Dynamics of broken symmetry nodal and anti-nodal excitations in Bi_{2} Sr_{2} CaCu_{2} O_{8+δ} probed by polarized femtosecond spectroscopy
Y. Toda, F. Kawanokami, T. Kurosawa, M. Oda, I. Madan, T. Mertelj, V. V. Kabanov, D. Mihailovic
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Dynamics of broken symmetry nodal and anti-nodal excitations in Bi Sr CaCu O δ probed by polarized femtosecond spectroscopy Y. Toda and F. Kawanokami
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan.
T. Kurosawa and M. Oda
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan.
I. Madan, T. Mertelj, V. V. Kabanov, and D. Mihailovic
Complex Matter Dept., Jozef Stefan Institute, Jamova 39, Ljubljana, SI-1000, Slovenia (Dated: November 7, 2018)The dynamics of excitations with different symmetry is investigated in the superconducting (SC)and normal state of the high-temperature superconductor Bi Sr CaCu O δ (Bi2212) using opticalpump-probe (Pp) experiments with different light polarizations at different doping levels. Theobservation of distinct selection rules for SC excitations, present in A and B symmetries, andfor the PG excitations, present in A and B symmetries, by the probe and absence of anydependence on the pump beam polarization leads to the unequivocal conclusion of the existence ofa spontaneous spatial symmetry breaking in the pseudogap (PG) state. Ultrafast pump-probe (Pp) spectroscopy has beenwidely used to investigate the high- T c superconductivityfrom various viewpoints [1–5]. Nonequilibrium studiesgive a unique insight into quasiparticle (QP) dynamics,revealing universal two-component QP dynamics associ-ated with the superconducting (SC) gap and pseudogap(PG) excitations in high- T c materials. The two typesof excitations were characterized by distinct relaxationtimes, temperature dependencies, and/or sign of the op-tical signal, depending on the material, doping level,photo-excitation intensity and the wavelengths of lightused in the Pp experiments [6–10]. The dependence onthe probe photon polarization of the two-component re-flectivity dynamics has also been reported [11]. However,the absence of a fundamental understanding of the opti-cal processes involved in Pp experiments so far preventedanalysis of the symmetry of excitations or detailed theo-retical analysis of the excitations on a microscopic level.Here, by performing a concise symmetry analysis of Ppexperiments on Bi2212 high temperature superconduc-tors and identifying the processes involved we open theway to investigations of the dynamics of states associatedwith hidden broken symmetry and local or mesoscopicsymmetry breaking in systems with competing orders.Generally, Pp experiments can be described as a twostep process. In the first step, the pump pulse excita-tion can be viewed as a process, which can be dividedinto a coherent stimulated-Raman (SRE) excitation [12]and an incoherent dissipative excitation (DE). In the DEprocess, the high energy (eV) photoexcited carriers cre-ate incoherent excitations by inelastic scattering on thetimescale of tens of fs resulting in a transient nonequilib-rium occupation of phonons (magnons, etc.) as well asa transient nonequilibrium QP occupation ∆ f near theFermi level. The information about the incoming photonpolarization is lost during this process. In the SRE pro- cess, various degrees of freedom are excited with a force F ( t ) ∝ P ( t ) , where P ( t ) is related to the temporal pro-file of the pump pulse. The symmetry of the coupling todifferent degrees of freedom is described via the appropri-ate Raman tensor π R [12]. For the nonsymmetric modesthe process should show characteristic dependence on thepump photon polarization.In pseudo-tetragonal ( D h ) symmetry, considered ap-propriate for the cuprates [ ? ], A and A as well asB and B excitations can be coherently excited by theSRE process for the photon polarizations lying in theCuO plane [13]. Totally symmetric A g excitation suchas DE can not coherently excite nonsymmetric modes.However, an additional possibility exists , where in thepresence of a local, dynamic or hidden symmetry break-ing nonsymmetric modes can be excited coherently alsoby the totally symmetric DE. This allows us to probesymmetry breaking by means of the Pp spectroscopy. A B B M G X x y (a) q Lens
HWP
PumpProbeDM Sample b (Bi-O) a Cu-O (b)
