Spin Dynamics in Cuprates: Optical Conductivity of HgBa2CuO4
J. Yang, J. Hwang, E. Schachinger, J. P. Carbotte, R.P.S.M. Lobo, D. Colson, A. Forget, T. Timusk
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Exchange Boson Dynamics in Cuprates: Optical Conductivity of HgBa CuO δ J. Yang , ∗ J. Hwang , † E. Schachinger , J. P. Carbotte , , R.P.S.M. Lobo , D. Colson , A. Forget , and T. Timusk , ‡ Department of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada Institute of Theoretical and Computational Physics,Graz University of Technology, A-8010 Graz, Austria Laboratoire Photons et Mati`ere, CNRS UPR5, LPS-ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 5, France CEA, IRAMIS, SPEC, 91191 Gif sur Yvette, France The Canadian Institute of Advanced Research, Toronto, Ontario M5G 1Z8, Canada. (Dated: November 27, 2018)The electron-boson spectral density function I χ (Ω) responsible for carrier scattering of the hightemperature superconductor HgBa CuO δ ( T c = 90 K) is calculated from new data on the opticalscattering rate. A maximum entropy technique is used. Published data on HgBa Ca Cu O δ ( T c =130 K) are also inverted and these new results are put in the context of other known cases. All spectra(with two notable exceptions) show a peak at an energy (Ω r ) proportional to the superconductingtransition temperature Ω r ≈ k B T c . This charge channel relationship follows closely the magneticresonance seen by polarized neutron scattering, Ω neutronr ≈ k B T c . The amplitudes of both peaksdecrease strongly with increasing temperature. In some cases, the peak at Ω r is weak and thespectrum can have additional maxima and a background extending up to several hundred meV. PACS numbers: 74.25.Gz, 74.62.Dh, 74.72.Jt
Superconductivity arises when electronic quasiparti-cles bind together into Cooper pairs which condense intoa macroscopic quantum state. The interaction whichdrives the pairing could be charge or spin polarizations,but in conventional metals, it is the polarization of thelattice of ions which provides the necessary attraction.There is also a smaller Coulomb repulsion. In the phononcase Eliashberg theory accounts for all the experimen-tal data with its central ingredient, the electron-phononspectral density α F (Ω) specifying the boson exchangepairing interaction [1]. Optical spectroscopy is a pow-erful tool that can be used to extract bosonic spectraldensity with high degree of accuracy in a variety of sys-tems where other techniques such as tunneling and angleresolved photoemission (ARPES) are difficult to apply.It has been used in conventional metals [2, 3] and in thehigh T c oxides [4, 5, 6, 7, 8, 9, 10] but its interpretationin the cuprates remains controversial. As emphasizedrecently by Anderson [11], the pairing interaction couldinvolve high energy virtual transitions across the Mottgap with energy set by the Hubbard U which is a largeenergy and the effective interaction would be instanta-neous on the time scale of interest. On the other hand,recent numerical work [12, 13] based on the t-J model,has shown that the main contribution to the pairing glueis provided by the spin fluctuations with characteristicenergies of at most a few hundred meV. Optical experi-ments provide a direct probe of this energy region.The mercury versions of the cuprates, HgBa CuO δ (Hg1201) and HgBa Ca Cu O δ (Hg1223) provide aunique opportunity to test these ideas. The one-copperlayer system Hg1201 shares a conventional transitiontemperature around 90 K with widely studied systemsYBCO and Bi2212, whereas the three-layer compoundHg1223 has a T c = 130 K, nearly 40 % higher. These dramatically different T c ’s lead, as we will show, to verydifferent bosonic spectra and place severe constraints onmodels of superconductivity in the cuprates.In this letter, we present new data on the optical con-stants for HgBa CuO δ (Hg1201). The bosonic spectraldensity I χ (Ω), recovered by maximum entropy inver-sion, is found to have a remarkable resemblance to pre-vious results for optimally doped Bi2212 [8]. For furthercomparison, using the same methods, we also