The strong, electronically induced anomaly in the spectrum of the
longitudinal optical (LO) phonons propagating along the main axes of the
CuO$_2$ plane is tentatively attributed to the oxygen-oxygen charge transfer
between the two oxygens in the plane. It is argued that this charge transfer
can be large and that it is strongly coupled to the zone boundary LO phonons.
The corresponding negative contribution to the free energy is quartic in the LO
phonon amplitude, making the LO phonon unstable through the first order phase
transition, with a concomitant domain structure.Comment: 3 pages, 2 figure

Barisic, Slaven

Kupcic, Ivan

Mrkonjic, Ivana

English

arXiv

Strong, electronically-induced anomaly in the spectrum of the longitudinal optical (LO) phonons propagating along the main axes of the CuO_2 plane is tentatively attributed to the oxygen-oxygen charge transfer between two in-plane oxygen atoms. It is argued that this charge transfer can be large and that it is strongly coupled to the LO dimerization of the oxygens. The corresponding negative contribution to the free energy is quartic in the LO phonon amplitude, indicating the instability via the first-order phase transition, with a concomitant domain structure

Barišić, Slaven

Mrkonjić, Ivana

Kupčić, Ivan

Repository of Faculty of Science, University of Zagreb

ISSN1330{0008CODENFIZAE4LETTER TO THE EDITORELECTRONICALLY INDUCED ANOMALY IN LO PHONON DISPERSIONOF HIGH-Tc SUPERCONDUCTORSSLAVEN BARI·SI¶C, IVANA MRKONJI¶C∗ and IVAN KUP·CI¶CDepartment of Physics, University of Zagreb, POB 331, 10002 Zagreb, Croatia∗E-mail address: ivanam@phy.hrDedicated to Professor Kseno Ilakovac on the occasion of his 70th birthdayReceived 28 September 2001; revised manuscript received 21 December 2001Accepted 7 January 2002 Online 6 April 2002Strong, electronically-induced anomaly in the spectrum of the longitudinal optical(LO) phonons propagating along the main axes of the CuO2 plane is tentatively at-tributed to the oxygen-oxygen charge transfer between two in-plane oxygen atoms.It is argued that this charge transfer can be large and that it is strongly coupledto the LO dimerization of the oxygens. The corresponding negative contribution tothe free energy is quartic in the LO phonon amplitude, indicating the instabilityvia the ¯rst-order phase transition, with a concomitant domain structure.PACS numbers: 74.25.Kc, 74.72.Bk, 74.72.Bn UDC 538.945Keywords: superconductivity, high-Tc cuprates, La2−xSrxCuO4, YBa2Cu3O6+x, LOphonons, oxygen-oxygen charge susceptibility, ¯rst-order phase transitionsThe discovery of the electron-induced anomaly in the dispersion of the oxy-gen longitudinal optical (LO) phonons, which propagate along the main axes ofthe CuO2 plane in La2−xSrxCuO4 (LSCO) [1, 2] and YBa2Cu3O6+x (Y123) [3],has renewed interest in the origin of this anomaly and its possible interplay withhigh-Tc superconductivity. Similar anomalies seem to occur in La2CaCuO4+x [4],Pb2Sr2(Ca,Y)Cu3O8 and Li1−xT2−xO4 [5]. The oxygen LO phonon is anoma-lous even in the superconducting perovskite Ba1−xKxBiO3 [6, 7]. On the otherhand, structural instabilities of a somewhat di®erent nature have been observed inLa2−xBaxCuO4 [8] and Tl2Ba2CaCu2O8 [9], where oxygen atoms move perpendic-ularly to the CuO2 planes.Common feature observed in oxygen LO phonons of LSCO and Y123 is theoccurrence of a break in the dispersion at 2¼=a(0.25,0), half-way between the originFIZIKA A (Zagreb) 10 (2001) 4, 169{176 169bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .(0,0) and the edge (0.5,0) of the CuO2 planar Brillouin zone (BZ). This break wastentatively attributed [1] to the (quasi)static longitudinal dimerization of the in-plane oxygens with wave vector [0.5,0] shown in Fig. 1a. The latter should producethe new BZ edge at [0.25,0] and, correspondingly, a gap in the LO dispersion.However, the (quasi)static dimerization has never been observed. This led to thespeculation [1] that the dimerization is highly disordered, with various, usuallyvery low estimates of the corresponding correlation length. However, the length inquestion should not be shorter than a few times 4a, if a gap of an LO phonon witha wavelength 4a is attributed to the 2a dimerization.Although the LO phonon anomaly depends strongly on doping, the electronicmechanism that produces it has not been understood until now.The conventional Peierls-like explanation is unlike, because it requires the ap-pearance of strong [0.5,0] charge density °uctuations, coupled to the lattice by alinear electron-phonon coupling.Although some phonon-induced features seem to exist [10] in the electron spec-tra, recent ARPES measurements [11-14] and their theoretical interpretations [15]have yielded, for the whole range of doping, Fermi surfaces of LSCO or Y123 whichdo not possess the nesting properties favouring [0.5,0] charge °uctuations. On theother hand, there is no apparent reason why the bare linear electron-phonon cou-pling would be particularly strong [16] for the LO dimerization.The aim of this note is to examine a mechanism which can explain the LOanomaly in LSCO and Y123. When the linear electron-phonon coupling is ruled out,the natural step is to consider quadratic electron-phonon coupling and charge °uc-tuations which couple through it to the lattice. Such a quadratic electron-phononcoupling was already used to explain the LTO/LTT transition in La2−xBaxCuO4[17], in which oxygen atoms move perpendicularly to the plane.When the LO dimerization mode is involved quadratically in the electron-phonon coupling, it can only couple to the homogeneous charge distribution withinthe unit cell. This charge distribution can be either symmetric or antisymmetricwith respect to the symmetry operation x *) y. We will show here that the antisym-metric charge distribution is large, that it is exactly the same as the one which leadsto the LTO/LTT instability in La2−xBaxCuO4 and that it explains satisfactorilythe LO instability in LSCO and Y123.The preceding symmetry analysis can be speci¯ed within the tight-bindingmodel, which distributes the holes over the Cu 3dx2−y2 orbital and the Ox;y 2px;yorbitals. This model is widely used in the theory of high-Tc superconductors eitherin the small U or in the large U limit, where U is the Coulomb repulsion at the Cusite. Only four single-particle parameters are usually used in addition to U , namely,the Cu and Ox;y site energies "d and "xp = "yp, the Cu-O overlap t and the Ox-Oyoverlap t′ [15].The variation of the crystal-¯eld potential of the ionic lattice, introduced by thedimerization modes with the amplitudes u¼;0 and u0;¼, leads in the ¯rst place tothe variation of the site energies, i.e. to the variation of ¢pd = "d +("xp + "yp)=2 and¢pp = "xp ¡ "yp. The variation of the oxygen site energies can be calculated in the170 FIZIKA A (Zagreb) 10 (2001) 4, 169{176bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .point-charge approximation, and written in the form [18, 19]"ip ¼2e2a[®ipuQ + ¯ipu2Q + ¢ ¢ ¢]; i = x; y (1)where Q = (¼; 0) or (0; ¼), ~uQ = 2uQ=a, uQ is the amplitude of the displacement,and a=2 is the distance between Cu and Ox.The parameters ®ip and ¯ip can be written in terms of two contributions asso-ciated with the ionic (¼; 0) displacements in even and odd CuO2 planes de¯nedaccording to Figs. 1. In general, they