Figure 1. (a) A schematic illustration of the two-color pump-probe setup for polarization-resolved measurements. Theprobe ( λ =800 nm) was variably polarized by a half-wave plate(HWP) and was combined with the pump ( λ =400 nm) by adichroic mirror (DM). The angle θ is measured relative to theCu-O bond axes. (b) The k -space selectivity of the probeaccording to the Raman-like process is indicated (from [13]). In the second step of the Pp experiment, the tran-sient change of reflectivity detected by the probe canbe described by the Raman-like process [12]. For QPexcitations the polarization P k due to a pump-inducedchange of ∆ f ( q ) is given by: P q k = P l R q kl E l ∆ f ( q ) ,where R q kl = ∂ǫ kl ∂f ( q ) is a Raman-like tensor, ǫ kl is the complex dielectric tensor, and E l is the l -th componentof the probe-pulse electric field.Assuming the pseudo-tetragonal structure ( D h pointgroup) for Bi2212, the photoinduced changes of the in-plane dielectric tensor components can be decomposedaccording to symmetry as [ ? ]: ∆ ǫ = (cid:20) ∆ ǫ A ∆ ǫ A (cid:21) + (cid:20) ∆ ǫ B − ∆ ǫ B (cid:21) + (cid:20) ∆ ǫ B ∆ ǫ B (cid:21) . (1)Taking the in-plane probe incident electric field as E = E (cid:0) cos θ sin θ (cid:1) , then to the lowest order the angle-dependenceof the photoinduced change of reflectivity R reduces to: ∆ R ( θ ) = ∂R∂ǫ h ∆ ǫ A + ∆ ǫ B cos(2 θ ) + ∆ ǫ B sin(2 θ ) i + ∂R∂ǫ h ∆ ǫ A + ∆ ǫ B cos(2 θ ) + ∆ ǫ B sin(2 θ ) i . (2)Here θ is defined in Fig. 1(a), and ǫ and ǫ are the realand imaginary parts of the in-plane dielectric constant,respectively. Rewriting (2) in a more compact form theangle-dependence of the transient reflectivity is: ∆ R ( θ ) ∝ ∆ R A + ∆ R B cos(2 θ ) + ∆ R B sin(2 θ ) . (3)Thus, in principle, by measuring the angle dependenceof ∆ R ( θ ) and using Eq. (3) we can separate the T -dependent dynamics of components associated with dif-ferent symmetries and consequently identify the statesinvolved [ ? ].Previous analysis in the cuprates have indicated thatthe electronic Raman scattering in the B symmetryprobes excitations in the nodal ( π /2, π /2) direction in the k -space, while the B scattering probes excitations inthe antinodal directions ( π /2,0) and (0, π /2) [13–15]. An A g symmetry component is also present, whose originis still highly controversial [13]. Recent studies suggestthe presence of the PG in the nodal direction [16] im-plying an s -wave symmetry, in contrast to the commonassumption of a PG with nodes, indicating that the PGsymmetry is still an open issue. A detailed symmetryanalysis of optical Pp experiments can therefore poten-tially give important new information on the symmetry,lifetime and temperature-dependence of nodal and anti-nodal excitations in the cuprates and other superconduc-tors with an enhanced bulk sensitivity with respect to thetime-resolved ARPES [2, 4].The optical measurements were performed on freshlycleaved slightly overdoped (OD, T c ≈
82 K) and under-doped (UD, T c ≈
69 K) Bi2212 single crystals grown bythe traveling solvent floating zone method. For optimal signal-to noise ratio we used a pump at E pu = λ pu =
400 nm) and probe at E pr = λ pr =
800 nm)from a cavity-dumped Ti:sapphire oscillator with a 120 fspulses and a repetition rate of 270 kHz (to avoid heating).The pump and probe beams were coaxially overlapped bya dichroic mirror and focused to µ m diameter spot onthe ab -plane of the crystal with an objective lens ( f = x and y point along the Cu-O bonds (Fig. 1(b)). The sam-ple orientation was checked by x-ray diffraction, in whichthe b -axis is determined by the direction of the multiplepeaks responsible for a one-dimensional (1D) superlatticemodulation.To highlight the main findings of the study, we willconcentrate on the presentation of the OD sample, whilenoting that the results on the UD sample are qualitativelysimilar. First we note that ∆ R is found to be independentof the pump polarization to less than ∼ , while theprobe polarization dependence of ∆ R is very temperaturedependent.The angular dependencies of ∆ R ( t ) at selected tem-peratures, obtained by rotating the probe polarizationfrom θ = 0 to ◦ at each temperature, are presentedin Fig.2. The upper panels (a)-(d) show the intensityplots of ∆ R ( θ ) together with the cross-sectional views at θ = 0 ◦ . The polar plots (e)-(h) show the angular depen-dence of the signal amplitude ∆ R ( θ ) .At the lowest temperature, where the SC signal is dom-inant, ∆ R ( θ ) is slightly elliptic, with the long axis closeto, but not coincident, with the Cu-O bonds direction(Fig.2 (a) and (e)). The amplitude of the signal in-creases with increasing F , showing saturation behaviornear F SCth = µ J/cm [ ? ]. With increasing F , an addi-tional fast relaxation signal with opposite sign with re-spect to the SC signal appears, which persists above T c and disappears around T ∗ . This component has beenpreviously assigned to the PG QPs [8, 17], where T ∗ ≃