invert pub-lished [14] optical constants in the three-layer Hg1223.The high-quality Hg1201 single crystal was grown by aflux growth technique. Our sample was a millimeter sizedplatelet with a well oriented ab -plane. It is slightly un-derdoped with a T c = 91 K. The real and imaginary partsof the optical conductivity σ ( T, ω ) ≡ σ ( T, ω )+ iσ ( T, ω )follow from the reflectivity. In analyzing optical data, weuse a memory function or optical self energy Σ op ( T, ω )instead of working with σ ( T, ω )[15, 16]. By definition, σ ( T, ω ) = ( iω p / π ) / [ ω − op ( T, ω )] where ω p is theplasma frequency. The optical self energy defined thisway plays a role analogous in optics (which involvesa two particle process) to the quasiparticle self energyΣ qp ( T, ω ) in ARPES [17, 18]. The optical scattering rateis 1 /τ op ( T, ω ) = − op ( T, ω ) and the optical effectivemass m op ( T, ω ) /m (with m the bare electron mass) isgiven by ω [ m op ( T, ω ) /m − ≡ − op ( T, ω ).Our results for the real part of the optical self en-ergy − op ( T, ω ), which provides an easy comparisonwith ARPES, often presented in terms of 2Σ qp ( T, ω ), areshown in Fig. 1 as a function of ω for eight temperatures.The lowest three temperatures are in the superconduct-ing state. The most prominent feature of the curvesis the peak around 100 meV seen at T = 29 K, whichprogressively disappears as the temperature is increased.To better see how the bosonic spectrum generates the
29K 50K 76K 101K 151K 200K 246K 295K
T=29 K I ()- op () ( m e V ) Hg1201 (T c =90K) (meV) - op () ( m e V ) (meV) -2 ( ) supercond. state -2 ( ) normal state (meV) I ( ) FIG. 1: (color online). Experimental results for the real partof the optical self energy − op ( T, ω ) as a function of pho-ton energy ω for eight temperatures. Inset, theoretical re-sults based on numerical solutions of the generalized Eliash-berg equations. (Solid blue superconducting and dashed blacknormal state.) The electron-boson exchange spectral densityused is shown as the dashed-dotted blue curve. The super-conducting gap value is ∆ = 22.4 meV. self energy we show in the inset of the figure a modelcalculation based on numerical solutions of the d -waveEliashberg equations. The input electron-boson spec-tral density I χ (Ω), shown as the dashed-dotted curve,consists of a large narrow peak (right hand scale) cen-tered at 56 meV followed by a dip and a long back-ground extending beyond 400 meV. The dashed curve, with the same bosonic spectrum , is the normal state re-sult for − op ( T, ω ) which is to be compared with thesolid curve in the superconducting state. The supercon-ducting gap obtained was 22.4 meV giving a gap to T c ratio, 2∆ /k B T c = 5.8. Note that, as theory predicts [19],the dashed curve in the normal state shows no visiblestructure at ω = Ω r = 56 meV. Instead there should bezero slope at ω = √ r for an Einstein spectrum. It isclear that boson structure is hard to see in normal state.However in the superconducting states the quasi-particleelectronic density of states acquires energy dependenceand this helps reveal the underlying boson structure asseen in the solid blue curve. The peak at ∼
100 meV isneither at Ω r nor √ r but is shifted upwards by theopening of the gap ∆ [19] but as we see, the position ofthe peak in I χ (Ω) cannot be read off the curve withoutthe knowledge of the value of the superconducting gap.In Fig. 2(b) we show results for the spectral density Hg1201 (T c =90K) (meV) / op () ( m e V ) A m p li t ud e r ( m e V ) Temperature (K) (b)(a) I () (meV) FIG. 2: (color online). Top frame, the optical scatteringrate 1 /τ op ( T, ω ) for Hg1201 v.s. ω for 8 temperatures (lightcurves). The wider curves are our maximum entropy recon-structions. Bottom frame, the electron-boson spectral func-tion I χ (Ω) v.s. Ω. The inset gives the peak position (bluetriangles) left scales as a function of temperature and the redsquares give the corresponding peak amplitude. I χ (Ω) of maximum entropy inversions augmented witha least squares improvement based on the full d -waveEliashberg equations [6]. Further applications are foundin Refs. [8, 9, 10]. The input to the inversion is the op-tical scattering rate 1 /τ ( ω ) = ω p π R e (1 /σ ( ω )). These areshown in Fig. 2(a) where the results of our inversions(heavy lines) are compared with the original data (lightlines). In all cases, the fit is very good. Recently, vanHeumen et al. [20, 21] have also presented optical data forHg1201 above T c which they analyze in terms of a spec-tral density represented by a set of histograms. Whilethey obtain fits which are of equal quality to ours andhave a background extending to >