can also depend on the interplanar phasecos('z) = ~uQ(odd)=~uQ(even) (as in Y123, Fig. 1b). Previous calculations [19] es-timated ®ip and ¯ip from the crystal-¯eld potential at the oxygen sites. The valuesof ®ip and ¯ip obtained in this way for the (¼; 0) mode in LSCO and Y123 are givenin Table 1. It should be noted that in LSCO (see Fig. 1a) a small linear term ®xpδnnδ(b)ppxyaCuOOxyu (pi,0)x(a)bcaCu OOxyu (pi,0)xc(c)bFig. 1. The (¼; 0) oxygen LO phonons in LSCO (a) and Y123 (b). Since in Y123there are two CuO2 planes per unit cell, the upper plane will be referred to as anodd, and the lower one as an even plane. The considered sites are placed in an evenplane. (c) The homogeneous component of the charge °uctuations induced by the(¼; 0) oxygen LO phonons.TABLE 1. (a) The electron-phonon coupling constants at the oxygen sites Ox (®xp ,¯xp ) and Oy (®yp, ¯yp ) for the (¼; 0) deformation, with 'z representing interplanarphase de¯ned by cos('z) = ~uQ(odd)=~uQ(even). The contributions of the odd planesare given for comparison in the parentheses. (b) The corresponding constants cal-culated for the LTT deformation of the La2CuO4 lattice.'z ®xp ¯xp ®yp ¯yp(a) Y123 0 0.0 1.5 2.4 0.9¼ 0.0 -0.2 1.5 0.9LSCO 0, ¼ 0.05 (0.05) 1.1 (0.0) 2.1 (0.0) 0.9 (0.0)(b) LTT 0.0 -0.9 0.0 -1.8FIZIKA A (Zagreb) 10 (2001) 4, 169{176 171bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .comes from the interplanar contributions [i.e from ®xp(odd)], while large linear term®yp is in the ¯rst place associated with the intraplanar terms [i.e. with ®yp(even)].Notice also that in this lattice the constants ®xp(even) and ¯ip(odd) vanish for thesymmetry reasons.Singling out the homogeneous component of ¢pp, it is found that±¢Hpp ¼ 2e2a (¯xp ¡ ¯yp )(u2¼;0 ¡ u20;¼);±¢Hpd ¼ 12 2e2a (¯xp + ¯yp )(u2¼;0 + u20;¼):(2)±¢Hpp couples to the homogeneous charge distribution between Ox and Oy, nHpp =(nxp ¡ nyp)H shown in Fig. 1b, adding a bilinear term to the Hamiltonian H0,H = H0 + nHpp±¢Hpp: (3)In most of the high-Tc superconductors (Y123 is an exception) ¢Hpp = 0 inthe non-deformed lattice and ±¢Hpp lifts the "xp = "yp degeneracy of the CuO2 unitcell in a way similar to the Jahn-Teller (JT) e®ect. Even in the presence of thehybridizations t and t′, the ±¢Hpp JT e®ect leads to large energetic gains. In theconventional JT e®ect, charge transfer between the two levels, whose degeneracy islifted, is described by a step function of the splitting ±¢Hpp. In general, the e®ectsof hybridizations t and t′ make this charge transfer analytic¹nHpp ¼ Âpp±¢Hpp; (4)where Âpp is the exact charge susceptibility for the Hamiltonian (3), i.e., Âpp isrelated to the exact correlation function hnHppnHppi for the Hamiltonian H0 of (3).The JT e®ect is now the consequence of the possibly large Âpp. Actually, singularÂpp in (4) indicates nonanalytic nature of the charge transfer ¹nHpp, analogous to theconventional JT e®ect.Free energy variation, corresponding to (4), is¢F = ¡12Âpp±¢Hpp2+ ¢ ¢ ¢ : (5)It should be noted that, according to Eqs. (2) and (4), ¢F of the Eq. (5) is quartic(of the fourth order) in the dimerization u¼;0. However, when Âpp is singular, ¢Fshould be evaluated with special care, as discussed below.Large Âpp is found every time when the Fermi level falls close to the two groupsof the quasi-degenerate electronic states associated with the a and b axes of theCuO2 plane, which are split by ¯nite ±¢Hpp. This is particularly the case whenFermi level falls close to the van Hove singularities at (0.5,0) and (0,0.5) pointsof the two-dimensional Brillouin zone, associated with single-electron propagationin the CuO2 planes. The single-particle picture can safely be