Spin Excitations and Phonons in Yba2cu306+X: a Status Report

ELSEVIER Physica B 234-236 (1997) 821-829

Spin excitations and phonons in YBa2Cu306+x: A status report

B. Keimer a'b'*, I.A. Aksay c, J. Bossy d, P. Bourges e, H.F. Fong a, D.L. Milius c, L.P. Regnault f, D. Reznik e, 1, C. Vettierg

a Department of Physics, Princeton University, Princeton, NJ 08544, USA b Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA c Department of Chemical Enoineerin 9, Princeton University, Princeton, NJ 08544, USA d Institut Laue-Langevin, 38042 Grenoble, Cedex, France e Laboratoire LOon Brillouin, CEA-CNRS, CEA-Saclay, 91191 Gij:sur-Yvette, France f Dbpartement de Recherche Fondamentale sur la Mati&e CondensOe, Service de Physique Statistique, Maonbtisme et Supraconductivitb, Groupe Magn~tisme et Diffraction Neutronique, CEA-Grenoble, 38054 Grenoble, Cedex 9, France g European Synchrotron Research Facility, Bofte Postale No. 220, 38043 Grenoble, Cedex, France

Abstract A review is made of recent developments in inelastic neutron scattering experiments on spin excitations and phonons in the high-temperature superconductor YBazCu306+x and its antiferromagnetic precursor YBa2Cu3062. These experiments include the detection of high-energy "optical" spin waves and the determination of the full spin Hamiltonian in YBa2Cu306.2, detailed investigations of the 40 meV magnetic resonance peak in the superconducting state of YBazCu3 07 and its precursors in underdoped YBa2Cu306+x, and experiments on the effect of superconductivity on phonon lifetimes in YBazCu3OT.

Keywords." Spin excitations; Phonons; YBa2Cu306+x

1. Introduction spin susceptibility z"(q, ~o) are a direct consequence of these interactions and provide incisive microscopic Neutron scattering is the only probe capable of constraints on theories of superconductivity. Since providing maps of the magnetic and lattice vibra- large, high-quality single crystals are available, the tional excitation spectra of materials over the entire YBazCu306+x family offers a unique opportunity to Brillouin zone. In complex materials such as the explore the evolution of the spin excitation spectrum high-Tc cuprates one cannot hope to arrive at a quanti- by neutron scattering as the material evolves from tative theory without experimental information about an antiferromagnetic insulator to a high-temperature these excitations. More specifically, strong electron- superconductor with increasing x. Much progress has electron interactions are now generally recognized as been made during the last decade [ 1-3], but this field central to the physics of these materials. Energy and is still in a stage of rapid development, mostly due to momentum-dependent enhancements of the dynamic the intrinsic difficulty of the experiments. This paper summarizes recent progress made by our groups * Corresponding author. which was made possible by the growth of very large I Present Address: Department of Physics, UCSD, La Jolla, single crystals at Princeton University. CA 92093, USA.

We are ultimately interested in the pairing mech- anism in these materials and have thus focused on o) the temperature evolution of the magnetic excitation spectrum through the superconducting transition. Early indications of effects of superconductivity on the spin excitation spectrum of YBa2Cu307 [4, 5] have recently been followed by detailed, quantita- tive studies [6, 7]. These experiments have revealed 4r "I~ a strikingly sharp magnetic resonance mode in the superconducting state which disappears in the normal b) state. This novel collective mode has stimulated much theoretical work, and its interpretation is still contro- Acoustic mode ~n ~n ~.,n n,~ n,n n.n n_n n~n versial. In an effort to discriminate between different v'~ v'~ v v ~-v ~v wv v-v v~ theoretical models we recently traced the evolution of n~n ~A n ~ n~ n~ A~A n.n n~n the resonance with doping into the deeply underdoped v'~ v-~ v v ~'~v v-v wv v-v v',v regime [8, 9].