400 meV as we have,they find that the height of the peak at 56 meV does notchange contrary to our findings for T = 246 and 295 K.This has been taken as evidence for coupling of the chargecarriers to phonons [22]. We can also get fits where thepeak height does not change but only if we use a biased Hg1223 (T c =130K) (meV) I () (b) / op () ( m e V ) (a) Hg1223 (T c =130K) (meV) FIG. 3: (color online). Same as Fig. 2 but for Hg1223( T c =130 K). maximum entropy inversion with the default model set tothe previous lower temperature solution instead of beingset to the constant of the unbiased inversion.In the inset to the lower frame of Fig. 2, we show thefrequency (left scale, triangles) of the prominent peak in I χ (Ω) as a function of temperature. As also noted byvan Heumen et al . [20], Ω r is fairly constant at ∼ T above 200 K. More importantly, the amplitude ofthe peak shows strong temperature dependence in the su-perconducting state and also above 200 K. Further, thewidth of the peak increases with increasing T . As notedfor Bi2212 [8], the shift in spectral weight into the peakat 60 meV can be interpreted to proceed through a trans-fer of spectral weight from high to low frequencies as thetemperatures is lowered. We note here that in contrasta bosonic function from the electron phonon interactionwould not have these properties: its amplitude, widthand center frequency would all be temperature indepen-dent and would not vary from one cuprate to another (asshown in Fig. 4 below).In Fig. 3 we display similar results for Hg1223 with T c = 130 K. In Fig. 3(a) we reproduce the optical scatter- r = 6.3 k B T c Hg1223 Hg1201 Bi2212 Tl2201 YBCO LSCO PCCO r ( m e V ) T c (K) FIG. 4: (color online). The optical resonance frequency Ω r as a function of T c . Bi2212 [6, 8], Tl2201 [5], YBCO [5, 6, 7],LSCO [9], and PCCO [10]. ing rate at four temperatures (with T = 15 K and 125 Kin the superconducting state) from the work of McGuire et al. [14]. Our results for I χ (Ω) are presented in Fig.3(b) and the quality of the data reconstruction is demon-strated by the heavy lines in Fig. 3(a) to be comparedwith the corresponding light line (data). We find signifi-cant residual (static impurity 1 /τ imp ≃
95 meV) scatter-ing rate in contrast to Hg1201 which is in the clean limit(1 /τ imp = 0). The superconducting state data (blue,dotted curve) for 1 /τ op ( ω ) at T = 15 K shows a peakaround 140 meV which is the indication of a gap in thecharge carrier density of states (DOS). Currently there isno known kernel which allows maximum entropy inver-sion of such data and so we do not show results in thisinstance. On the other hand, in the T = 125 K data thereis no signature of such a DOS-gap. Results are shown inFig. 3 b). The prominent peak at Ω r ∼ = 72 meV seenin the curve at T = 125 K is missing at higher T . Incontrast to the Hg1201 case no reasonable alternate fitscan be found for T = 225 K and 295 K which show asignificant peak amplitude at 72 meV.In Fig. 4, we place our results for Ω r , the frequency ofthe peak in I χ (Ω), in the context of other such resultsby plotting Ω r as a function of the superconducting T c for a number of cuprates. In all cases, I χ (Ω) has beenextracted from the optical data but not always using amaximum entropy technique. Some are fits to assumedforms including a broad background introduced in refer-ence [23] to model antiferromagnetic fluctuations, aug-mented with a resonance peak. Both methods give verymuch the same results as documented in reference [5].The heavy long dashed line is a least square fit to all theoptical data and gives Ω r ≈ k B T c . This is close to,but not quite, the position of the spin-one neutron res-onance obtained by He et al. [24, 25] where Ω neutronr ≈ k B T c represented in Fig. 4 as the dotted line.Several comments should be made about such a com-parison between charge excitations and the magnetic sus-ceptibility. First, the the neutron resonance plotted inFig. 