used in two limitsof the Hamiltonian H0 in (3), namely for free fermions (U = 0) and for in¯nite172 FIZIKA A (Zagreb) 10 (2001) 4, 169{176bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .U (U larger than ¢pd;¢pp; t and t′). In this case, for T ¼ 0 K, the real part ofthe generalized susceptibility, given in terms of the vertex functions g(k) and h(k),readsRefÂg;h("F )g = 1VXk¾g(k)h(k)@f ["L(k)]@"L(k): (6)Here f ["L(k)] is the Fermi function and "L(k) is the energy of the lowest occu-pied band [15]. The case g(k) = h(k) = 1 describes the density of states at the Fermilevel, nF , averaged appropriately over kBT , and g(k) = h(k) = 2@"L(k)=@¢pp thereal part of the intraband pp susceptibility, Âpp. Although it has been shown [15]that interband e®ects might also be important in high-Tc materials, for the sake ofsimplicity, we retain here only the intraband contributions to Âpp.For j"F ¡ "vH j À ±¢pp, and assuming that ¢Hpp = 0 at the outset, the suscep-tibility in Eqs. (4) and (5) takes the formÂpp = ¡c ¢ nF : (7)For the logarithmic van Hove singularity, with the electron-hole symmetry [15] withrespect to "vHnF ¼ 1"0log"0max[kBT; j"F ¡ "vH j] ; (8)and c is close to 1=5, as illustrated in Fig. 2. In the opposite limit[15] of the extended van Hove singularity, the electron-hole symmetry with0.5 1.5 2.5εF −εp (eV)00.10.20.30.40.5 −V0χg,h  [1/(eV)] 0.2nF −χpp(εF)Fig. 2. The density of states and the intraband pp susceptibility as functions ofthe Fermi energy "F , for the simplest case, ¢pd = 0:66 eV, t = 0:73 eV andt′ = ¢pp = U = 0.FIZIKA A (Zagreb) 10 (2001) 4, 169{176 173bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .respect to "vH is entirely broken. At T = 0,nF ¼ 1p"0("F ¡ "vH)(9)for "F > "vH , with the appropriate scale "0 [15]. The extension of Eq. (9) tokBT > ("F ¡ "vH) is straightforward [20].For ±¢pp À ("F ¡"vH); kBT , and in the symmetric case, it is easy to show thatEq. (5) should be modi¯ed by the replacementÂpp ! ¡ 1"0log"0±¢pp; (10)making ¢F resemble to the Peierls T = 0 result. However, on the contrary tothe Peierls' linear case, here ±¢pp is quadratic in u. In the opposite limit of theextended van Hove singularity [15], ¢F becomes linear [21] in ±¢pp, just like inthe conventional JT e®ect. Hence the linearity of ¢F for su±ciently large ±¢pp isrelated to the breakdown of the electron-hole symmetry at "vH .This reasoning can be applied to Y123 by the replacement ±¢Hpp ! ¢Hpp + ±¢Hppin Eqs. (4) and (5), assuming that the value of ¢Hpp in the non-deformed lattice issu±ciently small.The physical situation in the high-Tc superconductors is probably closer to thesymmetric limit of Eqs. (8) and (10) than to the asymmetric limit of Eq. (9) andits extension to the larger values of ±¢pp. In the former case, ¢F is quartic (orquartic Peierls-like) in u. Negative quartic terms are expected to lead to the ¯rst-order phase transition, as already found in Ref. [17] for the LTO/LTT transition inLa2−xBaxCuO4. The compelling feature of such explanation is that the two phaseswith u = 0 and a ¯nite u mix, i.e. the (disordered) domains of the dimerised phaseare expected to condense in the u = 0 phase. This might explain the di±culty inthe experimental observation of the dimerised domains, provided that the latterturns out to be small. The theoretical discussion of the domain size requires theinsight into the gradient terms quadratic and/or perhaps quartic in u, which is,however, beyond the scope of the present paper.Here, we only wanted to stress that the homogeneous charge transfer couplesappreciably to the LO dimerization because the electron-phonon coupling constantsof Table 1 are large and because the