Further, it has now become possible to extend ear- ~A ~A ~ A A ~ A~ A.A A_ A A~A lier investigations of low-energy spin excitations [ 1-3] V~ V~ g V b~'~V ~v'V V~V ~V V~ N~ A~ A.A A~A ~A ~A A A AJ~ to energies high enough to excite antiphase motion of VqV V'V V'V V~V V~ll q~ V ~? ~V spins in directly adjacent layers (the bilayers) [10]. Detection of such "optical" spin waves in the anti- Fig. 1. (a) Spin structure of antiferromagnetic YBa2Cu306+x ferromagnetic phase allows one to deduce the ex- (x < 0.4). (b) Sketches of acoustic and optical spin waves. change coupling between the bilayers, an important parameter in the Hamiltonian of these materials. Very recently, we have also observed such antiphase spin wavelength optical spin wave, spins located in directly excitations in the metallic regime [11]. adjacent layers precess in antiphase (Fig. l(b)). While Finally, while the intrinsic lifetimes of phonons acoustic spin waves have only a small gap due to ex- have been studied extensively at q -- 0 by Raman scat- change anisotropies and interbilayer exchange, the op- tering, insufficient energy resolution has until recently tical magnon branch has a much larger gap because prevented an extension of these studies to nonzero q. the exchange interactions between nearest-neighbor Such experiments are important because valuable in- spins within the same bilayer are very strong. Detailed formation about the electron-phonon coupling and the calculations [ 1] reveal that the optical magnon gap is nature of the pairing state can be extracted from the given by renormalization of phonon lifetimes below the super- ~ 2 ( l ) conducting transition temperature. We give a brief review of our recent work on this problem [12]. where JII ~ 120 meV is the nearest-neighbor superexchange in the CuO2 plane which is known from measurements of the slope of the dispersion of low- 2. Optical spin waves in antiferromagnetic energy acoustic spin waves (spin wave velocity) and YBa2Cu306.2 from two-magnon Raman scattering. The value of J±, the superexchange coupling between nearest neigh- For oxygen concentrations x~< 0.4, YBa2Cu306+x bors in the c-axis direction perpendicular to the CuO2 is an antiferromagnetically long-range ordered insu- planes, had been unknown until recently. (For the sake lator with a spin structure shown in Fig. l(a). As of simplicity, small corrections to Eq. (1) due to spin- there are two spins per chemical unit cell, the spin orbit coupling and superexchange coupling between wave spectrum consists of an optical and an acous- different bilayers as well as somewhat larger correc- tic branch. Pictorial representations of these two types tions due to quantum renormalization of the spin wave of spin excitation are given in Fig. l(b). For a long velocity are ignored in the present discussion.) B. Keimer et al. IPhysica B 234~36 (1997) 821429 823