140 K for OD and 240 K for UD samples, respectively,and is consistent with previous measurements [18]. Thereason for the PG signal appearing at higher F is that thePG component has a higher saturation threshold than the e ( deg . ) < R/R -4 R ( ) / R ( - ) < < < -4 -4 -4 T=10K T=80K T=90K T=270K (a) (b) (c) (d)(e) (f) (g) (h) b (Bi-O) a b (Bi-O) ab (Bi-O) a Cu-OCu-OCu-O e R max ( e ) Cu-O Cu-O -4 -4 -4 -4 -4 -4 -4 -4 Figure 2. (a)–(d) ∆ R ( θ ) /R transients at typical temperatures. (e)-(h) Polar plots of the maximum values of ∆ R ( θ ) /R . Thesolid lines indicate fits using Eq. (3). ∆ R ( θ ) / ∆ R at delay time of 10 ps is also shown (open circles with dashed fitting line)in (e). All data were obtained from the OD sample. Note that the Cu-O bonds directions are drawn horizontal and vertical,while the crystalline axes are along the Bi-O bonds, and are rotated nearly 45 ◦ from the Cu-O bonds. SC signal, and therefore becomes visible below T c withincreasing F [17].In the PG state above T c , but below T ∗ , the long axisis oriented along the crystalline axes ( θ ≃ ◦ ) (Fig.2 (c)and (g)).Above T ∗ , a signal with opposite sign to the PG (thesame sign as SC) becomes visible, which has been at-tributed to the electron energy relaxation in the metallicstate[19]. This high-temperature signal is almost inde-pendent of θ (Fig.2 (d) and (h)). - - T e m p e r a t u r e ( K ) Delay (ps) T e m p e r a t u r e ( K ) Delay (ps)
UD B - - OD A - - OD B - - OD B - - - - UD B (d) (e)(a) (b) (f)(c) Delay (ps)
UD A Figure 3. T -dependences of the B , B and A componentsof ∆ R corresponding to SC, PG and metallic state relaxationrespectively for OD (top) and UD (bottom) samples; Thevalues of the color bars indicate ∆ R/R × . In Fig. 3, we present the T -dependences of the ∆ R B /R , ∆ R B /R and ∆ R A /R obtained by fittingEq. (3) to experimental data. ∆ R B , and ∆ R B show B ( ×
7) B ( × A ( - a) ODb) UD A B ( ×
6) B ( × A ( - T (K) -2 -1 0 1 2 3 4 5 6 7 8 9 100246 B c) OD ∆ R / R ( - B -2 -1 0 1 2 3 4 5 6 7 8 9 10024681012 d) UD Delay (ps) ∆ R / R ( - B B Figure 4. T -dependences of the A , B and B compo-nent amplitudes for UD and OD samples. Note that all threecomponents begin to show an increase of the amplitude be-low T ∗ in the UD sample. In the case of the A g symme-try the PG contribution is superimposed on top of the nearlytemperature-independent signal, and has a negative sign. Thesolid and dashed lines are fits using Mattis-Bardeen [20] andKabanov [21] models, respectively. clear dominance of the SC and PG responses, respec-tively. ∆ R B is only visible below T c vanishing sharplyat T c . On the other hand, ∆ R B shows a gradual de-crease with increasing the temperature across T c and afaster sub-picosecond relaxation time, which is consis-tent with the general behavior observed for the PG QPs[6, 21]. The difference of the T -dependences between ODand UD samples reflects the systematic variation of thegaps with the doping level.In Fig. 4, we plot the amplitudes of different com-ponents for both OD and UD samples as a function oftemperature. ∆ R B and ∆ R A show dominant inten-sity below T c , and their T -dependences can be fit wellusing the Mattis-Bardeen formula [22, 23]. The ∆ R B component can be fit well by the Kabanov’s relaxationmodel [21, 24] which gives a T -independent ∆ PG = ∆ PG =41 meV for UD samples. Thevalues of ∆ PG in each sample are consistent with thevalues obtained from other experiments [25, 26].While the B component includes a signal extendingto ∼ T ∗ in the UD sample, the B component does notshow any measurable change at T c within the noise level.The absence of any pump polarization anisotropy hasimportant consequences. It rules out SRE as the exci-tation mechanism for any non-totally-symmetric modes.At the same time, the fact that the B and B responsesare observed by the probe means that they are somehowexcited by the pump pulse. Since DE can excite coher-ently only symmetric modes we are left with the only pos-sibility that the coherent B-symmetry modes are excitedbecause the underlying tetragonal