4 refers to the sharp peak that appears at q = ( π, π )whereas the bosonic function that governs the optical re-sponse is the q averaged local susceptibility. This arisesbecause the Fermi-surface to Fermi-surface electron scat-tering involves momentum transfers to boson excitationsthat span all momenta in the Brillouin zone, some involv-ing Umklapp processes. Where magnetic neutron scat-tering data for the q -averaged susceptibility are availablesuch as the Ortho II YBCO [26] or optimally dopedLSCO [27] the agreement between the neutron data andthe optical data is excellent: not only are the peaks inthe response at the same frequencies but also the tem-perature dependence of the amplitude of the peaks arein agreement [8, 9]. Secondly, it should be noted that insome cuprates, the resonance described involves only avery small fraction of the total weight seen in the localspin susceptibility as in LaSrCuO [9] and PrCeCuO [10].In YBCO . it is estimated to be 3% [26] and in somecases there is no resonance [5, 28], but it is always thelocal, Brillouin zone averaged spin susceptibility whichcontrols superconductivity. Finally, the two points at Ω r = 0 (not used in the fit to the data in Fig. 4) are forYBCO . [28] and overdoped Tl2201 [5]. In both cases,no optical resonance could be identified. The resonancemay enhance but is not essential for superconductivityin the cuprates and the scaling of the position of thepeak with T c shown in Fig. 4 must be the result and notthe cause of the rearrangement of the electronic DOS inthe superconducting state as suggested by several theo-rists [29, 30]. We also note here that recent dynamicalmean field calculations of the one-band Hubbard modelyield bosonic spectral functions very similar to what isshown in Figs. 2 and 3. [12, 13]In summary we find that in Hg1201 and Hg1223 opti-cal resonances are found in maximum entropy inversionsof the optical scattering, at 56 and 72 meV, respectively.However, when the temperature is increased towards 300K, the spectral weight under this resonance moves tohigher energy and broadens significantly, in contrast tothe findings of van Heumen et al. [21]. The optical res-onance scales with T c over a broad set of materials withΩ r ≈ . k B T c which is remarkably close to the energyof the spin one resonance seen in polarized neutron scat-tering, namely Ω neutron = 5 . k B T c leaving no doubt thatthe charge carriers are coupled to spin fluctuations, whilethere is no evidence for an important phonon contribu-tion.This work has been supported by the Natural Sci-ence and Engineering Research Council of Canada andthe Canadian Institute for Advanced Research. RPSML acknowledges support from the ANR grant BLAN07-1-183876 GAPSUPRA.Note added in proof.We have learned of a neutron scat-tering study 4 by Yu et al. [31], where a magnetic reso-nance in optimally doped Hg1201 is reported at 56 meV,exactly the same energy as the peak we found here. Re-cent Raman data 5 nd a superconducting gap 2?? veryclose to the values found in our calculations [32]. Note added in proof . -We have learned of a neutronscattering study by Yu et al [31], where a magnetic reso-nance in optimally doped Hg1201 is reported at 56 meV,exactly the same energy as the peak we found here. Re-cent Raman data finds a superconducting gap 2∆ veryclose to the values found in our calculations.[32] ∗ Current address: School of Materials Science and Engi-neering, Tianjin University, Tianjin 300072, P. R. China. † Current address: Department of Physics, University ofFlorida, Gainesville, Florida 32611, USA. ‡ Electronic address: [email protected][1] J.P. Carbotte, Rev. Mod. 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