The cross sections for neutron scattering from 3. Magnetic resonance peak in the superconducting acoustic and optical spin waves contain different dy- state of YBa2Cu307 namical structure factors which can be thought of as the Fourier transforms of the real space pictures of We will now discuss the spin excitation spectrum of Fig. l(b), quite analogous to the more familiar static superconducting YBa2Cu307 which is very different structure factors in X-ray or neutron diffraction. These from the one of the antiferromagnetic YBa2Cu306.2, are proportional to sin 2(nQ±zc) for acoustic magnons and then show in the next section how the spectrum and cos2(nQ±zc) for optical magnons, where Q± is evolves from one extreme to the other. YBa2Cu307 the momentum transfer perpendicular to the CuO2 has 39 different phonon branches, and painstaking sheets. We [ 10] have recently carried out high-energy work is necessary to separate the cross sections of op- magnetic scattering experiments on YBazCu306.2, tical phonons and spin excitations. Over the years, sev- and have exploited these different structure factors to eral groups have developed an arsenal of techniques separate optical from acoustic spin waves. We found to deal with this problem; these include comparison that the optical spin wave gap is Eop t ,-.o 70 meV, which of spectra taken in the same crystal in different oxy- translates into J±~10meV according to Eq. (1). genation states [3], detailed monitoring of the temper- This value is in good agreement with recent predic- ature dependence of the neutron cross section [2, 3], tions of band theory [13]. A subsequent report by phonon structure factor calculations [6], and spin po- Hayden et al. [14] appears consistent with this result. larization analysis [5, 6]. Together, these techniques As the superexchange coupling is generally propor- have resulted in a detailed, reliable picture of the spin tional to the square of the overlap of the electronic excitations of YBa2Cu3 07, and any remaining contro- wave functions, we have versies appear to have been resolved. The picture that has emerged is as elegantly simple as it is surprising. Rossat-Mignod and coworkers [4] had first discov- t± ~ - 0.28, (2) ered a peculiar enhancement of the magnetic cross ,. : VT- section of this compound around 40 meV at low temperatures, but the temperature dependence of this where t± and tll are the hopping matrix elements excitation was not determined. Further work on this between nearest-neighbor copper sites perpendicular mode was reported by Mook et al. [5]. Our recent ex- and within the CuO2 layers, respectively. The t's are periments [6, 7] have revealed the following remark- determined by quantum chemistry and are unlikely to able features of the 40 meV magnetic mode: First, the vary significantly upon changing the oxygen content excitation exists only at a single point in an (o9, q) x in YBa2Cu306+x. The anisotropy given by Eq. (2) diagram (Fig. 2): The width is resolution limited is therefore one of the fundamental parameters char- in energy and close to resolution limited in q, and acterizing the electronic state of the YBa2Cu306+x there is no evidence for dispersion of the mode. system. It is presumably no accident that the super- Second, the intensity of the sharp peak is nonzero conducting transition temperature, ire, of the copper only in the superconducting state. Some raw data oxide superconductors tends to increase substantially are shown in Fig. 3, and the temperature dependence as the number of consecutive CuO2 layers increases: of the spectral weight of the excitation is shown in Electron transfer processes between directly adjacent Fig. 4. A careful calibration allows us to extract the layers (the microscopic nature of which is still con- dynamical spin susceptibility, 7/', from the data in troversial) are probably responsible for enhancing Tc absolute units. Third, as also shown in Fig. 4, the with respect to typical single-layer materials. In some excitation energy remain constant (to within an ex- models interlayer electron transfer is even the driv- perimental error of <5%) at least up to T = 80 K ing force of the superconducting transition [15]. The (~0.8Tc, where Tc =93.0K). Finally, the dynami- energy scale for these processes, set by Eq. (2) and cal structure factor of the excitation as a function directly determined by our neutron scattering mea- of Q± is sin2(nQ±zc), identical to the structure surements, is thus an important part of a microscopic factor of acoustic spin waves in antiferromagnetic description of these materials. YBazCu306+r. 824 B. Keimer et al./ Physica B 234-236 (1997) 821-829

Magnetic Excitation Spectrum Magnetic Resonance 120 600 I I I I

100

8O 500

=.,.JE so ..-. 40o ..=, 40 ._= E 20 O

0 300 0.2 0.3 0.4 0.5 0.6 0]7 0.8 == o qx [units of 27r/a] 200 Fig. 2. (Lines) Experimentally determined dispersions of acous- i tic and optical spin waves in YBa2Cu306.2. (Dot) Location of the 40 meV resonance peak in the superconducting state of YBa2Cu307. lo0