point group symmetryis spontaneously broken below T ∗ .Formally, the B symmetry breaking of the pseudote-tragonal symmetry is already present at room tempera-ture in Bi2212 due to the weak inherent orthorhombicityof the underlying crystal structure from the BiO chainmodulation arising from the mismatch of Bi-O and Cu-O layers [27]. In the resulting D h point group symmetrythe a and b axes are rotated at ∼ ° with respect to theCu-O bonds. The presence of any coherent B sym-metry excitation, on the other hand, requires breakingof both the CuO -plane pseudotetragonal ( D h ) and theBi2212 D h symmetry [ ? ].In our data however, both symmetry breakings are sup-pressed at the room temperature appearing clearly below T ∗ , implying that the B g component is not simply a con-sequence of the underlying Bi2212 orthorhombicity. Thedata in Fig. 4 clearly show symmetry-breaking to oc-cur near T ∗ for both asymmetric components: B andB in the UD sample while in the OD sample the B component is clearly visible only in the superconductingstate. Furthermore, while the BiO chain ordering dis-cussed above can in principle cause the B symmetrybreaking effect it cannot cause the observed B symme-try breaking neither above nor below T c .Despite the d -wave SC order parameter corresponds tothe B symmetry the observed effect can not be linkeddirectly to the symmetry of the order parameter. TheSC order parameter is complex so any expansion of thedielectric constant in terms of the SC order parametercan only contain powers of | ∆ SC | , that are of the A g symmetry. This explains also the strong response of the SC state in the A g channel.The B symmetry breaking can therefore only be as-sociated with an underlying order oriented along the Cu-O bonds. This finding is consistent with previous STMmeasurements indicating the presence of stripe order ori-ented along one of the two orthogonal Cu-O bond direc-tions on the crystal surface [28–31]. The magnitude ofthe B g component significantly exceeds 10% of the A g magnitude. Since the optical probe penetration depth isof order of 100 nm [17] this indicates that the stripe or-der and the B g component are not limited to the surfaceand are present also in the bulk. The sensitivity of theB g component to the SC order also suggests that theinstability towards formation of the stripe order and thesuperconductivity are intimately connected.The distinct absence of the SC response in the B g channel is consistent with the sensitivity of the corre-sponding Raman vertex to the nodal ( π /2, π /2) directionin the k -space, where the SC gap has nodes. On the otherhand, the presence of the PG response in B g channel in-dicates that the PG response can, at least in part, beassociated with the nodal quasiparticles. Remarkably,this implies the absence of nodes in the PG, consistentlywith recent Raman results [16].Upon reduction of symmetry from tetragonal to or-thorhombic each of the B symmetry breakings of the4-fold axis can occur in two equivalent directions (e.g.along x or along y ). Since our experiment is stroboscopicand averages over many pulses B g and B g channels donot average out only if there is an underlying symmetrybreaking persisting between subsequent pulses separatedby 4 µ s. This can be imposed by extrinsic defect struc-ture or strain. In the case of Bi2212 it appears that thatthe underlying B g channel symmetry breaking can origi-nate in the weak orthorhombicity of the crystal, while theunderlying B g channel symmetry breaking is of extrin-sic origin amplified by the softness of the CuO planestowards stripe ordering or similar textures.We reiterate that the observed T -dependent in-planeprobe polarization anisotropy of the photoinduced reflec-tivity in Bi2212, and the concurrent absence of pump polarization anisotropy rules out stimulated Raman ex-citation as the external symmetry-breaking source.From this we conclude that the observed polarisationanisotropy indicates the presence of a bulk intrinsic spon-taneous breaking of both pseudotetragonal CuO -plane(D h ) and crystallographic Bi2212 (D h ) symmetry be-low T ∗ in both underdoped and slightly overdoped sam-ples.The detailed symmetry analysis of the Pp process inanalysis of electronic excitations presented here opens upnew possibilities for investigating the k -space anisotropycomplementary to Raman spectroscopy and new tech-niques like time-resolved ARPES. 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