corresponding susceptibility (7) is large. Thedisadvantage that the electron-phonon coupling is quadratic rather than linear inu is set o® by the possibility that the above phase transition is of ¯rst order, withthe domain structure expected at low temperatures. In this particular respect, theadvantage of the O-O charge transfer mechanism proposed here is that it does notactivate long-range repulsive Coulomb forces, since each unit cell remains neutral.In contrast to that, long-range Coulomb forces are involved in energy balance ofthe charged stripes, associated recently [22] with the explanation of LO phononanomaly. This repulsive Coulomb energy is adverse to the formation of the chargestripes [23], but not to the O-O (or Cu-O) charge transfer stripes.174 FIZIKA A (Zagreb) 10 (2001) 4, 169{176bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .Further investigations should, therefore, be focused on the nature of the low-temperature lattice and charge patterns, including in particular the charge-transferstripes since they gain JT energy and do not loose long-range Coulomb energy.AcknowledgementsUseful discussions with Lev Gor'kov, Jacques Friedel and Robert Comes aregratefully acknowledged.References[1] R. J. 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Hussainand Z.-X. Shen, Nature 412 (2001) 510.[11] M. C. Shabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D. A. Bonn, Ruixing Liang andW. N. Hardy, Phys. Rev. B 57 (1998) 6090.[12] M. C. Shabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D. A. Bonn, Ruixing Liang andW. N. Hardy, Phys. Rev. B 57 (1998) 6107.[13] A. Ino, C. Kim, T. Mizokawa, Z.-X. Shen, A. Fujimori, M. Takaba, K. Tamasaku,H. Eisaki and S. Uchida, J. Phys. Soc. Jpn. 68 (1999) 1496.[14] A. Ino, C. Kim, M. Nakamura, T. Yoshida, T. Mizokawa, Z.-X. Shen, A. Fujimori,T. Kakeshita, H. Eisaki and S. Uchida, cond-mat/0005370 (2000).[15] I. Mrkonji¶c and S. Bari·si¶c, cond-mat/0103057 (2001).[16] D. Mihailovi¶c and V. V. Kabanov, Phys. Rev. B 63 (2001) 4505.[17] S. Bari·si¶c and J. Zelenko, Solid State Commun. 74 (1990) 367.[18] S. Bari·si¶c and I. Batisti¶c, Int. J. Mod. Phys. B 1 (1988) 291.[19] S. Bari·si¶c, I. Kup·ci¶c and I. Batisti¶c, Int. J. Mod. Phys. B 3 (1989) 2051; I. Kup·ci¶c,S. Bari·si¶c and E. Tuti·s, Phys. Rev. B 57 (1998) 8590.[20] J. Labb¶e and J. Friedel, J. Physique 27 (1966) 153,303.FIZIKA A (Zagreb) 10 (2001) 4, 169{176 175bari·si¶c et al.: electronically induced anomaly in lo phonon dispersion of . . .[21] S. Bari·si¶c and J. Labb¶e, J. Phys. Chem. Sol. 28 (1967) 2477.[22] J. Tranquada, K. Nakajima, M. Braden, L. Pinchovius and R. J. McQueeney,cond-mat/0107635 (2001).[23] S. Bari·si¶c and I. Batisti¶c, J. Physique Lett. 46 (1985) 819.ELEKTRONSKI INDUCIRANA ANOMALIJA U DISPERZIJILONGITUDINALNIH OPTI·CKIH FONONA VISOKOTEMPERATURNIHSUPRAVODI·CAJaku, elektronski induciranu anomaliju u spektru longitudinalnih opti·ckih (LO)fonona, koji se ·sire uzdu·z glavnih osi ravnina CuO2, poku·savamo objasniti prijeno-som naboja kisik-kisik izmed{u dvaju kisikovih atoma u istoj ¶celiji. Tvrdimo da tajprijenos naboja mo·ze biti velik te da je sna·zno vezan s LO dimerizacijom kisikovihatoma. Negativni doprinos energiji je ·cetvrtog reda u dimerizacijskoj amplitudi,·sto ukazuje na nestabilnost sustava putem faznog prijelaza prvog reda s popratnomdomenskom strukturom.176 FIZIKA A (Zagreb) 10 (2001) 4, 169{176

Electronically induced anomaly in LO phonon dispersion of high - Tc superconductors

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Electronically Induced Anomaly in LO Phonon Dispersion of High - Tc Superconductors

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