This magnetic resonance peak is thus a novel and unusual signature of the superconducting state in YBa2Cu3OT. To our knowledge, similar phenomena 25 30 35 40 45 50 have not been observed in any other material. Two E (r.ev) very different explanations (with various modifica- tions) have been proposed for our experiment, and Fig. 3. Magnetic resonance peak in YBa2Cu307 at q = (r~, 7t) and at the maximum of the sin2(~Q±zc) dynamical structure factor there is as yet no consensus about its correct inter- as a function of energy at various temperatures. The phonon pretation. However, both explanations invoke and in background has been subtracted (from Ref. [6]). fact require a d-wave pairing state (or at least a gap function with sign reversal on the Fermi surface), which explains why this feature is not observed in conventional superconductors. to the Fermi surface. Z~~ is proportional to the joint The first explanation [6,16-20] attributes the density of occupied and unoccupied states and can be resonance peak to the creation of a quasielectron- calculated for the band structure of YBa2Cu3OT. The quasihole pair across the superconducting energy gap. first factor is the coherencefactor which is a conse- As the initial state is a singlet (Cooper pair) and the quence of the functional form of the BCS wave func- pair is created by magnetic scattering, the final state tion. It is apparent that the coherence factor is zero must be a triplet. The imaginary part of the dynamical on the Fermi surface (ek = ~k+q= 0) unless the gap susceptibility for this process is function has opposite sign at the points k and k + q where the two quasiparticles are created. In particular, Zot(q, og)= ~ (l _ ~k~l'+q-~- AkAk+q~ for a d-wave gap the coherence factor is maximum for E~Ek+q J q ~(Tz,~) which connects two maxima of [Akl with reverse sign. The energy to create a quasiparticle- × (fk+qT q-fk+ -- 1)~(Ek+q q- Ek --h09), quasihole pair with this relative momentum is 2Amax, (3) where Amax is the maximum of the d-wave gap function. As we have experimentally established that the where fk is the Fermi factor, Ak is the momentum- resonance peak vanishes in the normal state, we can dependent energy gap, Ek = v/A 2 ÷ e2 is the quasipar- indeed associate the resonance energy with the super- ticle energy, and ek is the band dispersion with respect conducting energy gap. The d-wave symmetry of the B. Keimer et al./Physica B 234-236 (1997) 821-829 825

Magnetic Resonance of the band structure are also capable of produc- 0.6 • . , . , . , . ing a peak in Z'o' [19]. The interlayer pair tunneling theory of Anderson and coworkers [20] can account -'~ 0.4 for most features of the experiment without ad hoc adjustment of parameters. In their picture, kinematic ~, 0.2 constraints associated with the non-BCS functional form of the interlayer gap equation are responsible 45 for the sharpness of the peak. --o • o oo A very different explanation has been suggested by A Zhang and collaborators [21]. Instead of a particle- E 30 hole excitation, they attribute the resonance to a particle-particle mode characteristic of the Hubbard O 15 t-. model. There are actually two such modes: The one W b) originally proposed consists of two electrons in a sin- 0 1 1 , | , | , gle state on the same site, moving with momentum 0 20 40 60 80 100 q = (rt, 7t). Creating this pair costs an energy of the Temperature (K) order of the on-site Coulomb repulsion, but once the Fig. 4. (a) Absolute spectral weight and (b) maximum energy of pair is created it has both a well-defined energy and the resonance peak as a function of temperature (from Ref. [6]). an infinite lifetime. The first property derives from the fact that for a tight-binding band the sum of the two particle energies, ek + ~k+q, has only one possi- gap function has now been firmly established by var- ble value for q = (n, n). The pair does not decay by ious quantum interference experiments, and our ex- scattering from other electrons, because such scatter- periments confirm this result in a completely different ing processes are forbidden by the Pauli principle. fashion. The second mode, and the one that is relevant for A detailed calculation [17] shows that Z~ot(q,e)) our experiment, is a particle-particle pair on nearest- evaluated at q=(n,n) as a function of he) only neighbor sites in a triplet state, the creation of which shows a step at an energy of 2Arnax, whereas we costs an energy of the order of Jl~. These particle- observe a sharp peak. However, the real part of the particle modes exist at all temperatures, but do not bare susceptibility, Z~, (which is not measured in couple to external probes in the normal state. In the a neutron scattering experiment) does show a peak superconducting state particle and hole states are at this energy. Several groups [16-18] have thus in- mixed, and the cross section for creation of the triplet voked a renormalization of Z0 by electron-electron mode by magnetic neutron scattering is proportional interactions which can be written in the RPA approx- to A2, where A is the superconducting energy gap. imation as There are open questions about both of these dif- z0(q, co) ferent explanations. In the particle-hole picture the ZRPA(q, e)) = 1 -- J(q)zo(q, e))" (4) resonance energy should follow the temperature dependence of the superconducting energy gap, which By calculating the imaginary part of this expression appears hard (but perhaps not impossible) to recon- one can convince oneself that the "Stoner factor" cile with Fig. 4(b). For the particle-particle picture, it in the denominator converts the peak in Z~ into has not been resolved how deviations from a nearest- a peak in ZRPA" if the interaction term J(q) is peaked neighbor tight binding band structure (which seem at q=(n, rt). In the strongly interacting limit the to be necessary to account for the observed Fermi quasielectron-quasihole pair may be thought of as surface) affect the resonance [24]. Another open ques- a "triplet exciton". By tuning the functional form of tion concerns the experimentally measured dynamical J(q), good agreement with the experimentally mea- structure factor of the resonance [~sinZ(rtQ±zc)] sured shape of the resonance peak can be achieved which has not yet been fully explained in either [16-18]. In an alternative scenario, nesting features picture. 826 B. Keimer et al./ Physica B 234-236 (1997) 821~829

