattering pro-cesses that we do not take into account. After numericalsolution of the equations of motion, the results can beinserted into Eq. (1), and a pump-probe difference Ra-man spectrum as a function of delay time can be readilycalculated.Figure 3(c) shows the calculated difference Ramanspectra at a fixed delay time of 3.3 ps and 15 ps. Theinset shows the corresponding measured data that havebeen corrected for the ratio of the pumped to the probedvolume [30]. We find a fair agreement considering thatwe have used no adjustable fitting parameters in the nor-malized plot of the Raman response. However, the relax-ation in the measured Raman spectra is faster as com-pared to the model calculations indicating a superposi-tion of different phonon modes that contribute to the re-laxation process. The electron-phonon coupling strengthfor the in-plane oxygen breathing and out-of-plane buck-ling modes [31] yields typical time constants of 4 ps and20 ps, respectively (not shown). From this we derivea roughly 80% contribution of the breathing mode inthe phonon mediated relaxation process [18, 27]. Fur-thermore, it is obvious that the second response below52 meV with a delayed onset after 5 ps and very fast de-cay time of less then 1.4 ps cannot be described withinour model. As mentioned above, this strongly suggestthat the hole-spin interaction and the inhomogeneous dis-tribution of holes within the copper-oxygen plane needs to be included in any theory that aims to fully understandthe time-resolved Raman response as shown in Fig. 2(b).In conclusion, we present a unique two-color Ramanexperiment revealing the ultrafast dynamics of the su-perconducting order parameter in Bi-2212 by employinga novel time-resolved pump-probe Raman experiment.Our results clearly demonstrate that the pair-breakingpeak in the Raman responses reacts on two different timescales. These time scales are equivalent to two differentcoupling mechanisms. Both couplings show the redistri-bution of spectral weight from the pair-breaking peak toa quasiparticle response within the gap. The first, fastresponse sets in at 2 ps and relaxes within 7.4 ps, thesecond response sets in at 5 ps and relaxes within 1.4 ps,respectively [see Fig. 2(b)]. 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