4. Magnetic resonance peak in underdoped 200 ...... YBa2Cu306+x YBa2Cus06.s 1,50 .= IOK-60K In the metallic but still underoxygenated regime of E 100 the phase diagram (for 0.4 < x < 1 ) antiferromagnetic Q 50 long-range order vanishes but low-energy spin exci- o tations are known to persist [2, 3]. These excitations o resemble acoustic spin waves in antiferromagnetic ~ -50 YBa2Cu306+x in that their cross section is maximum 200 : ." : : : I . I ' around the antiferromagnetic ordering wave vector YBa2Cu306.7 E 150 16K-80K ~[~ q--(rt, ~) and they follow the dynamical structure 0 factor which characterizes acoustic spin waves. How- 100 t.- ever, in contrast to antiferromagnetic spin waves 50 these excitations are broadened in momentum space, o 0 which implies a finite correlation length (typically only a few lattice spacings) in real space. Further, -50 the low-energy spectral weight of these excitations is -100 severely depressed, a phenomenon which has been 0 10 20 30 40 50 termed "spin pseudo-gap" and is also observed in Energy (meV) NMR experiments. Semiphenomenological models Fig. 5. Difference between the low temperature (< Tc) and high- which include the effects of both band structure and temperature (~> Tc) magnetic intensity at Q-(I,-- 1 ~,-5.4).1 The electron-electron interactions have been developed to data were adjusted for the Bose factor and for a weakly tem- account for the experimental observations [25]. perature-dependent featureless background at low energies in the In order to provide more experimental input for lower panel (from Ref. [8]). theories of the 40 meV resonance peak, we decided to search for analogous spectral weight enhancements in the superconducting states of underdoped 1200 0 mum YBa2Cu306+x [8]. To this end crystals with several -0.2 different oxygen concentrations were prepared. Fig. 5 ~_~1000 shows the difference between spectra measured above -0.8 800 iiiiiiiiii L.B and below Tc for two doping levels, YBa2Cu306.5 -"°0 ,0:1 .0 (Tc = 52 K) and YBa2Cu306.7 (Tc ---=67 K). Spin po- 600 larization analysis has revealed that the observed .-=_ ,P, enhancements of the magnetic cross section below . 100%. The data were therefore normalized to occurs, and smaller the fraction of the normal state the maximum susceptibility (from Ref. [8]). B. Keimer et al./Physica B 234-236 (1997) 821~29 827

tron scattering in conventional superconductors, and ,--,50>. by Raman scattering at q = 0 in high-temperature su- ,~E4o perconductors including YBa2Cu307 [30, 31]. It has recently become possible to achieve high enough energy resolution in neutron scattering experiments to measure intrinsic phonon lifetimes at 20 nonzero q. We [12] have focused on an oxygen vibration of Big symmetry at 42.5 meV which had been shown by Raman scattering to exhibit the largest superconductivity-induced line shape anomalies at "~ 0 0 20 40 60 80 100 q = 0. Harashina et al. [32] have also studied another Tc (K) oxygen vibration. In the normal state, the dispersion of the 42.5 meV phonon is completely fiat, and the Fig. 7. Energy of maximum superconductivity-inducedspectral intrinsic line width (~2 meV) is q-independent to enhancement as a function of the transition temperature Tc. within our resolution. In the superconducting state, For YBa2Cu306,7 and YBa2Cu307 the enhancement is resolu- Raman work has shown that at q = 0 the phonon tion-limited in energy, and the error bars are upper bounds on the intrinsic width of the resonance (from Ref. [8]). softens by almost 1 meV, and the line width broadens substantially. This can be understood on a qualitative level if 42.5 meV happens to just exceed 2Area×: In order to compensate for the loss of electronic states spectrum that is affected. This trend is followed also below the superconducting energy gap, there is a in Lal.86Sr0.14CuO4 where a very small enhancement pileup in the density of states above the gap which in below Tc ~ 35 K has recently been observed over a turn decreases the phonon lifetime via the electron rather wide energy range around 10-15 meV [26]. phonon interaction. By comparison of our recent data for YBazCu306.5 In a pioneering study Pyka et al. [33] had been able with previous work in the same range of concentra- to measure the effect of superconductivity on the dis- tion in the normal state, it is also clear that, while persion this phonon mode at nonzero q. Our recent the enhancement occurs sharply at Tc, it follows a experiments [ 11 ] had sufficient resolution to also de- trend already established in the normal state. This termine the lifetime of this phonon branch above and observation is presumably intimately related to recent below Tc. The lifetime is quite anisotropic (Fig. 8) and photoemission work in underdoped cuprates which is in fact much shorter at low temperatures along the indicates that a gap-like structure in the single-particle (1 0 0) crystallographic direction than along (1 1 0). density of states is already apparent at temperature One factor which is certainly relevant in explaining far above Tc [27-29]. this anisotropy is another coherence factor, analogous to the coherence factor entering Eq. (3). However, there is a crucial difference between the coherence 5. Phonon line shapes factor entering the cross section for magnetic neutron scattering and the one entering the phonon self-energy: Another, more indirect probe of the electronic state While magnetic neutron scattering is antisymmetric of YBa2Cu3 07 is the investigation of phonon energies under time reversal symmetry, electron-phonon scat- and lifetimes, especially their renormalization below tering is time-reversal symmetric, and the coherence the superconducting transition temperature. Electron- factor therefore has the opposite sign [34, 35]: phonon scattering contributes to the finite lifetimes of 1 + ekek+q + AkAk+q (5) phonons in the normal state, and depending on the EkEk+q magnitude of this contribution the lifetime of spe- cific phonons can be affected by the redistribution of This factor is maximum for q-vectors connecting two the electronic density of states in the superconduct- lobes of the gap function which have the same sign, ing state. This effect had been demonstrated by neu- that is, q-vectors along (1 00). The experimentally 828 B. Keimer et al./ Physica B 234-236 (1997) 821-829

MRSEC Program of the National Science Foundation under grant No. DMR-9400362, and by the Packard .... ! .... and Sloan Foundations. We also acknowledge the ~42.6°'Ila European "Human Capital and Mobility" program ~ 42.4 No. ERBCHRXCT930113 for financial support. 42.2 lOOK / ' 42 i: 50K /! References ~" 41.8 I*10K.... , . . . , . . .~....~--t ,-, , ~ , , ,~1 0.6 oa 0.2 0 0.2 0.4 0.6 [1] J.M. Tranquada, G. Shirane, B. Keimer, M. Sato and < [1,1/3] q [1/3,0] S. Shamoto, Phys. Rev. B 40 (1989) 4503; S. Shamoto,

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